or ILLINOIS 62 . 0.8 H 2.7 MECHANICS’ AND ENGINEERS’ POCKET-BOOK OF * TABLES, RULES, AND FORMULAS PERTAINING TO MECHANICS, MATHEMATICS, AND PHYSICS: INCLUDING AREAS, SQUARES, CUBES, AND ROOTS, ETC.; LOGARITHMS, HYDRAULICS, HYDRODYNAMICS, STEAM AND THE STEAM-ENGINE, NAVAL ARCHITECTURE, MASONRY, STEAM VESSELS, MILLS, ETC. ; LIMES, MORTARS, CEMENTS, ETC.; ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS, ETC., ETC. Fiftieth. Edition.. — One hundred and fifth Thousand. BY CHAS. H. HAS WELL, CIVIL, MARINE, AND MECHANICAL ENGINEER, MEMBER OF THE AM. SOC. OF CIVIL ENGINEERS, OF THE INSTITUTION OF CIVIL ENGINEERS, AND OF THE INSTITUTION OF NAVAL ARCHITECTS, ENGLAND, OF THE ENGINEERS’ CLUB, PHILADELPHIA, OF THE N. Y. ACADEMY OF SCIENCES, CORRESPONDING MEMBER OF THE AMERICAN INSTITUTE OF ARCHITECTS, OF THE BOSTON SOCIETY OF CIVIL ENGINEERS, AND ASSOCIATE MEMBER OF THE N. Y. MICROSCOPICAL SOCIETY, ETC. An examination of facts is the foundation of science. NEW YORK: HARPER & BROTHERS, PUBLISHERS, F E A N K I. I N SQUARE. 1 886 . MENSURATION. For Tuition and Reference, containing Tables of Weights and Measures; Mensura- tion of Surfaces, Lines, and Solids, and Conic Sections, Centres of Gravity, &c. To which is added, Tables of the Areas of Circular Segments, Sines of a Circle, Circular and Semi-elliptical Arcs, &c. By Chas. H. Haswell, Civil and Marine Engineer, &c. Fifth Edition. 12mo, Sheep, 90 cents. MECHANICS’ TABLES. Containing Areas and Circumferences of Circles, Sides of Equal Square ; Circum- ferences of Angled Hoops, angled Outside and Inside ; Cutting of Boiler-Plates. Covering of Solids, &c., and Weights of various Metals, &c. Miscellaneous Notes comprising Dimensions of Materials, Alloys, Paints, Lacquers, &c. ; U. S. Tonnage Act, with Diagrams, kc. By Chas. H. Haswell, Civil and Marine Engineer,, &c. Second Edition. 12mo, Cloth, 75 cents. Published by HARPER & BROTHERS, New York. Either of the above Works, or the Pocket-Book, will be sent by Mail, postage prepaid, on receipt of the price. Entered according to Act of Congress, in the year one thousand eight hundred and eighty-six, by HARPER k BROTHERS, In the Office of the Librarian of Congress, at Washington. £> 20 , S W'Z'i inscribed TO CAPTAIN JOHN ERICSSON, LL.D., AS A SLIGHT TRIBUTE TO HIS GENIUS AND ATTAINMENTS, AND IN TESTIMONY OF THE SINCERE REGARD AND ESTEEM OF HIS FRIEND, THE AUTHOR : it > # PREFACE To tlie Forty-fifth. Edition. The First Edition of this work, consisting of 284 pages, was submitted to the Mechanics and Engineers of the United States by one of their number in 1843, who designed it for a convenient reference to Rules, Results, and Tables con- nected with the discharge of their various duties. The Twenty-first Edition was published in 1867, consisted of 664 pages, and, in addition to the original design of the work, it was essayed to embrace some general information upon Mechanical and Physical subjects. The Tables of Areas and Circumferences of Circles have been extended, and together with those of Weights of Metals, Balls, Tubes, Pipes, etc., of this and some preceding editions were computed and verified by the author. This edition is a revision and an entire reconstruction of all preceding, embracing amended and much new matter, as Masonry, Strength of Girders, Floor Beams, Logarithms, etc., etc. To the young Mechanic and Engineer it is recommended to cultivate a knowledge of Physical Laws and to note re- sults of observations and of practice, without which eminence in his profession can never be attained ; and if this work shall assist him in the attainment of these objects, one great purpose of the author will be well accomplished. a • a/t a a a [ oT : .• : v- " f , •• i : ’j , u L - ■. . > > . ' * • : *" 1 ' 1 ' • • 1 '/ , ii I ; I) • iii '• : '• < v , : • I • '■i:; ' i'j-fvl I • l! _ ■ I » Qii.f ■ Cul-i Oi IJ-: : :r ; j ■ . • ' • • - ' ■j fr-ir .!'• . il UlfiihoM . ; Qi'i 'r. il /ii . • fioono'f >if! • .) i-?nr» ■ • 4/ ‘ w L ' v .1 ' J ■ 1 *■ • • ! . .-r t . ( t , j l- . l ‘ I;.. ' * lil.iiO - t , riifii'iJ .. I t *i< -V ‘ .olt) ; Hi::: ' ' *1 Jl ' v-. 1 00 ’ ' l * I [fir:,o iur-.i! Or; ,*> • l JR ci: Jo ' ’ C) . fo lo il’- > .. iiqi •'*> : . , ■ • K» j - — - - — INDEX. A. Page Abutments and Arches 604 Acids 188 Adulteration in Metals, Proportion of, \ in a Compound 216 Aerodynamics 614 Aerometry, Course of Wind 675 I ‘ Distance of Audible Sounds 674 “ Pneumatics 673-676 “ Resistance of a Plane Sur- “ face 675 I I Resistance to a Steam Vessel in Air or Water 91 1 ‘ ‘ To Compute Height of a Col- umn of Mercury to Induce an Efflux of Air 675 “ Velocity and Pressure of “ Wind 674, 91 1, 921 “ Volume of Air discharged “ Through an Opening , “ etc 674-676 “ Weight of Air 675 Aerostatics 427-431 “ Elevation by a Barometer 428 “ “ by a Thermometer 429 “ Velocity of and Sound. . . 428 “ Velocity of Air flowing into a Vacuum 428 Ages of Animals 192 Air, and Steam 737 “ Atmospheric 431-432 “ Consumption of. 432 “ Decrease of Temperature by Al- titudes 522 “ Expansion of. 520 “ Flow of in Pipes 745, 746, 909 “ Pressure and Resistance of . .... 648 u Resistance of different Figures in 646 “ Velocity Lost by a Projectile . .. . 648 “ Volume and Weight of Vapor in 68, 69 “ “0/ and Gas in a Furnace. 760 " " Pressure, and Density of. . 521 Pressure , Temperature , and Density of 522 Required per Hour , etc. . 525 194 Alcohol. . ‘ ‘ Elastic Force of Vapor of. . 707 “ Proportion of in Liquors. . 204 Ale and Beer Measures 45 Algebra, Symbols and Formulas.. 22, 23 Alimentary Principles 200 Alligation io 6 Alloys and Compositions 634-637 Almanac, Epacts and Dominical Let- ters , 1800 to 1901 73 Altitudes, Decrease of Temperature by 522 American Gauge 118, 120 Analysis of Organic Substances.. 190 “ of Foods and Fruits 201 “ of Meat , Fish , and Vegetables 200 Pag® Anchors and Kedges 174 “ Cables , Chains , etc 173, 174 “ Diameter of a Chain Cable. 175 “ Experiments on 175 “ Length of Chain Cables . ... 175 “ Number and Weight of . . . . 174 “ Resistance to Dragging 175 Ancient and Scripture Lineal Meas- ures 53 “ Weights 53 Angle and T Iron, Weight of. 130 Angles and Distances Corresponding to a Two foot Rule 160 Angles, To Describe , etc 222 Angles, To Plot and Compute Chord of 359 Animal and Human Sustenance 203 Animal Food 200-207 u Power 432-440 “ Birds and Insects 438 “ “ Camel 438 “ “ Crocodile 438 “ “ Day's Work. 434 “ “ Dog, 438 “ “ Horse . 435, 436, 437 , 439, 440 “ “ Llama 438 “ “ Men 433,434,438,439 u “ Mule and Ass 437 “ “ on Street Rails or Tramways 435 “ “ Ox 438 Animals, Proportion of Food for 205 Annuities no-in “ Amount of m 11 u at Compound Interest m “ Present Worth no 11 Yearly Amount that will Liquidate a Debt no Anti-attrition Metal 636 Apartments, Buildings, Ventilation of 524 Appendix 913 Aqueducts, Roads , and Railroads ... 178 Arc, To Describe 225, 227, 228 Arches and Abutments 604 Arches and Walls 602 Areas of Circles 231-236 “ by Logarithms. . .236, 252 ‘ ‘ Wh en Composed of an Integer and a Frac- tion 236 Area of a Circle, When Greater than any Contained in Tables 235, 252 Areas of Segments of a Circle. .267-269 “ of Circles , by Birmingham W.G 236 “ Greater than in Table 235 “ of Zones of a Circle 269-271 “ of Zones, To Compute 271 “ and Circumferences of Cir- cles by 10 ths and 12 ths. .243-257 u INDEX. Page Apothecaries’ or Fluid Measure 46 “ Weight 32 Arithmetical Progression 101 Artesian Well 179, 198 Ash 482 Asphalt 481, 689 Asphalt Composition 593 11 Pavement 690 Ass 437 Atmosphere, To Compute Volume of Vapor in ' 68 Atmospheric Air 431, 912 “ Proportion of Oxygen and Carbonic Acid at Various Locations 432 “ Carbonic Acid Exhaled by Man 432 Avoirdupois Weight 32 Axle, Compound 627 B. Babbitt’s Anti-attrition Metal 636 Baking of Meats, Loss by. 206 Balances, Fraudulent 65 Balloon, Capacity and Diameter of. . 218 Balls, Cast Iron and Lead 153 Balls, Lead, Weight and Dimension of 501 Barometer, Elevations by Readings. 429 Height of 429 11 Indications 429 “ Weather Glasses 430 “ Weather Indications . . . 431 Barrel, Dimensions of 30 Beam or Girder Trusses 823 “ General Deduction 824 Beams, Deflection of. 77°~777 ‘ k Dimensions of which a Struct- ure can Bear 644 “ Elements of Wrought Iron , Rolled 807 “ Elliptical Sided 826 “ Floor , Headers , Trimmers , etc. . . 835-838 u Formula of Transverse Stress 801 11 of Unsymmetrical Section , Neutral Axis and Strength of. 820 “ or Girders, Moments of . . .621, 622 u Shearing Stress 622 Bearings for Propeller Shafts 473 Beet Root and Beet Root Sugar 207 Beeves and Beef, Comparative Weights of. 35 Beils, Weight of. 180 Belt, Equivalent, ahd Wire Rope 167 Belting 9°7 Belts and Belting 44!-443 “ Width of 44 Bench Marks 85 Beton or Concrete 593 Birds 44° ‘ 1 and Insects 196, 438 Bissextile or Leap Year 70 Black and Galvanized Sheet Iron — 129 Black and Galvanized Sheet Iron, Weight of 129 Blast Furnace 5 2 9 Page Blast Furnace, Pipe of a Locomotive. 907 Blasting 443, 912, 913 ‘ ‘ Boring Holes in Granite . . 444 “ Charge of Gunpowder for. . 444 “ Effects 444 “ Weight of Explosive Mate- rials in Holes 444 Blasts and Draughts, Effects of 746 Blower and Exhausting Force 898 Blowers, Fan 447 Blowing Engines 445, 898 “ Memoranda 448 “ Power of etc 446 “ Pressure of Blast 447 “ Root's Rotary 449 “ To Compute Dimensions of a Driving Engine . . . 446 “ “ Elements of. 447 “ “ of a Fan-Blower 448 “ “ Power of a Centrifugal Fan 448 “ “ Volume of Air trans- mitted 447, 922 Blowing Off 726 Board and Timber Measure 61 Boiler, Steam 739 “ and Ship Plates 828 “ Areas and Ratio of Grate and Heating Surface , etc 741 “ Draught 739,744,745 “ “ and Blasts , Compara- tive Effect of. 746 “ “ Velocity of 746 “ EvaporativeCapacity of Tubes 742 “ Evaporation, Effects of for Different Rates of Combus- tion 743 “ Evaporation , Power of. 757 “ Fuel that may be Consumed.. 742 “ Heating Surfaces 740 I ‘ Loss of Pressure by Flow of Air in Pipes 745 ‘ ‘ Minimum Fuel Consumed per Square Foot of Grate 740 “ rower 760 “ Rate of Combustion 760 Relation of Grate , Heating Surface, and Fuel 741 ‘ ‘ Result of Experiments with a Steam Jet 746 II Results of Operation of 743 “ “ of Operation of Vari- ous Designs of Boiler 744 “ Riveting 755, 907 “ Safety Valves 746 “ Steam 739—745, 829 “ Steam Heating 526 “ Steam Room 748 “ Volume of Furnace Gas per Lb. of Coal 760 “ Weights of. 759 Boiling of Meats, Loss by 206 Boiling-Points 517 Bolts and Nuts, Dimensions and Weights of..... 156, 157 “ English Standard . . . 158 “ French Standard .. . 158 INDEX. Ill Page Bolts and Nuts, Square Heads 159 “ Tenacity of 198 “ Wrought-iron, Experiments on. 783 “ and Plates 749-75 7 Boriug aud Turning Metal 197 Instruments, Tempering of . . . 197 “ Wells 197 Brain, Weights of. 192 Brass, Sheet, Weight of. 142 Plates, Weight of 1 18, 1 19, 146 Wire, Weight of 120, 121 Brass 636 Castings, Weight of. 155 Tubes, Weight of 142 * ‘ Weight of. 136, 149 Braziers’ and Sheathing Sheets 155 Bread 207 Breakwaters 181 Breast- wheel 568 Brick Walls 603 Brickwork 597, 801 Brick or Compressed Fuel 907 Bricks .598, 599 “ Crushing Resistance of 908 “ Volume of and Number in a Cube Foot of Masonry 599 Bridge, Britannia Tubular 178 “ Highest 907 “ Iron 178 “ New York , Erie , and West- ern Railroad 178 ‘f New York and Brooklyn ... 178 “ Suspension 842 Bridge Plates and Rivets 830 Bridges 178 “ Lengths and Spans of . 181 “ Resistance of. 645 “ Suspension. Length of Span °f- 199 Suspension 178,842 Bridles or Stirrups, for Beams 838 British and Metric Measures, Com- mercial Equivalents of 906 Broccoli 207 Bronze 637 “ Malleable 907 Browning or Bronzing Liquid 874 Builders’ Measure 46 BuildingDepartment,.ReguiVeme/i!fso/; 907 B uilding Stones, Expans ion and Con- traction of ^4 Buildings, Walls of. ^9 “ Protection of 907 Buoyancy of Casks I92 Burns and Stings, Application for . . . 196 Buttress 696 C. Cabbage 20? Cables, Chain, Weight and Strength of 168 “• Chain, Breaking Strain and Proof of. 169 “ Circumference of I7I “ Galvanized Steel ^3 t 9*4 u To Ascertain Years of Coincidence 74 Churches and Opera-Houses 180 Circular Arcs, Length of. 260-262 Circular Measure .113, 114 “ Motion 618 Circulating Pumps 749 Circles, Areas of 231-236 “ “ by Logarithms 236 “ u by Wire Gauge .. . 236 “ Sides of a Square of . . . .258-259 Circumferences of Circles 237-242 “ “ by Logarithms. 242 “ “ by Wire Gauge 242 Cisterns and Wells, Excavation of . . . 63 “ u Capacity of 63 Civil Day 37, 7° Civil Year 7° Cloth Measure 27 Clouds 43° Coal, Anthracite 4 80 “ Average Composition of Heat of Combustion and Evapora- rative Power of. 4 8 6 11 Bituminous 479 “ “ Caking Splint or Hard Cherry or Soft 479 “ “ , Cannel 479 “ “ Chemical Composition , Varieties of. 479 “ Consumption of to Heat 100 Feet of Pipe 527 “ Effective Value of 908 “ Elements of Various 480 “ Fields , Areas of. 19 1 ‘‘ Japan 9°9 “ Measure 46 Coast and Bay Service 9° 8 Cocks, Composition, and Copper Pipes I 5° Cohesion 614 Coins, British Standards 38 “ To Convert U. S- to Bntish Currency 39 “ U. S., Weight and Fineness of. 38 Coins, Values of 39 “ Weight , Fineness , and Mint Value of Foreign 39 Coke 480 Cold, Extremes of in Various Coun- tries 191 “ Greatest . . . 908 College, Oxford 179 Collision or Impact 580-582 Color Blindness 195 Colors for Drawings 196, 913 Colors, Proportion of for Paints 66 Columns 180 Combination 112 Combustion 458-466 Composition and Equiva- lents of Gases 460 Chemical Composition of some Combustibles 461 Evaporative Power of 1 Lb. of a given Combus- tible 462 Heat of 463 Heating Powers of Com- bustibles 461, 462 Of Fuel 4 6 3-4 6 5 Rate of 760 Relative Evaporation of Combustibles. . . 465 “ Volumes of Gases or Products of per Lb. of Fuel 465 Temperature of. 462 To Compute Consumption of Fuel 446 1 “ Volume, of Air Chemically Consumed in Complete Combustion of 1 Lb. of Coal... 459 ‘ Volume of Air Required . 465 { Weigh t and Sped fi c Heats of Products of Combus- tion , etc • 462 Compass, Degrees , etc., of Each Point 54 Composition and Alloys 634-637 Compound Axle or Chinese Windlass 627 Compound Interest 108 Compound Proportion 95 Concretes, Limes, Mortars, and Cements 5 88 ~597 Concrete, Cements, and Mortar . .595, 914 or Belon 593 Cones 353 Conic Sections 3 8o_ 3 8 4 “ Conoid 3 8 ° “ Ellipse or Hyperbola , To De- termine Parameter of. . .380, 381 “ Ellipse , To Compute Area of Segment of 3 82 “ Hyperbola 3 8 3 “ “ To Compute Abscissce . . 383 “ “ Area of 384 “ “ Diameters 3 8 3>3 8 4 “ “ Length of any Arc 384 Parabola 3 8z “ To Compute Area of. . . 383 “ “ Ordinate or Absdssa 382 INDEX, Y Page Contractility 614 Copper. . .. 750 “ Braziers’ and Sheathing 131 ‘ ‘ Plates , Weight of . . . . 1 1 8, 1 1 9, 1 46 1 ‘ Pipes 1 50 “ Rods or Bolts , and Pipes, Weight of. i 4 8 “ Sheet, Weight of. ... 135 “ Tubes, Weight of. 140,144 “ Weight of. 136,155 “ Wire, Weight of. 120,121 “ Wire Cord .. 123 Copying 2g Cord, Copper Wire I23 Cordage, Friction and Rigidity of. Corn Measure Corrosion of Iron Steel 908 Corrugated Iron Hoof Plates, Weight of 131 Co- SECANTS AND SECANTS 403-4 1 4 “ . “ To Compute, etc. 414 Cosines and Sines 390-402 “ To Compute, etc. 401, 402 Cost, of Family of Mechanics in France.. Co-tangents and Tangents 416-426 _ “ “ To Compute, etc. 426 Cotton Factories 800 couple..,.,.... Couplings of Shafts 7Q 6 Coursing and Leaping 44 o Crane, Steam Dredging 8qq “ Wood y Page • 37 • 37 37? 70 Day, Marine or Sea. . ‘ ‘ Sidereal Solar and Civil. Day's Work. 4 _. Decimals * Deer Park, Copenhagen. Deflection Delta Metal Departures, Table of... Desiccation Cranes Crank. 900 ,• • • • I 79» 433, 455-457 Machinery of. 4 ^ 7 To Compute Dimensions of „ „, Post 456 Stress on 45 3 “ Stress upon Strut .... 45I “ Turning 433 Cream, Percentage of, in Milk. 205 Crocodile 43 g Crops, Mineral Constituents Absorbed 189 Croton Aqueduct. I? 8 Crushing Strength 76 4-760 Cube Measures 3 o Cube Root, To Extract . . ‘ [ gj Cubes, Squares, and Roots 272-302 “ To Compute and to As- certain, etc 300-302 Cucumber 207 Currency, To Convert U. S. to British 30 Current Wheel A/ 0 Curvature- and Refraction Cut Nails, Tacks. Spikes, etc k. Cutters ( Yachts ) g q5 Cycle, Dominical or Sunday Letter. . 70 Lunar or Golden Number 71 Of Sun 70 Cycloid, To Describe 228 Cyclones ’ g 75 Cylindrical Flues and Tubes, Hoilow 827 D. Dams, Embankments, and Walls. 700-703 Day, Astronomical, 92-94 ... 179 770-781 384, 9 X 3 • •• 54 Detrusive or Shearing Strength, 3 ^ _ _ . 782, 783 Dew Point, and To Ascertain 68 Diamond Weight ’ * ‘ 32 Diamonds, Weight of. * Diet , Daily, of a Alan . . 202 207 Digestion of Food ’ 20 g Dip of Horizon ’ g Q Discount or Rebate ’ * ’ ’ Ioq Displacement of a Vessel .’ 653 Distances, Steaming 86 “ Between Cities of U. S. . . . . . 184 u “ East and West 187 “ “ Principal Ports of World. 87 i “ ofU.S... 88 “ Various Ports of Eng- land, Canada, and V.S. 86 Distances and Angles Corresponding to a Two -foot Rule jgo Distances and Geographic Levelling. '. 56 _. “ “ Measures.. 54 Distemper Distillation P °B '.'.'.'.'.438,440 Domes and Towers. I7g I s 0 Dominical Letters, and Epacts ’ 73 “ or Sunday Letter 7Q Drainage of Lands 691 Drains, Diameter and Grade of to ^ Discharge Rainfall ’ qo 5 Draught 739/744, 746 Drawing and Tracing Paper 29 Drawings, Colors for ’ ’ ' ' * \ g § “ Dimensions of, for U. S. y Patents IQ g Dredger and Hopper Barge 8qq qoo Dredging, and Cost of \ g7 ‘ ‘ Machines ’ gqo Drilling *-* A/ Drowning Persons, Treatment of. . ! 187 Dry Measure qo J Dualin ia tk Cellulose 444 Dynamics. .616-620 ‘ Circular Motion 618 “ Decomposition of Forces. . . 620 “ Motion on an Inclined Plane 619 Uniform Motion 617 618 ‘ ‘ Work A ccum ulated in Mov- ing Bodies 619 By Percussive Force . 620 VI INDEX, E. Pa & e Earth .188, 198 “ Area and Population of .... . 100 “ Elements of Figure of 61 “ Motion of 7 ° “ Pressure of 695 “ Weight of per Cube Yard 468 “ Weights of ;• 33 Earthwork 4 ° 7 , 4 °° “ Bulk of Rock 468 ‘ ‘ Js umber of Loads and Vol- ume of per Day 908 “ Shovelling 9 ° 8 “ Volume of Transported per Day 9 ° 8 Easter Day . . 7 1 Ecclesiastical Year 7 ° Egyptian and Hebrew Measures 53 Elastic Fluids, Specific Gravity of.. 215 Elasticity * 95 , 614 “ Coefficients of 7 61 “ Modulus of. 7 62 “ Relative, of Metals 780 Electric and Gas Light 198 “ Light , Candle-Power of ... 908 Electrical Weights 34 Elementary Bodies * 9 ° Elevation by a Barometer 4 28 Elevations and Heights of Various Places i8 3 Elliptic Arcs, Length of 263-266 “ “ To Ascertain Length of. 266 Ellipse, To Describe , etc 226, 380 Embankment and Excavation 466 ‘ ‘ Walls and Dams . . 700-703 “ Weight of a Cube Foot of Materials 694 Endless Ropes *67 Engines and Machines 898 “ Elements of . “ and Sugar -mills , Weights of 908 Ensigns , Pennants , and Flags , U. S. . 199 Epacts, and Dominical Letters 73 Equation of Payments.. 109 Equilibrium, Angles of. 694 “ Of Forces 616 Ericsson’s Caloric 9°3 Establishment of the Port for Several Locations 8 5 Ether, Elastic Force of Vapor 707 Evaporation of Water 5 J 4 Evaporative Power of Tubes 5 r 3 Evolution 9 ° Excavation and Embankment of Earth and Rock 192 Exhausting Fan and Blower 898 Expansion 614 F. Fan Blowers 447 Fascines 690 Fellowship 99 Fence Wire Fig i°l Filter Beds Filtering Stone 9°9 Filters for Waterworks 184 Page Fire Bricks 600 “ Clay ........ 597 Fire-Engine, Steam 904, 909 Fish, Meat , and Vegetables , Analysis of Flags, Ensigns, and Pennants, U.S.. 199 Flax Mill 47 ^ Floating Bodies, Velocity of. 909 Flood Wave 9 12 Floors and Loads, Factor of Safety. 841 Weight of. 841 Flour ... ........ 207 “ Mills 9 °° Flues and Tubes *. . . - 747 ) 754 ? 827 “ Corrugated 9°9 “ or Furnaces 754 Fluid Measure 3°, 46 Fluids, Impact and Resistance of.. 577 “• Lamp and Gas 5 8 4 “ Percussion of 579 Flutter Wheel 57 * Fluxes for Soldering or Welding 636 Fly Wheel 45 1 Flying of Birds 44 ° Food, Animal 200-207 Digestion of 206, 914 Nutritive Constituents and Val- ues of. 202 Nutritive Equivalents of. .... . 205 Proportion Expended by Ani- mals 205 Foods, Analysis of .201, 203 Nutritious Properties of. 204 Relative Values of... .202, 204, 912 Thermometrical Powers and Mechanical Energy of 205 Forces, Composition and Resolu- tion of 615 Division of. .- 614 Equilibrium of. 616 Percussive 620 Decomposition of. 620 Foreign Measures and Weights. .48-52 Fortress Monroe J 79 Foundation Piles J 9 8 ’ 9°9 Fractions 89-91 Fraudulent Balances 65 Freeboard 666, 913 Friction 469-478, 57 662 “ and Rigidity of Cordage. . . 472 “ Application of Results 474 “ Bearings of Propeller Shafts 473 Coefficien ts of Axle 47 1 “ of Motion 470 “ of of Journals . 470 To Determine.. 471 47 ~« 47 8 47 8 Frictional Resistances Grain Conveyers. . Launching Vessels. Mechanical Effect of To Compute 47 1 of Bottom of Vessels 9°9 of Pivots 472 of Planed Surfaces 9°9 of Steam-Engines and Ma- chinery 475 of Walls and Earth 698 INDEX. vii Page Friction, Relative Value of Angles. . 472 “ Results of Experiments . 47 4, 475 “ Rolling 473 •* Steam-engine 478 k; Steamers 478 Tools 476 “ Value of Unguents 471 kk Wood-sawing 477 Frictional Resistances 475 of Machinery, Results of Experiments upon. . .475-478 Fkigorific Mixtures 193, 516 Fruits, Analysis of 201 kk Proportion of Acid and Sugar 203 Fuel v 479-487, 513 kl Area of Grate and Consump- tion of. 513 Ash 482 Average Composition of . . .485, 486 Briclc or Compressed 90 Elements of. 486 Lignite 481 Liquid 484 Miscellaneous 487 Peat 482 Produce of Charcoal. 481 Relative Values of. 483 Tan.. 482 Values, Weights, and Evapora- tive Power of. 483, 91 - 48! 523, 754 634 “ Wo- l Furnaces Fusible Compounds. G. Galvanized Sheet Iron, Thickness and IF eight of. 1 1 Charcoal Iron. . , “ Iron Wire Rope, “ Steel Cables Gas .. 585- and Electric Light “ Atmospheric Engine 587, “ Coal ‘ 1 Candies , etc “ Engines “ Flow of “ Mains, Dimensions of 11 Pipes 138,160, ‘ 1 Pipes , Thickness of. “ Steam and Hot-air Engines “ Temperature of '. “ Tubing, and Number of Burners, Regulation of. “ Volume of 586, “ Volume of Furnace per Lb. of Coal “ Weight of a Cube Foot of Gases, Expansion of Gauges , Wire 118- Gauging- Cask Geographic Levelling by Boiling-Point Geographic Measures and Dis- tances 54i Geographical and Nautical Meas- ures Geometrical Progression 103- Page Geometry 219-230 “ Angles 222 “ Arcs 227 Catenary 230 Circles . . 224 Cycloid and Epicycloid . . 228 “ Ellipse 226 ‘ ‘ Hexagon 223 “ Hyperbola 230 u Involute 229 “ Length of Elements 221 “ Lines 221, 222 u Octagon 223 “ Parabola. 229 “ Polygon 223 “ Rectilineal Figures 222 “ Spiral 230 Geostatics and Geodynamics 614 Gestation, Periods of. 192 Girder, Beam, etc., General Deduction, 824 Girders, Beams, Lintels, etc 822 Girders and Beams 805, 806 “ “ Centre of, and Ver- tical Distance of Centres of Crush- ing and Tensile Stress 819 (i ‘ £ Deflection of. .... . 840 “ “ Dimensions and Load of. 839, 840 “ “ Factors of Safety. 821, 841 “ “ Moment of Stress of. 621, 623 “ “ Trussed 823 “ “ Tubular 775 Glass Globes and Cylinders 831 124 *97 874 119 71 207 452 Glass, Window, Glazing Glues. Gold Sheet , Thickness of Golden Number or Lunar Cycle... Gooseberry Governors Grade, Reduction of, to Degrees. . . . 359 Grain, and Roots, Weights of. 34 “ Conveyers 478 ‘ ‘ Weights per Bushel 32 Graphic Delineation of Stress, with a Uniform Load, etc 623 “ Operation, Solution of Ques- tions by 905 Gravel 690 or Earth Roads 688 Gravitation.. 487-496 Accelerated and Retard- ed Motion . 494 Average Velocity of a Moving Body 495 Formulas to Determine “ the Various Ele- ments 490 “ “ of Retarded Motion 492 General Formulas for Accelerating and Re- tarding Forces 495 INDEX. Vlll Page Gravitation, Gravity and Motion on an Inclined Plane . 492, 493 “ Inclined Plane 493, 494 “ Miscell. Illustrations .. . 496 “ Promiscuous Examples. 489 “ Relation of Time, Space, and Velocities . . . .488-491 “ Retarded Motion.. . .492, 496 “ Space 4 8 9 “ To Compute Action of . . 488 it “ Velocity of a Fall- ing Stream of Water 49 6 tt “ Action of by a Body Projected Up- ward or Downward. . 490 “ Velocity 4 8 9 it “ due to a Given Height of Fall and Height due to Given Velocity 4 88 Gravity op Bodies • • • • 208 * ‘ at Various Locations at Level of Sea 4 8 7 “ Centre of. 605 “ Various Formulas for 488 Grecian Measures and Weights 53 Gregorian Calendar 7 °? 7 * Grindstones 47 8 Grouting 593 > 594 ? 59 8 Gudgeons and Shafts 79 °> 797 Gun Barrels , Length of. 19 8 Gun Cotton 443 Gun Metal, Weight of 136, 146, 149 Gunnery 49 . 7 " 5°3 “ Charge , Range , and Veloci- ty , To Compute 497, 499 ‘ ‘ Comparison of Forces oj a Charge in Various Arms . 502 1 1 Experiments with Ordnance. 498, 5 00 “ Lead Balls , Weight and Di- mensions of. 5 QI “ Number of Percussion Caps corresponding to B Gauge. 502 “ Penetration of Lead Balls in Small Arms 5 °° “ Penetration of Shot and Shell , etc 49 8 » 5 °° it Proportion, Powder to Shot. 502 “ Report of Board of Engi- neer s,U.S. A., for Fortifi- cations , etc • • 499 ‘ ‘ Velocity and Ranges of Shot and Shells 49 8 “ Windage and Waddings .. 501 Gunpowder 443 > 5 ° 2 “ Charge of. 444 ‘ ‘ Heat and Explosive Pow- er of , , etc 5°3 “ Proof of -402 tt Properties and Results of 503 “ Relative Strength of for Use under Water 503 Gunter’s Chain * 26 Gyration, Centre of. 009-01 1 t « Centre of of a Water -wheel 6 1 1 H. Pa g® Hammers Steam 179 Hancock Inspirator 901 Hawsers, Wire, and Ropes and Ca- bles, Comparison of. 169 Hawsers, Ropes, and Cables. . . . 169-172 “ u Strength and Cir- cumference of. . 171 “ “ Weight of. 172 Hay and Straw 19 8 Heat 504-529 Available Expended 909 Capacity for 5 ° 5 i 5°7 Communication 5 X 5 and Transmission of. .. 510 522 523 513 514 Condensation • • 5 1 5 Conduction or Convection of... 5 X 4 Congelation and Liquefaction. . 516 Decrease of Temperature by Al- titudes Degrees of Fahrenheit to Reau- mur and Centigrade , and Con- trariwise Desiccation Distillation . Effect upon Various Bodies 518 Evaporation 5 I2 > 5 * 3 > 5*4 Expansion and Dilatation of a Bar or Prism. 5*9 Expansion of Water 519, 520 “ of Liquids , Gases , and Air 5 2 ° Extremes of in Various Coun- tries J 9 * Frigorific Mixtures 193, 5 l6 Heating and Evaporating Water by Steam 5 11 1 Latent , 508,509 1 Latent, of Fusion 5°9 ‘ Latent, of Steam, To Compute.. 707 1 Mean Temperatures of Various T ncnliti •■»•»»•••*•* IQ2 ‘ Mechanical Equivalent of in Steam. 7°5 1 Melting and Boiling Points. .. . 517 ‘ Of Sun *93 • Perpetual Congelation or Snow Line I 9 2 ‘ Proper Temperature of Enclosed Spaces 526 it Quantities of, Transmitted from Water to Water through Plates, etc 5 11 “ Radiation of. 5°9 ‘ ‘ Radiating and Absorbing Power of Various Bodies 5 IQ “ Reduction of by Surfaces 525 “ Reflection 5 10 “ Refrigerator, To Compute Sur- face of •••• 5 12 “ Relative Power of Various Sub- stances 5 IQ “ Sensible 5°7 “ Specific 5 ° 5 > 5 ° 6 > 5°7 “ Temperature by Agitation 524 “ Temperature of a Mixture of Like and Unlike Substances, To Compute 5°6 INDEX. IX Page Heat, To Compute Volume or Pressure of a Constant Weight of Air or other Gas , etc., for a Given Temperature 522 “ To Reduce Degrees of Different Scales 523 “ Transmission of through Glass . 51 1 “ Underground Temperature 519 “ Volume , Pressure , and Density of Air at Various Tempera- tures 52 Heating, Air, Length of Pipe Required 525 11 by Hot Water 524 Hebrew and Egyptian Measures and Weights 53 Height and Elevation of Various Places 183 Heights, Corresponding to Boiling Points of Water 519 Hemp and Wire Rope, Circumference of for Rig- ging 172 “ “ General Notes 167 “ “ Relative Di- mensions of 172 “ “ Weight and Strength of 172 “ Weight of .. . 166 Hemp Rope, Iron and Steel 164 “ and Iron Wire 168 “ Circumference of 169 “ Destructive Strength of. 171 “ Iron and Steel , Relative Dimensions of. 168 I High Water, Time of. 74, 75 I Hills or Plants in an Acre. . . * 193 i Hoggin 690 | Hoisting Engines 901 Honey 207 Hoop Iron, Weight of 129, 131 Hopper Barge and Dredger 899, 900 Horizon, Dip of 60 Horizontal Wheels 572 Horse 43 6 Horse Power 441, 733, 758, 914 “ Tractive Power of ... . 436 “ Transmission of. 188 Horses, Age of 186 “ Labor of etc 435-437 “ Performance of. 439, 440 “ Transportation of 192 “ Weight of 35 Horseshoe Nails, Length of. i 33 Horseshoes and Spikes 152 Hulls of Vessels, Diameter of Rivets. 830 Human and Animal Sustenance 203 Hydraulic Radius or Mean Depth. “ Ram “ Cement Hydrostatic Press 561, 901 Hydraulics 529, 557 “ Canal Lochs 553, 555 “ Circular Sluices , etc 537 “ CoeffVts of Circular Open- ings or Sluices. 536 “ In Clack or Trap Valve or Cock. . 546 A* 552 561 591 Page Hydraulics, CoeffVts of Friction. 544-546 “ u of Resistance in Bent or Angular Circular Pipes , Valve Gates , or Slide Valves. . . 545 “ Computation of Volume of Discharge 533 “ Curvatures, Radii of .. . 544 “ Curves and Bends 545 •“ •“ Coefficients of Re- sistance 545 “ Cylindrical Ajutage 549 “ Discharge from a Notch. 541 “ ‘‘ from Conduits or Pipes 530 “ “ • from Vessels not Receiving any Supply 538 “ “ from Vessels of Communication 541 ■“ • “ from Irregular - shaped Vessels. 542 ‘‘ “ of Water in Pipes for any Length and Head , etc., 547, 548 “ “ or Efflux from Va- rious Openings and Apertures. 532 “ “ under Variable Pressures 540 “ Experiments on Dis- charge of Fluids, from Reservoirs , Conduits , or Pipes 529, 531 “ Flow and Velocity in Rivers , Canals , and Streams 550, 552 “ 11 in Lined Channels. 551 “ “ of Water in Beds. . 542 “ Forms of Sections of Canals 543 “ Friction in Pipes and Sewers 543, 544 “ “ of Liquids 531 “ Heigh t of a Jet, To Com- pute 913 “ Jets d’Eau 550 “ Miner's and Water Inch 557 “ Miscel. Illustrations .556, 557 “ Prismatic Vessels 539 ‘ - Rectangular Weir . . . 532-535 “ Reservoirs or Cisterns . . 541 “ Sluice Weirs or Sluices.. 535 “ Submerged or Drowned Orifices and Weirs.. 553 u To Compute Depth of Flow over a Sill , etc. . . . 534 “ Fall of a Canal or Conduit to Conduct and Discharge a Given Quantity of Water per Second . 914 u “ Head and Discharge of Water in Pipes of Great Length . . . 914 X INDEX. Page Hydraulics, To Compute Head or Height of Water from Surface of Supply to Centre of Discharge 544 << “ Vertical Height of a Stream Projected from a Pipe 549 u “ Volume of Water flowing in a Riv- 543 Page Hydrodynamics, Water -wheels, Di- mensions of Arms 571 “ Wldtelaw's Wheel. . . 576 “ Short Tubes , Mouth pieces , and Cylindri- cal Prolongations or Ajutages 53 6 ~537 “ Triangular , Trapezoid- al , Prismatic Wedges , Sluices, Slits, etc 538 u Variable Motion 543 « Velocity of Water or of Fluids S3 1 “ Vena Gontracta 52Q ‘ < Weirs or Notches 5 39i 9 1 9 Hydrodynamics . 558-580 “ Barker's Mill 5 77 “ Boyden Turbine 574 “ Breast Wheel 568-570 “ Centrifugal Pump . . 579 “ Current Wheel 570 “ Flutter Wheel 571 “ Fontaine Turbine. . . 574 “ Friction of Journals or Gudgeons 571 u Horizontal Wheels. . 572 u Hydrostatic Press. . . 561 “ Hydraulic Ram, 561, 562 n Impact and Reaction Wheels •• 57 6 “ Impulse and Besist- ance of Iluids. 577, 578 “ Jonval Turbine 575 “ Memoranda 57 1 “ Overshot Wheel . . 563-566 “ Percussion of Fluids. 579 “ Pipes, Elements of . . 561 u “To Compute Thickness of etc... 560 “ PonceleV s Wheel 567 u Pressure and Centre of 558-560 “ Reaction Wheel 576 “ Tangential Wheel. .. 576 u Tremont Turbine... . 576 “ Turbine and Water Wheels , Compari- son Between 579 “ Turbines 57 2 ~57 6 “ Undershot Wheel 566 “ Victor Turbine 576 “ Water Power 562 “ Water- Pressure En gine 579 “ Water Wheels 563 a “ Diameter and Journal of a Shaft , etc — 57 Hydrometers “ To Compute Strength of a Spirit Hygrometer “ To Ascertain Dew-point. “ To Compute Volume of Vapor in Atmosphere. Hyperbola, To Describe \ I. Ice 912 and Snow 849 ‘ ‘ Boats and Speed of. 896, 909 “ Strength of. *95 Impact and Reaction Wheels 576 “ or Collision 580 Impenetrability ^95 Inclined Plane 619, 628 Incubation of Birds, Periods of 192 Indicator 7 2 4 Inertia, Moment of a Revolving Body, To Compute 609 “ Moment of, to Ascertain Ap- proximately 659 “ of a Revolving Body, To Compute 616 “ of a Solid Beam 819 Ink, Chinese or India 9°7 Inks 8 75 Insects and Birds I 9 & Interest, Simple and Compound. 107, 109 Involute, To Describe 22 9 Involution 9 6 Iron - 6 37- 6 4o Bolts in Wood , Tenacity of. 198 Bridge , and Iron Pipe Bridge. . 178 Cast Iron.. Pipes, Weight of 637 48 Wrought Iron 639 Rope, Hemp , Iron , and Steel, . . 164 * ‘ Hemp and Steel , U l timate Strength and Safe Load of. «<>5 “ Variable Motion 543 Steel and Hemp Rope, Relative Dimensions of 168, 172 Wire and Hemp Rope 168 ‘ * Gauge , Weight and Length. 163 “ “ Weight of. Irregular Bodies, Volume of .120, 124 J. Jewish Measures. 53 .1 Ui Kill Jumping, Leaping, etc K. Hedges and Anchors, Weight and Khorassan, or Turkish Mortar.. . INDEX. xi L. Page LABOR 433, 434, 436, 468 Lacquers 875 Laitance . 593 Lakes, Areas, Depth , and Height of. '. 181, 182 Lamps, Fluids, and Gas 584 Land Measure 29 Larry ing 59 8 Laths 603 Latitude x 9 8 Latitude and Longitude 76-80 Launching Vessels, Friction of. 478 Lead 640 “ Encased Pipe, Weight of 151 “ Measure ... 32 “ Pipe 831 “ Pipe , Weight of 139, 150 “ Plates , Weight of 146 “ Weight of 136,151 Lead and Cast - iron Balls, Weight “ “ and Volume of. 153 “ “ To Compute Weight of 155 Leap or Bissextile Year 7 o Leaping, Jumping, etc 439, 440 Leaves 207 Lee -way or Drift of a Vessel 910 Legal Tenders 38 Lenses and Mirrors 670 Levelling, Geographic 55, 56 Lever 624, 626 Lifting 430 Ll ? t HT -- ;•••••; i95, 583-587 Decomposition of 583 “ Gas, Volumes and Temperature °f 585, 587 ‘ 1 Gas and Electric . ........... “ Intensity of. ^5 “ Loss of by Use of Globes. ..... 584 11 Refraction • 5 8 4 ii Relative Intensity , Consump- tion, and Cost of Various Modes of Illumination 584 “ Standard of. OT o Lightning 9 . ° Lignite 479, . 8l Limes, Cements, Mortars, and Con- cretes 588-59 7 “ Cements, and Mortars, Ex- periments of Gen. Gillmore. 596 “ and Cements 3 94 “ Asphalt Composition 5 g 3 “ Concrete or Beton 59 3 “ Conclusions from Experiments 590 ‘ ‘ General Deductions cg6 “ Pozzuolana ** 5 g g “ Transverse Strength sg 6 u Turkish Plaster or Hydraulic Cement t- QI Lines, To Draw, Bisect , etc. 221 Liquid Measure 30,31 46 Liquids, Expansion of '$ 20 “ Volume of at Boiling-Points 518 Liquors, Proportion of Alcohol in. .. 204 _ . Proof of Spirituous 218 Lithro-fracteur 443 Locomotive “Experiment” go 2 “ Axles g IO Locomotives, Operation of. .681-685, 912 “ Adhesion 681,685 Tractive Power 681 “ Train Resistances. .682, 920 Log Lines 2 - Logarithm of a Number 23 Logarithms 305-310 “ Hyperbolic 331-334 “ . of Numbers. ....... .311-330 Longitude, To Reduce to Time 54 “ and Latitude 7 6 “ Lengths of a Degree 60 “ of Observatories in Time. 80 Luminous Point IQ5 Lunar Cycle or Golden Number. . . 1 . 7I Lunar Month 70 M. Macadamized Roads 687, 690 Machines and Engines, Elements of 898 Magnetic Variation 57-59 “ Bearing of N.Y. . 184 Magnetism 614 Malleable Castings * ’ ’ 6 39 Cast Iron. ............... 785 Manganese Bronze ’ . 832 Manures ’ [* " * jgg Marine Day ’ ’ 37 Marine Steamers and Engines. .886, 887 “ Auxiliary Freight 887 “ Fire boat 887 ‘ ‘ Freight and Passenger 886, 887 “ Iron Cruiser.. 886 a u Freight and Passenger Propellers 886 “ Steel Launch 887 Marine Steam Vessels and En- GINES 887-891 Composite Yachts. 888 “ Ou tier ] 889 “ Ferry Boat 889 “ Iron Yachts ’ 888 “ Light Draught. ” .* 889 “ side Wheels 889, 890 u Steel Launch 887 “ “ Yachts .’888,889 “ Torpedo Boats, Iron, Steel, and Composite 889 u Wood Side Wheels , Passengers and Deck Cargo . . . 890, 891 “ 11 Propellers 891 “ “ Towing 891 Masonr y 197, 597-605, 9x3 Arch, To Compute Depth of . . 605 “ Brick , Stone , and Granite, 595-600 “ . Designation of .602-603 “ Estimate of Materials and Labor, etc. 6 04 “ Rubble 6 0 j \\ % 01 } e .600, 603 Technical Terms c Q7 t Q 8 Mason’s and Dixon’s Line 188 Mastic Q Materials, Strength of 761-841 Matter.. xii INDEX. Page Mean Proportion 94 Measures, Ale and Beer 45 “ Apothecaries' 47 “ Avoirdupois 3 2 “ Board and Timber 6i “ British and Metric , Com- mercial Equivalents of J 906 “ Builder's 46 “ Circular n3> JI 4 Cloth 2 7 “ Coal 33 “ Copying 2 9 “ Corn J 9 8 “ Cube 3° “ Dry 3° £ ‘ Foreign of Value 39"43 u Foreign Memoranda 43 u Geographic , and Distances 54 kC Geographical and Nauti - cal 26 u Grain 3 2 “ Grecian 53 “ Gunter's Chain 26 “ Hebrew and Egyptian . . . 53 “ Jewish 53 “ Land 2 9 u Liquid 3 1 u Men and Women 35 “ Metric 27-33, 36, 44, 46, 47 “ u Equivalent Value , U. S. , and Old and New U.S 28, 30, 33 u “ power and Work. 36 u “ Temperatures 37 1 u “ Velocities 37 I u “ Volumes 36546 u “ Weights and Press- ures 36 a Miscellaneous. . . .27, 29, 31, 46 ‘ ‘ Nautical 3° £k Old and New TJ. S. Ap- proximate Equivalents. 33 “ of Length 26, 44 « of Offal in a Beef and Sheep. 35 “ of Paper 2 9 “ of Surface 29,44 it of Timber, Local Stand' ds 62 “ of Time 37 tt of Value 3 8 ‘ ‘ of Volume •' • • • 3°^ 45 “ of Weight. 3 2 > 47 “ Pendulum 2 7 “ Roman Long 53 “ Ropes and Cables 26 tt Scripture and Ancient. . . 53 “ . Shoemaker's 2 7 « Timber , English 62 “ Troy 47 “ Vernier Scale 27 “ Wine and Spirit 45 “ Wood Measures and Weights ..26-35 a English and French, 44. 45 “ Foreign 4 8 "53 Meat, Analysis of, and of Ftsh and Vegetables 2 °9 “ To Preserve x 9 6 Mechanics.. Page Meats, Roasting of. • • 2 °6 Mechanical Centres 605-614 Centre of Gravity. . .605-608 “ of Gyration of a Water-wheel 61 1 ‘ ‘ of Gyration . . . 609-61 1 Elements of Gyration. ..610 Ratio of Gyration 609 Mechanical Powers 624-634 Compound Axle or Chi - nese Windlass 627 Inclined Plane 628 Lever 624-626 Pulley 632 Rack and Pinion 628 Screw 630 “ Differential 632 Wedge 630 Wheel and Axle 626 . m 4 .614-623 Accumulated Work 619 Decomposition of Forces. . 620 ‘ ‘ Dynamics 616-620 “ Moment 614 “ Moments of Stress on Gir- ders 621-623 *• Motion on an Inclined Plane 619 a Solid, Fluid, and Aeri- form Bodies 614 11 Uniform Motion 617 tt Work by Percussive Force 620 Melting-Points 5*7 Memoranda • * 9°7i 9 12 “ Cast and Wrought Iron and Steel. 8 3 2 Men., 433-435 Mensuration of Areas, Lines, feUR- fages, and Volumes, 335"37 8 Any Figure of Revolution, 35 8 5 376 “ Plane Figure 359 Arc and Chord, etc., of a Circle ..^.343-345 Area Bounded by a Curve. 342 Capillary Tube 35 8 Cask Gauging 377 Chord of an Angle 359 Circle “ Section of 34® “ Segment of. 346 Circular Zone 349 Cones 363 Cube and ParaUelopipedon 360 Cycloids 35 2 Cylinder 35°, 3 6 3 “ Sections 357 Ellipsoid, Paraboloid, and Hyperboloid of Revolu- tion 357 > 375 Helix {Screw) 354 Irregular Bodies 377 “ Figures 34 1 Links • • *3535 37° Lune 35 2 of Areas, Lines, and Sur- faces 335~359 INDEX. xm Page Mensuration op Areas, Lines, Sur- faces, and Volumes. 360-378 “ Parallelograms 335 “ Polygons 338,343 “ Polyhedrons 362 “ Prism . 350,360 “ Prismoid 351, 361 “ Pyramids 354, 365 “ Reduction of Ascending or Descending Line to Hori- zontal Measurement 359 “ Regular Bodies, 340, 341,362, 364 “ Rings 353 368 “ Sphere 347,367. ‘ ‘ Spheroids or Ellipsoids . 348, 368 ‘ ‘ Spindles 355 , 370-374 Spirals 355 ‘ To Ascertain Area of any Plane Figure 359 “ To Compute, Chord of an Angle 359 “ To Plot Angles 359 “ Trapezoid 338 “ Triangles 335-338 “ Ungulas 35 x “ Useful Factors 343 “ Wedge 350, 361 Mercurial Gauge 9 IO Meta- Centre of Hull of a Vessel , To Compute 659, 913 Metal Products ofU.S. 910 Metals, Alloys and Compositions . . 634 “ Comparative Quality of Va- rious 821 “ Lustre of X94 “ To Compute Weight of 131 “ To Compute Weight of by Pattern 217 Milk, Nutritive Values and Constitu- ents of 202 “ Percentage of Cream 205 “ Relative Richness of of Several Animals 207 “ To Detect Starch in 196 Mineral Constituents from an Acre of Soil... 189 Mineral Waters, Analysis of etc.. 850, 851 Minerals, Hardness of..... X93 Miner’s Inch cc 7 lining ] 445 Mining Engines and Boilers 901 Mining Ropes x6 5 Mirage 195, 669 Mirrors and Lenses 670 Miscellaneous Elements 188-198, 906 Mixtures 871-879 Operations and Illus- trations 879-885 Mississippi River, Silt in 910 Models, Strength of Molasses 207 Molding and Planing a 7 6 Molecules, Velocity , Weight , and Vol- ume of I94 Momentum !.!!!!]. 105 Monoliths 179 Page Month, Lunar 70 Months, Numbers of 74 Moon’s Age, To Compute 74 Mortar 590, 592, 595 Mortars, Limes, Cements, and Con- cretes 588-597 Motion, Accelerated and Retard- ED * 494, 495 Motion of Bodies in Fluids 645 “ Pressure , Velocity, Time, etc., To Compute 648 i 1 Resistances of Areas and Dif- ferent Fig' sin Water or Air. 646 Mountains and Passes, Heights of . . . 182 Mowing 433 “ Machine 910 Mule 437 Mural Efflorescence 593 N. Nails, Length and Number of ... 153, 154 “ . and Spikes, Retentiveness of. .. 159 National Road x 7 g Natural Powers X98 Nautical Measure 3Q Naval Architecture 649-667 “ Angles of Courses and Sails . . 665 ‘ 4 Heel and of Steady Heel 664, 665 “ Area of Sails, etc 663, 664 “ Centre of Effort, To Compute Location of. 659 “ “ of Gravity of Hull, En- gines, etc 658 “ Centres of Lateral Resistance and Effort. 658 “ Course and Apparent Course of Wind 666 u t Curve and Coefficients of Dis- placement. 657 “ Displacement, and its Centre of Gravity 653, 655 “ Elements of a Vessel 653, 660 “ of Capacity and Speed of Several Types of Steamers 660 “ Experiments upon Forms of Vessels 649 1 1 Freeboard 666 ‘ 1 Lee-way ‘ 666 “ Masts, Location of 66 4 “ Memoranda 667 ‘ ‘ Meta-Centre of Hull of a Vessel, To Compute 659, 913 “ Moment of Inertia and Meta- Centre 659 “ . Pitch of Propeller and Slip of Side Wheels 662 ‘ ‘ Plating Hulls 667 “ Power Utilized in a Steam- vessel 662 “ Relative Positions of Lateral Resistance and Centre of Effort. 659 Resistance of Bottoms of Hulls, 663 “ “ to Wet Surface of Hull, To Compute 653 XIV INDEX. Page Naval Architecture, Rudder Head . 667 “ Sailing, Ratio of Effective Area of Sails , eic., to Wind 663 “ “ Power of a Vessel 665 “ Sails, Area and Trimming of 664, 665 ‘ ‘ Surface , Bottom , and Immersed Hull , To Compute 653 44 Stability 649, 650 44 “ Results of Experiments upon 649, 650 “ “ To Compute Statical and Dynamical 651 “ 11 To Determine Mean of of Hull of a Vessel.. 650 “ To Compute Displacement, Ap- proximately, and Co- efficients of 655 “ “ Elements of Power Re- quired to Careen a Vessel 652 “ 44 Power Required in a Steam - vessel. Speed and IP, and Coeffi- cient of. 661 “ Wind, Effective Impulse of. . . 665 Needle, Magnetic Variation of. 57 44 Decennial Variation of 58 44 Variation of it in U. S. and Canada 59 Neutral Axis of a Beam 820 New and Old Style 37' 7° Niagara, Falls of. *9 8 Nitro-Glycerine 443 Non-conductibility of Materials 911 Notation 2 5 Number of Direction •• 7* Numbers, Properties and Powers of. % 98 “ 4 th and 5 th Powers of. . 303, 304 44 To Compute 4 th, $th, and 6th Power, and 4th and 5 thRootof 304 Organic Substances, Analysis of by Weight 1 9° Orthography of Technical Words and Terms * 021-928 Oscillation and Percussion, Centres of..., 612-614 44 Centre of in Bodies of Various Figures 613 4 4 To Compute Centre of, 6 1 2 , 6 1 4 Overshot Wheel 5 6 3 Nutritive Equivalents,tfuma« Milk, 1 205 0 . 179 Obelisks Observatories, Latitude and Longi tude Oceans and Seas, Depths and Areas. . 182 Offal, Weight of, in a Beef and Sheep . 35 Oil , Yield of , from Seeds 189 “ Calce and Vegetables, Nutritious Properties of. 2 °4 44 Proportions of , in Air-dry Seeds. 203 Oils, Petroleum, Schist, and Pine- Wood 484 Old and New Style 37 ? 7° Onion 2 °7 Opera-Glasses 671 Opera-Houses. . Optics -668-671 44 Elements of Mirrors and Lenses 670 44 Refraction 668,669 44 To Compute Dimensions or Vol- ume of an Image 668 Ordnance, Energy of. 9 JO Painting 66 Parabola, To Describe 229 Park, Deer *79 Parsnips 2 °7 Passages of Steamboats 896 44 Steamer and Sailing Vessels. 897 Passes, Mountains, and Volcanoes. . . 182 Pavement, Asphalt 690 4 4 Block Stone 689, 690 44 Granite 690 44 Rubble Stone. . .1.. 689 44 Telford 688 44 Wood 689,690 Pavements, Roads, and Streets. . .686-690 Payments, Equation of 109 Peat 4 82 Pendulum Measure 27 Pendulums 45 2 44 Centre of Gravity of.... 453 44 Lengths and Number of Vibrations of. 453, 454 44 Time of Vibration . . 454 Pennants, Ensigns, and Flags, U.S.. . 199 Percussion and Oscillation, Centres of 612-614 Performances of Men, Horses, etc. 438 Perimeter of a Figure 9 12 Permutation 100 Perpetuities 112 Petroleum, Elastic Force of Vapor. . 707 44 Evaporative Effects of. . . 910 Pile Driving 433i 671-673, 902 44 Coefficient of Resistance of Earth, To Compute 672 44 Pneumatic 672 44 Ringing Engine 672 44 Sheet Piling 672 44 Sinking 673 44 To Compute Weight of Ram 672 Piles, Foundation 198, 909, 912 44 Extreme Load a Pile will Bear 913 44 Retaining Walls of Iron 196 Piling of Shot and Shells 65 Pillar at Delphi *79 Pipes 747 44 Lead and Tin, Weight of. 139 4 4 Riveted Iron and Copper, Weight of • H 8 44 Steam, Gas, and Water 138 44 To Compute Thickness of. 560 a 44 Weight of Metal. . . 147 4 4 Tin , Weight of. 44 Lead Encased Pivots, Friction of..... 80 151 151 593 472 INDEX. XV Position . Potato . . . Page Planing Cast Iron and Molding 476 Plank Roads 688 Plants or Hills, in an Acre 193 Plaster, Turkish 591 Plastering 197, 604 Plate Bending 476 Plates, Test of and Bolts.. 749, 753 “ Thickness of, by Wire Gauges 121 “ To Compute Thickness of . . . . 751 “ Wrought- Iron Shell .... 750 Plating Iron Hulls 667 Ploughing 433 Pneumatics. — Aerometry 673-676 Pointing 598 Poisons, Antidotes and Treatment of. 185 Poles and Spars 62 Poncelet’s Wheel 567 Population, Comparative Demity of and Number of Persons in a House in Different Cities 910 “ of Earth 188 of Principal Cities 187 9 8 > 99 207 POWER AND WORK, METRIC 36 “ and Mechanical Energy of io Grains of Various Substances when Oxidized in Human Body 205 Motive 910 Movers and Transmitters of.. 797 of a Quantity , To Ascertain Value of. 359 “ Required to Draw a Vessel up an Inclined Plane 910 “ To Sustain a Vehicle on an In- clined Road 845 “ Transmission of. 167 Powers, Natural I9 8 “ of first 9 Numbers 98 “ of 4 th and $th Numbers. 303 “ of 6th Number , etc 304 Probability 114-117 Progression . 101-105 Proof of Spirituous Liquors 218 Propellers 73 o, 886-891 Propeller Steamers, Ordinary Distri- bution of Power in g n Properties of Numbers g8 Proportion 94 _? 6 Pulley 433 Pumping Engines 73 8 902-903 Pumps, Fire 738 Centrifugal.. 01 x •• Worthington , 738 Circulating , 740 Pushing or Drawing Pyramids, Statues, etc 178 Q. Quartermasters, Service Train of . . . . 198 R. Race-Courses, English, Lenqth of.... c 2 Rack and Pinion 628 Railroads ...].!!! 178 Rails, Iron and Steel * * gi 2 „ . Par* Rails, Tangential Angles for Chords and Curves 677, 678 “ To Define a Curve 677 * ‘ To Determine Elevation of Out- er Rail 4 679 Railways 677-685 “ Curves by Offsets 678 ‘ ‘ Elevation of Rail 678, 679 Operation of Locomotives . 681-685 “ Points and Crossings 678 “ Radii of Curves 679 “ Rise per Mile and Resist- ance to Gravity 679 Sidings 677 “ To Compute Weight of Rail 679 ‘ ‘ Load a Locomotive will Draw up an Inclination 680 “ “ Maximum Load Drawn up the Maximum Grade it can Attain. . . 680 “ “ Resistance of Grav- ity 679 ‘ ‘ Traction , A dhesion , etc 680 “ Velocity of Trains 680 Railway Trains gn Rainfall, Volume of. 850 Reaction and Impact Wheel 576 Reaping..... 433 Rebate or Discount 109 Reciprocals 3 q 4 Rectilineal Figures, To Describe. . . . . 222 Refraction. 668 “ of Light 584 Refrigerator, Surface of 5I2 Rendering 59 8 Resilience of Woods 763 Retaining Walls ! ! ] 695 “ of Iron Piles 196 Revetment Walls 694-700 “ Surcharged 699 “ To Compute Elements of 696 Ringing Engine 672 River Steamboat gj 3 River Steamboats and Engines. .892, 914 “ Traction on 848 Rivers, Current of X g 3 “ Descent of Western 188 ‘ ‘ Flow of Water in c c 0 , c c 1 “ Lengths of ...183 Obstruction in 55! Riveted Joints, Comparative Strength of etc. . .751, 752, 755, 829, 829 Experiments on 783 RiVET,!iG 755-757 _ . 829, 830 Diameter of etc 756, 829 Memoranda 830 Roads I7 g Streets, and Pavements, 686-690 690 689 Rivets. Bituminous . Concrete Construction. Corduroy . . . . 686 688 XVI INDEX. Page Roads, Getters , Fillers , and Wheelers , Proportion of, \ in Different Soils £88 “ Macadamized 687, 690 “ Materials , etc 689 “ Metalling 690 “ Miscellaneous Notes 690 “ Planlc 688 “ Rolling, Sprinkling, etc 690 “ Ruts 687 “ Sweeping, Watering, and Wash- ing 690 “ . Telford . 688 Roadway, Construction of 687 Rock • 467 Rock and Earth, Excavation and Embankment of T 9 2 Rocks. Bulk of 468 “ ' Weight of, per Cube Yard 468 Roman Calendar 7 1 “ Indiction , To Compute 71 “ Long Measures 53 Roof Plates, Corrugated, Weight of... 131 Roofs of Buildings *79 “ Wooden i8 9 Root, To Compute, of an Even Power. 98 “ . To Extract any • 97 Roots and Grains, Weights of 34 Roots, Square and Cube 272-302 To Compute 4 tli and 6 th 301 Ropes 7 82 ‘ and Cables 26 and Chains of Equal Strength. 165 Cables, Chains, etc 163-175 “ Endless. ..... . *67 “ Equivalent, and Belts 167 ‘ ‘ Hawsers and Cables 169-172 “ 1 ‘ To Compute Strain 0/170, 1 7 1 u u u Circumference of 171 “ “ “ Weight of. 172 “ Hemp and Wire 167 “ “ Iron, and Steel 164,168 “ “ Weight and Strength of . 172 “ Mining. *65 ‘ To Compute Stress, Tension , and Deflection 166 u “ Circumference of, etc 169 “ Transmission of Power. 167 “ Wire 161-172 “ u Experiments on 169 Rowing 433 Rubble Stone Pavement 689 Rule of Three 95 Running.. 43 s ? 44° Safety Valves 74 6 , 9 12 Sago..... 2 °7 Sailing ••• ° • * “ . Vessels, Iron 894, 095 Sails, Propulsion and Area of. 663 “ . . Trimming of. 665 Saline Saturation 7 26 “ Matter in Sea Water 7 2 7 Page Sandstones *93 Saw Mill 9°4 Sawing Stone and Wood 196 Saws, Circular i 97 j 477 > 9 11 “ Vertical and Band 477 Scale or Sediment in Boilers 726 Scales, To Divide a Line , etc 221 “ Weighing without 66 Scarfs, Resistance of. 841 Screw 630 Cutting 477 Differential 632 Propeller, Pitch and Speed of . 662 Screw Propeller, Friction of Engines 663 Scripture and Ancient Measures.. 53 Sea Depth l8 4 Seas, Depth and Area 182 Secants and Cosecants 4°3“4 I 4 “ “ To Compute Degrees , Minutes, etc 414 Seeds, Humber of, in a Bushel, and per Sq. Foot per Acre 193 Proportion of Oil in Air -dry . 203 Segments of a Circle, Area of. .267, 268 44 ‘‘ To Compute Area of 268 Sewers • • • * 6 ^ 1 ’ 6 9 2 ‘■4 Drains, Diameter and Grade of, to Discharge Rainfall. . 906 “ Pipes and Sewage 692 Shaft, Bearings for Propeller 473 Shafts 77 8 > 793? 794> 79 6 , 797 “ and Gudgeons 79 ° Deflection of. 77° “ Supports of 796 Shearing or Detrusive Strength, 782, 783 Sheathing and Braziers’ Sheets 155 and Braziers' Copper 131 Nails, Weight of Sheet Iron, Galvanized — Sheet Piling Shingles..,.. Ship and Boiler Plates. . . . Shoemaker’s Measure, Shot and Shell, Piling of 05 Shot, Chilled and Drop 90b “ No., Diameter, and Numbers Of 9°6 Shrinkage of Castings. 2 i« Shrouds, Hemp and Wire 173 Side Wheels, Area of Blades 662 # “ Friction of Engines.. 662 “ Slip of. 662 1 Sides of Equal Squares 258,259 \ Silt, in Mississippi River 9 10 Silver Sheet, Thickness of. IX 9 < Simple Interest io 7 . Simpson’s Rule, To Compute Area.. . 344 . “ Volume of an Irreg- ular Body 8 7 ° Sines and Cosines 39°"4 02 “ To Compute 4® 1 u “ Number of Degrees, etc 4° 2 Sixth Power of a Number, To Com i35 .124, 129 .... 672 . . . . . 63 828 Sand. , 599 pule. 3°4 INDEX. xvii Pape Skating 439 Slackwater, Canal , etc., Traction on. 848 Slaking 594 Slate, To Compute Surface of. 64 Slates and Slating 64 Slates, English 64 “ Weight per 1000 and Number Required to Cover a Square 64 Slide Valves 731, 733 Smelting of Ore 445 Slotting 477 Smoke Pipes and Chimneys 748 Snow-flakes 195 “ Line of Perpetual Congelation. 192 “ Melted 195 Snow and Ice 849 Solar Day and Year 70 Solders 634, 636 Soldering 875 Sound 195 “ To Compute Velocity of 428 Soundings, To Reduce to Low Water. 60 Spars and Poles 62 Specific Gravity, To Ascertain, of a Body Heavier or Lighter than Water 209 “ of a Body Soluble in Water 209 “ of a Fluid 209 Specific Gravity and Weight. .208-215 ‘ ‘ To Compute Weight of a Body 215 “ “ Proportions of Two In- gredients in a Com- pound , etc 216 “ Weights and Volumes of Va- rious Substances in Ordi- nary Use 216 Spikes and Nails, Retentiveness of. . . 159 Spikes, Ship, Boat, and Railroad. i 5 2 i i 54 and Horseshoes 152 “ and Nails 159 “ Wrought - Iron Nails and Tacks. 154. Spiral, To Describe % §3^ Spires v . . x %>' 2l8 ( 191 7.79 Spirituous Liquors, Dilution Per Cent] “ Proof of ... %VN “ Proportion of Alcohol. Springs, Deflection of. v . ....... . Square and Cube Root, To Compute^ of a Higher Number than cop tainted in fable. “ Ta Ascertain, of a Number consisting of Inte gers and Decimals “ of Decimals v , , Square PvOOT, To Extract.... . . Square, To Ascertain One that has Same Area as. a given Circle. Squares, Cubes, and Square and Cube Rgqts , .272- M , To Compute Square and Cube Roots of Roots, of Whole Numbers, and of Integers and, Pecimals . . . Squads, Sides of Equal, in Area to a Circle 258, Pape Stability 693-703 “ Dynamical 651 “ Dynamical Surface 651 “ of Earth... 695 “ of Models 649 “ To Ascertain , of a Body . . 693 “ To Careen a Vessel 652 “ To Compute Statical 651 “ Equilibrium of Walts 701 “ “ Stability 701 “ To Determine Measure of Hull of a Vessel , etc 650 Staging, Coach 44 o Staining 876 Starch, Proportion of in Vegetables. . 205 Stars, Velocity of. I9 8 Statics 6x5-617 “ Composition and Resolution of Forces. 615 “ Equilibrium of Force 616 Statues, Pyramids, etc 178 Stay Bolts, Diameter , etc 754 Steam 704-727 “ and Air. Mixture of. 737 “ Average Pressure , etc 71 1, 7I2 “ Blowing off of Saturated Water 726 “ Boilers 829 “ “ Zinc Foil in n I2 “ Clearance, Effect of 7^ ‘ ‘ Compound Expansion 720-7 24 “ Conclusions on Actual Effi- ciency of. 724 “ Condensing Surface , Experi- ments on qjj “ Density or Specific Gravity of 706 “ Expansion 7IO “ Effects of. . 713,718,719 ‘ ' Points of. 7 1 2 “ Feed Water , Gain in, at High Temperatures, To Compute. . 719 “ Gain in Fuel, To Compute 725 u Gaseous and Total Heat of. .. 710 “ Ifammers i 79 “• Heat of Saturated 705 “• Heating Co. of N. Y. 904 Indicator ... 724 “• Injector , . . . . . . 73 6 “ Latent Heat of. 70 . 7 “ Mean Pressure, To Compute . . 713 “ Mechanical Equivalent of. .. . 705, “ - Pipes , Gas, etc , 138 ‘ ‘ Poin t .of Cutting off, and Press- ure at 7, I(t> Pressure q/., * 705, 710, 7/11 u Weight , and Temper- ature ...... 705 “ Properties of, of Maximum Density 7^7- Ratio of Expansion ^q, u Relative Effect of Equal Vol- ume Of. 7x4; u - Saline Saturation in Boilers:. ‘ * Saturated 704, 708, 7 1 C u • Scale or Sediment * Removal off 72S “ Specific Gravity. of . , , . , . .704, 706* “• Superheated 7^, “ Temperature of. 703; XV111 INDEX. Page Pagt Steam, Temperature of Water m a Condenser 7°7 tt To Compute Volume of Water Evaporated per Lb. of Coal 725 tt “ Consumption of Fuel ... . 725 “ Total Effect of 1 Lb. of 7*4 it “ Heat of Saturated 707 t< Velocity of to Compute 7 IO > 9*3 “ Volume of a Cube Foot of Wa- ter in 7°6 « tt of Cylinder, to Compute. 715 “ “ of to Raise a Given Vol- ume of Water 7°6 « t. of Water Contained in.. 706 a “ of Water to Raise or Re- duce to any Required Temperature 7°7 “ Weight of. 7°5 “ Wire Drawing 7 l8 “ Woolf Engine . 7 22 Steamboats, River, and Engines . 892, 893 ‘ 1 Passages of 896 “ Wood Side Wheels 892 “ “ Ferry Boat 893 “ “ Passenger and Light Freight 893,894 a “ Stern Wheels .. : 893,894 ■Steam-Engine 727-760 ‘ ‘ and Boilers , Cost of per Day. 904 “ Area of Feed Pump 73 6 a a injection Pipe 735 “ Boiler 73^-745 a a Evaporative Power oj . 757 a a Weights of 759 a Circulating Pumps 749 “ Condensing • 7 2 7 “ Elements and Capacity of Steam Pumps 73 8 a Evaporation '• 747 a Fire a Flues and Tubes. 747 a Friction of Side Lever 478 a General Rules 728, 730 a IP, to Compute , etc 733, 734 a Injection Pipe , to Compute Volume of Flow 735 a Eon-condensing 728 a Plates and Bolts 749~753 a Practical Efficiency of 737 a Propeller , to Compute Thrust of 73i a Propellers 73 °j 73 1 a Relative Cost of 757 a Results of Operation of. . .737, 9 21 a Riveting -753-757 a a General Formulas. . 757 a Safety Valves 74 6 , 9 12 a Smoke Pipes and Chimneys . 748, 749 a Stay Bolts , Rods , etc 753, 754 a Steam Room 748 “ Slide Valves 73* tt a To Compute and ^4s- certain Lap, Breadth of Ports, Portion of Stroke, Lead , etc. 731-733 a Volume of Circulating Water 735 Steam-Engine, Volume of Injection and Feed Water 735, 736 a Volume of Water Required to be Evaporated 734 a Water Surface 748 Water- Wheels 730 Weights of 758, 759, 9” Steam Fire-Engine 9°4 Steam Pumps, Elements and Capaci- ties of. 738 Steam- Vessel, Power Utilized in 662 Steam- Vessels, Resistance to, in Air and Water 9 11 Steamer u Great Eastern” 179 Steamers, Friction of Screws 478 Steaming Distances 86 Steel 640-643, 750, 783, 787, 827 a Cables, Galvanized ... 163 a Columns , Crushing Weight of . 768 < ‘ Hemp and Iron Ropes, Relative Dimensions of 168 a Hemp, and Iron Wire Rope. . . 164 a Locomotive Tubes 138 a Manufacture of. 642 “ Plate 8 3 o a Plates, Weight of. 118, 119, 146 “ Relative Dimensions of 172 ‘ ‘ Rolled and Bar, Weight of. . 1 34, 1 35 a Rope , Hemp and Iron, Round and Flat 164 a weight of. 136.149 t ‘ Wire , Weight of 20, 1 2 1 Sterling, Pound 3 8 Stings and Burns, Application for. . . 196 Stirling’s Mixed Iron 7 8 5 Stirrups or Bridles, for Beams 838 Stone and Ore Breakers 9°3 Stone, Expansion and Contraction Of- a Hauling 4° 8 a Masonry 595 a Resistance of , to Freezing 184 a Sawing 196,904? a Voids in a Cube Yard 690 . Stones, Cements, etc 766 Streams, Flow of Water in 55° Street Rails or Tramways 4351 9° 2 Streets, Roads, and Pavements.. 686, 690 Strength of Materials 761-841 Crushing Strength 764-769 “ Comparative Value of Long Solid Columns 7 6 9 : a of Cements, Stones, etc 596 ; a of Columns . to Compute } Weight of. 7 68 “ of i-inch Cubes 767^ 1 ‘ of Various Materials . . 765, 766. a Resistance of Rivets 9° 8 - 1 a Riveted Joints 828,829 a Safe Load of Columns , Arches , Chords , etc., of Cast Iron 766 a Weight borne with Safety by Cast-Iron Column. . . 768 a Woods, Destmictive Weight of Column of. 769 INDEX. xix Page Strength of Materials, Crushing Strength of Woods , Rel- ative Value of Various . 769 “ Wrought- Iron Cylinder and Rectangular Tubes 767 Deflection 770-781 “ and Distributed Weight for Limits of to Compute. . . 778 ‘ ‘ Cast-Iron Bars and Beams, 777, 778 “ Continuous Girders or Beams of Wood 772 “ Formulas for Beams. . .771, 772 “ General Deductions 779 “ Mill and Factory Shafts . . . 779 “ of a Shaft from its Weight alone 778 “ of Bars, Beams, Girders, etc 770-782 “ of Rectangular Bars or Beams of Cast Iron 777 “ of Wrought- Iron Bars 773 “ of Wrought -Iron Rolled Beams 774 “ Rails 776 “ Relative Elasticity of Vari- ous Materials 780 “ Results of Experiments . . . . 780 “ Shafts of Wrought Iron 778 “ To Compute, and Compar- ative Strength of Cast- Iron Flanged Beams. . 778 “ “ and Weight that may be Borne by a Rectan- gular Bar or Beam of Cast Iron 777 u 11 Maximum Load that may be Borne by a Rectangular Beam. 773 “ 11 of and Weight Borne by a Rectangular Bar or Beam 773,774 u “ of Cast-Iron Flanged Beams 777 “ Woods 772 “ Wrought Iron and Woods. . 772 “ Wrought - Iron Bars or Beams .773, 774 “ Wrought - Iron Riveted Beams 774, 775 “ Wrought - Iron Tubular Girders 775 Detrusive or Shearing Strength .782, 783 “ Comparison between it and Transverse 782 “ of Woods 782 Results of Experiments. 782,783 “ Riveted Joints, Cast Iron , Treenails, and Woods . . . 783 “ Shearing 783 ‘ ‘ To Compute Length of Sur- face of Resistance of Wood to Horizontal Thrust 782 “ Wrought and Cast Iron Riv'd Joints , Steel, Tree- nails, and Woods 783 Page Strength of Materials, Elasticity and Strength 761-764 “ Coefficients of 761 “ Comparative Resilience of Woods 763 I ‘ Extension of Cast-Iron Bar 762 “ Modulus of Cohesion 763 “ “of Elasticity .... .762, 763 “ “ of Elasticity and Weight 763 “ of Elasticity, Height of, to Compute . . . 763 u “ of, to Compute 762 “ Weight a Material will Bear without Permanent Alteration 763 Tensile Strength 784-789 “ Elements Connected with Resistance of Various Bodies 786 “ Malleable Iron 785 “ Manganese Bronze 832 “ of Cast and Wrought Iron, 78 4, 785 “ of Tie-rods 787 II of Wrought Iron 785 ‘ ‘ Ratio of Ductility and Mal- leability of Metals 787 ‘ ‘ Steel , Bars and Plates . . 787, 788 “ Stirling's Mixed Iron 785 “ Various Materials 788-790 Torsional Strength 790-797 “ Couplings 796 “ Hollow Shafts 792, 794 “ Journals of Shafts, etc 796 “ Metals and Woods 793 “ Mill and Factory Shafts . . 797 “ Minimum and Maximum Diameter of Shafts, For- mulas for 796 1 1 of Various Materials 793 “ Shafts and Gudgeons. . . 790-795 ‘ 1 To Compute , of Shafts 794 “ “ Diameter of a Shaft to Resist Lateral Stress 791 “ “ “ of Shafts of Oak or Pine 793 “ “ “ of a Centre Shaft. 794 “ “ “ of Solid and Hollow Shafts. .791, 792, 794 Transverse Strength 799-841 “ Bars, Beams, Cylinders, etc 801-805 “ “ Girders, or Tubes, Comparative Value of. 824 “ Bow-string Girder 812 “ Brick-work 801 “ Bridge Plates and Rivets . . 830 “ Cast-Iron 814-817 “ “ and Woods .'... 798 “ “ Girders and Beams. 813 “ Channel and Deck Beams , and Strut Bars 808 “ Comparative Qualities of Various Metals 8q,. XX INDEX. Page | Strength of Materials, Transverse , Comparative Strength and Deflection of Cast-Iron Beams 8o 9 “ Cylinders, Flues, and Tubes, ^ u . Cylindrical and Elliptical Beams or Tubes 810 “ Dimensions and Propor- tions of Wrought -Iron Flanged Beams 809 “ Elastic StrengthofWrought- Iron Bars 808 “ Elements of Rolled Beams. 807 “ Flanged Beams , Compara- tive Strength and Deflec- tion of. 809 “ Flanged Hollow or Annu- lar Beams of Symmetri- cal Section 815 a Form and Dimensions of a Symmetrical Beam or Girder 825 “ General Deductions 824 “ General Formulas for De- structive Weight of Solid Beams of Symmetrical and Unsymmetrical Sec- tion 816, 817 “ Girders and Beams of Un- symmetrical Section 810 “ « Beams , Lintels , etc., 822-826 11 Iron and Steel Rails 812 “ Memoranda •• 8 3° “ “ Cast and Wrought Iron 832 Pag? Strength of Materials, Transverse , To Compute Section of Flange of a Girder or Shaft of Cast Iron 817, 818 “ “ Ultimate Strength of Homogeneous Beams 820 “ T missed Beams or Girders , 823, 824 “ Unequally Loaded Beams. . 810 “ Wrought - Iron Inclined Beams, etc.. 81 1 « “ Plate Girders 81 1 u “ Rectangular Girders or Tubes 809 Working Strength and Factors of Safety 7 Sl Strength of Models ; • °44 u To Compute Dimensions oj a Beam, etc 644 « “ Resistance of a Bridge from a Model 645 Stress, Moment of. 621-623 Stucco : 591 Suez Canal, Via Sugar Cane, and Beet Root 207 Sugar-Mill Rollers 9 11 Sugar Mills ••••••'•* 9°3 Sulpliuret of Carbon, Elastic Force of Vapor of 7°7 Sun 168 “ Heat of J 93 Sunday Cycle or Cycle of the Sun — 70 “ or Dominical Letter 7° 69 Sun-dial, To Set - Tron .... 032 ! Surcharged Revetments 699 " S" **:: lit 1 59 6 Sustenance, ing-man 2 °7 Sweet Potato 2 °7 Swimming * * 439 Symbols, Algebraic, and Formulas. 22 of Various Figures of Cast Iron 800 ‘ 1 Materials. . . 799-801 “ Metals 799 Rectangular , Diagonal , or Circular Beam or Shaft. 817 Relative Stiffness of Mate- rials 79 8 Rivets and Plates. . .. .828, 830 Solid and Hollow Cylinders of Various Materials 801 Steel Bars 8x7 u plates 83° To Compute Centres of Grav- ity and of Crushing and Tensile Strength of a Girder or Beam 819 “ Destructive Weight or Loads, Borne by Rolled Beams or Gir- ders or Riveted Tubes 805-807 ‘ “ Inertia, Moment of of a Solid Beam 819 » “ Neutral Axis of a Beam of Unsymmetrical Section Tacks, Nails , Spikes, etc. Iron 820 Wrought 754 Tan 482 Tangential Wheel • 57° Tangents and Co tangents 410-426 41 “ To Compute. 426 Tannin, Quantity of in Substances. . . 190 Tee and Angle Iron, Weight of ...... . 13° Teeth of Wheels 859-861 “ Involute 8 59 Telegraph Wire, Span of £79 Telescopes, Opera-Glasses, etc • * • 071 Telford Roads 688 > 6 9° Temperature *95 “ by Agitation . 5 2 4 “ Decrease of .by Altitude. 522 “ of Enclosed Spaces 526 u 0 f Various Localities.. . 192 “ To Reduce Degrees of ■ Different Scales 523 INDEX. XXI Page Temperature, Underground 519 Temperatures, Metric 37 Tempering Boring Instruments 197 Tenacity of Iron Bolts in Woods 198 Tensile Strength 784-700 Terne Plates 124 Terra Cotta 602 Test of Plates of Iron 749 Theatres and Opera-Houses 180 Thermometers 523 Throwing Weights 439 Thrust, To Compute Weight of a Body, To Sustain 693 Tidal Phenomena 75 Tide Table, Coast of U. S. 84 Tides ig s “ of A tlantic and Pacific 19 1 , 198 ‘ ‘ of Pacific Coast 85 “ Rise and Fall of Gulf of Mexico 85 ‘ { Time of High Water. 74,75 Tie-rods .787 Time, after Apparent Soon, before Moon next passes Meridian. 75 “ Difference of 81-83 u Measures of. . . . 37 “ New Style 37 “ Sidereal and Solar 37 “ To Compute Difference of, be- tween New York and Green- wich 83 “ To Reduce to Longitude 54 Timber, Comparative Weight of Green, 217 “ Measure, and to Compute Volume of ...61, 62 “ Waste in Hewing or Sawing, 62 “ and Board Measure 61 “ and Woods 865-870 “ Impregnation of 868 Seasoning and Preserving. . . 865 Strength of 870 644 and Lead Pipes, Weight of. 139 “ Plate, Marks and Weights 139 Tolerance, of Coins 38 Tonite 443 Tonn age, Approximate Rule 176 “ Builder" 1 s Measurement . .. . 176 1 1 Corinthian and New Thames Yacht Club jyy “ Freight or Measurement . . . 177 “ of Suez Canal I77 “ of Vessels 175-177 “ Royal Thames Yacht Club. 177 “ To Compute x 7 3 “ Units for Measurement, and Dead Weight Cargoes 176 ‘ ‘ Weight of Cargo 1 7 7 Tools 476 Torsional Strength 790-797 Towers and Domes x8o Towing, Erie Canal and Hudson River .................... .... 103 Traction 843-849 “ Ascending or Descending Page Traction, Friction of Roads 847 “ Grade 847 “ Omnibus 844 u on Common Roads 843,844 “ Resistance of a Car 849 “ of a Stage Coach 848 “ of Gravity and Grade 847 “ on Common Roads 843-845 “ Results of Experiments on . 843 “ To Compute Power neces- sary to Sustain a Vehicle on an Inclined Road. 845, 846 “ Various Roads and Vehi- 845 845 198 9i5 i93 Tin . cles Wagon Train, Service , of a Quartermaster. . Tramways or Street Rails. .. 435 84* Transportation, Canal “ of Horses and Cattle. 192 Transverse Strength 708-841 Treadmill 433 Treenails 783 Trees, Large, in California 184 Trigonometry, Plane 385-389 Trim, Change of, in a Vessel 65s Tripolith 41 Trotting 439 Troy Measure 32 Truss, Iron 178 Tubes, and Flues 747, 827 “ and Pipes, To Compute Weight of 147 “ Brass , Weight of. 142 “ Copper Drawn, Weight of. 140, 144 “ Lap Welded Iron Boiler 137 “ or Girders 809 “ Seamless Copper 140, 144 u Steel Locomotive . . . / 138. “ Wrought-iron 143, i4«t Tubular Bridge x-g Tunnels, Lengths of. x jg, Turbines ’ ^ Boy den 574 “ CompaHson with Water- wheels. . 5791 Downward-flow 574 Fontaine 574 Fourneyron 37^ High - Pi'essure, Operaticm °f- Inward-flow Jonval Low-pressure Outward-flow Poncelet . 574- 57S 57S 575 575 574- Ratio of Effect to Power 577 an Elevation Canal, Slackwater , River. Swain Tremont, Victor. Turkish Plaster and Mortar 591 592, Turning “ and Boring Metal 197 575 576 576 and 846 Turnips 2D7 Turpentine, Elastic Force of Vapor °f' * JO? XXII INDEX. U. Page Underground Temperature 5*9 Undershot-Wheel 5°6 Unguents, Value of 47 1 Uniform Motion 6x7 V. Value of Coins : — * • : • * • “ and Weight of Foreign Coins . Vapor in Atmosphere, Volume of... “ Weight of •••••• ‘ « Elastic Force of A Icohol , Ether , JSulphuret of Carbon , Petroleum , and Turpentine 7°7 Variable Motion 6l 7 Variation of Magnetic Needle 57 “ Decennial, of Needle 50 * ‘ of in U. S. and Canada ... 59 Varnishes 876 Vegetable Marrow 207 Vegetables, Analysis of Meat and Fish , 200 « and Oil -cake, Nutritious Properties of 204 “ Proportion of Starch in . . 205 “ Tubers 2 °7 Vegetation, Limit of * 9 2 Velocities, Metric 37 Ventilation 5 2 4 •“ of Mines 449 Vernier Scale 27 Vessels, Elements of 653 “ Hulls of 830 Veterinary * Volcanoes, Height of ........... “ Power of Volume and Weight of Various Sub- stances Volumes Page Washington Aqueduct * 7 8 Water 849-852 Boiling-Points of 851 Density of 5 2 o Deposits of 852 Expansion of 5*9 Fresh and Sea 849, 851 Inch 557 Motors , Ratio of Effective Power 503 Power 502 Pressure Engines 579 Rainfall and Volume of 850 Resistance of, to an Area of One Sq . Foot 646 Velocity of a Falling Stream of. f 6 Volumes of. °49 Weight of 852, 920 Wheels • • • *5 6 3j 73° “ Compared with Tur bines “ Overshot ‘ ‘ Undershot Waterfalls and Cascades Watermelon Water Pipes, Cast-Iron “ Dimensions, etc. .. .138, 139 Water-Wheel, Centre of Gyration. . 611 Water-Wheels, Diameter and Journal of a Shaft,* tc 581 “ Dimensions of Arms 57 1 Waves of the Sea 852, 853 579 563 566 184 207 H 7 W. Walking • - 433 , 43 8 Wall, Chinese *79 Walls and Arches 602 “ Centre of Gravity of 7 02 “ Dams and Embankments 700 “ Elements of ‘ 1 Friction of. 698 “ Moment of 7 °* a “ of Pressure 098 “ of Buildings i8 9 “ Retaining, of Iron Piles 190 “ Revetment 694 “ Stability of. 7 ° 2 Warehouses, Brick Walls for 003 Warming Buildings 527-528 1. By Hot-air Furnaces or Stoves 5 2 8 “ By Hot Water.. 5 2 4 “ By Steam 5 2 7 u Coal Consumed per Hour . 527 “ Furnaces 5 28 « Illustrations <>f Heating. . . 5 2 7 Open Fires 528 M Volume of Air Heated by Radiators , Consumption of Coal , Areas of Grate and Heating Surface, of Boiler, etc.. • 528 Tidal. Velocity of Weather-Foretelling Plants Glasses Indications Wedge Weighing without Scales. . Weight , Avoirdupois 853 853 . 185 • 430 • 43 i . 630 . 66 A 32,47 and Diameter of Cast-Iron Balls J 53 and Dimensions of Lead Balls 5 oi and Dimensions of Wr ought- iron Bolts and Nuts . .156-158 and Fineness of U. S. Coins. 38 and Marks of Tin Plates. . . 139 and Mint Value of Foreign Coins 4 ° and Strength of Hemp and Wire Ropes * 7 2 and Strength of Iron Wire , etc I2 4 and Strength of Stud -link Chain Cables 168 Angle and T Iron 125, 130 Apothecaries' 3 2 > 4 7 Bells 180 Brain J 9 2 Brass 136 } 1 49 “ Plates M 6 “ Sheet and Tubes cor- responding to Iron . 142 “ Wire 20, 121 Cast Iron *49 INDEX. XX111 Page Weight, Cast-Iron and Lead Balls . . 153 “ “ Bar or Rod 131 “ “ Pipes or Cylinders 132 “ 44 Plates 146 4 4 Composition S heath ing Nails 135 44 Copper 136 44 “ Rods or Bolls 148 44 44 /Seamless Tubes 144 • 4 4 4 Sheet 135 44 Conjugated Roof Plates 13 44 Cube Foot of Embankments, Walls , etc 694 44 Diamond, and Diamonds . 32, 193 44 Electrical 34 44 Flat Rolled Iron 126-128 44 Foods , to Furnish Nitroge- nous Matter 202 44 Galvanized Sheet Iron 124 “ Grain 32 44 44 and Roots 34 44 Green and Seasoned Timber 217 44 Gun Metal 149 44 Hemp and Wire Rope 166 44 Hexagonal, Octagonal, and Oval 135 44 Horses 35 44 Ingredients , that of Com- pound being given 218 4 4 Iron, Steel, Copper , and Brass Plates. 1 1 8, 1 19 44 4 4 44 Wire. . 1 20-1 2 1 44 Lead 32 “ 44 Encased Tin Pipes .. . 151 “ “ Pipes 139,150 4 4 4 4 Plates: 146 “ 44 Sheet 151 44 Men and Women 35 44 Metal by Weight of Pattern. 217 44 Metals of a Given Sectional Area 149 44 Molecules, Weight , etc 194 44 of Articles of Food Consumed in Human System to De- velop Power of Raising 140 Lbs. to a Height of 10000 Ft. 204 44 of Beef and Cattle 35 44 of Cast and Wrought Iron. . 155 44 44 4 4 44 Steel , Copper , and Brass. 136 44 of Earths 33 “ of Offal 35 “ of Sq. Foot of Slating 64 44 Rocks , Earth , etc 468 44 Rolled and Bar Steel 134, 135 44 Round Rolled Iron 126 44 Sheet Iron 129 4 ‘ Steam ■ 7°5 44 Steel 136, 149 4 4 4 4 Plates 134, 146 44 Tin Pipe 139, i 5I 44 To Ascertain , of a Solid or Liquid Substance 217 44 To Compute, of an Elastic Fluid 217 44 Various Materials 155 44 Substances in Bulk 217 “ 44 per Cube Foot 217 Page Weight, Wire and Hemp Rope 166 44 Wood 33 44 Wrought and Cast Iron 155 4 4 4 4 Iron. .125, 126, 136, 149 4 4 4 4 Plates 146 44 Sheet and Hoop. . . 129 4 4 4 4 Tubes 143-145 44 4 4 Tubes and Plates, 145, 146 44 Zinc Sheets 123, 146, 151 4 4 4 4 4 4 and Dimensions °f - *r< I5 1 Weights and Measures 26-35 44 and Volumes of Various Substances 216 4 4 English and French 44 44 Foreign 48 44 Grecian 53 44 Hebrew and Egyptian 53 44 Measures of 32, 47 “ Metric 33,36 4 4 Miscellaneous 33 44 of Steam-Engines. .758, 759, 91 1 44 Roman 53 Well, Artesian 179, 198 44 Boring i 97 Wells or Cisterns, Excavation of, etc 63 Welding 786 44 Cast Steel, Composition for. . 634 44 Fluxes for 636 Wheel and Axle 626 44 and Pinion, Combinations or Complex Wheel Work 628 Wheel Gearing. 854-861 44 Circumference of 857 “ General Illustrations 858 44 Pitch , Diameter , Number of Teeth , Velocity, etc 855, 857 44 Revolutions of 858 44 Spur Gear 91 x 4 ‘ Teeth of 859 44 To Compute Diameter of. . . . 857 4 4 4 4 W of a Tooth 861 4 4 4 4 Velocities of. 856,857 Wheels, Proportions of 862 44 Teeth of. 859-862 Whitewash or Grouting 594 Wind, Course of. 675 44 Effective Impulse of 665 4 4 Force of. 674 44 Pressure of. 91 x 44 Velocity and Pressure 674 Winding Engines 476, 862, 863 44 To Compute Diameter of a Drum 862 4 4 4 4 Number of Revo- lutions 863 Windlass 433 44 Chinese 627 Windmills • 863-865 44 Results of Operation of 865, 921 44 To Compute Elements of . . 864 Window Glass I2 4 Wine and Spirit Measures 45 Wire Gauge, French 123 “ Standard of Great Britain 122 XXIV INDEX. Page Wire Gauges 122 “ Iron Gauge, Weight and Length of 163 Wire, Length of. 124 “ Rope 161,473 “ and Equivalent Belt 167 “ and Hemp, General Notes 167 “ Cables, Galvanized Steel 163 “ Endless 167 “ Fence , Weight and Strength of. 164 “ Results of an Experiment with Galvanized 161 “ “ of Experiments on, at U. S. Navy Yard 169 “ Transmission of Power of ... . 167 “ Ropes, Hemp, Iron, and Steel, Relative Dimensions of. ... . 168 1 ‘ and Hemp Rope, Iron and, Steel, Relative Dimensions of 172 “ and Hemp Ropes, Weight and Strength of. 172 u and Tarred Hemp Rope, Haw- sers, Cables , Comparison of. . 169 “ Rope , Circumference of, to Com- pute 169 “ “ for Standing Rigging, Circumference of, to Compute 172 “ Shrouds 173 Woods 481, 765, 769, 782, 783 “ Bituminous or Lignite 479 “ Coefficients for Safety 835 “ Detrusive Strength of. 782 “ Floor Beams 835 “ Measure 47 “ Pavements 689,690 u Relative Value of their Crush- ing Strength and Stiffness combined 769 “ Safe Statical Loads for 834 “ Sawing 196 “ Weights of. 33 Wood and Timber 865-870 “ Creosoting, Effects of. 869 “ Decrease by Seasoning ... . 869 Page Wood and Timber, Defects of 866 “ Durability of Various 869 ‘ ‘ Impregnation of. 868 ‘ ‘ Proportion of Water in. . . 869 “ Seasoning and Preserving , 866, 868 “ Selection of Trees 865 “ Strength of. 833, 870 “ Transverse Strength of, to Compute 833 “ Weight of Oak and Yellow Pine per Cube Foot 870 Work 432 “ Accumulated in Moving Bod- ies, etc 619 “ and Power , Metric 36 Works of Magnitude 178, 179 Wrought Iron 639, 765, 768, 773, 785 “ Crushing Weight of Columns 768 “ Deflection of Bars , Beams, etc. 773-775 “ “ of Rails 776 “ Plates and Bolts 749 “ Plates, Weight of. 118, 119 “ To Compute Weight for 125 “ Weight of. 155 “ Wire, Weight of.. 120,121 Yam 207 Year, Bissextile or Leap 70 “ Civil 70 “ Solar 70 Years of Coincidence 74 Zinc 644 “ Plates, Weight of 146 “ Sheets, Thickness and Weight of, 123, 151, 152 Zinc Foil in Steam Boilers 912 Zones of a Circle, Areas of 269-271 Addenda. Page Asbestos 9*3 Asphalt, Mortar and Concrete. ..... 913 Blowing Engine, Friction of Air in Pipes 9 21 Boilers, Cylindrical Shells 751, 752 “ Fiued , Arched, or Circular Furnaces 754 “ Girders 754 “ Plates , Straps, and Stays. . 753 Chain Cables, Stowage of 913 Concrete, CoigneVs 914 Distances, Velocities, and Accel- erations 9 1 ** Flexible Paint for Canvas 915 Gas, Natural 9*3 Gauges, Steam, Vacuum, and Hy- drostatic 9 1 ^ Hose, Delivery and Friction of. 919 Page Ice or Cold Producing Machines.. 922 Jarrah Wood 913 Light, Penetration of, in Water 915 Locomotive, Brakes 920 Materials, Non-conducting 914 Ocean. Depths of. 912 Pumps, Direct Acting 738 Safety Valve, Adjustable Pop 918 Shafts 914 Siphon. Steam 918 Steam Heating 913 Steel Guns 913 Temperature, Conductivity of 914 Tramways or Steel Railroads 915 Troops, Marine Transportation of . . 914 Water, Friction of, in Pipes 922 Weirs, Gauges of 919 Windmills 921 EXPLANATIONS OF CHARACTERS AND SYMBOLS Used in Formulas , Computations , etc., etc. = Equal to, signifies equality ; as 12 inches = 1 foot, or 8 X 8 = 16 x 4. -f Plus , or More , signifies addition ; as 4 + 6 -f- 5 = 15. — Minus , or Less, signifies subtraction ; as 15 — 5 = 10. X Multiplied by, or Into, signifies multiplication; as 8 x 9=72. a X d, a.d, or ad, also signify that a is to be multiplied by d. -r - Divided by, signifies division ; as 72 9 = 8. : Is to, :: /So is, : 7 b, signifies Proportion, as 2 4 :: 8 : 16; that is, as 2 zs /o 4, 50 is 8 to 16. signifies Therefore or Hence, and v Because. Vinculum, or Bar, signifies that numbers, etc., over which it is placed, are to be taken together ; as 8 — 2 -j- 6 = 12, or 3 x 5 + 3 = 24. . Decimal point , signifies, when prefixed to a number, that that number lias some power of 10 for its denominator; as .1 is — , .is is — etc. Difference , signifies, when placed between two quantities, that their difference is to be taken, it being unknown which is greater. V Radical sign, which, prefixed to any number or symbol, signifies that square root of that number, etc., is required ; as Vg, or Va+b. The degree of the root is indicated by number placed over the sign, which is termed index of the root or radical ; as V , V , etc. > H , < L signify Inequality , or greater, or less than, and are put between two quantities ; as a r | b reads a greater than b, and a L b reads a less than b. ()[] Parentheses and Brackets signify that all figures, etc., within them are to be operated upon as if they were only one ; thus, (3 + 2) x 5 = 25 ; [8 — 2] x 5 = 30. " ’ ± =F signify that the formula is to be adapted to two distinct cases, as c ^v==c, either diminished or increased by Here there are expressed two values : first, the difference between c and v ; second, the sum of c and v. In this and like expressions, the upper symbol takes preference of the lower. p or 7r is used to express ratio of circumference of a circle to its diameter = 3.1416; L p = .785 4, and ^p = .523 6. signify Degrees, Minutes, Seconds, and Thirds. U\lnlhlp ri ° r t0 a figUre ° r figUreS ’ signif y> in tienoting dimensions, Feet a' a" a'" signify a prime, a second, a third, etc. 1, 2, added to or set inferior to a symbol, reads sub 1 or sub 2 and is used to designate corresponding values of the same element, as h, hz,h 2 , etc. b i; ^ d t ® d K 0r set * u Pf‘°r to a number or symbol, signify that that mini- pft bv 1“ l\S 7 iX’£i 'tS 2X5- “ b “ i, y , h ; 22 ALGEBRAIC SYMBOLS AND FORMULAS. i 1 etc., set superior to a number, signify square^ or cube root, etc., of the number; as 2^ signifies square root of 2; also 3 , a , 3 , 3 , etc., set superior to a number, signify two thirds power, etc., or cube root of square, or square or cube root of 4th power, or cube root of sixth power; as 8*= VW or = (v / 8) 2 . 1.7 3.6^ e tc., set superior to a number, signify tenth root of 17th power, etc. .02 .059 set superior to a number, signify hundredth root of 2d power, or thousandth root of 59th power, the numerator indicating power to which quantity is to be raised, and denominator indicating root which is to be ex- tracted. CO signifies Infinite, as - or a quantity greater than any assignable quan- tity. Thus, - = oo signifies that o is contained in any finite quantity an in- 0 „ . a a , finite number of times : - — a % — = 10a, etc. ac signifies Varies as. Thus, M oc D X,V signifies that mass of a body in- creases or diminishes in same ratio as product of its density and \olume, or S oc t 2 , signifies S varies as t 2 . / signifies Angle. -L Perpendicular. A Triangle. □ Square , as D inches ; "and g| cube, as cube inches. Notes.— D egrees of temperature used are those of Fahrenheit. g is common expression for gravity = 32. 166, 2 g = 64. 33, v 2 g — 8.02 jeet. 0 signifies Dead Flat , denoting dimensions or greatest amidship section of hull of a vessel. ALGEBRAIC SYMBOLS AND FORMULAS. I representing length, h' representing h prime, v representing versed sine, , b “ breadth , c “ chord, h ‘ h sub, d “ depth, a “ area, sin. ‘ sine, h “ height , r “ radius , g gravity. = sum of length and breadth divided by depth. ^ = product of length and breadth divided by depth. — = difference of length and breadth divided by depth. d p b 3 = product of square of length and cube of breadth. VJl = square root of length divided by cube root of breadth. V b = square root of sum of length and breadth divided by depth. d (x). Each of these notations is read, a is a function of x. If in such function of x the value of * is assumed to commence with o and to increase uniformly the notation indicating rate of increase is dx, and is read the differential of X.” Differentiation, d is its symbol, and it is the process of ascertaining the ratio existing between the rate of increase or decrease of a function of a variable and the rate of increase or decrease of the variable itself. If V = 3Z 2 ,V or its equal 3X 2 is the function of *, and * is the independent variable while the exponent of the variable or the primitive exponent is 2. By the operation of Calculus, such expressions are differentiated by ui- minishing the exponent of the variable by unity, multiplying by the pnm- iti This indicates *¥ f la ‘ ion w between the differential of y, the function of x and the differential of x itself. Assume that a- increasing at rate of 3 per second becomes 4; that 1^* - 4 , QriH j r/. — ^ . hence dv — 6 X 4 X 3 — 72. That is, if x is increasing at rate of 3 per second, at the time that x = 4 , the function itself is increasing at rate of 72 per second. # e To differentiate an expression of two or more terms, it ls ne ^^ sar ^. t ? differentiate them separately and connect the results with the signs with which the terms are connected. __ , Thus, differentiating u — 3 x 2 - 2 x, we have du — d (3 x - 2 x)—Sxdx Assuming a = 4 and d x= 3, we have d u — (6 X 4 “ 2 ) X 3 = ^ ^is indicates that when x = 4, and is increasing at rate of 3 per second, the func- t“or ^ - a *, is at 4 kme instant incasing at rate of 66 per second. Intearation. Its symbol / was originally letter S, initial of sum, the symbol of an operation the reverse of differentiation; and when the oper- ation of integration is to be performed twice, thrice, or more times, it written f f , f f f 5 etc. _ .. , Bv the operation of Calculus, expressions are integrated by increasing the ; exponent of the variable by unity, dividing by the new exponent, and do ta Hence, integrating the differential 6 x dx , we have / 6 at d x = 3 x . Thi= result is the function, the differential of which is 6xdx^ j To integrate an expression of two or more terms, it is necessary , grate the terms separately and connect the results with the signs with which the terms are connected. . , r (A-jr—zdx) Thus integrating (6x— 2) dx,wo, have f (6 x— 2) dx— f {ox d x is the function the differential of which is (6.x — 2) dx or (6a; — 2 x ° ) d x. Note. -A quantity with the exponent °, as *° or 3 0 , is equal to unity. NOTATION. 25 The operation of summation may also be illustrated in use of the sym- bol / . Assuming x = 4, the former of the preceding results becomes / 6 x d x = 3 x 2 = 48, the latter / (6 x — 2) d x = 3 x 2 — 2 x = 40. Here 2; is assumed to commence at o and to continue to increase by in- finitely small increments of d x until it becomes 4. The summation is the addition of all these values of x from o to 4. Arithmetically . — The first formula may be written 6 (x + x + x" -f- etc.) dx. If then x is to advance from o to 4 by in- crements of 1, we have 6 (o + 1 + 2 + 3 -f- 4) x 1 = 60, which exceeds 48. H dx is assumed to be .5, the result is 54. The correct result is obtained only when d x is taken infinitely small. By Arithmetic this is approximated, but it is reached by the operations of Calculus alone. The second formula may be written (6 \x + x" -f- x"‘ -}- etc.] — 2 [ x° ' -f- x°" + x°'" etc.] ) dx. Assuming x = 4, and dx = i, we have (6 [1 4- 2 4- 3 4-4J - 2 [1 4- 1 4- 1 4- 1]) x 1 = 52, which exceeds 40. If d x= .25, the result would be 43, and if .125 it would be 41.5, ever approaching but never reaching 40, so long as a finite value is assigned to d x. A, Delta , when put before a quantity, signifies an absolute and finite in- crement of that quantity, and not simply the rate of increase. 2, Sigma, signifies the summation of finite differences or quantities. Thus, 2 y 2 Ax = (, y ' 2 -f- y " 2 4- y "' 2 4- etc.) A x. Assume y = 6, y“ = 8, y " = 4, and A x the common increment of x = 5, then 2 y 2 A x = (36 -j- 64 4- 16) X 5 = NOTATION. 1 = 1. 20 = XX. 1 000 = M, or CIO. 2 = II. 30 = XXX. 2 000 = MM. 3 = HI. 40 = XL. 5 000 = V, or 100 . 4 = iv. 50 = L. 6 000 = VI. 5 = V. 60 = LX. 10 000 = X, or CCIOO. 6 = VI. 70 = LXX. 50 000 = L, or IOOO. 7 = VII. 80 = LXXX. 60 000 = LX. 8 = VIII. 90 = xc. 100 000 = C, or CCCIOOO. 9 = ix. 100 = c. 1 000 000 = M, or CCCCIOOOO. 10 = X. 500 = D, or 10 . 2 000 000 = MM. as ^C < ~“^2oo^ a character is re P eate ^ } so many times is its value repeated, A less character before a greater diminishes its value, as IV = V — I. A less character after a greater increases its value, as XI = X -f- 1 . For every 0 annexed to ID the sum as 500 is increased 10 times. number'is 'doubted!* ^ ^ ° f 1 “ "‘ any timeS as 0 is on the ri 8 ht > the A bar, thus over any number, increases it 1000 times. Illustration i.— 1880, MDCCCLXXX. 18 560, XVmDLX. rp 2 ’ „ ^ ^°°‘ ~ 5 00 X 2 = IOOO. IOD =r 500 X IO =: 5000. _ 1 JJ — 5000 X 2=10000. 1000 = 500 x iox 10 = 50000. CCCI 000 — 50000 x 2 = IOOOOO. 26 CHRONOLOGICAL ERAS. — MEASURES AND WEIGHTS. CHRONOLOGICAL ERAS AND CYCLES FOR 1884. The year 1884, or the 109 th year of the Independence of the United States of America , corresponds to The year 7392—93 of the Byzantine Era) “ 6597 of the Julian Period; “ 5644-45 of the Jewish Era ; “ 2660 of the Olympiads, or the last year of the 665th Olympiad, commenc- ing in July (1884), the era of the Olympiads being placed at 775-5 years before Christ, or near the beginning of July of the 3938th year of the Julian Period; “ 2637 since the foundation of Rome, according to Varro; n 2196 of the Grecian Era, or the Era of the Seleucidae; “ 1600 of the Era of Diocletian. The year 1301 of the Mohammedan Era, or the Era of the Hegira, begins on the 7th of February, 1884. ' , ^ The first day of January of the year 1884 is the 2,409,178th day since the com- mencement of the J ulian Period. Dominical Letters F, E I Lunar Cycle or Golden Number 4 Epact 3 I Solar Cycle 17 Homan Indiction was a period of 15 years, in use by the Romans. The precise time of its adoption is not known beyond the fact that the year 313 A-D. was a first year of a Cycle of Indiction. Julian Period is a cycle of 7980 years, product of the Lunar and Solar Cycles and the Indiction (19 X 28 X 15), and it commences at 4714 years B.C. 6^3 (given year — 1800) = year of Julian Period, extending to 3267. Note. — if year of Julian Period is divided by 19, 28, 15, or 32, the remainders will respectively give the Lunar and Solar Cycles , the Indiction , and the Year qf the Dionysian. MEASURES OF LENGTH. Standard of measure is a brass scale 82 inches in length, and the yard is measured between the 27th and 63d inches of it, which, at tem- perature of 62°, is standard yard. Lineal 12 inches = 1 foot. 3 feet = 1 yard. 5.5 yards = 1 rod. 40 rods = 1 furlong. 8 furlongs = 1 mile. Inch is sometimes divided into 3 barleycorns , or 12 lines. A hair’s breadth is .02083 (48th part) of an inch. 1 yard = .000 568, and 1 inch = .000015 8 of a mile. Inches. Feet. Yards. Rods. Furl. 36= 3 - 198= 16.5= 5-5- 7 92O = 660 = 220 = 40. 63 360 = 5 280 = I 760 = 320 — 8. G-unter’s Chain. 7.92 inches = 1 link. 100 links = 1 chain, 4 rods, or 22 yards. 80 chains — 1 mile. Ropes 1 fathom = 6 feet. and Cables. I 1 cable’s length =120 fathoms. Greographical and Nautical. 1 degree, assuming the Equatorial radius at 6974532.34 yards, as given by Bessel, = 69.043 statute miles = 364 556 feet. 1 mile = 2028.81 yards or 6086.44 feet. 1 league = 3 nautical miles. MEASURES AND WEIGHTS. 27 Log Lines. Estimating a mile at 6086.43 feet, and using a 30" glass, 1 knot = 50 feet 8.64 inches. | 1 fathom = 5 feet .864 inches. If a 28" glass is used, and 8 divisions, then 1 knot = 47 feet 4 inches. | 1 fathom = 5 feet 1 1 inches. The line should be about 150 fathoms long, having 10 fathoms between chip and first knot for stray line. Note. — This estimate of a mile or knot is that of U. S. Coast Survey, assuming equatorial radius of Earth to be 6974532.34 yards and a meter to be 39.36850535 inches of the Troughton scale at 62°. Cloth.. 1 nail = 2.25 inches. | 1 quarter = 4 nails. | 5 quarters = 1 ell. Pen cl 11 lmxL . 6 points = 1 line. J 12 lines = 1 inch. Shoemakers’. No. 1 is 4.125 inches, and every succeeding number is .333 of an inch. There are 28 numbers or divisions, in two series or numbers— viz., from 1 to 13, and 1 to 15. IVI iscellaneous. 12 lines or 72 points = 1 inch. I 1 hand = 4 inches. 1 palm = 3 inches. | 1 span = 9 inches. 1 cubit = 18 inches. ■Vernier Scale. V ernier Scale is divided into 10 equal parts ; so that it divides a scale of ioths into iooths when two lines of the two scales meet. Metric, by _A.ct of Congress of Jnly 28 , 1866 . Unit of Measurement is the Meter, which by this Act is declared to be 39.37 ins. Denominations. Meters. Inches. Feet. Yards. Miles. Millimeter . IOO •°394 •3937 3-937 39-37 393-7 Centimeter — Decimeter i° .328083 3.28083 32.80833 328.083 33 3280.83333 — Meter ' 1.093 61 10.936 11 109.361 11 1093.611 11 — Dekameter IO — Hektameter IOO — Kilometer IOOO* — Myriameter I OOOO* .62137 6.2137 In Metric system, values of the base of each measure — viz., Meter Liter Stere Are, and Gramme— are decreased or increased by following prefix. Thus, ’ ’ Milli, 1000th part or .001. Centi, 100th “ ,01. Deci, 10th part or .1. I Deka, 10 times value. Myria, 10000 times value. Hekto, 100 times value. Kilo, 1000 “ Note —The Meter, as adopted by England, France, Belgium Prussia and Russia of WortT ^ in a d ? a Po4; R Clarke > R E., F.R.S., 1866, Which at 32° in terms ^, Il P P i en f 1 u Sta 'xr dar ^ at 6z ° 18 39-37° 432 inches or 1.09362311 yards its leo-al France 6111 ^ MetnC Act of 1864 being 39-37° 8 inches, the same as adopted °i 11 Coro? 1 wa? comparison and the one formerly adopted by the U. S. Ordnance T 39 ‘ 370 797 J mches ’ or 3-280899 7 6 feet, and the one adopted bv the U. S. Coast Survey, as above noted, is = 39. 368 505 35 inches. P tne 28 MEASURES AND WEIGHTS. Equivalent Values in Metric Denominations of UJ. S. Denominations. Value in Meters. 1 1 Denominations. Values in Meters. .0254 . 304 800 6 .9144018 [ Rod 5.029 209 9 XT’ Furlong 201. 168 396 Yard 1 Mile 1609.347 168 i Chain = 20 meters . 1 Furlong . . . = 200 . “ 5 Furlongs . . . = 1 kilometer . Approximate Equivalents of Old and Metric XJ. S. Measures of Eengtli. i Kilometer . . . . = .625 mile. 1 Mile = 1.6 kilometers. 1 Pole or Perch . = 5 meters. x Root =3 decimeters or 30 centimeters. 1 Metre =3.2808 33 feet = 3 feet 3 ***• and 3 eighths. n Meters =12 yards. | 1 Decimeter ... =4 inches. 1 Millimeter . . = 1 thirty-second of an inch. To Convert Meters into Inches.— Multiply by 40; and to Convert Inches into Meters. — Divide by 40. Approximate rule for Converting Meters or parts, into Yards— Add one eleventh or .0909. Inches Decimally — Millimeters. Milli- Inches. Milli- meters. Inches. Milli- I meters. Inches. Milli- I meters. ] Inches. 1 Milli- meters. .01 • 2 5 .2 5.08 .48 12.2 ' .76 l 9 .3 .02 • 5 1 .22 5-59 . -5 12.7 •78 19.8 .03 .76 .24 6. 1 •52 13.2 .8 20.3 .04 1.02 .26 6.6 •54 13-7 .82 20.8 •°5 1.27 .28 7 - 11 •56 14.2 .84 21.3 .06 1.52 •3 7.62 •58 14.7 .86 21.8 .07 1.78 •32 8-13 .6 15.2 .88 22.4 .08 2.03 •34 8.64 .62 15-7 •9 22.9 .09 2.29 • 3 6 9.14 .64 16.3 .92 23-4 . 1 2-54 .38 9-65 | .66 16.8 .94 23-9 .12 3-°5 •4 10.2 i .68 17-3 .96 24.4 .14 3-56 .42 10.7 •7 17.8 .98 24.9 .16 4.06 •44 11. 2 .72 18.3 1. 25-4 .18 4-57 .46 n -7 1 •74 18.8 127 152.4 177.8 203.2 228.6 254 2794 304.8 foot. Inches in Fractions = Millimeters. Eighths. Six- teenths. Thirty- SecOnds. Milli- meters. Eighths. Six- teenths. Thirty- seconds. Milli- meters. J3 w Six- teenths. t e .- 0 JS « E- 1 as Milli- meters. J Eighths. ST| ' 1 •79 9 7 .!4 17 13- 5 13 1 i-59 5 — 7-94 9 14-3 3 2.38 11 8-73 *9 15- 1 1 3*7 3 — — 9-52 5 — — !5. 9 7 5 3-97 13 10.32 21 16.7 i5 3 4.76 7 — 11. 11 11 — I 7-5 2 7 5- 56 6- 35 4 — 15 11. 91 12.7 6 - 23 18.3 *9 8 - H S Bv means of preceding tables equivalent values ot incnes ana mumneie s, equivalent values of inches in centimeters, decimeters, and meters, may be ascertained by altering position of decimal point. Illustration —Take 1 millimeter, and remove decimal point successively by one figure to the right; the values of a centimeter, decimeter, and meter become 1 millimeter. . . . 1 centimeter. . . . Ins. I 1 decimeter 3.94 1 meter 39-4 .32 inch = 8.13 millimeters. 3.2 inches = 81. 3 MEASURES AND WEIGHTS. 2 9 MEASURES OF SURFACE. 144 square inches = 1 square foot. | 9 square feet = 1 square yard. Architect's Measure , 100 square feet = 1 square. Land. 30.25 square yards = 40 square rods = 4 square roods ) 10 square chains J 640 acres = 1 square rod. 1 square rood. 1 acre. 1 square mile. Yards. Rods. Roods. 1210 . 4840 = l6o. 3 097 600 = 102 400 = 2560. 208.710326 feet, 69.570109 yards square, or 220 by 198 feet square = 1 Acre. Paper. 24 sheets = 1 quire. | 20 quires = 1 ream. | 21.5 quires = 1 printer’s ream. 2 reams = 1 bundle. | 5 bundles = 1 bale. Drawing. Cap , . 13 X 16 inches. Demy Medium . . . . , u • Royal ll Super-royal . . . 19 x 27 n Imperial . . . . . u Elephant . . . . Peerless 'll Columbier .... 23 X 34 inches. Atlas ........ 26 X 34 li Theorem 28 X 34 u Doub. Elephant, 27 X 40 11 Antiquarian . . . 3 1 X 53 u Emperor 40 X 60 u Uncle Sam .... 48 X 120 ll 18 x 52 inches. Tracing. Double Crown 20 X 30 inches. Double D. Crown . . 30 X 40 “ Double D. D. Crown, 40 X 60 Mounted on cloth, 38 ins. in width. Grand Royal 18 X 24 inches. Grand Aigle 27 X 40 “ Vellum Writing, 18 to 28 ins. in -width. Miscellaneous. 1 sheet = 4 pages. 1 quarto =8 “ 1 octavo = 16 “ 1 duodecimo = 24 pages. 1 eighteenmo = 36 “ 1 bundle = 2 reams. 1 piece wall-paper, 20 ins. by 12 yards. 1 “ “ “ French, 4.5 sq. yards. Roll of Parchment = 60 sheets. Copying. 100 Words = 1 Folio. Metric, by ^Act of Congress of Jnly 28, 1866. Unit of Surface is Are or Square Delcameter. A square meter (39.37?) = 1549.9969 sq. ins., but by this Act is declared to be 1550 sq. ins. Denominations. Sq. Meters. Sq. Inches. Sq. Feet. Sq. Yards. Acres. Centimeter .0001 .01 1. 100. JO 000. •155 15-50 1550. .107638 10.763 888 1076.388 88 1.196 119.6 II 960. .02471 2.471, Decimeter Centare or ) Square Meter j Are Hectare 30 measures and weights. Denominations. Sq. Inch . Foot Yard Rod Sq. Meters'. 1 1 Denominations. .00064516 Sq. Chain . .09290323 “ Rood.. .83612907 ^ Acre.. 25.292904 II u Mile..!. Sq. Meters. Sq. Hectares. Sq. Ares. 404.686 47 1011.716 175 4046.864699 V f — . 404 686 258.999 34 4.046 865 10. 117 162 40.468647 25 899-934 °74 Approximate ^ ^ 5. 5 square centimeters %** inch.^ j « ? measures of volume. C ,1 „ a » nn measures 231 cube ins., and contains 8.338 882 2 Standard g Xroy grains of distilled water, at temper- Standard bushel is the Winchester, which contains 215042 “ -r"? "Set'S ,„X“”hS s,r.h« »». "»« <■> - -» ^ “»"■ equal 2747.715 cube ins. for a true cone. \ struck bushel contains 1.24445 cube feet Liqnid. 4 gills = 1 pint. 2 pints = 1 qnart. 4 quarts = i gallon. 2 pints = 1 quart. 4 quarts = i gallon. 2 gallons = 1 peck. 4 pecks = 1 bushel. Dry. Cube Ins. 28.875 57-75 231. Cube Ins. 67.2006 268.8025 537 - 6°5 2150.42 Gills. Pints. 8 . 3 2 = 8. Pints. Quarts. Galls. 8 . 16 = 8. 64 — 32 = 8. Cnloe. Inches. 46656 1728 cube inches = i foot. I 27 cube feet = i yard. | . . Note-A cube foot contains 2200 cylindrical inches, or 3300 spherical inches. Din id. 60 minims = i dram. 8 drams = i ounce. 16 ounces = i pint. 8 pints = 1 gallon. Minims. Drams. Ounces. 480. •7 680 = 128. 6l 24O = 1024 = 128. ISTantical. j. =r ii; cube feet. 1 ton displacement in salt water __ ^ « u 1 “ registered internal capacity Dimensions of a Barrel. Diameter of head, , 7 ins. ; hung, , 9 ins. ; length, 28 ins. ; volume, 7 68 9 cube ina = 3.5756 bushels. MEASURES AND WEIGHTS. 31 Miscellaneous. I ? ub , e * oot 74805 gallons. 1 bushel 9.309 18 gallons. 1 chaldron _ 36 bushels, or 57-244 cube feet. I “I d l of i W ” 0d : c»be feet. 24.75 cube feet. 1 load hay or straw = 36 trusses. i quarter = 8 bushels. 1 Barrel Galls. 1 Tierce Butt of Sherry Pipe of Port Pipe of Teneriff'e •••35X50. .. . 108 • 115 Butt of Malaga •••33X53* •• • 105 Puncheon of Scotch Whisky. . no to %o Puncheon of Brandy 34X52. .no to 120 1 uncheon of Rum t 0 no Hogshead of Brandy 28X40.. 55 to 60 Pipe of Madeira Q2 Hogshead of Claret 46 oft ®p?St, i or°Pun\h f ^ QUarter cask iS 0De f0Urth ’ and an 0ctave is eighth T t Act of Congress of July 28, 1S66 brut or Base of Measurement is a cube Decimeter or Liter, which is declared to be 01.022 cube ms . Cube Measures. Denominations. Values. Cube Inches. Cube Centimeter “ Decimeter .001 cube milliliter 1 cube liter . .061 022 61.022 “ Meter Kiloliter or stere. . Cube Feet. I Cube Yards. Denominations. Milliliter Centiliter Deciliter Liter Dekaliter Hektoliter Kiloliter ) or Stere j * * * Dry IVTeasnres, | Cube Ins. Quarts. cube centimeter. 1 “ decimeter meter. * Or .227 gallon. .061 .6102 6.1022 61.022 .908* 9.08 • 0 35 3i3 657 — 35 - 3 I 3 657 I 1.308 Bushels. [Cube Yards. •1135 c * *35 [ *35 • 2 83 75 2-8375! 28.375 + 3-531 365 7 ^be feet. .1308 1.308 to cc, is used instead of in. Metric Denominations of TJ. S. Dry Measures. Equivalent Values Denominations. Inch . . . Pint... Quart . . Gallon . Peck. . . Bushel. Centiliters. Deciliters. .0881 •3524 .110 125 .4405 .881 3-524 Denominations. Milliliter . . . Centiliter. . . Deciliter Liter Dekaliter. . . Hektoliter. . Kiloliter ) or Stere j Liters. Dicfuid. 3VIeasu.res. Liters. 1.10125 4-405 8.81 35-24 Dekaliters. 11. 0125 44-05 352.4 .001 .01 100 1000 •27 2-7 27 Ounces. •338 3-38 33-8 •21134 2.1134 21.134 Quarts. 1.0567 10.567 Gallons. .26417 2.6417 26.417 264.17 32 measures and weights. Approximate E ^“™ u *^ S 0 f Volume of Old and Metric TJ. S. ; liters. i cube foot . . = 28. 3 liters. ’ cube meter.... ,.^.33^^ \ u kiloliter = 2240 lbs. nearly of water. measures OE WEIGHT. Standard tilled water weighed in air, at v39- j > A cube inch of such water weighs 252.693? g™“S. ^voircTupois. 16 drams = i ounce. 16 ounces = i pound. 1 12 pounds = 1 cwt. 20 cwt. = 1 t° n * Drams. Ounces. Pounds. 256. 28672= I79 2 - 573 440 = 35 840 — 2240. \ dram = M 3-343 75 grains Troy, or 53-5 *"»■* i stone =14 pounds. Troy. I Grains. Dwt. 24 grains = i dwt. 20 dwt. = 1 ounce. 12 ounces = i pound. 7000 Troy grains 437.5 “ “ 27.343 75 Troy grains !75 Troy pounds 175 u ounces •r “ ounce pound 480. 5760 = 240, — 1 lb. avoirdupois. a — I OZ. = i dram u = 144 ibs. ;; = 192 OZ. = 4 8ogrs. “ .822 857 lb. ?. Troy . 20 grains 3 scruples 8 drams 12 ounces 45 drops Grains. Scruples. Drams. 60. 480 = 24. 3760 = 288 — 96. avoirdupois pound = t.215 270 .A.potliecaries = 1 scruple. = 1 dram. . — 1 ounce. = 1 pound. 45 arops = 1 teaspoonful or a fluid dram. The SulTouhct^ata are the same as in Troy weight. Diamond. . __ I 4 grains = 3.2 grains. 1 gram- i 6 parts j carat =4 grams. 16 parts = .0 gram. \ 15.5 carats = i Troy ounce. Dead. Shlutdfo}S.5 8 tots feet in width and from 30 to 35 feet to length. G-rain. Standard Weights per Bushel. ^ Lbs. I n * ^nd c8 I Rve ... 56 ' Oats 32 1 Barley .... 48 Wheat.... 60 1 Corn 56 and 58 l nye 5 MEASURES AND WEIGHTS, 33 Miscellaneous. COAL. Anthracite i cube foot = 1.75 broken. “ 50 to 55 lbs. per cube foot. 44 41 to 45 cube feet = 1 ton broken. Bituminous 70 to 78 lbs. per heaped bushel. a 40 to 50 lbs. per cube foot. “ Cumberland 53 “ “ “ “ 44 Cannel 50.3 lbs. per cube foot. 44 Welsh 43 cube feet = 1 ton. 44 Lancashire 44 “ “ = 1 “ 44 Newcastle 45 “ “ =1 “ 44 Scotch 43 “ 44 = 1 44 44 R. N. allowance 48 “ “ = 1 “ Charcoal, hardwood 18.5 lbs. per cube foot. 44 pine wood . 18 44 44 “ “ WOOD. Virginia pine . . 2700 lbs. = 1 cord. | Southern pine . 3300 lbs. = 1 cord. EARTH. River sand . . 21 cube feet ~ 1 ton. I Marl or Clay, 28 cube feet — 1 ton. Coarse gravel, 23 44 44 = 1 44 | Mold 33“ 44 = 1 44 Metric, L>y J^ct of Congress of July SB, 1866. Unit of Weight is the Gram, which is weight of one cube centimeter of pure water weighed in vacuo at temperature of 4 0 C., or 39. 2 0 F., which is about its tem. perature of maximum density = 15.432 grains. Denominations. Values. Grains. | Ounces. | Lbs. Ton. Milligram 1 cube millimeter _ — — Centigram 10 44 4 4 •154 32 — — — Decigram .1 44 centimeter z -543 2 — : — — Gram 1 4 4 4 4 15-432 •035 27 — — Dekagram Hektogram 10 4 4 4 4 1 deciliter •3527 3-527 .220 46 Kilogram or Kilo . . x liter — 35-27 2.2046 — Myriagram 10 44 — — 22.046 — Quintal 1 hektoliter — — 220.46 .098 419 Millier or Tonneau. 1 cube meter — — 2204.6 .984 196 Kilogram — 2.679 *7 Troy , or 2 lbs. 8 oz. 3 dwts. .3072 grain. Equivalent Values in NEetric Denominations of XT. S. Denominations. Grams. Dekagrams. Denominations. 1 Grams. Kilograms. Grain .0648 1.296 1-5552 1.77187 3.888 Ounce 28.3502 31.1042 453.6028 373.2504 .028 35 .031 1 •4536 •373-25 1016.057 28 Scruple Pennyweight Drachm 44 (Apoth.) 17.7187 38.88 44 Troy Pound 4 4 Troy Ton Approximate Equivalents of Old. and New IT. S. Measures of Weight. The ton and the gram are at nearly equal distances above and below the kilogram. Thus, 1 ton . . . . — 1 016057.28 grams. | 1 kilogram = 1000 grams. 1 gram is nearly 15.5 grains (about .5 per cent. less). 1 kilogram about 2.2 pounds avoirdupois (about .25 per cent. more). 1000 kilograms, or a metric ton, nearly 1 Engl, ton (about 1.5 per cent. less). 34 MEASURES AND WEIGHTS. Electrical. ( British Association.') T?^«iQtanoe —Unit of resistance is termed an Ohm , which represents resist- ant of a column ‘of mercury of i sq. millimeter in section and 1.0486 meters m length, at temp. o° C. Equivalent to resistance of a wire 4 millimeters in diameter 1 and 100 meters in length. 1 000 000 Microhms . . . . . . . . .— absolute electro-magnetic units. T ohm 10000000 “ “ ‘ 1 ooo 000 Ohms .* : = i Megohm or id* “ Electro-motive Eorce.-Unit of tension or difference of potentials is 1 . e — 54 56 54 56 ' 56 ’ 56 60 ■ 56 » 5^ » 56 . • — • 5< > 5 C » — — . — — 5 C > 5 C > 5o 45 45 i 45 i 45 — * — 6c > 5^ i 60 1 60 6c > 6c > 60 6c i — -6c ) 6c > 60 -1- ■ — 20 | — - 2C > 20 1 3° 60 48 60 47 60 56 34 60 46 48 50 60 56 60 28 50 60 56 60 MEASUKES AND WEIGHTS. 35 XV eight of Men. and. Women. Average weight of 20000 men and women, weighed in Boston, 1864, was — men, 141.5 lbs. ; women, 124.5 lbs. Average of men, women, and chil- dren, 105.5 lbs. XVeight of Horses.— (XT. S.) Weight of horses ranges from 800 to 1200 lbs. WEIGHT OF CATTLE. To Compute Dressed XVeight of Cattle. Rule. — Measure as follows in feet: 1. Girth close behind shoulders, that is, over crop and under plate, immediately behind elbow. 2. Length from point between neck and body, or vertically above junction of cervical and dorsal processes of spine, along back to bone at tail, and in a vertical line with rump. Then multiply square of girth in feet by length, and multiply product by factors in following table, and quotient will give dressed weight of quarters. Condition. Heifer, Steer, or Bullock. Bull. Condition. Heifer, Steer, or Bullock. Bull. Half fat 3*15 3-36 Very prime fat . . . 3-64 3.85 Moderate fat Prime fat 3-36 3-5 3-5 3-64 Extra fat 3-78 4.06 Illustration.— Girth of a prime fat bullock is 7 feet 2 ins., and length measured as above 4 feet 5 ins. 7' 2" = 7.i7, and 7.17 2 51.4, which x 4' 5" and by 3.5 — 794-5 lbs. Exact weight was 799 lbs. Note. — 1. Quarters of a beef exceed by a little, half weight of living animal. 2. Hide weighs about eighteenth part, and tallow twelfth part of animal. Comparative 'Weigh. ts of Live Beeves and. of Beef. Lbs. Per cent. Lbs. Bullocks 2800 72 to 78 Rnl looks Heifers 2600 Heifers I 55 ° Bullocks 2600 | 70 to 76 Ru Hocks I 55 ° 1260 1200 Heifers 2400 2400 2100 | 66 to 70 Heifers Bullocks Rnl locks . Heifers | 64 to 68 Heifers Bullocks 2100 Rnl locks 1050 980 950 Heifers 1800 63 to 66 Heifers Per cent. 61 to 64 58 to 61 57 to 58 50 to 56 "Weight of Offal in a Beef and. Sheep. BEEF. Lbs. Hide and Hair 56 to 98 Tallow 42 “ 140 Head and Tongue . 28 “ 40 Feet. " y SHEEP. Lbs. 8 to 16* BEEF. Lbs. SHEEP. Lbs. Kidneys, Heart,) . , , . Liver, etc....’! 31 to 62 6 to. Stomach, Entrails, etc., 126 u 196 Blood 42 u 56 32 36 MEASURES, WEIGHTS, PRESSURES, ETC. To Compute Equivalents of Old. and. New TJ . S. and of Metric Denominations. By Act of Congress , July 28, 1866. Rule. — Divide fourth term by second, multiply quotient by first term, and divide product by third term. Or, Ascertain relative ratio of first and second terms, and multiply result by ratio of third and fourth terms. Note. — When result is required in French or other Metric denominations than those of U.S., use exact denominations, as, 61.025 387 for 61.022, 39.370432 for 39.37? etc. Example 1.— If one gallon (1st), per sq. foot, yard, acre, etc. (2d) ; how many liters, ( 3 d), per sq. foot, yard, acre, etc. (4th) ? _L x 231 -4- 61.022 .=3.7851 liters or 3.7848 litres. 1 Or, — = 1.604, and =2.3598; hence, 1.604 X 2.3598 = 3.7851 liters - 5 144 61.022 Note.— In computing ratios, first term is to be dividedby second, and fourthby third. Example 2.— If one ton per cube foot, how many kilograms per cube decimeter? 6l, ? 1 2 * * * ? -X 2240-4-2.2046 = 35.881 liters , or 35.882 litres. 1728 MEASURES. By Act of Congress of U. S. By Metric Computation. 1 Liter per sq. foot, etc. = .2642 Gallon per sq.foot , or .264 2 gallon. 1 Liter per sq. meter . = .0245 Gallon per sq.foot , or .024 5 gallon. 1 Gallon per sq. foot . = 40.746 Liters per sq. meter , or 40.745 4 litres. 1 1 Sq foot per acre . . . = .2296 Sq. meters per hectare , or 2.29609 metres . WEIGHTS AND PRESSURES. By Act of Congress of U. S. Per sq. inch. Lev sq. inch. — .1929 Lb. = 6.6679 Kilograms, z= 2.54 Centimeters, = 453.6029 Grams, 1 Centimeter 1 Atmosphere . 1 Inch mercury 1 Pound 1 Kilogram . J — o-#- -r — t * ~ - Note — -*o ins. of mercury at 62° = 14 7 lbs - P er inch >' hcnce > 1 j 6 * * * * * * = 2 - °4°8 ins . , and a centimeter of mercury = 30 = . 3937 for U. S. computation, and 30= . 393 704 32 for French or Metric. By Metric Computation . • .19292 lb. 6.667 8 kilogrammes. _r 2.54 centimetres. or 453.592 6 grammes. — ^ ^ "T 7 J = 317.4624 Lbs. per sq.foot , or 317.465 POWER AND WORK. 1 Horse - powe r = Cheval or Cheval - vapeur = 4500 k X 772 = 33 000-4- (4500 X 2.2046 X 39.37 -4- I2 ) — 1.01388 chevaux. 1 Cheval or Chev al-vapeur (7 5 kxm per second) = horse-power. (4500 X 2.2046 X 39-37 -M2) -4- 33 000 = .9863 horse-power. By A ct of Congress of TJ. S. By Metric Computation . 1 Foot-pound = Kilogrammeter kxm — 7.233 foot-lbs. ; hence, 1— (2.2046x3.280833) = .13826 Kilogrammeter , or .13825 kilogrammeire . 1 Cube foot per IP = .0279 Cube meter per cheval . or .0279 cheval. 1 Pound “ “ . . = .447 38 Kilogram per cheval. or .447 38 kilogramme. 1 Cube meter per cheval = 35-8038 Cube feet per IP, or 35.8058 IP. I 1 i <. i i \ PRESSURES, ETC. — MEASURES OF TIME. 37 TEMPERATURES. i Caloric or French unit = 3.968 Heat-units , and 1 heat-unit = 1 -r- 3.968 = .252 c alone. 1 U. S. Mechanical equivalent ( 772 foot -lbs. ) = 772 ■— 7.233 = 106.733 Kilo gr ammeters and 106.733 kilogrammetres. 1 French Mechanical equivalent (423.55 k X m) = 3.280833 X 2.2046 X 423.55 = 3063.505 foot-lbs., or 3063.566 foot-lbs. Metric. 1 Heat-unit per pound = .5556 Kilogram , or .5556 kilogramme. 1 Heat-unit per sq. foot = .2715 Caloric per sq. meter , or .271 %per sq. metre. VELOCITIES. 1 Foot per second, minute, etc. = .3047 Meter per second, or .3047 metres. 1 Mile per hour =1 .447 “ u u or .447 “ MEASURES OF TIME. 60 thirds = 1 second. I 60 minutes = 1 degree. 60 seconds = 1 minute. | 30 degrees = 1 sign. 360 degrees = 1 circle. True or apparent time is that deduced from observations of the Sun, and is same as that shown by a properly adjusted sun-dial. Mean Solar time is deduced from time in which the Earth revolves on its axis, as compared with the Sun ; assumed to move at a mean rate in its orbit, and to make 365.242 218 revolutions in a mean Solar or Gregorian year. Sidereal time is period which elapses between time of a fixed star being in meridian of a place and time of its return to that place. Standard unit of time is the sidereal day. Sidereal day = 23 h. 56 m. 4.092 sec. in solar or mean time. Sidereal year , or revolution of the earth, 365 d. 5 h. 48 m. 47.6 sec. in solar or mean time = 365.242 24 solar days. Solar day, mean = 24 h. 3 m. 56.555 sec. in sidereal time. Solar year (Equinoctial, Calendar, Civil or Tropical) =365.242 218 solar days, or 365 d. 5 h . 48 m. 47.6 sec . Civil day commences at midnight. Astronomical day commences at noon of the civil day, having same designation, that is, 12 hours later than the civil day. Marine or sea day commences 12 hours before civil time or 1 day before astronomical time. New Style was introduced in England in 1752. Note —In Russia days are reckoned by Old Style, and are consequently 12 days behind Gregorian record. D 38 MEASURES OF VALUE. 32 til MEASURES OF VALUE. 10 mills = 1 cent. I 10 dimes = 1 dollar. 10 cents = 1 dime. | 10 dollars = 1 eagle. Standard of gold and silver is 900 parts of pure metal and 100 of alloy in 1000 parts of coin. Fineness expresses quantity of pure metal in 1000 parts. Remedy of the Mint is allowance for deviation from exact standard fineness and weight of coins. Nickel cent (old) contained 88 parts of copper and 12 of nickel. Bronze cent contains 95 parts of copper and 5 of tin and zinc. Pure Gold 23.22 grains = $1 00. Hence value of an ounce is $20.67.183+. Standard Gold, $18.60.465+ per ounce. » WEIGHT, FINENESS, ETC., OF U. S. COINS. GrOld.. Denomination. Weigh of Coin. t of Pure Metal. Denomination. Weigh of Coin. t of Pure Metal. Dollar Oz. •053 75 •134 375 .161 25 Grs. 25.8 64-5 77-4 Grs. 23.22 58-05 j 69.66 Half Eagle Eagle Oz. .268 75 •537 5 1-075 Grs. 129 258 5x6 Grs. 1 16. 1 232.2 464.4 Quarter Eagle . . Three Dollar . . . Double Eagle. . . Silver. r)i me 1 .08037=; ! 38.58 I 34.722 II Half Dollar ! .401 875 20 Cent .16075 77.16 69.444 1 Trade Dollar. .875 Quarter Dollar . j .200937 5 | 96.45 | 86.805 || Silver Dollar . . . | -859375 192.9 420 412.5 173.61 378 37i- 2 5 Weight. Copper. Tin and Zinc. Weight. Copper. Tin and Zinc. One Cent Two Cents . . . Grains. 48 96 Per cent. 95 95 Per cent. 5 5 Three Cents. Five Cents. . Grains. 30 * 77.16 | Per cent. 75 1 75 Per cent. 25 25 r rolerance. — Gold, Dollar to Halt Eagle, .25 grams, regies, —Silver, 1.5 grains for all denominations. — Copper, 1 to 3 cents, 2 grains, 5 cents, 3 grains. „ , Le Jal Tenders. — Gold, unlimited. - Silver. Dollars of 412.5 grams u+mited; for subdivisions of dollar, *10. (Trade dollars [420 grams] are not legal tender.) — Copper or cents, 25 cents. Note. -Weight of dollar up to 1837 was 416 grains, thence to 1873, 412.5- Weight of $1000, @ 4x2.5 S r - =859.375 OK . British standards are: Gold , fg of a pound,* equal to 11 parts pure gold and 1 of alloy ; Silver, fit of * pound, or 37 parts pure silver and 3 of alloy = A 9 Troy C ounce of standard gold is coined into £3 i 7 » ; ounce of tandard silver into 5.. 6 d. 1 lb. silver is coined into 66 shillings. Copper is coined in proportion of 2 shillings to pound avoirdupois. £ Sterling (1880) $486.65; hence ^ of this — value of 1 penny 2.027 708 33 ce nts. __ * A pound is assumed to be divided into a 4 equal parts or carats, hence the pro- portion is equal to 22 carats. FOREIGN MEASURES OF VALUE. 39 To Compute "Value of Coins. Rule. —D ivide product of weight in grains and fineness, by 480 (grains in an ounce), and multiply result by value of pure metal per ounce. Or, Multiply weight in ounces by fineness and by value of pure metal per ounce. Example i. — When fine gold is $20.67.183-}- per oz., what is value of a British sovereign? By following tables, p. 40, Sovereign weighs .2567 oz., and .2567 X 480 = 12^ 216 grains, and has a fineness of .9165. H J Hence, 123.216 x 916 5 480 X 20. 67. 1 83+ = $4.86.34. Example 2.— When fine silver is $1. 15. 5 per oz., what is value of U. S. Trade dollar? By table, p. 40, Dollar weighs .875 oz. and has a fineness of .900. Hence, .875 x -900 X 1-15-5 = 90-95625 cents. nr 5 X r^ P - LE 3 - —A 4-Florin (Austrian) we : ghs 49.92 grains and has a fineness of .000. What is its value ? y 49.92x.900_ „ „ , ■ —■ — X 20.67.183+= $1.93.49. To Convert LJ. S. to Britisli Currency and. Contrari- wise. Rule i.— Divide Cents by 2.027 7 i~ (2.027 708 33), or, Multiply by .493 12- (.493 1 18 26), and result is Pence. 2. Multiply Pence by 2.02771— , or divide by .49312—, and result is Cents. Example. — What are 100 cents in pence? 100 X -493 12 — — 49 - 3 i 2 — pence = 45. 1.312 d. 2. What is a Pound sterling in cents? 20 X 12 = 240 pence, which x 2.027 71— — $4 86.65. FOREIGN MEASURES OF VALUE. Weight, Fineness, and Mint Values of Foreign Silver and. CL old Coins. By Laws of Congress, Regulations of the Mint, and Reports of its Directors. # Current Value of silver coins is necessarily omitted, as the value of silver is a variable element. Hence, in order to compute current value of a silver coin, the price of fine or a given standard of silver bein" known, 0 Proceed as per above rule to compute value of coins. The price of silver should be taken as that of the London market for British standard (925 fine), it being recognized as the standard value and governing rates in all countries. Example.— If it is required to determine value of a Mexican dollar in cents. Weight 867. 5 oz. .903 fine. Value of Silver in London 52.75 pence per ounce - 100.9616+ cents. Then — - 5X9 3 = .846 867— and 106.9616 x .846 867 = 90.5822 cents. 925 40 foreign measures of value. Weight and Mint Values of Foreign Coins. Countries given in Italics have not a National Coinage. Country and Denomination. Weight. Fine- ness. Oz. Thous’s. Arabia. Piastre or Mocha Dollar. . Argentine Republic. Dollar = ioo Centisimos (Employs South American and Foreign Coins.) Australasia. Same as British. Australia. Sovereign, 1855 Pound, 1852 Austria. Kreutzer (copper) Florin, new Dollar, “ 4 Florins Ducat Souverain Belgium. Same as France. Bolivia. Centena Dollar, new Doubloon, 1827-36 Brazil. Rei. Milreis . Double Milreis 20 Milreis, 1854-56 . Moidore, 4000 Reis . Canada. Mil, sterling. . Cent 20 Cent, currency . 25 u “ Penny “ Shilling “ Dollar, sterling 4 “ =20 shillings, currency Pound “ Cape of Good Hope. Same as British. Central America. 4 Reals Dollar 2 Escudos Doubloon ante 1834 Chili. Centaro Dollar, new 10 Pesos. - Doubloon China. Cash, Lc 10 Cents, Leang Dollar Cochin China. Mas, 60 Sapeks 10 Mas, 1 Quan Pure I Silver Current or or Gold. Nominal. Grains. | Cents. 83.14 — 50.69 Value. Gold. U. S. British. .256.5 .281 •397 •59 6 .104 .112 •363 916 916.5 900 900 900 986 900 £ 8. d. .801 .867 .028.8 .82 •575 .261 900 870 i7 J -47 257-47 •75 346.03 362.06 — 4- 85-7 5- 32-37 1.93.49 2. 28. 3 6-75-4 19 11.5 : 1 10. 5 711 . 9 4-6 7 9-* i5-59‘3 3 4 1 — — -547 — 916.66 12.67 — 9 i8 -5 9 I 7-5 914 •54-59 393.6 — — 10.90.6 4.92 — — — 1. 01 — •i5 .187 5 9 2 5 9 2 5 66.6 83-25 / 2 7 26.92 2 4 9.84 1 o 2.63 •05 •5 — — I — 1-52 .027 .866 .209 .869 .801 •492 .867 .087 .866 l 75 850 853-x 833 900.5 900 870 901 901 ii-34 353-33 346.22 37-9 8 374-63 6.75 67.52 3-97-43 3-99-97 3.68.8 14.96.39 9- I 5-4 15-59-3 •75 4 2 16 4 16 5.25 15 1.88 3 1 5-97 -45 1 17 7-45 3 4 1 .07 3-33 9-33 FOREIGN MEASURES OF VALUE. 41 Weigh, t and. Mint Values. Country and Denomination. Weight. Fine- ness. Pure Silver or Gold. Current or Nominal. V A V V I G U. S. E. old. British. Cuba. Same as Spain. Colombia. Centaro Oz. Thous ,s . Grains. Cents. 1. 01 $ c. £ 9. d. Peso, new .801 900 844 870 346-03 '•5 4 Escudos 7-55-5 15-59-3 1 11 0.58 Doubloon, old .867 Costa Rica. ' Same as Mexico. Denmark. Mark, 16 Skilling 8.94 3 4 1 4-39 Crown 7 900 877 895 26.8 2 Risrsdaler .927 390.23 13- 22 10 Thaler 7.90 1 12 5.6 East Indies. See Hindostan and Japan. Ecuador. Centaro 1. Ol Peso .801 900 346.03 •5 England. Penny 2, Q2 j Groat 26.82 80.99 79-°3 201.8 161.44 1 Shilling, new. . .182.5 .178 # AC A. C 9 2 5 ‘ ‘ average 9 2 4-5 9 2 5 9 2 5 9 2 5 916.5 916.5 Half Crown Florin •3 6 3-6 .256.7 .256.2 Sovereign or Pound, new . . . “ “ average. Egypt. Piastre, 40 Paras - 4.86.65 4-85-1 04.9 100 100 7 c t 14-5 Guinea, Bedidlik . 27 £ /DD 875 875 875 10 6. 84 Pound . 27 C 5- 0.52 Purse, 5 Guineas • ^ J O i-375 .032 4.97.4 25. 2.6 1 0 5-3 France. Centime 5 2 10. 2 Sou, 5 Centimes » l6l .2 . 1 Franc, 100 Centimes .l6l .804 .207.5 900 1. 01 •5 5 Francs °9-55 347-76 20 Francs, Napoleon, new . . . 25 Francs 20 centimes =£1 Stg. Germany. Groschen, 10 Pfenning 899 2.38 3.85.8 15 10.26 Mark, 10 Groschen .012.8 .128 . CO c 900 900 23.8 2.38.24 I - I 75 10 Marks 11.74 Thaler 257.04 9 9-5 Ducat •0 yo 986 2.28.38 i9-3 9 4-63 9-5 Greece and Ionian Islands.. Same as France. Drachma, 100 Lepta 5 Drachmas .010.4 900 900 900 I 20 Drachmas .719 .185 310.61 ■ — Pound 44.2 14 i-75 1 0 9.6 Guatemala. Same as Mexico. Guiana , British, French, and Dutch. Same as that of their Countries. Hanse Towns. Mark .012.8 900 5- 6. 11 Holland. Cent 23.8 11.74 •4 — .2 * 2.027 7 1 cents. D* 42 FOREIGN MEASURES OF VALUE, Wei glit and Mint Values. Country and Denomination. Holland. Florin or Guilder, ioo cents. io Guilders Hindostan. Rupee Honduras. Same as Mexico. Italy. Same as France. Lira, ioo Centimes Scudo Indian Empire. Pic, nominal Anna “ Rupee,* 16 Annas io Rupees, and 4 Annas Moliur, 15 Rupees Japan. Sen Itzebu, new Yen, 100 Sen Weight. .021.6 .215 •374 .16 .864 •375 •375 Thous’s, 900 899 916. s 835 900 9 i6 -5 9 i6 -5 .279 .866.7 .053-6 377-^7 372.98 336-25 Cobang, old -2^9 57^ ' new 362 568 20 Yen 1.072 900 Java. Same as Holland. Liberia. U. S. Currency. Mciltci 12 Scudi = 1 Sovereign Mexico. Peso, new “ Maximilian Doubloon, new 20 Pesos, Republic Morocco. Ounce, 4 Blankeels .... 10 Ounces, Mitkeel Naples. Scudo 6 Ducati Netherlands. Same as Holland. New Brunswick. Same as Canada. Newfoundland. Same as Canada. New Granada. Dollar, 1857 803 Doubloon, Popayan -867 Norway. Alike to Denmark. Mark, 24 Skillingen . . . . Nova Scotia. Same as Canada. Persia. Keran, 20 Shahis 10 Keran, Toman Paraguay. Foreign coins. * .092 76 of a X Stg., nominal value = 2 shillings sterling. .867. .861 .867. 1. 081 Fine- ness. Pure Silver Current or Nominal. 164.53 65. 12 373-24 900 900 165 119.19 374-4 9°3 902.5 870.5 873 830 996 896 858 •25 3-03 Value. Gold. $ c. 40.49 3-99-7 4.86.65 6. 84.36 99.72 3-57-6 4.44 19.94.4 21.63 22.81 1 8 16 5.11 1 10.5 4.86.65 15. 6.1 I9-5I-5 5- 4-4 15-37-8 .125 i-5 I -5 •5 4 1. 18 14 8.35 18 2.96 . 1 11 . 6 4 3 4 4 o 1.88 2.4 1 o 8.75 3 3 3-39 10.66 11.25 FOREIGN MEASURES OF VALUE, 43 Weight and. Mint Values. Country and Denomination. Weight. Fine- ness. Pure Silver or Gold. Current or Nominal. Value G < U. S. Peru. Dollar, 1858 Oz. .766 Thous ?s . 900 900 868 912 912 835 Grains. 341.01 346.46 Cents. $ c. Sol...'...'! Doubloon, old .867 .308 •095 15-55-7 5. 80. 66 10.8 Portugal. Coroa, 1838, 10000 Reis — — Roumania. 2 Lei 129.06 Russia. Copek •77 100 Copek, Rouble .667 875 916.6 835 277-73 5 Roubles 3-97-6 Sandwich Islands. U. S. Currency. Sardinia. Lira .16 65.12 Spain. Centimo .19 100 Centimo, Peseta . 16 8 3 7 900 64.13 345-6 Dollar, 5 Peseta .8 100 Reals .268 .270.8 4.96.4 10 Escudos 896 896 750 75o 900 20 Reals vellons=i U.S. Dollar. Sweden. Riksdaler, 100 Ore 28 5- I -5 Rixdollar • 2 73 1.092 .104 393.12 • Carolin, 10 Francs 1-93-5 Switzerland. Same as France. St. Domingo. Gomdes, 100 Cents 6-33 11.83 Tunis. Piastre, 16 Karubs 5 Piastre Ctt 220.38 25 Piastre O A A . l6l 090.5 900 2.99.5 Turkey. Piastre, 40 Paras 4-39 20 Piastre •77 9 QT 830 915 900 306.77 100 Piastre, Medjidie 4.36.9 Tuscany. Zecchino, Sequin Tripoli. 20 Piastres, Mahbub 74-3 2 -3 I -3 Uruguay. Dollar, 100 Centimes West Indies, British. Same as England. Venezuela. Centaro Bolivar, 1 Franc — — _ 7 Memoranda. France.— Bronze coins 9.5 copper, 4 tin, and 1 zinc. Hanse Towns. —Monetary system same as that of German Empire. Switzerland.— The Centime is termed a Rappe. Spain.— 25 Peseta piece is 19.5. 9.5 d. Stg. ; Real vellon was 2.5 d. Stg. Italy. — All coins same weight and fineness as those of France. Malta.— 7 Tari and 4 Grani ■= 1 Shilling Sterling. Egypt.— A Para — .061 5 d. Sterling, and 97.22 Piastres = 1 Sovereign. Indian Empire.— 1 Lac Rupees=:£ioooo Sterling. In Ceylon, Rupees 3 3 ii-22 2 4 5-5 .38 16 4.8 •095 1 o 4.8 I o 7.32 7 11.42 3-125 _5-83 12 3-7 2.16 18 o 9 6.1 3 0.89 : 100 Cents. 44 ENGLISH AND FRENCH MEASURES AND WEIGHTS. ENGLISH AND FRENCH MEASURES AND WEIGHTS. MEASURES OF LENGTH. English. — Imperial standard yard is referred to a natural standard, which is a pendulum 39.1393 ins. in length vibrating seconds in vacuo in London, at level of sea ; measured between two marks on a brass rod, at temperature of 62°. Note. — In consequence of destruction of standard by fire in 1834, and difficulty of replacing it by measurement of a pendulum, the present standard is held to be about 1 part in 17 230 less than that of U. S., equal to 3.67 ins. in a mile. Miscellaneous. Land —Woodland pole or perch or Fen = iS feet. Forest pole — 21 Irish mile = 2240 yards. | Scotch mile ...*. = 1984 yards. Sea. — IO cables, or 1000 fathoms, or 6086.44 feet, or 1.152 8 statute miles = 1 Admiralty or Nautical mile or knot. 3 miles = 1 league. 60 Nautical or 69.168 Statute miles or 20 Leagues — 1 Mean length of a minute of latitude at mean level of the sea =1.1508 statute miles. . A ^ Nautical mile is taken as length of a minute at the Equator. Nautical fathom is 1000th part of a nautical mile, and averages about .0125 longer than the common fathom. French.— Standard Metre or unit of measurement is defined as the ten millionth part of the terrestrial meridian, or the distance from the Equator to the Pole, passing through Paris. .Actual standard is a plat- inum metre, deposited in the Palais des Archives, Paris. Metric Length, in. Inches, Feet, etc. Denomination. Metres. Inches. Feet. Yards. 100 1 ooo 10000 .039 37 •393 7 3-937 °4 39- 370 43 3. 280 87 32.808 69 328.086 9 3280. 869 1 Millimetre. . . 1 Centimetre . . 1 Decimetre . . . 1 Metre 1 Dekametre . . 1 Hektometre. . 1 Kilometre . . 1 Myriametre. . Note.— For length of metre see p. 27. Old. Measure. 1 Terrestrial league = 4.444 kilometres. 1 Nautical league . = 5.555 . “ 1 Arpent 900 sq. toises. 1.093 62 10.93623 109.36231 1 093.623 1 10936.231 Miles. .621 38 6.21377 i Toise = 1.949 metres. 1 Mille = 1.949 kilometres. 1 Nocud (knot). = 1.855 “ MEASURES OF SURFACE. English.— Same as that of United States of America. Miscellaneous. Builders. i superficial part = 1 ^uare inch - 12 parts — 1 2WC "* 12 inches == square foot. Boards . — Boards 7 inches in width are termed battens, 9 inches deals, and 12 inches planks. ENGLISH AND FRENCH MEASURES AND WEIGHTS. 45 French. IMetric Surfaces in. Square Inches, Feet, etc. Denomination. Sq. Inches. Sq. Feet. Sq. Yards. Sq. Acres. i Square millimetre .001 55 •155 003 I K. ^OO 300 1 “ centimetre 1 “ decimetre .107 641 10.764 104 1076.410358 1 “ Metre or Centiare 1 ‘ ‘ dekametre or are 1 “ hektometre or hectare 1 u kilometre 1550.0309x6 1. 19601 119.601 15 1 1 960. 1 1 5 09 .024 711 2.471 098 247. 109 816 24 710.981 6 1 “ myriametre* — — — * Equal 38.610 908 sq. miles. Old. System. 1 square inch = 1.135 87 inches. 1 toise = 6.394 6 feet. 1 arpent (Paris) = 900 square toises = 4089 square yai'ds. 1 arpent (woodland) = 100 square royal perches = 6108.24 square yards. MEASURES OF VOLUME. Imperial gallon measures 277.123 cube ins., but by Act of Parliament 1825 its volume is 277.274 cube ins., equal to 10 lbs. avoirdupois of distilled water, weighed in air, at temperature of 62°, barometer at 30 inches. 6.2355 gallons in a cube foot. Imperial bushel , 18.5 ins. internal diameter, 19.5 external, and 8.25 in depth, contains 2218.192 cube ins., and when heaped in form of a right cone, at least .75 depth of the measure, must contain 2815.4872 cube ins. or 1.6293 cube feet. Grain . — 1 quarter = 8 bushels or 10.2694 cube feet. Vessels. — 1 ton displacement = 35 cube feet ; 1 ton freight by measure- ment = 40 cube feet. 1 ton internal capacity = 100 cube feet , and 1 ton ship - builders = 94 cube feet. English standard No. 5 is .008 grain heavier than the pound, and U. S. pound is .001 grain lighter than English. Wine and Spirit Measures. 4 quarts (231 cube ins.) = .8333 Imperial gallon. 10 gallons = 1 anchor. 18 31*5 42 63 84 126 2 pipes or 3 puncheons (15 imperial) , 26.25 35 52.5 70 105 = 1 runlet. = 1 barrel. = 1 tierce. — 1 hogshead. = 1 puncheon. = 1 pipe or butt . = 1 tun. 4 quarts (28c cube ins.) . . = 1.017 9 gallons = 1 firkin = 9.153 2 firkins = 1 kilderkin . . . = 18.306 -Ade and Beer Measures. Imp’l gall’s. Imp’l gall's. 2 kilderkins = 1 barrel = 36.612 54 gallons = 1 hogshead = 54.918 108 “ = 1 butt . . . . = 109.836 46 ENGLISH AND FRENCH MEASURES AND WEIGHTS. Apothecaries’ or Fluid. Measures. T ,i rfm = 1 grain. I 4 drachms = 1 tablespoon. ^ol's ; ; ; ; — 1 drachm. I 2 ounces (875 grains) = 1 wineglass. Coal Measures. 1 sacks = 1 chaldron . 50 pounds . . . . = 1 cube foot. 88 “ = 1 bushel, g bushels . . . . = 1 vat. , ( 1 London or 80 or 84 pounds ^ Newcastle bushel. go or 94 tk = 1 Cornish 93 pounds . . . . — i Welsh bushel. 3 heaped bush. = i sack. 10 sacks = 1 ton. 1 chaldron = 58.6548 cube ft. 5.25 chaldrons . . = i room. 1 London chaldron =26.5 cwts. 1 Newcastle “ = 53 “ i ton =44-5 cube feet. i room = 7 ions. 21 chaldrons = i score. i barge or keel . . = 21.2 tons. 1 last corn 1 ton water 1 dicker hides .... 1 last hides 1 barrel tar ...... 6 bushels wheat . . 1 clove 1 score •'* 1 sack flour 1 truss straw NLiscellaneoms. 1 truss old hay = 50 pounds. 1 “ new “ = 60 “ 1 bushel oats = 4° u 1 “ barley . . . . = 47 “ 1 “ wheat = 60 “ 1 cube vard new hay = 84 “ 1 “ “ old “ = 126 “ 1 quintal = 100 1 boll = 140 “ 1 sack wool = 364 ‘ = 80 bushels. = 35-9 cube feet. = 10 skins. z=z 20 dickers. — 26.5 gallons. =z i sack flour. = 7 pounds. = 20 “ = 28.2 “ = 36 35.9 cube feet = 1 ton water. Liquid. 1 wine gallon = 231 cube ins. 1 beer “ =282 “ “ 1 litre = .220 09 gallon. 1 gallon = 4-544 1 cube foot . . = 6.2321 gallons. 1 anker . . . . = 8.333 1 hogshead wine . . = 5 2 -5 gallons . 1 “ beer . . . = 54-9 l8 “ 1 puncheon wine . . = 70 1 pipe or butt wine = 105 1 u “ “ beer = 109.836 “ 1 tun ton water 62° = 224 gallons. Builders. 1 solid part = 12 cube ins 12 “ parts = 1 “ inch. 12 “inches ” • . = 1 cube foot. 1 load timber, rough = 40 “ feet. x « u hewn = 50 1 “ lime ....... = 32 bushels. 1 “ sand . 36 “ 1 square sq. feet. bundle laths 1 rod brickwork . 1 rood masonry . Batten, in section Deal, ? u Plank, “ “ = 120 laths. — 306 cube feet. = 648 “ “ = 7 X 2.5 ins. = 9X3 “ = 11X3 Metric Volumes in. Cube Indies, Feet, etc. Denominations. Litres. Gills. Pints. Quarts. Gallons. Bushels. Quarters Cent i 1 i t.rp. .01 .0704 .0176 — — — — Decilitre . 1 • 7°43 . 1761 — — Litre * 1 7.0429 1.7607 .8804 .2201 Dekalitre 10 — 8. 8036 2.2009 .275 11 •3439 3 - 43^9 Hectolitre 100 — — — 22.OO9I 220.0908 2.751 J 3 Kilolitre 1000 — — — 27 - 5 II 35 * Equal 61.025 24 cube ins. ENGLISH AND FRENCH MEASURES AND WEIGHTS. 4.7 Wood NTeasnre. i Stere or cube metre = 35.3150 cube feet or 1.308 cube yards. 1 Voie de bois (Paris) = 70.6312 cube feet ; 1 voie de charbon (charcoal) = 7.063 cube feet ; 1 corde = 4 cube metres = 141.26 cube feet. MEASURES OF WEIGHT. British. — 1 Troy grain = .003 961 cube inches of distilled water. 1 Troy pound =22.815 689 cube inches of water. 1 Avoir, drachm = 27.343 75 Troy grains. 16 drachms, or 437.5 grains 16 ounces, or 7000 grains -Avoirdupois. 8 pounds , = 1 ounce. 14 28 . = 1 ton = 1 stone (for meat). = 1 stone. = 1 quarter. = 1 cwt. = 1 pound. 20 hundredweights . The gram, of which there are 7000 to the pound avoirdupois, is same as Troy grain, of which there are by the revised table 7000 to the Troy pound. Hence Troy pound is equal with the Avoirdupois pound. In Wales, the iron ton is 20 cwt. of 120 lbs. each. Troy. 24 grains = 1 dwt. 20 pennyweights, or\ 437-5 grams j • • - 1 o sq. ft. Noosfia, Arabic J 3 8 cul J- ins - Gudda. ^ galls. 3 *6f- Tomand Other Measures like those of Egypt. Argentine Confederation, Paraguay, and TJ ruguay. Fanega Arroba 25-35 16 s - Quintal 101,4 Also Decimal System in Argentine Con- federation and Paraguay. Australasia. Land Section 80 acres - Other Measures same as English. Austria. Zoll 1 *037 1 ins. Fuss 1 0371 it Meile 24 000 It. Klafter, quadrat 35- 8 54 sq. yds. Cube Fuss ft ' Achtel 1.692 galls. Eimer u Mptyp 1.6918 bush. Unze 8642 grains. Pfund (1853, 500 grammes), 1.2347 lbs. Centner 123.47 Also Decimal System. Antwerp. Fuss h\ s . Corde 24.494 cub ft. Bonnier 3.2507 acres. Also Decimal System. , 18.205 ins. Babylon Pachys Metrios Baden. Fuss Klafter 5-9°55 ft- Stundcn 4 88 ° 5 “ s - Morgen ^ galls. . 1 1268 bush. Malter ; ; ; ; ; ; ; ; 1 . 1023 ibs. Also Decimal System. Pfund . foreign measures and weights. 49 Bagdad. Guz 31-665 ins. Barbary States. Pic, Tunis linen 18.62 ins. “ “ cloth 26.49 “ “ Tripoli 21.75 “ Batavia. Foot . . Covid . El 12.357 ms. 27 “ 27.75 “ Bavaria. Fuss ii-49 ins - Klafter 5-745 36 It. Ruthe 3.1918 yds. Meile 8060 Ruthe, quadrat 10. 1876 sq. yds. Morgen orTagwerk 8416 acre. Klafter, cube 4. 097 cub. yds. Eimer 15.058 56 galls. Soheffel 6.119 “ Metze 1.0196 bush. Pfund 8642 grains. Also Decimal System. Belgium. Meile 2.132 yds. Also Decimal System. Benares. Yard, Tailor’s 33 ins. Bengal, Bombay, and Cal- cntta. Moot 3 ins. Span 9 “ Ady, Malabar 10.46 ins. Hath 18 “ Guz, Bombay 27 “ “ Bengal 36 “ Corah, minimum 3.417 ft. Coss, Bengal 1.136 miles. “ Calcutta 1-2273 “ Kutty. 9.8175 sq. yds. Biggah, Bengal 3306 acre. Bombay 8114 “ Seer, Factory 68 cub. ins. Covit, Bombay 12.704 cub. ft. Seer, Bombay 1.234 pints. Parah 4. 4802 galls. Mooda 1 12.0045 u Liquids and Grain measured by weight. Bohemia. Foot, Prague 11.88 ins. “ Imperial 12.45 “ Also same as Austria. Bolivia, CHili, and Bern. • 33 Vara Fanegada, Gallon. . . . Fanega. . . Libra Arroba 25. Originally as in Spain; now Decimal System in Chili and Peru. 333 ms. .5888 acres. -74 gall •572 “ .014 lbs. ■ 36 .927 inch. . 11.128 ins. Brazil. Palmo, Bahia 8.5592 ins. Vara 3. 566 ft. Braca 7-132 “ Geora 1.448 acres. Also same as Portugal , and sometimes as in England. Buenos Ayres. Vara 2. 84 ft. Legua 3.226 miles. Suertes de Estancia .... 27 000 sq. varas. Also same as Spain * Burmah. Paulgat 1 men. Dain 4.277 yds. Viss 3.6 lbs. Taim 5.5 “ Saading 22 u Also same as England. Canary Isles. Onza Pic, Castilian Almude 0416 acre. Fanegada 5 “ Libra 1.0148 lbs. Also same as Spain. Cape of Good Hope. Foot n. 616 ins. Morgen 2. 1 16 54 acres. Also same as in England. Ceylon. Seer 1 quart. Pariah 5.62 galls. Also same as in England. China. Li 486 inch. Chih, Engineer’s 12.71 ins. “ or Covid 13-125 “ “ legal 14. 1 “ Chang 131-25 “ “ legal 141 “ P« 4-°5 ft- Chang, fathom 10.0275 ft. Li 486 yds Pu or Rung 3.32 sq. yds. King, 100 Mau 16.485 acres. Tau 1. 1 3 galls. Tael i-333 oz. Catty 1.333 lbs. Cochin China. Thuoc or Cubit 19.2 ins. Sao 64 sq. yds. Mao 1.32 acres. Hao 6. 222 galls. Shita 12.444 u Nen 8594 lb. Colombia and Venezuela. Libra 1. 102 lbs. Oncha 25 “ Also Decimal System. 50 foreign measures and weights. Hentnarli,* Greenland, Ice- land, and Norway. Tomme 1-0297 ins. Fod — Favn, 3 Alen 1.0297 ft. 6.1783 “ Mil'"?. “7" 4-6&> 55 mites. “ nautical o 6lOJ2 «ik Ank er 8.0709 galls. " nKtr .a 7 8 bush. Skeppe y o u ^ingkar;...;.....,. Lispund i7'3 6 7 u Centner * Also Decimal System. Hungary. Fuss...:.. 30-67 “ 9.139 y ds - Also as in Vienna. Elle. Meile . Guz Cowrie Indian. Empire. 27.125 ins. i sq. yd. Sen ... . . • ' • ' 61025 39 <^ ms u 2.204737 IDS. Uniform standard of multiples of the Sen adopted in 1871. Ecuador. Decimal System. Genoa, Sardinia, and Turin. Palmo. 9-8076 ins. Piede, Manual, 8 oncie. . . 13- 4 88 u “ Lipraudo, 12 •“ ...20.23 Trabuco orTesa IO,I rL miles Starello 9 8o 4 ac ™- Giomaba 9394 Italy. IMilan and "Venice. Decimal System. The Metre is termed Metra ; the Are, Ara ; the Stere, Stero; the Litre, Litro; the Gramme, Gramma, and the Tonneau, Tonnelata de Mare. Naples and Two Sicilies. Palmo mi Germany. The old measures of the different States differ very materially ; generally , how- ever , Foot. Rhineland 12. 357 Meile 4.603 miles. Decimal System made compulsory in 1872. Greece. Stadium 6155 niile. Also Decimal System. , r . ,• i.mo6 miles. MSltego:::::::::::'.::'.:: .7467 acre. Moggia ° « Pezza, Roman Roman States. Old Measure. Foot to* “ Architect’s.. n-73 Braccio.. 30-73 u Palmo 8-347 Miglio 1628 yds. Quarta 1. 1414 acres. Lucca and Tuscany. Guinea. Jachtan . , 12 ft. Hamburg. Fuss 11.2788 ins. Klafter 5-6413 “-• - - Morgen 2.386 acres. Cube Fuss - 83 1 1 CU V; ft ’ Pie Palmo Braccio Passetto Passo Miglio Quadrato Saccato Cube Fuss. Tehr 99-73 ,7 Viertel i-594 7 g alls - Pfund (500 grammes) . . . 1-102 32 lbs. Ton 21 35- 3 lbs. Also Decimal System. .6476 acre. Hanover Fuss i1 - 5 , 5 ' Morgen Hindostan. Borrel '“f- 2.387 u Kobe 2 9-o65 Coss 3-65 f - Tuda 1.184 cub. ft. Candy ..,.14.209 . 11.94 ms. • XI -49 “ . 3.829 ft. • 5-74 “ . 1.0277 miles. . .8413 acre. . i-3 2 4 u Sun, 1. 193* ins. ,5* ins. Japan 303 03 -Metre. . . Shaku, 3.0303 Metres. . . . n-93°5* Jo, 30.303 ‘ •••• 9-942 1 * ft. R, e , n A88o 5 • “ 8 %| t mnek - Kai-ri 6080 feet t Hiro ••••• 4 - 97 ** 4 1* feet. Momme : . 3-756 521 7 g™™™^ Hiyaku-me o' 8 q S i7 Kwam-me 8.28171 ^ Hiyak-kin - 132-507 3 2 u Man’s load S7-97 2 u Koku ^‘q 68 ! 1 “ Hiyak-koku 33126.8308 * These are as equivalent as they are prac 1 cable of reduction. + Admiralty knot. FOREIGN MEASURES AND WEIGHTS. 51 Java. Duim 1.3 ins. Ell 27.08 Rjong 7.015 acres. Ran 328 galls. Tael 593-6 grains. Sach 61.034 lbs. Pecul 122.068 “ Catty... 1.356 u IVTadras. -Ady — ... 10.46 ins. Covid 18.6 “ Guz. , ■ 33 Culy 20.92 ft. League 3472 yds. 338 galls. M areal 2.704 “ Tola 180 grains. Seer 625 lbs. Viss 3.086 “ Maund .....24.686 u Malabar. 10.46 ins. iVlalaeoa. Hasta or Covid Depa Orlong . 18.125 ms. . 6 ft. . 80 yds. Palmo. Pie. . . . Canna. , Salma. , NXalta. Foot Kot, silk. , Fathom . Also as in Sicily. ^Moldavia. . 10.3125 ins. . 11.167 “ • 82.5 “ 4.44 acres. . 8 ms. . 24.86 ins. . 8 ft. Molucca Islands. Covid 18.333 ins. Morocco. Tom in Cadee. Cubit. Muhd. . . 2.81025 ms. . 20.34 ins. Kma, on.:: :::::: ::::::: ll% 35e ^ Rotal or Artal I-I2 ibs. Liquids other than oil are sold by weight. Mysore. Angle Haut Guz Netherlands. Elle *--v 39-370 432 ins. Decimal System since 1817. 2.12 ms. 19. 1 11 38.2 Gereh . JPersia. Gueza, common ’ 2t - “ Monkelrcr 37 . 5 375 ms. Archin, Schah ........... 31.55 ins. “ Arish 38.27 “ * Parasang 6076 yds. Chenica 80. 26 cub. ins. Artaba 1 . 809 b u sh. Mi seal grains. E ate i 2.1136 lbs. Batman Maund 6.49 “ Liquids are measured by weight. Trewice . IPoland. . 14.03 ms. Precikow j ns> Pretow 4.7245 yds. Mile, short 6075 yds Morgen 1.3843 acres. Portugal and Mozambique. foot ins. M ] ii la 1.2788 miles. Almude 3.7 galls. ff D £ a 1.488 bush. Alguieri 3.5 « Libra..: 1. 012 lbs. Also Decimal System. Prussia. 12.358 ins. Rathe. v . 4. n 92 yds. M ei le 2 4 ooo feet. Quadrat Fuss 1.0603 sq. ft. MR 1 ^ 631 03 acre. Cobe Fu SS 1.092 cub. ft. Scheffel 1.5121 bush. p"f n e rt r 7-559.galls. iouna.. 7217 grams. Zollpfund 1.1023 lbs. Centner 113-43 tbs. K-nssia. Vershok 1.75 ins. f o°t 12 ins. Arschine u Rhein Fuss. 1.03 ft. Sajene ... £ ^. rst 3500 - Dessatiila 5-5574 miles. uessatina ; . 2.4954 acres. Vedro 2. 7049 galls. 1.4426 bush. Tschetwert 5.7704 “ E™? d 6 3i7 grains. 90285 lbs. Decimal System adopted in 1872. Siam. S^R 9-75 ins. Ken 39 u dod ; 09848 mile. Roeneng 2.462 miles. Silesia. I ™? ins. S 4.7238 yds. M eile 7086 yds. Mor S en 1.3825 acres. 52 FOREIGN MEASURES AND WEIGHTS. Singapore. Hasta or Cubit 18 ms. Dessa “• Orlong 80 y ds - Smyrna. pj c 26.48 ins. Indise 2 4- 6 4 8 “ Berri 1S28 yds. Spain, Cuba, Malaga, Ma- nilla, Guatemala, Hondu- ras, and. Mexico. Pie 11.128 ins. Vara 33-3 8 4 “ Milla ... .865 mile. Legua, 8000 varas Fauegada Vara, cubo Cuartilla Arroba, Castile Fanega Libra Tonelada Also Decimal System. 4. 2151 miles. . 1.6374 acres. . 21.531 cub. ft. .. .888 gall. • • 3-554 galls- . . 1.5077 bush. . . 1. 0144 lbs. 2028.2 lbs. Stettin. Fuss Foot, Rhineland. Elle Morgen , 11. 12 ms. • 12.357 “ . 25.6 ins. . 1.5729 acres. Sumatra. Jankal or Span 9 ins. Elle - *8 ‘ Hailoh 36 Fathom ° Tung 4 yds. Surat. Tussoo, cloth 1.161 ins. Guz, “ 27.864 1 Hath 20.9 Covid . . .. i8 -5 Biggah acre. Sweden. Fot 11.6928 ins. Ref ’. y ds - Faden 5.845 ft. League 3- 35^4 miles. Meile 6 - 6 4i7 Tunnland 1.2198 acres. Anker 8.641 galls. Spann 1.962 bush. Centner 1 12.05 lbs. Also Decimal System. Switzerland. Fuss, Berne n.52 ms. “ n -54 “ Vaud 1 1- 81 “ Klafter 5-77 .. Meile 4.8568 miles. Juchart, Berne 8 5 acre. Maas 2.6412 pints. Eimer S^^galls M alter 4.1268 bush. Ffund 1 -1023 lbs. Also Decimal System. Tripoli. Pik, 3 palmi 26.42 ins. Almud 4 cu>>. ins. Killow 2 ° 2 3 Barile 14.267 galls. Temer 7383 bush. Kottol 768° §»'“*• Turkey. Pic, great 27.9 ins. “ small .....27.06 ‘ Berri I - 828 y d .®- Alma 1. 1 54 galls. Also Decimal System. W urtemberg Fuss FUp. Meile Morgen 8146.25 yds. 7703 acre. Cube Fuss. . . . . .83045 cub. ft Eimer Scheflfal . - - - 4.878 bush. Pound Fuss Zurich. FUft Klpft^r ^.0062 ft. 4.8568 miles. Jachart. ..... 808 acre. 1 Cube Klafter. 144 cub. ft. LENGTHS Course. Miles. OF ENGLISH RACE-COURSES. NEWMARKET. Across the Flat Beacon Cambridgeshire Cesare witch Round Rowley Mile Summer Course Two-year old, new . . Yearling 1.292 4.206 1. 136 2.266 3-579 1.009 2 .702 .277 Course. DONCASTER. Circular Fitzwilliam Red House St. Leger Cup Course EPSOM. Craven Derby and Oaks. . i Metropolitan I -9 I 5 1 •7 11 1.825 2.634 1.25 i-5 2.25 Course. • goodwood. Cup Course LIVERPOOL. New Course New Castle — Oxford YORK. Stakes Course. . Two-mile Miles. 2-5 1.5 1.796 x-75 1.923 SCRIPTURE MEASURES. ANCIENT WEIGHTS. 53 SCRIPTURE AND ANCIENT LINEAR MEASURES. Scripture. 9 12 inch. I Span, 3 palms IO inc: Palm, 40 digits 3.648 ins. | Cubit, 2 spans u * Fathom, 4 cubits. . 7 feet 3 . 552 ins. Hebrew and. Egyptian. Nahud cubit i. 475 feet. I Babylonian foot T T . n fppt Royal “ 1 . 721 6 “ Hebrew “ 1,140 leet - Egyptian finger 06145 “ I “ cubit. ..!!*!!! 1 Hebrew sacred cubit 2.002 feet. 1. 212 i.8i 7 Digit Pons (foot) Cubit Pythic or natural foot . Attic or Olympic “ . Grrecian. 7554 inch. oo 73 feet. 1332 “ 814 foot. 009 feet. Ancient Greek foot ) (16 Egyptian fingers) ) ••••• -9841 foot. Arabian foot 1.095 feet. Stadium 604.0^5 “ Olympic stadium 606.29 “ Mile, 8 stadium 48^ feet Alexandrian or Phileterian stadium (600 Phil. feet) = 7 o8 65 feet Volume. Keramion or Metretes 3.488 gallons. ' Jewish. Cubit.... 1.824 feet. Sabbath day’s journey 3648 “ I Mile, 4000 cubits ?2 g6 feet I Day’s journey 33 . i64 m fl e s. Roman. Long Measures. Digit 7 2 3 73 ins. Uncia (inch) q 6 7 “• Pes (foot) 11.604 “ Passes "I 505 *!** r assus 4*835 tf Mile, milliarium 4842 “ ANCIENT WEIGHTS. Hebrew and Egyptian. Attic obolus Troy grains. { «•’? Denarius, Roman Troy grains. j 51-9* “ drachma X 9-*t ( 51 - 9 * U Shekel. . . Nero ( 62. 5f Lesser mina ( 69 1 Ounce . . . (4I5-I* Greater mina Egyptian mina Drachm. . ( 43 i- 2 $ Ptolemaic “ . Libra Alexandrian “ y D • • • Pound. . . . . 12 Roman ounces. Obolus Talub Talent (60 minae) Obolus, ancient . Gramme ......! Drachma. 50 u great .’! !!.’!!! 69 Grrecian Troy grains. " 33 57 i 5 . 56 lbs. avoirdupois. Mina ‘ ‘ great . . Talent “ Attic . 47 I Troy ounces. . .. 10.41 ... 14.472 . . 625.19 . . 868.32 Roman. 0unce 416.82 grains. | Pound. 10.41 ounces. * Christian!. + Arbuthnot. E* X Paucton. 54 GEOGRAPHIC MEASURES AND DISTANCES. GEOGRAPHIC measures and distances. rp oil nee Longitude into Time. • and seconds by 4, and product is th .6 time. , . . o ‘31^ v 4 = fh. 22 w. 4®* Example.— Required time corresponding to 50 31 • 50 31 X 4 3 To Reduce Time into Longitude. Rule Reduce hours to minutes and seconds, divide by 4, and quo- tient is the longitude. Or, Multiply them by 15. EXAMPLE. -Required longitude corresponding tc 5 h. 8 m. 11. 2 s. = 308m. 11.2s., which . 4 — 77 45-5 • Or, multiplying by 15: S h - 8w * I1 * 2S * X 15 = 77° 2 45-5 • Table of Departures for — * ' Course. Course. Departure. 3.5 points. 4 “ •773 •7°7 Distance run of 1 IVtile. Departure. II Course. | Departure. 4.5 points. 5. 5 points. 6 Thus if a vessel holds a course of 4 points that is without leeway, for distance 0f o“a ™sse e iIaUi“g a l e N 7 E 7 upo^a' c'oursTof 6 points for 100 miles will make 38.3 froox 083) miles of longitude. _ . ^ tioo x r l point of tiie Miinntes, and Seconds of eacn ro Compass with. Meridian. « • " Sin. A.* Degrees North. South. N.N.E N.N.W N.E. by N. . N. W. by N. . Points. N.E. . N.W. N.E. by E. .. N.W. by W. . . E.N.E W.N.W E. by N W. by N East or West. S.S.E s.s.w S.E. by S. ... . S.W. by S.... S.E S.W S.E. by E.... S.W. by W... E.S.E. .. W.S.W. . E. by S.. W. by S.. East or West. 2 48 45 5 37 30 8 26 15 11 i 5 14 3 45 16 52 30 19 4 1 *5 22 30 25 18 45 27 7 30 30 56 15 33 45 36 33 45 39 22 30 42 11 15 45 47 4 8 45 50 37 3° 53 26 *5 56 15 59 3 45 61 52 30 64 41 15 67 30 70 18 45 73 7 3° 75 56 i5 7 8 45 81 33 45 84 22 30 87 11 15 90 | Cos. A.* Tan. A.* .9988 .0491 .995 2 .0985 .9891 .1484 .9808 .1989 •97 .2504 .9569 •3034 .9415 •3578 .9 2 39 .4142 •9°4 •4729 .8819 •5345 •8577 •5994 • 8315 .6682 .8032 .7416 •773 .8207 .7409 .9063 .7071 1 •6715 1-103 •6344 1.218 •5957 1.348 •555 6 1.497 •5141 1.668 .4714 1.871 •4275 2.114 .3827 2.414 •3368 2-795 .2903 3.296 .2429 3-94i .195 5.027 .1467 6.741 .098 10.153 .0489 20.555 .0000 03 * A. re ■presenting course or points from the meridian. GEOGRAPHIC LEVELLING. 55 GEOGRAPHIC LEVELLING. Curvature and Refraction. Correction for Curvature of Earth, to be subtracted from reading of a levelling-staff, is determined as follows : Divide square of distance in feet from level to staff, by Earth’s Equa- torial diameter — viz., 41 852 124 feet. Or, Two thirds of square of distance in statute miles equal the cur- vature in feet. Correction for Refraction is to be added to reading, and as a mean may be taken at about one sixth of that for curvature. Correction for Curvature and Refraction combined, is to be subtracted from reading on staff. Formulas of Capt. T. J. Lee , U. 8 . Engineers . D 2 . D 2 — = correction for curvature , — m = correction for refraction, and D 2 (r — 2 m) — = correction for curvature and refraction. D representing distance , R radius of earth , and m a coefficient of refraction = 075 all in feet. /J ’ Illustration. A distance is 3 statute miles, what is correction for curvature and refraction? 5280 X s (1 — 2 x .075) 4I g 52 124 = • 8s X 5-996 = 5-09 7 feet Approximately , — D 2 = curvature in feet. 3 Revelling Toy Roiling Boint of Water. To Compute Height Above or Below Bevel of Sea. 517 (2i2°-T) + (212 0 - T Y = Height. Illustration.— What is height of an elevation, when boiling point of water is 182° ? 517 X 212° 182° + 2I2°-i82° 2 - 5 I 7 X 30+ 30 2 = 1 6 410 feet. Corrections for Temperature to he made in Connection with Formula. Temp.j Temp. Temp. C ^T Temp. Temp. C ?"n? Temp. .972 .976 .98 .984 .988 .992 .996 r. 004 1.008 1. 012 1. 016 1.02 1.024 1.028 1.032 1.036 .041 54 56 58 60 62 64 66 68 70 | 1.046 1.05 1.054 1.058 1.062 1.066 1. 071 I -°75 •°79 1.083 1.087 r.091 [.096 c. 1 [. 104 [. 108 [. 112 [. 116 90 92 94 9 o 98 100 102 104 106 1. 124 1. 128 1. 132 1.136 1. 14 r - 1 44 1.148 1. 152 ' ' 7 I 1UU Illustration. Ass u m e temperature in preceding illustration to have been 8o-. Then 16410X 1.1 = 18051/eet and Differences Let ween. True 56 GEOGRAPHIC LEVELLING AND DISTANCES. *?**??~* If It > d> d\ o « f in-OD^COVO <*00 VOCO^ 2 £- 2 “ Si'S ^SRSiaWS-SS^S fi 1 n w-. SR'S *A*feUft&rsfS* i.i 1 III IS lla 5^| 111 lisill I lllifl I aS&a j suit's | | ^ H M CO ^t-^O w CO g it f«S.$&-a8 8.<8 8-8 g>8 ! j naiif n f f n f tittissill js ft ri i it j I jl ij ! i — -M : •S' I i«%rs«Ksr:j BfifiUllin I |i : i — 5-2 ill & «5s»f ij It; if W u,vo j , s ?5 !£ GEOGRAPHIC LEVELLING. — MAGNETIC VARIATION. 57 Illustration. — Curvature of Earth independent of refraction is computed at .667 foot = 8.004 ins. for 1 geographical mile, and as refraction on land is taken as .104 foot or 1.248 ms., and on ocean at .099 foot or 1.188 ins., relative visible dis- tances of an object, including curvature and refraction, for an elevation of .667 foot is 1.09 miles on land, and 1.08 miles at sea. I “ “ I.33 “ “ “ “ j o 2 “ a 9 feet “ 4 “ “ “ “ 3 8 u « u 1 mile “ 104.03 “ “ “ “ 103.54 “ “ “ Difference between two levels in feet is as square of their distance in miles. Illustration —At what elevation can an object be seen, at surface of ocean when it is 2 miles distant ? * I 2 : 2 2 :: .568 : 2.272 feet '== 2 feet 3.25+ ins. . 5^ erence between two distances in miles is as square root of their heights m feet. 0 Illustration i. — At an elevation of 9 feet above level of sea, at what distance can an object be seen upon its surface? ’ bianco V- 568 = .754:: 1 : : if 9 ; 3. 98 miles. 2 AT If a I 5 an , at the ^re-topgallant mast-head of a vessel, 100 feet from water sees another and a large vessel “hull to,” how far are the vessels apart? ’ A large vessel’s bulwarks are at least 20 feet from water Then, by table, 100 feet = 11.27 20 “ 5-93 Distance 19.20 miles. When an observation for distance is taken from an elevation, as from a light-house, a vessel’s mast, etc., of an object that intervenes between observer and horizon, or contrariwise, observer being at a horizon to elevated object, distance of observer from intervening object can be determined by ascertaining or estimating its elevation from horizon and subtracting its distance from whole distance between observed and pomt from which observation is taken, and remainder will give distance of object from observer. & fa —Top of smoke-pipe of a steamer, assumed to be 50 feet above sur- fr0m aQ 100 ^et; whatTs ffi 100 feet = 13 27 50 “ ■ 9* 38 Distance 3.89 miles. D Refraction = -S« ^ for land and .563 D= for sea MAGNETIC VARIATION OF NEEDLE. -Needle reached a Westerly maximum in 1660 and then varied to East until 1800, when it reversed to West. ’ Easf trf o ’ w"~ f r T t0 1815 variation ranged from n° 15' 24 27 West, when it receded gradually to 21 0 in 186^. Jamaica (W. I.). — No variation from year 1660. Diurnal Variation.— There is a small diurnal variation bein- greatest 3 r; zw&ztnzr*' «• 58 MAGNETIC VARIATION OF NEEDLE. Variation in U. &— Professor Loomis concludes that the Westerly variation is increasing and Easterly diminishing in every part of United States- that this change occurred between 1793 and 1819, and that present annual change is about 2' in Southern and Western States, from 3' to 4' in Middle States, and 5' to 7' m Eastern States. Rules for computation of variation are empirical, except in each particular locality, as the annual and diurnal variations of the needle, added to local attraction, render it altogether unreliable. Decennial Variation of Needle. Mr. Schott , U. S. Coast and Geodetic Survey. From January i, 1790, to January 1, 1880. Location. Halifax, N. S Quebec, Can Portland, Me Burlington, Vt. . . . Newburyport,M’s. Portsmouth, N. H. Rutland, Vt Salem, Mass Boston, Mass Cambridge, Mass. . 1 790. | 1 800. | 1810. wT w. w. i5-9 1820. W. Hartford, Conn. . . 5-2- New Haven, Conn. 4.8 New York, N. Y. . 4.29 Philadelphia, Pa. . 2.4 Baltimore, Md. . . . — Albany, N. Y — Buffalo, N. Y .14 E. Erie, Pa. •03 Cleveland, 0 2.2 Detroit, Mich - Washington, D. C. .1 Acapulco, Mex.. . . 7.2 Charleston, S. C.. . Havana, Cuba — Kingston, W. I — 6-3 San Diego, Cal 11 Savannah, Ga — Mobile, Ala — Key West, Fla . — Monterey, Cal. ... 11. 4 Mexico, Mex 7.1 New Orleans, La. . ■ 7 San Bias, Mex ■ 7-4 1 San Francisco, Cal . 12.8 Sitka, Alaska . — Vera Cruz, Mex. . • 8.37 E. •35 1830. 1840.] 1850. | i860. W. 7- 1 W. w. w. 1870. w. 8.95 I 9-32 IV 8 8.64 9-33 10.03 i' 6-73 7-43 8.31 9.09 5. 46 5 5-8 5-43 6.24 5-99 6.77 6.67 4-47 4.91 5-59 6-34 2.28 2.71 3-33 4. 11 .8 1.2 x -7 2.4 5-79 6.32 6.97 7-7 • 3 •74 i-33 2.05 E. E. • 8 3 •43 •17 •25 E. E. i-5 1.05 .6 ' I4 ! 2.9 2-55 2.09 1.56 W. W. W. w. .6 1 1.49 1.99 E. E. E. E. 8.68 8.88 8.91 8.79 4.04 3-44 2.78 2.12 5 6.22 6. 12 5-94 5-7 1 5-4 5 4.6 4.2 11. 6 11. 9 12.2 12. 54 4.8 4-5 4.14 3-65 7-3 7.2 7-1 7 6.9 6.52 6.03 5-47 13-3 13-9 14.44 14-95 8.6 8.8 8.9 8.76 8.1 8.2 8.14 7-94 8 8.61 8.84 8.97 9.9 1 14.42 14.92 x 5-38 15-78 1 27. 89 28.48 1 28.88 29.08 . 8.66 1 12 9.4 8 9.42 1 9.14 i3- 15 11.97 11.49 12.8 i-5 7-99 8.18 2.23 1.07 3.08 8.8 4.86 15.42 8.48 7.61 8.91 16. 11 29.08 16.36 28.88 7-i5 16. 52 28.5 For variation in other locations in unirea oiawb *uu see treatises of J. B. Stone, C.E., New York, and Heller and Brightly, Philadelphia, 1878. MAGNETIC VARIATION OF NEEDLE. Table for Reducing Observed. Daily Variation, of Needle to Mean Variation of* the Day. * TJ. S. Coast and Geodetic Survey, 1878. Needle East of Mean Mag- Needle West of Mean ft Season. netic Meridian. Meridian. A.M. A. M. A.M. A.M. A.M. A.M. NOON. P.M. P.M. P. M. h. h. h. h. h. h. h. h. ~hT 6 t 7 1 8 / 9 f 10 II Noon. 1 2 3 Spring 3 4 4 3 I I 4 5 5 4 Summer. 4 5 5 4 I 2 4 6 5 4 Autumn 2 3 3 2 — 2 3 4 3 2 Winter 1 i 2 2 I — 2 3 5 2 Variation of* Needle U.S. Location. Astoria, TV. T Augusta, Ga Austin, Tex Bismarck, Dak Chicago, 111 Cincinnati, 0 Colorado Springs, Col. , Columbia, S. C Columbus, 0 Deadwood, Dak Denver, Col Detroit, Mich Duluth, Min Galveston, Tex Green Bay, TVis Houston, Tex Indianapolis, Ind Jackson, Miss Jacksonville, Fla Kansas, Kan Keokuk, la Little Rock, Ark Louisville, Ky Milwaukee, \75s. at Locations in United States and Canada, 1S775. Coast and Geodetic Survey. EAST. Variation. 30 28 15 6 55 18 45 38 Location. Augusta, Me Bangor, Me Batavia, N. Y, Belfast, Ale Bridgeport, Conn. . Calais, Me Concord, N. H Dover, Del Fall River, Mass. . . Hamilton, Can Harrisburg, Pa.. . . . Hudson, N. Y. Lewiston, Me Lowell, Mass Montpelier, Vt Montreal, Can New Bedford, Mass, New London, Conn. Newark, N. J. 48 WEST. Montgomery, Ala Natchez, Miss Nebraska, Neb New Orleans, La Olympia, W. T Omaha, Neb Oregon City, Or Paducah, Kan Portland, Or Port Townsend, TV. T.. Sacramento, Cal Salt Lake City, Utah. . San Antonio, Tex Santa Barbara, “ ... Santa Fe, N. Mex Springfield, 111 St. Augustine, Fla. . . . St. Louis, Mo. St. Paul, Minn Tallahassee, Fla .* Toledo, 0 Topeka, Kan Vincennes, Ind Yazoo, Miss 14 16 34 4 40 i5 22 8 18 12 11 42 4 12 10 30 2 55 4 18 8 14 48 11 15 12 5 12 20 10 30 9 15 7 18 Newburgh, N. Y Newport, R. I Norfolk, Va Ogdensburgh, N. Y. . Oswego, N. Y Ottawa, Can Pittsburgh, Pa Raleigh, N. C Richmond, Va Rochester, N. Y. Saratoga, N. Y Stamford, Conn. . . . Syracuse, N. Y Toronto, Can Trenton, N. J Troy, N. Y Utica, N. Y. Wilmington, Del Wilmington, N. C. . . Variation. 5 2 7 26 11 20 6 55 23 *7 4 *7 9 17 J 4 58 13 18 6 3 2 55 6 30 10 30 4 14 4 35 25 8 38 28 24 48 60 GEOGRAPHIC LEVELLING. — BASE LINE. — SOUNDINGS. Dip of Horizon. Approximate, 57.4 V H =. dip in seconds, varying with temperature of air. H representing height of observer's eye in feet. .667 « 2 = H : ' .498 s" = H : 1.42 VH = *: 1.23 VH = n. n representing distance in geographical miles and s in statute. Multi- plier. Angle. Multi- plier. Angle. Multi- plier. Angle. Multi- plier. Angle. Multi- plier. Angle. 1 45 0 2.5 68 11 4 75 58 5-5 79 42 8 82 52 1.5 56 18 3 7 i 34 4-5 77 2 9 6 Bo 32 9 83 40 2 63 26 3-5 74 4 5 78 41 7 81 52 10 84 17 uptruuLuri. — oci dvakcj-lu vw — o — ; « multiplied by number opposite to it. Illustration. — When sextant is set at 8o° 32', and horizontal distance from ob- ject in a vertical line is 100 feet, what is its height? 100 X 6 = 600 feet. By Trigonometry: 1 : 100 *.*. 5.997 (tan. angle) : 599-7 f eet - \{ To Reduce a Sounding to Dow NV'ater. 180 1'\ 1 zp cos> 2 zrJLj = h'. h representing vertical rise of tide, and h! sound U ina or depth at low water, both in feet; t time betiveen high and low water, and J L _ 180 1 0 t> time from time of sounding to low water, in hours. — cos. when -j- <90 7 and 4* cos. when >90°. Illustration. — Low water occurring at 3.45, and high water at 10.15 p.m ; , a sounding taken at 5.30 p.m. was 18.25 feet; what was depth at low water, vertical rise being 10 feet? h = 10 feet ; V — 5 h. 30 m. — 3 h. 45™. = ih. 45™- = 1.75 hours. t=z ioh. 15 m. — 3 h. 45 m. = 6 h. 30WI. — 6.5 hours. 10/ 180X1.75' Then “ ( 1 4 - cos - 6-5 3/1. 45WI. = On. 30/rt. _ O. 5 iivui a. — 5(1 — 48° 27' 24") = 5 X (1— .663 186) = 1.684 07 feet Sounding 18.25 /eei — Reduction 1.68407/^=16.56593/^. Ijeii"tlis of a Degree of Longitude on parallels of Dati- tvTde, for each, of its Degrees from Equator to Dole. Lat. Miles. Lat. Miles. Lat. Miles. Lat. Miles. Lat. Miles. Lat. Miles. i° 59-99 1 6° 57- 6 7 3 i° 5 i -43 46 ° 41.68 6i° 29.09 76 ° 14.52 2 59.96 17 57-38 32 50.88 47 40.92 62 28.17 77 x 3-5 3 59.92 18 57.06 33 50.32 48 40.15 63 27.74 78 12.48 4 59-85 x 9 56.73 34 49-74 49 39-36 64 26.3 79 n -45 5 59-77 20 56.38 35 49- 1 5 50 38.57 65 25-36 80 10.42 6 59.67 21 56.01 36 48.54 5 i 37-76 66 24.4 81 9 - 3 8 7 59.55 22 55-63 37 47.92 52 36.94 67 23-44 82 8.35 8 59. 42 2 3 55-23 38 47.28 53 36.11 68 22.48 83 7 - 3 1 9 59.26 24 54.81 39 46.63 54 35-27 69 21.5 84 0.27 10 59 - 09 25 54-38 40 45.96 55 34 - 4 i 70 20.52 85 5-23 1 1 58.89 26 53-93 41 45.28 56 33-45 7 1 x 9-53 86 4 - 1° 12 58. 69 27 53-46 42 44-59 57 32.68 72 18.54 87 3- J 4 13 58.46 28 52-97 43 43.88 58 3 i -79 73 x 7-54 88 2 14 58.22 29 52.48 44 43.16 59 30.9 74 16.54 89 1.05 15 57-95 30 51.96 45 42-43 60 30 75 15-53 90 .oo Note. — Degrees of longitude are to each other in length as Cosines of their latitudes. FIGURE OF EARTH. — BOARD AND TIMBER MEASURE. 6 1 Elements of* F’igu.re of* tlie Eartla. Capt. A. R. Clarke , 1866. Feet. Major semi-axis of Equator (longitude 15 0 34' E.) 20926 350 Minor “ “ “ ( “ 105 0 34' E.) 20919972 Polar “ “ 20853429 Equatorial semi-axis 20 926 062 Diameter, Miles. 3963.324. 3 962. 1 15. 3 949-513- 3963.269. 24 898.562. 7916. BOARD AND TIMBER MEASURE. BOARD MEASURE. In Board Measure, all boards are assumed to be 1 inch in thickness. To Compute Measure or Surface. When all Dimensions are in Feet. Rule. — Multiply length by breadth, and product will give surface in square feet. When either of Dimensions are in Inches . Rule. — Multiply as above, and divide product by 12. When all Dimensiom are in Inches . Rule. — Multiply as before, and divide product by 144. Example. — What are number of square feet in a board 15 feet in length and 16 inches in width ? 15X16 = 240, and 240 -r- 12 = 20 feet TIMBER MEASURE. To Compute "Volume of Round Timber. When all Dimensions are in Feet. Rule. — Add together squares of diameters of greater and lesser ends, and product of the two diameters ; multiply sum bv .7854, and product by one third of length. Or, a + «' + a" X ; = V, and c 2 -f c' 2 -f c x c' X .07958 X - = Y. a and , . 3 3 a representing areas of ends , a' area of mean proportional, l length and c and c circumference of ends. Note.— Mean proportional is square root of product of areas of both ends. Illustration. — Diameters of a log are 2 and 1.5 feet, and length 15 feet. 2 2 + i- 5 2 — :4 ~f~ 2 - 2 5 4“ 2 X 1.5 = 9.25, which x .7854 and ^=36.32 cube feet. 3 When Length in Feet, and A.reas or Circumferences in Inches. Rule. — Proceed as above, and divide by 144. When all Dimensions are in Inches. Rule. — Proceed as before, and divide by 1728. N ote. — Ordinary rule of Hutton, Ordnance Manuat of U. S., and Molesworth, of 4 ’ ?' VG u a result of ab °ut .25 less than exact volume, or what it would b( 11 me log was hewn or sawed to a square, c representing mean circumferences. F 62 BOARD AND TIMBER MEASURE. To Compute Yolume of Squared Tiirfber. When all Dimensions are in Feet. Rule. — Multiply product of breadth by depth, by length, and product will give volume in cube feet. When either Dimension is in Inches. Rule. — Multiply as above, and divide product by 12. When any two Dimensions are in Inches. Rule. — Multiply as before, and divide by 144. Example.— A piece of timber is 15 inches square, and 20 feet in length; required 15 X 15 X 20 its volume in cube feet. 144 - = 31.25 cube fed. Allowance Is to be made for bark, by deducting from each girth from .5 inch in logs with thin bark, to 2 inches in logs with thick bark. ALeasnres of Tircfber. — [English.) 100 superficial feet » are . of planking ) u 120 deals = 1 hundred. 50 cube feet of squared ) __ j oa( j timber J 40 feet of unhewn timber = 1 load. 600 superficial feet of inch planking = 1 load. Deals. Deals. — Boards exceeding 7 ins. in width, and if less than 6 feet in length, are termed deal ends. Battens are similar to deals, but only 7 inches in width. Balk . — Roughly squared log or trunk of a tree. Blanks are boards 12 ins. in width. JLocal Standards. Country. Long. Broad. Thick. Volume. Country. Long. ! Broad. Thick. Volume. Ft. Ins. Ins. Cub. ft. Ft. Ins. Ins. Cub. ft. Russia and Norway . . 12 9 3 2.25 Prussia . . 12 II i-5 i-375 Christiana II 9 1.25 •859 Sweden . . . 14 9 3 2.625 Quebec. . . 12 11 2-5 2.292 100 Petersburg!! standard deals equal 60 Quebec deals. SPARS AND POLES. Pine and Spruce Spars , from 10 to 4.5 inches in diameter inclusive, are to be measured by taking, their diameter, clear of bark, at one third of their length from abut or large end. Spars are usually purchased by the inch diameter ; all under 4 inches [ are termed Poles. Spars of 7 inches and less should have 5 feet in length for every inch of diameter, and those above 7 inches should have 4 feet in length * for every inch of diameter. IjOss or "Waste Oak, English . . “ African.. “ Dantzic . . “ American in Herwing or Sawing of Timber. [C. Mackrow.) 200 per cent. 100 “ “ 50 “ “ 10 “ “ Yellow Pino from planks. . 10 per cent. Teak *5 “ ‘‘ Elm, English 200 “ u American 15 ‘ CISTERNS. SHINGLES. 63 CISTERNS. Capacity of Cisterns in Cube IFeet and. GJ-allons. For each 10 Inches in Depth. Diam. [ Cub. ft. | Gallons. Diam. Cub. ft. | Gallons. | Diam. | Cub. ft. Feet. 2 2.618 19.58 Feet. 9-5 59.068 441.8 Feet. 17 189.15 2-5 4.091 30.6 10 65.449 489.6 I 7-5 200.432 3 5-89 44.07 10.5 72.158 539-78 18 212.056 3-5 8.018 59-97 11 79.194 592.4 *9 236.274 4 10.472 78.33 n-5 86.558 647-5 20 261.797 4-5 13-254 99.14 12 94.248 705 21 288.632 5 16.362 122.4 12.5 102.265 764.99 22 316.776 5-5 19.798 148.1 13 1 10.61 827.4 23 346.23 6 23.562 176.24 13-5 119.282 892.29 24 376.992 6-5 27.652 206.84 14 128.281 959-6 25 409,062 7 32.07 239.88 14-5 137.608 1029.38 26 442.44 7-5 36.816 275-4 i5 147.262 1101.6 27 47 1 -! 3 8 41.888 3I3-33 15-5 I 57- 2 43 1176.26 28 513.126 8-5 47.288 353-72 16 167.552 I2 53-37 29 550.432 9 53014 396-55 16.5 178.187 J 33 2 -93 !l 30 589.048 Gallons. 1414.94 1 499 -33 1586.28 1767,45 1958.3 2159.11 2369.64 2589.97 2820.09 3059-8 3309-67 3569-17 3838.44 4ii7-5i 4406.08 Excavation and Lining of Wells or Cisterns For each 10 Inches in Depth. Feet. 3 3 - 5 4 4 - 5 5 5 - 5 6 6 - 5 7 7- 5 3 c .2 Bricks. Masonry. J .2 Bricks. Num- Laid 8 inches 1 foot I g g Num- Laid 1 W ber. dry. thick. thick. s W ber. dry. Cub. ft. 126 Cub. ft. Cub. ft. Cub. ft. Feet. Cub. ft. Cub. ft. 12. 29 5-24 6.4 IO -47 8-5 63.29 356 14.83 I 5- 2 9 18.62 J 47 168 6. 11 6.98 7.27 8. 14 11.78 13.09 9 9-5 69. 89 76.81 377 398 15-71 16.58 22.27 26.25 188 209 7-85 8.73 9.02 9.89 14.4 I 5 - 7 I 10 10.5 84.07 9 i - 6 5 419 440 17-45 18.33 3 °. 56 230 9.6 10.76 17.02 11 99-56 461 19^2 35-2 251 10.47 11.64 i8 -33 12 116.36 503 20.94 40. 16 272 n -34 12.51 19.63 13 134.46 545 22. 69 45-45 2 93 12.22 13-38 20.94 14 153-88 586 2 4-43 51-07 3 i 4 13.09 14-25 22.25 1 5 174.61 628 26.18 57.02 335 13.96 15-13 23-56 16 196.64 670 2 7-92 1 8 inches thick. Masonry. Cub. ft 16 16.87 I 7-75 18.62 19.49 20.36 22. 11 2 3-85 25.6 27-34 29. 09 Cub. ft. 24.87 26. 18 27.49 28.8 30.11 31.4a 34-03 36-65 39-27 41.89 44-51 brick vi 7 « h v d • curD ? re . taken at dimensions of ordinary oricK \iz., 8 by 4 by 2.25 ins. = 72 cube ins. J hn lr liS 0 T? Uting ™ ml r er . of bricks required, an addition of 5 per cent should ** *- ' - excavation SHINGLES. » 1 *"» h - . <• 7 abut. ’ t0 llght ‘ 2 5 inch ^ point and .3125 inch at than'^V^rshinHpT 863 H c ° uraes of about 8 inches - so that less quired 3 p^ s ;:aoV:ffi? ed * ^ shingles are re- Shingles, alike to Slates, are laid upon boards or battens. 6 4 SLATES AND SLATING. SLATES AND SLATING. A Square of Slate or Slating is ioo superficial feet. Gauge is distance between the courses of the slates. Lap is distance which each slate overlaps the slate lengthwise next but one below it, and it varies from 2 to 4 inches. Standard is assumed to be 3 inches. # . Margin is width of course exposed or distance between tails 01 the slates. , Pitch of a slate roof should not be less than 1 in height to 4 of length. rp 0 Compute Surface of a Slate 'when laid, and Num- ber of Squares of Slating. Rule. — Subtract lap from length* of slate, and half remainder will give length of surface exposed, which, when multiplied by width of slate, will give surface required. Divide 14 400 (area of a square in inches) by surface thus obtained, and quotient will give number of slates required for a square. Example. — A slate is 24 X 12 inches, and lap is 3 inches; what will be number required for a square ? 24 _ 3 = 21, and 21-4-2 = 10.5, which X 12 = 126 inches; and 14400-4-126 = xi 4. 29 slates. Dimensions of Slates. Ins. Ins. i Ins. | American. Ins. ] Ins. Ins. [ Ins. 14 X 7 14 X 8 14 x 9 14 X IO 16 X 8 16 x 9 16 X 10 18 X 9 18 X 10 | 18 X II l8 X 12 1 20 X IO 20 X II 20 X 12 22 X II 22 X 12 22 X 13 24 X 12 24 x 13 24 X 14 24 X 16 English. Ins. Ins. Ins. Doubles 13 X 10 12X 8 Marchioness . . 22X22 u Small doubles . u 13X 7 11X 6 10X 5 12x10 13 x IO 18X 10 Ladies - 14X 8 14X 12 15 x 8 Duchess Imperial Rags 24 X 12 30 X 24 36X24 Plantations . . j Viscountess . . . Countess .... i6x 8 16 X 10 20 X 10 Queens Empress Princess 36X24 26 X 15 24XI4 Thickness of slates ranges from .125 to .3125 of an inch, and their weight varies from 2 to 4.53 lbs. per sq. foot. Weight of One Square Foot of Slating. .125 in. thick on laths 4 75 lbs. -25 in. thick on laths. . • • • • • • 9- 2 5 lbs* “ “ “ “ 1 in. boards.. 6.75 “ “ ‘‘ u 1 m boards. . 11.25 .1875 in. thick on laths 7 “ -3 I2 5 in - thmk on laths. . .... zz. 5 V 5 u u a , in . boards. 9 “ “ “ “ “ 1 in. boards, 14.10 Slate weighs from 167 to 181 lbs. per cube foot, and in consequence of laps, it requires an average of nearly 2.5 square feet of slate to make one 01 slating. Weights per 1000 and Number Required to Cover a Square. I Lb*. | No. I] I Lbs< Doubles 13 X 6 1680 480 j| Countess. . .20X10 6720 Ladies 15 X 8 I 2800 I 240 1 1 Duchess . . . 24 X 12 1 44 «° * Length of a slate is taken from nail-hole to tail. SHOT AND SHELLS. FRAUDULENT BALANCES. 65 PILING OF SHOT AND SHELLS. To Compute NTxiinUer of Sliot. . Triangular Pile. Rule.— Multiply continually together, number of shot m one side of bottom course, and that number increased by 1 and a«-ain bv 2 , and one sixth of product will give number. 0 ing 30 shotr What iS nUmbGr ° f Sh0t in a trian £ ular P il€ V each side of base contain- To Detect Tliem .— After an equilibrium has been established between weight and article weighed, transpose them, and weight will preponder- ate if article weighed is lighter than weight, and contrariwise if it is heavier. To Ascertain True Weight. Rule. — Ascertain weight which will produce equilibrium after article to be weighed and weight have been transposed • reduce these weights to same denomination, multiply them together and square root of their product will give true weight. 24 lbs. 8 oz. =24.5 lbs. Then 32 X 24.5 — 784, and ^784 = 28 lbs. Or, when a represents longest arm , I A greatest weight, and 0 lt shortest arm, | B least weight. or W^L^B^ndV- VA b" 6 *’ multiplying these two e ^ one sixth of product be?ng A f6 > and+ qUired number of shells in an oblon « Pile, numbers in base course being 16 and 7? 16X3-7-1X7X7 + 1 _ 2352 — 392 shells. 6 ' y^yMuiuwa, FRAUDULENT BALANCES. Hence, a _ 10, 6 = 8. 75, or 28 2 = 32 x 24. 5, and V32 X 24.5 = 28. Then 32 : 28 ;; 10 : 8.75. 3 66 WEIGHING WITHOUT SCALES. PAINTING. NV'eiglxing witliovit Scales. To Ascertain Weiglit of a Bar, Beam, etc., by Aid of* a, known. "Weiglit. Operation.— Balance bar, etc., over a fulcrum, and note distance between it and end of its longest arm. Suspend a known weight from longest arm, and move bar, etc., upon fulcrum, so that bar with attached weight will be in equilibrio : subtract distance between the two positions of fulcrum trom longest arm first obtained ; multiply this remainder by weight suspended, divide product by distance between f ulcrums, and quotient will give weight. Example. — A piece of tapered timber 24 feet in length is balanced over a fulcrum when n feet from less end; but when the body of a man weighing 210 * lbs - ' s sus- pended from extreme of longest arm, the piece and weight are balanced when ful- crum is 12 feet from this end. What is weight of the timber? I3 _ I2 _ Ij and 13 — 1 = 12 feet Then 12 X 210-^1 = 2520 lbs. PAINTING. 1 pound of paint will cover about 4 square yards for a first coat and about 6 yards for each additional coat. Colors. White Lead. Red Lead. Red Ochre. >• u I Spanish 1 Brown. Colors. White Lead. Lamp- black. | Red 1 Lead. White 100 100 - - - Lead Red 98 2 50 Green 25 1 - - 75 - Chocolate. . — 4 > fe 50 96 These are the colors alone, to whicn Doueci nnseeu on, nmuge, .japan , and spirits turpentine are to be added according to the application of the paint. Lamp-black and litharge are ground separately with oil, then stirred into the ^Thus for black paint: Lamp-black 25 parts, litharge 1, Japan varnish 1, boiled lin- seed oil 72, and spirits turpentine 1. Tar Baint. — Coal tar 9 gallons, slaked lime 13 lbs., turpentine or naphtha 2 or 3 quarts. Superficial A Gallon of Paint will cover feet. On stone or brick, about . On composite, etc., from . On wood, from . ■go to 225 300 “ 375 375 “ 525 A Gallon of Paint will cover On well-painted surface or iron One gallon tar, first coat “ “ “ second coat .. . Superficial feet. 600 90 160 Boiled. Oil Raw linseed oil 91 parts, copperas 3, and litharge 6. Put litharge and copperas in a cloth bag and suspend in middle of a kettle Boil oil four hours and a half over a slow fire, then let it stand and deposit the sediment. Wliite Baint. Inside work. Outside work. I . , Inside work. Outside work. White lead in oil . . 80 80 Raw oil 9 Boiled oa .’. . . . . 4.5 9 I Spirits torpentme 8 4 New wood- work requires i lb. to square yard for three coats. Coats for ioo Square Yards New White Pine. Inside. White lead. Raw oil. - Turpen- tine. Drier. Lbs. Pts. Pts. Lbs. Priming 16 — 6 •25 2d coat 15 3-5 I *5 •25 3 d “ 1 13 2-5 1-5 •25 Outside. Priming — 2d and 3d \ coats j | White lead. Lbs. 18.5 Raw oil. Boiled oil. Turpen- tine. lb. of drier with priming and coating for outs.de. HYDROMETERS. 67 HYDROMETERS. IT. S. Hydrometer (Tralle’s) ranges from o (water) to 100 (pure spirit) ; it has not any subdivision or standard termed “Proof,” but 50, upon stem of instrument, at a temperature of 6o°, is basis upon which com- putations of duties are made. In connection with this instrument, a Table of Corrections, for differences in tem- perature of spirits, becomes necessary; and one is furnished by the Treasury De- partment, from which all computations of value of a spirit are made. Illustration. — A cask contains 100 gallons of whiskey at 70 0 , and hydrometer sinks in the spirit to 25 upon its stem. Then, by table, under 70 0 , and opposite to 25, is 22.99, showing that there are 22 qq gallons of pure spirit in the 100. Commercial Hydrometer (Gendar’s) has a “Proof” at 6o°, which is equal to 50 upon U. S. Instrument and its gradations, run up to 100 with it, and down to 10 below proof, at o upon U. S. Instrument ; or o of the Commercial Instrument is at 50 upon U. S. Instrument,’ from which it progresses numerically each way, each of its divisions being equal to two of latter. In testing spirits, Commercial standard of value is fixed at proof; hence any difference, whether higher or lower, is added or subtracted’ as case may be, to or from value assigned to proof. . A scale 0 *' Corrections for temperature being necessary, one is fur- nished with a Thermometer. Application of Thermometer.— Elevation of the mercury indicates correction to be added or subtracted, to or from indication upon stem of hydrometer. When elevation is above 6o°, subtract correction ; and when below, add it. Illustration.— A hydrometer in a spirit indicates upon its stem 50 below proof and thermometer indicates 4 above 6o° in appropriate column. Then 50 — 4 = 46 = strength below proof To Compute Strength, of a Spirit, or Volume of its Pure Spirit, by Commercial Hydrometer, and. Convert it to Indication of a TJ. S. Hydrometer. TFTien. Spirit is above Proof. Rule. — A dd 100 to indication, and divide sum by 2. re mai nder^byl ** below Proo f Rule. — Subtract indication from 100, and divide pofTon of^iure^pWt'does'it contain f°°^ ^ * CommerciaI Hydrometer; what pro- 11 -f- 100 — 2 = 55.5 per cent. To Compute Strength, etc., by a TJ. S. Hydrometer. When Spirit is above Proof Rule. -Multiply indication by 2, and subtract 100. U bd0W Pr °°f' RCLE ' ~ MultipI ^ indication by 2, and subtract it Example.— A spirit is 55.5; what is its per centage above proof? 55- 5 X 2 — 100 == 1 r per cent Commercial practice of reducing indications of a hydrometer is as follows: nu P 1 V. er .P^ gMlons of spirit by per centage or number of degrees above 2 X7e. and qU ° tient Wi “ *"> “»“** of ganonlThl £22 Illustration.— 50 gallons of whiskey are n per cent, above proof. Then 50 X iz zoo = 5. 5, which added to 50 = 55. 5 gallons. 68 HYGROMETER. HYGROMETER. D e w-p oint .—When air is gradually lowered in its temperature at a constant pressure, its density increases, and ratio of increase is sensibly same for the vapor as for the air with which it is combined, until a point is reached at which the density of the vapor becomes equal to the maximum densitv corresponding to the temperature, ^ „ , This temperature is termed dew-point of given mass, and any further re- duction of it will induce the condensation of a portion of the vapor m form of dew, rain, snow, or frost, according as temperature of surface is above or below freezing point. Mason’s or like Hygrometer. To Ascertain Dew-point. Rule. — Subtract absolute dryness from temperature of air, and remainder is dew-point. Example.— Temperature of air 57 0 , and absolute dryness 7 0 . Hence 57 0 — 7 0 = 50 0 dew-point. To Ascertain Absolute Existing Dryness. Rule —Subtract temperature of wet bulb from temperature of air, as indicated by a dry bulb, add excess of dryness from following table, multiply sum by 2, and product will give absolute dryness in degrees. Example.— Temperature of air 57 °, wet bulb 54°- Then 57 0 — 54 0 = 3 0 , and 3 0 -f • 5 ° (from table) X 2 = 7 0 absolute dryness. Observed jExcess of j Dryness. | Dryness, j Observed Dryness. Excess of Dryness. Observed] Excess of! Dryness, j Dryness, j • 5 .083 5 •833 9-5 1-583 I . 166 5-5 .9165 10 1.666 1-5 .2495 6 1 10.5 1-7495 2 •333 6-5 1.083 11 1-833 2-5 * •4 i6 5 7 I. 166 n-5 1.9165 3 •5 7-5 1.2495 12 2 3.5 . 583 8 i -333 12.5 2.083 4 .666 8-5 1.4165 13 2. 166 4-5 •7495 1 9 i -5 I 3-5 2.2495 Observed Dryness. Excess of 1 Dryness. Observed Dryness. Excess of Dryness. 14 8-333 18.5 , 3- o8 3 14-5 2.4165 19 3.166 15 2-5 19-5 3-2495 15-5 2.583 20 3-333 16 2.666 20.5 3-4i65 16,5 2-7495 21 3-5 17 2.833 21-5 3-583 17-5 2.9165 22 3.666 18 3 22.5 3-7495 To Compute Volume of Vapor in Atmosphere. By a Hygrometer. When temperature of atmosphere in shade , and of dew-point are given. If temper- ature of air mid dew-point correspond, which is the case when are alike and air consequently saturated with moisture, then m table opposite to temperature will be found corresponding weight of a cube foot of vapor in grams. Illustration. -Assume temperature of air and dew-point 70° Then opposite tomnerature weight of a cube foot of vapor = 8. 392 grains. But if temperature of air is different from dew-point, a correction is necessary to obtain exact w T eight. Illustration.— Assume dew-point 7 o° as before, but temperature .o^ 1 ^ s ^ ade 80° then the vapor has suffered an expansion due to an excess of 10 , which re ^InLble^rcm^ections for 10° is 1.0208. Then divide 8.392 grains at dew-point- viz., 70 0 by correction corresponding to degrees of absolute dryness— viz., 10 . 8,392 =8.221 grains of existing vapor , which, subtracted from weight of vapor col-responding to temperature of 80°, will give number of grains required for satu- ration at that temperature. . n.333 grains at temperature of 8o°— 8.221 contained in the air = 3.112 required for saturation. - . * For tabic, see Mason’s as published by Pike & Sons, New York, and compared with Sir John Leslie’s and l’rofcssor Daniel ? s. HYGROMETER. SUN-DIAL. — CHAINING. 69 To ascertain relations of these conditions on natural scale of humidity (complete saturation being 1000), divide weight of vapor at dew-point by weight at tempera- ture of air, and quotient will give degrees of saturation. Illustration.— Dew-point = 70 0 , weights 8.392. Then 8.392-^-11.333 (at 8o°) = .7405 degrees of humidity; saturation ~ 1000. To Compute Weight of Vapor in a Cube Foot of Air. See Pressures, Temperatures, Volumes, and Density of Steam, p. 708. Thus, Required weight of vapor in a cube foot of saturated air at 21 2 0 . At a temperature of 212 0 density or weight of 1 cube foot of air = .038 lb. If density is required for any temperatures not in table, see rule, p. 706. Humidity .— Condition of air in respect to its moisture involves amount of vapor present in air and ratio of it to amount which would saturate it at its temperature, and it is this element which is denoted by term humidity, and it is expressed as a per centage; thus, if weight of vapor present is .7 of that required for saturation, the humidity is 70. Dry Air is air, humidity of which is below zero, but it is customary to term it dry when its humidity is below the average proportion. Note. —Air in a highly heated space contains as much vapor (when weight of it is equal) as a like volume of external air, but it is drier as its capacity for vapor is greater. SUN - DIAL. To Set a Su.11-d.ial. Set column on which dial is to be placed perpendicular to horizon. Ascertain by spirit level that upper surface is perfectly horizontal ; screw on plate loosely by means of centre screw, and bring gnomon as nearly as practicable to its proper direction wn a bright day set dial at 9 a.m. and 3 f.m. exactly, with a correctly regulated watch; observe difference between them, and correct dial to half difference. Pro- ceed in same manner till watch and dial are found to agree perfectly. Then fix plate firmly in that situation, and dial will be correctly set. This is obvious; for, if there were any defects, the Sun’s shadow would not agree w;th time indicated by watch, both before and after he passed meridian. Take care, however, to allow for equation of time, or you may set dial wrong. Best day in the year to set a dial is 15th of June, as there is no equation to allow for and no error can arise from change of declination. A dial may be set without a watch by drawing a circle around centre, and marking spot where top of shadow of an upright pm or piece of wire, placed in centre, just touches circle in a.m., and again in p m A une should be drawn from one spot to the other, and bisected exactly; then a line drawn from centre of dial through that bisection will be a true meridian line on which the XII hours’ mark should be set. ’ CHAINING OVER AN ELEVATION. I C = L, and C = cos. angle. I representing length of line chained , C cos. angle of elevation with horizon, and L Length of line reduced to horizontal. horizSXanTjf “ Sth ° f “ eIeVati ° n at ““ angle of 3 °° V is 100 f <*‘; "hat is By Table of Cosines, 30° 17' = .863 54. Hence, 100 X .863 54 = 86. 354 feet. To set out a Right Angle with a Chain, Tape-line, etc. and^nfor Chain 0r fe fu 0f Iine for base ’ 30 links or feet for perpendicular, and 50 for hypothenuse, or in this ratio for any length or distance. Useful Numbers in Surveying. For Converting Multiplier. Converse. For Converting Multiplier. Converse. Feet into links.. Yards “ “ I-5I5 4-545 .66 .22 Square feet into acres. . Square yards “ “ .. .000022 9 . 000 206 6 43 56o 4 840 3 70 CHRONOLOGY. CHRONOLOGY. Solar day is measured by rotation of the Earth upon its axis with respect to the Sun. _ . . „ , , Motion of the Earth, on account of ellipticity of its orbit, and o± perturba- tions produced by the planets, is subject to an acceleration and retardation. To correct this fluctuation, timepieces are adjusted to an average or mean solar day ( mean time ), which is divided into hours, minutes, and seconds. In Civil computations day commences at midnight, or A.M., and is divided into two portions of 12 hours each. In Astronomical computations and in Nautical time day commences at M., or 12 hours later than the civil day, and it is counted throughout the 24 hours. Solar Year, termed also Equinoctial , Tropical , Civil , or Calendar Y ear, is the time in which the Sun returns from one Vernal Equinox to another; and its average time, termed a Mean Solar Year , is 365.242218 solar days, or 365 days , 5 hours , 48 minutes , and 47. 6 seconds. Year is divided into 12 Calendar months, varying from 28 to 31 days. Mean Lunar Month , or lunation of the Moon, is 29 days, 12 hours, 44 minutes, 2 seconds, and 5.24 thirds.* Bissextile or Leap Year consists of 366 days; correction of one year in four is termed Julian ; hence a mean Julian year is 365.25 days. ^ In vear 1*82 error of Julian computation of a year had amounted to a period ot 10 days, which, by order of Pope Gregory VIII., was suppressed in the Calendar, and 5th of October reckoned as 15th. Error of Julian computation, .00776 days, is about 1 day in i a8.7oyeMS, and adop- tion of this period as a basis of intercalation is termed Gregorian Calendar j or JSew Style, t Julian Calendar being termed Old Style. Error of Gregorian year (365.2425 days) amounts to 1 day in 3571.4286 years. New Style was adopted in England in 1752 by reckoning 3d of September as 14th. Bv an English law, the years 1900, 2100, 2200, etc., and any other 100th year, ex- cepting only 8 every 400th year, commencing at 2000, are not to be reckoned bissex- tile years. Dominical or Sunday Letter is one of the first seven letters of and is used for purpose of determining day of week corresponding to an> g^ven date In Ecclesiastical Calendar letter A is placed opposite to 1st day ^f^no’ B to second- and so on through the seven letters; then the lettei which falls oppo- site to first Sunday in year will also fall opposite to every following Sunday in tha ^ Note.— In bissextile years two Dominical letters are used, one before and the other after the intercalary day. In Ecclesiastical Year the intercalary day is reckoned upon 24th February; hence 24th and 25th days are denoted by same letter, the dominical letter bein e se back one place. In Civil Year the intercalary day is added at end of February, the change of letter taking place at 1st of March. Dominical Cycle is a period of 400 years, when the same order of dominical letters and days of the week will return. Cycle of the Sun, or Sunday Cycle , is the 28 years before same onjei ot letters return to same days of month, and it is considered as ha\mg commenced 9 years before the era of Julian Calendar. To Compute Cycle of tlie Sxxn. Rule.— Add 9 to given year; divide sum by 28; quotient is number of cycles that have elapsed, and remainder is number or years of cycle. Note —Use of this computation is determination of dominical letter for any given year of Julian Calendar for each of the 28 years of a cycle. * Ferguson. t Now adopted in every Christian country except Russia and Greece. CHRONOLOGY. 7 I By adoption of Gregorian Calendar , order of the letters is necessarily interrupted by suppression of the century bissextile years in 1900, 2100, etc., and a table of Do- minical letters must necessarily be reconstructed for following century. Lunar Cycle , or Golden Number , is a period of 19 years, after which the new moons fall on same days of the month of Julian year, within 1.5 hours. Year of birth of Jesus Christ is reckoned first of the Lunar Cycle. To Compute Lunar Cycle, or Grolden Number. Rule.— Add 1 to given year; divide sum' by 19, and remainder is Golden Number. Note. — I f o remain, it is 19. Example.— What is Golden Number for 1879 ? i8 79 + 1 -L 19 = 9 8 ? aud remainder = 18 = Golden Number. Epact for any year is a number designed to represent age of the moon on 1st dav of January of that year. See table, p. 73. * To Compute tlie Roman Indiction. Rule.— Add 3 to given year; divide sum by 15, and remainder is Indiction. Note. — I f o remain, Indiction is 15. March” 667 * °f Direction is the nlimber of da y s that Easter-day occurs after 21st of Easter-day is first Sunday after first full moon which occurs upon or next after 21st of March; and if full moon occurs upon a Sunday, then Easter-day is Sundav after and it is ascertained by adding number of direction to 21st of March It is therefore March N 21, or April N — 10. Illustration. — If Number of Direction is 19, then for March, i 0 4 _ 2I =4D and 40 — 31=9 = 9 th of April; ; y 1 again for April, 19 — 1 0 = 9 — 9 th of April. Note. Moon upon which Easter immediately depends is termed Paschal Moon. Full Moon is 14th day of moon, that is, 13 days after preceding day of new moon. Days of* tlie Roman Calendar. Calends were the first 6 days of a month, Nones following 9 days, and Ides remain ing days. ’ In March, May, July, and October, Ides fell upon 15th and Nones began upon 7 th In other months Ides commenced upon 13th and Nones upon 5th. For Roman Indiction and Julian Period see p. 26. Chronology. Creation of World (according to Julius Africanus, Sept. 1, 5508- Samaritan Pentateuch, 4700; Septuagint, 5872 ; Josephus, 4658 ; Talmudists Sea liger, 395o; Petavius, 3984; Hales, 5411). d 576. Money coined at Rome. 562. First Comedy performed at Athens. 480. First recorded Map by Aristagoras 420. First Theatre built at Athens. 336. Calippus calculates the revolution of Eclipses. 320. Aristotle writes first work on Me- chanics. 310. Aqueducts and Baths introduced in Rome. 306. First Light house in Alexandria. 289. First Sun-dial. 267. Ptolemy constructs a Canal from the Nile to the Red Sea. 224. Archimedes demonstrates the Prop- erties of Mechanical Powers and the Art of measuring Surfaces, Sol- ids, and Sections. 219. Hannibal crossed the Alps. 219. Surveying first introduced. 202. Printing introduced in China. B. C. 4004. 234 8 2247. 2203. 2090. 1920. 1891. 1822. 1490. 1240. 1180. 1120 . 753 - 640. 605. Deluge (according to Hales, 3154). Bricks made and Cement first used. Tower of Babel finished. Chinese Monarchy. First Egyptian Pyramid and Canal. Gold and Silver Money first intro- duced. Letters first used in Egypt. Memnon invents the Egyptian Al- phabet. Crockery introduced. Axe, Wedge, Wimble, Lever, Masts and Sails invented by Daedalus of Athens. Troy destroyed. Mariner's Compass discovered in China. Foundation of Rome. Thales asserts Earth to be spherical. Geometrjr, Maps, etc., first intro- duced. 7 2 CHRONOLOGY. B.C. 198. Books with leaves of vellum first introduced by Attalus. 170. Paper invented in China. 168. An eclipse of the Moon which was predicted by Q. S. Gallus. 162. Hipparchus locates the first degree of Longitude and the Latitude at Ferro. A.D. 69. Destruction of Jerusalem. 79. Destruction of Herculaneum and Pompeii. 214. Grist-mills introduced. 622. Year of Hegira, commencing 16th July ; Glazed windows first intro- duced into England in thiscent’y. 667. Glass discovered. 670. Stone buildings introduced into Eng- land. 842. Lands first enclosed in England. 933. Printing said to have been invented 1 by the Chinese. 991. Arabic Numerals introduced. 1066. Battle of Hastings. mi. Mariner’s Compass discovered. 1180. Destruction of Troy. Mariner’s Compass introduced in Europe. 1368. Chimneys first introduced into Rome from Padua. 1383. Cannon introduced. 1390. Woollens first made. 1434. Printing invented at Mayence. 1460. Wood-engraving invented and First Almanac. 1471. Printing in England by Caxton. 1477. Watches first introduced at Nurem- berg. 1492. America discovered. 1497. Vasco de Gama discovers passage to India. 1500. Variation of Mariner’s Compass ob- served. 1522. F. de Magellan circumnavigates the Globe. 1530. Incas conquered by Pizarro. 1545. Needles first introduced. 1586. Potato introduced into Ireland from America, 1590. Telescopes invented by Jansen and used in London in 1608. 1616. Tobacco first introduced into Vir- ginia. 1620. Thermometer invented by Drebel. 1627. Barometer invented. 1629. First Printing-press in America. 1639. First Printing-office in America at Cambridge. 1647. Otto Van Gueriche constructed first electric machine. 1650. Railroads with wooden rails intro- duced near Newcastle. 1652. First Newspaper Advertisement. 1704. First Newspaper in America. 1705. Blankets first made at Bristol, Eng- land. 159. Clepsydra, or Water- clock, invent- ed. 146. Carthage destroyed. 70. First Water-mill described. 51. Caesar invaded Britain. 45. First Julian Year by Caesar. 8. Augustus corrects the Calendar. A.V. 1752. Benjamin Franklin demonstrated identity of the electric spark and lightning, by aid of a kite. 1752. New Style, introduced into Britain; Sept. *3 reckoned Sept. 14. 1753 - First Steam-engine in America. 1769. James Watt— First design and pat- ent of a Steam-engine with sepa- rate vessel of condensation. 1772. Oliver Evans— Designed the Non- condensing Engine. 1792. Ap- plied for a patent for it. 1801. Constructed and operated it. 1774- Spinning-jenny invented by Robert Arkwright, 1776. Iron Railway at Sheffield, England. 1783. First Balloon ascension, and Vessel’s bottoms coppered. 1700 Water-lines first introduced in mod- els of Vessels in the U. S. 1797. John Fitch— Propelled a yawi-boat by application of Steam to side- wheels, and also to a screw'-propel- - ler, upon Collect Pond, New York. 1807. Robert Fulton — First Passenger Steamboat. 1824. Compound marine steam-engines : first introduced by James P. Al- t lan, New York. 1825. Introduction of steam towing by Mow r att, Bros. & Co., of New York, by steam boat “ Henry Eckford,”, New York to Albany.* 1826. Voltaic Battery discovered by Alex. > Volta, and First Horse-railroad. 1827. First Railroad in U. S., from Quincy to Neponset. 1829. First Lucifer Match and first Loco- motive in America. 1830. Liverpool and Manchester Railroad. opened. First Steel Pen and first Iron Steamer. 1832. S. F. B. Morse invents the Magnetic Telegraph. { 1836. Robert L. Stevens first burned An ' thracite Coal in furnace of boiler, of steamboat “Passaic.” 1840. First steam-boiler constructed for burning Anthracite Coal in steam- boat “North America,” N. Y. 1844. Telegraph line from Washington to Baltimore, Md. 1846. First complete Sewing-machine. Elias How r e, inventor. 1866. Submarine Telegraph laid from Valencia to Newfoundland, N.S. * Witnessed by author. CHRONOLOGY. 73 Dates oT Day- of Week, corresponding to Day deter- mined "by following Table. February, March, November. February,* August. May. January, October. January,* April, July. September, December. June. z 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3 i • Thus, if Monday is the day determined by the year given, the following dates are the Mondays in that year. Epacts, Dominical Letters, and an Almanac, from 1SOO to 1901. Use of Table. — To ascertain day of the week on which any given day of the month falls in any year from 1800 to 1901. Illustration. — The great fire occurred in New York on 16th of December, 1835; what was day of the week ? Opposite 1835 is Sunday; and by following table, under December, it is ascertained that 13th was Sunday; consequently , 16th was Wednesday. Years. Days. Dom Let- ters. l w Years Days. Dom. Let- ters. cl • W Years. Days. Dom, Let- ters. 4 e* 1 w 1800 Saturday. E .4 1834 Saturday. E 20 1868 Sunday.* ED 6 1801 Sunday. D 15 1835 Sunday. D I 1869 Monday. C 17 1802 Monday. C 26 1836 Tuesday.* CB 12 1870 Tuesday. B 28 1803 Tuesday. B 7 1837 Wednesd. A 23 1871 Wednesd. A 9 1804 Thursday.* AG 18 1838 Thursday. G 4 1872 Friday.* GF 20 1805 Friday. F 29 1839 Friday. F 15 1873 Saturday. E 1 1806 Saturday. E 11 1840 Sunday.* ED 26 1874 Sunday. D 12 1807 Sunday. D 22 1841 Monday. C 7 1875 Monday. C 23 CO 00 Tuesday. * CB 3 1842 Tuesday. B 18 1876 Wednesd.* BA 4 1809 Wednesd. A 14 i8 43 Wednesd. A 29 1877 Thursday. G 15 1810 ; Thursday. G 25 i8 44 Friday. * GF 11 1878 Friday. F 26 1811 { Friday. * F 6 1845 Saturday. E 22 1879 Saturday. E 7 1812 Sunday.* ED *7 1846 Sunday. D 3 1880 Monday.* DC 18 1813 Monday. C 28 1847 Monday. C 14 1881 Tuesday. B 29 1814 i Tuesday. B 9 1848 Wednesd.* BA 25 1882 Wednesd. A 11 1815 i Wednesd. A 20 1849 Thursday. G 6 1883 Thursday. G 22 1816 Friday.* GF 1 1850 Friday. F 17 1884 Saturday.* FE 3 1817 Saturday. E 12 1851 Saturday. E 28 1885 Sunday. D 14 1818 Sunday.' D 23 1852 Monday.* DC 9 1886 Monday. G 25 1819 Monday. C 4 i8 53 Tuesday. B 20 1887 Tuesday. B 6 1820 Wednesd.* BA *5 i8 54 Wednesd. A 1 1888 Thursday.* AG 17 1821 Thursday. G 26 i8 55 Thursday. G 12 1889 Friday. F 28 1822 Friday. F 7 1856 Saturday.* FE 23 1890 Saturday. E 9 1823 Saturday. E 18 i8 57 Sunday. D 4 1891 Sunday. D 20 1824 Monday.* DC 29 j 1858 Monday. C 15 1892 Tuesday. * CB j 1825 Tuesday. B 11 1859 Tuesday. B 26 1893 Wednesd. A 12 1826 Wednesd. A 22 I i860 Thursday.* AG 7 1894 Thursday. G 23 1827 Thursday. G 3 1 1861 Friday. F 18 1895 Friday. F 4 1828 Saturday.* FE H 1 1862 Saturday. E 29 1896 Sunday.* ED 15 1829 Sunday. D 25 j 1863 Sunday. D 11 1897 Monday. c 26 1830 Monday. C 6 1864 Tuesday. * CB 22 1898 Tuesday. B 7 1831 Tuesday. B 17 1865 Wednesd. A 3 1899 Wednesd. A 18 l8 32 j Thursday.* AG 28 1866 Thursday. G 14 1900 Thursday. 29 i8 33 1 Friday. F 9 I 1867 Friday. | F 1 25 1901 j Friday. F j 11 In leap-year, January and February must be taken in columns marked *. CHRONOLOGY.— '-MOON’S AGE. — TIDES. 74 To ^Ascertain Year or Years of Coincidences of a given Day of the Week with, a given Day of a Month. Look in preceding table and ascertain day of week opposite to year of occurrence, and every year in which same day is given will be year of coin- cidences required. Illustration.— If a child was born on Saturday, 19th Sept. 1829, when could and can his birthdays be celebrated, that occurred or are to occur on same day of week and date of month ? Opposite to 1829 is Sunday, and in preceding table the Sundays for September of that vear were 6th, 13th, 20th", hence, if 20th was Sunday, the 19th was Saturday. Hence, every year in table opposite to which is Sunday are the years of the coin- cidence required, as 1835, 1840, 1846, 1857, 1863, 1868, 1874, 1885, etc. moon’s age. To Compute Moon’s -A_ge. Rule. — To day of month add Epact and Number of month; from sum subtract 29 days, 12 hours, 44 min. and 2 sec., as often as sum exceeds this period, and result will give Moon’s age approximately at 6 o’clock a.m. in United States, east of Mississippi River. Numbers of the Months. d. h. January 5 February 1 22 March 9 d. h. April 1 21 May 2 8 June.. 3 19 d. h. July 4 7 August 5 18 September ... 7 5 d. h. October 7 16 November 9 4 December .... 9 15 Example.— Required age of Moon on 25th February, 1877? Given day 25 + epact 15 -{-number of month 1.22 = 41 d. 22 li. —29 d. 12 h. 44 m. 2 sec. = 12 d. 9 h. 15 min. 58 sec. In Leap-years add 1 day to result after 28th February. To Compute Age of Nfoon at Mean .Noon, at any other r Location than that Griven. Rule. — Ascertain age, and add or subtract difference of longitude or time, according as place may be West or East of it, to or from time given. Or , when time of new Moon is ascertained for a location, and it is required to ascertain it for any other, add difference of longitude or time of the place, if East, and subtract it if it is West of it. Moon’s Southing, as usually given in United States Almanacs, both Civil and Nau- tical, is computed for Washington. To Compute Tiine of High-water by jAid of A.xnerican N autical Almanac. Rule. — Ascertain time of transit of Moon for Greenwich, preceding time of the high-water required. ( For any other location (west of Greenwich), multiply the time in column “ diff. for one hour” by longitude of location west of Greenwich, expressed in hours, and add product to time of transit. Note.— I t is frequently Necessary to take the transit for preceding astronomical day, as the latter does not end until noon of day under computation. Example. — Required time of liigh-water at New York on 25th of August, 1864. Longitude of New York from Greenwich = 4 Ji. 56 m. 1.65 sec., which, multiplied by 2. 17 min . , the difference for 1 hour = 10.71 min. for correction to be added to time of transit, to obtain time of transit at New York. TIDES. MOON’S SOUTHING. Time of transit, 18 h. 38.8 m . ; then 18 h. 38.8 m. -f 10.71 m. = Time of transit at New York, 24 d. 18 h. 50 m. Establishment of the Port, 8 13 7 5 18 hours 49.51 min. Note. - 25 d. 3 h. 3 m. = time of high-water. p.H. Civil mine 0 ° f 25th at 3 A ' 3 Astron <>mical computation = 25 th at 3 h. 3 m. To Compute Time of High-water at Full and Change of Moon. Time of High-water and Age of Moon on any Day being given. Rule. Note age of Moon, and opposite to it, in last column of following tabie, take time, which subtract from time of high-water at this ara of Moon added to 12 h. 26 m ., or 24 h. 52 m., as case may require (when sum to be subtracted is greatest), and remainder is time required. Example.— What is time of high-water at full and change of Moon at New York? Time of high-water at Governor’s Island on 25th of Jan. 1864, was q h 20 m a m civil time. Age of Moon at 12 m. on that day was 16 d. 8 h. 59 m. 9 nn 2 p P°f e ‘° 16 in following table, is 13 h. 28 m., and difference between 16 d Th 12 ^ ( - 1 r I ? or I3 - 5 * 3 ~ l3 * 28) is 25 ^ hence ? ^ « h. = 25 m. , 16 a. 8h.j 9 m. — 16 d. _ 8 h. 59 m. = 18.71 or 19 w., which, added to 13 h. 28 w. = 13 h. ™ en 9 12 \ 26 (aS Su ?> t0 be subtrac ted is greater than time) _ 13 h 47 m. = 21 n. 46 w. — 13 h. 47 m. = 7 7i. 59 m. 7 J This is a difference of but 13 minutes from Establishment of Port. Time after apparent Noon before Moon next passes Meridian, Age at Noon being given. Days 1-5 Moon at Meridian. Age of Moon. Moon at Meridian. Age of Moon. Moon at Meridian. H. M. P. M. Days. H. M. P. M. Days. H. M. P. M. 0 6 5 3 12 10 6 25 6-5 5 28 12.5 10 31 50 7 5 53 13 10 56 1 16 7-5 6 19 J 3-5 11 21 I 41 8 6 44 H 11 47 A. M. 2 6 8.5 7 9 * 4-5 12 12 2 31 9 7 34 15 12 37 2 57 9-5 7 59 i 5-5 13 2 3 22 10 8 25 16 13 28 3 47 10.5 8 50 16.5 *3 53 4 12 11 9 i 5 17 14 28 4 38 1 II *5 9 40 J 1 W -5 H 43 1 Days. 18 18.5 J 9 i9-5 20.5 21 21.5 22 22.5 23 23-5 15 8 *5 34 *5 59 16 24 16 49 W i5 17 40 5 18 30 18 56 19 21 19 46 Age of Moon. Moon at Meridian. Days. H. M. A. M. 24 20 II 24-5 20 37 25 21 2 25-5 21 27 26 21 52 26.5 22 17 27 22 43 27-5 23 8 28 23 33 28.5 23 58 29 24 24 29-5 I 24 48 Tidal IPlienomena. tJK 6 elevati °n of a tidal wave towards the Moon slightlv exceeds tint n f the P The Sun & 1 f. tens . ,t y of diminishes from Equator to the Poles the 1 actiTo^th^Cn^urwith 1 TTSE" «“ sea lowing on boZiwTlrnEmh C ° mbinCd aCti ° n ° f the Sun aDd Moon wh ™ they are •J&S StST.g’ iMKasiS Sr s* » «» ;6 LATITUDE AND LONGITUDE, LATITUDE AND LONGITUDE. Latitude and Longitude of Principal Locations and Observatories. Compiled from Records ofU. S. Coast and Geodetic Survey and Topograph- ical Engineer Corps , Imperial Gazetteer , and Bowditclis Navigator. Longitude computed from Meridian of Greenwich. A represents Academy; Az., Azimuth ; A. S., Astronomical Station; C., College; Cap , Ctopitol; Ch. , Church ; C. YL , City Hall ; C. S., Coast Survey ; Cl Court-house; Cy .'Chimney; F.S., Flagstaff; G.S Geodetic Station ; Hos. . Hospital ; L. Light- house ; Obs ..Observatory; S.H., State-house; Sp. Spire ; Sq., Square; Station • T Telearanh ; T. H. , Toten JTaM ; U. , University; L n. , I mow ; B. , Baptist , Co Episcopal ; P.', rSesby. ; and M.Ch, JfcO. U/mrc^S. Location. Latitude. 1 NORTH AND SOUTH N. AMERICA. 0 t n Acapulco Mex. 16 50 19 Albany, P.Ch N.Y. 42 39 3 Ann Arbor Mich. 42 16 48 Annapolis . Md. 38 58 42 Apalachicola, F.S. Fla. 29 43 30 Astoria, F.S 46 11 19 Atlanta, C. H ..Ga. 33 44 57 Auburn N.Y. 42 5.5 Augusta ..Ga. 33 28 Augusta, B.Ch.. . . Me. 44 18 52 Austin .Tex. 30 16 21 Balize ..La. 29 8 5 Baltimore, Mon’t . Md. 39 i7 48 Bangor, Tho’s Hill. Me. 44 48 23 Barbadoes, S.Pt. . W.I. 13 3 Barnegat, L .N. J. 39 46 Bath, W.S.Ch... ..Me. 43 54 55 Baton Rouge 30 26 Beaufort, Ct .N.C. 34 43 5 Beaufort, E.Ch. . . S.C. 32 26 2 Belfast, M.Ch... . . Me. 44 25 29 Benicia, Ch . Cal. 38 3 5 Benington ...Vt, 42 40 Bismarck, S. S . . .Neb. 46 48 Boston, L Mass. 42 19 6 Boston, S.H “ 42 21 30 Brazos Santiago. .Tex. 26 6 Bridgeport Conn. 41 10 3° Bristol .R. I. 41 40 11 Brooklyn, C. H. . .N.Y. 40 41 31 Brownsville, S.S. ..Tex. 26 Brunswick, C.Sp. .Me. Buffalo, L N.Y. Burlington N. J. Burlington, C Vt. B u rl i n gton, P ub. Sq., Ia. Bushncll Neb. Cairo Til. Calais, C.S. Obs. . . Me. Callao, F.S Peru Cambridge, Obs. ‘Mass. Camden S.C. Campeachy . .Yucatan Location. 3i 43 54 2 9 42 50 40 4 52 44 28 52 40 48 22 41 i3 54 36 59 48 45 11 5 S. 42 2.2 52 34 17 19 49 W. 99 49 9 73 45 24 83 43 76 29 84 59 123 49 42 84 23 22 76 28 81 54 69 46 37 97 44' I 2 89 1 4 76 36 59 68 46 59 59 37 74 6 69 49 91 18 76 39 48 80 40 27 69 19 122 9 23 73 18 100 38 70 53 6 71 3 3° 97 12 73 11 4 71 16 5 73 59 27 97 30 r 81 29 26 69 57 24 78 59 74 52 37 78 10 91 6 25 103 52 57 89 11 14 67 16 5 77 !3 71 7 43 80 33 9° 33 Latitude, j Longitude. NORTH AND SOUTH N. AMERICA. Canandaigua N.Y. Cape Ann, S. L. .Mass. Cape Breton Va. Cape Canaveral. . .Fla. Cape Cod, L. P. L. . . Ms. Cape Fear N.C. Cape Flattery, L..W.T. Cape Florida, L. . . Fla. Cape Hancock, Colo. R. Cape Hatteras, L. , N.C. Cape Henlopen, L. , Del. Cape Henry, L Va. Cape Horn, S. Pt., Her- mit’s Island Cape May, L N. J. Cape Race N. S Cape Sable N.S. Cape Sable, C.S. . . Fla. Cape St. Roque, Brazil Carthagena. . . . . .N.G. Castine Me. Cedar Keys, Depot Isl. Chagres N.G. Charleston , C. Ch. , S. C. Charlestown, Mon. , Ms. Cheboygan, L . . . Mich. Chicago, C.Ch 111. Chickasaw Miss. Cincinnati, Obs 0. Cleveland, Hos “ Colorado Springs.. Col. Columbia, S.H S.C. Columbus, Cap O. Concord, S. H N. H. Corpus Christi. . .Tex. Council Bluffs. .Neb. T. Crescent City, L. .Cal. Cumberland Md. Darien. W.H Ga. Davenport, S. S la. Dayton 0. „ Deadwood. S. S. . . Dak. ',44 22 Decatur, S. S Tex. 133 10 42 54 9 42 38 11 45 57 28 27 30 42 2 33 48 48 23 25 39 54 46 16 35 35 i5 2 38 46 6 36 55 3° S. 53 $ 38 55 48 46 39 24 43 24 25 6 53 S. 5 28 N. 10 26 44 22 30 29 7 3° 9 20 32 4 6 44 42 22 36 45 40 9 41 53 48 35 53 30 39 6 26 41 30 25 38 50 33 59 58 39 57 40 43 12 2 27 47 1 3° 44 34 39 39 i4 31 21 54 41 32 39 44 W. 0 / n 77 17 70 34 10 59 48 5 80 33 70 9 48 77 57 124 43 54 80 9 2 124 1 45 75 3° 54 4 7 67 16 74 57 18 53 4 3 65 36 81 15 35 17 75 38 68 45 83 2 45 80 1 21 79 55 39 71 3 18 84 24 37 87 37 47 88 6 .25 84 29 45 81 40 30 104 49 8 8123 82 59 40 71 29 97 27 2 95 48 124 11 22 78 45 25 81 25 39 90 38 84 11 103 34 97 30 LATITUDE AND LONGITUDE. 77 Latitude and Longitude — Continued. Location. | Latitude. Longitude.) Location. Latitude. Longitude. NORTH AND SOUTH AMERICA. Denver, S. H.Sp. . Col. Des Moines, C. H . . . la. Detroit, St. P. Ch. , M ich. Dover Del. Dover N. H. Dubuque Ia. Duluth, S. S Min. Eastport, Con.Ch. . Me. Eden ton, C.H....N. C. Elizabeth City, Ct. “ Erie, L Penn. Eureka, M. Ch Cal. Falls St. Anth’y..Minn. Fernandina, A. S. , Fla. Florence Ala. Fort Gibson Ind. T. Fort Henry Tenn. Fort Laramie. .Neb.T. Fort Leavenworth, Ks. Frankfort Ky. Frederick Md. Fredericksburg, E.Ch., Ya. Fredericton N.B. Galveston.Cath’l..Tex. Georgetown Ber. Georgetown, E.Ch., S.C. Gloucester, U. Ch. . . Ms. Grand Haven, S. S., Mich. Halifax, Obs N.S. Harrisburg Penn. Hartford, S. H. . .Conn. Havana, Moro . . .Cuba Hole in the Wall, L. Bahamas Holmes’s Hole, Ch., Ms. Hudson N.Y. Huntsville Ala. Indianapolis Ind. Indianola,G.S Tex. Jackson Miss. Jacksonville, M. Ch. Fla! Jalapa Mex. Jefferson City Mo. Jersey City, Gas Ch’y. Kalama, M.Ch. . . W.T. Keokuk, S. S Ia. Key West, T. Obs., Fla. Kingston Jamaica Kingston. C. H. . .C. W. Knoxville Tenn. La Crosse, Ct.S. . . Wis. Lancaster Penn. Lavaca. A. S Tex. Leavenworth, S.S.. Ks. Lexington Ky. Lima. . . Peru N. 39 45 . 35 42 19 46 39 10 43 i3 42 29 55 46 48 44 54 i5 36 3 24 36 1 7 58 42 8 43 40 48 11 44 58 40 30 40 18 34 47 13 35 47 35 36 30 22 42 12 10 39 21 ] 38 14 39 2 4 38 18 6 46 3 29 18 17 32 22 33 22 8 42 36 46 43 5 44 39 4 40 16 41 45 59 23 9 25 5i 5 41 27 13 42 14 34 36 39 55 28 32 28 32 23 W. o / io 4 59 33 93 37 16 83 2 23 75 30 70 54 9° 39 57 92 8 66 59 14 76 36 3 76 13 23 80 412 124 9 4 93 10 30 81 27 47 87 41 40 95 i5 10 88 3 40 104 47 43 94 44 84 40 77 18 77 27 38 66 38 15 94 47 26 64 37 6 79 16 49 70 39 59 NORTH AND SOUTH AMERICA. Lockport N.Y. Los Angeles Cal. Louisville Ky. Lowell, St. Ann’s Ch., Mass. Machias, Th .Me. Macon, Arsenal Ga. Madison, Dome.. .Wis. Marblehead, L ..Mass. Martinique, S.P’t.W.I. Matagorda, G.S. . .Tex. Matamoras. .... Matanzas Cuba Memphis, S.S. . .Tenn. Mexico Mex. Milwaukee Mich. xMinneapolis, U.C.,Min. Mississippi City, G. S., Miss. Mobile, E. Ch Ala. Monterey, Az. S. . .Cal. Montevideo. . .Rat Is’d N. 42 38 46 44 43 ’32 5° 25 43 4 33 42 3° *4 J 4 27 28 41 29 25-52-50 23 3 35 7 J 9 25 45 43 2 24 44 58 38 30 22 54 30 41 26 36 35 S. 34 53 N. 30 19 43 19 30 38 36 40 43 28 46 26 40 23 24 33 31 17 58 44 8 35 59 43 58 50 40 2 36 28 37 36 39 2 9 38 6 S. Little Rock Ark. 1 34 40 32 22 45 45 31 37 4 47 4i 23 24 41 16 57 36 9 33 25 5 3i 34 4i 5 5 41 38 10 4.1 18 28 41 21 16 29 57 46 40 42 44 35 6 41 30 42 48 30 39 39 36 41 29 12 36 50 47 4i 2 50 4i 33 35 6 28 44 45 37 47 3 4i 15 43 43 28 32 45 23 8 57 9 39 16 2 30 20 42 18 73 32 56 89 12 70 2 24 70 5 57 86 49 3 77 21 2 91 24 42 101 21 24 70 55 36 72 55 45 72 90 74 2 4 5 29 3 28 Montgomery, S. H., Ala. Montreal C. E. Mound City 111. Nantucket, L. . . . Mass. Nantucket, S. Tower, 86 18 I Mass. 63 35 Nashville, U Tenn. 76 50 Nassau, L N. P. 72 40 45 Natchez Miss. 82 21 23 Nebraska, Junction of Forks of Platte Riv. 77 10 6 New Bedford, B. Ch. 70 35 59 Mass! 7346 New Haven, Col., Conn. 86 57 New London, P.Ch. 86 5 New Orleans, Mint, La. 90 3 8 1 New York , C. H. , N. Y. Newbern, E. Sp. . .N.C. 81 39 14 Newburg, A. Sp., N.Y. 96 54 30 Newburyport,L.,Mass. 92 8 New Castle. E.Ch., Del. 74 224 Newport, Sp R. ' 122 50 39 Norfolk, C. H Va. 91 25 Norwalk Conn. 81 48 31 Norwich 76 46 Ocracoke, L N.C. 76 28 37 < )gdensburg, L. . . N. Y. 83 54 Did Point Comfort, Va. 91 14 48 Olympia Wash.T. 762033 Omaha, P. Ch. .. .Neb. 96 37 21 Oswego, S. S N.Y. 9458 Ottawa Can. 84 18 Panama, Cath’l. ..N.G. Parkersburg W. Va. 77 6 Pascagoula Miss. Pensacola, Sq’re..Fla. 30 24 33 92 12 I Petersburg, C. H. . .Va.|3 7 13 47 77 24 16 G* W. o" L ,J 78 46 [18 14 32 85 30 71 19 2 67 27 21 83 37 36 89 24 3 7° 5o 39 60 55 95 57 56 97 27 50 81 40 90 7 99 5 6 87 54 4 93 H 8 1 57 2 28 121 52 59 56 13 77 5 74 33 70 52 28 75 33 48 71 18 49 76 7 22 73 2 5 35 72 7 75 58 51 75 30 76 18 6 122 55 95 56 14 76 35 5 75 42 79 2 7 17 34 12 32 45 87 12 53 78 LATITUDE AND LONGITUDE, ZLatitncLe and JLjon.gitu.cLe— Continued. Latitude. Longitude. Location. Latitude. NORTH AND SOUTH N. AMERICA. 0 1 u Philadelphia, S. H., Pa. 39 56 53 Pike’s Peak, S.S. .Col. 38 48 Pittsburg Penn. 40 32 Plattsburg, Sp — N.Y. 44 4 i 57 Plymouth, Pier . . .Ms. 41 58 44 Point Hudson W.T. 48 7 3 Port au Prince. . . W. I. 18 33 Port Townshend, A.S., Wash. T. 48 6 56 Portland, C.H Me. 43 39 28 Portland, S. S. ..... .0. 45 30 Porto Bello N. G. 9 34 Porto Cabello, Mara- caibo 10 28 Portsmouth, L. . .N. H. 43 4 16 Prairie du Chien..Wis. 43 2 Princeton, S.Cap., N. J. 40 20 40 Providence, U. Ch. , R. I. 41 49 26 Provincetown, Sp., Mass. 42 3 Puebla de los Angelos, Mex. 19 J 5 Quebec, Citadel. .Can'a 46 49 12 Queenstown.... “ 43 9 Raleigh, Square.. N.C. 35 46 50 Richmond, Cap Va. 37 32 16 3 .. Rio de Janeiro, S. Loaf. 22 56 N. Rochester, R. H. .N. Y. Rockland, E. Ch . . . Me. Sackett’s Harbor, N. Y. Sacramento. ..... .Cal Salem, So Mass. Salt Lake City, Obs., Utah Saltillo Mex. San Antonio Tex. San Buenaventura, G. S Cal. San Diego, A. S — “ San Francisco, C. S. Station Cal. San Jose, Sp “ San Luis Obispo. . “ San Pedro “ Sandusky, L 0. Sandy Hook, L. . . N. J. Santa Barbara, M. Cli., Cal. Santa Clara, C.Ch.. Santa Cruz, F. S. . Santa Fe N. Mex. Savannah, Sp.. Ga. Schenectady N.Y. Sherman, R. R. D.,Wy. Shreveport, S. S I .a. Smithville, G.S. . .N.C. Springfield Mass. Springfield, S.II 111. Springfield, S. S “ St. Augustine. ... . .Fla. 1 17 Location. 43 44 6 43 55 38 34 41 42 31 40 46 25 26 22 29 25 22 34 15 46 32 43 68 37 48 37 19 50 35 10 38 33 43 20 41 32 3° 40 27 40 34 26 37 20 49 3 6 57 3 1 35 41 6 32 4 52 42 48 41 7 50 32 30 33 54 58 39 47 57 42 6 W. 75 9 104 59 80 2 73 26 54 70 39 12 122 44 33 72 16 3 122 44 58 70 i5 122 27 30 79 4° 68 7 70 42 34 9 1 8 35 74 39 55 71 24 19 70 11 18 98 71 12 15 79 8 78 38 77 26 43 9 77 5i 69 6 52 75 57 121 27 44 7° 53 58 hi 53 47 101 4 45 98 29 15 19 *5 5 6 17 9 40 122 23 19 121 53 39 120 43 31 18 16 82 v: - 1 74 19 42 42 [21 26 56 122 I 29 106 81 5 26 73 55 105 23 33 93 45 78 1 89 39 20 72 36 NORTH AND SOUTH AMERICA. St. Augustine, P. Ch., Fla. St. Bartholomew, S. Point W. I. St. Christopher, N. Pt., W. I. St. Croix, Obs “ St. Domingo “ St.Eustatia,Town. “ St. Jago de Cuba, En- trance W. I. St. John N. B. St. Joseph .Mo. St. Louis, W.U.. . St. Mark’s, Fort.. Fla. St. Martin’s, Fort,W. I. St. Mary's, M. H. . .Ga. St. Paul Minn. St. Thomas, Fort Ch’n, W. I. St. Vincent’s, S. Point, W. I. Staunton Va. Stockton, S.S Tex. Stonington, L. . .Conn. Sweetwater River, Mouth of. ...Neb. T. Sydney, S. S N. S. Syracuse . .N.Y. Tallahassee Fla. Tampa Bay, E. Key “ Tampico, Bar Mex. Taunton, T. C.Ch., Mass. Tobago, N. E. P’r.W I. Toronto Can. Trenton, P.Ch . . .N. J. Trinidad, Fort...W. I. Troy, D. Ch N.Y. Tuscaloosa Ala. Utica, Dut.Ch N.Y. 29 53 J 17 53 3° 7 24 17 44 30 18 29 17 29 19 58 45 i4 23 3 i3 38 8 3 30 9 18 5 30 43 12 44 52 46 29 48 3c 81 35 Valparaiso, Fort. .Chili Vanda lia 111. Vera Cruz Mex. Vicksburg, S. S.. .Miss. Victoria Tex. Vincennes Ind. Virginia City, S.S.,M.T Washington. . .Capitol Watertown, Ars’l. .Ms. West Point N.Y Wheeling Va. Wilmington, E. Ch.. N. C. Wilmington, T.H. .Del. Worcester, Ant. H. . Ms. Yankton, S.S Dak. Yazoo Miss. York Penn | Yorktown Va. 13 9 38 8 51 30 50 41 19 36 42 27 18 46 12 43 3 30 28 27. 36 22 15 30 54 11 20 43 39 35 40 13 10 10 39 42 43 33 12 43 6 49 38 50 19 11 52 32 23 28 46 57 S 8 43 45 20 38 53 20 42 21 41 23 26 40 7 34 J 4 39 44 27 42 16 42 45 33 5 39 58 37 13 W. / n 81 18 41 62 56 54 62 50 64 40 42 69 52 63 75 52 66 3 30 109 40 44 90 12 4 84 12 30 63 3 81 32 53 95 4 54 64 55 18 61 14 79 4 i5 102 50 71 54 107 45 27 60 12 76 9 16 84 3 6 82 45 15 97 5i 5i 71 5 55 60 27 79 23 21 74 45 5° 61 32 73 2 16 87 42 75 13 77 71 41 89 2 96 8 36 9° 54 97 1 87 25 112 3 36 71 9 45 73 57 1 80 42 77 5 6 38 75 33 3 71 48 13 97 3° 90 20 76 40 7 6 34 LATITUDE AND LONGITUDE. 79 Latitude and Longitude— Continued. Location. EUROPE, ASIA, AFRICA, AND THE OCEANS. Aleppo Alexandria, L Algiers, L Amsterdam Antwerp Archangel Athens Barcelona. Latitude. Batavia, Obs Beucoolen, Fort, Su’a. Berlin, Obs. . . Bombay, F. S. Botany Bay, C. Roads , Bremen Bristol Brussels, Obs. Bussorah Cadiz. Cairo Calais . . . Calcutta . Candia. . . Canton . . Cape Clear. Cape of G. Hope. Obs. Cape St. Mary, Mad'r. Ceylon, Port Pedro . . Christiana Congo River Constantinople, St. S. . Copenhagen Corinth Cronstadt Dover Dublin Edinburgh Falkland Islands, Helena, Obs. St. N. 36 11 31 12 36 47 52 22 51 13 64 32 37 58 4i 23 S. 6 8 3 48 N. 52 30 16 18 56 S. 34 2 N. 53 5 5i 27 50 51 3 30 3 ° 36 32 30 3 50 58 22 34 35 31 23 7 51 26 S. 33 56 25 f 9 49 59 55 S. 6 8 N. 41 1 55 41 37 54 59 59 8 Longitude. E. o / 37 10 29 53 3 4 4 53 4 24 40 33 23 44 106 50 102 19 13 23 45 72 54 151 13 8 49 W. 2 35 E. 4 22 48 XV. 6 18 E. 31 18 1 5i 8 20 25 8 13 14 W. 28 45 45 7 80 23 10 43 Location. EUROPE, ASIA, AFRICA, AND THE OCEANS. Genoa Gibraltar. . . . Glasgow . . . . Greenwich . Hamburg . Havre Hawaii or Owyhee.. . Hongkong . Honolulu . . Hood Isl’d,Gallapagos. Hood’s Island, Mar quesas Jeddo or Tokio. Jerusalem Leghorn, L Leipsic Leyden Lisbon Liverpool, Obs. . Madras Madrid . Fayal. S. E. Point Feejee Group, Ovolai Obs Florence Funchal, Madeira. . Geneva 53 23 12 55 57 S. i5 55 N. 38 30 S. 1 7 4i N. 28 59 !2 34 22 52 29 47 43 46 32 38 46 11 59 6 20 30 3 12 5 45 28 42 E. *78 53 11 16 W. 16 55 E. 6 9 15 Majorca, Castle. Malaga Malta, Valetta. Manila Marseilles Messina, L. Mocha Moscow Muscat Naples, L N. 44 24 36 7 55 52 51 28 38 53 33 Longitude. 22 16 30 21 18 12 1 23 S 9 26 N. 35 40 3i 48 43 32 51 20 20 52 9 28 38 42 53 24 *4 4 9 40 25 39 34 36 43 5 22 4 16 New Castle New, t . Hebrides, Table Island Niphon, Cape Idron, Japan ' Odessa Palermo, L Paris, Obs Pekin Plymouth Port Jackson. .N.S.W. Porto Praya, Cape Verd Islands Prince of Wales Island. 35 54 J 4 36 43 18 38 12 3 20 55 40 23 37 49 50 54 58 S. 5 28 N. 34 36 3 46 28 38 8 48 50 13 39 54 50 21 S. 35 5 i 32 N. 14 54 S. 10 46 E. 9 58 6 W. i55 54 E. 114 14 45 157 30 36 W. 89 46 E. r 38 57 i39 40 37 20 10 18 12 22 4 29 15 W. 9 9 3 e. 80 15 45 W. 3 42 E. 2 23 W. 4 26 E. *4 30 121 2 5 22 15 35 43 12 35 33 58 35 14 16 W. *37 E. 1 67 7 38 50 35 30 44 13 22 2 20 16 28 W. V 51 18 w. 23 3 E. 142 12 8o LATITUDE AND LONGITUDE. Latitude and. 3L.ongitri.de— Continued. Location. JI Longitude. EUROPE, ASIA, AFRICA, AND THE OCEANS. Queenstown Rome, St. Peter’s. Rotterdam Santa Cruz Ten’fe Scilly, St. Agnes, L Senegal, Fort Sevastopol . Seville Siam Sierra Leone. Singapore. Smyrna — N. 5i 47 ' 4 1 54 5 54 28 28 49 54 16 1 44 37 36 59 H 55 S. 8 30 N. 1 17 38 26 Southampton . S. St. Helena ••••I I 5 55 W. °8 /g " 12 27 16 16 6 21 16 32 E. 33 30 5 58 ' 100 W. 13 18 E. 103 50 27 7 W. 1 30 5 45 Location. EUROPE, ASIA, AFRICA, AND THE OCEANS. St. Petersburg Suez v*'** ■ Surat, Castle Sydney N.S.W. Tahiti or Otaheite Tangier Toulon Tripoli Tunis, City. . . Venice Vienna Warsaw, Obs. Wellington... New Z’d Yokohama Zanzibar Island, Sp. . N. 0 'a ‘ 59 56 29 59 11 S. 33 33 4 2 7 45 N. 35 47 43 7 34 54 36 47 40 50 48 13 52 13 S. 41 14 N. 35 26 S. 6 28 Longitude. E. O / // 30 19 32 34 72 47 151 23 W. 149 30 E. 5 54 5 22 13 11 10 6 14 26 16 23 2129 i74 44 139 39 39 33 0"bs© i*'V’atoi*ios. — 2Vo£ included in pvevious Tuble. Longitude given in Time. Albany, Dudley . . Alleghanv, Penn. . Birr Castle, Earl of Rosse Cambridge, U. S. . . Cambridge, Eng... Cape of G. Hope.. Copenhagen, Un’y. Crescent City, A. S. , Cal Dublin Edinburgh Florence. Geneva. . Longitude. Georgetown, U.S. . Gibbes’s, Charles- ton, U. S Greenwich . Hamburg . Leipsic . . . Leyden . . . N. i ‘ // ■ iti 42 39 49-55 40 27 36 53 5 47 42 22 52 52 12 51.6 S. 33 56 3 N. 55 40 53 4 1 44 43 53 23 13 55 57 23-2 43 46 4*-4 46 11 59.4 38 53 39 32 47 7 51 28 38 53 33 5 51 20 20.1 52 9 28.2 53 24 47-8 Liverpool . . . L. M. Rutherfurd, New York 14° 43 49 W. m. s 4 54 59- 52 5 20 2.9 31 40.9 4 44 30-9 E. 22.75 1 13 55 50 19.8 W. 8 16 49.1 25 22 12 43.6 E. 45 3-6 24 37-7 W. 5 8 12.5 5 19 44-7 E. 39 54- 1 49 28.5 17 57-5 W. 12 o. 11 4 55 57 Madras Marseilles. . Mitchell’s, Cin.,0. Moscow Munich, Bogenli’n Palermo Portsmouth . Quebec Rome, College. . . Salt Lake City, Utah San Francisco, Sq., Cal Latitude. | Longitude. • N. 1 u m 13 4 8.1 43 *7 5° 39 6 26 Santiago de Chili. St. Croix, W. I St. Petersburg, A. . Stockholm Sydney TiflVs, Key West. Fla Unkrechtsberg, 01- mutz Washington. . . West Point, N.Y. . 55 45 19 8 48 8 45 38 6 44 5° 48 3 46 48 30 41 53.52-2 40 46 4 37 47 55 S. 33 26 24.8 N. 17 44 30 59 56 29.7 59 20 31 S. 33 5 n 41 ' 24 33 31 49 35 4° 38 53 39 41 23 26 E. h. m. s. 5 20 57.3 21 29 W. 5 37 59 E. 2 30 16.96 46 26.5 53 24-17 W. 4 23 9 4 44 49.02 E. 49 54-7 W. 7 27 35- 1 8 9 38.1 4 42 18.9 4 18 42.8 E. 2 1 13-5 1 12 24.8 10 4 59.86 W. 5 27 i4-i E. 5 8 12.03 4 55 48 DIFFERENCE IN TIME, 8l DIFFERENCE IN TIME. Difference in Time at Following Locations. Longitude computed both from New York and Greenwich . Exact Difference of Time between New York and Greenwich is 4 h. 56 m. 1.6 sec., but in following table 2 seconds are given when the decimal in any reduction exceeds .5 seconds. Location. Acapulco Albany Alexandria. . Egypt Algiers Amsterdam . Antwerp Apalachicola Astoria Atlanta Auburn Augusta Ga. Augusta Me. Austin Baltimore Bangor Barbadoes, S. Ft. Barnegat, L Bath Baton Rouge. . . Beaufort N.C. Beaufort S. C. Belfast Benicia Berlin Bismarck Bombay, F.S, Boston, S.H. , Bremen Bridgeport . . . Brooklyn, N. Yard. Brunswick Me. Brunswick Ga. Brussels Buenos Ayres Buffalo, L Burlington Ia. Burlington N. J. Burl ngton Vt. Bushnell Neb. Cadiz Cairo Cairo 111. Calais Me. Calcutta Callao Cambridge . . Mass. Canton Cape Girardeau Cape of Good Hope. Cape Horn Cape May Cape Race C.irthageoa Castinc F representing Fast , and S Slow . New York, j Greenwich. Location. h. m. 1 43 15 S. 1 F. 6 55 34 5 8 18 5i6,S 5 1338 43 54 S. 3 19 *7 4i 32 9 50 31 34 16 55 F 1 34 55 S. 10 26 20 54 F. 57 34 22 S. 16 46 F. I 9 10S. 10 38 26 39 20 1 F. 3 12 36 S, 5 49 37 F. 1 46 30 S. 9 47 38 F 11 47.6 5 3i 18 3 17 4 1 6 12 29 56 S. 5 13 30 F. 2 34 19 54 S. 8 24 3 29 16 38 59 30- 4 30 5o F. 1 14 43 S. 2 6 57 F. 49 22 12 50 S. II 30 F. 2 28 58 2 10 S. 9 57 F. 26 58 2 56 S. 23 4 6 F. 6 30 S. 21 2 F. h. m. 6 39 i7 S. 4 55 1 59 32 F. 12 16 19 32 17 36 5 39 56 S. 8 15 19 5 37 33 5 5 52 5 27 36 4 39 6 6 30 27 5 6 28 4 35 8 3 58 28 4 56 24 4 39 16 6 5 12 5 6 39 5 22 40 4 36 1 8 8 38 53 35 F. 6 42 32 S. 4 51 36 F. 4 44 14 S. 35 16 F. 4 52 44 S. 4 55 58 4 39 5o 5 25 58 17 28 F 3 53 28 S. 5 15 56 4 26 4 59 30 5 12 40 6 55 32 25 12 5 12 F. 5 56 45 S. 4 29 4 5 53 20 F 8 52 S. 4 44 3i 32 56 F. 58 12S. 13 55 F. 29 4 S. 58 58 3 32 16 5 2 32 4 35 New York. Cedar Keys Chagres Charleston Charlestown Cheboygan Chicago Chickasaw Cincinnati Cleveland Colorado Springs. . Columbia Columbus Concord Constantinople Copenhagen Corpus Christi Council Bluffs Crescent City Darien Davenport Dayton Deadwood Denver Detroit Dover Del. Dover. N. H. Dublin Dubuque Duluth Eastport Edenton Edinburgh E 1 i zabe th C i ty , N. C. Erie *. Eureka Falls St. Anthony. . Fernandina Fire Island, L Florence Ala. Fort Gibson. Fort Henry. .Tenn. Fort Laramie Fort Leavenworth. Frederick Fredericksb’g. . Va. Fredericton. . .N.B. Funchal Galveston Geneva Geneva N.Y. Genoa Georgetown. . .Ber. Georgetown. . .S. C. Gibraltar, A. 36 9S. 24 3 23 4i 11 48 F. 4i 37 S. 54 30 56 24 4i 57 30 40 2 3 15 28 7 35 57 10 6 F. 6 51 58 5 4<5 18 1 33 47 S. 1 27 10 3 20 44 29 4i 1 12 30 40 42 1 58 30 2 3 57 36 76 35 58 12 26 F. 4 30 4° 1 6 38 S. 1 12 10 28 6 F. 10 24 S. 4 43 14 F. 8 52 S. 24 15 3 20 37 1 16 40 29 50 3 10F, 54 45 S. 1 24 59 56 13 2 3 9 1 22 54 13 10 *3 49 29 29 F. 3 48 22 1 23 8 S. 5 20 39 F. 12 14 S. 5 3i 34 F. 37 33 21 6 S. 4 34 34 F. 5 32 11 S. 5 20 5 5 19 43 4 44 13 5 37 38 5 50 31 5 52 26 5 37 59 5 26 42 6 59 17 5 24 8 5 3i 59 4 45 56 1 55 56 F 50 16 6 29 48 S. 6 23 12 8 16 45 5 25 43 6 2 32 5 36 44 6 54 32 6 59 58 5 32 10 5 2 4 43 36 25 22 6 2 40 6 8 32 4 27 56 5 6 26 12 48 5 4 54 5 20 17 8 16 39 6 12 42 5 25 51 4 52 51 5 5° 47 6 21 1 5 52 15 6 59 n 6 18 56 5 9 12 5 9 5i 4 26 33 1 7 40 6 19 10 24 37 F. 5 8 16 S. 35 32 F. 4 18 28 S. 5i 7 7 21 28 82 DIFFERENCE IN TIME. Difference Location. New York. Greenwich. t i. m. 8 . 1 i. m. 8 . Glasgow 4 38 58 F. 17 4S. : Gloucester 13 22^ 4 42 40 Grafton 24 5 S. 5 20 7 Grand Haven 49 10 5 45 12 Greenwich 4 56 1 .6 — Halifax 41 42 F. 4144° Hamburg 5 35 54 39 52 F. Harrisburg 11 18 S. 5 7 20 S. Hartford 5 i 9 F. 4 50 43 Havana, Morro. . . 33 24 S. 5 29 26 Havre 4 56 26 F. 24 F. Hawaii or Owyhee 5 27 34 S. : 10 23 36 S. Hongkong : i2 27 I F. 7 36 59 F- Honolulu : 15 27 30 10 31 28 Hudson 1 12 4 54 40 S. Huntsville 51 46 S. 5 47 48 Indianapolis 48 18 5 44 20 Indianola 1 30 2 6 26 4 Jackson L 1 4 30 6 0 32 Jacksonville.. .... 30 35 5 26 37 Jalapa 1 31 36 6 27 38 Jeddo or Tokio . . . 14 16 2 F. 9 20 F. Jefferson City. . . ; 1 12 30 S. 6 8 32 S. Jersey City 8 4 5 6 i° Jerusalem 7 25 22 F. 2 29 20 F. Kalama 3 i 5 21 S. 8 11 23 S. Keokuk 1 9 38 6 5 40 Key West 31 *3 5 27 14 Kingston Can. 9 53 5 5 54 Kingston Jam. 11 2 5 7 t Knoxville 39 34 5 35 36 La Crosse 1 8 58 6 4 59 La Guayra 27 54 f • 4 28 8 Lancaster 19 21 s. 5 *5 22 Lavaca 1 30 27 6 26 29 Leavenworth 1 23 40 6 19 52 Leghorn 5 37 i4 F- 41 12 F. Lexington 41 10 S. 5 37 12 S. Lima 12 22 5 8 24 Lisbon 4 19 26 F. 36 36 F. Little Rock 1 12 46 S. 6 8 48 Sj Liverpool 4 44 2 F. 12 Lockport 19 2 b. 5*5 4 Los Angeles 2 56 10 7 52 18 Louisville 45 58 T 7 5 42 ^ Lowell. IO 45 F. 4 45 16 Machias Bay 26 12 4 29 49 Macon. 38 28 s. 1 1 35 4 41 14 F 4 38 18 5 34 3 ° Madison . . . 5 57 36 Madrid.. Malaga 14 48 17 44 M • 5 54 2 • 13 10 58 I Manila 848 Maracaibo 9 2 4 47 S Marblehead, L. . . 12 41 4 43 Marseilles . 5 17 20 21 28 F Martinique Matagorda Matamoras. Matanzas . 52 22 . 1 27 50 S • 1 33 50 . 1 3 ° 38 4 3 4 oS . 6 23 52 6 29 51 5 26 40 Memphis . 1 4 26 6 0 28 Mexico . 1 40 19 6 36 20 in Time — Continued. Location. Monterey. Montreal. . . Montserrat Moscow Mound City Nantucket Naples Nashville Nassau Natchez Nebraska New Bedford New Haven New London New Orleans New York. Newbern Newburg Newbury port New Castle New Castle. . . Del, Newport Norfolk Norwalk Norwich Ocracoke. ....... Odessa Ogdensburg. .. . . Old Point Comfort Olympia Omaha.. Oswego . . — Ottawa Paducah Palermo Panama Paris Parkersburg. Pekin ...... Pensacola. . . Petersburg. . Philadelphia Pike’s Peak. Pittsburg . . . Plattsburg . . Plymouth Plymouth.. .Mass. Port Au Prince, St. Domingo Port Townshend. . Portland Portland Porto Praya. Porto Rico. . Portsmouth. few York. Greenwich. . m. 8. 1 1. m. 8. 55 35 S. 5 5i 3 6 S - 1 16 55 6 12 57 1 6 5 56 8 56 7 5 52 9 18 22 F. 4 37 40 3 n 3 oS - 8 7 32 1 11 10 F. 3 44 52 49 10 S. 5 45 12 1 50 F. 4 54 12 47 i4 4 8 48 7 18 14 2 22 12 f. 1 26 S. 5 56 28 S. 15 38 F. 4 40 24 5 53 6 57 4F. 51 15 s. 5 47 I 6 b - 13 23 5 9 2 4 1 9 37 6 5 39 1 49 24 6 45 26 12 19 F. 4 43 42 4 19 4 5i 43 7 40 4 48 22 1 4 12 S. 6 14 — 4 56 1.6 12 18 5 8 20 1 4 56 2 12 32 F. 4 43 30 4 49 34 6 28 6 13 S. 5 2 15 10 46 F. 4 45 15 9 8S. 5 5 9 2 19 F. 4 53 42 7 34 4 48 28 7 54 S. 5 3 55 6 58 58 F. 2 2 56 I 5 58 S. 5 2 £ 9 11 5 5 12 3 i5 38 8 11 40 1 27 43 6 23 45 .Or. io 19 6 46 58 22 5 49 30 F. 1 21 48 S. 5 5 22 F. 30 15 S. 12 41 54 F. 52 50- s. 13 35. 4 34 6 2 3 50 24 6 2 14 F, 4 39 26 13 25 16 34 3 14 58 S. 15 2 F. 3 13 48 S. 3 23 50 F. 33 26 13 11 5 6 20 5 2 48 5 54 2 4 ca 28 F. 5 17 49 s. 9 20 F. 5 26 37 S. 7 45 52 F. 5 48 52 S. 5 9 37 5 3 6 * 2 6 59 52 5 20 8 $ 4 4 2 37 4 39 28 8 11 4 4i 8 9 50 1 32 12 4 22 36 4 4 2 5i DIFFERENCE IN TIME, 83 Difference in Time- Continued. Location. New York. Greenwich. Location. New York. Prairie du Chien Princeton. . Providence Provincetown. . . Quebec Queenstown, L. Raleigh Richmond Rio de Janeiro. . Rochester. . Rockland . . Rome Rotterdam . Sackett’s Harbor, Sacramento Salem Salt Lake City. . Saltillo San Antonio San Buenaventura San Diego San Francisco, C. S. S. • San Francisco, P San Jose Sandusky. . . Sandy Hook. Santa Barbara. . . . Santa Clara Santa Cruz. Santa Cruz. Ten’ fe Santa Fe Savannah Schenectady Seville Sherman Shreveport Siam Sierra Leone Singapore Smithville Smyrna Southampton Springfield 111. Springfield. .Mass. St. Augustine St. Croix, Obs St. Helena St. .Jago de Cuba. . St. John St. Joseph Mo. 833S. 2 38 10 24 F. 15 16 11 13 4 42 46 18 31 S. 13 43 2 3 26 F. 15 22 S. 19 34 F. 5 45 50 5 13 58 7 46 S. 3 9 49 12 26 F. 2 31 34 S. 1 48 17 1 37 55 312 2 52 37 3 13 32 3 13 5o 3 n 33 34 47 iF. 3 2 49 S. 3 9 46 3 12 4 3 5o 58 F. 8 4 S. 28 22 F. 4 3^ io 5 33 S. 1 18 58 1 36 2 F. 4 2 50S. 1 51 22 F. 16 3 S. 6 44 30 F. 4 5o 2 2 36 S. 5 39 F - 29 13 s. 37 19 F - 4 33 2 7 26 S. 31 48 F. | 2 22 41 S. | A. 6 4 34 S, 4 58 40 4 45 37 4 40 45 4 44 49 33 16 5 14 32 5 9 44 2 52 36 5 11 24 4 3 6 2 7 49 48 F. 17 56 5 3 48 S. 8 5 51 4 43 36 7 27 35 6 44 19 6 33 57 7 57 4 7 47 39 8 9 33 8 9 51 8 7 35 5 30 49 4 56 1 7 58 50 5 8 8 1 5 4 7 4 5 5 24 22 4 55 40 23 52 7 1 34 6 15 6 40 F. 53 12 S. 6 55 20 F. 5 12 5 S. 1 48 28 F. 6 S. 5 58 37 4 50 24 5 25 15 4 18 43 23 5 3 28 4 24 14 7 18 43 St. Louis St. Mark’s St. Mary’s St. Paul St. Petersburg. . . St. Thomas, Fort. Staunton Stockholm Stonington Suez. ; Sweetwater River, Mouth of. Sydney N.S. Sydney. . .N.S.W. Syracuse Tahiti or Otaheite. Tallahassee Tampa Bay Tampico Bar Taunton Toronto Toulon Trenton Tripoli Troy Tunis Turk’s Island Tuscaloosa Utica Valparaiso Vandalia Venice Vera Cruz Vicksburg Victoria Tex. Vienna Vincennes . . . Virginia City. Warsaw Washington, Obs. . West Point. Wheeling Wilmington. .Del. Wilmington. .N. C. Worcester Yankton Yazoo Yeddo Yokohama York York town A. m. «. 4 47 S. 40 48 30 10 11 24 18 6 57 18 F. 36 20 20 15 S. 6 8 26 F. 8 26 7 6 18 2 15 8 2 15 I34 ~ 8 35 S. 14 44 2 F. 42 22 S. 34 59 1 35 25 11 38 F. 21 32 S. 5 17 3 ° F - 3 2 S. 5 48 36 F. 3 54 5 36 26 11 22 54 46 S. 4 5o 9 18 1 6 5 53 46 F. 1 28 33 S. 1 7 34 1 32 2 6 1 34 F. 53 38 S. 2 32 10 6 20 11 F. 12 10 S 14 F. 26 46 S. 6 11 15 45 8 49 F. 1 33 58 S. 1 5 18 14 14 42 F. 14 14 43 10 38 S. io 14 j (Greenwich. A. m. s. 6 48 S. 5 36 50 5 26 12 6 20 20 2 1 16 F. 4 19 4 i S. 5 16 17 1 12 25 F. 4 37 36 S. 2 10 16 F. 7 11 2 S. 4 48 10 5 32 F. 5 4 37 S. 9 48 F. 5 38 24 S. 5 3 i 1 6 31 27 4 44 24 5 1 7 33 21 28 F. 4 59 3 S. 52 44 F. 4 52 9 S. 40 24 F. 4 44 40 S. 5 5 o 48 5 52 4 46 44 5 56 8 57 44 F. 6 24 34 S. 6 3 36 6 28 4 1 5 32 F. 5 49 40 S. 7 28 12 1 24 9 F. 5 8 12 S. 4 55 48 5 22 48 5 2 12 5 11 47 4 47 13 6 30 6 1 20 9 18 40 F. 9 18 41 6 40 S. 6 16 fo Compute Difference of Time between ISTew York and Greenwich and any Location not given in Table. Rule. — Reduce longitude of location to time, and if it is W of as- sumed meridian it is Slow ; if E., it is Fast. If difference for New York is required, and it exceeds 4 h. z6 m. 2 »sec subtract this time, and remainder will give difference of time, S. • and it it (4 h. 56 m 2 sec.) does not exceed it, subtract difference from it, and remainder will give difference of time, F. 8 4 TIDES. TIDES. Tide-Table for Coast of United States, Showing Time of High-water at Full and New Moon , termed Establish- ment of the Fort , being Mean Interval between Time of Moon's Transit and Time of High-water. (H. S. Coast and Geodetic Survey.) Locations and Time. Spring. J Neap. 1 COAST FROM EASTPORT C TO NEW YORK. h. m. Feet. ] Feet. Eastport Me. 11 30 15 ( Campo Bello*.... 11 25 ( Portland “ 11 25 9.9 7.6 1 Cape Ann* 11 30 11 Portsmouth N.H. 11 23 9.9 7.2 1 Newburyport. . .Mass. II 22 6.6 ] Salem “ II 13 10.0 7.6 . Cape Cod* “ 11 3 ° 6 ] Boston Light .. . “ II 12 10.9 8. 1 Bostont “ II 27 10.3 8-5 ' Nantucket “ 12 24 3-6 2.6 Edgartovvn “ 12 T6 2-5 1.6 Holmes’s Hole. . “ n 43 1.8 13 ; Tarpaulin Cove . “ 8 4 2.8 1.8 < Wood’s Hole, n. side. 7 5 o 4-7 3 -i N. Bedford (Dump-) A 6 2 * 8 1' ing Rock) J 7 57 New York? N.Y. 8 13 5-4 3-4 “ Albany* “ 3 30 1 LONG ISLAND SOUND. Newport R. I. 7 45 4.6 3 -i Point Judith “ 7 32 3-7 2.6 Montauk Point. . N.Y. 8 20 2.4 1.8 Watch Hill R.I. 9 3 -i 2.4 Providence* 8 25 5 Stonington Ct. 9 1 3.2 2.2 Little Gull Isl’d. N.Y. 9 38 2.9 2-3 New London Ct. 9 28 3 -i 2.1 New Haven “ 11 16 6.2 5-2 Bridgeport “ 11 11 8 4-7 Oyster Bay N.Y. 11 7 9.2 5-4 Sand’s Point “ 11 13 8.9 6.4 New Rochelle. . . u 11 22 8.6 6.6 Throg’s Neck. . . “ 11 20 9.2 6. 1 Hell Gate* u 9 35 6 COAST OF NEW JERSEY. Cold Spring Inlet, N. J. 7 32 5-4 3-6 Sandy Hook N.J. 7 29 5-6 4 Amboy “ 8 15 5 Cape May Landing “ 8 19 6 4-3 Egg Harbor*. ... . “ 9 34 5 DELAWARE BAY AND RIVER. Delaware Break watei r 8 4-5 3 Higbee’s (Cape May). , • 8 33 6.2 3-9 Egg Isl’d Light . .N.J • 9 4 7 5 -i New Castle Del • 11 53 6.9 6.6 Philadelphia Penn • 13 44 1 6.8 5 -i Locations and Time. CAROLINA, GEOR-GIA, AND FLORIDA. Beaufort . Charlestonll (C. H. ) Wharf S.C. j Fort Pulaski Ga. Savannah “ St. Augustine Fla. Cape Florida ... . “ Key West “ Tampa Bay “ Cedar Keys “ WESTERN COAST. .Cal. San Diego San Pedro Cuyler’s Harbor . San Luis Obispo.. Monterey South Farallone . San Francisco . . . Mare Island Benicia Ravenswood Bodega. Humboldt Bay. . . Astoria Or. Nee-ali Harbor, Wash. Port Townshend “ 13 i 5 t Refem to rise and fall of tide alone. MISCELLANEOUS. Bay of Fundy*. .N.S. Blue Hill Bay*. . St. John’s* Kingston* Jam. Halifax* ...N. S. Pensacola* Fla. Galveston* Tex. + t § li see p. 85. Feet. 1.6 60 30 1.8 1.6 M ., n i n ter vaV has been increased 12 A. =6 min. (hnlf « menu lunar day) for aom. nnrU ir Def Note.- aware TIDES. 85 Bench. UVIarks referred, to in preceding Table. t Boston. — Top of wall or quay, at entrance to dry-dock in Charlestown navy- yard, 14.76 feet above mean low- water. t New York. —Lower edge of a straight line, cut in a stone wall, at head of wooden wharf on Governor’s Island, 14.56 feet above mean low- water. § Old Point Comfort, Va. — A line cut in wall of light-house, one foot from ground, on southwest side, n feet above mean low- water. II Charleston, S. C. — Outer and lower edge of embrasure of gun No. 3, at Castle Pinckney, 10.13 feet above mean low-water. Establishment of the Port for several Locations in Europe, etc. Port. | Time. || Port. Time. Port. Time. Amsterdam Antwerp Beachy Head . . Eng. Belfast Bordeaux Bremen Brest Harbor Bristol. Bristol Quay Cadiz Calais Calf of Man Cape St. Vincent. . . . h. in. 3 4 25 11 50 10 43 6 50 6 3 47 7 21 6 27 1 40 11 49 n 5 2 30 Chatham Cherbourg Clear Cape Cowes Dover Pier Dublin Bar Funchal Gravesend Eng. Greenock Holyhead Hull Eng. Land’s End Lisbon h. m. 1 2 7 49 4 10 46 11 12 11 12 11 30 1 i4 8 10 11 6 29 3 57 2 30 Liverpool London Bridge Newcastle Portsmouth D.-yard, Eng.' Quebec Ramsgate Pier Rye Bay Eng. Sheerness Sierra Leone Southampton. .Eng. Thames R.,mo’th “ Woolwich “ h. n 41 8 10 27 11 20 57 8 15 11 40 12 2 15 Rise and Fall of Tides in Grnlf of Mexico. Locations. C § be .5 m Neap. Locations. Mean. Spring. Feet. Feet. Feet. Feet. Feet. St. George’s Island.. . .Fla. 1. 1 1.8 .6 Isle Derniere . . . . La. 1.4 1.2 Fort Morgan ( Mobile ) Entrance to Lake Cal - 1 Bav) Ala. j 1 1-5 •4 PftS'ifM] La j i -5 i -4 Cat Island Miss. 1.3 1.9 .6 Aransas Pass 1. 1 1.8 Southwest Pass La. 1. 1 1.4 •5 Brazos Santiago. . . u •9 1.2 Tides of G-nlf of IVIexico. On Coast of Florida, from Cape Florida to St. George’s Island, near Cape San Bias, the tides are of the ordinary kind, but with a large daily inequality. From St. George's Island, Apalachicola entrance, to Derniere Isle, the tides are usually of the single-day class, ebbing and flowing but once in 24 (lunar) hours. At Calcasieu en- trance. double tides reappear, and except for some days about the period of Moon’s greatest declination, tides are double at Galveston, Texas. At Aransas and Brazos Santiago the single-day tides are as perfectly well marked as at St. George’s, Pensa- cola. Fort Morgan, Cat Island, and the mouths of the Mississippi. For some 3 to 5 toys, however, about the time when the Moon’s declination is nothing, there are generally two tides at all these places in 24 hours, the rise and fall being quite small. Highest high and lowest low waters occur w j hen greatest declination of Moon happens at full or change. Least tides when Moon’s declination is nothing at first or last quarter. Tides of Bacifxc Coast. On Pacific coast there is, as a general rule, one large and one small tide during each day, heights of two successive high-waters occurring, one A.M., and other P. M. of same 24 hours, and intervals from next preceding transit of Moon are very different. These inequalities depend upon Moon’s declination. When Moon’s de- clination is nothing, they disappear, and when it is greatest, either North or South, they are greatest. The inequalities for low w T ater are not same as for high, though they disappear, and have greatest value at nearly same time. When Moon’s declination is North, highest of twm high tides of the 24 hours oc- curs at San Francisco, about 11.5 hours afte* Moon’s southing (transit); and w T hen declination is South, lowest of the two high tides occurs about this interval. Lowest of two low-waters of the day is the one which follows next highest high- water. IT 3 86 STEAMING DISTANCES, STEAMING DISTANCES. Distances between various Forts of* United States and Canada. By Lake, River, and Canal. Locations. Lake 1 and 1 River. | Canal. Total. Miles. Miles. Miles. Duluth to Buffalo. . . 1024 1 1025 Chicago to Buffalo . . 925 — 925 Duluth to Oswego. . . 1133 27 1160 Chicago to Oswego.. 1034 26 1060 Duluth to New York, via Buffalo 1166 353 1519 via Oswego 1294 233 1527 Duluth to Montreal . 1289 72 1361 Chicago to New York, via Buffalo . 1067 352 1419 Locations. Chicago to NewYork, via Oswego Chicago to Montreal. Buffalo to Colborne, via Welland Canal. Buffalo to New York. Welland Canal to Montreal Montreal to Kingston Ottawa to Kingston Lake and River. ii95 190 142 3°4-5 126. 25 232 7 1 26.77 352 7°-5 120 126.25 Total. .427 1261 26.77 494 375 246.25 126.25 Miles. Alexandria. . . Amsterdam . . Barbadoes . . . Batavia Bermudas . . . Bombay Boston Bremen Bristol Buenos Ayres Cadiz Calcutta . . iNT.Y, 4893 3291 1855 8972 682 8522 356 3428 2979. 6010 3 I2 5 9350 Miles. Lond. 3 102 262 3.812 1 1 492 3 H 2 10703 3030 408 ’ 50 1 6 280 x ns Cape Race. . Cowes Funchal . . . Galway .... Gibraltar . . Glasgow . . . Halifax .... Havana Hobart Town. . Kingston, Jam. Lima xi 531 1 Madras . N. Y. 1 004 3092 2 760 2 720 3 260 2913 59° 1 161 9 i8 7 1456 10050 8707 2249 200 1 3°3 72 1 325 765 2 706 4i97 11 368 4 305 10 149 10888 Miles. New Orleans. Norfolk Pensacola . . . Philadelphia. Quebec Queenstown . Rio Janeiro. . St. Johns Southampton Swan River. . Tortugas . . . Washington N.Y. 1790 308 1623 262 1360 2780 4970 1064 3 io 3 8480 1151 461 Halifax to Liverpool St. Thomas St. Johns, N. F. . Quebec to Glasgow Liverpool to Boston Quebec Philadelphia Callao Fastnet Cape Race Aspinwall Port Said Melbourne. . . . Rio Janeiro San Francisco.. via Panama. . . via Tehuantepec Miles. 2563 x 563 520 2563 2 955 2855 3147 11 379 283 1992 4650 3 290 13290 5125 13 800 7 378 6400 Poets. Liverpool to Havana Portland Baltimore N. Orleans to Havana Cape Race to Fastnet Halifax Boston St. Johns, N. F., to Quebec Boston Greenock Bermudas to Nassau. Panama to San Juan del Sud . Gulf of Fonseca. . . Acapulco. ......... Manzanilla 4100 2770 3400 57° 1711 457 835 891 890 1848 804 570 739 1416 1724 Poets. Panama to San Diego Monterey San Francisco San Francisco to San Juan del Sud. Acapulco Manzanilla San Diego Monterey Humboldt Columbia R. Bar. . Vancouver. Portland Port Townshend . . Victoria Yokohama Honolulu Honolulu to Callao. . Distances between various Ports and New York and London. Not included in preceding Table. Miles. Lond. 4 730 3 447 4 654 3 404 3080 55i 5200 2 214 211 10 661 4 182 3612 Distances between various Forts of England, Canada, United States, etc. Not included in preceding Table. Miles. 2897 3I9 8 3240 i 2685 j 1841 i 1543 474 105 200 53° ‘ 638 650 732 7i5 4750 2080 5H5 > = J g g a : § : o «^.S ^ :■© ■ •Ialisg5^gl3f ea -.|- :-|l oSxSg=o«$a 3 2 ^-jriS 3 : ls!iill&-a2 l^!!”S STEAMING DISTANCES. Puerto Bello | 1558 I 1410 I 97 2 I XI 4 2 fractions. 89 FRACTIONS. A Fraction, or broken number, is one or more parts of a Unit. Ti T USTRATION -12 inches are 1 foot. Here, 1 foot is unit, and 12 inches its parts ; inches therefore, are one fourth of a foot, for 3 is fourth or quarter of 12. A Vulvar Fraction is a fraction expressed by two numbers placed one above the other, with a line between them ; as, 50 cents is the i of a dollar. Upper number is termed Numerator , the lower Denominator. Terms of a fi ac- tion express numerator and denominator; as, 6 and 9 are terms of -9. ‘ A Proper fraction has numerator equal to, or less than denominator; as, T , etc. An Improper fraction is reverse of a proper one; as, f, etc. A Mixed fraction is a compound of a whole number and a fraction; as, 5§-, etc. A Compound fraction is fraction of a fraction ; as, i of f, etc. A Complex fraction is one that has a fraction for its numerator or denominator, or both; as, J. or JL, or * r or etc. ’ 6 4 S 6 Note.— A Fraction denotes division, and its value is equal to quotient, obtained by dividing numerator by denominator; thus, - is equal to 3, and 5 is equal to 45. Fied. notion of Fractions. To Compute Common Measure or greatest Number tlx at will divide Two or more IsT umbers without a Remainder. Ruj.e. Divide greater number by less; then divide divisor by remainder; and so on, dividing always last divisor by last remainder, until there is no remainder, and last divisor is greatest common measure required. Example i.— W hat is greatest common 936) 1908 (2 measure of 1908 and 936 ? i8 7 2 36) 936 (26 7 2 216. Hence 36. 2.— How many squares can there be obtained in an area of 90 by 160 feet? Here 10 is greatest common measure. Hence, A 6 / = 16, and = 9; therefore 16 X 9 = J 44- To Compute least Common. ^Multiple of* Two or more ISTiimhers. Rule —Divide given numbers by any number that will divide the greatest num- ber of them without a remainder, and set quotients with undivided numbers in a ^JDiv^de Second line in same manner, and so on, until there are no two numbers that can be divided; then the continued product of divisors and last quotients will give common multiple required. 5) 40 . 50 . 25 5 ) 8 . 10 ♦ 5 Example. — What is least common multiple of 40, 50, and 25 ? 2) 8 . 4 . 1 . 1. Then 5X5X2X4X1X1 = 200. To Reduce a Fraction to its Lowest Term. R rLE Divide terms by any number or series of numbers that will divide them vithout a remainder, or by their greatest common measure. Example.— Reduce £§{j- of a foot to its lowest terms. It-S- - 10 = it - 8 = A - 3 = J. or 9 *«»• 90 FRACTIONS. To Reduce a Mixed Fraction to its Equivalent, an Im- proper Fraction. Rule. — Multiply whole number by denominator of fraction and to product add numerator; then set that sum above denominator. Example i. — Reduce 2^# to a fraction. 23 X f 2 = ~ . 663 2.— Reduce inches to its value in feet. 123 - 4 - 6 = 20 | = 1 foot 8^ ins. To Reduce a Complex Fraction to a Simple one. Rule. — Reduce the two parts both to a simple fraction, multiply numerator of re- duced fraction by denominator of reduced denominator, and denominator of numer- ator fraction by numerator of denominator fraction. Example.— Simplify complex fraction - 2|= f 4 * = V 8 X 5 =40 3 X 24 = 72 5_ 9 ’ To Reduce a NVliole Number to an Equivalent Fraction having a given Denominator. Rule.— M ultiply whole number by given denominator, and set product over said denominator. Example. — Reduce 8 to a fraction, denominator of which shall be 9. 8X9 = 72; then result required. To Reduce a Compound Fraction to an Equivalent Simple one. Rule. — Multiply all numerators together for a numerator, and all denominators together for a denominator. Note. — When there are terms that are common, they may be cancelled. Example.— Reduce of | of § to a simple fraction. i X 4 X 3 = J4 = 4- 0r ’ i x f x f — 4 » hy cancellin 9 2 ’s and 3 ’s. To Reduce Fractions of different Denominations to Equivalents liaving a Commoix Denominator. Rule. — Multiply each numerator by all denominators except its own for new nu- merators; and multiply all denominators together for a common denominator. Note. — In this, as in all other operations, whole numbers, mixed or compound fractions, must first be reduced to form of simple fractions. 2. When many of denominators are same, or are multiples of each other, ascertain their least common multiple, and theu multiply the terms of each fraction by quo- tient of least common multiple divided by its denominator. Example. — Reduce -|, and to a common denominator. 1 X 3 X 4= 12) 2X2X4 = 16 3X2X3 = 18 ) 2 X 3 X 4 = 24 — 1 2 16 _ 18 2 4 — 2 4 — T4> 0r 3 T» A and A* Addition. Rule. — If fractions have a common denominator, add all numerators together, <*, and place sum over denominator. Note. — If fractions have not a common denominator, they must be reduced to one. Also, compound and complex must be reduced to simple fractions. Example 1. — Add ^ and | together. J + f = f = I* FRACTIONS. 91 Subtraction. Rule. —Prepare fractions same as for other operations, when necessary; then subtract one numerator from the other, and set remainder over common denom- inator. Example.— between -J Multiplication. Rule.— Prepare fractions as previously required; multiply all numerators to- gether for a new numerator, and all denominators together for a new denominator. Example i. — What is product of and ^ f X = g^g- — 2. —What is product of 6 and | of 5 ? f X f of 5 = f X ^ = - 6 g°- = 20. Division. Rule.— P i ipare fractions as before; then divide numerator by the numerator, and denominator by the denominator, if they will exactly divide; but if not, invert the terms of divisor, and multiply dividend by it, as in multiplication. Example i.— Divide 2 ^ 5 by 2 ¥ 5 - 4 - f = f = if. _ nivi/lo 5 2 5 2 — 5 V 15 . — 15 y 5 7 5 — 2 5 j 1 2.— Divide j ny ^ . 15 — j X -y- — 44 281 400 1 44 1 281 282 1 1 19 00 Square Roots of Fractions. Rule.— Reduce fractions to their lowest terms, and that fraction to a decimal, and proceed as in whole numbers and decimals. Note.— When terms of fractions are squares, take root of each and set one above the other ; as is square root of Example.— What is square root of T 9 ^ ? .866 025 4. To Compute 4 t,li or 8tli Root of a IN' umber, etc. Rule. —For the 4th root extract square root twice, and for 8th root thrice, etc. To Extract Cube Root. Rule. — From table of roots (page 272) take nearest cube to given number, and term it the assumed cube. Then, as given number added to twice assumed cube, is to assumed cube added to twice given number, so is root of assumed cube to required root, nearly ; and by using in like manner the root thus found as an assumed cube, and proceeding in like manner, another root will be found still nearer; and in like manner as far as may be deemed necessary. Example. — W hat is cube root of 10 517.9? Nearest cube, page 272; 10648, root 22. 10648. 10 517.9 2 2 21296 21035.8 10 517.9 10648. 31 813.9 : 3 1 683.8 :: 22 : 21.9 -f-. To Ascertain or to Compute the Square or Cube Roots of Roots, Whole Numbers, and of Integers and Decimals, see Table of Squares and Cubes, and Rules, pp. 272, 300. To Extract any Root whatever. Let P represent number. I Let A represent assumed power, r its root. n “ index of the power. | R li required root of P. Then, as sum of n -{- 1 x A and n — 1 X P is to sum of n -}- 1 X P and n — 1 X A. so is assumed root r to required root R. Illustration. — What is cube root of 1500? Nearest cube, page 272, is 1331, root n. P = 1500, n = 3, A = 1331, r = n ; then, n 1 XA = 5324, n- J- 1 X P = 6000 n — 1 X P — 3000, n — 1 X A = 2662 8324 8662 :: 11 I : 11.446-}-. 98 EVOLUTION. PROPERTIES OF NUMBERS. POSITION. To Compute tlie Root of an Even Rower greater tlian any given in Table of Square and. Cube Roots. Rule. — Extract square or cube root of it, which will reduce it to half the given power; then square or cube root of that power; and so on until required root is ob- tained. Example i. — Suppose a 12th power is given ; the square root of that reduces it to a 6th power, and the square root of 6th power to a cube. 2. — What is biquadrate, or 4th root, of 2 560000? y/2 560 000 = 1600, and 1600 = 40. Note. — F or other rules for extraction of roots see pp. 301-4. PROPERTIES OF NUMBERS. 1. A Prime Number is that which can only be measured (divided without a re- mainder) by 1 or unity. 2. A Composite Number is that which can be measured by some number greater than unity. 3. A Perfect Number is that which is equal to the sum of all its divisors or ali- quot parts ; as 6 |, -|, •§. 4. If sum of the digits constituting any number be divisible by 3 or 9, the whole is divisible by them. 5. A square number cannot terminate with an odd number of ciphers. 6. No square number can terminate with two equal digits, except two ciphers or two fours. 7. No number, the last digit of which is 2, 3, 7, or 8, is a square number. Rowers of tlie first UNTine NTnixTbers. 1st. 2d. 3d. 4 th. 5th. 6th. 7th. 8th. I 9 th. I I 1 1 1 1 1 1 1 2 4 8 16 32 64 128 256 512 3 9 27 81 243 729 2 187 6561 19683 4 16 64 256 1 024 4096 16384 65 536 262 144 5 25 125 625 3125 15625 78 125 39° 625 1953125 ~6~ 36 216 1296 7776 46656 279936 1 679616 10 077 696 7 49 343 2401 16 807 117649 823 543 5 764 801 40 353 607 8 64 512 4096 32768 262 144 2 097 152 16777 216 134 217 728 9 81 7 2 9 6561 59 °49 1 53i 44i 4 782 969 1 43046721 387 420 489 POSITION. Position is of two kinds, Single and Double, and it is determined by number of Suppositions. Single ^Position. Rule. — Take any number, and proceed with it as if it were the correct one; then, as result is to given sum, so is supposed number to number required. Example i. — A commander of a vessel, after sending away in boats J, J, and ^ of his crew, had left 300; what number had he in command? Suppose he had 600. A of 600 is 200 A of 600 is 100 -Jof6ooisi5o 450 150 : 300 *. *. 600 : 1200 men. POSITION. FELLOWSHIP. 99 2. — A person asked his age, replied, if £ of my age be multiplied by 2, and that product added to half the years I have lived, the sum will be 75. How old was he ? 37.5 years. Double ^Position. Rule.— Assume any two numbers, and proceed with each according to conditions of question; multiply results or errors by contrary supposition; that is, first posi- tion by last error, and last position by first error. If errors are too great, mark them and if too little, — . Then, if errors are alike , divide difference of products by difference of errors; but if they are unlike , divide sum of the products by sum of errors. Example i. — A asked B how much his boat cost; he replied, that if it cost him 6 times as much as it did, and $30 more, it would have cost him $300. What was price of the boat? Suppose it cost him. . 60 30 6 times. 6 times. 360 and 30 more 180 and 30 more 90 90 39° 3°° 30 2d position. 2700 5400 90— 60 1st position. 5400 180) 8100 (45 dollars. 2.— What is length of a fish when the head is 9 inches long, tail as long as its head and half its body, and body as long as both head and tail ? 6 feet. FELLOWSHIP. Fellowship is a method of ascertaining gains or losses of individuals engaged in joint operations. Single PCellowsLip. Rule. — As the whole stock is to the whole gain or loss, so is each share to the gain or loss on that share. Example. — Two men drew a prize in a lottery of $9500. A paid $3, and B $2 for the ticket; how much is each share? 5 : 9500 3 : 5700, A’s share. 5 : 9500 : : 2 : 3800, B’s share. Double IPellowslxip, Or Fellowship with Time. Rule.— Multiply each share by time of its interest; then, as sum of products is to product of each interest, so is whole gain or loss to each share of gain or loss. Example.— A cutter’s company take a prize of $10000, which is to be divided ac- cording to their rate of pay and time of service on board. The officers have been on board 6 months, and the crew 3 months; pay of lieutenants is $100, ensigns $50, nnd crew $10 per month; and there are 2 lieutenants, 4 ensigns, and 50 men; what is each one’s share ? 2 lieutenants $100 = 200 x 6 = 1200 4 ensigns 50 = 200X6 = 1200 50 men 10 = 500 X 3 = 1500 3900 Lieutenants 3900 : 1200 10000 : 3076. 92-S- 2 = 1538.46 dolls. Ensigns 3900 : 1200 ” 10000 : 3076. 92 -4- 4= 769.23 “ Men 3900:1500:110000:3846.16-7-50= 76.92 u 3 ioo PERMUTATION. ti PERMUTATION. Permutation is a rule for ascertaining how many different ways any given number of numbers of things may be varied in their position. Permutation of the three letters abc , taken all together , are 6 ; taken two and two, are 6; and taken singly, are 3. Rule.— Multiply all the terms continually together, and last product will give result. Example i. — How many variations will the nine digits admit of ? 1X2X3X4X5X6X7X8X9 = 362 880. 2 . How many years would there be required to elapse before 10 persons could be seated in a varied position collectively, each day at dinner, including one day in every 4 years for a leap year ? 9935 years, 42 days. When only part of the Numbers or Elements are. taken at once. Rule.— Take a series of numbers, beginning with number of things given, decreasing by 1, until number of terms equals number of things or quantities to be taken at a time, and product of all the terms will give sum required. Example i.— How many changes can be made with 2 events in 5? 5 — 1 == 4, and 4X5 = 2 terms. Hence, 5X4 = 20 changes. 2.— How many changes of 2 will 3 playing cards admit of? 3 — 1 = 2, and 2X3 = 2 terms. Hence, 2X3 = 6 changes. 2 . How many changes can be rung with 4 bells (taken 4 and 4 together) out of 6 ? 4 _ 1 = 3, and 3X4X5X61=4 terms or changes. Hence, 3X4X5X6 = 360 changes. When several of the Elements are alike. Rule.— Ascertain the permutations of all the numbers or things, and of all that can be made of each separate kind or division; divide number of permutations of whole by product of the several partial permutations, and quotient will give number of permutations. Example. —How many permutations can be made out of the letters of the word persevere (9 letters, having 4 e’s and 2 r’s)? 1X2X3X4X5X6X7X8X9 = 362880; 1 X 2 X ^ X 4 = 24 for the e’s; 1 x 2 = 2 for the r s, and 24X2 = 48. Hence, 362880 = 48 = 7560. Or, Add logarithms of all the terms together, and number for the sum will give result. Example 1.— How many permutations can be made with three letters or figures? Log. 1 =*00 2 = . 301 03 3 = . 4.77 121 3 .778151 3 = log. of number 6. 2 . How many variations will 15 numbers in 16 places admit of? Add logarithms of numbers 1 to 16 and take logarithm of their sum— viz. , 13. 320 661 97 = 20 922 789 888 000. Number of positions of the blocks in the “ 15 puzzle ” is as above for their 16 permutations. IPerrriuitatioiTS, Whereby any questions of Permutation may be solved by Inspection , number of terms not exceeding 20. I 1 ! 5 120 9 362 880 13 6 227 020 800 2 2 6 720 10 3 628 800 M 87 178 291 200 3 6 7 5040 11 39 916 800 i5 1 307 674 368 000 4 24 8 40320 12 479001 600 16 20 922 789 888 000 355 687 428 096 000 6 402 373 705 728 000 121 645 100 408 832 000 2 432 902003 176 640000 ARITHMETICAL PROGRESSION. IOI ARITHMETICAL PROGRESSION. Arithmetical Progression is a series of numbers increasing or de- creasing by a constant number or difference ; as, i, 3, 5, 7, 12, 9, 6, 3. The numbers which form the series are designated Terms; the first and last are termed Extremes , and the others Means . When any three of following elements are given , the remaining two can he ascer- tained— x iz. , First term, Last term, Number of terms, Common Difference , and Sum of all the terms. When Last term , Number of terms, and Sum of series are given. Rule. — From quotient of twice sum of series, divided by number of terms, subtract last term. ing 1st , l last , n number of and S sum of all terms , and d common difference. Illustration. — A man travelled 390 miles in 12 days, travelling 60 miles last day. How far did he travel first day ? When First term , Common Difference , and Number of terms are given. Rule.— Multiply the number of terms less 1, by common difference, and to product add first term. Example.— A man travelled for 12 days, at the rate of 5 miles first day, 10 second, and so on ; how far did he travel the last day ? When First term , Number of terms, and Sum of series are given. Rule. — Divide twice sum of series by number of terms, and from quotient subtract first term. Illustration. — A man travelled 360 miles in 12 days, commencing with 5 miles first day ; how far did he travel last day ? When Common Difference and Extremes , or First and Last term, are qiven Rule.— Divide difference of extremes by common difference, and add 1 to quotient] Example —A man travelled 3 miles first day, 5 second, 7 third, and so on till he went 57 miles in one day; how many days had he travelled at close of last day? When Sum of series and Extremes are given. Rule.— D ivide twice sum of series by sum of first and last terms. To Compute First Term, ^ — = 65, and 65 — 60 = 5 first term. To Compute Last Term. 12-1X5 = 55 ? and 55 + 5 = 60 miles. , S , din — 1) and _ — l. n 2 360 X 2 — — — = 65, and 65 — 5 — 60 miles. To Compute Number of Terms. 57 ~ 3 2 — 27, and 27 -f- 1 = 28 days. day; how many days was he travelling? Illustration. -A man travelled 840 miles, walking 3 miles first day and <7 last' iv: how manv davs was travail, n«9 00 J 0 840 X 2 1680 3 + 57 60 -7 — = 28 days. 102 ARITHMETICAL PROGRESSION. Or, To Compute Common Difference. Extremes are of terms. I -|- a X l — ® When Number of terms and Extremes are given. Rule.— Divide difference of extremes by i less than number of terms. 2 S — 2 an n (n — i) ’ 2 S — l — a * and i nl- n (n — i) = & ILLCSTRATION.-Extremes are 3 and .5, and number of terms 7 ; what is common difference ? 15 - 3 (7 -») = and X = ' 2 com ' d if - To Compute Sum of the Series or of all Terms. When Extremes and Number of terms are given. Rule.— Multiply number of terms by half sum of extremes. l-\-aX{l — a ) , l-\~ a . Or, 2 a-\- d (n — i) X -5 n \ ^ ' 2 ’ and 2 i-(dXn-i) X -5 n = s * Illustration.— How many times does hammer of a clock strike in 12 hours? 12 X 12 + 1 = 156, and 156-7-2 = 78 times. To Compute any Number of Arithmetical Means or Terms 'between two Extremes. p _ Subtract less extreme from greater, and divide difference by 1 more than number of means or* terms required to be ascertained, and then proceed as in rule. To Compute Two Arithmetical Means or Terms between two given Extremes. er, will give means. Example 1.— Compute two arithmetical means between 4 and 16. j6 — 4-7-3 = 4 com.dif \ 4 + 4 = 8 one mean. _ 4 = 12 second mean. 2.— Compute four arithmetical means between 5 and 30. 30 — 5 = 25, and 25 -T- 4 + 1 = 5 > = c ™ l - d V } c -4- 5 = 10 = 1st mean, 10+5 = 15 = ^ “ *5 + 5 — 20 = 3d mean. 20+5 = 25 = 4^ “ ^Miscellaneous Illustrations. * Earner bavins been purchased upon following terms— viz.: $5000 upon 1st. How many months must elapse before final payment? ■ 2 d What was amount of purchase-money, or sum of series ? Here aro first and last terms— viz., 500 and 5000, and common difference , 500. Hence , To compute number of terms and amount of purchase, 5000 - 500 = 500 = 9, and 9 + 1 = 10 = number of terms or months , and 10 X 5000+500 _ IO x 2750 = $ 27 500, amount of purchase. from first stone? , . r First term 2, last term 200, and number of terms 100. Hence, joo X = 10 100 yards. GEOMETRICAL PROGRESSION. 103 _ If in the sinking of curb of a well, $3 is to bo given for first foot in depth, $5 for second, $7 for third, and increasing in like manner to a depth of 20 feet, what would it cost? First tei'm 3, common difference 2, and number of terms 20. Hence, 20 — 1 X 2 + 3=41, last term. 4 If a contractor engaged to sink a curb to depth of 20 feet for $400, and the contract was annulled when he had reached a depth of 8 feet; how much had he earned ? . 400 — 20 — number of terms. But inasmuch as 400 may be divided into 20 terms in arithmetical proportion in many different ways, according to value of 1st term, it becomes necessary to assume the value of the first foot as value of 1st term. Assuming it at $5, the required proportion will be, 1st term 5, number of terms 20, sum of ser ies 400. Then, 5 -f- x 7 = i6yg- = 1 si term -{-product of common difference and 8th term less 1, which added to 5 — 21N-, and X 4 = half number of terms for which cost is sought = 84^ dollars , sum earned. Geometrical Progression is any series of numbers continually in- creasing by a constant multiplier, or decreasing by a constant divisor, as 1, 2, 4, 8, 16, etc., and 15, 7.5, 3.75, etc. The constant multiplier or divisor is the Ratio. When any three of following elements are given , remaining two can be computed , viz. : First term, Last term, Number of Terms, Ratio , and Sum of all Terms. When Ratio, Last Term , and Number of Terms are given. Rule. — Divide last term by ratio raised to a power denoted by number of terms less 1. S sum of all terms, and r ratio. Illustration. — Last term is 4374, number of terms 8, and ratio 3; what is first term ? When First Term and Ratio are Equal. Rule.— W rite a few of leading terms of series and place their indices over them, beginning with a unit. Add together the most convenient and least number of indices to make the index to term required. Multiply terms of the series of these indices together, and product will give term required. Or, Multiply first term by ratio raised to a power, denoted by number of terms less 1. Example 1.— First term is 2, ratio 2, and number of terms 13; what is last term? Then, 5-1-5 + 3 = 13 = sum of indices, and 32 X 32 X 8 = 8192 == last term. Or, 2X2 1 = 8192. Also by inspection of table, page 105, 13th term = 8192. Then, 3 + 41 X — = 440, sum of all terms, or cost of curb. Hence, 400 — 5 X 20 X 2 600 20 X (20 — 1) 380 -- — 1 11 common difference. o 1 y GEOMETRICAL PROGRESSION. To Compute First Term Or, — — and rl — S (r — 1) = a. a representing 1st term, l last, n number of, To Compute Last Term. Indices, 1234 5 Terms, 2, 4, 8, 16, 32. 2> The price of 12 horses being 4 cents for first, 16 for second, and 64 for third, and so on; what is price of last horse? Indices, 1234 Terms, 4, 16, 64, 256. Then, 4 + 4 + 4 = 12 — sum of indices , and 256 X 256 X 256 = 256 3 — $ 167 772 10. When First Term and Ratio are Different. Rule.— W rite a few of leading terms of series and place their indices over them, beginning with a cipher. Add together the most convenient indices to make an index less by 1 than term sought. Multiply terms of these series belonging to these indices together, and take product for a dividend. Or Raise first term to a power, index of which is 1 less than number of terms multiplied; take result for a divisor; proceed with their division, and quotient will give term required. Example 1.— First term is 1, ratio 2, and number of terms 23; what is the last term? Indices, 01234 5 Terms, 1, 2, 4, 8, 16, 32. Then 5 + 5 + 5 + 5 + 2 = 22 = sum of indices , and 32 X 3 2 X 3 2 X 3 2 X 4 = 4 194 304, and 4 194 304 + the 5 th power (6 — 1) of 1 = 1 = 4 *94 3Q4- Or, 1 X 2 2 3 — 1 = 4 194 304. By inspection of table, page 105, 23d term = 4 194 3°4- 2 —If 1 cent had been put out at interest in 1630, what would it have amounted to in 1834, if it had doubled its value every 12 years? 1834 — 1630 = 204, which -4- 12 = 17, and 17 + 1 — 18 = number of terms. Indices, 01234 7 Terms, 1, 2, 4, 8, 16, 128. Then, 7 + 4 + 3 + 2 + 1 = 17, and 128 X 16 X 8 X 4 X 2 X 1 = 131 072, andi 3 io 7 2 -4- 1 , the 4th power (5 — 1) of 1 = $ 1 3 10 7 2 * When First Term , Ratio , and Sum of the series are given. Rule. — F rom sum of series subtract quotient of first term subtracted from sum of series, divided by ratio. O r ,fl X V*-i=t Example.— First term is 2, ratio 3, and sum of series 2186; what is last term? 2186 — — - = 2186 — 728 = 1458, last term. To Compute TsTnm'ber of* Terms. When Ratio First , and Last Terms are given. Rule.— D ivide logarithm of quo- tient of product of ratio and last term, divided by first term, by logarithm of ratio. Or, log. (a + S r — 1) — log- a . log. I — log, a log. r and log. I — log, (r l — - r log. (S - _ S) -a) — log. (S — l) + 1 = n. + > Example. — Ratio is 2, and first and last terms are 1 and 131072; what is num- ber of terms ? log. 2 X 131 °g = log. 262 144 = 5.418 54, and 5.418 54 log. of 2 = = lS - To Compute Sum of Series. When First Term, Ratio , and Number of Terms are given. Rule.— R aise ratio to a power index of which is equal to number of terms, from which subtract i ; then divide remainder by ratio less i, and multiply quotient by first term. GEOMETRICAL PROGRESSION". I0 5 Or, Illustration i.— First term is 2, ratio 2, and number of terms 13; what is sum of series ? % 2 13 1 = 8192 — 1 = 8191, and 8191 -4- (2 — 1) = 8191, and 8191 x 2 = 16 382. 2. — If a man were to buy 12 horses, giving 2 cents for first horse, 6 cents for second, and so on, what would they cost him ? $5314.40. To Compote Ttatio. When First Term , Last Term , and Numbers of Terms are given. Rule.— D ivide last term by first, and quotient will be equal to ratio raised to power denoted by 1 less than number of terms; then extract root of this quotient. S^ — a Illustration.— First term is 2, last term 4374, and number of terms 8; what is ratio ? 4374 8— 1. — £= 2187, and 2187 — 3, ratio. Miscellaneous Illustrations. 1. What is 9th term in geometrical progression 3, 9, 27, 81, etc.? and what is sum of terms? 1st term = 3, number of terms 9, and ratio 3. Hence, by rule to compute last term, 1st term and ratio being equal — Indices, 1234 Terms, 3, 9, 27, 81. Then, 2 -f- 3 -I-4 = 9 == sum of indices, and 9 X 27 X 81 = 19 683 = last term. By rule to compute sum of terms— 39 — 1 19682 — _ ^ X 3 — — - — = 9841 x 3 = 29 523, sum of terms. 2. First term is 1, ratio 2, and last term 131 072 ; what is sum of series? 131 072 X 2 — 1 = 262 143, and 262 143 -4- 2 — 1 = 262 143. 3. What are the proportional terms between 2 and 2048? 4 -f- 2 = 6, and 6 — 1 = 5, and 5 /2048 V— = 4 - Hence, 2 : 8 : 32 : 128 : 512 : 2048. 4. Sum of series is 6560, ratio 3, and number of terms 8; what is first term ? 3 — 1 2 - = 6560 X — 2, first term. 6560 X - 3 8 — 1 “ ' ' 6560 Geometrical Progressions, Whereby any questions of Geometrical Progression and of Double Ratio may be solved by Inspection , number of terms not exceeding 56. 16384 29 268435456 43 4398046511104 32768 30 536870912 44 8796093022208 65536 3 1 1073741824 45 17592186044416 131072 32 2147483648 46 35184372088832 262144 33 4294967296 47 70368744177664 524288 34 8589934592 48 14° 737 4 8 8 355 32B 1048576 35 17179869184 49 281474976710656 2097152 36 34 359 738 368 50 562949953421312 4 I 94 3°4 37 68719476736 51 1125899906842624 8388608 38 137 438 953 472 52 2251799813685248 16777216 39 274877906944 53 4503599627370496 33 554 432 40 549755813888 54 9007199254740992 67108864 41 1099511627776 55 18014398509481984 134217728 42 2199023255552 56 36028797018963968 Illustrations. — 12th power of 2 = 4096, and 7th root of 128 — 2. I 15 2 16 4 17 8 18 1 6 *9 32 20 64 21 128 22 256 23 512 24 1024 25 2048 26 4096 27 8192 28 io6 ALLIGATION. ALLIGATION. Alligation is a method of finding mean rate or quality of different ma- terials when mixed together. To Compute Mean. IPrice of* a Mixture. When Prices and Quantities are known. Rule. - Multiply each quantity by its ra t.692 77 2- 85433 $•02559 3.207 13 3- 399 56 3-60353 81974 04873 Vnr nnv other Kale or jreriuu . — number tor logarithm will give tabular amount as above Rate. i Time. || Per cent. 1 69.68 2 35 3 23-44 II Rate. Per cent. 4 5 6 Rate. 17.67 14.21 Per cent. 7 8 9 10.34 9.01 8.04 30 7.27 3- 8 2.64 1 - 4 Years. p er Cent. Per Cent. •5 1 1- 5 2 2- 5 3 3- 5 4 4- 5 5 5- 5 6 1. 015 1.0302 1-0457 1.0614 1.0773 1.0934 1. 1098 1.1265 1. 1434 1.1604 1. 178 1.1956 1.02 1.0404 1.0612 1.0824 1. 1041 1. 1262 1.1487 1. 1717 1. 1951 1. 219 1.2434 1.2689 5 Per Cent. of* 13 Years. 3 I 4 ! 5 1 6 Years. p er Cent. Per Cent, j Per Cent, j Per Cent. 1.025 1.0506 1.0769 1.1038 1.1314 I- 1597 1.1887 1.2184 1.2489 1.2801 1. 3121 3449 6 | Per Cent. I 1.03 1.0609 1.0927 i-i255 i-i593 1. 1941 1.2299 1.2668 1.304 8 1-3439 1.3842 [.4258 6.5 7 7-5 8 8.5 9 9-5 10 10.5 11 n-5 12 | 1. 2134 1. 2317 1.2502 1.269 1. 288 1.3073 1.3269 1.3469 1.3671 1.3876 1.4084 4295 1.2936 i. 3 i 95 1-3459 1.372 8 1.4002 1.4282 1.4568 1,486 I- 5 I 57 1 546 1-5769 .6084 1. 37 8 5 i- 4 i 3 1.4483 1.4845 1.5216 1.5597 i- 59 8 7 1.6386 1 6796 1.7216 1.7606 1.8087 1.4684 1 5102 i -558 1.6047 1.6528 1.7024 1-7535 1. 8061 1.8603 1.9161 x -9736 2.0356 ) 1 . 1950 - , Illustration.— What is amount of $500 at semi-annual interest o 5 per compounded for 10 years? e Tabular number 1.6386. Then, 500 X 1.628 89 - $ 8x4-44- 5 - To Compute Interest on any Given Sum A - */ A For a Period of Years. P (1 -f r) n = A ; = p; — W (l + ’*) n V’ v p and log- A-i°g- p = n P repr esenting principal, r rate per cent, per annum , n numberlf years] and A amount of principal and interest. DISCOUNT OR REBATE. EQUATION OF PAYMENTS. IO9 Illustration.— Assume as preceding, $500 at 5 per cent, for 10 years. 500 X 1.05 10 = 500 X 1.628 89 = 1814.44.5, amount, — 500, principal. 500 X 1.05 V For any Period. — Assume elements of preceding case, interest payable semi- J .05 , annually. 10 X 2 = 20, number of payments ; —— — 025, rate. Then, 500 X i.o25 2 °= 500 X 1-638 62 = $ 819.31. When term of payments and rate are not given in table. DISCOUNT OR REBATE. Discount or Rebate is a deduction upon money paid before it is due. To Compute lie To ate upon any Su.m. Rule.— Multiply amount by rate per cent, and by time, and divide product by sum of product of rate per cent, and time, added to 100. Example 1.— What is discount upon $ 12075 for 3 years, 5 months, and 15 days, at 6 per cent. ? 2. — What is present value of a note for $963.75, payable in 7 months, at 6 per cent. ? 6 rate. 7 months = T 7 ^- of 1 year = 6 X 7-4-12 = 3.5, and 3.54-100 = 103.5 = 1.035. 963-75 I .° 35 = $93 i * i 6. To Compute tlie Sum for a given Time and. Rate, to yield a Certain Sum. Rule.— Divide given sum by proceeds of $ 1 for given time and rate. Example.— For what sum should a note be drawn at 90 days, that when dis- counted at 6 per cent, it will net $ 200? Discount on $ 1 for 904- 3 days at 6 per cent.= $ .0155. Hence, $1 — .0155 = .9845, proceeds, and $200 -4-. 9845 = $203. 14. 9. EQUATION OF PAYMENTS. Rule. — Multiply each sum by its time of payment in days, and divide sum of products by sum of payments. Example.— A owes B $300 in 15 days, $60 in 12 days, and $350 in 20 days; when is the whole due ? Illustration. — Assume $1000 for 30 years, at 7 per cent, half-yearly. log. 4- 1 = .014 940 3, and log. .014 940 3 X 3° X 1000 = $ 2806.78. 2 3 years 5 months and 15 days = 3. 4574 years. 12 075 X 6 X 3- 4574 _ 2 50 48S. 63 _ 2Q74 ^ $ 100 4- (6 X 3-4574) 120.7444 = 2074.53 = $2074.53. 300 x 15 = 4500 60 X 12 = 720 350 X 20 = 7 000 710 ) 12220 (17 4~ days. K IIO ANNUITIES. ANNUITIES. To Compute Amount of Annuity. When Time and Ratio of Interest are Given. Rule. — Raise the ratio to a power denoted by time, from which subtract i ; divide remainder by ratio less i, and quo- tient, multiplied by annuity, will give amount. Note.— $ i added to given rate per cent, is ratio, and preceding table in Compound Interest is a table of ratios. Example.— What is amount of an annual pension of $ioo, interest 5 per cent., which has remained unpaid for four years? 1.05 ratio; then 1.054— 1 = 1.21550625 — i = . 215 506 25, and .215 506 25 = (1.05 — 1). 05 = 4.310 125, which x 100 = 1431.01.25. To Compute Present 'VVprtli of an. Annuity. When Time and Rate of Interest are Given. Rule.— Ascertain amount of it for whole time; divide by ratio, involved to time, and result will give worth. Example.— What is present worth of a pension or salary of $500, to continue 10 years at 6 per cent, compound interest? $ 500, by last rule, is worth $6590.3975, which, divided by 1.06 10 (by table, page 108, is 1. 790 84) = $ 3680.05. Or, Multiply tabular amount in following table by given annuity, and product will give present worth. Illustration l— As above; 10 years at 6 per cent. = 7. 360 08, and 7.36008 X 50° = 3.68.004 dollars. 2. What is present worth of $150 due in one year at 6 per cent, interest per annum ? •943 39 X 150 = 1141.50.85. Present ‘Worth. of an Annuity of $1, at A, £>, and 6 Per Cent. Compound Interest for Periods under So Years. Years. 4 Per Cent. 5 Per Cent. 6 Per Cent. Years. 4 Per Cent. 5 Per Cent. 6 Per Cent. 1 • 9 61 54 .95238 •943 39 *3 • 9.98562 9-393 57 8.85268 2 1.886 09 1.85941 I -833 39 14 10. 563 07 9.898 64 9. 29498 3 2-775 1 2,72325 2.673 01 15 11.11843 10.37966 9.71225 4 3.6299 3-545 95 3-465 1 16 11.651 28 10.837 78 10.105 89 5 4-452 03 4.32948 4.212 36 17 12. 166 26 11.27407 10.477 26 6 5.242 15 5-07569 4.9x732 18 12.65926 11.689 58 10.827 6 7 6.002 03 5-78637 5- 582 38 J 9 I 3 -I 33 88 12.085 32 11.158 11 8 6.731 76 6.463 21 6.209 79 20 13.59029 12.462 21 11.469 92 9 7 - 43 6 4 7. 107 82 6. 801 69 21 14.029 12 12.821 15 11.76407 10 8.11085 7.72173 7.36008 22 14.451 12 13-163 12.041 58 11 8.76044 8.306 41 7.886 87 23 14.856 82 13.48807 12.303 38 12 9-38505 8.86325 8. 383 84 24 15.24695 13.798 64 12.55035 For a Rate of Interest and Term of Years not given in either Table . '== A. Notation as preceding. (i + r)»J Illustration. — Take $ 1 at 4 per cent, for 24 years. Log. 1.04 = .017033, which X -24 = .408 799. log. .408 799 = 2.5633 = ratio raised to power of 24. Then, — X (1 ~r~) = 25X1— 39° 122 = $ I 5- 2 4- 695- .04 V 2.5633/ To Compute Yearly Amount tliat will Tiiqnidate a Delot in a Given Number of Years at Compound Interest. P r a. Illustration. — What is amount of an annual payment that ( 1 -j- r ) n — 1 will liquidate a debt of $100 in 6 years at 5 per cent, compound interest? ANNUITIES. I I I (i -f. 05) 6 per table, page i°8, 100 X .05 (f-^t -ps) 6 __ 5 X T -34 ^7 __ ^ = i-34- (1 + .05) 6 — 1 I, 34 1 *34 When Annuities do not commence till a certain period of time , they are said to be in Reversion. To Compute Present Worth of an. Annuity in Reversion. r cle Take two amounts under rate in above table — viz., that opposite sum of two given times and that of time of reversion; multiply their difference by an- nuity, and product will give present worth. Example.— What is present worth of the reversion of a lease of $40 per annum, to continue for 6 years, but not to commence until end of 2 years, at rate of 6 per cent. ? 6 -f- 2 = 8 years 6. 209 79 2 “ 1.833 39 4.37640 X 40 = 1175.05.6. Amount of Annuity of Si, etc.. Compound Interest, from 1 to SO Years. 4 Per Cent. 2.04 3.121 6 4. 246 46 5.41632 6.632 97 7.898 29 9.2x423 10. 582 79 12.006 11 5 Per Cent. 2.05 3-I52 5 4.310 12 5-525 63 6.801 91 8.14201 9. 549 1 1 11.026 56 12.577 " 6 Per Cent. 2.06 3-1836 4.374 62 5-637 09 6.97532 8-393 84 9.897 47 1 1- 49*32 13.18079 7 Per Cent. 2.07 3.2149 4- 439 94 5- 750 74 7-*53 29 8.654 02 10.259 8 11.97799 13.816 45 4 Per Cent. 13-486 35 15.025 8 16. 626 84 18.291 91 2Q. 023 59 21.82453 23.69751 25.64541 27.671 23 29.77808 5 Per Cent. 14.206 79 15-9*7*3 17.712 98 19.59863 21.57856 23-657 49 25.84037 28.132 38 30.539 33-o65 95 6 Per Cent. 14.97164 16.869 94 18.882 14 21.01507 23.275 97 25-672 53 28.212 88 30.90565 33-75999 36.785 59 7 Per Cent. 15.7836 17.88845 20. 140 64 22.55049 25.129 02 27.88805 30. 840 22 33.99903 37-378 96 40.995 49 Illustration. — What is amount of $ 1000 for 20 years at 5 per cent.? 5 per cent, for 20 years = 33.065 95 ; hence, 1000 X 33.06595 = $33.06.595. To Compute Amount of an Annuity for any Reriod and Rate. Rule. — From table for Compound Interest, page 108, take value for rate per cent, for 1 year, and raise it to a power determined by time in years, from which subtract 1, divide remainder by rate, and quotient multiplied by annuity will give amount required. Example.— What will an annuity of $ 50, payable yearly, amount to in 4 years, at 5 per cent. ? By table, page 108, 1.054 = 1.2155. 1. 2155 — 1 -4- (1.05 — i) = 4.3i > and 4.31 X 50 = 1215.50. For Half-yearly and Quarterly Payments. Multiply annuity for given time by amount in following table: Rate per cent. Half-yearly. Quarterly. Rate percent. Half-yearly. Quarterly. 3 1.007445 1.011 181 5-5 1.013 567 1.020 395 3-5 1.008 675 1. 013 031 6 1.014781 1.022 227 4 1.009 902 1. 014 877 6-5 i-oi5 993 1.024 055 4-5 1. 011 126 1. 016 729 7 1. 017 204 1.025 88 5 1. 012 348 1. 018 559 7-5 1. 018 414 1.027 704 Illustration i. — Annuity as determined in previous case = $215. 50. Hence, 215.50 X 1.012348 from above table = $218. 16 for half yearly payments. 2. A person 30 years of age has an annuity for 10 years, present worth of it being $1000, provided he may live for 10 years. What is annuity worth, assuming that 60 persons out of every 3550, between the ages of 30 and 40, die annually? 3550 — 600 (-60 X xo) = 2950 would therefore be living. And, 3550 : 2950 :: 1000 = 1830.98. 1 1 2 PERPETUITIES. — COMBINATION. PERPETUITIES. Perpetuities are such Annuities as continue forever. To Compute Value of a Perpetual Annuity. Rule.— D ivide annuity by rate per cent., and multiply quotient by unit in pre- ceding table. Example.— What is present worth of an annuity for $ ioo, payable semi-annually, at 5 per cent. ? 100-^.05 = 2, and 2 X 1.012348, from preceding table = 2.024.70. To Compute Value of a ^Perpetuity- in Reversion. Rule.— Subtract present worth of annuity for time of reversion from worth of annuity, to commence immediately. Example.— What is present worth of an estate of $50 per annum, at 5 per cent., to commence in 4 years ? 50 - 4 - .05 — 1000 $50, for 4 years, at 5 per cent. = 3. 545 95 (from table, page no) X 50= 177.2975 822.7025 which in 4 years, at 5 per cent, compound interest, would produce $1000. COMBINATION. Combination is a rule for ascertaining how often a less number of num- bers or things can be chosen varied from a greater, or how many different collections may be formed without regard to order of each collection. Combinations of any number of things signify the different collections which may be formed of their quantities, without regard to the order of their arrangement. Thus, 3 letters, a, b, c, taken all together, form but one combination, abc . Taken two and tzoo, they form 3 combinations, as ab , ac , be. Note.— Class of the combination is determined by number of elements or things to be taken ; if two are taken, the combination is of 2d class, and so on. Rule. — Multiply together natural series 1, 2, 3, etc., up to the number to be taken at a time. Take a series of as many terms, decreasing by 1, from number out of which combination is to be made, ascertain their continued product, and divide this last product by former. Example i. — How many single combinations, as ab, ac, may be made of 2 letters out of 3? lX2 = 2 = 6 = 3X2 6 2 3 ' 2. — How many combinations may be made of 7 letters out of 12? ix 2 x 3X4X5X6X7 _ 5040 and 3 99i68o _ 12 x II X IO x 9 X 8 x 7 x 6 3991 680’ 5040 3. — How many different hands of cards may be held, as at whist, combinations 13 out of 52 ? 635 013 559 600. When two Numbers or Tilings are Combined. Rule.— M ultiply together natural series 1, 2, 3, etc., to one less term than number of combinations; ascertain their continued product, and proceed as before. Example. — There are 3 cards in a box, out of which two are to be drawn in a re- quired order. How many combinations are there? Here there are 2 terms ; hence, 2 — 1 = 1, and — - — = — = 6 -4- 1 = 6. 3X2 6 To Compute 1ST umber of* "Ways in whioh any Number of Distinct Objects can be Divided among any Number. Rule.— M ultiply together numbers equal to number given, as often as objects are to be divided among them. Example.— I n how many different ways can 10 different cards be divided among 3 persons? 3X3X3X3X3X3X3X3X3X30* 3 IO = 59049. COMBINATION. — CIRCULAR MEASURE. “3 Combinations with. Repetitions. In this case the repetition of a term is considered a new combination. Thus, 1 2, admits of but one combination, if not repeated; if repeated, however, it admits of three combinations, as 1, 1 ; 1, 2; 2, 2. r ule> To number of terms of series add number of class of combination, less 1 ; multiply sum by successive decreasing terms of series, down to last term of series; then divide this product by number of permutations of the terms, denoted by class of combination. Example.— H ow many different combinations of numbers of 6 figures can be made out of 11? j! j_ (6 _ j) — — sum of number of terms, and number of class, less 1. 16 X 15 X 14 X 13 X 12 X u = 5 765 760= product of sum, and successive terms to last term. , . iX2XsX4X5X6 = 720 permutations of class of combination. ^,”£5760= 8008. 720 ‘V'ariations -with Repetitions. Every different arrangement of individual number or things, including repeti- tions, is termed a Variation. Class of Variation is denoted by number of individual things taken at a time. Rule.— R aise number denoting the individual things to a power, the exponent of which is number expressing class of variation. Example i. — How many variations with 4 repetitions can be made out of 5 fig- ures? 54 = 625. 2. — How many different combinations of 4 places of figures can be made out of the 9 digits ? . , , 12 x II X 10 X 9 II 880 9 + (4 - X) = 12, and JX2X3X4 = = 495- Coin "bin eition. without Repetitions. Rule.— F rom number of terms of series subtract number of class of combination, less 1 ; multiply this remainder by successive increasing terms of series, up to last term of series; then divide this product by number of permutations of the terms, denoted by class of combination. Example 1 — How many combinations can be made of 4 letters out of 10, exclud- ing any repetition of them in any second combination ? 10 — (4 — 1) =: 7 = number of terms — number of class, less 1. 7X8X9X10 = 5040 —prod, of remainder 7, and successive terms up to last term. iX2X3X4 = 24 = permutations of class of combination. Then, - 2io . 24 2 - — How many combinations of the 5th class, without repetitions, can be made of 12 different articles? 12 — (5 — 1) = 8, and l X 9 X 10 X 11 X 12 _ 85 040 1X2X3 X 4 x 5 • = 792 - CIRCULAR MEASURE. Unit of Circular Measure is an angle which is subtended at centre of a circle by an arc equal to radius of that circle, being equal to l8o ° 371416 — 57.296°. Circular measure of an angle is equal to a fraction which has for its numerator the arc subtended by that angle at centre of any circle, and for its denominator the radius of that circle. CIRCULAR MEASURE. — PROBABILITY. 1 14 To Compute Circular Measure of an -A-iigle. Rule. — M ultiply measure of angle in degrees by 3.1416, and divide by 180. Example. — What is circular measure of 24 0 10' 8"? 24 ° 10' 8" X 3- _ 87008 x 3.1416 _ g 180 180X60X60 -° 4 21 To Compute Measure of an -A_iigle, its Circular Measure being Given. Rule.— Multiply circular measure of angle by 180, and divide by 3.1416. PROBABILITY. Probability of any event is the ratio of the favorable cases, to all the cases which are similarly circumstanced with regard to the occurrence. If an event have 3 chances for occurring and 2 for failing, sum of chances being 5, the fraction f will represent probability of its occurring and is taken as measure of it. Thus, from a receptacle containing 1 white and 2 black balls, the probability of drawing a white ball, by abstraction of 1, is i; prob- ability of throwing ace with a die is : in other words, the odds are 2 to 1 against first, and 5 to 1 against second. If m -f n — whole number of chances, m representing number which are favorable, andn unfavorable . Therefore — ^ — =z probability of event m-\-n Probabilities of two or more single events being known, probability of their oc- curring in succession may be determined by multiplying together the probabilities of their events, considered singly. Thus, probability of one event in two is expressed by -J; of its occurring twice in succession, X or ; of thrice in succession, ^ X ^ X or -g, etc. Illustration i.— If a cent is thrown twice into the air, the probability of its fall- ing with its head up, twice in succession, is as 1 to 4. Thus, it may fall: 1. Head up twice in succession. \ 2. Head up 1st time and wreath 2d time, f „ 1 _ _ J_ _ times 3. Wreath up 1st time and head 2d time. » ’ i-f-3 ’ ^ .25 * 4. Wreath up twice in succession. ) These are the only results possible, and being all similarly circumstanced as to probability, the probability of each case is as 1 to 4. or odds are as 3 to 1. * Probability of either head or wreath being up twice in succession is as 1 to 1, or chances are even, because 1st and 4th cases favor such a result; probability of head once and wreath once in any order is as 1 to 2, because 2d and 3d cases favor such a result; and probability of head or wreath once is as 3 to 4, or odds are as 3 to 1, be- cause 1st, 2d, and 3d, or 2d, 3d, and 4th cases favor such a result. Note. — 1 to 2 is an equal chance, for i out of 2 chances = 1 to 1, being an equal chance ; again, 1 to 5 is 4 to 1, for 1 out of 5 chances is 1 to 4. 2.— If there are 4 white balls and 6 black in a bag, what is the chance of a person drawing out 2 black at two successive trials? This is a combination without repetition. Hence, 6 — (2 — 1) = 5, and 5 X 6 _ : = 11 which x 2 for successive trials = — or — . 1 X 2 2 1 7 ' 2 15 3. — Suppose with two bags, one containing 5 white balls and 2 black, and the other 7 white and 3 black. Number of cases possible in one drawing from each bag is (5 + 2) X (7 + 3) =7 X 10 = 70, because every ball in one bag may be drawn alike to one in the other. PROBABILITY. 115 Number of cases which favor drawing of a white ball from both bags is 5 X 7 = 35i for every one of the 5 white balls in one bag may be drawn in combination with every one of the 7 in the other. For a like cause, number of cases which favor drawing of a white ball from 1st bag and a black one from 2d is 5 X 3 = !5 ; a black ball from 1st bag and a white ball from 2d is 7 X 2 = 14; and a black ball from both is 3 X 2 = 6. Probability, therefore, of drawing is as ;L*LZ. — — — — i to 1, a white ball from both bags. - ^ - = — = — = 3 to u, 7 o 70 2 7° 7° I 4 a white ball from 1st , and a black from 2 d. ^ = y = 1 to 4, a black ball from 1st, and a lohite from 2 d. “ = 3 to v 32, a black ball from e V 1-4-5 Y 7 20 both. L 5X3 + 2 X7 _. 2 9_^ to 4Ij a w hite ball from one , arid a black from other , 70 70 - = — = 10 to 30, a white ball 49 49 , 1 , 3 2 9 5X7 + 5X3+ 2 X7 for both 2d and 3 d cases favor this result ; hence, — + — = — = — = — = 32 to 3, at least one white ball , for the 1st, 2d, and 3d cases favor this 70 35 result ; hence, + + — + — = —• ’ 2 14 5 35 Again, if number of white and black balls in each bag are same, say 5 white and 2 black, 5 + 2 X 5 + 2 = 49, then probability of drawing is as 5X5 — ?*> = 25 to 24, a white ball from both. 5 X - - 49 49 from 1 st and a black from 2d. — — = 10 to 39, a black ball from 1 st, and a J 49 49 white from 2d. - + = 4 to 45, a black ball from both. 49 49 4 when two dice are thrown, probability that sum of numbers on upper sides is any given number, say 7, is as follows : As every one of the six numbers on one die may come up alike to, or in combi- nation with the other, number of throws is 6 X 6 = 36. ! i and 6 1 2 “ 5> ; and as these numbers may be upon either die, there are 3 x 2 = 6 throws in favor of the combination of 7 ; hence . 6 1 probability of throwing 7 is — = -g-, or as 1 to 5. 5 . —Probability of a player’s partner at Whist holding a given card is as follows: Number of cards held by the other 3 players is 3 X 13 = 39 i probability, there- fore, that it is held by partner is +, but it may be one of the 13 cards which he holds; hence probability is — X 13 = — = — , or as 1 to 2 - ’ 39 39 3 6. —Probability of a player’s partner at Whist holding two given cards is as follows: 30 X 38 Number of combinations of 39 things, taken 2 and 2 together, is — = 741 ; therefore, probability that these 2 cards are in partner’s hand is 39 x 3 8 — 1 X 2 39 X 19 = _i_ = x to 740 ; but they may be any 2 cards in partner’s hand; therefore, since 741 ^ ^ number of combinations of 13 cards, taken 2 and 2 together, is = -— == 78, 78 2 probability required is ~ = — , or as 2 to 17. 1X2 Similarly, probability that he holds any 3 given cards is as — , or as 22 to 681. Probabilities at a game of Whist upon following points are: 9 to '7, that one hand has tiuo honors , and two hands one ; 9 to 55, that two hands have each two honors ; 3 to 29, that each hand holds an honor ; 3 to 13, that one hand has three honors , and one hand one ; 1 to 63, that four honors are held by one hand. 7 . if 3 half-dollars are thrown into the air, probability of any of the possible com* binations of their falling is determined as follows : Hence, ^= - 12 5 = 1 to 7 in favor of 3 heads. “ 2 heads and 1 tail. “ 1 head and 2 tails. “ 3 tails. And in like manner, if 5 were thrown up, probability of any of their possible combinations would be determined as follows : / I , i\5 /i\S, 5 /iN 5 , 5 X 4 / 1 \5 , 5 X 4 X 3 /i\ 5 . 5X4X 3 X2/i\S 17 + T/ = (+ + Tw + U j + o< 7^3 W + *x 2 x 3 x 4 W , 5X4X3X2X1 /_^\5 t lX2X3X4XS \2/ Hence, ^ 5 = .03125 = 1 to 31 in favor of 5 heads; Y (“) 5 = -15625 = 5 to 27 “ “ 4 heads and 1 tail ; f$i(v) = - 3125 = 10 1022 “ “ 3 heads and 2 tails 5 X 4 X 3 / 1 \ 5 u I — )=.3i2 5 = 10 to 22 “ 1X2X3V2/ J 3 “ 2 heads and 3 tails , SX 4 X 3 X 2 /i \5 . xX 2 X 3 X 4 (-) =-15 S ^' 7 “ 1 head and 4 tails ; 5X4X3X2Xi/iy = i. IX2X3X4X5W J 5 0 “ 5 tails. All Wagers are founded upon the principle of product of the event, and contingent gain, being equal to amount at stake. Illustration i. — S uppose 3 horses, A, B, and C, are entered for a race, and X wagers 12 to 5 against A, n to 6 against B, and 10 to 7 against C. If A wins, X wins 6 + 7 — 12 = 1. “ B “ X “ 5 + 7 — 11 = 1. “C “ X “ 5 + 6 — 10=1. Hence, X wins 1, whichever horse wins, from having taken field against each horse at odds named. ~ . . . . ( A are 5 to 12 ) ( T V in favor of A > Odds given in fa- 1 _ .. *; LL r ; corresponding probabil- 1 V u vorof *JB “ 6 “ u l ity te B, 7 10 / l IT and — + — + — = — = 1.06 = 1.06 to 1 in favor of taker of odds. 17 17 17 17 PROBABILITY. II 7 2. —Odds given upon first seven favorite horses for Oaks Stakes of 1828 were so great, that probability in favor of taker of the odds when reduced was as follows : 1st, 5 to 2 : 2d, 5 to 2 ; 3d, 4 to 1 ; 4th, 7 to 1 ; 5th, 14 to 1 ; 6th, 14 to 1 ; 7th, 15 to 1 ( 4 X 3 X 16 = 192 5. , 3 16 73 ’ I X 7 X 16 = 112 3 X 7 X 3 — 6 3 7 x 3 x 16 336 — 267 -r- 336 = 1.092 = 1.092 to 1, in favor of. taker of odds, yet neither of the horses upon which these odds were given won. 3 ._If odds are 3 to 1 against a horse in a race, and 6 to 1 against another horse in a second race, probability of 1st horse winning is J, and of other i Therefore probability of both races being won is ^g-, and odds against it 27 to i,or 1000 to 37.037. Odds upon such an event were given in 1828 at 1000 to 60, or 16.67 to 1. 4 —Two persons play for a certain stake, to be won by winner of three games or results. One having won one and the other two, they decide to divide the sum, proportionate to their interest. How much of it should each one receive? Operation.— If winner of two games should win game to be played, he would be entitled to the whole sum ; if he lost, he would be entitled to half of it. Now as - = — , half of which : , or share one event is as probable as the other, -- -f- - of winner of two games. When events are wholly independent, so that occurrence of one does not affect that of the other, probability that both will occur is product of proba- bilities that each will occur. Note.— It is indifferent whether events are to occur together or consecutively. Illustration i. — Assume three boxes, each containing white and black balls as 6 white, 5 black; 7 white, 2 black; 8 white, 10 black. What is chance of drawing from them a white, black, and a white ball? Probabilities are — , — , and product of which = 6 ~^ 2 ~^~ 8 11 9 18 297 : 17.625 tO I. 2. — A gives an answer correctly 3 times out of 4, B 4 times out of 5, and C 6 out of 7. What is probability of an event which A and B declare correct and C denies? Operation. — Compound probability that A and B answer correctly and C denies (all 3 of which are in favor of event) is X — X — = — = 4 5 7 I 4° 35 Compound probability that A and B deny and C is correct (all 3 of which are 6 _ 3 140 70’ Then correct, divided by sum 3 / 3 , 3\ _ .8714 _ 355 986 282 303 ,223 89I .177 548 .140 796 .111 66 .088 548 .070 221 .055 685 .044 164 .035 026 .027 772 .022 026 .017 468 .013 851 .OIO 989 .008 712 .006 909 .005 478 .OO4344 .OO3445 .002 734 .002 167 .OOI 719 .OOI 363 .OOI o8l .OOO 857 I .OOO 679 7 .OOO 539 I .OOO 427 5 .OOO 338 9 .000 268 8 .000 213 2 .000 169 1 .000 134 1 .000 106 3 .000 084 45 .000066 87 ,000 053 04 .ooo 042 05 Lbs. .640513 .507 946 .402 83 • 3*9 45 1 .253 342 .200911 •159 323 .126 353 .100 2 .079 462 .063 013 .049 976 .039 636 .031 426 .024 924 .019 766 .015 674 .012 435 .009 859 .007 819 .006 199 i .004916 ! .003 899 ! .003 094 1 .002 452 ! .001 945 ! .OOI 542 .001 223 .000 969 9 .000 769 2 .000 609 9 .000 483 7 .000 383 5 .000 304 2 .000 241 3 .000 191 3 .000 151 7 .000 120 4 .000 095 6 .000 075 7 .00006003 .000 047 58 Brass. j .000 033 36 .000 037 75 ! .000 026 44 I .000 029 92 Lbs. .605 176 .479 908 .380 666 .301 816 .239 353 189 818 150 522 .119376 .094 666 .075 075 .059 545 .047 219 .037 437 .029 687 .023 549 .018 676 .014 809 .011 746 .009 315 .007 587 .005 857 .004 645 .003 684 .002 92 .002 317 .001 838 .001 457 .001 155 .0009163 .000 726 7 .0005763 .000457 .000 362 4 .000 287 4 .000 228 .000 180 8 .000 143 4 .000 113 7 .000090 15 .000 071 5 .000056 71 j .00004496 .000 035 66 I .000 028 27 Specific Gravities 7-774 Weights of a Cube Foot . . 485.87 “ “ Inch . . .2812 7.847 490.45 .2838 8.88 | 8.386 554.988 | 524 - 16 .3212 | .3033 Specific Gravities to determine the computations of these weights were made by author for Messrs. J. R. Browne & Sharpe, Providence, R. I. 121 WEIGHTS OF IKON, STEEL, COPPER, ETC. 'W' rought Iron, Steel, Copper, and Brass Wire. Birmingham Wire Gauge, f. full, 1. light. No. of Gauge Thickness. I Iron. Per Lini | Steel. sal Foot. Copper. Brass. Inch. Lbs. Lbs. Lbs. Lbs. oooo •454 or xg- £ .546 207 •SSI 36 .623 913 .589 286 ooo . 4 2 5 „ .478 656 .483 172 •546 752 .516407 oo .38 orf-f. .382 66 .386 27 •437 099 .412 84 o •34 or if. •30634 .309 23 •349 921 •330 5 i *3 •2385 .240 75 .272 43 •257 31 2 .284 .213 738 •215 755 .244 146 .230 596 3 .259 or if. .177 765 .179 442 .203 054 .191 785 4 .238 .150 107 •151 523 .171 461 .l6l 945 5 1 .22 .128 26 .129 47 .146507 •138 376 6 .203 or J f. , .109 204 .110 234 .124 74 .117 817 7 .18 or T \ 1. .085 86 .086 667 .098 075 .092 632 8 .165 or i 1. .072 146 .072 827 .082 41 .077 836 9 .148 or 1 f. .058 046 •058 593 .066 303 .062 624 IO •i 34 .047 583 .048 032 •054 353 .051 336 ii ! .12 or 1. .038 16 .038 52 •043 589 .041 17 12 ! .109 .031 485 .031 782 .035 964 .033 968 13 [ -095 or ^ 1. .023 916 .024 142 .027 319 .025 802 . 14 ! .083 .018 256 .018 428 .020 853 .019 696 15 .072 .013 728 .013 867 .015 692 .014 821 16 i -065 .011 196 .01 1 ^302 .012 789 .012 O79 17 ! .058 .008 915 .008 999 .010 183 .OO9 6l8 18 ! •°49 or ^ 1. .006363 .006 423 .007 268 .006 864 19 .042 .004 675 .004 719 •00534 •005 O43 20 •035 .003 246 .003 277 .003 708 .003 502 21 i .032 .002 714 .002 739 .003 1 .002 928 22 .028 .002 078 .002 097 .002 373 .002 24I 2 3 •025 or ,L. .001 656 .001 672 .001 892 .OOI 787 24 .022 .001 283 .001 295 .001 465 .OOI 384 25 .02 or - 6 V .001 06 .001 070 .001 211 .OOI I44 26 .Ol8 .000 858 6 .000 866 7 .000 980 7 •OOO 926 3 27 • Ol6 .000 678 4 .000 684 8 .000 774 9 .OOO 731 9 28 .OI4 .0005194 .000 524 3 •000 593 3 .OOO 560 4 29 .013 .000 447 9 .000 452 1 .0005116 •OOO 483 2 30 .012 .000 381 6 .000 385 2 •0004359 | .OOO4II 7 31 • OI or TOT .000 265 .000 267 5 .000 302 7 .OOO 285 9 32 .009 .000 214 7 .000 216 7 .000 245 2 •ooo 231 6 33 •008 .000 169 6 .000 171 2 •000 193 7 .OOO 183 34 .OO? .000 129 9 .000 131 1 I . .000 148 3 •ooo 140 I 35 •°°5 or a-J o .000 066 25 i .000 066 88 .000 075 68 •ooo 071 48 36 .004 or ttIs .000 042 4 .000 042 8 .000 048 43 .OOO O45 74 Thickness of Plates. No. Inch. No. Inch. No. Inch. No. Inch. 1 •3125 9 •156 25 17 •056 25 25 .023 44 2 .281 25 10 .140625 18 •05 26 .021 875 3 .25 11 .125 !9 •043 75 27 .020312 4 • 2 34 375 12 .112 5 20 •037 5 28 .018 75 5 a .218 75 13 .1 21 •034 375 29 .017 19 0 .203 125 14 .0875 22 •031 25 30 •015 625 7 8 •!87 5 .171 875 15 16 •075 .062 5 23 24 r .028 125 .025 3 1 32 .014 06 .012 5 122 WIRE GAUGES, WIRE GAUGES. (English.) Warrington ( Hylands Brothers). No. Inch. No. | Inch. No. Inch. No. • Inch. | No. Inch. 7/0 6/0 5 /° 4/0 3/ ° 2/0 No. 1 % % 11/ /as Inch. 0 1 2 3 4 5 No. .326 •3 .274 •25 .229 .205 Sir Jose Inch. 6 7 8 9 10 10.5 ph W) No. .191 .174 •159 .146 •133 .125. litworth * Inch. 11 12 13 14 15 16 tfc Co. No. .117 .1 .09 .079 .069 .0625 ,’s. Inch. 17 18 19 20 21 22 No. | •053 .047 .041 .036 •0315 .028 Inch. 1 .001 14 .0.14 34 •034 85 .085 240 .24 2 .002 15 .015 36 .036 90 .09 260 .26 3 .003 16 .016 38 .038 95 .09 280 .28 4 .004 17 .017 40 .04 100 .1 300 •3 5 .005 18 .018 45 •045 no .11 325 •325 6 .006 19 .019 50 •05 120 .12 350 •35 7 .007 2Q .02 55 •055 135 •135 375 •375 8 .008 22 .022 60 .06 150 .15 400 •4 9 .009 24 .024 65 .065 165 .165 425 •425 10 .01 26 ,026 70 .07 180 .18 450 •45 11 .011 28 .028 *75 •075 200 .2 475 •475 12 .012 3<=> •03 80 .08 220 .22 500 •5 13 .013 32 .032 1 1 Sir Joseph Whitworth, in 1857, introduced a Standard ire-uauge, rang- ing from half an inch to a thousandth, and comprising 62 measurements. It commences with least thickness, and increases by thousandths or an inch up to half an inch. Smallest thickness, ^ of an incl b i % No ‘ 1 I ’ ® 0 ' 2 is 2 and so on, increasing up to No. 20 by intervals of two ? No. 20 to No. 40 by yow*’ and from No * to No *. 100 by ,T^o?' +1 ihe thicknesses are designated or marked by their respective numbers m thou- sandths of an inch. . This gauge is entering into general use m England. of Grreat Britain, UNTew Standard. Wire Gauge 1884,. 7/0 6/0 5/° 4/0 3/° 2/0 Inch. J No. Inch. |j No. Inch. j No. • S 8 .160 I 22 .028 36 .464 9 .144 23 .024 37 •432 10 .128 24 .022 38 . 4 . 11 .Il6 25 .02 39 .372 12 • io 4 26 ,Ol8 40 •348 13 .092 27 .0164 41 •324 | 14 .08 28 .OI48 42 • 3 I 15 .072 29 .OI36 43 .276 16 .064 1 30 .0124 44 .252 17 .056 31 .OIl6 45 .232 ' l 18 .048 32 . .0108 46 .212 1 19 .04 1 33 .OI 47 .192 20 .036 i 34 .0092 48 .176 1 21 .032 II 35 .0084 49 No. 50. .001 inch. Inch. .OO76 .0068 .006 .0052 .OO48 .OO44 .OO4 •OQ36 .OO32 .0028 .0024 .002 .OOl6 .0012 WIRE GAUGES. GAS PIPES AND WIRE COED. 1 23 French. ( Jauges de Fils de Fer). French wire-gauges, alike to the English, have been subjected to variation.- Following table° contains diameters of the numbers of the Limoges gauge. Wire-Gauge (Jauge de Limoges). Number. Millimetre. Inch. I O •39 .OI54 I •45 .OI77 2 •56 .0221 3 .67 .0264 4 •79 .0311 5 •9 •0354 6 I.OI .0398 7 1. 12 .O44I 8 1.24 .O488 Number. Millimetre. Inch. Number. ' Millimetre. Inch. 9 i -35 •0532 18 3-4 .134 10 1.46 •0575 19 3-95 .156 11 1.68 .0661 20 4-5 .177 12 1.8 .0706 21 5 -i .201 13 1.91 .0752 22 5-65 .222 !4 2.02 •0795 23 6.2 .244 15 2.14 .0843 24 6.8 .268 16 2.25 .0886 1 7 2.84 .112 For GJ-alvanizeid Iron Wire. Number. Millimetre. Inch. I Number. Millimetre. Inch. Number. Millimetre. | Inch. I .6 .0236 9 1,4 .0551 17 3 - .Il8 2 •7 .0276 10 i -5 .O59I 18 3-4 •134 3 .8 •0315 11 1.6 .063 19 39 •154 4 •9 •0354 12 1.8 .O709 20 4.4 •173 5 1. •0394 13 2. .0787 21 4.9 •193 6 1. 1 •0433 14 2.2 .0866 22 5-4 .213 7 1.2 •0473 15 2.4 •0945 23 5-9 .232 8 i -3 .0512 16 2.7 .106 For "Wire and. Bars. Mark. Millimetre. Mark.| Millimetre. P 5 7 12 I 6 8 13 2 7 9 14 3 8 10 i 5 4 9 11 16 5 10 12 18 6 11 [ Mark. Millimetre. Mark. Millimetre. Mark. Millimetre. 13 20 19 39 25 70 14 22 20 44 26 76 15 24 21 49 27 82 l6 27 22 54 28 88 17 30 23 59 29 94 18 34 24 64 30 100 Thickness of Gras Bipes. Diameter. Thickness. II Diameter. Thickness. I Diameter. Thickness. 1.5 t0 3 • 2 5 8 to 10 •5 14 to 15 •75 4 “6 •375 II 12 “ 13 .625 1 16 “ 48 .875 Copper W r ire Cord. Circumference and Safe Load. Inch. Inch. Inch. Inch. Inch. Inch. Iris. Ins. Circumference 25 .375 .5 .625 .75 1 1.125 1.25 Safe load in Lbs 34 50 75 112 168 224 336 448 Zinc— sheets. Thickness and "Weight per Square IToot. Inch. I Inch. | Inch. .0311 = IO OZ. *0534 = 14 oz. .0686 = 18 OZ. .O457 = 12 OZ. I .o6lI = l6 OZ. I 10761 = 20 OZ. 124 WEIGHT AND STRENGTH OF WIRE, IRON, ETC. WEIGHT AND STRENGTH OF WIRE, IRON, ETC. "Weight and. Strength, of "Warrington Iron Wire. Manufactured by Rylands Brothers. (England.) Weight per ioo Lineal Feet. No. Diame- ter. Weight Breaking An- nealed. Weight. Bright. No. Diameter. Weight. Breaking An- nealed. Weight. Bright. Gauge. Inch. Lbs. Lbs. Lbs. Gauge. Inch. Lbs. Lbs. Lbs. 7/0 X 64.46 3490 5233 9 .146 5*5 298 447 6/0 % 56.66 3066 4603 10 •133 4*43 247 370 5/0 % 49-36 2673 4OOO 10.5 .125 4*03 2l8 327 4/0 % 4253 2303 3457 11 .117 3*53 I 9 I 288 3/0 % 36.26 1963 2945 12 .1 2.66 145 217 2/0 . X 3O.46 1653 2473 13 .09 2.1 113 169 O .326 27.36 i486 2226 14 .079 1.6 87 130 I •3 2 3-3 1257 1885 i 5 .069 1.23 66 99 2 .274 19.36 IO46 1572 16 .0625 .96 53 77 3 •25 16.13 873 1309 17 •053 •73 39 59 4 .229 13*53 732 1098 18 .047 •56 3 * 46 5 .209 11.26 6lO 9 i 3 19 .041 •43 23 35 6 .191 9.4 509 763 20 .036 •33 18 27 7 • I 74 7.8 422 633 21 •031 25 .26 14 21 8 •159 6-53 353 5 i 9 22 .028 .2 11 16 To Compute Length of IOO Pounds of Wire of a (Given Diameter. Rule. — Divide following numbers by square of diameter, in parts of an inch, and quotient is length in feet. 37.68 for wrought iron. I 33.42 for copper. I 28 for silver. 37.45 for steel. | 34-41 for brass. | 15.3 for gold. 13.64 for platinum. Window Glass. Thickness and "Weight per Square Foot. No. Thickness. Weight. No. Thickness. Weight: No. Thickness. Inch. Oz. Inch. Oz. 26 Inch. 12 •059 12 17 .083 17 .125 G .063 13 .091 19 •154 15 .071 15 21 . 1 21 36 .167 16 .077 16 24 .III 24 42 .2 W eight. Oz. 26 32 36 42 Terne IPlates. Terne Plates — Are of iron covered with an amalgam of lead. Thickness and Weigh, t of Galvanized Slieet Iron. Sheet 2 Feet in Width by from 6 to 9 Feet in Length (M. Lejferts). £6 No. 29 28 27 Weight per Sq. Foot. Wire Gauge. Weight per Sq. Foot. Wire Gauge. Weight per Sq Foot If Weight per Sq. Foot. Wire Gauge. Weight per Sq. Foot. Wire Gauge. Weight per Sq. Foot. Oz. No. Oz. No. Oz. No. Oz. No. Oz. No. Oz. 12 26 15 2 3 20 20 27 17 36 14 53 13 25 l6 22 22 19 30 16 42 13 6l 14 24 l8 21 24 18 35 15 46 12 70 WEIGHTS OF METALS. 125 Wrought Iron. "Weight of Square Rolled. Iron, From .125 Inch to 10 Inches . one foot in length. Side. Weight. Side. Weight. Side. , | Weight. Side. Weight. Tn. Lbs. Ins. Lbs. Ins. Lbs. Ins. Lbs. .125 •053 2.125 I 5-263 4* I2 5 57 - 5 W 6.25 I32.O4 .25 .211 •25 I 7 .II 2 .25 61.055 ‘5 142.816 •375 •475 •375 19.066 •375 64.7 •75 154.012 •845 .5 21.12 •5 68.448 7 165.632 .625 1.32 .625 23.292 .625 72 305 •25 177.672 •75 1.901 •75 25-56 •75 76.264 >5 190.136 .875 2.588 .875 27-939 .875 80.333 •75 203.024 1 3-38 3 30.416 5 84.48 8 216.336 .125 4.278 .125 33 -oi .125 88.784 •25 230.068 .25 5.28 •75 35-704 •25 93.168 •5 244.22 •375 6-39 •375 38.503 •375 97-657 •75 258.8 .5 7.604 •5 41.408 •5 IO2.24 9 273.792 .625 8.926 •625 44.418 .625 106.953 •25 289.22 •75 10352 •75 47-534 •75 111.756 •5 305 056 •875 11.883 .875 50.756 .875 116.671 -75 321-33 2 13-52 4 54.084 6 121.664 10. 327.92 Illustration.— What is weight of a bar 1.5 inches, by 12 inches in length? In column 1st. find 1.5; opposite to it is 7.604 lbs., which is 7 lbs. and .604 of a lb. If lesser denomination of ounces is required, result is obtained as follows: Multiply remainder by 16, point off the decimals, and the figures remain- ing on left of the point will give number of ounces. Thus, .604 of a lb. = .604 X 16 = .9.664 = 7 lbs. 9.664 ounces. To Compute Weight for less than a Foot in Length. Operation.— What is weight of a bar 6.25 inches square and 10.5 inches long? In column 7th, opposite to 6.25 is 132.04, which is weight for a foot in length. 6.25 X 12 inches = 132.04 6 ins.=.5 =66.02 3 “ =.25 =33 -oi 1.5“ = .125 = 16.505 *i5-535 "WeigHt of -A.ngle Iron, From 1.25 to 4.5 Inches, one foot in length. Thickness measured in Middle of each Side. L Equj Sides. il Sides Thick- ness. i . Weight L Unec Sides. iUAL Sll Thick- ness. DES. Weight. L Unequ Sides. al Side Thick- :s. Weight. Ins. Inch. Lbs. Ins. Inch. Lbs. Ins. Inch. Lbs. I.25XI.25 •1875 i-5 3 X2.5 •375 6.25 6 X3-5 .625 18 1-5 XI.5 •1875 2 3-5x3 •4375 7-75 6 X4-5 .625 20 1-75X1.75 •25 3 3-5X3 •4375 9.6 y 2 X 2 •25 3-5 4 X3 •5 11 2.25X2.25 •3125 4-5 4 X3-5 •5 II-5 2 X 2.375* •375 5-5 2.5 X2.5 •3125 5 4 X3-5 •5 n-75 2.5X2.875 •375 6-5 3 X 3 •375 7 4-5X3 •5 n-75 3-5 X 3-5 •4375 10.5 3-5 X3.5 •4375 9 5 X3 •5 12.65 4 X3.5 1 •4375 I 3 4 X 4 •5 12.5 5 X 3 •5625 13-7 •75 4-5 X4.5 •5 14 5-5 X 3-5 •5 14-5 4 X3.5 •75 I 3-5 4-5 X4-5 •5625 16 5-5 X 3-5 •5625 15.6 * Tbi» column gives depth of web added to the thickness of base or flange. L* 126 WEIGHTS OF METALS. "Weight of Round Rolled. Iron, From .125 Inch to 12 Inches in Diameter . one foot in length. Diameter. Weight. Diameter. Weight. Diameter. Weight. Diameter. | Weight. Lbs. Ins. Lbs. Ins. Lbs. Ins. Lbs. .125 .041 2 IO.616 4-375 50.815 7-5 149.328 •25 .165 .125 II.988 •5 53-76 •75 l' 59-456 .3125 •259 •25 13-44 .625 56.788 8 169.856 •375 •373 •375 14-975 •75 59-9 •25 180.696 •4375 .508 •5 16.588 5 66.35 •5 191.808 •5 .663 .625 18.293 .125 69.731 •75 203.26 •5625 .84 •75 20.076 •25 73 -I 72 9 215.04 .625 1.043 .875 21.944 •375 76.7 •25 227.152 .6875 1.254 3 23.888 •5 80.304 •5 239.6 •75 1-493 .125 25.926 .625 84.001 •75 252.376 .875 2.032 •25 28.04 •75 87.776 10 265.4 1 2.654 ♦375 30.24 6 95-552 •25 278.924 .125 3-359 •5 3 2 - 5 12 •25 103.704 •5 292.688 •25 4- *47 .625 34.886 •375 107.86 •75 306.8 •375 5.019 •75 37-332 •5 112.16 11 321.216 •5 5-972 .875 39.864 .625 116.484 •25 336.OO4 .625 7.01 4 42.464 •75 120.96 •5 351-104 •75 8.128 .125 45-174 7 130.048 •75 366.536 .875 9-333 •25 47-952 •25 139-544 12 382.208 Weight of Iflat Rolled Iron, From .5X.125 Inch to 5.5 X 4-5 Inches* one foot in length. Thickness. Weight. Thickness. Weight. Thickness, Weight. Thickness. Weight. Inch. Lbs. Ins. Lbs. Ins. Lbs. Ins. Lbs. .5 .875 1.25 1.5 .125 .211 •75 2.217 •5 2.112 •75 3.802 •25 .422 .875 2.583 .625 2.64 .875 4-435 •375 •634 •75 3.168 1 5.069 •5 •845 1 -875 3.696 1.125 5-703 .125 .422 1 4.224 1.25 6.337 .625 •25 .845 1. 125 4.752 1-375 6-97 .125 •25 .264 .528 -375 •5 1.267 1.69 1.375 1.625 •375 •792 .625 2.1 12 .125 .58 .125 .686 •5 1.056 •75 2-534 • 2 5 1.161 •25 1.372 .625 1.32 .875 2.956 •375 1.742 -375 2.059 .75 1.125 •5 .625 2.325 2.904 •5 .625 2.746 3-432 .125 • 3 l6 .125 •475 •75 3-484 •75 4.H9 •25 •633 •25 •95 .875 4-065 •875 4.805 •375 •95 •375 1.425 1 4.646 1 5-492 •5 1.265 •5 1. 901 1. 125 5.227 1. 125 6.178 .625 1.584 .625 2-375 1.25 5.808 i -5 6.864 •75 1.9 •75 2.85 i -375 6.389 1-375 7-551 .125 .875 •369 .875 1 3-326 3.802 1.25 .125 1.5 •633 i -5 8.237 1.75 •25 •738 •25 1.266 .125 •739 •375 1. 108 .125 .528 -375 1.9 •25 1.479 •5 i *477 •25 1.056 •5 2-535 •375 2.218 .625 1.846 •375 1.584 .625 3-!68 •5 2-957 * For weights of square bars 6ee preceding page. WEIGHTS OF METALS. 127 Thickness. Weight. Thickness. Weight. Thickness. Weight. Thickness. Weight. Ins. Lbs. Ins. Lbs. Ins. Lbs. Ins. Lbs. 1.75 1.125 2.5 2.875 .625 3.696 1-5 10.772 1.25 10.56 .125 1.2 1 5 •75 4-435 I.625 11.67 1-375 11.616 •25 2.429 .875 5.178 1-75 12.567 1-5 12.672 •375 3.644 1 5 - 9 i 4 1.875 13465 I.625 13.728 •5 4.858 1. 125 6.653 2 14.362 1-75 14.784 .625 6.072 1.25 7-393 O 05 1.875 15.84 -75 7.287 1-375 8.132 tC.tio 2 16.896 •875 8.502 i -5 8.871 .125 •95 2.125 17-952 1 9.716 1.625 9.61 -25 1.9 2.25 19.008 1.125 10.931 1.875 •375 2.851 2-375 20.064 1.25 12.145 •5 3.802 9 f ?95 1.375 13.36 .125 .792 .625 4.752 i -5 14-574 • 2 5 1.584 •75 5.703 .125 1. 109 1.625 15.789 -375 2.376 .875 6.653 . 25 . 2.218 1-75 17.003 •5 3- i 68 1 7.604 •375 3.327 1.875 18.218 .625 3.96 1. 125 8.554 •5 4.436 2 19.432 •75 4-752 1.25 9-505 .625 5.545 2.125 20.647 .875 5-544 i -375 10.455 •75 6.654 2.25 21.861 1 6.336 i -5 11.406 .875 7.763 2.375 23.076 1. 125 7.129 1.625 12.356 I 8.872 2.5 24.29 1.25 7.921 i -75 13-307 1. 125 9.981 2.625 25.505 1-375 8-713 1-875 14.257 1.25 11.09 2.75 26.719 i -5 9-505 2 15.208 1-375 12.199 3 1.625 10.297 2.125 16.158 1-5 13.308 1 *75 11.089 9 875 1.625 14.417 .125 1.267 0 i -75 15.526 .25 2-535 & .125 I.003 1.875 16.635 -375 3.802 .125 ■845 •25 2.006 2 17.744 •5 5.069 .25 I.689 •375 3-009 2.125 18.853 •625 6.337 •375 2-534 •5 4.013 2.25 19.962 •75 7.604 •5 3-379 .625 59 i 6 2.375 21.071 .875 8.871 .625 4.224 -75 6.019 2-5 22.18 1 10.138 •75 5.069 .875 7.022 1. 125 11.406 •875 5.914 1 8.025 A. itJ 1.25 12.673 1 6.758 1. 125 9.028 .125 I.l 62 1-375 13.94 1-125 7.604 1.25 10.032 •25 2.323 i .5 15.208 1.25 8.448 i -375 11-035 •375 3485 1.625 16.475 1-375 9.294 i -5 12.038 •5 4.647 i -75 17.742 i -5 IO.I38 1.625 13.042 .625 5.808 1.875 19.01 1.625 IO.983 i -75 14.045 •75 6.97 2 20.277 i -75 II.828 1.875 15.048 .875 8.132 2.25 22.811 1-875 12.673 2 16.051 1 9.294 2-5 25.346 2.125 2.125 . 17.054 1. 125 10.455 2.75 27.881 2.25 18.057 1.25 II.617 q nc .125 .898 2.5 1-375 12.779 0.4.0 •25 1-795 i -5 13-94 .125 1-373 •375 2.693 .125 1.056 1.625 15.102 .25 2.746 •5 3 - 59 i •25 2.112 i -75 16.264 •375 4 -II 9 .625 4.488 -375 3.168 1.875 17425 •5 5.492 •75 5.386 -5 4.224 2 18.587 .625 6.865 •875 6.283 .625 5.28 2.125 19.749 •75 8.237 1 7.181 -75 6.336 2.25 20.91 .875 9.61 1-125 8.079 .875 7-392 2.375 22 O72 1 10.983 1.25 8.977 1 8.448 2-5 23.234 1.125 12.356 I -375 9.874 1. 125 9504 2.625 24-395 ; 1.25 13-73 128 WEIGHTS OF METALS. Thickness. | Weight. | Thickness. Weight. Thickness. Weight. 1 Thickness. Weight. Ins. Lbs. Ins. Lbs. Ins. Lbs. Ins. Lbs. 3.25 3.75 4.5 5 1-375 15.102 1-875 23.762 -75 II.406 3-25 54 - 9*6 i -5 16.475 2 25-346 1 15.208 3-5 59-14 1.625 17.848 2.25 28.514 1.25 19.OI 3-75 63-365 i -75 19.221 2-5 31.682 1.5 22.8l2 4 67.589 i -875 20.594 2.75 34-851 i -75 26.614 4-25 71.813 2 21.967 3 38.019 2 30.415 4-5 76.038 2.25 24.712 3-25 41.187 2.25 34-217 4-75 80.262 2.5 27.458 3-5 44-355 2-5 38.OI9 5.25 2-75 30.204 a 2-75 41.82 4 436 3 32.95 3 45.623 .25 .125 1.69 3-25 49.425 •5 8.871 0. 0 •25 338 3-5 53.226 •75 13-307 .125 1.479 •5 6-759 3-75 57.028 1 17.742 •25 2.957 •75 IO.I38 4 60.83 1.25 22.178 •375 4-436 1 I 3 - 5 I 8 4-25 64.632 i -5 26.613 •5 5 - 9 i 4 1.25 I6.897 4 75 i -75 3 I -°49 .625 7-393 i -5 20.277 3 t. 1 2 35-484 •75 8.871 i -75 23.656 •25 4.013 2.25 39 * 9 2 .875 10.35 2 27.036 •5 8.026 2-5 44-355 1 11.828 2.25 30.415 -75 12.036 2.75 48.791 1. 125 13-307 2-5 33-795 1 16.052 3 53.226 1.25 14-785 2-75 37- I 74 1.25 20.066 3-25 57.662 1-375 16.264 3 40-554 i -5 24.079 3-5 62.097 i -5 17.742 3-25 43-933 I -75 28.092 3-75 66-533 1.625 19.221 3-5 47 - 3 I 3 2 32.105 4 7O.968 i -75 20.699 3-75 50.692 2.25 36.118 4-25 75-404 i -875 2 22.178 23.656 4.25 2-5 2-75 40.131 44.144 4-5 4-75 79-839 84.275 2.25 26.613 .125 1-795 3 48.157 5 88.71 2-5 29-57 •25 3 - 59 i 3-25 52.17 2-75 32.527 -5 7.181 3-5 56.184 0.0 4.647 3 35.485 -75 10.772 3-75 60.197 .25 3-25 38.441 1 14.364 4 64.21 •5 9.294 0 1.25 17.953 4-25 68.223 •75 13-94 2.75 i -5 21.544 4-5 72-235 1 18.587 .125 1.584 i -75 25.135 a 1.25 23-234 •25 3.168 2 28.725 O 1-5 27.881 •375 4.752 225 32.31:6 •25 4.224 i -75 32.527 •5 6.336 2-5 35-907 -5 8.449 2 37-174 .625 7.921 2-75 39-497 •75 12.673 2.25 41.821 •75 9-505 3 43.088 1 16.897 2-5 46.468 .875 n.089 3-25 46.679 1-25 21.122 2-75 51.114 1 12.673 3-5 50.269 i -5 » 25.346 3 55 - 76 i 1. 125 14.257 3-75 53-86 i -75 29-57 3-25 60.408 1.25 15.841 4 57-45 2 33-795 3-5 65055 1-375 17-425 4 K 2.25 38.019 3-75 69.701 i -5 19.009 2-5 42.243 4 74-348 1.625 20.594 .25 3.802 2-75 46.468 4-25 78-995 i -75 22.178 1 -5 7.604 1 3 50.692 4-5 83.642 Illustration.— What is weight of a bar of iron 5.25 ins. in breadth by .75 inch in thickness? In column 7, as above, find 5.25; and below it, in column, .75; and opposite to that is 13.307, which is 13 lbs. and .307 of a pound. For parts of a pound and of a foot, operate according to rule laid down for table, page 125. WEIGHT OF SHEET AND HOOP IRON. 129 "Weight of* SHeet Iron. {English. D. K. Clark.) Per Square Foot (at 480 lbs. per Cube Foot). Thickness. Weight. As by Wire-gauge used in South Square Staffordshire, England. Inch. .0125 .OI4I .OI56 .0172 .Ol88 .0203 .0219 .0234 .025 .0281 •0313 Lbs. •5 .562 .625 .688 •75 .813 .875 •938 1 I * I 3 1.25 ,qi F< in 1 ton. No. 4480 3986 3584 3256 2987 2755 2560 2388 224O 1982 1792 Thickness. Inch. •0344 •0375 .0438 •05 •0563 .0625 •075 .0875 .1 .1125 .125 Weight. Lbs. 1.38 i -5 1- 75 2 2.25 2 - 5 3 3 - 5 4 45 5 Square Feet in 1 ton. No. 1623 1493 1280 1120 996 896 747 640 560 498 448 Thickness. Weight. Inch. .1406 .1563 .1719 .1875 .2031 .2188 •2344 •25 .2813 • 3 I2 5 Lbs. 5-63 6.25 6.88 7-5 8.13 8.75 9-38 10 11.25 12.5 Square Feet in 1 ton. No. 398 358 326 299 276 256 239 224 199 179 Width. Ins. .625 •75 .875 19 18 ■Weight of* Hoop Iron. Per Lineal Foot. Width. {English.) Weight. Lbs. .067 .0875 .I2l6 .1636 Ins. 1. 125 1.25 i -375 I *5 W. G. Weight. Width. W. G. Weight. No. Lbs. Ins. No. Lbs. 17 .21 i *75 14 .484 16 .27 2 13 •634 15 •33 2.25 13 .714 15 •36 2-5 12 .91 Weight of 33 lack and. Galvanized Sheet Iron. (Morton's Table, founded upon Sir Joseph Whitworth Sf Co. s Standard Bir- mingham Wire-Gauge .) (D. K. Clark.') Note.— Numbers on HoltzapffePs wire-gauge are applied to thicknesses on Whit- worth gauge. Gauge and Weight of Black Sheets. Approximate number of Sq. Ft. in i ton. Black. (Galvanized. Gauge and Weight of Black Sheets. Approximate number of Sq. Ft. in i ton. Black. | Galvanized. No. Inch. Lbs. Sq. Ft. Sq. Ft. No. Inch. Lbs. Sq.Ft. Sq.Ft. I •3 12 187 185 17 .06 2.4 933 876 2 .28 II . 2 200 197 18 •05 2 1120 IO38 3 .26 IO.4 215 212 19 .04 1.6 1400 I274 4 .24 9.6 233 229 20 .036 1.4 1556 1403 5 .22 8.8 254 250 21 .032 1.28 1750 1558 6 .2 8 280 275 22 .028 1. 12 2000 1753 7 .18 7.2 3 11 304 23 .024 .96 2333 2004 8 .165 6.6 339 331 24 .022 .88 2545 2159 9 •15 6 373 363 25 .02 .8 2800 2339 10 •135 5-4 4 i 5 403 26 .Ol8 ‘72 3 111 2553 11 .12 4.8 467 452 27 .Ol6 .64 35 oo 2808 12 .11 4.4 509 49I 28 .014 •56 4000 3122 13 •095 3-8 589 566 29 •013 •52 4308 3306 14 .085 3-4 659 63O 30 .012 .48 4667 35 13 15 .07 2.8 800 757 31 .01 •4 5600 4017 16 .065 . 2.6 862 813 32 .009 •36 6222 4327 130 WEIGHT OF ANGLE AND T IRON, "Weigh-t of Englisli Single and T Iron. { D . K . Clark .) ONE FOOT IN LENGTH. Note. — W hen base or web tapers in section, mean thickness is to be measured. , , Sum of Width and Depth in Inches. Thick- □ess. i -5 1 .625 i -75 1-875 2 2.125 2.25 2.375 2.5 2.625 275 Inch. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. .125 •57 .62 .68 •73 .78 •S 3 .88 •94 •99 I.04 I.09 •1875 .81 .89 .97 1.05 I* 13 1. 21 I.29 1-37 i .45 1.52 1.6 •25 I.04 I * I 5 1.25 1-36 I.46 1.56 I.67 I.77 1.88 I.98 2.08 •3125 I.24 i -37 i -5 1.63 I.76 I.89 2.02 2.15 2.28 2.41 2-54 2.875 3 3.125 3.25 3-375 3-5 3.625 375 3-875 4 4-25 .125 1. 14 1.2 1.25 1-3 1*45 1. 41 I.46 I. 5 I 1.56 1.62 I.72 •i« 7 S 1.68 I.76 1.84 1.91 1.99 2.07 2.15 2.23 2.3 2.38 2-54 •25 2.19 2.29 2.4 2-5 2.6 2.71 2.8l 2.92 3-02 3.13 3-33 •312S 2.67 2.8 2-93 3.06 3.19 3-32 345 3.58 3 . 7 i 3.84 4.1 •375 3 -i 3 3.28 3-44 3-59 3-75 3 . 9 i 4.06 4.22 4.38 4-53 4.84 •4375 3-57 3-75 3*93 4.11 4.29 448 4.66 4.84 5.02 5-2 5-56 4-5 4-75 5 5-25 5-5 575 6 6.25 6.5 6-75 7 1875 2.7 2.85 3 - QI 3- j 6 3-32 3.48 3-63 3-79 3-95 4.1 4.26 25 3-54 3-75 3 - 9 6 4.17 438 4-58 4*79 5 5.21 542 5.63 3125 4-36 4.62 4.88 5.14 5-4 5.66 5 92 6.18 6-45 6.71 6.97 375 5.16 5-47 5.78 6.09 6.4I 6.72 703 7-34 7.66 7-97 8.28 4375 5-92 6.29 6.65 7.02 7-38 7-75 8.11 8.48 8.84 9.21 9-57 5 6.67 7.08 7-5 7.92 8-33 8-75 9.17 9.58 10 10.42 10.83 5625 7-38 7-85 8.32 8.79 9.26 9-73 10.2 10.66 11. 13 12.6 12.07 7.25 7-5 7-75 8 8.25 8.5 8-75 9 925 9-5 975 25 5.83 6.04 6.25 6.46 6.67 6.88 7.08 7.29 7-5 7.71 7.92 3125 7-23 7-49 7-75 8.01 8.27 8-53 8.79 9.05 9 . 3 i 957 9-83 375 8.59 8.91 9.22 9-53 9.84 10.16 10.47 10.78 11.09 1 1. 41 11.72 4375 9-93 10.3 10.66 11.03 n -39 11.76 12.12 12.49 12.85 13.22 I 3-58 5 11.25 11.67 12.08 12.5 12.92 I 3-33 13-75 14.17 14.58 15 15-42 5625 12.54 13.01 13.48 13-94 I4.4I 14.88 15-35 15.82 16.29 16.76 17.23 625 13.8 14.32 14.84 15-36 15.89 16.41 16.93 1745 17.97 18.49 19.01 1 0 10.5 i i 1 1.5 1 2 12.5 13 » 3-5 '4 ' 4-5 •5 375 12.03 12.66 13.28 13-91 14-53 18.31 4375 13-95 14.67 15.4 16.13 l6.86 17-59 19.04 19.77 20.5 21.22 5 15-83 16.67 17-5 18.33 19.17 20 20.84 21.67 22.5 23.34 24-17 5625 17.7 18.63 19.57 20.51 21-44 22.38 23.31 24.25 25.19 26.12 '27.06 625 19-53 20.57 21.61 22.66 23-7 24.74 25.78 26.83 27.87 2S.91 2995 75 23-13 24.38 25.63 26.88 28.13 2937 30.63 31.88 33.13 34 3s 35 63 1 2 ' 2-5 13 13.5 '4 '5 16 »7 18 19 20 .625 23-7 24.74 25.78 26.83 27.87 29-95 32.03 34.12 36.2 38.28 40.36 •75 28.13 29-37 30.63 31.88 33-13 35.63 38.13 40.63 41.13 43.63 46-13 .875 32.45 33 - 9 1 35.36 36.82 38.28 I41.19 44.12 47.02 49-95 52.87 55-78 1 36.67 38.33 40 41.67 43-33 46.67 50 53-33 56.67 60 [63-33 Note.— A merican rolled is slightly heavier. WEIGHT OF HOOP IRON. CAST IRON. — METALS. I 3 I "Weigh. t of Hoop Iron. (D. K. Clark.) ONE FOOT IN LENGTH. As by Wire-gauge used in South Staffordshire (England). Width in Inches. Thickness. .625 •75 .8751 ' 1.125 ! '-25 1 I 1 -375 | '•5 1 1.625 i '•75 2 No. Inch. Lb. Lb. Lb. Lb. Lb Lb. Lbs. Lbs. | Lbs. Lbs. Lbs. 21 •0344 .0716 .0861 .1 .115 .129 • T 44 •158 .197 .201 .229 20 .0375 .0781 .0938 .IOQ .125 .141 • i 56 .172 .188 .203 .219 •25 19 .0438 .0911 .109 .128 .146 .164 .182 .2 .2I9 1 .238 •257 .292 iS .05 .IO4 .125 a 46 .167 .188 .208 .229 •25 .271 .292 •333 17 .0563 .117 .141 .164 .188 *211 •234 .258 .281 •305 .328 •375 16 .0625 •13 .156 .182 .208 .234 .26 .286 •313 •339 •365 .417 15 •075 .156 .188 .219 •25 .281 • 3 i 3 •344 •375 •307 •438 •5 14 .0875 .183 .219 .256 •293 •329 .366 .402 •438 •475 .512 .585 13 ii .208 •25 .292 •333 •375 .416 .458 •5 •543 •584 .667 12 .1125 •234 .281 .328 •375 .422 .469 .516 •563 .609 .656 •75 II .125 .26 •313 •365 .417 .469 .521 •573 .625 .677 .729 •833 IO .1406 •293 •352 .41 .469 •527 .586 .645 •703 .762 .82 •938 9 •1563 .326 •391 •456 .522 •587 .652 • 7 I 7 •783 .848 • 9 I 3 1.04 8 .1719 •358 •43 .501 •573 .644 .716 .788 •859 • 93 i 1 I * I 5 7 •1875 •391 .469 •547 .625 •703 .781 •859 •938 1.02 1.09 1.25 6 .2031 •423 .508 •593 .677 .762 .836 •931 1.02 1. 1 1. 19 i -35 5 .2188 •456 •547 .638 .729 .82 .912 1 1.09 1. 19 1.28 1.46 4 .2344 .488 .586 .683 .781 .879 •977 1.07 I.!7 1.27 i -37 1.56 CAST IRON. To Compute "W eight of a Cast Iron Bar or Rod. Ascertain weight of a wrought iron bar or rod of same dimensions in preceding tables, or by computation, and from weight deduct ^,-tli part. Or, As .1000 : .9257 :: weight of a wrought bar or rod : to weight re- quired. Thus, what is weight of a piece of cast iron 4 X 3.75 X 12 inches? In table, page 128, weight of a piece of wrought iron of these dimensions is 50.692 lbs. Then, 1000 : .9257 :: 50.692 : 46.93 lbs. Braziers’ and. Sheathing Copper. Braziers’ Sheets, 2X4 feet from 5 to 25 lbs., 2.5 X 5 feet from 9 to 150 lbs., and 3X5 feet and 4X6 feet, from 16 to 300 lbs. per sheet. Sheathing Copper, 14 x 48 inches, and from 14 to 34 oz. per square foot. Yellow Metal, 14 x 48 inches, and from 16 to 34 oz. per square foot. "W eight of Corrugated Iron Roof Blates. per square FOOT. (Birmingham Gauge.) No. Black. Galvanized. | ' No. Black. Galvanized. No. Black. Galvanized. Oz. Oz. Oz. Oz. Oz. Oz. 20 26 29 23 20 22 25 l6 18 22 22 24 1 24 18 20 , 26 14 l6 METALS. To Compute "Weight of Metals of a ny Dimen- sions or Form. # By rules in Mensuration of Solids (page 360 ), ascertain volume of the piece, multiply it by weight of a cube inch, and product will give weight in pounds. i3 2 WEIGHT OF CAST IRON PIPES. "Weiglit of Cast Iron IPipes or Cylinders. From i to 70 Inches in Internal Diameter . ONE FOOT IN LENGTH. Diameter. Thickn. Weight. Diameter. Thickn. Weight. Diameter. Thickn. Weight. Ins. I Inch. •25 Lbs. 3.°6 Ins. 4-75 Inch. •375 Lbs. 18.84 Ins. II Inch. .875 Lbs. IOI.85 •375 5-05 •5 25.72 n -5 •5 58.81 1.25 .25 3.68 .625 32.93 .625 74.28 • 3 I2 5 4-79 •75 40.43 •75 90.06 •375 5.97 5 •375 19.76 .875 106.13 1.5 •375 6.89 •5 26.95 12 •5 61.26 •4375 8.31 .625 34-46 .625 77-34 •5 9.8 •75 42.27 •75 93-73 i -75 •375 7.81 5-5 •375 21.59 *875 110.42 •4375 9-38 •5 29.4 12.5 •5 63-71 •5 11.03 .625 37-52 .625 80.4 2 •375 8-73 •75 45-95 •75 97-4 •4375 io -45 6 •375 2343 .875 114.71 .5 12.25 •5 3 i,8 6 13 •5 66.16 2.25 •375 965 .625 40-59 .625 8347 •4375 11.52 •75 49.62 •75 101.08 •5 13.48 6.5 •375 25.27 •875 1 19 2.5 •375 10.57 •5 34-31 I 3*5 •5 68.61 •4375 12.6 .625 4365 .625 86.53 .5 14.7 •75 53-3 •75 104.76 2.75 •375 11.49 7 •5 36.76 •875 123.29 •4375 14.67 •5625 41.7 14 •5 71.06 .5 15*93 .625 46.71 .625 89.6 3 •375 12.4 •75 56-97 •75 108.43 •5 I 7- I 5 7-5 •5 39.21 •875 127.58 .625 22.2 •5625 44-45 14-5 •5 73-51 •75 27-57 •625 49-77 .625 92.66 3-25 •375 .5 I 3-32 18.38 8 •75 •5 60.65 41.66 •75 •875 112.11 131.87 .625 23-74 •5625 47.21 15 •5 75-96 •75 29.4 .625 52.84 •625 95-72 3-5 •375 .5 14.24 19.6 9 •75 •5 64.32 46.56 •75 •875 115.78 136.16 .625 25.27 5625 52.72 15-5 •5 78.47 •75 3 I - 2 4 .625 58.96 •625 98.78 3-75 •375 15.16 •75 7 i>6 7 •75 119.46 .5 20.83 9-5 •5 49.01 16 •875 140.44 .625 26.8 •5625 55-48 •625 101.85 •75 33-°8 •625 62.06 •75 123.14 4 •375 .5 16.08 22.05 10 •75 •5 75-35 5 i -45 16.5 •875 1 144-73 166.63 .625 •75 28.33 34 - 9 2 •625 •75 65.09 79-°3 •625 •75 104.9 126.75 4*25 •375 17 .875 93 - 2 ; •875 149.02 .5 23.28 10.5 •5 53 - 9 1 1 I 7 I -53 .625 29.86 •625 68.15 17 .625 107.97 •75 36.76 •75 82.7 •75 130.48 4-5 •375 .5 17.92 23.88 ji •875 •5 97-56 56-36 •875 1 153-3 176-43 .625 3 i -4 .625 71.21 17-5 .625 111-03 •75 3859 •75 86.38 •75 134 - J 6 WEIGHT OF CAST IRON PIPES. 133 Diameter. Thickn. Weight. Diameter. Thickn. Weight. Diameter. Thickn. | Weight. Ins. Inch. Lbs. Ins. Ins. Lbs. Ins. Ins. Lbs. 17-5 .875 157-59 29 • 7 /> 218.7 40 .875 350.56 I 181.33 .875 256.23 I 4OI.86 18 .625 II4.I I 294.05 1. 125 453.46 •75 I 37-84 30 •75 226.05 1.25 505 - 4 I •875 161.88 .875 264.8 42 .875 367.69 1 186.23 I 303.86 I 421.45 19 .625 120.23 1.125 343-22 1. 125 472.52 •75 145-19 31 •75 233 . 4 I I.25 529.87 .875 170.46 .875 273.38 44 .875 384.88 1 196.03 1 313.66 I 44I. I 20 .625 126.35 1. 125 354-24 1. 125 497.58 •75 152.54 32 •75 240.75 I.25 554.42 .875 179.03 .875 281.95 46 •875 402.01 1 205.84 1 323.46 I 460.07 21 .625 132.48 1.125 36 S -27 1. 125 519.64 •75 159.89 33 •75 248. 11 I.25 578.88 •875 187.61 •875 290.53 48 •875 419.17 1 215.64 1 333.26 I 480.29 22 .625 138.61 1.125 376.29 1. 125 541.69 •75 167.24 34 •75 255 46 I.25 603.44 .875 196.19 •875 299.ll 50 .875 436.43 1 225.44 1 343-06 I 499.89 23 .625 144-73 1. 125 387.33 1. 125 563.75 •75 174-59 35 •75 262.81 I.25 627.93 .875 204.76 .875 307.68 52 .875 453-49 1 235-24 1 352.87 I 519.5 24 .625 150.86 1.125 398.35 1. 125 585-81 •75 181.95 36 •75 270.16 I.25 654.42 .875 213-34 .875 316.26 55 •875 479-23 1 245.04 1 362.67 I 548.9 25 .625 156.98 1. 125 409.28 1 . 125 618.91 •75 189.3 1.25 456.37 I.25 689.21 •875 221.92 37 •75 277.51 58 I 578.29 1 254-85 .875 324.84 1. 125 651.96 26 .625 163 11 1 372.47 I.25 725-93 •75 196.65 1.125 420.4 1-375 800.22 •875 230-5 1.25 468.65 60 1 597-92 1 264.65 38 •75 284.86 1. 125 674.01 27 .625 169.23 .875 333.41 1.25 750.45 •75 204 1 382.27 1-375 827.17 •875 239.07 1.125 431.41 65 1 646.93 1 274-45 1.25 480.89 1. 125 729.18 28 .625 I 75-36 39 •75 292.21 1.25 811.73 •75 211-35 •875 341.97 1-375 894.6 •875 247.65 1 392.08 70 1 69592 1 284.25 1.125 442.44 1.25 872.98 29 .625 181.49 1.25 493.14 i -5 1051.25 Equivalent Length of Pipe for a SocTcet. 7 + — = d representing diameter of pipe and l length in inches. Additional weight of two flanges for any diameter is computed equal to a lineal foot of the pipe. Note.— T hese weights do not include any allowance for spigot and socket ends. 2.— For rule to compute thicknesses of pipes, flanges, etc., see page 560. M 134 WEIGHT OF FLAT ROLLED BAR AND SQUARE STEEL. NVeiglit oF Flat Foiled. Far Steel. {D. K. Clark.) From .5 Inch to 8 Inches in Width, one foot in length. Width in Inches. Thick- ness. •5 .625 | -75 1 .875 1 1 | 1.25 | Inch. Lb. Lbs. j Lbs. Lbs. j Lbs. I Lbs. % • 4 2 5 •533 .64 •743 •85 I.06 /16 •531 .665 .8 • 9 2 9 I.06 i -33 % .638 •798 .96 1. 11 1.28 i -59 % •744 • 93 i 1. 12 i -3 I.49 1.86 y .85 1.06 1.28 1.49 i -7 2.13 y 1.2 1.44 1.67 1. 91 2*39 % — i *33 1.6 1.86 2.12 2.66 % — — 1.76 2.04 2-34 2.92 % — — 1.92 2.23 2-55 3- I 9 % — — — 2.41 2.76 3-45 X — — — 2.6 2.98 3 - 7 2 % — — — — 3 -i 9 398 1 — — — — 3-4 4-25 Width in Inches. Thick- ness. Inch. X | % % % 3 1 3-25 3-5 4 1 » 5 1 5 1 Lbs. Lbs. Lbs. Lbs. I Lbs. | Lbs. I 2.55 2.76 2.98 3-4 j 3.82 4.26 3.19 3-45 3-72 4 - 25 , 4 - 78 , 5-32 3-83' 4 !4 4.46 5 -i 1 5 - 74 ! 6.38 4.46 4-83 5.21 5 - 95 , 6.7 7*44 5 1 5-53 5-95 6.8 7.66 8.5 5-74 6.22 6.69 7- 6 5 8.6 9-56 6.38 6.91 7-44 8-5 9-56 10.6 7.01 7.6 8.18 9-35 10.5 11 *7 7-65 8.29 8.93 10.2 n -5 12.8 8.29 1 8.98 9.67 11. 1 12.4 13.8 8.93 i 9-67 10.4 11.9 13-4 14.9 9.56110.4 11.2 12.8 14-3 15-9 10.2 In. 1 11.9 13.6 15-3 17 Lbs. I Lbs. Lbs. 4-68 5-1 i 5*5 2 5.84] 6.38. 6.9 7,02! 7.66 8.28 8.181 8.92 9.66 9.36; io .2 10.5 JII.5 12.8 ■Weight of Rolled Square Steel. „ Tnah to 6 Inches Square, one foot in length. Side. [Weight. Inch. .125 .1875 •25 • 3 I2 5 •375 •4375 •5 •5625 .625 .6875 Lbs. •053 .119 .212 •333 .478 .651 .85 1.08 i -33 1.61 Side. | Weight. Side. Weight. Side. Weight. Side. Ins. Lbs. Ins. Lbs. Ins. Lbs. Ins. •75 I.92 1-375 6.43 2.125 15-4 3-75 .8125 2.24 1-4375 7-°3 2.25 17.2 4 •875 2.6 1 - 5 - 7-65 2-375 19.2 4-25 •9375 3°6 1.5625 8-3 2-5 21.2 4-5 1 3-4 1.625 8.98 2.625 23-5 4*75 1.0625 3-83 1.6875 9-79 2-75 25-7 5 1.125 4-3 i *75 10.4 2.875 28.2 5-25 1.1875 4-79 1.8125 11. 2 3 30.6 5-5 1.25 5 - 3 i 1-875 11.9 3-25 35-9 5-75 i- 3 I2 5 5.86 2 13-6 3-5 41.6 6 Weight. Lbs. 47.8 54-4 6l.4 68-9 76.7 85 93-7 102*8 112.4 122 4 WEIGHT OF ROLLED STEEL, SHEET COPPER, ETC. 1 3 5 eiglit of Round Rolled. Steel. From .125 Inch to 12 Inches Diameter . one foot in length. Diam. Weight. Diameter. Weight. Diameter. ] Weight. Diam. Weight. Diam. Weight. Inch. Lbs. In 9 . Lbs. Ins. Lbs. Ins. Lbs. Ins. Lbs. .125 .0417 •875 2.04 I.625 7-05 2.875 22 5-75 88.3 •1875 •0939 •9375 2.35 I.6875 7.61 3 24.I 6 96.I •25 .167 1 2.67 i -75 8.18 3.25 28.3 6.5 1 13.2 •3125 .26 1.0625 3 1.8125 8.77 3-5 32.7 7 130.8 •375 •375 1. 125 3-38 1.875 9-38 3-75 34-2 7-5 136.8 •4375 • 5 ii 1.1875 3-76 2 10.7 4 42.7 8 170.8 •5 .667 1.25 4.17 2.125 12 4-25 48.3 8.5 193.2 •5625 •845 1-3125 4.6 2.25 13.6 4-5 54-6 9 218.4 .625 1.04 1-375 5.05 2-375 I 5 -I 4-75 60.3 9-5 24I.2 .6875 1.27 1-4375 5.18 2.5 16.7 5 66.8 10 267.2 •75 i -5 i -5 6.01 2.625 18.4 5-25 73-6 11 323 .8125 1.76 1.5625 6.52 2-75 20.2 5-5 80.8 12 352-8 Weiglit of Hexagonal, Octagonal, and. Oval Steel. ONE FOOT IN LENGTH. HEXAGONAL. OCTAGONAL. OVAL. Diam. Diam. Diam. Diam. over Sides. Weigh t.J over Sides. Weight. over Sides. Weight. over Sides. Weight. Diam. over Sides. Area. Weight. Inch. Lb,. Ins. Lbs. Inch. Lbs. Ins. Lbs. Ins. Sq. In. Lbs. % .414! I 2.94 % •396 I 2.82 %x% .251 .853 A •736 *A 3-73 A .704 ^A 3-56 %xy 2 •344 1. 1 7 % I - I 5 % 4.6 A 1. 1 4.4 1 xx .446 I.52 a 1.66 I % 5-57 % 1.58 5-32 .697 2-37 % 2.25 *A 6.63 A 2.l6 6-34 iXxM .884 3 "Weiglit of a Square Foot of Slieet Copper. Wire Gauge of Wm . Foster Sf Co, (England.) Thickness. | Weight. Thickness. Weight. Thickness. Weight. W. G. Inch. Lbs. W. G. Inch. Lbs. W. G. Inch. Lbs. I .306 14 II .123 5.65 21 •034 1-55 2 .284 13 12 .109 5 22 .029 i -35 3 .262 12 13 .098 45 23 .025 1. 15 4 .24 II 14 .088 4 24 .022 1 5 .222 10.15 15 .076 3-5 25 .019 .89 6 •203 9-3 l6 .065 3 26 .017 •79 7 ,l86 8.5 17 •057 2.6 27 •015 •7 8 .l68 7-7 l8 .049 2.25 28 .013 .62 9 •153 7 19 .O44 2 29 .012 •56 10 .138 6.3 20 .038 i -75 30 .Oil •5 'W'eigh.t of Composition SHeatliing ISTails. No. Length. Number in a Pound. No. Length. Number in a Pound. No. Length. Number in a Pound. No. Length. Number in a Pound. I Inch. •75 29O 4 Ins. 1-125 201 *7 Ins. 1. 125 184 IO Ins. I.625 IOI 2 •875 260 5 I.25 I99 8 I.25 168 II i *75 74 3 1 212 6 I 190 9 r *5 no 12 2 64 Ij6 WEIGHT OF IRON, STEEL, COPPER, ETC. Weight of Cast and Wrought Iron, Steel, Copper, and Brass, of a given Sectional Area. Per Lineal Foot. Sectional Wrought Cast Iron, j Steel. Copper. 1 Lead. Brass. ( 5 un-metal. Area. Iron. Sq. Ins. .1 Lbs. .336 Lbs. •313 Lbs. •339 Lbs. .385 Lbs. .492 Lbs. •357 Lbs. .38 .2 .671 .626 .677 .771 .984 .713 •759 .3 1.007 •939 1.016 I.156 I.476 I.07 1 . 139 .4 1-343 1.251 i -355 1.542 I.967 I.427 I- 5 I 9 • 5 1.678 1.564 1.694 I.927 2.461 1.783 1.894 .6 2.014 1.877 2.032 2.312 2-953 2.I4 2.279 .7 2-35 2.19 2.371 2.698 3-445 2.497 2.658 .8 2.685 2.503 2.71 3083 3-937 2.853 3-038 .Q 3.021 2.816 3-049 3-469 4.429 3.21 3.418 I 3-357 3.129 3387 3-854 4.922 3.567 3.798 1. 1 3.692 3-442 3-726 4-24 5.414 3-923 4-177 1.2 4.028 3-754 4.065 4.625 5-906 4.28 4-557 1.3 4.364 4.067 4.404 5-01 6.398 4.636 1 4-937 1.4 4.699 4-38 4.742 5.396 6.89 4-993 5 - 3 I 7 1.5 5-°35 4-^93 5.081 5 - 78 I 7-383 5-35 5.696 1.6 q.371 5.006 5-42 6.167 7.875 5.707 6.076 1.7 5.706 5 - 3*9 5-759 6.552 8.367 6.063 6.456 1.8 6.042 5 ^32 6.097 6-937 8.859 6.42 6.836 1.9 6.378 5-945 6.436 7-323 9-351 6.777 7-215 2 6.714 6.258 6-775 7.708 9-843 7-133 7-595 2.1 7.049 6-57 7.H4 8.094 10.33 7-49 7-97 2.2 7 - 3 8 5 6.883 7-452 8.474 10.83 7-847 8-35 2.3 7. 7 21 7.196 7.791 8.864 11.32 8.203 8-73 2.4 8.056 7-509 8.13 9-25 11. 81 8.56 9.11 2 -5 8.392 7.822 8.469 9.635 12.3 8 - 9 i 7 9.49 2.6 8.728 8.135 8.807 10.02 12.8 9-273 9.87 2.7 9.063 8.448 9.146 IO.4I 13.29 963 10.25 2.8 9-399 8.76 9-485 IO.79 13-78 9.98 10.63 2.9 9-734 9-073 9.824 II. l8 14.27 10.34 11. 01 3 10.07 9.386 10.16 II.56 14.76 10.7 ii-39 3.1 10.41 9.699 10.5 n -95 15.26 11.06 11.77 3.2 10.74 10.01 10.84 12.33 15-75 11.41 12.15 3.3 11.08 10.32 11. 18 12.72 16.24 n-77 12.53 3.4 11.41 10.64 11.52 13.1 16.73 12.13 12.91 3.5 n-75 10.95 11.86 13.49 17.22 12.48 13.29 3.6 j 12.08 11.26 12.19 13-87 17.72 12.84 13.67 3.7 ! 12.42 11.58 12-53 14.26 18.21 13.2 14.05 3-8 I 12.76 11.89 12.87 14.64 18.7 13-55 14-43 3.9 I 13.09 12.2 13.21 15.03 I 9* I 9 I 3-9 I 14*01 4 1 13.43 12.51 13-55 15.42 19.69 14.27 15- J 9 4 * 1 13.76 12.83 13.89 15.8 20.18 14.62 15-57 4.2 14.1 i 3- I 4 14.23 16.19 20.67 14.98 15-95 4-3 4.4 1443 14-77 13-45 13-77 14.57 14.91 16.57 16.96 21.16 21.65 15-34 15.69 16.33 16.71 4.5 15. 11 14.08 15.24 17.34 22.15 16.05 17.09 4.6 4*7 15-44 15.78 14-39 14.7 15.58 15.92 17.73 18.11 22.64 23-13 16.41 16.76 17-47 17.85 4.8 16. 11 15.02 16.26 •18.5 23.62 17.12 18.23 4.9 16.45 I 5 v 33 16.6 18.88 24.12 17.48 18.61 5 16.78 15.64 16.94 19.27 24.61 17*83 18.99 IRON BOILER TUBES, 137 Lap Welded Charcoal Iron. Boiler Tubes. Standard Dimensions. National Tube Works Company . Length per Diam iter. In- i a <£ Circum Ex- erence. In- Tran Ex- sverse Ar In- eas. Sq.l of Su: Ex- Foot rface. In- Weight per Foot. ternal. ternal. H ternal. ternal. ternal. ternal. Metal. ternal. ternal. Ins. Ins. No. Ills. Ins. Sq. Ins. Sql Ins. Sq. Ins. Feet. Feet. Lbs. I .86 .072 15 3- I 4 2.69 .78 •57 .21 3.82 446 .71 1. 125 .98 .072 15 3-53 3.08 •99 •76 .24 3-39 3-89 .8 1.25 i.n .072 15 3-93 3-47 1.23 .96 .27 3.06 3-45 .89 I.3 2 I-I 5 .083 14 4.12 3-6 1-35 1.03 •32 2.91 3-33 I.08 1-375 1. 21 .083 14 4-32 3-8 1.48 I - I 5 •34 2.78 3 -i 6 1.5 i -33 .083 14 4.71 4.19 1.77 1.4 •37 2-55 2.86 1.24 1.625 i -43 .095 13 5 -i 4 - 5 i 2.07 1.62 .46 2-35 2.66 i -53 *•75 1.56 .095 13 5-5 4.9 24 1. 91 •49 2.18 2-45 1.66 1.875 1.68 •095 13 5-89 5.29 2.76 2.23 •53 2.04 2.27 1.78 2 1. 81 •095 13 6.28 5-69 3-14 2.57 •57 1. 91 2. 11 1. 91 2.125 i -93 •095 13 668 6.08 3-55 2-94 .61 1.8 I -97 2.04 2.25 2.06 .095 13 7.07 647 3-98 3-33 .64 *•7 1.85 2.16 2-375 2.16 .109 12 7.46 6.78 4-43 3-65 •78 1.61 1.77 2.61 2-5 2.28 .IO9 12 7-85 7- I 7 4 9 1 4.09 .82 1.53 1.67 2-75 2-75 2-53 .109 12 8.64 7-95 5 94 | 5-03 •9 i -39 i- 5 i 3-04 2.875 2.66 .109 12 9 °3 8.35 6.49 5-54 •95 i -33 1.44 3-i8 3 2.78 .109 12 9.42 8.74 7.07 ! 6.08 •99 1.27 I -37 3-33 3-25 3.01 .12 II 10.21 9.46 8.3 7 * 12 1. 18 1. 17 1.26 3-96 3-5 3.26 .12 II 11 10.24 9.62 8 - 3 S 1.27 1.09 1. 17 4.28 3-75 3-51 .12 II 11.78 11.03 11.04 9-68 1-37 1.02 1.09 4.6 4 3-73 •134 IO 1 12.57 11.72 1257 | 10-94 1.63 •95 1.02 5-47 4-25 3 98 •134 IO 13-35 12.51 14.19 : I2 -45 i -73 •9 .96 5.82 4-5 4.23 -I .34 lo 14.14 13.29 15.9 1 I4 °Z 1.84 .85 •9 6.17 4-75 448 - 134 IO 14.92 14.08 17.72 15.78 I -94 .8 .85 6.53 5 4-7 .148 9 i 5 - 7 i 14.78 19.63 1 17-38 2.26 •76 .81 7-58 5-25 4 95 .148 9 16.49 I 5-56 21.65 19.27 2.37 •73 •77 7-97 5-5 5-2 .148 9 17 28 16.35 23.76 21.27 2.49 •7 •73 8.36 6 5-67 .165 8 18.85 17.81 28.27 25-25 3.02 .64 •67 10.16 7 6.67 .165 8 21.99 20.95 38.48 34 94 3-54 •55 •57 11.9 8 767 .165 8 25.13 24.1 50.27 46.2 4.06 .48 •50 13-65 9 8.64 .18 7 28.27 27.14 63.62 58.63 4.99 .42 •44 16.76 10 9-59 .203 6 31.42 30.14 78 54 72.29 6.25 .38 •4 20.99 11 10.56 .22 5 34-56 33-17 95-03 87.58 7-45 •35 •36 25-03 12 n -54 .229 4-5 37-7 36.26 H 3 - 1 104.63 8.47 •32 •33 28.46 13 12.52 .238 4 40.84 39-34 : 132.73 123.19 9-54 .29 •3 32.06 14 13 5 1 .248 3-5 ! 43.98 42.42 15394 143.22 IO.7I .27 .28 36 15 1448 .259 3 47.12 45 5 176.71 164.72 11.99 •25 .26 40.3 16 15-43 .284 2 50.26 48.48 201.06 187.04 14.02 .24 •25 47.11 17 16.4 •3 1 5341 5 i -52 226.98 211.24 15-74 .22 •23 52.89 18 ! 17-32 •34 0 5655 54-41 1 254 47 235.61 1 18.86 .21 .22 1 63.32 Note. — In estimating effective heating or evaporating surface of Tabes, as heating liquids by steam, superheating steam, or transferring heat from one liquid or one gas to another, mean surface of Tubes is to be computed, M* 138 STEAM, GAS, AND WATER PIPE. Iron Welded. Steam, Gras, and Water Pipe. Standard Dimensions. National Tube Works Company. Diameter. In- ternal. i Ex- i ternal. Ii Ins. Ins. .125 •4 •25 •54 •375 .67 •5 .84 •75 1.05 1 1*31 1.25 1.66 i -5 1.9 2 2-37 2*5 2.87 3 3-5 3-5 4 4 4-5 4-5 5 5 5-56 6 6 62 7 7.62 8 8.62 9 9.62 10 10.75 11 n -75 12 12.75 13 14 14 i 5 i 5 16 16 17 17 18 STEEL LOCOMOTIVE TUBES. Lap Welded Semi-Steel Locomotive Tubes Standard Dimensions. National Tube Works Company. Diameter. § Circumference. Transverse Areas. Ex- ternal. In- ternal. H ® & 3 Ex- ternal. In- ternal. Ex- ternal. In- I ternal. Metal. Ins. I Ins. •834 Ins. .083 No. 14 Ins. 3 * I 4 2 Ins. 2.62 Sq. Ins. •785 Sq. Ins. .546 Sq. Ins -239 1.25 i -5 1.75 I.084 1 - 3 ! 1.532 .083 .095 .109 14 13 12 3 - 9 2 7 4 - 7 12 5 - 498 3 - 405 4* II 5 4- 813 I.227 I.767 2.405 •923 I.348 I.843 • 3°4 .419 .562 A 4 Q 2 1.782 .IO9 12 6.283 5-598 3 - I 4 2 2-494 •O40 2.25 2.032 .IO9 12 7.069 ; 6.384 3-976 3-243 •733 Q~.Q 2.5 2.26 .12 II 7-854 7-1 4.909 4.01 1 .090 2-75 3 2.51 2.76 .12 .12 II II 8.639 9-425 7.885 8.67 5-94 7.069 4.948 1 5-983 •992 1.086 Lengtl Sq. I of Sui Ex- ternal. h per r oot rface. In- ternal. Weight per Foot. Feet. Feet. | Lbs. 3.82 4-58 | .81 3-056 3-524 I.03 2.546 2.916 I.42 2.183 2-493 I.9I 1. 91 2.144 2.2 I.698 1.88 2.49 1.528 1.69 3-05 I.389 1.522 3-37 1 1-273 i 1-384 3.68 WEIGHT OF LEAD AND TIN PIPE AND TIN PLATES. 1 39 'W'eiglit of Lead, and Tin Lined 3?ipe per Foot. From .375 Inch to 5 Inches in Diameter . ( Tatham Bros.) WASTE-PIPE. 1 BLOCK-TIN PIPE. Diam. Weight. Diam. | Weight. | | Diam. Weight. Diam. Weight. [ Diam. Weight. Ins. Lbs. Ins. Lbs. Inch. Lb. Inch. Lbs. Ins. Lbs. i*5 2 4 8 •375 •3594 .625 •5 I.25 I.25 2 3 4-5 6 •375 •375 .625 .625 I.25 i-5 3 3-5 4-5 8 •375 •5 •75 .625 1-5 2 3 5 5 8 •5 •375 •75 •75 i-5 2.5 4 5 5 10 •5 •5 1 •9375 2 2.5 4 6 5 12 •5 .625 1 1.125 2 3 WATER-PIPE. From .375 Inch to 5 Inches in Diameter. Diam. Thick- ness. Weight. Diam. Thick- ness. Weight. Diam. Thick- ness. ■Weight. Diam. Thick- ness. Weight. Inch. Inch. Lbs. Ins. Inch. Lbs. Ins. Inch. Lbs. Ins. Inch. Lbs. •375 .08 .625 .625 •25 3-5 I.25 .19 4-75 2-5 •3*25 *4 •375 .12 I •75 .1 1.25 I.25 •25 6 2-5 •375 17 •375 .l6 1.25 •75 .12 *•75 *•5 .12 3 3 •1875 9 •375 .19 1-5 •75 .l6 2.25 *•5 .14 3-5 3 •25 12 •375 •34 2.5 •75 .2 3 i-5 *7 4-25 3 •3125 16 •5 .07 •0545 •75 •23 3-5 *•5 .19 5 3 •375 20 •5 .09 •75 •75 •3 4-75 *•5 •23 6-5 3?5 •*875 9-5 •5 .11 1 1 .1 *•5 *•5 •27 8 3-5 •25 *5 •5 •13 1.25 1 .11 2 *•75 •13 4 3-5 •3*25 18.5 •5 .16 i-75 1 .14 2.5 *•75 •*7 5 3-5 •375 22 •5 .19 2 1 •17 3-25 *•75 .21 6-5 4 •1875 12.5 •5 •25 3 1 .21 4 *•75 •27 8-5 4 •25 16 .625 .08 .0727 1 .24 4*75 2 •15 4-75 4 •3*25 21 .625 .09 1 1 •3 6 2 .18 6 4 •375 25 .625 •13 i-5 1.25 .1 2 2 .22 7 4-5 •1875 14 .625 .16 2 1.25 .12 2-5 2 .27 9 4-5 •25 18 .625 .2 2.5 1.25 .14 3 2.5 •1875 8 5 •25 20 .625 .22 2-75 1.25 .16 3-75 2-5 •25 11 5 •375 3* Marks and Weight of* Tin-plates. (English.) Mark or Brand. Plates per Box. Dimensions. Weight per Box. Mark or Brand. Plates per Box. Dimensions. Weight per Box. No. Ins. No. No. Ins. No. 1 Cor 1 Com. 225 13.75X10 112 DXXXX 100 16.75X12.5 189 2 C 225 13.25X 9-75 105 SDC T C V T T 168 3 c 225 12.75X 9 5 08 SDX . . 200 A 0 A U 188 H C 225 13. 75X 10 I IQ SDXX . 200 15 X 1 1 T C V T T 209 230 H X 225 13 75X10 ■ y 157 SDXXX 200 AH 15 Xu 1 X 225 13.75X10 140 SDXXXX. . . . 200 15 XII 251 2 X 225 I 3- 2 5X 9-75 133 SDXXXXX. . 200 15 Xu 272 3 X 225 12.75X 9 5 126 SDXXXXXX. 200 15 X 11 293 1 XX 225 I3-75 Xio 161 Leaded IC. . . 1 12 20 X 14 112 1 XXX. ..... 225 13.75X10 •182 “ IX... 1 12 20 X14 140 I xxxx. ... 225 13.75X10 203 ICW 225 13.75X10 112 I xxxxx . . 225 13.75X10 224 IX w 225 13.75X10 140 i xxxxxx. 225 13.75X10 245 CSDW 200 15 Xu 168 DC 100 l6. 7S X 12. K 08 CTTW DX 100 16.75X 12.5 126 XIIW IOO x 6-75X 12-5 105 126 DXX 100 16. 75X 12.5 14.7 TT. .. 16. 75 X 12.5 DXXX 100 16.75X12.5 X T / l68 XTT 45° 450 I 3-75X 10 i3-75Xio 126 When the plates are 14 by 20 inches, there are 112 in a box. 140 WEIGHT OF COPPER TUBES. Weigh-t of Seamless Drawn Copper Tubes American T' ube Works. (Boston.) BY EXTERNAL DIAMETER. ONE FOOT IN LENGTH. Stubs’’ W. G. From .25 Inch to 12 Ins.— f full, l light. 13 I No. | 20 19 18 17 16 | 15 14 Ins. V32 / 3/64 / 3 / 64 / V16 l l /i6f 5/64 l 5 / 64 / Diamet’r. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. •25 .09 .1 .12 •13 .14 •15 •17 •375 .14 .16 .19 •23 .24 .26 .29 •5 .2 •23 •27 • 3 1 •34 •37 .42 .625 •25 .29 •34 •4 •44 .48 •55 •75 •3 •36 .42 •49 •54 •59 .67 .875 •36 .42 49 .58 .64 •7 .8 1 • 4 1 .48 •57 .67 •74 .81 •93 1-125 .46 •55 .64 .76 •83 .92 1.05 1.25 •52 .61 • 7 i .84 •93 1.03 1. 18 1.375 •57 .68 •79 •93 I -°3 1. 14 I * 3 I 1.5 .62 •74 .86 1.02 1 • I 3 1.25 i -43 1.625 .68 .8 •94 1. 11 1.23 1.36 1.56 i -75 •73 .87 1. 01 1.2 i -33 1.47 1.69 1-875 .78 •93 1.09 1.29 i -43 1.58 1.81 2 .84 1 1. 16 i -37 i -53 1.69 1.94 2.125 .89 1.06 1.24 1.46 1.63 1.8 2.07 2.25 •94 I * I 3 I * 3 I i -55 i -73 1 . 91 2.19 2375 1 1. 19 i -39 1.64 1.82 2.02 2.32 2-5 1.05 1.25 1.46 i *73 1.92 2.13 2-45 2.625 1. 1 1.32 i -54 1.82 2.02 2.23 2-57 2-75 1. 16 1.38 1.61 1.9 2.12 2-34 2-7 2.875 1. 21 i -45 1.68 1.99 2.22 2-45 2.83 'X 1.26 I- 5 1 1.76 2.08 2.32 2.56 2-95 3- 2 5 i -37 1.64 1. 91 2.26 2.52 2.78 3 - 21 3 5 1.48 1.77 2.06 2.43 2.72 3 346 3-75 1.58 1.9 2.21 2.61 2.92 3.22 3 - 7 i 4 1.69 2.02 2.36 2-79 3 -ii 3-44 3-97 4.25 1.8 2.15 2.51 3-14 3-31 3.66 4.22 4-5 1.9 2.28 2.65 3-32 3 - 5 i 3.88 4-47 4.75 2.01 2.41 2.8 3-49 3 - 7 i 4.1 4-73 5 2.12 2-54 2-95 3-67 3 - 9 i 4-32 4.98 5.25 2.23 2.66 3 -i 3.85 4.11 4-54 5-23 5-5 2-34 2.79 3-25 3.85 4-3 4.76 5-49 5-75 2.44 2.92 3-4 4,02 4-5 4.98 5-74 6 2-55 3-°5 3-55 4.2 4-7 5-2 5-99 6.25 2.66 3.18 3-7 4.38 4.9 5 - 4 i 6.25 6-5 2.76 3-31 3-85 4-55 5 -i 5-63 6-5 6-75 2.87 3-44 4 4-73 5-3 5-85 6-75 7 2.98 3 - 5 6 4 -i 5 4 . 9 i 5-49 6.07 7.01 7.25 3.09 3-69 4*3 5.09 5-69 6.29 7.26 7.5 3.19 3.82 4*45 5.26 5-89 6.51 7 - 5 i 8 3 4 1 4,08 4-74 5.62 6.29 6-95 8.02 8-5 3.62 4-33 5-04 5-97 6.68 7-39 8.52 g 3-83 4-59 5-34 6-33 7.08 7-83 9°3 9-5 4-05 4-85 5-^4 6.68 7.48 8.26 9-54 10 4.26 5 - 11 5-94 7-03 7.87 8.7 10.05 10.5 4-47 5-37 6 24 7-39 8.27 9- I 4 10.55 11 4.69 5.62 6-54 1 7*74 8.67 9-58 11.06 11.5 4.9 5.88 6.84 ! 8.1 9.06 10.02 11-56 12 5 - 11 6.13 7- I 3 18.45 9.46 10.45 1 12.07 12 3/32 / j 7/64 Lbs. .18 •32 •47 .61 .76 •9 1.05 1.19 i-34 1.48 1.63 i-77 i 1.92 | 2.06 j 2.21 2.35 2*5 2.64 2.79 2.93 3.08 3.22 3-37 3.66 3-95 4.24 4*53 4.82 5 - 11 5-4 5-69 5-98 6.27 6.56 6.85 7.14 7 43 7.72 8.01 8.30 8 - 59 9.17 9- 75 10.33 10.91 11.49 12.07 12.65 1323 13.81 Lbs. .19 •35 •52 .69 •85 1.02 1. 18 I *35 1.52 1.68 1.85 2.02 2.18 2.35 2.51 2.68 2.85 3.01 3 - lS 3-35 3 - 5 i 3.68 3-84 4.18 4 i 5 I 4.84 5 -i 7 5 - 5 i 5-84 6.17 6.5 6.84 7.17 7-5 7-83 8.17 8.5 8.83 9.16 9-5 9-83 10.49 11.16 11.82 12.49 1315 13.82 14.48 I 5 -I 5 15.81 11 1/8 1 Lbs. .19 •37 •56 •74 .92 1. 11 1.29 1.47 1.65 1.84 2.02 2.2 2.39 2.57 2.75 2.93 3 - 12 3-3 3-48 3 - 6 7 3.85 4 - 03 4.22 4.58 4 - 95 5 - 3 i 5.68 6.05 6.41 6.78 7 -i 4 7 - 5 i 7.87 8.24 8.61 8.97 9 34 9-7 10.07 10.44 10.8 n -53 12.26 13 13-73 14.46 1519 I 5-92 16.66 17.29 WEIGHT OF COPPER TUBES. No. 10 9 8 7 1 6 5 4 3 2 Ins. 9/64 1 9/64 / n/64 l 3/16 1 13/64 V32 / i 5/6 4 / r /4 / 9/32/ Diamet’r. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. •375 •4 .41 .42 -44 — — — — — •5 .61 .64 .67 •71 •73 •75 .76 — — .625 .81 .86 .92 •99 1.04 1.09 1. 12 I * I 3 1. 18 •75 1. 01 1.09 I - I 7 1.26 i-35 1.42 I.49 i-53 I.6l .875 1.22 I -3 I 1.42 i-53 1.66 1.76 1.85 1.92 2.04 1 1.42 i-54 1.67 1.81 1.97 2.09 2.21 2.32 2.48 1. 125 1.63 1.78 i-93 2.08 2.28 2.43 2.58 2.71 2.9I 1.25 1.83 2 2.18 2.36 2-59 2.76 2.94 3- 11 3-34 x -375 2.03 2.22 2-43 2.63 2.9 3-i 3-3 3-5 3-77 i-5 2.24 2.44 2.68 2.91 3.21 3-43 3-67 3-9 4.21 1.625 2.44 2.67 2.93 3 -i 8 3-52 3-77 4-03 4.29 4.64 I -75 2.65 2.89 3 -i 8 3-45 3.83 4.11 4-39 4.69 5-07 i-875 2.85 3.12 3-44 3-73 4.14 4.44 4.76 5.08 5-5i 2 3.06 3-34 3-69 4 4-45 4.78 5.12 5-48 5-94 2.125 3.26 3-57 3-94 4.28 4-75 5-n 5.48 5.87 6-37 2.25 3-46 3-8 4.19 4-55 5.06 5-45 5-84 6.27 6.81 2-375 3-67 4.02 4.44 4.82 5-37 5.78 6.21 6.66 7.24 2-5 3.87 4-25 4.69 5-i 5.68 6.12 6.57 7.06 7.67 2.625 4.08 4-47 4-95 5-37 6 6-45 6.93 7-45 8.1 2-75 4.28 4-7 5-2 5-65 6-3 6-79 7.29 7-85 8-54 2.87s 4.48 4.92 5-45 5.92 6 61 7.12 7.66 8.24 8.97 3 4.69 5-i5 5-7 6.2 6.92 7.46 8.02 8.64 9.4 3-25 5-i 5-6 6.2 6-74 7-54 8.13 8.75 9-43 10.27 3-5 5-5 1 6.05 6.71 7.29 8.16 8.8 9-47 10.22 11. 14 3-75 5-9i 6-5 7.21 7.84 8.78 9-47 10.2 II.OI 12 4 6.32 6-95 7.71 8-39 94 10.14 10.92 n.8 12.87 4.25 6-73 7-4 8.22 8.94 10.02 10.81 11.65 12.59 13.73 45 7.14 7-85 8.72 9.49 10 64 11.48 12.37 13-38 14.6 4-75 7-55 8-3 9.22 10.04 11.26 12.16 131 14.17 15.46 5 7.96 8-75 9-73 10.58 11.88 12.83 13.83 14.96 1633 5.25 8.36 9.21 10.23 11. 13 12.49 w 14.55 15-75 17.2 5-5 8-77 9.66 10.73 11.68 13.11 14.17 15.28 16.54 18.06 5-75 9.18 10. 1 1 11.24 12.23 13-73 14.84 16 17-33 18.93 6 9-59 10.56 11.74 12.78 14-35 15-51 16.73 18.12 19.79 6.25 10 II.OI 12.24 I 3-33 14.97 16.18 17.46 18.91 20.66 6-5 10.41 11.46 12.75 -3-88 i5 59 16.85 18.18 19.7 21-53 6-75 10.82 11.91 13-25 14.42 16.21 17-52 18.91 20.49 22.39 7 11.22 12.36 13-75 14-97 16.83 18.19 19.63 21.28 23.26 7-25 11.63 12.81 14.26 15-52 17-45 18.86 20.36 22.07 24.13 7-5 12.04 13.26 14.76 16.07 18.07 19-54 21.08 22.86 25 7-75 12.45 I3-7I 15.26 16.62 18.68 20.21 21.81 23-65 25 86 8 12.86 14.17 15-77 17.17 19-3 20.88 22.54 24.44 26.72 8.25 13-27 14.62 16-27 17.71 19 92 21.55 23.26 25-23 27-59 8.5 13-67 15-07 16.77 18.26 20.54 22.22 23-99 26.02 28.45 8.75 14.08 I5-52 17.28 18.81 21.16 22.89 24.71 26.81 29.32 9 14.49 15-97 17.78 19.36 21.78 23-56 25-44 27.6 30.18 9-25 14.9 16.42 18.28 I 9-9 I 22.4 24.23 26.17 28.39 31-05 9-5 i5-3i 16.87 18.79 20.46 23.02 24.9 26.89 29.18 31.92 9-75 15-72 17.32 19.29' 21.01 23.64 25.57 27.62 29.97 32.78 10 16.12 17-77 19-79 21-55 24.26 26.24 28.34 30.76 33.65 10.5 16.94 18.68 20.8 22.65 25-5 27.59 29.79 32.34 35-38 11 17.76 19.58 21.81 23-75 26.73 28.93 31-25 33-92 37-n Ix *5 18.57 20.48 22.81 24.84 27.97 30.27 32.7 35-5 38.84 12 *9-39 21.38 23.82 25-94 29.21 3I.6l 34- 1 5 37.08 40.58 141 19 / 64 / Lbs. I.63 2.09 2.55 3 346 3 - 92 4 - 38 4 - 83 5 - 29 5-75 6.21 6.66 7.12 7-57 8.04 8.49 8.95 9.41 9.87 10.78 n-7 12.61 13-53 14.44 I 5-3^ l 6.27 17.19 I8.I 1902 *9-93 20.85 21.76 22.68 23-59 24.51 25.42 26.34 27.25 28.17 29.08 30 30.91 31-83 32.74 33 - 66 34- 57 35- 49 37-32 39-15 40.98 I 42.81 142 WEIGHT OF COPPER AND BRASS TUBES, ETC. By Internal Diameter. Add following Units to Weights for External Diameter in preceding tables. No. | 1 2 3 ! 4 | 5 6 1 | 7 l 8 I 9 10 2.21 i -97 1.66 1-38 1. 18 I. Ol .78 .67 •53 •43 No. ! 11 12 13 14 15 16 17 00 rH 19 20 r- i -35 .29 .22 •I? •13 .11 .08 .06 | •05 •03 Illustration. — What is weight of a copper tube 6 ins. in internal diameter, No. 3 gauge, and one foot in length? By preceding table 6 ins. external, No. 3 gauge = 18.12, and 18.12 1 .66 = 19.78 lbs. WEIGHT OF BRASS TUBES. To Compute ”VV r eiglit of Brass Tubes. American Tube Works. (Boston.) Rule. — Deduct 5 per cent, from weight of Copper tubes. Example. — What is weight of a brags tube 6 ins. in external diameter, No. 3 gauge, and one foot in length? By preceding table 6 ins. = 18.12, from which deduct 5 per cent. = 17.21 lbs. By Internal Diameter. Rule. — Proceed as above for internal diameter of copper tube, and deduct 5 per cent. Example.— Weight of a copper tube 6 ins. internal diameter, No. 3 gauge, and 1 foot in length = 19.78 lbs. Hence, 19.78 — 5 per cent. = 18.79 lbs. Note.— Diameter of Tubes, as for Boilers, is given externally, and that for Pipes internally. Weights of English as given by D. K. Clark are essentially alike to the 'preceding. Brass Tubes Corresponding with, and ITitted for Iron Tubes or Pipes. American 'F'u.'be Works. (Boston.) WEIGHT PER LINEAL FOOT. Diameter of Iron Pipe. Diameter of Iron Pipe. Diameter of Iron Pipe. Weight. Internal. External. Weight. Internal. External. W eight. Internal. External. Inch. Ins. Lbs. Ins. Ins. Lbs. Ins. Ins. Lbs. .125 •375 •25 I 1-3125 1-7 3 3-5 8-3 •25 •5625 •43 1.25 I.625 2-5 3-5 4 IO.9 •375 .6875 •63 i-5 1-875 3 4 4-5 12.7 •5 .8125 •9 2 2-375 4 5 5-5 15-7 •75 1.0625 1.25 2.5 2.875 4.87 "W eiglit of SHeet Brass. one square foot. {Iloltzapjj eV s Gauge.) Thickness. Weight. Thickness. Weight. Thickness. Weight. I Thickness. Weight. No. Inch. Lbs. No. Inch. Lbs. No. Inch. Lbs. No. Inch. Lbs. 3 •259 IO.9 9 .148 6.23 15 .072 3-03 21 .032 i-35 4 .238 IO 10 •134 5^4 16 .065 2.74 22 .028 1. 18 5 .22 O.26 11 .12 5-05 17 .058 2.44 23 .025 1.05 6 •203 8.55 12 .109 4-59 18 .049 2.06 24 .022 .926 7 .18 7-58 13 •095 4 19 .042 1.77 25 .02 .842 8 .165 6-95 14 .083 3-49 20 •035 i-47 WEIGHT OF WROUGHT IRON TUBES, 143 "Weiglit of Wrought Iron Tubes. (English.) EXTERNAL DIAMETER. ONE FOOT IN LENGTH. HoltzapjfeVs Wire-Gauge, f full , l light. No. - - 4 5 1 6 7 8 9 Ins. • 3125 .281 .238 .22 .203 .18 .165 . 148 5/16 9/32 * 5 / 64 / 7/32 r 3 / 64 3/16 l n/64 l 9/64 / Diam. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. 7 21.9 19.8 16.9 IS .6 14*5 12.9 11. 8 10.6 7-5 23-5 21.3 I 8 .I 16.8 15-5 13.8 12.7 11.4 8 25.2 22-7 19-3 17.9 16.6 14.7 i 3-5 12,2 8-5 26.8 24.2 20.6 19.I 17.6 15*7 14.4 12.9 9 28.4 25-7 21.8 20.2 18.7 16.6 15-3 13-7 9-5 30.1 27.1 23.1 2I.4 19.8 17.6 16. 1 14-5 10 3 i -7 28.6 24-3 22.5 20.8 18.5 17 15-3 No. 7 8 9 10 1 1 12 13 '4 '5 Ids. .18 .165 .148 •134 .12 .109 •095 .083 .072 3/16 l 11/64 l 9/64/ 9/64 l 1/8 l 7/64 3/32/ S/64 / 5/64 1 Diaoi. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. I 1-55 I.44 I.32 1.22 I. II 1.02 •9 •797 •7 1. 125 I.78 1.66 I - 5 I 1-39 I.26 1. 16 i -3 .906 -794 I.25 2.02 1.88 1.71 i -57 I.42 Ji 3 I - I 5 I.OI .888 J -375 2.25 2.09 1.9 i -74 1.58 I -45 1.27 1. 12 •983 i -5 2.49 2.31 2.1 1.92 i -73 i -59 1.4 1.23 1.08 1.625 2.72 2.52 2.29 2.09 1.89 i -73 1.52 i -34 1. 17 i -75 2.96 2-74 2.48 2.27 2.05 1.87 1.65 i -45 1.27 i -875 3-19 2.96 2.68 2-45 2.21 2.02 1.77 1.56 1*36 2 3-43 3 -i 7 2.87 2.62 2.36 2.16 1.9 1.67 1-45 2.125 3.67 3-39 3.06 2.8 2.52 2.3 2.02 1.78 i -55 2.25 3-9 3-6 3.26 2.97 2.68 2-44 2.14 1.88 1.64 2-375 4.14 3.82 3-45 3 -i 5 2.83 2-59 2.27 1.99 1.74 2-5 4-37 4.04 3-65 3-32 2-99 2-73 2-39 2.1 1.83 2.625 4.61 4.25 3-84 3-5 3 -i 5 2.87 2.52 2.21 i -93 2-75 4.84 4-47 4-03 3.67 3 - 3 i 3.02 2.64 2.32 2.02 2.875 5.08 4.68 4.23 3-85 3-46 3.16 2.77 2-43 2,11 3 5-32 4.9 4.42 4.02 3.62 3-3 2.89 2-54 2.21 3-25 5-79 5-33 4.81 4-37 3-94 3-59 3 -i 4 2-75 2.4 3-5 6.26 5.76 5-2 4.72 4-25 3.87 3- 39 2.97 2-59 3-75 6-73 6.19 5-58 5-07 4-57 4.16 3-64 3 -i 9 2.77 4 7.2 6.63 5-97 5-43 4.88 4.44 3-89 3-4 2.96 4.25 7.67 7.06 6.36 5-78 5-2 4-73 4 -i 3 3.62 3.15 4-5 8.14 7-49 6-45 6.13 5 - 5 i 5.01 4-38 3-84 3-34 4-75 8.61 7.91- 7 -i 3 6.48 5.82 5-3 4-63 4.06 3-53 5 9.08 8-35 7-52 6.83 6.13 5.58 4.88 4.27 3-72 5-25 9-56 8.79 7.91 7.18 6.44 5-87 5 .i 3 4.49 3-9 5-5 10 9.22 8-3 7-53 6.76 6.15 5-38 4.71 4.09 5-75 10.5 9^5 8.68 7.88 7.07 6.44 5-63 4-93 4.28 6 11 IO. I 9.07 8.23 7-39 6-73 5-87 5 -i 4 4-47 6.25 11.4 10.5 9.46 8.58 7-7 7.01 6.12 5.36 4.66 6-5 11.9 10 9 9-85 8-93 8.02 7-3 6-37 5-58 4.85 6-75 12.4 11.4 10.2 9.28 8-33 7-58 6.62 5-79 5-03 7 12.9 11.8 10.6 9-63 8.64 7.87 6.87 6.01 5.22 7-25 13-3 12.2 11 9.99 8.96 8.15 7.12 6.23 5 * 4 i 7-5 13.8 12.7 11.4 10.3 9.27 8.44 7-37 6-45 5-6 7-75 14-3 I 3 * 1 11.8 10.7 9-59 8.72 7.62 6.66 5-79 8 14-7 13-5 12.2 11 9-9 9.01 7.86 6.88 5 - 9 8 \ 144 WEIGHT OF COPPER TUBES. ■Weight of Seamless Drawn Copper Tubes. (English.) For Diameters and Thicknesses not given in preceding Tables. (D. K. Clark.) INTERNAL DIAMETER. ONE FOOT IN LENGTH. HollzapjiJeV s Wire-Gauge, f full , l light. Specific Weight = 1.16. Wrought Iron = i. No. 0000 1 000 00 0 No. 0000 000 00 0 •454 . 4 2 5 .38 •34 •454 .425 •38 -34 Ins. 29/64 27/64 / 3/8/ n/32 29/64 27/64 / 3/8/ n(tm Lbs. Lbs. Lbs. Lbs. Diam. Lbs. Lbs. Lbs. Lbs. •75 — 4-5 5-75 34-2 3 x -9 28.3 25.2 .875 • 5-79 5.02 6 35-6 33-2 29-5 26.2 1 8.02 7*36 6-37 5*53 6-5 3 8 -4 35-8 31.8 28.3 1. 125 8.71 8 6.95 6.05 7 41. 1 3 8 -3 34 -i 30.3 1.25 9.4 8.65 7-52 6-57 7-5 43-9 40.9 3 6 -4 32.4 I *375 IO.I 9*3 8.1 7.08 8 46.6 43-5 38-7 34-5 1.5 10.8 9 94 8.68 7.6 9 52.1 48.7 43-3 38.6 1.625 11. 5 10.6 9.26 8.12 10 57-7 53 - 8 47-9 42.7 1 .75 12. 1 11. 2 9-83 8.63 11 63.2 59 52.5 46.8 1.875 12.8 11.9 10.4 9* I 5 12 68.7 64.2 57-2 5 1 2 13.5 12.5 11 9.66 1 3 74.2 69-3 61.8 55 - 1 2.125 14.2 13-3 1 1. 6 10.2 14 79-7 74-5 66.4 59-2 2.25 2-375 2.5 14.9 15-6 16.3 13.8 14 - 5 1 5 - 1 12. 1 12.7 13-3 10.7 11.2 n.7 15 16 x 7 85.2 90.7 9 6 *3 79.6 84.8 90 7 i 75-6 80.2 63-4 67.7 71.8 -A 2.625 2.75 17 17.7 15.8 16.4 x 3-9 14-5 12.2 12.8 18 19 101.8 107.3 95 -i 100.3 84.9 89-5 76 80.1 3 3.25 19.1 20.4 17.7 19 15.6 16.8 13.8 14.8 20 21 112.8 118.3 105.5 110.7 94.1 98.7 04.2 88.3 3.5 21.8 20.3 17.9 15-9 22 123.8 115.8 103-3 9 2 -5 3 - 75 4 4 - 25 4.5 23.2 24.6 25-9 2 7-3 21.6 22.9 24.2 2 5*4 19.1 20.2 21.4 22.5 16.9 17.9 19 20 23 24 26 28 129.3 134.8 146 157-2 120.9 126.1 136.4 146.7 107.9 112.6 121.8 I 3 I 9O.O 100.6 108.8 117.1 4.75 28.7 26.7 23-7 21 30 168.4 I 57 - 1 140.2 125.4 5 30. 1 28 24.8 22.1 32 179.6 167.4 149*5 I 33 - 6 5-25 5-5 31*5 32.8 29-3 1 30-6 26 27.1 23.1 1 24.1 34 3 6 190.7 201.9 177.7 1 188 158.7 1 167.9 i 4 x -9 i 5 °- 1 For Diameters from 13 to 24 Inches. No. 1 | 2 3 4 5 6 7 8 9 10 Ins. •3 * 9 / 64 / .284 9 / 32 / •259 V 4 / .238 1 5 / 6 4 / .22 7/32/ .203 13/64 .18 3/16 l .165 n/64 1 .148 9/64/ •134 9/64 l Diam. x 3 14 15 16 17 18 19 20 21 22 23 24 Lbs. 48.5 52.1 55-8 59-4 63 66.7 70.3 74 77.6 81.3 84.9 88.6 Lbs. 45-8 49-3 52.7 56.2 59-6 63.1 66.5 70 73-4 76.9 80.3 83.8 Lbs. 41.7 44.9 48 51.2 54-3 57-4 60.6 63-7 66.9 70 73-2 7 6 -3 Lbs. 38-3 4I.2 44.1 46.9 49.8 52.7 55.6 58.5 61.4 643 67.2 70.1 Lbs. 35-3 38 40.7 43-4 46 48.7 5 i -4 54 56.7 59-4 62.1 64.7 Lbs. 32.6 35 -i 37-6 40 42.5 45 47-4 49.9 52.4 54-9 57-3 59-8 Lbs. 28.8 31 33-2 35-4 37-5 39-7 41.9 44.1 46.3 48.5 50.7 52-9 Lbs. 26.4 28.4 30-4 32.4 34-4 3 6 -4 384 40.4 42.4 44.4 | 46.4 1 48.5 Lbs. 23.6 254 27.2 29 30.8 32.6 34-4 36.2 38 39-8 41.6 43-4 Lbs. 21.4 23 24.6 26.3 27.9 29-5 3 1 - 2 32.8 34-4 36 37-7 39-3 WEIGHT OF COPPER AND WROUGHT IRON TUBES. I45 For Diameters from 13 to 24 Inches. No. " 12 13 '4 15 J6 17 18 19 20 Ins. .12 . 109 •095 .083 .072 .065 b Ln 00 .049 .042 •035 1/8 l 7/64 3/32/ 5/64/ 5/64 l x /i6/ x /i6 l 3/64/ 3/64 1 V32 / Diam. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. 13 19.I I 7‘4 I 5 .I 13.2 II.4 IO.3 9.2 7-77 6.65 5-55 14 20.6 18.7 16.3 I4.2 12.3 II. I 9.9 8.37 7.16 5-98 15 22.1 20 17.4 15.2 13.2 II.9 10.6 8.96 7.67 6.4 l6 23-5 21.3 18.6 l6.2 14.I 12.7 n -3 9*56 8.18 6.82 17 25 22-7 19.7 I7.2 14.9 13-5 12. 1 10.2 8.69 7.27 IS 26.4 24 20.9 18.2 15.8 14*3 12.7 10.7 9.2 7.69 19 27.9 25-3 22 I9.2 16.7 I 5 - 1 13-4 11 *3 9.71 8.12 20 29-3 26.6 23.2 20.2 17.6 15-9 14. 1 11.9 10.2 8.54 21 30.8 27.9 24-3 21.3 18.4 16.6 14.8 12.5 IO.7 8.96 22 32.3 29-3 25-5 22.3 19-3 17.4 15-5 I 3 • 1 II . 2 9-39 23 33-7 30.6 26.7 23-3 20.2 18.2 16.2 13-7 II . 8 9.81 24 35-2 3i-9 27.8 243 21. 1 !9 16.9 14-3 12.3 10.2 "Weigh, t of* Wrought Iron Tubes. (English.) For Diameters and Thicknesses not given in 'preceding Tables. (D. K. Clark.) INTERNAL DIAMETER. ONE FOOT IN LENGTH. HoltzapjfeVs Wire-Gauge, f full , l light. No. 4 5 1 6 1 7 Ins. 5/8 9/16 Thickni J /2 ESS IN tl 7/ 16 v’CIIES. 3/8 5/ 16 V 4 .238 *5/64/ .22 7/32/ .203 | J 3/6 4 | .18 3/16 l Diam. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. 19 128.5 II5.2 102. 1 89.I 76.I 63.2 50.4 48 44-2 40.8 36.2 20 135 I 2 I.I IO7.3 93-6 80 66.5 53 50.4 46-5 42.9 38 21 i 4 i -5 127 112.6 98.2 83.9 69.7 55-6 52.9 48.8 45-1 39-9 22 148.1 I32.9 117.8 102.8 87.9 73 58.3 55-4 51 . 1 47.2 41.8 23 154.6 138.8 123. 1 107.4 91.8 76.3 j 60.9 57-9 53-4 49-3 43.7 24 161.2 I44.7 128.3 112 95-7 79.6 63-5 60.4 55-7 51.5 45-6 26 174-3 156.5 138.8 121.1 IO3.6 86.1 68.7 65.4 60.3 55-7 49-3 28 ! 187.4 168.3 149.2 130-3 III.4 92.7 74 70.4 64.9 60 53-1 30 ' 200.4 l8o 159.7 139-5 1 19.3 99.2 79.2 75-4 69.5 64.2 56.8 32 ! 213.5 I9I.8 170.2 148.6 I27. 1 105.7 84.4 80.4 74.I 68.5 60.6 34 j 226.6 203.6 180.6 157-8 135 112.3 89.7 85-4 78.7 72.8 64.4 36 239-7 215-4 191.1 167 I42.9 118.8 94 9 90.4 83.4 77 68.1 No. 8 9 (0 1 1 1 2 13 '4 '5 16 17 ! '8 .165 .148 • 134 .12 . 109 .095. .083 .072 .065 .058 .049 Ins. 11 / 64 l 9/64/ 9/64 1 Vs l 7/64 3/32/ 5/64/ 5/64 1 Vi 6/ V16 1 3/64/ Diam. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. 19 33 - f 29.7 26.9 24 21.8 *9 16.6 I4.4 13 1 1.6 9.78 20 34-8 31.2 28.3 25-3 22.9 20 17-5 I 5 .I 13.7 12.2 IO.3 21 36.6 32.8 29.7 26.5 24.I 21 18.3 15-9 14-3 12.8 10.8 22 38.3 34-3 31 . 1 27.8 25.2 22 19.2 16.6 15 13-4 n -3 23 40 35-9 32.5 29.I 26.4 23 20.1 17.4 15.7 14 11.8 24 41.8 37-4 33.9 30-3 27-5 24 20.9 I8.I 16.4 14.6 12.6 26 45-2 40-5 36.7 32.8 29.8 26 22.6 19.7 17.7 15.8 13-4 28 48.7 43-6 39 5 35-3 32.1 28 24.4 21.2 I9.I 17 14.4 30 52.1 46.7 42.3 37-8 34-4 30 26.1 22.7 20.5 18.3 15.4 32 55-5 49.8 45 -i 40.4 36.7 32 27.9 24.2 21.8 19-5 16.5 34 59 52.9 48 42.9 39 34 29.7 25.8 23.2 20.7 17.5 36 62.4 56 50.8 45-4 4 i .3 36 31.4 27.3 24.6 21.9 18.6 I46 WEIGHT OF IRON, STEEL, COPPER, ETC. "Weight of a Square JF'oot of "Wronght and. Cast Iron, Steel, Copper, Lead, Brass, and Zinc Blates. From .0625 to 1 Inch in Thickness, Thickness. Wrought Iron. Cast Iron. Steel. Copper. Lead. Brass. Gun- metal. Zinc. In^h Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. .0625 2.517 2.346 2.54I 2.89 3.691 2.675 2.848 2-34 .125 5 035 4-693 5.081 5-781 7.382 5-35 5.696 4.68 •1875 7-552 7-039 7.622 8.672 II.074 8.025 8-545 7.02 •25 IO.O7 9.386 IO.163 II.562 I 4-765 10.7 n -393 936 • 3 I2 5 12.588 n -733 12.703 14-453 18.456 13-375 14.241 n -7 •375 15.106 14.079 15 244 17-344 22.148 16.05 17.089 1404 •4375 17.623 16.426 17-785 20.234 25-839 18.725 19.938 16.34 .5 20.341 18.773 20.326 23-125 29-53 21.4 22.786 18.72 • 56 2 5 22.659 21.119 22.866 26.016 33.222 24.075 25-634 21.06 .625 25.176 23466 25.407 28.906 36 - 9 j 3 26.75 28.483 23-4 .6875 27.694 25.812 27.948 31-797 40.604 29.425 3 i- 33 i 25-74 •75 30.211 28.159 30.488 34.688 44.296 32.1 34 -W 9 28.68 .8125 32.729 30-505 33.029 37-578 47.987 34-775 37.027 30.42 .875 35-247 32-852 35-57 40.469 51.678 36.656 39-875 32.76 •9375 37 - 7 6 4 35-199 38.11 43-359 55-37 39-331 42.723 35 -i 1 40.282 37-545 40.651 46.25 59.061 42.8 45-572 37-44 From One Twentieth Inch to Tivo Inches in Thickness. Thickness. Wrought Iron. Cast Iron. Steel. Copper. Lead. Brass. Uun- metal. Zinc. Inch. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. •°5 2.OI4 I.877 2.033 2.312 2-593 2.14 2.279 I.872 .1 4.028 3-754 4-065 4.625 5.906 4.28 4-557 3-744 .15 6.042 5.632 6.098 6.938 8.859 6.42 6.836 5-616 .2 8.056 7.509 8.13 9- 2 5 II.812 8.56 9.H4 7.488 •25 IO.O7I 9.386 IO.163 11.562 14-765 IO.7 n -393 9-36 •3 12.085 11.264 12.195 13-875 17.718 12.84 13.672 11.232 •35 14.099 13.141 14.228 16.187 20.671 14.98 15-95 I 3- I °4 .4 16.II3 15.018 16.26 18.5 23.624 17.12 18.229 14.976 •45 18.127 16.895 18.293 20.812 26.577 19.26 20.507 16.848 .5 2O.I4I 18.773 20.325 23.125 29-53 21.4 22.786 18.72 •55 22.155 20.65 22.358 25-437 32.484 23-54 25.065 20.592 .6 24.169 22.527 24-391 27-75 35-437 25.68 27-343 22.464 •65 26.183 24.409 26.423 30063 38.39 27.82 29.622 24.336 .7 28.197 26.281 28.456 32.375 41-343 29.96 31-9 26.208 •75 30.211 28.154 30.488 34-687 44.296 32.I 34-179 28.08 .8 32.226 30-035 32-521 37 47.249 34-24 36458 29.95 .85 34-24 31.912 34-553 39-312 50.202 36.38 38.736 31.824 •9 36.254 33-79 36.586 41.625 53-154 38.52 41.015 33.696 •95 38.268 35-668 38.628 43-937 56.108 40.66 43-293 35.568 .t 40.282 37-545 40.651 46.25 59.061 42.8 45-572 37-44 1. 125 45 - 3 I 7 42.238 45-732 52.031 66.443 48.15 51.268 42.12 1.25 50.352 46.93 1 50.814 57-813 73-826 53-5 56-965 46.8 i- 3 I2 5 52.87 49.278 53-354 60.703 77 . 5 W 56.17 59813 49.14 1-375 55-387 51.624 55-895 63-594 81.209 58.85 62.661 51.48 1-4375 57 - 9°5 53 - 97 1 58.436 66.484 84-9 6 i .53 65-51 53-82 1.5 60.422 56337 60.976 69-375 88.591 64.2 68.358 56.16 1.5625 62.94 58663 63-517 72.266 92.283 66.88 71.206 58.5 1.625 65-458 61.011 66.058 75156 95-974 69-55 74-054 60.84 i -75 7°-493 65.704 71.139 S0.93S 103.356 74-9 79-751 65.52 1.075 75-528 70.397 76.22 86.719 1 10.739 80.25 85-447 70.2 2 80.564 75-09 81.3 9 2 -5 118.122 85.6 9 I - I 44 74-88 WEIGHT OF CAST IRON WATER PIPES AND TUBES. 1 47 Standard. Cast Iron "Water Pipes. {English.) For a Head of 200 Feet. Diameter. Thickness. Depth of Socket. Thickness of Socket. Packing. -*. 1 TJ .“5 os es S PS p. ! 1 P I Thickness. 1 Depth of Socket. Thickness of Socket. Packing. Weight per Yard.* Lead Joint. Ins. Inch. Ins. Inch. Inch. Lba. j Lbs. Ins. Inch. Ins. Inch. Inch. Lbs. Lbs. 3 •3 I2 5 3-5 .625 • 2 5 36 .8 8 •4375 3-75 .625 •375 113 3-3 4 •3 I2 5 3 .625 •25 51 , 1.2 9 •4375 3-75 •75 •375 128 4.6 5 •375 3 .625 •375 61 2 IQ •5 4 •75 •375 l68 4.9 6 •375 3-75 .625 •375 75 2 -7 II •5 4 •75 •375 175 5 3 7 •375 3-75 .625 •375 85 2.9 12 .5625 ! 4 •875 •375 213 5-7 * Measured as laid. To Compute "W eight of* IVIetal Pipes. 1)2 __ ^ D and d representing external and internal diameters in inches , and C coefficient. Cast Iron 2.45. Wrought Iron 2.64. Brass 2.82. Copper 3.03. Lead 3.86. To Compute AW eight of jVLotal Tubes and Pipes per Pineal Foot. From .5 Inch to 6 Inches Internal Diameter. Diam. Area of Plate. Diam. Area of Plate. Diam. Area of Plate. Diam. Area of Plate. Ins. Sq. Foot. Ins. Sq. Foot. Ins. Sq. Feet. Ins. Sq. Feet. .5 .1309 i-3 T2 5 •3436 2-75 .7199 4-5 I.I781 •5625 •H73 1-375 •36 2 875 .7526 4.625 1.2108 .625 .1636 1-4375 •3764 3 •7854 4-75 1.2435 .6875 .18 i-5 •3927 3125 .8l8l 4875 I.2763 •75 .1964 1.625 4254 3-25 .8508 5 I.309 .8125 .2127 i-75 .4581 3-375 .8836 5-125 I-34I7 .875 .2291 i-875 .4909 3-5 •9 i6 3 5-25 1-3744 •9375 •2454 2 •523b 3-625 •949 5-375 1.4072 1 .2618 2.125 •5543 3-75 .9818 5-5 1-4399 1.0625 .2782 2.25 .587 4 1.0472 5-625 1.4726 1.125 •2945 2-375 .6198 4.125 1.0799 5-75 1.5053 1.1875 •3105 2-5 •6545 4-25 1.1126 5.875 i-538i 125 .3272 2.625 .6872 4-375 I-I454 6 1.5708 Application, of Ta/ble. When Thickness of Metal is given in Divisions of an Inch. To internal diameter of tube or pipe add thickness of metal ; take area of the plate in square feet, from table for a diameter equal to sum of diameter and thickness of tube or pipe, and multiply it by weight of a square foot of metal for given thickness (see table, page 146), and again by its length in feet. Illustration. — Required weight of 10 feet ot copper tube 1 inch in diameter and .125 of an inch in thickness. 1 4-. 125 = 1. 125 X 3.1416-4- 12^.2945 square feet for 1 foot of length. Weight of 1 square foot of copper .125th of an inch in thickness, per table, page 135, =5.781 lbs. ; then, .2945 (from table above) X 5.781 X 10 = 17.025 lbs. When Thickness of Metal is given in Numbers of a Wire-Gauge. To internal diameter of tube or pipe add thickness of number from table, pp. 120 or 121 ; multiply sum by 3.1416, divide product by 12, and quotient will give area of plate in square feet. Then proceed as before. I48 WEIGHT OF IRON" AND COPPER PIPES, BOLTS, ETC. Illustration. — Required weight of 10 feet of copper pipe 2 inches in diameter and No. 2 American wire-gauge in thickness. 2-}-. 257 63 X 3. 1416 -r- 12 = 2.25763 X 3.1416-f- 12 = .591 square feet; then, .591 X 11.6706 (weight from table, page 118) =6.897 lbs. "Weight of Riveted Iron and. Copper Pipes, From 5 to 30 Inches in Diameter. ONE FOOT IN LENGTH. Diameter. Thickness. Iron. Copper. Diameter. Thickness. Iron. Copper. Inch. Lbs. Lbs. Ins. Inch. Lbs. Lbs. 5 .125 7.12 8.14 9 •25 25.OI 28.58 •1875 10.68 12.21 •25 26.33 3O.O9 •25 14.25 16.28 10 •25 27-75 3 I -7i 5-5 .125 7.78 8.89 10.5 •25 29.19 33-22 •1875 11.66 13-33 11 •25 30-49 34.85 •25 I5-56 17.78 12 •25 33- I 3 37.86 6 .125 8.44 9.64 13 •25 35-88 41 •1875 12.65 14.46 14 •25 38.52 44.02 •25 16.88 19.29 i5 •25 41.26 47-15 6.5 .125 9- 1 10.4 16 •3I 2 5 51-57 58.94 •1875 13-65 15.6 •25 43-9 So- 1 7 .2 5 18.2 20.8 •3125 54-87 62.71 7 .125 9.78 II. 18 17 •25 46.53 53 -i 8 .1875 14.68 16.78 •3125 58.17 66.48 •25 19-57 22.37 18 •25 49.17 56.2 7-5 .125 10.49 11.99 •3 I2 5 61.47 70.25 •1875 15-73 17.98 20 •3 I2 5 68.07 77-79 •25 20.89 23.87 24 •3i 2 5 81 -33 9 2 -95 8 .1875 16.7 19.08 25 •3i 2 5 84-57 96.65 •25 22.26 2544 28 •3 I2 5 94-56 107-95 8.5 •25 23-59 26.96 30 •3i 2 5 IOI.I4 H5-59 Above weights include laps of sheets for riveting and calking. Weights of the rivets are not added, as number per lineal foot of pipe depends upon the distance they are placed apart, and their diameter and length depend upon thickness of metal of the pipe. Weight of Copper Rods or Bolts, From .125 Inch to 4 Inches in Diameter. ONE FOOT IN LENGTH. Diameter. Weight. Diameter. W eight. Diameter. Inch. Lbs. Ins. Lbs. Ins. .125 •047 .8125 1.998 1-5 .1875 .106 .875 2.318 •5625 •25 .189 •9375 2.66 .625 •3 I2 5 .296 1 3-03 •75 •375 .426 1.0625 3-42 .875 •4375 •579 .125 3-831 2 -5 -757 .1875 4.269 .125 •5625 •958 •25 4.723 .25 .625 1.182 •3 I2 5 5.21 •375 .6875 t-43 1 -375 5-723 -5 •75 i- 7°3 •4375 6.255 .625 Weight. Diameter. Weight. Lbs. Ins. Lbs. 6.8II 2.75 22.891 7-39 .875 25.OI9 7-993 3 27.243 9.27 .125 29-559 IO.642 .25 31.972 12.108 •375 34.481 13.668 •5 37.081 I5-325 .625 39-777 17-075 •75 42.568 18.916 .875 45-455 20.856 4 48.433 WEIGHT OF METALS. 149 "Weigh, t of Metals of a Griven Sectional Area. From .1 Square Inch to 10 Square Inches. PER LINEAL foot. (. D . K. Clark.) Sect. Wrought Iron. x * Cast Iron. Steel. Crass. Gun- metal. Sect. W rought Iron. Cast Iron. Steel. Brass. Gun- metal. Area. • 9375 * 1.02. 1.052. 1.092. Area. 1. • 9375 - 1.02. 1.052. 1.092. Sq.Ius. Lbs. Lbs. Lbs. Lbs. Lbs. Sq.Ins. Lbs. Lbs. Lbs. Lbs. Lbs. .1 •33 •31 •34 •35 •36 5 -i 17 i 5-9 17-3 17.9 18 6 .2 .67 .62 .68 •7 •73 5-2 17-3 16.3 17.7 18.2 18 9 •3 1 •94 1.02 !‘°5 1.09 5-3 17.7 16.6 18 18.6 i 93 •4 i -33 1.25I 1.36 i -43 1.46 5-4 18 16.9 18.4 18.9 19.7 •5 1.67 1.56 i -7 i -75 1.82 5*5 18.3 17.2 18.7 19*3 20 .6 2 1.88 2.04 2. 11 2.18 5-6 18.7 17-5 19 19.6 20.4 •7 2.33 2.19 2.38 2.46 2.55 ! 5-7 !9 17.8 19.4 20 20.8 .8 2.67 2.5 2.72 2.81 2 . 9 1 5-8 19-3 18.1 19.7 20-3 21. 1 •9 3 2.81 3.06 3.16 3-28 5-9 19.7 18.4 20.1 2O.7 21.5 1 3-33 3 -i 5 3 4 3-51 3-64 6 20 18.8 20.4 21 21.8 1. 1 367 3-44 3-74 3-86 4 6.! 20.3 19.1 20.7 2I.4 22.2 1.2 4 3 75 4.08 4.21 4-37 j 6.2 20.7 19.4 21. 1 2I.7 22.6 i -3 4-33 4.06 4.42 4-56 4-73 6 -3 21 19.7 21.4 22.1 22.9 1.4 4.67 4-38 4.76 4 - 9 1 5 * 1 6.4 21.3 20 21.8 22.4 23-3 i-5 5 4.69 5 -i 5.26 546 ; 6 -5 21.7 20.3 22.1 22.8 2 . 3-7 1.6 5-33 5 5-44 56 i 5-82 1 6.6 22 20.6 22.4 23.I 24 I *7 567 5 - 3 i 5-78 5 - 9 6 6.19 6.7 22.3 20.9 22. 8 23-5 24.4 1.8 6 5 63 6.12 6 31 6.55 6.8 22.7 21.3 23.1 23-9 24.8 1.9 6.33 5-94 6.46 6.66 6.92 j 6.9 23 21.6 23-5 24.2 25.1 2 6.67 , 6-25 | 6.8 7.01 7.28 7 23.3 21.9 23.8 24.6 25-5 2.1 7 6.56 7.14 7-36 7-64 7 -i 23.7 22.2 24.1 24.9 25.8 2.2 7 33 ! 6.88 ! 7.48 7.72 8.01 7.2 24 22.5 24-5 25-3 26.2 2-3 7-67 | 7 19 7.82 8 07 8-37 7-3 24-3 22.8 24.8 25.6 26.6 2.4 1 8 7 5 8.16 8.42 8.74 7-4 24.7 23.1 25.2 26 26.9 2-5 ! 8, 33 7.81 8-5 8.77 9.1 7-5 25 23-4 25-5 26.3 27-3 2.6 1 8.67 8.13 884 9.12 9.46 7.6 25-3 23.8 25-9 26.7 27.7 2.7 i 9 844 9.18 9-47 9-83 7-7 25*7 24.1 26.2 27 28 2.8 ! 9 33 8.75 952 9.82 10.2 7-8 26 24.4 26.5 27.4 28.4 2.9 1 9.67 9 06 9-86 10.2 10.6 7*9 26.3 24.7 26.9 27.7 28.8 3 10 938 10.2 10.5 10.9 8 26.7 25 27.2 28.1 29.1 3 -i 1 io -3 969 io -5 10.9 11 *3 8.1 27 25-3 27-5 28.4 29-5 3-2 10.7 10 10.9 11. 2 n. 7 8.2 2 7-3 25.6 279 28.8 29.9 3-3 11 10.3 1 1. 2 11.6 12 8-3 27.7 259 28 2 29.1 30.2 3-4 n -3 10.6 11.6 11.9 12.4 8:4 28 26.3 28.6 29*5 30-6 3*5 I f *7 10.9 11 *9 12.3 12.7 8-5 28.3 26.6 28.9 298 30 9 3-6 12 n -3 12.2 12.6 i3-i 8.6 28.7 26.9 29.2 30.2 3i-3 3-7 12.3 11.6 12.6 13 13-5 8.7 29 27.2 29.6 30-5 317 3-8 12.7 11.9 12.9 i3-3 13.8 8.8 29*3 27-5 29.9 30-9 32 3-9 13 12.2 13-3 13-7 14.2 8.9 29.7 27.8 30-3 3 1,2 324 4 13-3 12.5 13-6 14 14.6 9 30 28.1 30.6 31.6 32.8 4.1 13*7 12.8 13-9 14.4 14.9 9.1 30-3 28.4 30-9 3i-9 33- 1 4.2 14 I 3* 1 14-3 14.7 15-3 9.2 30-7 28.8 3 i -3 3 2 -3 33-5 4-3 14-3 13-4 14.6 i5-i 15-7 9-3 3i 29.1 3 j *6 32.6 33-9 4.4 14.7 13.8 15 15-4 16 9.4 3i-3 29.4 32 33 34-2 4-5 15 14.1 i 5-3 15.8 16.4 9-5 3 i -7 29.7 32.3 33-3 34-6 4.6 15-3 14.4 15.6 16.1 16.7 9.6 32 30 32.6 33-7 34-9 4-7 15-7 14*7 16 16.5 1 7. 1 9*7 32*3 30-3 33 34 35-3 4.8 16 15 16.3 16.8 17-5 9.8 32-7 30.6 33-3 34-4 35-7 4.9 16.3 x 5-3 16.7 17.2 17.8 9.9 33 30-9 33-7 34-7 36 5 16.7 15-6 17 17-5 18.2 10 33-3 3i-3 34 35 -i 36.4 150 LEAD PIPES. — COPPER PIPES AND COCKS. “Weigh, t of Lead Pipe. (English.) ONE FOOT IN LENGTH. Diam. Thick- ness. Weight. Diam. Thick- ness. Weight.. Diam. j Thick- ness Weight. Diam. Thick- ness. Weight Inch. Inch. Lbs. Ins. Inch. Lbs. Ins. Inch. Lbs. Ins. Inch. Lbs. •5 .097 •93 I .136 2.4 i -75 .166 5 3 •275 14 .112 I.07 .156 2.8 .199 6 3-5 .225 13 .124 1.2 .2 3-73 .228 7 •273 16 .146 I.47 .225 4.27 .256 8 4 •257 17 .625 .089 I 1.25 •139 3 2 .178 6 • 3 I2 5 20.5 .IOI I * I 3 .l6 3-5 .204 7 •327 22 .121 1.4 .18 4 .231 8 4-25 •3125 22.04 .14 2 •193 4-33 .266 9-33 4-5 .232 17 •75 .112 1.6 i *5 .156 4 2.5 .2 8.4 •295 22 .147 1.87 .179 4.67 .227 9.6 • 3 I2 5 23*25 .l 8 l 2.13 .224 6 .261 11.2 4-75 • 3 I2 5 24-45 .215 2.4 •257 7 3 .218 11. 2 5 • 3 I2 5 25.66 Dimensions of Copper Pipes and Composition Cocks. From 1 Inch to 23 Inches in Diameter. •s 4 Flange Diameter. Thick- Bolts. s.e-3 Flange Diam. Thick- Bolts. Diar Pi; and < Pipe. Cock. ness. No. | Diam. .5 -a 0 § Pipe. ness. No. Diam. Ins. Inch. Inch. Ins. Ins. Inch. Inch. I 3-375 3-5 •375 3 •5 9 12.75 .625 9 .625 I.25 3-625 3-75 •375 3 •5 9- 2 5 13-125 .625 10 .625 1.5 3-875 4-25 •375 3 •5 9-5 13-375 .6875 10 .625 1-75 4.125 4-375 •4375 4 •5 9-75 13-625 .6875 10 .625 2 4-375 4-75 •4375 4 •5 10 I 3-875 .6875 10 .625 2.25 4.625 5-25 •4375 5 •5 10.5 14-5 .6875 10 .625 2-5 4-875 5-5 •4375 5 •5 11 15 .6875 10 .625 ■ 2-75 5-25 5-75 •4375 5 •5 11 -5 15-625 •75 10 •75 'Z 6 6.25 •5 5 .625 12 16.125 •75 10 •75 3.25 6.125 6.625 •5 6 .625 12.5 16.625 •75 10 •75 3-5 6.375 6.875 •5 6 .625 13 17-25 •75 10 •75 3*75 6.625 7-25 •5 6 .625 i 3-5 I 7-875 •75 10 •75 4 6.875 7-375 •5 6 .625 14 i 8.375 •75 10 •75 425 7-125 7.625 •5 6 .625 14-5 18.875 •75 10 •75 4-5 7-375 8.25 •5 6 •625 i 5 19-5 •75 10 •75 4-75 7*625 8-5 •5 6 •625 15-5 20 •75 10 •75 5 8 9 •5 6 •625 16 20.5 •75 10 •75 5.25 8.25 9- 2 5 •5 6 .625 16.5 21.125 •75 10 •75 5.5 8.5 9-5 •5 6 •625 17 21.625 •75 11 •75 5.75 9 9-875 •5 6 •625 i 7-5 22.125 •75 11 •75 6 9.25 .625 8 •625 18 22.75 •75 11 •75 6 25 9-75 .625 8 •625 18.5 23-25 •75 11 •75 6.5 10 .625 8 •625 19 23-75 •75 12 •75 6.75 10 .625 8 •625 19-5 24-375 •75 12 •75 7 10.5 .625 8 •625 20 24.875 •75 12 •75 7.25 10.75 .625 8 •625 20.5 25-375 •75 13 •75 7.5 11. 125 .625 8 •625 21 26 •75 13 •75 7-75 n -375 .625 8 •625 21.5 26.5 •75 13 •75 8 11.625 .625 9 •625 22 27 •75 13 •75 8.25 12 .625 9 •625 22.5 27.625 •75 14 •75 8.5 12.25 .625 9 •625 23 28.125 •75 14 •75 8.75 12.5 .625 9 l -625 WEIGHT OF SHEET LEAD, LEAD AND TIN PIPES, ETC. I 5 I "W^eigh-t of Slieet Lead. PER SQUARE FOOT. Thickness. Weight. Thickness. Weight. Thickness. Weight. Thickness. Weight. Inch. Lbs. Inch. Lbs. Inch. Lbs. Inch. Lbs. .OI7 I .068 4 .118 7 .169 IO .034 2 .085 5 •135 8 .186 II .051 3 .IOI 6 .152 9 .203 12 "Weigh/t of Tin Pipe. ONE FOOT IN LENGTH. Diam. External. THICK inch. :ness. % inch. Diam. External. THICK inch. :ness. % inch. Diam. External. THICKN. 3^ inch. Diam. External. Inch. Lb. Lbs. Ins. Lbs. Lbs. Ins. Lbs. Ins. .25 .148 — 1.25 I.O95 I- 4 I 7 2.25 5-04 3- 2 5 •5 •384 .472 i -5 I.328 1.732 2.5 5-67 3-5 •75 .62 .787 i -75 I.564 2.O47 2-75 6-3 3-75 1 .856 I. IO3 2 1.802 2.362 3 6-93 4 THICK??. 3^ inch. Lbs. 7-56 8.19 8 82 945 Diameter. Ins. •375 •5 .625 •75 1 1.25 i -5 Light Weights. For Si 50 feet and under. apply of Water Head 51 to 250 feet. * 251 to 500 feet. Lbs. Lbs. Lbs. I. ,bs. Lbs. Lbs. I 1-5 2 2-5 to 4 3 to 4-5 3-5 to 5 2 2.5 3 3*5 u 5 4 u 6 4-5 7 3 3-5 4 4-5 u 7 5-25 u 8 6 9 3-5 4 4-5 5-5 ' u 8 6 9 7 || 10 4-5 5 5-5 7-25 IQ 8 “ 11 9 u‘ 12 6.5 7 8 9 u 12.5 10 “ 14 12 16 8 9 10 11 u 16 12.5 » 18 14 21 11 13 — 16 a 23 18.5 u 26 21 U 30 Dimensions and. "Weiglit of Sh.eet Zinc. (Vielle-Montagne . | PER SQUARE FOOT. No. Thickness. 2X.5 metres ; area, 1 square metre. 6.56X1.64 feet ; area, 10.76 square feet. 2X .65 metres ; area, 1.3 sq. metres. 6.56X2.13 feet ; area, 13.99 square feet. 2X.8 metres ; area, 1.6 sq. metres. 6 56X2.62 ft. ; area, 17.22 square feet. Weight. Millim. Inch. Kilom. Lbs. Kilom. Lbs. Kilom. Lbs. Lbs. 9 • 4 1 .Ol6l 2.9 6-39 3-7 8.16 4.6 IO.I4 •589 10 • 5 i .0201 3-45 7.61 4-45 9.81 5-5 12.12 •704 11 .6 .0236 4-05 8-93 5-3 11.68 6-5 14-33 .832 12 .69 .0272 4-65 IO.25 6.1 13-45 7-5 16.53 .96 13 .78 .0307 5-3 11.68 6.9 15.21 8-5 18.74 I.088 14 .87 .0343 5-95 I 3 - 1 ? 7-7 16.94 9-5 20.94 I.2l6 15 .96 .0378 6.55 14.44 8-55 18.85 10.5 23-15 i -344 16 1. 1 •0433 7-5 16.53 9-75 21.5 12 26.46 i -536 17 1.23 .0485 8-45 18.63 10.95 24.14 13-5 29.97 1.74 18 1.36 •0536 9-35 20.61 12.2 26.9 15 33-07 1.92 19 1.48 .0583 10.3 22.71 13-4 29.54 16.5 3638 2.112 20 1.66 .0654 11.25 24.8 14.6 32.19 18 39.68 2.304 21 1.85 .0729 12.5 27.56 16.25 35-82 20 44.09 2.56 22 2.02 •0795 13-75 30 - 3 i 17.9 39-46 22 48.5 2.816 23 2.19 .0862 15 33 07 19-5 42.99 24 52.91 3-073 -24 2-37 •0933 16.25 35-82 21.1 46.52 26 57-32 3-329 25 2.52 .0992 17*5 38.58 22.75 50.15 28 61.73 3-585 26 2.66 .IO47 18.8 41.44 24.4 53-79 3 1 68.34 3-969 152 WEIGHT OF SHEET ZINC. SPIKES, HORSESHOES. Ta"ble— (Continued). Special Sizes for Sheathing Ships. No. Thickness. Dimension 1 1. 15 X .35 metres ; area, .402 sq. metre. 3.77 X 1. 15 feet ; area, 4.33 sq. feet. j of Sheets. 1.3 X .4 metres ; area, .52 sq. metre. 4.26 X 1. 31 feet ; area, 5.6 sq. feet. Weight per Sq. Foot. Mi Him. Inch. Kilom. Lbs. Kilom. Lbs Lbs. 15 .96 .0378 2.65 5-84 3-4 7-5 1-344 16 1. 1 •0433 3 6.6l 3-9 8.6 i-S 36 17 I.23 .0485 34 7-5 4.4 9-7 1.74 18 I.36 •0536 3-75 8.27 4.9 10.8 1.92 19 I.48 •0583 4-15 9- I 5 5-35 11.79 2.112 20 1.66 .0654 4-55 10.03 5-85 12.9 2.304 Note.— A deviation of 25 dekagrammes, or about half a pound, more or less, from the proper weight of each number of sheet, is allowed. Nos. 1 to 9 are employed for perforated articles, as sieves, and for articles de Paris. Nos. 10 to 12 are used in manufacture of lamps, lanterns, and tin-ware gen- erally, and for stamped ornaments. The last numbers are used for lining reservoirs, and for baths and pumps. Skip and Railroad Spikes. DIMENSIONS AND NUMBER PER POUND. (P. C. Page, MOSS .) Sliip Spikes. X In. Sq. | « In. Sq. I In. Sq. Y In. Sq. In. Sq. | % In. Sq. Y In.Sq. .fl •S’a .3 R-3 a •Sr 5 C ”3 ,3 .C-O .R-3 4 e-i ti R 3 ’ . s 0 0 R ►3 S 9 1 ^ o' | Zn* ■ be q 71 A O O ^ P-l 1 J ° ° c? 0 ) ° © Ins. Ins. Ins. Ins. Ins. | Ins. Ins. 3 19 3 IO 4 5-4 5 3-4 6 2.2 ! 8 1.4 IO .8 3-5 15.8 3-5 9.6 4-5 5 5-5 3-i 6-5 .2 9 1.2 15 .6 4 I 3. 2 4 8 5 4.6 6 3 7 1.9 j 10 1. 1 — — 4-5 12.2 4-5 6 5-5 4.2 6-5 2.8 7-5 1,8 11 I — — 5 10.2 5 5-8 6 4 7 2.6 8 J -7 — — — — — — 6 5-2 6.5 3-2 7-5 2.4 8-5 1.6 — — — — 8 2.2 9 i-5 — — — — 10 1.4 — — — — Railroad Spikes 5 inch square X 5.5 ins. 2 per lb. “ “ 5625 “ u x 5.5 “ 1.6 “ Spikes and Horseshoes. length and number per pound. (Z7. Burden , Troy , N. F.) Length. Boat £ c . — 0 -3 Spikes. Ul R ►3 No. in Lb. Length. No. in c/3 § *3, Length, jjf No. in Lb. Hook Hea Length. d. c . — x> Horse "Sc 3 shoes. R . "" 3 6 *-3 £ Ins. Ins. Ins. Ins. Ins. Ins. 3 i7-5 6-5 4.78 4 8 7-5 2.5 4 X.375 5-55 I .84 3-5 14.68 7 3.62 4-5 6-5 8 I.74 i 4-5 X. 4375 4.14 2 •75' 4 12-57 7-5 3-37 5 4-37 8-5 I.63 5 X .5 2.52 3 •65 4-5 9-2 8 2-95 5-5 4-3 9 i-55 5-5 X. 5 2.41 4 •56 5 7.2 8-5 2.9 6 4.2 10 I - I 5 ! 5.5 x. 5625 1.87 5 •39 5-5 6-3 9 2.1 6-5 3-77 — — 6 X .5625 I -7 2 — — 6 4-97 10 1.98 7 2-75 — — ! 6 X .625 1 I, 3 8 — — CAST IRON AND LEAD BALLS. NAILS. I 53 Wei glit and. Volume of Cast Iron and Lead Balls. From 1 Inch to 20 Inches in Diameter . Diameter. Volume. Cast Iron. Lead. ; Diameter. Volume. Cast Iron. | Lead. Ins. Cube Ins. Lbs. Lbs. Ins. Cube Ins. Lbs. Lbs. I .523 .136 .215 9 381.703 99-51 156.553 i-5 I.767 .461 .725 9-5 448.92 1 17.034 184.121 2 4.189 I.092 I.718 TO 523-599 136.502 214.749 2.5 8.l8l 2.133 3-355 10.5 606.132 158.043 248.587 3 14-137 3*685 5-798 II 696.91 181.765 285.832 3-5 22.449 5-852 9.207 11 -5 796.33 207.635 326.591 4 33-51 8.736 13-744 12 904.778 235.876 371.096 4-5 47-713 12.439 19.569 12.5 IO22.656 266.647 419.512 5 65 45 17.063 26.843 13 II50.346 299.623 471.806 5-5 87.114 22.721 35-729 14 1436.754 374.563 589.273 6 1 13.097 29.484 46.385 15 I767.I45 460.696 724.781 6-5 143-793 37-453 58.976 16 2144.66 559- 1][ 4 879.616 7 179-594 46.82 73-659 17 2572.44 670.717 1055.066 7-5 220.893 57-587 90.598 18 3053.627 796.082 1252.422 8 268.082 69.889 109.952 19 359 1 -363 936.271 1472.97 8.5 321.555 83.84 I3T.883 20 4188.79 1092.02 1717.995 Note. — T o compute weight of balls of other metals, multiply weight given in table by following multipliers: For Wrought Iron 1.067. I Brass .1.12. Steel 1.088. I Gun-metal 1.165. Wei glit and Diameter of Cast Iron Balls. Weight. Diameter. Weight. Diameter. Weight. Diameter. Weight. Diameter. Weight. Diameter. Lbs. Ins. Lbs. Ins. Lbs. Ins. Lbs. Ins. Lbs. Ins. I I.94 12 4-45 50 7.16 224 II. 8 1344 21.44 2 2-45 14 4.68 56 7-43 336 13-51 1568 22.57 3 2.8 l6 4.89 60 7-6 448 I4.87 1792 23.6 4 3.08 18 5-09 70 8.01 560 16.02 2016 24.54 ,5 3-32 20 5-27 80 8-37 672 17.02 2240 25.42 6 3-53 25 5.68 90 8.71 784 I7.9I 2800 27.38 7 3-72 28 5-9 IOO 9.02 896 18.73 3360 29.I 8 3.89 30 6.04 112 9-37 1008 19.48 3920 30.64 9 4.04 40 6.64 l68 10.72 1120 20.17 4480 32.03 B eng tli of Horseshoe IS'ails. By Fumhers. No. 5 1.5 Ins. I No. 7 “ 6 1.75 “ I “ 8 1.875 Ins. I No. 9 2.25 Ins. 2 “ | “10 2.5 “ Lengths of Iron UNTails, and Number in a Ifb. Size. | L’gth. No. Size. L’gth. No. Size. L’gth. No. I Size. L’gth. No. Size. L’gth. No. 3 d ) Ins. 1-25 420 5 d . Ins. i -75 220 8 d. Ins. 2-5 IOO 126 ?. Ins. 3-25 52 30 cl Ins. 4 24 4 1 1 -5 270 6 2 175 10 3 65 20 | 3-5 28 40 4.25 20 154 NAILS, SPIKES, TACKS, ETC, "W^ronght Iron Cut Nails, Tacks, Spikes, etc* ( Cumberland Nail and Iron Co.) Lengths and Number per Lb, Ordinary- Size. Length. 3 fine 3 4 5 6 7 8 20 30 40 50 6o 4 5 6 8 6<* 7 8 No. per Lb. Size. Finishing. Length. No. per Lb. Ins. .875 I.0625 I.0625 1-375 i-75 2 2.25 2.5 2.75 3-5 3- 75 4- 25 4- 75 5 5- 5 Light 1-375 1- 75 2 Brads. 2 2- 5 2- 75 3- 125 Fence 2 2.25 2.5 2.75 3 716 588 448 336 216 166 118 94 72 50 32 20 17 14 373 272 196 163 96 74 50 96 66 56 50 40 6^ 8 10 12 20 30 40 WH WHL 6 ts OO • 3 I ^5 3.16 546.875 10480 . 7701 2.64 655 cast f * O/y A English. (D. K. Clark.) Wrought iron Cast iron Steel Copper plates Gun-metal. . . 7.698 7.217 7.852 8.805 8.404 .278 .26 .283 • 318 •304 3-6 480 Tin 7.409 .268 3-74 3-84 45o Zinc 7.008 .253 3-95 3-53 489.6 Lead 11.418 .412 2.43 3-15 549 Brass, cast. . . 8.099 .292 3-42 2.02 524 “ wire.. 8.548 .308 3-24 462 437 712 505 533 WROUGHT AND CAST IRON. To Compute Weight of' Wrought 01* Oast Iron. Rule.— Ascertain number of cube inches in piece; multiply sum by .2816* for wrought iron and .2607* for cast, and product will give weight in pounds. Or, for cast iron multiply weight of pattern, if of pine, by from 18 to 20, accord- ing to its degree of dryness. Example.— What is weight of a cube of wrought iron 10 inches square by 15 inches in length ? 10 X 10 X 15 X .2816 = 422.4 lbs. COPPER. To Compute Weight of Copper. Rule.— A scertain number of cube inches in piece; multiply sum by .32118* and product will give weight in pounds. Sheathing and. Braziers’ Sheets. For dimensions and weights see Measures and Weights, pages 118-121, 131, 142. LEAD. To Compute Weight of Lead. Rule. — A scertain number of cube inches in piece; multiply sum by .41015 * and product will give weight in pounds. Example. — What is weight of a leaden pipe 12 feet long, 3.75 inches in diameter and 1 inch thick? Bl J Pule in Mensuration of Surfaces , to ascertain Area of Cylindrical Rings. Area of (3.75 -f- 1 -|- 1) — 25.967 “ “ 3.75 — 11. 044 ^ 00 Difference, 14.923 {area ofnng) x 144 (12 feet) = 2148.912 X .410 15 = 881.376 lbs. n * BRASS. To Compute Weight of Ordinary Brass Castings. Rule.— A scertain number of cube inches in piece; multiply sum by .2022 * and product will give weight in pounds. * y ’ crnvitv «f tL°Ltt«i b , e !i nch a8 E ere 5 ivei ? are for the ordinary metals; when, however, the specific & rnit e s\re 9 g1v r em° n “ kn ° Wn > the we * hfc of a tube * » ^uld DIMENSIONS AND WEIGHTS OF BOLTS AND NUTS. Dimensions and Weights of Wrought Iron Bolts and. JNTvits. SQUARE AND HEXAGONAL HEADS AND NUTS. rtoagh, and from .25 Inch to 4 Inches in Diameter , Square Head and USTirt. Diameter of Bolt. Ins. .25 .3125 •375 •4375 •5 .5625 .625 .6875 •75 .8125 .875 1 1.125 1.25 1-375 i-5 1.625 i-75 1.875 Width. Head. I Nut. Ins. .36 •45 •54 .63 .72 .82 •9 1 1 1.09 1. 18 1.27 1.45 1.63 1.81 1.99 2.17 2.36 2- 54 2.72 2.9 3.08 3.26 3- 44 3.62 3.81 3- 99 4.17 4- 35 4.71 5- 07 5-44 5.8 Diagonal. Head. Ins. •49 .58 .67 .76 .84 •94 1.03 1. 12 1. 21 i-3 i-39 i.57 1- 75 1.94 2.12 2 - 3 2.48 2.66 2.84 3° 2 3.21 3- 39 3-57 3-75 3-93 4.11 4.29 447 4.84 5-2 556 5-9 2 Ins. •51 .64 .76 .89 1.02 1. 16 I.29 1. 41 1- 54 1.67 1.8 2.05 2- 3 2.56 2.81 3*07 3- 34 3-59 3- 85 4- 1 4*35 4.61 4.86 5.12 5- 49 5-64 5-9 6.15 6.66 7-i7 7.69 8.2 Depth. Head. I Nut. Ins. .69 .82 •95 1.07 1. 19 1- 33 1.46 1.58 1.71 1.84 1.96 2.22 2.47 2- 74 3 3- 25 3*51 3- 76 4.02 4.27 4- 54 4- 79 5- 05 5-3 5-56 5.81 6.07 6.32 6.84 7*35 7.86 8.37 Ins. •25 •3 •34 •4 .44 .48 •53 .58 •63 .67 .72 .81 •9 1 1. 1 1. 18 1.28 i-37 1.46 1.56 1.65 i-75 1.84 1.94 2.03 2.12 2.22 2.31 2.5 2.68 2.87 3.06 .25 •3 I2 5 •375 •4375 •5 •5625 .625 .6875 •75 .8125 •875 1 1. 125 1.25 i-375 i-5 1.625 i-75 1.875 Weight. Head and Nut. , Bolt jper Inch. 2.125 2.25 2.375 2-5 2.625 2- 75 2.875 3 325 3- 5 3-75 4 Lbs. .024 •043 .068 .104 •145 .204 •273 •356 •454 •565 .696 1.013 1.416 1.923 2.543 3- 234 4- 105 5.087 6.182 7-49 1 8.936 10.543 12.335 14-359 16.549 18.897 21-545 24.464 30.922 38-39 1 47.168 56.882 Threads per Inch. Lbs. .014 .022 .031 .042 .055 •07 .086 .104 .124 .145 .168 .22 .278 •344 .416 •495 .581 .674 •773 .88 •993 i-ii3 1.24 1-375 i-5i5 1.663 1.818 1.979 2.323 2.694 3093 3-518 20 18 16 14 13 7 7 6 6 5.5 5 5 4-5 4-5 4-5 4-375 4-25 4 4 3-75 3-5 3-5 3-25 3 3 2.125 2.25 2.375 2.5 2.625 2.75 2.875 3 3-25 3-5 3-75 4 j"-' o w Finished. — Deduct .0625 from diameters of bolts and depths of all heads and nuts. Screws with square threads have but one half number of threads of those with triangular threads. Note.— The loss of tensile strength of a bolt by cutting of thread is, for one of 1.25 ins. diameter, 8 per cent. The safe stress or capacity of a wrought iron boit and nut may be taken at 5000 lbs. per square inch. Preceding width, depth, etc., are for work to exact dimensions, whether forged or finished. To Compute Weiglit of a I3olt and Nut. Operation. — Ascertain from table weight of head and nut for given di- ameter of bolt, and add thereto weight of bolt per inch of its length, multi- plied by full length of its body from inside of its head to end. Note.— Length of a bolt and nut for measurement , as such, is taken from inside of head to inside of nut, or its greatest capacity when in position. DIMENSIONS AND WEIGHTS OF BOLTS AND NUTS. 157 Illustration. —A wrought iron bolt and nut with a square head and nut is 1 inch in diameter and 10 inches in length; what is its weight? Weight of head and nut 1.013 lbs. “ bolt per inch of length .22 X 10 = 2.2 “ 3.213 “ Hexagonal Head ancl ISTnt. Diameter of Bolt. Wi Head. 1th. Nut. Diag Head. onal. Nut. Di Head. epth. Nut. Weig Head and Nut. ;ht. Bolt per Inch. Threads per Inch. Ins. Ins. Ins. Ins. Ins. Ins. Ins. lbs. Lbs. No. •25 •375 •5 •43 .58 •25 •25 .022 .014 20 •3125 •4375 .5625 •5 •65 •3 •3125 .037 .022 18 •375 .5625 .6875 .65 •79 •34 •375 .062 .031 l6 •4375 .625 •75 •72 .87 •4 •4375 .094 .O42 14 •5 •75 •875 .87 1 •44 •5 •134 •055 13 •5625 .8125 •9375 •94 1.08 .48 •5625 .18 .07 12 .625 •9375 1.0625 I.08 1.23 •53 .625 •249 .086 II .6875 1 1. 125 1. 16 i -3 .58 •6875 .318 .IO4 II •75 1. 125 1-2.5 i -3 1.44 •63 •75 •413 .124 IO .8125 1.25 1-375 1.44 i -59 •67 .8125 .522 •145 IO .875 1-3125 1-4375 1.52 1.66 •72 .875 •639 .168 9 1 i -5 1.625 i -73 1.88 .81 1 •931 .22 8 1. 125 1.6875 1.8125 1-95 2.09 •9 1.125 I.299 .278 7 1.25 1-875 2 2.17 2.31 1 1.25 1-759 •344 7 1-375 2 2.1875 2.31 2-53 1. 1 1-375 2.263 .416 6 i -5 2.25 2-375 2.6 2.74 1. 18 i -5 2.958 •495 6 1.625 2.4375 2.5625 2.81 2.96 1.28 1.625 3-741 .581 5-5 i -75 2.625 2-75 3-03 3-i8 i -37 I -75 4-654 •674 5 1-875 2.8125 2-9375 3-25 3-39 1.46 i -875 5-675 •773 5 2 3 3-125 3-46 3.61 1.56 2 6.854 .88 4-5 2.125 3-1875 3-3125 3.68 3-83 1.65 2.125 8.163 •993 4-5 2.25 3-375 3-5 3-9 4.04 i -75 | 2.25 9.658 1-113 4-5 2-375 3-5625 3-6875 4.11 4.26 1.84 2-375 11.263 1.24 4-375 2-5 3-75 3-875 4-33 4-47 1.94 2.5 I 3 -I 49 i -375 4-25 2.625 3-9375 4.0625 4-55 4.69 2.03 2.625 I 5 .I 5 i- 5 i 5 4 2.75 4-125 4-25 4-77 4.91 2.12 2-75 17.285 1.663 4 2.875 4-3125 4-4375 4.99 5.12 2.22 2.875 I 9 . 75 I 1.818 3-75 3 4-5 4.625 5-2 5-34 2.31 3 22.378 1.979 3-5 3-25 4-875 5 5-63 5-77 2-5 3-25 28.258 2.323 3-5 3-5 5-25 5-375 6.06 6.21 2.68 3-5 35 -o 8 i 2.694 3-25 3-75 5-625 5-75 6-5 6.64 2.87 3-75 43 -I 78 3-093 3 4 6 6.125 6.93 7.07 3.06 4 51.942 3-518 3 Finished. — Deduct .0625 from diameters of bolts and depths of all heads and nuts. For Wood or Carpentry, Head and Nut (Square), 1.75 diameter of bolt. Depth of Head, .75, and of Nut , .9. Washer. — Thickness, .35 to .4 of diameter of bolt, on Pine 3.5 diameter, and Oak 2.5. English. Molesworth gives following elements of Thread of Bolts : Angle of thread , 55 0 . Depth of thread = Pitch of screw. Number of threads per Inch. — Square , half number of those in angular threads. Depth of thread . — .64 pitch for angular and .475 for square threads. 0 158 DIMENSIONS AND WEIGHTS OF BOLTS AND NUTS. ITrencli Standard Bolts and iNoxts. (Armengautf s.) HEXAGONAL HEADS AND NUTS. Equ Diamete of Bolt. late X 0 . «I ** Threads cl* per Inch. ^ iangu Thick Head. lar 1 ness. Nut. Breadth ^ across Flats. a l Safe Tensile Stress. S Diameter of Bolt. quare c -a' f ,2 Threads ^ per Inch. — — os ad. Is? Safe Tensile Stress. Min. Ins. Ins. No. Ins. Ins. Ins. Lbs. Mm. Ins. Ins. No. Ins. Lbs. 5 .2 .13 18.I .24 .2 •55 44 20 •79 .O72 6.57 1.82 717 7-5 •3 .22 16 •3 •3 .68 99 25 .98 .081 5-97 2.01 I I42 IO •39 • 3 1 14.I .38 •39 .88 178 30 1. 18 •093 5-4 2.22 1635 12.5 •49 •39 12.7 •44 •49 1.04 277 35 1.38 .1 4-93 2.4I 2 2l8 15 •59 .48 ii -5 •52 •59 1.2 400 40 i -57 .106 4-53 2.63 2 912 I 7-5 .69 .58 10.6 .58 .69 1.4 545 45 1.77 .114 4.2 2.85 3^4 20 •79 .66 9.8 .66 •79 i -5 713 50 1.97 .128 3 - 9 1 3-07 4547 22.5 .89 .76 9.1 .72 .89 1.68 902 55 2.17 •13 3-65 3-3 5288 25 .98 .84 8-5 .8 .98 1.84 1 120 60 2.36 .14 343 3-5 6540 3 ° 1. 18 1.02 7-5 •94 1. 18 2.16 1635 65 2.56 •15 3-23 3-7 7 660 35 1.38 1.2 6.7 1.08 1.38 2.48 2 218 70. 2.76 .158 3.06 3 - 9 2 8893 4 ° 1.58 1.4 6 1.22 1.58 2.8 2912 75 2-95 .166 2. 92 4 -i 3 10 214 45 1.77 1.56 5-5 1.36 1.77 3-2 3674 80 3 -i 5 .174 2.76 4-36 11 603 50 1.97 1-74 5 -i i -5 1.97 3-44 4 547 85 3-35 .183 2.63 4-58 13 100 55 2.17 1.92 4-7 1.64 2.17 3-76 5288 90 3-54 .192 2.51 4.78 14 794 60 2.36 2.08 4.4 1.74 2.36 4.08 6540 95 3-74 .2 2.41 5 16352 6 5 2.56 2.26 4.1 1.92 2.56 4.4 7 660 100 3-94 .209 2.31 5.22 18 144 70 2.76 2.44 3-8 2.06 2.76 4-7 8893 105 4 -i 3 .22 2.22 5*43 20000 75 2 -95 2.6 3-5 2.2 2-95 5 10214 no 4-33 .226 2.13 5.66 21950 80 13-15 2.78 1 3*4 2-34 3 -i 5 5-35 11 468 115 4-53 •23 2.06 5-87 23990 English. Bolts and HSTnts. (Whitworth's.) Hexagonal Heads and Nuts, and. Triangular Threads. Diame Bolt. S’ Base of r* Thread. Threads per Inch. De Head. ipth. Nut.- Width of Head and Nut. Diam Bolt. eter. Base of Thread. Threads per Inch. Dep Head. th. Nut. Width of Head and Nut. — Ins Inch. No. Inch. Ins. Ins. Ins. Ins. No. Ins. Ins. Ins. .125 •093 40 .109 .125 •338 1.25 I.067 7 I.O94 1.25 2.048 •1875 .134 24 .164 •1875 •448 1-375 I.l6l 6 1.203 1-375 2.215 .2187 •25 .186 24 20 .219 •25 •525 i -5 1.625 1.286 I.369 6 5 i* 3 12 1.422 i -5 1.625 2.413 2.576 .3125 •375 .241 •295 l8 l6 •273 .328 •3125 •375 .601 •709 i *75 i -875 I.494 1-59 5 4-5 I- 53 1 1.641 i -75 1.875 2.758 3.018 •4375 •346 14 .383 •4375 .82 2 1-715 4-5 i -75 2 3- T 49 .5 •393 12 •437 •5 • 9*9 2.125 1.84 4-5 1.859 2.125 3-337 •5625 •456 12 •492 •5625 1. 01 1 2.25 i -93 4 1.969 2.25 3-546 .625 .508 II •547 .625 I.IOI 2-375 2.055 4 2.078 2-375 3-75 .6875 • 57 1 II .601 .6875 1. 201 2-5 2.18 4 2.187 2-5 3-894 •75 .622 IO .656 •75 1.3 01 2.625 2.305 4 2.297 2.625 4.049 .8125 .684 IO • 7 11 .8125 i *39 2-75 2.384 3-5 2.406 2-75 4.181 .875 •733 9 .766 .875 1.479 2.875 2.509 3-5 2.516 2.875 4-346 •9375 1 •795 •84 9 8 .82 .875 •9375 1 i -574 1.67 3 3-25 2.634 2.84 3-5 3-25 2.625 3 4 - 53 1 1. 125 I.942 7 .984 1. 125 1.86 3-5 3.06 3-25 1 ~ RETENTION OF SPIKES AND NAILS. 159 Sqxiare Heads and. N'nts. ( Whitworth's .) Diameter. Threads per Inch. Diameter. Threads per Inch. Diameter. Threads, per Inch. Bolt. Base of Thread. Bolt. Base of Thread. Bolt. Base of Thread. Ins. Ins. No. Ins. Ins. No. Ins. Ins. No. 3-75 3-25 3 4-5 3-875 2.875 5-25 4-4375 2.625 4 3-5 3 4-75 4.0625 2-75 5-5 4.625 2.625 425 3-75 2.875 5 4-25 2-75 6 4-875 2-5 "Weight of Heads and Nuts in Lbs. ( Molesworth .) Hexagonal , 1.07 D 3 . Square , 1.35 3 D 3 . D representing diameter of bolt in inches. Fieteii.tiven.ess of NV' rou.glrt Iron Spikes and. UNTails. Deduced from Experiments of Johnson and Bevan. SPIKES. 43 Vi O O .J Spike. Wood. 1 a, g-s ill 0 3 £ HI Remakks. ea O Qm 0^73 as is Ins. Ins. Ins. Lbs. Square Hemlock) •39 3-5 1297 1.58 Seasoned in part. V it >H Chestnut •37 •38 3-5 1873 2.l6 Unseasoned. It * Yellow pine •375 •375 3-375 2052 2-37 Seasoned. it * White oak •375 •375 3-375 39 IQ 4-52 a It Locust •4 •4 3-5 5967 6-33 a Flat narrow. . Chestnut •39 •25 3-5 2223 3-93 Unseasoned. “ “ White oak •39 •25 3-5 3990 7-05 Seasoned. Locust •39 •25 3-5 5673 9-32 ti “ broad . . Chestnut •539 .288 3-5 2394 2.66 Unseasoned. it n White oak •539 .288 3-5 533° 5-7i Seasoned. it u Locust •539 .288 3-5 7040 7.84 ti Square) £• Hemlock) •4 •39 3-5 1638 i-75 Seasoned in part. Chestnut) •4 •39 3-5 1790 1.81 Unseasoned. “ J Locust) •4 •39 3-5 3990 4.17 Seasoned in part. Round and) grooved. . J Ash Diam. .5 3-5 2052 2.21 Seasoned. u U U •5 3-5 2451 2.41 U it White oak U .48 3-5 3876 3-2 u * Burden’s patent. t Soaked in water after the spikes were driven. NAILS. Depth of Insertion. Force required to draw it. Pressure required Nail. Length. Pine. Hemlock. Elm. Oak. Beech. to force them into Pine. Ins. Ins. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Sixpenny 2 I 187 3 12 327 507 667 235 tt 2 i-5 327 539 571 675 889 400 a 2 2 530 857 899 1394 1834 6lO G-eneral Remarks. With a given breadth of face, a decrease of depth will increase retention. In soft woods, a blunt - pointed spike forces the fibres downwards and backwards so as to leave the fibres longitudinally in contact with the faces of the spike. l6o ANGLES AND DISTANCES. DISTANCES AND ANGLES. To obtain greatest effect, fibres of the wood should press faces of the spike in direction of their length; thus, a round blunt bolt, driven into a hole of a less diameter, has a retention equal to that of any other form, when who y dr The’ retei^tlon^of^ a spike, whether square or flat, in unseasoned chestnut, from two to four inches in length of insertion, is about 800 lbs. per square inch of the two surfaces which laterally compress the faces of the spike. When wood was soaked in water, after spikes were driven, order : of their retentive power was Locust, White oak, Chestnut, Hemlock, and 1 ellow Pme. Threads per Inch . . . 1 - I2 S 1 •25 I •375 I •5 I •75 I 1 I j 1.25 I 1-5 I 1-75 1 1 28 1 19 1 19 1 14 1 14 1 11 | | 11 1 11 1 11 1 ANGLES AND DISTANCES. Angles and. Distances corresponding to Opening . ® Rule of Two Feet. of a Angle. | Distance. Angle. O Ins. 0 I .2 19 2 • .42 20 3 .63 21 4 .84 22 5 1.05 23 6 I .26 24 7 1.47 25 8 I .67 26 9 1.88 27 10 2.09 28 11 2-3 29 12 2.51 30 13 2.72 31 14 2.92 32 i 5 3 -i 3 33 16 3-34 34 17 3-55 35 18 3-75 36 Distances and 1 Distance Distance. Ins. 7.61 7.81 8.01 8.2 8.4 8.6 8.8 8.99 9.18 938 9-57 9.76 9-95 10.14 10.33 10.52 10.71 10.9 Angle. 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 7 1 72 Distance. Ins. 11.08 II.27 11-45 II.64 11.82 12 I2.l8 12.36 12.54 12.72 12.9 I3.O7 13-25 1342 13.59 13.77 13.94 i4.II Angle. | Distance. 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 Ins. I 4.28 I 4.44 14.61 14.78 14.94 i 5 .II 15.27 15-43 15.59 1575 15-9 16.06 16.21 16.37 16.52 16.67 16.82 16.97 Rule of Two Feet. to Opening of a Distance. Ins. .25 •375 •5 .625 •75 .875 1 1.25 i -5 1 - 75 2 2.25 2.5 2 - 75 Angle. Distance. 1. 12 I. 48 2.24 2-59 3*35 4.12 448 5.58 7* 1 8.22 9-34 10.46 II . 58 13.1 Ins. 3 3-25 3.5 3 - 75 4 4.25 4 - 5 4 - 75 5 5.25 5 - 5 5-75 6 6.25 Angle. Distance. 14.22 15.34 16.46 17.58 1 9 .11 20.24 21.37 22.5 24.4 25.16 26.3 2744 28.58 30.12 Ins. 6.5 6 - 75 7 7.25 7 - 5 7-75 8 8.25 8.5 8.75 9 9.25 95 9-75 Angle. 3 I2 6 3 2 - 4 33 - 54 35.09 36.24 37*4 38.56 40.12 41.28 42.46 44.2 45-2 46.38 47.56 Distance. Ins. 10 10.25 10 - 5 10.75 11 11.25 11- 5 n-75 12 12.25 12.5 12.75 13 13.25 Angle. 49 * x 4 50.34 51.54 53.14 54*34 55-54 57 .i 6 58.38 60 61.23 62.46 64.1 65-36 67.02 Distance. ] Angle. Ins. 13 - 5 13.75 14 14.25 14 - 5 14-75 15 15.25 15.5 15*75 1 6 16.25 16.5 16.75 68.28 69-54 71.22 72-5 74.2 75.5 77*22 78-54 80.28 82.2 83-36 85.14 86.52 88.32 WIRE ROPE. 161 WIRE ROPE. Wire rope of same strength as new Hemp rope will run on sheaves of like diameter ; but greater diameter of sheaves, less the wear. Short bends should be avoided, and wear increases with the speed. Adhesion is same as that of hemp rope. It should not be coiled, but should be wound as upon a reel. When substituting wire rope for hemp, it is well to allow for former same weight per foot which experience has approved of for latter. As a general rule, one wire rope will outlast three of hemp*. To guard against rust, stationary rope should be coated once a year with linseed- oil, or well painted or tarred. Running rope in use does not require any protection. Where great pliability is required, centre or core of rope should be of hemp. Annealing wire, in rendering it more pliable than when unannealed, reduces its elasticity and consequent strength from 25 to 50 per cent. Running rope is made of finer wire than standing rope. For safe working load, deduct one fifth to one seventh of ultimate strength, according to speed and vibration. It is better to increase load than speed, as it increases wear. Standing rigging of a vessel of wire rope is one fourth less in weight than when of hemp. Rope of 19 wires to a strand is more pliable than one of 7 and 12 wires, and hence it is better suited to operation over small drums, for hoisting, etc. Ultimate strength of iron ropes is 4480 lbs. for each pound in weight per fathom, and for galvanized steel 6720 lbs. Strength per square inch of section of a rope is about 53 per cent, of an equal section of solid metal of same tensile strength per square inch. Steel ropes may be one third less in weight than iron for same load. Their durability is much greater, especially when required to run rapidly over sheaves. Hemp should be one third heavier than iron. Steel wire No. 14 W. G. = .083 inch, weight 2 lbs. per yard, will bear a stress of 2000 lbs. The combined sectional area of the wires in a cable is to the area of the cable as 1 to 1.3. Hence, to ascertain areas of the wires in a cable multiply diameter by .77, and for areas of the voids, multiply area of cable by .23. In short transmissions, it is necessary to connect rope quite taut, and an additional diameter of two numbers of rope must be given to it. In long transmissions, when there is an insufficiency of height to admit of a proper deflection of rope, and it becomes necessary to connect it very taut, an additional diameter of one number of rope must be given to it. When distance exceeds 350 feet, transmission should be divided into two or more equal lengths by aid of intermediate wheels. Rope Nos. 7 and 8 (. Roebling's ) are made with Nos. 1 and 2 as strands, and twisting six of them around a hemp centre. Results of an Experiment with Galvanized Wire. A strand of 2-inch wire rope broke with a strain of 13 564 lbs., and a piece of a like rope, when galvanized, withstood a strain of 14 796 lbs. be- fore breaking. 0 * 1 62 WIRE ROPES. Elements of Running and Standing Wire Rope. J. A. Roebling's Sons Co., New York. 19 Wires in a Strand. Diam. Circum. j Iro> Breaking Weight. r. Safe Load. Circum. of Hemp Rope. Weight per Foot. Ins. Ins. Lbs. Lbs. Ins. Lbs. • 5 1-5 6960 I OOO 3-5 •35 .5625 I.625 • 8 480 I 500 4 •44 1 .625 2 10 260 2 5OO 4-5 •7 •75 2.25 17 280 3 500 5 ; .88 .875 2.75 23 000 5 ooo 6 1.2 1 3- I2 5 32 000 6000 7 i -58 1. 125 3-5 40 000 8 ooo 8 2 1.25 4 54000 II ooo 9-5 2.5 i-5 4-375 70 000 14 ooo io-75 3-65 1.625 5 88 000 18 ooo 12 4* 1 x -75 5-5 108 000 22 OOO 13 5-25 2 6 130 000 26 OOO 14-5 6.3 2.25 6.75 148 000 30000 15-5 8 It Break mg Weight. Lbs. II OOO 13 OOO 18 OOO 26 OOO 40 OOO 48 OOO 60000 78 OOO IIO OOO 128 OOO 156 OOO 200 OOO 260000 Cast Steel. Circum. of Hemp Rope. Safe Load. Lbs. 2 OOO 3 000 4 OOO 6006 8 000 10000 12 000 16 000 22 000 26 000 34000 40000 44 000 Wires in a Strand. Diam. Cast Steel. Circum. of Hemp Rope. Safe Load. Lbs. I 050 1 25O 2 OOO 2 4OO 2 500 3 500 45OO 6000 7 000 10000 13 000 16 000 20 000 25 000 32 000 Weight per Foot. Ins. 2.375 3 3- 75 4- I2 5 4- 75 5 5- 5 6- 5 7- 2 5 8- 5 10 10.75 12 13 15 Lbs. .125 .16 .19 .23 •31 .41 .51 .68 .86 1. 12 i.5 1.82 2.28 2.77 3- 37 r A 021; 72 OOO i 10 cue /O 0*0/ ^ Note. -When made with wire centre instead of hemp, weight is ,o per cent. more. Galvanized Charcoal Iron Wire. Rigging and Derrick Grays. 12 Wires in a Strand. "Vessels’ Circum. Circum. of Hemp Rope. Breaking Weight. Ins. Ins. Lbs. 3 6 24OOO 3-25 6-5 28000 3-5 7 32 OOO 3-75 7-5 40 OOO 4 8 460OO 4-25 8.5 52 OOO Safe Load. Lbs. 6000 7 OOO 8000 10000 II 500 13000 Weight Circum. 1 Breaking Safe per Foot. Circum. of Hemp Rope. W eight. Load. Lbs. Ins. Ins. Lbs. Lbs. 1-33 4-5 9 60 000 15000 x -58 4-75 9-5 66000 1.6 500 I.83 5. 10 7OOOO 17 500 2 5-25 10.5 80 000 20 000 2.46 5-5 11 86000 21 500 2.66 6 12 IOO OOO 25000 I Weight I P ttr F oot. Lbs. 3 3- 46 366 4.12 4.46 4- 83 WIRE ROPES AND CABLES. 163 Galvanized Charcoal Iron. "Vessels’ .Rigging and Derrick Grays. J. A. Roebling's Sons Co ., New York . 7 Wires in a Strand. Circum. Circum. of Hemp Rope. Breaking Weight. Safe Load. Weight per Foot. Circum. Circum. of Hemp Rope. Breaking Weight. Safe Load. Weight Foot. Ins. Ins. Lbs. Lbs. Lbs. Ins. Ins. Lbs. Lbs. Lbs. I 2 4000 I OOO .125 3-5 7 32 000 8000 I.79 1.25 2.5 5000 I 250 •25 3-75 7-5 40 000 10 OOO 2 i -5 3 7000 1750 •334 4 8 460OO II 500 2.46 i -75 3-5 IO OOO | I 2500 .427 4-25 8-5 52 000 13000 2.67 2 4 14000 3 500 .583 4-5 9 60 000 15000 3 2.25 4-5 16000 4000 .708 4-75 9-5 66000 16500 3-42 2-5 5 18000 4 500 •875 5 10 70 000 17500 3.66 2-75 5-5 20 000 5000 1. 1 7 5-25 10.5 80 000 20 000 4.08 3 6 24OOO 6000 i *33 5-5 11 86000 21 50O 4.41 3-25 6-5 28 000 7 000 1.58 6 12 IOO OOO 25 000 4.81 Grange, 'Weight, and Length of Iron "Wire. No. 6/0 5 /o 4/0 3/0 2/0 1/0 Diann. Weight per 100 Feet. = % -a — c ro 3 S 3 | Area. Gauge. Diam. Weight per 100 Feet. Weight of one Mile. 63 lbs. Bundle. Area. Inch. Lbs. Lbs. Feet. j Sq. Inch. No. Inch. Lbs. Lbs. Feet. Sq. Inch. .46 56.I 2962 1 12 .166 19 3:6 •063 1.05 55 6000 .003 II 7 •43 49.OI 2588 129 .145 22 17 •054 •77 41 8 182 .002 29 •393 4O.94 2l62 154 .121304 18 •047 .58 31 IO862 .OOI 734 .362 34-73 1834 l8l .102 921 19 .041 •45 24 I4OOO .OOI 32 • 33 1 29.04 1533 217 .086 O49 20 •035 •32 17 19687 .OOO 962 •307 27.66 1460 228 .O74O23 21 .032 •27 14 23333 .OOO 804 .283 21.23 1 121 296 .062 9OI 22 .028 .21 II 30 000 .000 61 5 .263 18.34 968 343 •054 325 23 .025 •175 9.24 360OO .000 491 .244 I 5-78 833 399 .046 759 24 •023 .14 7-39 45 000 .000415 .225 13-39 707 470 i -039 76 25 .02 .116 6.124 54 310 .000314 .207 ir -35 599 555 •033653 26 .018 •093 4.91 67 742 .000 254 .192 9-73 5 i 4 647 .028952 27 .OI7 •083 4.382 75 903 .000227 .177 8.03 439 759 .024 605 28 .Ol6 •074 3907 85 135 •OOO 201 .162 6.96 367 905 .020612 29 .015 .061 3.22 IO3 278 .OOO 176 .148 5.08 306 1086 j .017 203 30 .OI4 •054 2.851 1 16 666 .OOO I54 •i 35 4-83 255 r 3°4 •014 3 L 3 31 •0135 •05 2.64 126000 .OOO I33 .12 3.82 202 1649 .011 309 32 •013 .046 2.428 136956 .OOO I32 .105 2.92 154 2158; .008 659 33 .Oil •037 I -953 170270 •OOO O95 ,092 2.24 118 2813 .006647 34 .OI •03 1.584 210000 .OOOO78 .08 1.69 89 3728 : .005 026 35 •OO95 •025 1.32 252 000 .OOO 07I .072 i -37 72 4598 : .004071 36 .009 .021 1.161 286 363 .OOO064 Galvanized Steel Cables for Suspension Bridges. Diameter. Ultimate Strength. Weight per Foot. Diameter. Ultimate Strength. Weight per Foot. Diameter. I Ultimate Strength. Weight per Foot. Ins. Lbs. Lbs. Ins. Lbs. Lbs. Ins. Lbs. Lbs. i -5 130 000 3-7 1-875 200 OOO 5-8 2-375 360OOO IO 1.625 150 OOO 4-35 2 220 000 6-5 2.5 400 000 II .3 i -75 19OOOO 5-6 2.25 310000 8.64 2.625 44OOOO 13 IRON, STEEL, AND HEMP HOPE. "Weight and. Strength, of Single Strand and Cable laid Fence "Wire. [F. Morton & Co.) Length per iooo lbs* Of a Of Rope. Strands. No. Single Wire of equal Diameter. No. No. Inch. 3 2A 8 .159 4 2 7 .174 7 I 6 .191 7 O 5 .209 1 Strand. Feet. 20 090 14 730 13 I2 5 Feet. 15 270 12 79O IO 580 8928 1 No. Single Wire of equal Diameter. Len per io< Of a Strand. gth x> lbs. Of Rope. No. Inch. Feet. Feet. I OO 4 .229 83OO 7366 3/0 3 •25 8036 6228 U/O 2 •274 7500 1 5/0 1 •3 5090 4286 No. and diameter of wire is that of Ryland s Bros., pp. 122 4. Hemp, Iron, and. Steel. (K. f bewail <& Co.) ROUND. HEMP. | Weight Circumference. per 1 Foot. Ins. Lbs. 2.75 •33 3-75 .66 4-5 .83 5-5 1. 16 6 i -5 6.5 1.66 7 2 7-5 2.33 8 2.66 8.5 3 9-5 3.66 IO 4-33 11 5 12 5.66 Dimensions. 4 X .5 3 - 33 ] 5 Xi.25 4 5-5 X 1.375 4-33 5 - 75 XI .5 4.66 6 X 1.5 5 7 X 1.875 6 8.25X2.125 6.66 8.5 X2.25 7-5 9 X2.5 8.33 9-5 X 2-375 : 9- I 6 10 X2.5 10 IRON. cumference. J Weigh J per I Foot. STEEL. I 1 Circumference. Weight £o\. Ins. Lbs. Ins. Lbs. I .16 — — i -5 • 2 5 I .16 1.625 •33 — i -75 .42 i -5 •25 1.875 •5 — — 2 .58 1.625 •33 2.125 .66 1-75 .42 2.25 •75 — 2.375 .83 1.875 •5 2-5 .92 — 2.625 1 2 .58 2.75 1.08 2.125 .66 2.875 1. 16 2.25 •75 3 1.25 — — 3 I2 5 i .33 2.375 .83 3* 2 5 1.41 — 3.375 i .5 2.5 .92 3-5 1.66 2.625 1 3-625 1.83 2.75 1.08 3-75 2 — 3.875 2.16 3- 2 5 1*33 4 2-33 — — 4.25 2.5 3*375 i -5 4*375 2.66 — — 4-5 3 3.5 1.66 4.625 333 1 3-75 2 Tensile Strength. Ultimate Strength. Lbs. 672 I 008 1344 1 680 2 016 2 352 2 688 3024 33 60 3696 4032 4368 4704 5040 5 376 5672 6048 6 720 7 392 8064 8736 9408 10080 10 752 12096 13440 FLAT. Dimensions. 2.25 x .5 2.5 x .5 2.75 X.625 3 X .625 3.25 X.625 3.5 X.625 3 - 75 X.6875 4 X .6875 4- 25 X.75 4-5 X .75 4.625X.75 Dimensions. 1.85 I — — 4928 2.16 — — 5824 2-5 — — 6 720 2.66 2 x .5 1.66 7 168 3 2.25 x. 5 1.83 8064 3 * 33 2.25 X - 5 2 8960 3.66 2.5 x .5 2.16 9850 4.16 2-75 X - 375 2-5 11 200 4.66 3 X .375 2.66 12544 5-33 3.25 X . 375 3 14 336 5.66 1 3-5 X .375 3*2 3 I 5 2 3 2 Lbs. 4480 6 720 8960 11 200 13440 15 680 17920 20 160 22400 24640 26680 29 120 3*36° 33600 36840 38 080 40320 44800 49 280 53 76° 58240 62 720 67 200 71 680 80640 89600 44800 51520 60480 62 720 71 680 80640 89600 100800 112000 125440 134400 ROPES AND CHAINS. 165 From preceding tables following results are determined: Ultimate Strength Safe Load per Lb. Weight per per Lb. Weight per per Square of Circum- Foot. Foot. ference in Inches. Lbs. Lbs. Lbs. Hemp 15 OOO 4550 IOO Iron 22 OOO 4500 600 Steel ( 3OOOO f 6000 r 1000 (45 500 \ 8000 (1300 ROUND AND FLAT MINING ROPES. (MM. Harmegnies , Dumont <& Co., Anzin, France.) For a Depth of 400 Metres or 440 Yards, Round. Flat. No. Diameter. Weight per Foot. Safe Load. No. of Strands. Width. Thick- ness. Weight per Foot. Safe Load. Ins. Lbs. Lbs. Ins. Ins. Lbs. Lbs. 17 ♦51 2.l6 560 9 2.4 •55 2 3 3 6 0 16 •59 1.66 1120 6 . 2.8 •59 2.13 4032 15 •63 1.26 1680 / 3-2 •63 2.66 4 480 14 • 7 i 1 2240 3*2 .67 3 5600 13 .83 .83 3360 ! 3-5 •79 3-33 6 720 12 .98 .66 4480 f 4-3 .67 3.66 7840 II 1. 1 •5 5600 0 3-9 .83 4 8 960 IO i -3 •33 6720 8 4-7 •79 4-33 10080 8 5 -i .87 5-33 11 200 Ropes and. Ch.ai.ns of Equal Strength. Diameter of Iron Chain. CIRCI Hemp Rope. UMFEREJs Crucible Steel Rope. CE. Charcoal lion Rope. Steel Rope. WEI Iron Rope. GET PER ] Hemp Rope. 700 T. Iron Chain. Safe Load. Ins. Ins. Ins. Ins. Lbs. Lbs. Lbs, Lbs. Tons. .218 75 2.75 — I — .14 •34 •5 •3 •25 3 — 1. 18 .21 .46 •65 •4 .281 25 3-5 I i -39 •17 .28 .67 .81 •5 •312 5 4-25 I.26 i -57 •25 •33 •75 .96 .6 •375 4-5 i -45 1.77 •3 •45 •S3 1.38 .8 •437 5 5 i -57 1.97 •35 •57 1. 16 I.76 1 .468 75 5-5 1.77 2.19 •45 •7 1.2 2.2 1-3 •5 5-75 1.96 2.36 •59 •83 1.6 2.63 1.5 .625 6-75 2.36 2-75 .85 1.08 2 4 - 2 i 2-3 .6875 7-75 2-75 3- I 4 1. 1 i -43 2.65 4-83 3 -i •75 8-75 2-95 3-53 1.28 1.8 3-35 5-75 3-8 •875 9-75 3-i4 3-93 i -45 2-3 4.6 7-5 4.8 •937 5 10.5 3-53 4-32 1.83 2.94 4.92 9-33 5-9 1.062 5 “"•75 3-93 4.71 2-33 3-56 5.83 10.6 7 1. 125 “•75 4-32 5 -i 2.98 4 6.2 11.9 8.2 1.25 14-75 4.71 5-5 3-58 4.8 8.7 14-5 9-5 1-375 15-25 4.81 5-89 3-65 5-6 9 17.6 11 i-5 I 5-75 5 -i 6.28 4.04 6.3 10. 1 20 !2.5 1.625 17-75 5-8 7.07 5.65 7-95 13-7 22.3 15-9 i -75 19-5 6-35 7.85 6-5 9.81 16.4 24-3 19.6 By experiments of U. S. Navy, hemp rope of this circumference has a breaking weight 0/71 309 lbs., and a wire rope of 5.34 ins. has equivalent strength. 1 66 WEIGHT, STRESS, AND TENSION OF ROPES. Circum- ference. Weight of Hemp and. Wire Rope. In Lbs. per Fathom. Hemp. Wire. (Molesworth.) Common. Good. Ins. 1 i -5 i -75 2 2.25 2*5 2.75 3 3-25 3-5 3-75 4 Lbs. .18 .41 •55 .72 .91 1.13 1.36 1.62 1.9 2.21 2.53 2.88 Lbs. .24 •54 •74 .96 1.22 i -5 1.82 2.16 2.54 2.94 3-38 3-84 Iron. Steel. Lbs. .87 I.96 2.66 3-48 4.4 5-44 6.58 7-83 9- I 9 10.66 12.23 13.92 Lbs. .89 2 2 - 73 3- 56 4- 5i 5- 56 6 - 73 8.01 9*4 10.9 12.52 14.24 Circum- ference. Heu Common. 4 P. Good. Ins. Lbs. Lbs. 5 4-5 6 5-5 5-45 7.26 6 6.48 8.64 6-5 7.61 10.14 7 8.82 11.76 7-5 10.13 13,5 8 II.52 15 - 3 ^ 8-5 13-05 17-34 9 14.58 19.44 10 18 24 12 26 34-56 15 4O.52 54 To Compute Stress upon a Rope set at an Inclination. Rule.— M ultiply sine of angle of elevation by strain in lbs., add an allow- ance for rolling friction and weight of rope, and multiply by factor of safety . Factor of safety. -Vox standing rope 4, for running 5, and for inclined planes from 5 to 7. an^=.^ wldchis tUelddld rolb^Tric^rand'weightof rop^assurnedm be n; hence, 3 Factor of safety assumed at 6, consequently 1375 X 6 =; 8250 lbs. , capacity or break- inq weight or stress of rope. „ A . By table, page 162, 8200 lbs. is breaking weight of a wire rope of 7 strands, .625 inch in diam. To Compute Tension of a Rope. = L r representing velocity of rope in feet per minute, H> horses' power, and t tension in lbs. Illustration -Assume wheel 7 feet in diameter, revolution 140 per minute, and BP as per preceding table, 29.6. Then 2 9- 6 X 33 ooo __ 976800 _ m 7 X 3-1416 X r 4 o 3079 To Compute Operative Reflection, of a Rope. 1)2 w X) representing distance between centres of wheels or drums m feet^J weight of rope in feet per lb ., t tension , or power required to produce required power or tension of rope when at rest , and d deflection m feet. Illustration. -Take elements of preceding case: diam. of wire rope of 7 strands = . 5625 inch, and by table, page 162, w = .41 lb. , and D — 300 teet. 300 2 X -4 1 Then 10.7 X 3*7- 2 : 10.87 feet. Capacity . — At the Falls of the river Rhine there is a wire rope in operation that transmits the power of 600 horses for a distance exceeding one mi . TRANSMISSION OF POWER AND EQUIVALENT BELT. 1 6 / Endless Ropes. Wire Ropes, when practicable and proper for application, can be used for transmission of power at a less cost than belting or shafting. Transmission of Power. Diameter of Wheel. Revolu- tions per Minute. Diameter of Rope. Horse Power. Diameter of Wheel. Revolu- tions per Minute. Diameter of Rope. Horse Power. Diameter of Wheel. Revolu- tions per Minute. Diameter of Rope. Horse Power. Feet. 4 80 Ins. •375 3-3 Feet. 7 IOO Ins. .5625 21. 1 Feet. II I40 Ins. .6875 I32.I 4 IOO •375 4.1 7 140 .5625 29.6 12 80 •75 99-3 4 120 •375 5 8 80 .625 22 12 IOO •75 1 24. 1 4 14O •375 5-8 8 IOO .625 27-5 12 I40 •75 173-7 5 80 •4375 6.9 8 140 .625 38.5 13 80 •75 122.6 5 IOO •4375 8.6 9 80 .625 4 i -5 13 IOO •75 153-2 5 120 •4375 10.3 9 IOO .625 5 i -9 A 3 120 •75 183.9 5 140 •4375 12.1 9 140 .625 72.6 14 80 .875 148 6 80 •5 10.7 10 80 .6875 58.4 14 IOO .875 176 6 IOO •5 13-4 10 IOO .6875 73 14 120 .875 222 6 120 •5 16.1 10 140 .6875 102.2 15 80 .875 217 6 140 •5 18.7 11 80 .6875 75-5 15 IOO •875 259 7 80 •5625 16.9 11 IOO .6875 94.4 15 120 .875 300 Wire Rope and Ecini valent Belt. Li substituting wire rope for an ordinary flat belt, the diameter is deter- mined by rule in practice for estimating power transmitted by a belt — viz., One horse power for every 70 square feet of running belt surface per minute. Thus, a belt 15 inches wide running at rate of 1400 feet per min- ute, its power would be equal to (1400 X 15) — (70 X 12) = 25 horses’ power. The same result is obtained by the use of a wire rope .5625 inch in diam- eter, running over a wheel 6 feet in diameter, making 130 revolutions per minute. Average life of iron wire rope with good care is from 3 to 5 years, and that of steel rope is greater. Wear increases rapidly with velocity. Greneral Notes. — Kemp and Wire Ropes. White Hope, 2 inches in circumference, of different manufactures, parted at a stress of from 4413 to 6160 lbs. Specimens of Italian, Russian, and French manufacture parted with an average stress of 5128 lbs. = 1633 lbs. per square inch of rope. Bearing capacity of a hemp rope is proportional to its thickness, number of its strands, slackness with which they are twisted, and quality of the hemp. Hemp and Wire Ropes . — Ultimate Strength is 2240 lbs. per lb. per fathom for round hemp, 4480 lbs. for iron, and 6720 to 7840 lbs. for steel. Working Load is 336 lbs. per lb. weight per fathom for round hemp, 672 lbs. for iron, and 1120 lbs. for steel. Or, .83 times square of circumference in inches for round hemp, 5 times square of circumference for iron, and 9 times square of circumference for steel. (D. K. Clark.) Steel Ropes may be one third less in weight than iron for like working load, and Hemp Ropes should be one third heavier than iron for like work- ing load. 1 68 ropes and chains. IRON WIRE AND UNITED STATES NAVY HEMP ROPE. Wire 6 Strands , Hemp Core. Rope 4 Strands. WIRE. Circumference. 4-937 4-375 3-5 3- i8 7 2.75 2.5 2-375 Nominal.j 4.9 4-5 3-3 6 2.98 2.68 2.45 2.4 2.06 Core. Ins. 2*35 2.25 i-57 i-57 1.27 1.17 .78 .78 .78 •39 Wires. No. 108 108 114 114 114 114 114 114 42 114 Breaking Weight. | Circui Actual. Lbs. Ins. 187400 1 12 104 050 11 65409 ! 10.5 55 316 10 34 480 9 5 28 606 1 9 21 846 8 * 5 15692 8 15718 7 ‘ 5 IO925 II 7 HEMP. Lbs. 75 966 77 633 76933 70 533 58 766 56 466 42 866 3 8 500 40 OCX) 32 166 Weight and Strength, of Stud-lmk Cham Cable. (English.) Dimensions. Width of Link. Dimensions. Diarn. of each Side. Ins. -4375 •5 .5625 .625 .6875 •75 .875 Length of Link. 1. 125 1.25 1-375 Tns. 2.625 3 3-375 3- 75 4- i2 5 4- 5 5- 25 6 6.75 !' 7-5 ; 8.25 Width of Link. Ins. i-575 1.8 2.025 2.25 2.475 2.7 3-i5 3- 6 4- 05 I 4-5 4-95 Weight Lbs. 11 -3 13-4 17.2 21 25-4 30.2 41.2 53-8 69 84 101.6 Tons. 3- 5 4- 5 5- 5 7 8*5 10.125 13-75 18 22.75 28.125 34 Dimensions. Diam. of each Side. Ins. i-5 1.625 i-75 1.875 2.125 2.25 2-375 2.5 2.75 Length of Link. Ins. 9 9-75 10.5 11.25 Weight per Fathom. 12 12- 75 13- 5 14.25 15 16.5 Ins. 54 5- 85 6- 3 6- 75 7.2 7- 65 8.1 8- 55 9 9- 9 Admiralty Proof-stress (adopted by Lloyds’). Lbs. 121 142 164.6 189 215 242.8 276.2 303-2 336 406.6 Tons. 40.5 47-5 55-125 63-25 72 81.25 9 i - i2 5 101.5 112.5 136.125 No™ I. -Safe Working-stress is taken at half Proof-stress, 3.82 tons per sq. inch "^r« and Safe Working - stress for close-link chains are respectively i two-thirds of those of stud-link chains. . g/rnioth , Proof-stress averages 72 per cent, ultimate strength, and Ultimate Strength. avlmg^?iot e per Iquafe inVof section of rod or one s.de of a hnk. Weight of close-link chain is about three times weight of bar from which it is made, for enual lengths. . Karl von Olt comparing weight, cost, and strength of the three materials, of cost of chains. Load of Cliains. ( ilolesworth ). Diameter Safe Worliiiv Diameter of Iron. ROPES AND CHAINS. I69 Breaking Strain and Proof of Chain Cables. Diam. of Chain. Breaking Strain. Diam. of Chain. Breaking Strain. Diam. of Chain. Breaking Strain. Diam. of Chain. Breaking Strain. Ins. Lbs. Ins. Lbs. Ins. Lbs. Ins. Lbs. I 67 700 I.1875 92 940 i -5 143 IOO 2 243 180 I.0625 75640 I.25 102 l6o 1.625 165 920 2.125 272 580 I.I25 84 IOO i -375 12 1 84O i -75 2l6 120 2.25 303 280 Proof-stress is 50 per cent, of estimated strength of weakest link and 46 per cent, of strongest. Comparison of Wire Ropes and- Tarred Hemp Hope, Hawsers, and Cables. COARSE LAID. FINE LAID. Diam- eter. Circum. Safe Load. Three Strands. ^ 0 -0 Four “ Strands. a Three ^ Strands. «. Cables. Is Diam- eter. Safe Load. Four 0 Strands. Haws’rs. 2 2 rd 05 Cables. 2 -o t. a jz as Hi C/3 Ins. Ins. Lbs. Ins. Ins. Ins. Ins. Ins. Lbs. Ins. Ins. Ins. •25 l ’ 78 425 I .25 — — — •5 1875 3.12 2.87 — •3125 690 2-43 2.25 3-32 — •5625 2 420 3-56 3-25 4.87 •375 1.25 825 2.68 III 5 3-5 — .625 2900 3-93 3.62 5-25 •5 i -375 1 600 2.87 3- 8 7 — -75 4320 4.81 4-37 6-37 :l875 x -75 2 800 3.81 3-5 5.18 — .875 5700 5-5 5 7.25 2.125 3800 4-75 4-25 6.12 — 1 8200 n 8.81 6.25 8-75 •8 75 2-375 2.625 4400 6 150 5- 25 6.12 4.87 5-75 l 8.62 8 1-125 1.25 10 IOO 13600 £.06 9-5 11 I 3 8 400 6.62 6.12 8.62 \:L S 17500 21 800 10 9-75 12.5 1.25 3-75 1340 ° 8.81 8.5 10.93 10.93 11. 18 10.93 — 1-375 4-25 16 800 9.87 9-56 12.25 12.12 27000 12.5 12.12 — l\l 5 4-625 20 160 io-75 IO -5 13 13. 12 1-875 32 500 — — — 5 24600 — ix . 87 11 56 n-75 2 37000 — — — In above table, determination of circumference of rope, etc., is based upon Breaking Weight or Tensile resistance of wire being reduced by one fourth, and ultimate resistances of rope, etc., are reduced one third. Result of Experiments upon Wire Hope at XT. S. Navy Yard, Washington. (J A. Roebling y s Sons.) Circumference. Se- Weight Breaking Circumference. £ « c p a> 1U-S Breaking Actual. Nom- inal. *li ll* /oot. Weight. Actual. Nom- inal. jT « 5 s .553 Weight. Ins. Ins. No, No. Lbs. Lbs. Ins. Ins. No. No. Lbs. Lbs. 4-9375 4.9 J 9 II 3-14 65 409 2-375 2.4 7 13 .14 15718 4-375 4-5 19 13 2.15 55 3*6 2.1875 2.12 7 14 .11 14478 3-9375 3 - 9 1 19 14 2.0875 44420 2 2.06 19 19 .1 IO925 3-5 3-36 19 14 1-1525 34840 1-9375 1.9 7 14 .1 IO Il8 3-1875 2.98 19 15 1.09 28606 i -75 1.85 7 17 .07 7 880 2-75 2.68 19 17 1.0275 21 846 1-4375 1-45 19 20 .06 5687 2.6875 2.56 7 13 1.0225 l88lO 1.3125 1-31 7 18 •05 4428 2-5 2-45 19 18 .14 15692 1. 125 I.II 7 19 -035 3 729 To Compute Circumference of Wire Hope with Hemp Core, of Corresponding Strength to Hemp Hope, and of Hemp Hope to Circumference of "Wire Hope. Rule i. — M ultiply square of circumference of hemp rope by .223 for iron wire and .12 for steel, and extract square root of product. 2. — Multiply square of circumference of hemp-core wire rope by 4.5 for iron wire and 8.4 for steel wire. Example.— What are the circumferences of an iron and steel wire rope corre- sponding to one of hemp-core, having a circumference of 8 ins. ? V8 2 x .223 = 3. 78 ins. iron , and V8 2 X . 12 = 2. 77 ins. steel. P 170 ropes, hawsers, and cables. ROPES, HAWSERS, AND CABLES. Rones of hemp fibres are laid with three or four strands of twisted fibres, and are made up to a circumference of 12 ins., and those of four strands up to 8 ins. are fully 16 per cent, stronger than those of three strands. Hawsers are laid with three or four strands of rope. Cables are laid with but three strands of rope. Hawsers and Cables, from having a less Propor- tionate number of fibres, and from the irregularity of the resistance ot their fibres in consequence of the twisting of them, have less strength than ropes, difference varying from 35 to 45 per cent., being greatest with least circum- ference, and those of three strands up to 12 ins. are fully 10 per cent, strong- er than those having four strands. Tarred ropes, hawsers, etc., have 25 per cent, less strength than white ropes ; this is in consequence of the injury fibres receive from the high tem- perature of the tar, viz. 290°. Tarred hemp and Manila ropes are of about equal strength, and have from 25 to 30 per cent, less strength than white ropes. White ropes are more durable than tarred. The greater degree of twisting given to fibres of a rope, etc., less its strength, as exterior, alone resists greater portion of strain. Ultimate strength of ropes varies from 7000 to 12000 lbs. per square inch of section, according as they are wetted, tarred, or dry. One sixth oi ulti- mate strength is a safe working load = 1166 to 2000 lbs. per square inch. Units for computing Safe Strain tliat may he Horne by USTevv Liopes, Hawsers, and CaUles. (U. S. Navy.) Descrip- tion. Circumference. Wh 3 strands. Rope ite. 4 strands. s. Tar 3 9tr’ds. red. 4 str’ds. Haw: White. 3 str?ds. 3 ERS. Tarred. 3 str’ds. Cab: White. 3 str’ds. LE9. Tarred. 3 str’ds. Ins. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. White 2 -5 to 6 II40 1330 — — 600 — “ — u 6 u 8 IO9O 1260 — — 570 — 510 it 8 U 12 1045 880 — — 530 — 530 — u 12 u 18 . — — — 550 — 550 — a 18 u 26 — — — — — — 560 — Tarred 2-5 u 5 — — 855 1005 — 460 — — u 5 a 8 — — 825 94O — 480 — — ii 8 u 12 — — 780 820 - — : . 505 — 505 u 12 11 18 — — — — — — — 525 a 18 u 26 — — — — — ’ — — 550 Manila 2-5 a 6 8lO 950 — , — 440 — — u 6 u 12 760 835 — — 46s — 510 u 12 u 18 — — — — — — 535 — u 18 u 26 — ' — — 1 "T • 560 — Illustration. — What weight can be borne with safety by a Manila rope of 3 strands, having a circumference of 6 inches? ( See Rule, page 167.) 6 2 X 760 = 27 360 lbs. When it is required to ascertain weight or strain that can be borne by ropes , etc., in general me, preceding Units should be reduced from one third , to two thirds, in order to meet their condition or reduction of their strength by chafing and exposure to weather. Molesworth’s table is based upon a reduction of three fourths. •Illustration. — What weight can be borne by a tarred hawser of 3 strands, 10 inches in circumference, in general use ? io 2 X (505 — 505 - 5 - 3) = 100 X 366-67 = 33 667 lbs. ROPES, HAWSERS, AND CABLES. 171 Destructive Strength, of Tarred Hemp Hopes (D. K. Clark.) 109 . 3 3 - 5 4 4 - 5 5 Diam. Reg Common Cold. ister. Russian Warm. Circum. Diam. Reg Common Cold. iater. Russian Warm. Ins. Lbs. Lbs. Ins. Ins. Lbs. Lbs. •95 7 390 8620 5-5 I *75 24800 29 120 1. 11 II 200 II 760 6 1.91 28985 33 150 1.27 13 IOO 15 340 6-5 2.07 34 030 40550 i*43 16330 I944O 7 2.24 40 320 47041 i-59 19580 23990 8 2-54 52480 61 420 Specimens furnished by National Association of Rope and Twine Spinners , As tested by Mr. Kirkaldy. Rope. Circum- ference. Weight per Lb. Extreme Strength. Breaking Weight perlb.per Fathom. Extensk at Stra pe 1000 lbs. >n in 50 in 3s per lb. r Fathom 2000 lbs. s. Length Weight of 3000 lbs. Russian rope ... 48 thr’ds. Machine yam. . . 50 “ Hand-spun yam, 51 “ Ins. 5.26 5*37 5-39 Lbs. .926 .891 1.006 Lbs. II 088 II 514 18278 Lbs. 1933 2152 3024 Ins. 5-29 4-53 4.46 Ins. 6.56 5 - 9 i Ins. 6.63 Breaking Strength of Tarred Hemp Hopes. Ins. 3 3*5 4 4-5 5 Ins. •95 1. 11 1.27 i-43 Old Method. Common Hemp. Lbs. 5 056 7466 8 780 10300 I *59 I 33 2 8 Lbs. 6248 8668 10460 12432 15 859 By Register. Cold. Lbs. 7 392 II 200 13 IO4 16330 20496 Warm. Lbs. 8624 II 760 17810 19443 23990 Ins. 1- 75 1.91 2.07 2.24 2- 54 Old Method. Common Hemp. Lbs. 15 456 18 144 20518 22938 26 680 Best Russian. Lbs. 18414 21 6 lO 23 6 lO 27462 32 032 (Mr. Glynn.) By Register. Cold. Warm. Lbs. 24 797 28986 34630 40320 52483 Lbs. 29 120 33150 40 544 47 040 6l 42O To Compute Strain that may- he home with safety by new Hopes, Hawsers, and Cables. Deduced from experiments of Russian Government upon relative strength of different Circumferences of Ropes , Hawsers , etc. U. S. Navy test is 4200 lbs. for a White rope of three strands of best Riga hemp, of 1.75 inches in circumference (= 17000 tbs. per square inch of fibre), but in preceding table (page 166) 14000 lbs. is taken as unit of strain that may be borne with safety. Rule.— S quare circumference of rope, hawser, etc., and multiply it by Units in table. To Compute Circumference of a Hope, Hawser, or Cable for a Given Strain. Rule.— D ivide strain in pounds by appropriate units in preceding table, and square root of product will give circumference of rope, etc., in ins. Example i. — S tress to be borne in safety is 165 550 lbs. ; what should bo circum- ference of a tarred cable to withstand it ? 165 552 4- 55° — 301, and y/301 = 17. 35 ins. 2.— What should be circumference of a Manila cable to withstand a strain, in general use , of 149 336 lbs. ? Assuming circumference to exceed 18 ins., unit = s6o. *49 33 6 4 - (560 — 560 - 4 - 3) = 400, and y/400 = 20 ins. ROPES, hawsers, and cables. 1 7 Z XTrixTcrGlPPS* gtlld O 3/13 A 0 S • To Compute Weight of it by appropriate unit in Ru le. Square f r JXct wi'u fihe per f oof in lbs.: following table, and product wtj 3-strand Hemp °32 ■°i I '°3‘ f.JjJSSd tarred Hemp, .048 — ~ 3-strand tarred Hemp, .042 .04 - 4 * trand Manila °35 -°34 -34 istrand Manila 032 .03 • 3 1 of material. fathoms? ^ x Q34 _ 3 . 4) and 120 X 6 X 3-4 = 2 +4 8 lbs ' ■ ^ ,d Strength- of Hemp and Wire Ropes. Weigh ai ( Molesworth .) n _ w C 2 fc = L; C 2 x = S; and v /jt= C - values of y , ar, asd k, , } k V 1 * k | HOPES. 1 y — — - .131 .117 Warm register, hemp Manila hawser 1 1 - x 77 •155 • 235 .22 •037 .87 .207 •15 .025 1 -89 — 1 .6 .1 | Steel • • Hawser, hemp Tarred hawser, hemp “ cable, u cold register, ~ - " Q r Wire Rope l0C Tr 0 fSSl g B^ (H. S. Navy-) Rule. — T o length of ® ast d b ^e *— * < 0M > “ 4 ■ , “™ of product. . For Mizzen, take .74 of Fore and Mam. masl of a TesS el , 6 < . Head “ 7 $~ “ Breadth of beam, 45 feet. 58 + 45^4S = , 58i andv(3. 5 8X^ = V.oM6 = »o,iina Then if circumference of *“ ir0U wire "rope ofT'h'fanda steel rope of 3.25+ >“ s - Washington , gave for flex- for\ensile strength a like loss of ° ent ' of Hemp Rope and Iron and Steel Relative Dimension^ ^ ope . (U. S. Navy.) Circumference in Inches. o e ne 11 n-75 x 3-5 ‘5 . - r.25 6.5 7-75 8 -5 9-5 1 6 7 Hemp. 2.5 3-125 4 4-5 5 5 ^ 4 4-5 5 5 5 5 . 2S Iron . . 125 x - 62 5 J ell i. 8 7 s »»5 2 *5 2 *75 3- 2 5 3-5 Steel.. .875 X - I2 5 *-5 I>C)2:> 7b ANCHORS, CABLES, ETC. *73 ANCHORS, CABLES, ETC. Anchors, Chains, etc., for a Given Tonnage. ( American Shipmasters’ Association.) SAILS. I Tonnage computed ns per Rule.* Bo With- out Stock. wers. Admi ralty Test. Anciioi Inc Stream IS. luding S . J Kedge tock. 2 d Kedge | Diameter Ce "So C [AIN CAB! Admi- ralty Test. le.— Six Weig Stud. m. ;ht per F Short Link. athom. Eng- lish.f 75 100 125 150 175 200 250 3°o 350 400 45 o 500 600 700 800 900 1000 1200 1400 1600 1800 2000 2500 3000 Lbs. 6l6 728 840 952 IO36 1120 1288 1456 1624 1848 I 9°4 2016 2352 2688 3024 3248 3584 3808 4032 4256 4480 4704 5040 5376 Tons. 7 8 9 10 11 12 13 14 15-5 17 18.5 20 22 24 26 28 295 3 i 32.5 34 35-5 37 39 41 Lbs. 168 196 224 280 336 392 448 504 560 616 672 784 896 1008 1120 I232 *344 J 456 1568 1680 1792 1904 2128 2353 Lbs. 84 1 12 112 I40 l68 I96 224 252 280 308 336 392 448 504 560 6l6 672 738 784 840 896 952 1120 1232 Lbs. 1 12 126 I 4 ° 154 168 196 224 252 280 308 336 364 392 420 448 504 560 6l6 j Ins. .8125 •875 •9375 1 1.062=5 1. 125 i -i875 1 1.25 1-3125 I * 3 I2 5 I, 375 *•4375 i -5 *•5625 1.625 i * 68 7 5 I *75 i -875 1-9375 2 2 2.0625 2.125 2.1875 Paths. 90 105 105 120 120 120 135 135 150 150 165 165 l 8 o l 8 o l 8 o l 8 o 180 180 180 180 ISO l 8 o l 8 o 180 Tons. II *3 15 17-5 20 22.5 25 28 3 i 3 i 37 40 44 47 5 i 55 59 63 67 72 72 81 86 96 | Lbs. 40 44 5 i 59 66 75 82 9 i 100 100 115 120 132 145 156 162 i 75 189 205 219 240 Lbs. 42 48 55 63 70 79 88 98 106 106 118 35 48 54 68 84 102 122 T 4 3 166 191 217 244 t Brown, Lennox, & Co. To Compute Tonnage. ^ 10 icv/uinmt Hawsers and Warps to be 90 fathoms in length. Sh.roo.ds. a tonnage of 7S ’ ia ' greater thanfoTsquIre-riggeci? 01 ' 1 ' 25 t0 1 mch in diameter progressively 174 ANCHORS, CABLES, ETC. (. American Shipmasters 1 Association.') STEAM. [ Tonnage computed as per Rule preceding. With- out a Stock. | era. gjjs NCHORS Inclu g 03 ding Stc 80 £ ck. •sf Diam- eter. c u c !hain C < ~ ABLE.— STI Diam. Stream. [ID. Weigl 03 lit per ] 1| O’. Fath. fc£ rJ ws Lbs. Tons. Lbs. Lbs. Lbs. Ins. Faths. Tons. Ins. Lbs. Lbs. IOO 336 4.9 1 12 — — .6875 105 8.1 •5 — — 25 150 448 6.4 I96 — — .8125 120 11.9 •5625 40 42 35 200 6l6 7.6 224 — — •875 120 13-8 •5625 44 48 — 250 672 8.2 280 — — •9375 120 15-8 .625 5 i 55 48 3 °° 8l2 9-5 308 — — 1 120 18 .625 59 63 54 35 o 924 10.4 336 — — 1.0625 120 20.3 .6875 66 70 — 400 1120 12 532 252 — 1. 125 135 22.8 .6875 75 79 68 450 1344 13-9 560 280 — 1.1875 135 25-4 •75 82 88 — 500 1512 15.2 672 336 — 1,25 150 28.1 •75 9 i 98 84 600 1708 16.7 738 364 — 1-3125 150 3 i .8125 100 106 — 700 1876 18 784 392 — 1-375 165 34 .8125 115 118 104 800 2026 19 896 448 224 1-4375 165 37-2 .875 120 — — 900 2352 21.6 1008 504 252 i -5 l8o 40-5 •875 132 — 122 1000 2632 23-5 1120 560 280 1-5625 l8o 44 •9375 145 — — 1200 2856 25.2 II76 588 308 1.625 l8o 47-5 •9375 156 — 143 1400 3 iq 8 26.9 1232 6l6 308 1.6875 l8o 51.2 1 162 — — 1600 336 o 28.6 1344 672 336 i -75 l8o 55 -i 1 175 — 166 1800 3534 30.1 1456 738 364 1.8125 l8o 59 -i 1.0625 189 — — 2000 3808 31.6 1512 . 766 364 1-875 l8o 633 1.0625 205 — 191 2300 4088 33-4 1568 784 392 1-9375 l8o 67.6 1-125 215 — — 2600 4256 34-5 1624 8l2 392 2 27O 72 1-125 240 — 217 3000 4480 35-7 1680 840 420 2.0625 27O 76.6 1.1875 — — — 3500 4592 37 1792 896 476 2.125 27O 81.3 1.1875 — — 244 4000 4816 38 i960 952 504 2.1875 27O 86.1 i 1-25 — — — 4500 5040 39-2 2128 1064 532 2.25 27O 91. 1 ji -25 — — — 5000 5264 41 2352 1120 560 2.3125 27O 96 1 1-3125 — — — * Brown, Lennox, & Co. ANCHORS AND KEDGES. (U. S. Navy.) To Compute "Weight of a Bower Anchor for a Vessel of a given Character and. Rate. Rule. — Multiply approximate displacement in tons, by unit in following table, and product will give weight in lbs., inclusive of stock. TJ nits to determine Weights and IN' umber of A.ncliors or Kedges. Displacement of Vessel in Tons. Unit. Bower. Sheet. Stream. Kedge. Displacement of Vessel in Tons. Unit. Bower. Sheet. Kedge. Over 3700 i -75 2 2 I 4 Over 1500 . . 2-5 2 2 3 “ 2400 2 2 2 I 3 “ 900 . . 2-75 2 I 3 “ 1900 2.25 2 2 I 3 900 and under 3 2 I 2 Example. — T onnage of a bark-rigged steamer is 1500. 1500 x_2. 5 = 3750 lbs . , weight of anchor. Bower and Sheet Anchors should be alike in weight. Stream Anchors and Kedges are proportional to weight of bowers. Thus, Stream. Anchor .25 weight. Kedges. — If 1, .125 weight; if 2, .16 and .1 weight; if 3, .16, .125, and .1 weight. ANCHORS, CABLES, ETC. — TONNAGE. i75 To Compute Diameter ot a Chain. Cable correspondin°' to a Given Weight of Anchor. (U. S. Navy.) Rule.— C ut off the two right-hand figures of the anchor’s weight in lbs., multiply square root of remainder by 4, and result will give diameter of chain in sixteenths of an inch. Example. — The weight of an anchor is 2500 lbs. V 25.00 X 4 = 20 sixteenths — 1.25 ins. Note.— Diam. of a messenger should be. 66 that of the cable to which it is applied. Lengths of Chain Cables for each Anchor. (V. S. Navy.) Weight of Anchor. Bower. Sheet. Stream. Weight of Anchor. Bower, Sheet. Stream. Lbs. Under 800 Over 800 “ 1200 “ 1600 Fathoms. 60 90 90 105 Fathoms. 60 90 90 105 1 Fathoms. 60 60 75 75 Lbs. Over 2000 u 3000 “ 5 °°° “ 75 oo Fathoms. 120 120 120 135 Fathoms. 120 120 120 135 Fathoms. 90 90 105 105 ANCHORS. jFi om Experiments of a Joint Committee of Representatives of Ship- owners and Admiralty of Great Britain. An anchor of ordinary or Admiralty pattern, Trotman or Porter’s im- proved (pivot fluke), Honiball, Porter’s, Aylin’s, Rodgers’s, Mitcheson’s, and Lennox s, each weighing, inclusive of stock, 27000 lbs., withstood without injury a proof strain of 45000 lbs. Breaking weights between a Porter and Admiralty anchor, as tested at Woolwich Dock-yard, were as 43 to 14. Comparative Resistance to Dragging. Trotman s dragged Aylin’s, Honiball’s Mitcheson’s and Lennox’s ; Aylin’s mid Mitcheson’s dragged Rodgers’s ; and Rodgers’s and Lennox’s dragged TONNAGE OF VESSELS. To Compute Tonnage of Vessels. For Laws of United States of America, with amendments of 1882 relative to Steam-vessels, see Mechanics’ Tables, with rule and illustrated diagrams by Chas. H. Haswell, 3d edition, Harper & Bros., New York, 1878. Rnglish Registered. Tonnage. ( New Measurement.) *?, ivide length of upper deck between after-part of stem and fore-part of stern- post into 6 equal parts, and note foremost, middle, and aftermost points of division Sr Ure n dei f S at th a ese i three pointS in feet and tentlls of a f00t ; also de P th s from under side ot upper deck to ceiling of limber-strake; or in case of a break in the p PP h er H de ?, k ^ fr f ° m a Iin ® fetched in continuation of the deck. For breadths, divide each depth into 5 equal parts, and measure the inside breadths at following points ;; 7 A Q ; 2 and ' 8 fr ° m , u PP cr deck of foremost and aftermost depths; and from fmm off from . u PP® r deck of amidship depth. Take length at half amidship depth from after-part of stem to fore part of stern-post p ‘Tf? a ™idship depth add foremost and aftermost depths for sum of depths and add together foremost upper and lower breadths, 3 times upper breadth with lower breadth at amidship, and upper and twice lower breadth at after division ior sum oj oreadths. bv^'^hiT^r SUm of depths sum of breadths, and length, and divide product °y 35°°, which will give number of tons. 1 If the vessel has a poop or half-deck, or a break in upper deck, measure inside ^bulSd b mnh t ni “I’h hei , g h ht ° f 6Uch part thereof as ma >' be included within tne bulkhead, multiply these three measurements together, divide product bv 02 a and quotient will give number of tons to be added to result as above ascertained.' 4 ’ TONNAGE OF VESSELS. 176 For Open Vessels. —Depths are to be taken from upper edge of upper strake. For Steam Vessels.— Tonnage due to engine-room is deducted from total tonnage computed by above rule. To determine this, measure inside of the engine-room from foremost to aftermost bulkhead; then multiply this length by amidship depth of vessel, and product by inside amidship breadth at .4 of depth from deck, and divide final product by 92.4. The volume of the poop, deck-houses, and other permanently enclosed spaces, available for cargo or passengers, is to be measured and included in the tonnage, but following deductions are allowed, the remainder being the Register tonnage. Deductions.— Houses for the shelter of passengers only; space allotted to crew (12 square feet in surface and 72 cube feet in volume for each person); and space occupied by propelling power. Approximate H vile. Gross Register.— Tonnage of a vessel expresses her entire cubical volume in tons of 100 cube feet each, and is ascertained by following formula : — Gross tonnage, and L B — c = Register tonnage. L representing length IOO 100 of keel between perpendiculars, B breadth of vessel, and, D depth of hold, all in feet. 13 nil tiers’ Measurement. (L r .6 B)XBX : 5 A^ Tonnage , 94 Fore-perpendicular is taken at fore part of stem at height of upper deck. Aft-perpendicular is taken at back of stern-post at height of upper deck. In three-deckers, middle deck is taken instead of upper deck. Breadth is taken as extreme breadth at height of the wales, subtracting differ- ence between thickness of wales and bottom plank. Deductions to be made for rake of stem and stern. Iron Vessels. ( Girth + B) ead —\ x length == Gross tonnage. 10000 \ 2 / Length measured on upper deck, between outside of outer plank at stem and the after-side of stern-post and rabbet of stern-post, at point where counter-plank crosses it Girth measured by a chain passed under bottom from upper deck at extreme breadth, on one side, to corresponding point on the other. Register tonnage = - follows : Ships of usual form Clippers and Steamers LxBxD X C. C representing a coefficient for vessels as ( 2 decks. . . .6<$ (3 “ .68 I Yachts above 60 tons 5 I Small vessels { A";": Units for Measurement and. Dead-weiglit Cargoes. (C. Mackrow , M. S. N. A.) To Compute Approximately for an Average Length of Voijage the Measure- ment Cargo , at 40 feet per Ton , which a Vessel can carry. Rule. — M ultiply number of register tons by unit 1.875, and product will give approximate measurement cargo. To Compute Approximately Deacl-iveight Cargo in Tons which a J essel can carry on an Average Length of Voyage. Rule.— Multiply number of register tons by 1.5, and product will give approximate dead-weight cargo required. With regard to cargoes of coasters and colliers, as ascertained above, about 10 per cent, may be added to said results, while about 10 per cent, may be deducted in cases of larger vessels on longer voyages. TONNAGE OF VESSELS. 177 In case of measurement cargoes of steam-vessels, spaces occupied by ma- chinery , fuel, and passenger cabins under the deck must be deducted from space or tonnage under deck before application of measurement unit thereto! J. J, n case ' °l dead-weight cargoes, weight of machinery, water in boilers, and appiStofd^dgwLr dead weight,as ascertoincd above lj y sdectio“”r f ° r Pr ° ViSi0nS ’ St ° reS ’ CtC - are allowed - To Ascertain Weight of Cargo for an Average Length of Voyage. ( Moorsom. ) ton^ d " Ct ‘r a ^-°f SpacCS °^ Pafenger accommodations from net register tonnage, and multiply remainder by 1.5. Average space for each ton weight of cargo on such a voyage 67 cube feet. Freight Tonnage or Measurement Cargo. anfi Cargo is 40 cube feet of s P ace for cargo, and it is about 1.875 times net register tonnage less that for passenger space! Royal Thames Yacht Club. sSSrS All fractional parts of a ton are to be considered as a ton Measurements to be taken either above or below main wales. L — BxBx.S B — — Tons. L representing length and B breadth , in feet Corinthian and New Thames Yacht Clnb. board; multiply length breadth ^nlfde^nh T ule »f n ^ de P th to to P of covering tient will give tonnage ’ ’ “ d depth divide result by 200, and quo" LXBXI) — - '= Tons. 200 Snez Canal Tonnage. all^co°ve^ sTch a is d nooD b f IOW and uppermost deck, cook^deck, and wheel houses, and all^K^ spaces for steward for crew ’ not includi ng except captain; galleys, cook houses etc Ut n’«5Pd accommodations for officers, spaces above uppermost dec^d^S f °, r cr T ew ’ aud Closed Sp C L“es P rS £ ST -r d, or owner exceed^o^er cent oTgro^toimage! 0 ™ “ USt deduction tor Propelling power I 7 S "WORKS OF MAGNITUDE. WORKS OF MAGNITUDE. .American. Aqueducts, Roads, and Railroads. ** " 38.134 idles in length er of masonry g fee t in diameter. Stone arch Aqueduct , Washington. y 25 f ee t rise, over Cabin John’s Creek, 220 feet spa , si- 5 Cum berland to Illinois Town, National m”li^ Macadamized for a width of 30 feet. ^^mS^Chicago to Cairo, length 365 miles, Centralia to Bun- leith 344 miles, total 709 miles. Bridges. Suspension Bridge, Niagara River.-Wire, Span ro 4 2 feet ro ins. Suspension Bridge , iVeie ins. ; of each land span 930 feet , Jsq feet; width 85 feet; number approach 1562 feet 6 ins . ; % ins . each 9 consisting of 6300 parallel steel of cables 4; diameter of banned to a solid cylinder; ultimate strength wires No. 7 gauge, closely laid P f oun d a tion below high water, Brooklyn, of each cable n 200 tons; depth 01 to t i; ne iao vc;q feet; towers at roof 45 feet— New York 7 8 feet; »*; clear h , e i ght course 136X53 f ee f? total height . , * - 0 o joe feet; height of floor of bridge in centre of riyer span abov . g g rade of roadway 3 feet in 100; anchor- Iron Pipe Bridge over [tnch'thick" 0 These pipes conveying themselves over the great span, but support a street road and railway. M d.- 3 ns each 375 Iron Bridge over Kentucky River near Shakers Ferry, feet, and 275-5 feet above oww ^ ^ Western Railroad across the Kinzua.- Of^ror? lengthei^o^St ; ^central span 30. feet in height. Jron Truss. —Cincinnati and Southern Railway, over Ohio River, $ feet. Foreign* Pyramids, Statues, etc. • pyramid of Clteops, Egypt.-l^ 1 ^®® Snld^engfh 568^ *fl ™S le w g 527 00 built 2170 years B.C. . . -Rronze* height of horse 17 feet; of man “>*> » wide > and 17 h,gh ’ wclBhing hfeZ New Vork Harbor. - jo ^ " - ** " ference 87 feet; face 8.5 feet; circumference of thumb 3.5 me . Colossus of Rhodes. — Height, 105 feet. Bridge. . . m r , t> . w „„ of iron with a double lino of Railway, 964 feet in Britannia Tubulat Bi idge. ’ r i, weight 3658 tons, length, with two approaches of 230 feel each - w e g 3 5 WORKS OF MAGNITUDE. 179 NT oiiolith.s. Obelisk at Karnak , Egypt.— Of granite, 108 feet 10 ins. : pedestal 13 feet 2 ins • height 400 tons. J * Obelisk in Central Park , N. Y.-Of granite, 68 feet n ins.; weight 168 tons. raids of Egypt lry ’ WashiD S ton — Some stones o f , are heavier than any in the Pyra- Steam Hammers. . workshops of Herr Krupp, at Essen, there is a steam hammer weighing so tons i°^.3 metres; and at Creusot there is a hammer weighing between 7? and 80 tons having a fall of 5 metres. s g cen 75 Crane. tons! CreUSOt there is a steam crane havin ® a capacity to lift and revolve with 150 Chimney's. J. Townsend’s chemical works, Glasgow, diameter at foundation so feet- at ton 12 feet 8 ins. ; height from foundation 488 feet; from ground 474 feet. % t0p New York Steam Heating Co., 220 feet in height. Filial*. At a gate near Delhi is a wrought-iron pillar having diameters of t * ?r>c< feet in its height above ground and ,, in S P a t its I«“eXmted ^rt,m the re- font 0f Tf XCaVatl ^ >n p at its base to be 60 feet in lei3 gth or height and to weigh n tons. Its period of structure is assigned to the 3 d or 4 th century A.D. S 7 Roofs. Midland Railway Station, London. 240 ft. I Union Railway Station Glasgow Tr .- ft Imperial Riding-School, Moscow. 235 “ | Grand Central Station, N. Y. & . . . ! 200 “* Diameters of Domes Domes. I Feet, j Domes. Feet. ’I Domes. Feet. Capitol, Washington Glasgow W. Railw’y | I2 4-75 I 198 | St. Paul’s, London . St. Peter’s, Rome. . XI2 *39 1 Midl’nd Rail’y, Lon. Great North’n, Eng. 240 210 Tunnels. Feet. Tunnels. Blaizy 13455 4280 25031 y'Vf _____ _ _ Blue Ridge. . . crunpowcicr. Mu. • . Sutro. . 4 . Hoosac Semmering . . . | Feet. 3 6 5 oo Thames and Medway, n 880 feet. Tunnels. Nerthe. Nochistongo. . . . _ _ RiquiVel Weebawken, 4000 feet. *5 153 21 659 18623 Mont Cenis 7.5 miles 242 yards, rises 1 in 45, and descends r in 2000. - - 1 1 1 iu 2000. mum grade 2.7 feet per 100.^ Miscellaneous. Fortress Monroe , Old Point Comfort, Ya. —Largest fortress. 60 ^tiS:~ Spa “ ° Ver r ‘ Ver KiStnah between B " Sectanagran, Deer Park , Copenhagen. — 4200 acres. Alfred!” 1 * C ° Ue9e ’ EDgland — Largest. University; said to have been founded by Rome - Widtb ° f «6 feet; of the cross 25. feet; total len^tmtso^Ues. 25 feet baSe ’ 15 at top; height > with a parapet of 5 feet, 20 feet; of disctogeTf^glltons^wday. 1 deptb ’ tem P erature of water 99 o ; volume I l 80 BELLS, CHURCHES, COLUMNS, TOWERS, ETC. Bells. Y Lbs. TV'eiglits of Bells Bells. I Lbs. Bells. Lbs. 120000 Oxford, “ Great Tom,” Eng Olmutz, Bohemia. Rouen, France. .. St. Paul’s, Eng. . . St. Ivan’s, Moscow 17024 40 320 40000 42 000 127 830 St. Peter’s, Rome. Yipnna 18000 40 200 35620 24080 13000 10233 28 560 443 772 30 800 28670 Westm’ster, “Big Ben,” England. York State House, Phila. Pekin Lewiston, Me. . . Montreal, Can. . . Moscow, Russia. Erfurt, Saxony. . Notre Dame, Paris| Rangoon, Burmah, 201 600 lbs. Capacity of Principal Churches and Opera Houses. Estimating a person to occupy an Area 0/19.7 Ins. Squaie. C lvu.rcli.es . St. Peter’s 54 000 Milan Cathedral 37 000 St. Paul’s, Rome 32000 St. Paul’s, London 25 600 St. Petronio, Bologna 24400 Florence Cathedral 24300 Antwerp Cathedral 24000 St. Sophia’s, Constantinople 23000 St. John, Lateran 22900 Notre Dame, Paris 21 000 Pisa Cathedral 13000 St. Stephen’s, Vienna 12400 St. Dominic’s, Bologna 12000 St. Peter’s, Bologna 11 400 Cathedral of Sienna 11 000 St. Mark’s, Venice 7 000 Opera Houses and. Theatres Carlo Felice, Genoa 2560 Opera House, Munich 2370 Alexander, St. Petersburg 2332 San Carlos, Naples 2240 Imperial, St. Petersburg 2160 La Scala, Milan 2113 Academy of Paris 2092 Teatro del Liceo, Barcelona 409° Covent Garden, London 2684 Opera House, Berlin 1636 New York Academy 2526 “ “ Windsor 34°° Philadelphia Academy 3 12 4 Chicago “ 3°°° Heights of Columns, Towers, Homes, Spires, Locations. CHIMNEYS. Townsend’s Glasgow. , St. Rollox “ Musprat’S. Liverpool. GasWorks Edinburgh New England Glass Co . Boston . Steam Heating Co New York. COLUMNS. Alexander St. Peters’g Bunker Hill Mass City London... July Paris. . . . Napoleon _“•••• Nelson’s Dublin. . Nelson’s London . Place Vendome Paris Pompey’s Pillar Egypt... Trajan J^me . 7 Washington Wash gton York London . . . TOWERS AND DOMES. Babel Balbec ••••••• 7 ‘ ' Capitol Wash gton Cathedral Antwerp . . u Cologne... u Cremona.. <1 ** Escurial. . etc. Feet. TOWERS AND DOMES. Cathedral Florence . . u Magdeb’rg a Milan about 12 feet thick; Hearting, second-class rubble (of stones .5 to 2 tons weight), about 6 feet thick; Nucleus, of quarry rubbish. Algiers. Depth of water 50 feet; rubble base carried up to 33 feet from surface of Saso'Ai T 1 °f la ^ b don blocks 2 5.5to..s\\ch; slopes of rabble base 1 to 1 , Outei slope of beton blocks 1.25 to 1 ; Inner slope of beton blocks 1 to 1. f Said (Suez Canal). — Concrete blocks, 10 cubic metres each, composed of 1 for by mnnlh« w ° f SaUd ’ miXGd With Sea water ; 4 days in the molded dried bemnTfinr^ before being put in position. In some instances the composition of ballasting k ' 33 ,me ° F cemeut t0 ,66 sand and broken stone, about the size of :^ hbl * °\ B ! oc7c F i lBn 9 - Proportion of interstices to volume of breakwater fin- rubble of * t0 5 tons, .25; second-class rubble of .5 to 2 tons 2 * third-class rubble, quarry chips, etc., .16; beton blocks, 15 to 25 tons 33 ’ ' Noth. — For force of water, see Waves of the Sea, page 853. Q 1 82 LAKES, OCEANS, SEAS, MOUNTAINS, ETC. Areas, Depths, and Heights of Great Northern hakes of TJ nited. States. Lakes. Length. Breadth. Mean Depth. Height above Sea. Area. Erie Miles. 250 Miles. 80 Feet. 200 Feet. 564 Sq. Miles. 6000 Hnuon 200 160 120 574 20000 Michigan 360 180 109 900 5 8 7 20000 Ontario 65 500 234 6000 Superior* 400 160 288 635 32000 Elevation Above Tide-water at Albany. — 'Late Erie 570.6 feet; Hudson River 2.46 feet. Mean Depths and Areas of the Oceans and Seas. {Herr Krummel) ' Fathoms. Atlantic 2013 Archipelago 487 Azof. — Baltic Sea 36 1 Black Sea — a Behring’s Straits 550 c Caspian Sea — ' China (East) Sea 66 c Dead Sea — 4 English Channel, etc. 47 Area Sq. Miles. I Fathoms. Gulf of Mexico IOOI “ “ St. Lawrence 160 Indian 1829 Japan 1200 Mediterranean 729 North Sea 48 North Ice Sea. 845 Persian Gulf 20 Pacific 3887 j Red Sea 243 29 5H 2 75 3 046 600 8800 159 690 1 56 000 864 555 120000 472 210 37° 78 416 Mean depth of Ocean surrounding land 1877 fathoms = 2. 19 miles. land = 1377 feet, or one eighth that of Ocean. Heiahts of Mountains, Volcanoes, and Passes above Level of* Sea. Area Sq. Miles. I 765910 101 075 28369 595 383 205 I 109 230 210 505 5 264 600 90 100 60 343 690 170820 Mountains. | Feet. EUROPE. Azores Pico Barthelemy, France Ben Lomond. ..... Ben Nevis Elbrus, Caucasus. . . Guadarama, Spain. Hecla Ida Jungfrau, Switz’d. Mont Blanc “ Cenis Mont d’ Or, France Mulahassen,Gren’a. Nepbin, Ireland Olympus Parnassus Plynlimmon, Wales. The Cylinder, Pyr. . Wetterhorn Ararat Caucasus Dhawalagheri Geta, Java Mount Lebanon. . . Mountains. 7613 7 365 3240 4380 17776 8 520 5147 4960 13 725 15 797 6 780 6 510 11 663 2634 6510 6000 2463 10930 12 154 17 100 16433 28077 8 500 12000 MountEverest (Him- alaya, highest) Mount Libanus. Petcha Sinai AFRICA. of Atlas Compass, Cape Good Hope.. Dianai Peak, St He- lena Kilimanjaro .... Ruivo, Madeira Teneriffe Peak. 29003 9 523 15000 7496 10400 10000 2 700 20000 5 160 12 300 AMERICA. Aconcagua (highest in America) Blue Mount, Jam’a. Catskill Chimborazo.. Correde, Potosi .... Crows’ Nest, High- lands, N. Y. . . Great Peak, New Mexico Mauna Rauh,Owy’e 23910 8000 3804 21 441 16036 1 370 19788 18400 Mountains. Mount Pitt Mount Washington. Nevado de Sorata. . Orizaba Potosi Sierra Nevada Tahiti White Mountains . . VOLCANOES. Cotopaxi Etna Hecla Popocatapetl Sahama St. Helen’s, Oregon. Vesuvius 9 6$ 25 248 18879 18000 15 700 IO895 6 230 18887 IO874 5 000 17784 22 350 3320 3 930 PASSES. Cordilleras j Mont Cenis “ Cervis. .. ... . Pont d’ Or St. Bernard, Great. “ Little.. St. Gothard Simplon 13525 15225 6778 11 100 9 8 43 8 172 7 x 9 2 6808 6578 CANAL LOCKS, ELEVATIONS, AND RIVERS. Dimensions of Canal Locks. — (U. S.) Albemarle and Chesapeake. . Black River, Crook’d L’ke, Chenango, Chemung, and Genesee I Valley J Chesapeake and ) Delaware j 90 Length of Canal. Miles. 14 f 77 8 97 33 113-75 Champlain Cayuga and ) Seneca j Delaware and ) Raritan j Dismal Swamp. . . Erie Falls of Ohio, Ky. Oswego Welland, Canada. . 18 24 i7-5 18 80 18 45 Feet. 5 7 7 5-5 7 2-60 4 14 183 Length of CanaL Miles. 66.75 24-75 43 44 352 3 8 28 Length of vessel that can be transported is somewhat less than lengths of locks. Suez Canal. — Width 196 to 328 feet at surface, 72 at bottom, and 26 deep Length 99 miles. ’ Hei.gh.ts of obtained Elevations, and various -Places and Points above the Sea. Locations. Aconcagua, Chili . . Antisana, highest established eleva- tion (Farmhouse) . . Bal lo on ( G a y Lussac) “ (Green, 1837) “ (Glaisherand Coxwell) Brazil, Quito, and Mexico plains. . Condor’s flight Eagle’s Feet. 23910 *3 434 22 900 27 000 37000 6000 8 000 29 500 16 500 Everest, Himalaya. | 29003 Locations. Geneva city Geneva Lake... Gibraltar Humboldt’s highest elevation Tsthmus of Darien.. Jungfrau, Switz’d. . La Paz, Bolivia. . . . Laguna, Teneliffe. . . London, city Madrid Mexico, city of Mont Blanc, Alps.. . Feet. 1 220 1 096 1 439 19 400 645 13725 12 225 2 OOO 64 2 200 7 525 J 5 797 Locations. Mont Rosa, Alps . . Mount Adams Mount Katahdin . . . Mount Pitt Mount Washington. Paris, city Pont d’ Oro, Pyr’s. . Posthouse, Ap. , Peru Potosi, Bolivia Quito St. Bernard’s Mon’y Vegetation White Mountain . . . Feet. 15 155 5 930 5 36a 9 549 6426 ii5 9 8 43 I 4 377 13223 13 5 oo 8 040 17000 6 230 Lengths of Rivers. Rivers. Miles. Rivers. Miles. Rivers. Miles. EUROPE. Ganges 1514 3040 1800 176 2762 1160 Kansas Hoang Ho La Platte 850 Danube 1800 Indus Maekenyie Dnieper 1243 400 1035 780 Jordan 2440 Douro Lena Missouri *35° Dwina Tigris . , 3°3° 1480 Elbe Yenesei and Se- Ohio and Allegheny Potnm n r*. Garonne 442 545 420 760 5io 450 250 tin lenga 358o Red 420 Loire Yang-Tse 1520 Po 33*4 n , 2300 Rhine AFRICA. irwio (jrnin&o.. •••••• St. Lawrence 1800 Rhone Gambia 2172 620 Seine Niger Tennessee. ........ Shannon Nile 2400 790 Tagus 4000 SOUTH AMERICA. Thames 220 NORTH AMERICA. Tiber 190 630 2400 Arkansas p, ' 4000 Vistula Colorado 2070 1050 lSSGCJUIDO . • • 520 900 1600 Volga, Russia Columbia Orinoco ASIA. Connecticut 410 420 Platte Delaware Rio Madeira 2300 Amoor 2500 1786 Hudson and Mo- hawk 2300 Euphrates 325 | Rio Negro. ........ TTrn orn o xr 1650 1100 1 84 SEA DEPTHS, BUILDING STONES, ETC. Large Trees in. California. “ Keystone State. ”— Calavera Grove, is 325 feet in height. « Father of the Forest.” -Felled, is 385 feet in length, and a man on horseback can ride erect 90 feet inside of its trunk. « Mother of the Forest”- Is 315 feet in height, 84 feet in circumference (26.75 feet in dfameter) inside of its bark, and is computed to contain 537 000 feet of sound 1 inch lumber. Sea - Deptlls . | Feet. I Feet. Feet. Baltic Sea 1 Adriatic English Channel. . . Straits of Gibraltar. 1 Eastward of “ 120 13° | 3 °° too 1 3000 Coast of Spain West of St. Helena. Tortugas to Cuba . . Gulf of Florida Off Cape Florida. . . 6 000 27 000 4 200 3720 1950 Off Cape Canaveral. “ Charleston “ Cape Hatteras. . “ Cape Henry “ Sandy Hook 26 000 fe 2400 4200 3120 4200 2400 et. “ “ Pacific 250 miles off Cape Cod, no bottom at 7800 feet. Cascades and “W aterfalls. 29000 Arve, Savoy. . . Cascade, Alps . Cataracts of the Nile. Chachia, Asia Foyers, Scotland . . Garisha, India Gavarny, Pyrenees Feet. 1600 2400 3 ° 34 40 362 197 1000 1260 Genesee, N. Y 100 Lidford, England 1 100 Lulea, Sweden 1 600 Mohawk 68 ( 5 ° Missouri | ]8o (94 250 800 Location. Niagara Great Fall Passaic Potomac Ribbon, Yosemite) Valley ) Ruican, Norway — Staubbach, Switz'd. . Tendon, France... 164 152 74 74 3300 800 798 125 Montmorenci . . . Nant d’Apresias. Yosemite Valley 2600 feet. E^aixsion and Contraction of Building Stones for eacli Degree of Temperature. {Lieut W. H. C Bartlett, U. S. E.) For One Inch. Sandstone 000 009 532 Whitepine 00000255 For One Inch. 1 Granite 000004825 Marble 000005668 | Resistance of Stones, etc.. to tlie Effects of Freezing. Various experiments show that the power of stones, etc., to resist effects of freez- ing is a fair exponent of that to resist compression. Magnetic Bearings of ISTew York. The Avenues of the City of New York bear 28° 50' 30" East of North. Filters for Waterworks. x square yard of filter for each 840 U. S. and 700 Imp’l gallons in 24 hours • formed of 2.5 feet of fine sand or gravel and 6 inches of common sand or shells. Led off by perforated pipes laid in lowest stratum. Distances between ISTew York, Boston, Bliiladelpkia, Baltimore, and Western Cities of TJ. Assuming Boston as standard, New York averages .2 per cent, nearer to these cities, Philadelphia 18 per cent., and Baltimore 22 per cent. Between New York and Chicago the line of the Penns>dyania Railroad 8 47 ™^ shorter than that by the Erie and its connections, 50 miles sbo f ter N. Y. Central and Hudson River and its connections, and 114 miles shorter than that by the Baltimore and Ohio and its connections. For Distances between these and other cities of the U. S., see page 88. WEATHER-PLANTS, ANTIDOTES, ETC. i8 5 "Weather- foretelling iPlaxits. ( Hanneman .) If Rain is imminent. — Chickweed,* Stellaria media ; its flowers droop and do not open. Crowfoot anemone, Anemone ranunculoides ; its blossoms close. Bladder Ketmia, Hibiscus trionum ; its blossoms do not open. Thistle, Carduus acaulis; its flowers close. Clover, Trifolium pratense , and its allied kinds, and Whitlow grass, Draba verna ; all droop their leaves. Nipple- wort, Lampsana communis ; its blossoms will not close for the night. Yel- low Bedstraw, Galium verum ; it swells, and exhales strongly ; and Birch, Betula alba , exhales and scents the air. Indications of Rain. — Marigold, Calendula pluvialis ; when its flowers do not open by 7 A. M. Hog Thistle, Sonchus arvensis and oleraceus ; when its blossoms open. Rain of short duration.— Chickweed, Stellaria media ; if its leaves open but partially. if cloudy. — Wind-flower, or Wood Anemone, A nernone memorasa ; its flowers droop. Termination of Rain. — Clover, Trifolium pratense ; if it contracts its leaves. Birdweed and Pimpernel, Convolvulus and Anagallis arvensis; if they spread their leaves. Uniform Weather. — Marigold, Calendula pluvialis ; if its flowers open early in the A. M. and remain open until 4 l\ M. Clear Weather. — Wind-flower, or Wood Anemone, Anemone memorasa ; if it bears its flowers erect. Hog Thistle, Sonchus arvensis and oleraceus ; if the heads of its blossoms close at and remain closed during the night. Treatment and. -Antidotes to Severe Ordinary Poisons. Antidotes in very small doses. Chloroform and Ether . — Cold affusions on head and neck, and ammonia to nostrils. Antidote. — Camphor, petroleum, sulphur. Toadstools . — (Inedible mushroom). Antidote.— Same as for chloroform. Arsenic or Fly Powder. — Emetic ; after free vomiting give calcined mag- nesia freely. If poison has passed out of stomach, give castor oil. Antidote.— Camphor, nux vomica, ipecacuanha. Acetate of Lead (Sugar of lead). — Mustard emetic, followed by salts, Large draughts of milk with white of eggs. Antidote. — Alum, sulphuric acid alike to lemonade, belladonna, strychnine. Corrosive Sublimate (Bug poison). — White of eggs in 1 quart of cold water, give cupful every two minutes. Induce vomiting without aid of emetics. Soapsuds and wheat flour is a substitute for white of eggs. Antidote. — Nitric acid, camphor, opium, sulphate of zinc. . Phosphorus Matches— Rat Paste . — Two teaspoonfuls of calcined magne- sia, followed by mucilaginous drinks. Antidote. — Camphor, coffee, nux vomica. Carbonic Acid (Charcoal fumes), Chlorine , Nitrous Oxide , or Ordinaiy Lu.?.— Fresh air, artificial respiration, ammonia, ether, or vapor of hot water. Antidote. — Camphor, coffee, nux vomica. Belladonna (Nightshade). — Emetic and stomach pump, morphine and strong coffee. Antidote.— Camphor. Opium. Stomach pump or emetic of sulphate of zinc, 20 or 30 grains, or mustard or salt. Keep patient in motion. Cold water to head and chest. Antidote. Strong coffee freely and by injection, camphor, ether, and nux vomica. Strychnine (Nux vomica). — Stomach pump or emetic, chloroform, cam- phor, animal charcoal, lard, or fat. Antidote. Wine, coffee, camphor, opium freely, and alcohol in small doses. Vegetable Poisons .— As a rule, an emetic of mustard and drink freely of warm water. J * Spreads its leaves about 9 A. M., and they remain open until noon. Q* veterinary. 1 86 "V'eterinary. u „„ Cnthartic Ball Cape Aloes, 6 to io drs. ; Castile Soap, i dr. ; Spirit of wSrf dr" Simp to form a ba P ll. If Calomel is required, add from 20 to 50 grains. ’ Daring its operation, feed upon mashes and give plenty of water. g eP 3 a dr t s le iDx ( ; a aid 1 giveTn a P quartof g 4 rueT For Calves, 1 on^third will be sufficient. , „ , nnimn t dr • Gineer 2 drs.; Allspice, 3 drs., and Caraway Seeds, <1 drs ° ? all powdered. ’ Make into a ball with sirup, or give as a drench m gruel. cordial strong Ale° or’ (frae?, r pint. Give every morning till purging ceases. For Sheep 2 A Iterative — Ethiop’s Mineral, .5 oz. ; Cream of Tartar, 1 oz. ; Nitre, 2 drs. Divide into from 16 to 24 doses, one morning and evening in all cutaneous di.ea.es. Diuretic Ball- Hard Soap and Turpentine, each 4 drs. ; Oil of Jumper, 20 drop , ^SSS^SBaaaisossas^^ 5rrt wsssasaai ssa&vami «s Tallow, each , lb. ; Turpentine .5 lb. Melt and mix. Repeat’every 5 houra’till it o 3 pSs°“ ° f * * — fuI or two of common salt. Give twice a week if required. . Distemper Powder - Antimonial Powder, ^^3, 0^4 ^ 5 , STay'^K 1 -5 gr. to , gr. of Digitalis, and every 3 or J'Tfrd"^ . ei^ris : eral days between each dose. Age of Horses. To Ascertain a Horse’s Age. A foal of six months has six grinders in each jaw, three in each side, and also six nippers or front teeth, with *Tn front' Weth begin to decrease, and he l.as four At age of one Jf sffie Ca one of permanent and remainder of milk set. grinders upon each side 1 first milk grinders above and below, and front At age of two years be Jos 8 horses of eight years of age. j casts his two front uppers, and in a i short time after the ‘« ““t and about four ^ a half his nippers < ar^Vormanent^by 8 re^lactog ^ ^remaining two corner teeth ; tushes then appear, . an lt h flvt Thorne ha^^ bt'tushes, and there is a black-colored cavity in centre of all ' cavity is obliterated in the two front lower nippers At six, this matK ; c ; y and tushes blunted; and at eight, that At seven, cavities of next two are ini be a ed cavities in nippers DISTANCES, POPULATION, DEOWNING, ETC. 1 87 Distances between ^Principal Cities of ICast and. West. In Miles. Cities. Burlington, la. Chicago Cincinnati Cleveland Columbus, 0. . Detroit Indianapolis . . Kansas City. . . j Boston. New York. Phila- delphia. [ Balti- more. Cities. Boston. New York. Phila- delphia. Balti- more. 1216 1106 1030 995 Louisville 1161 870 794 706 1009 900 823 802 Memphis I 43 8 1247 1171 1083 927 743 667 576 Milwaukee 998 947 908 887 671 580 504 483 Omaha 1503 1393 I 3 W 1294 807 623 547 512 St. Joseph 1478 1356 1280 1223 724 673 682 661 St. Louis 1212 1050 973 917 951 810 735 700 St. Paul 1418 1308 1232 1211 1487 1324 1248 1192 Toledo 784 693 617 59 6 Population of Principal Cities ( 1 SS 2 ). London 3832440 Paris 2225910 Berlin 1222500 New York 1206299 Vienna 1103 no St. Petersburg. . . 876 570 Philadelphia 847 170 Moscow 611970 Constantinople . . 600 000 Chicago 583 185 Brooklyn 566663 Hamburg 410 120 Naples 403 no Lyons 372 890 Madrid 367 280 Boston 362 839 Buda-Pesth 360 580 Marseilles 357530 St. Louis 350518 Warsaw 339400 Baltimore 332313 Milan 321440 Amsterdam 317 010 Rome 300470 Lisbon.. 246300 Palermo 244990 Copenhagen 234850 San Francisco. . . . 233959 Munich 230200 Cincinnati 225 139 Bucharest 221 800 Dresden 220820 New Orleans 216190 Florence 169000 Stockholm 168770 Brussels 161 820 Cleveland 160146 Pittsburgh 156389 Buffalo 155 134 Antwerp 150650 Washington 147293 Cologne 144770 Frankfort 136820 Newark 136508 Venice 132830 Louisville 123758 Jersey City 120722 Detroit 116340 Milwaukee.. 115587 Providence 104857 Rouen 104 010 Treatment of Drowning Persons. Practice adopted by Board of ITealtli, Ne w York. Place patient face downward, with one of his wrists under his forehead. Cleanse his mouth. If he does not breathe, turn him on his back with shoulders raised on a support. Grasp tongue gently but firmly with fingers covered with end of a hand- kerchief or cloth, draw it out beyond lips, and retain it in this position. To Produce and Imitate Movements of Breathing . — Raise patient’s extended arms upward to sides of his head, pull them steadily, firmly, slowly, outwards. Turn down elbows by patient’s sides, and bring arms closely and firmly across pit of stomach, and press them and sides and front of chest gently but strongly for a mo- ment, then quickly begin to repeat first movement. Let these two movements be made very deliberately and without ceasing until patient breathes, and let the two movements be repeated about twelve or lifteen times in a minute, but not more rapidly, bearing in mind that to thoroughly fill the lungs with air is the object of first or upward and outward movement, and to expel as much air as practicable is object of second or downward motion and pressure This artificial respiration should be maintained for forty minutes or more when the patient appears not to breathe; and after natural breathing begins, let same motion be very gently continued, and give proper stimulants in intervals. What Else is to he Done , and What is Not to he Done , while the Movements arc being Made . — If help and blankets are at hand, have body stripped wrapped in blankets, but not allow movements to be stopped. Briskly rub feet and legs, press- ing them firmly and rubbing upward, while the movements of the arms and chest are in progress. Apply hartshorn, or like stimulus, or a feather within the nostrils occasionally, and sprinkle or lightly dash cold water upon face and neck. The legs and feet should be rubbed and wrapped in hot blankets, if blue or cold or if weather is cold. ’ What to Do when Patient Begins to Breathe . — Give stimulants by teaspoonful two or three times a minute, until beating of pulse can be felt at wrist, but be careful and not give more of stimulant than is necessary. Warmth should be kept up in teet and legs, and as soon as patient breathes naturally, let him be carefully removed to an enclosure, and placed in bed, under medical MISCELLANEOUS ELEMENTS. 1 88 MISCELLANEOUS ELEMENTS. Earth. Polar diameter 7890.3 miles. Mean density or specific gravity of mass 5.672. Mass 5 37 ° Zo 000 000 io 9 ^, 000 tons. Apparent diameter as seen from Sun .7 seconds. Sun. Heat of Sun equal to 322 794 thermal units per minute for each sq. foot of pho- t0 Dhimeter of Su^S^ooo miles, tangential velocity 1.25 miles per second or 4.41 times greater than that of the Earth. Distance from Earth 91.5 to 92 millions of miles. Mason and. Dixon’s Dine. 39O ^3' 26. 2>" N. mean latitude. 68.895 miles. Divisions. Area. Population. Divisions. Area. Population. America. Europe Sq. Miles. 14 491 000 3 760000 16 313000 10936 000 95 495 5oo 315 929000 834 707 000 2°5 679 OOO Ocean ica. . . Greenland \ Iceland j Total. Sq. Miles. 4 500 OOO 4031 060 82000 Africa 50000000 M55 9 2 3 500 Austria ) Hungary ) China France. . . , . 38 000 000 .434626000 37 000 000 Countries. Germany 43900000 India, British ..240298000 Great Britain. .34000000 Canada 3839000 (Russia.. 66000000 Mexico 9485000 (Territories .. .22 000000 Brazil 11106000 ( United States 50000000 I (Turkey....... 8866000 (Indians 300000 | ( ‘ in Asia. .16 320 000 About one thirtieth of whole population are born every year, and nearly an equal Dumber die 6 in same time; making about one birth and one death per second. Earlier authority estimated population at 1 288000000, divided as follow s. Caucasians 360 000 000 Mongolians 552000000 Ethiopians 190 000 000 Asiatics 60000000 Mohammedans . 190 000 000 Pagans 300 000 000 Catholics 1 250000000 Rom. & Greek ) Malays and , 177000000 Indo-Amer’s ) 11 Protestants 80000000 Israelites 5000000 Descent of Western Divers. Slope of rivers flowing into Mississippi from East is about 3 inches per mile; 1 D M ean^descTn t* of* Oh k) River from Pittsburgh to Mississippi, 975 miles, is about 5.2 inK ^per mfle; and that of Mississippi to Gulf of Mexico, 1x80 miles, about 2.8 inches. Transmission of Horse Dower. T nn?est and perhaps most successful, wire rope transmission is one at han«?en at Falls of the Rhine. Here, power of a number of turbines, amounting to over’ 600 IP, is conveyed across the stream, and thence a mile to a town, where it 1 S Mm toes *0? *Fa km* Sweden, a power of over 100 horses is transmitted in like manner for a distance of three miles. Acids. Acetic Acid (Vinegar), acid of Malt beer. etc. Tartaric Acid, acid of Grape tin ns. Lactic Acid, acid of Milk , Millet beer, and Cider. IVf anures. Relative Fertilizing Properties of Various Manures. Peruvian Guano . . . . i | Horse 048 | Farm-yard 0298 Human, mixed 069 | Swine 044 | Cow • * ‘ 59 w Or, 1 lb. guano = 14.5 human, 21 horse, 22.5 swine, 33.5 farm-yard, and 38.5 cow. Relative Value , Covered and Uncovered , on an Acre of Ground. Cowed 1 1 tons 1665 lbs. potatoes, 61 lbs. wheat, 215 lbs. straw Uncovered.'.'.'. 7 “ *397 “ “ 6l *5 “ ‘ 156 MISCELLANEOUS ELEMENTS, I89 Yield, of* Oil of* Several Seeds. PerCent. I Per Cent. I Per Cent. I Per Cent. I PerCent Poppy. . 56 to 63 I Castor . . 25 I Sunflower. 15 | Hemp. 14 to 25 I Linseed. n to 22 Thickness of* Walls of* Buildings. (English.) ( Molesworlh .) Outer Walls. Maximum Height of Wall. Width of Footings. Ground Floor. Mi ist Floor. inimum 2 d Floor. Width < 3*1 Floor. >f Walls 4 th Floor. 5 th Floor. I 6 th | Floor. Feet. Ins. Ins. Ins. Ins. Ins. Ins Ins. “ l~ g ' 1 st class dwelling. 85 38.5 21.5 21.5 i7-5 17-5 17-5 13 I 3 2 d u 70 30-5 17-5 *7-5 17-5 13 13 13 3 d “ 52 30.5 I 7-5 13 13 13 13 4 th “ 38 21.5 13 13 8.5 8.5 — — Party Walls. 1 st class dwelling. 85 38.5 21.5 21.5 i7-5 17-5 17-5 13 13 2 d “ “ 70 30-5 17-5 17-5 17-5 13 13 1.3 3 d “ “ 52 30-5 I 7*5 13 13 13 8.5 4 th “ u 38 21.5 13 8-5 8.5 8.5 — — by half a brick. Warehouses M w-^ m Width 1st Class. of Wall. For a height of 36 feet from ins. topmost ceiling 17.5 Warehouses 2d Class. of Wall. For a height of 22 feet below ins. topmost ceiling 13 For a height of 40 feet lower . . 21.5 ; For a height of 36 feet lower . . 17, 24 feet lower . . 26 For footings 43.5 3d Class. For a height of 28 feet below topmost ceiling 13 For a height of 16 feet lower . . 17.5 For footings. 1 7-5 8 feet lower . . 21.5 For footings 34.5 dtli Class. For a height of 9 feet below topmost ceiling 8.5 For a height of 13 feet below . . 13 30.5 ; For footings 21.5 Wooden Roofs. (English.) Span in Feet. Principal Beam. Tie Beam. King Posts. Queen Posts. Small Queens. Straining Beam. Struts. 20 4 X 4 9X4 4X4 — — — 3 X 3 25 5X4 10 X 5 5x5 — ■f— — 5 X 3 30 6x4 11 X 6 6 X 6 — — — 6 X 3 35 5X4 11 X 4 — 4X4 — 7X4 4 X 2 45 6X5 13 X 6 — 6 x 6 — 7X6 5 X 3 50 8 x 6 13 X 8 — 8 x 8 8 X 4 9X6 5 X 3 55 8x7 14 x 9 — 9 X 8 9X4 IO x 6 5-5 X 3 60 8 x 8 15 X 10 — 10 X 8 IO X 4 11 X 6 6 X 3 Mineral Constituents absorbed or removed fr, Acre of Soil b y several Crops. (Johnson.) om an Crops. Wheat, 25 bushels. Barley, 40 bushels. Turnips, 20 tons. Hay, 1.5 tons. Crops. Wheat, 25 bushels. - Barley, 40 bushels. Turnips, 20 tons. Hay, 1.5 tons. Potassa Lbs. 29.6 Lbs. Lbs. 47.1 8.2 Lbs. Q ry Lbs. Lbs. Lbs. Lbs. Soda I 7-5 JO. ^ oUipnunc I Acid. . . } • ‘ fib Torino 10.6 2.7 z 3-3 9. 2 Lime 3 12.9 10.6 5- 2 W 9.2 29.9 19.7 7-i 46-3 44-5 7. I 16 129.5 2.4 Magnesia Silica. . . 118. 1 3- 6 247.8 4.I 78.2 Oxide of Iron. Phosphoric ) 2.6 20.6 2. 1 /• 1 .6 Alumina. Acid ] 25.8 i5- 1 Total 210 213 4 2 3 209 190 miscellaneous elements. Average Quantity of Tannin in Several Substances, ( Morjit . ) Catechu. Per Cent. Bombay 55 Bengal 44 Kino 75 Nutgalls. Aleppo 65 Chinese Oak. Old, inner bark Oak. Per Cent. Young, inner b’k 15.2 “ entire b’k. 6 “ spring- ) cut bark ) “ root bark. 8.9 Chestnut. Amer. rose, bark 8 14.2 Horse, “ 2 Sassafras , root bark 58 Alder hark 36 per cent. 69 Sumac. Per Cent. Sicily and Malaga 16 Virginia 10 Carolina 5 Willow. Inner bark 16 Weeping 16 Sycamore bark 16 Tan shrub “ 13 Cherry-tree 24 To Convert Chemical Formulae into a Mathematical Expression. 1? ULE Multiply together equivalent and exponent of each substance, and product Win give proportion A compound by weight. Divide .ooo by sum of ttor praduete, and multiply this quotient by each of these products, and products w ill give re spective proportion of each part by weight in 1000. Example. — Chemical formula for alcohol is C A H 6 0 2 . Required their propor- tional parts by weight in 1000? Ca Carbon == 6. 1 X 4 = 2 4- 4 ) . He Hydrogen = 1X6= 6 /X21.55 0 2 Oxygen = 8X2 = 16 ) 1000 -f- 46.4 =21.55 Symbols and. 525-82 129.3 344-8 999.92 by weight. Elementary Bodies, with, their Equivalents. Body. | Symb. | Equiv. Aluminium.. . Antimony.. . . Arsenic Barium Bismuth. .... Boron ... Bromine. . . . . Cadmium Calcium Carbon Chlorine Chromium. . . Cobalt Columbium. . Copper Fluorine Glucinum — Analysis Body. 13-7 64.6 37-7 68.6 7i-5 11 78.4 55-8 20.5 6.1 35-5 26.2 29-5 184. 8 3i-7 18.7 6.9 Body. | Symb. Equiv. Gold Hydrogen Iodine Iridium Iron Lead Lithium Magnesium . . Manganese. . . Mercury Molybdenum. Nickel Nitrogen Osmium Oxygen Palladium — Phosphorus. . 196.6 1 126.5 98- 5 28 io3-7 7 12.7 26 200 47-9 29-5 14.2 99- 7 8 53-3 15-9 Body. Symb. Equiv. Platinum Pt 98.8 Potassium . . . K 39 - 2 Rhodium R 52.2 Selenium .... Se 40 Silicon Si 22 Silver Ag 108.3 Sodium Na 23-5 Strontium.... Sr 43-8 Sulphur S 16. 1 Tellurium Te 64.2 Tin Sn 58-9 Titanium Ti 24-5 Tungsten . . . . W ¥ Uranium . . . . U 60 Yttrium Y 32 Zinc Zn 32-3 Zirconium . . . Zr 34 of certain Organic Substances "by Weight. Albumen Alcohol Atmospheric air Camphor Caoutchouc .... Casein Fibrin Gelatine Gum Hordein Lignin Hydro gen. 52-9 52- 7 73-4 87.2 59*8 53- 4 47-9 42.7 44.2 52.5 7-5 12.9 10.7 12.8 7-4 7 l 9 6.4 6.4 5-7 Oxy- gen. Nitro- gen. 23-9 34-4 77 . 15.6 11. 4 19.7 27.2 50.9 47.6 41.8 15-7 23 21.4 19.9 17 1.8 Morphine Narcotine Oil, Castor Linseed. . . . Spermaceti. Quinine Starch Strychnine Sugar Tannin Urea Hydro- gen 72.3 65 74 76 78 Q 75-8 44.2 76.4 42.2 52.6 18.9 6.4 5-5 10.3 ii-3 11. 8 7-5 6.7 6.7 6.6 3-8 9-7 Oxy- gen. 16.3 27 i5-7 12.7 10.2 8.6 49.1 11. 1 43- 6 26.2 8.1 7s 45-2 MISCELLANEOUS ELEMENTS, 191 Dilution Per Cent. Necessary to Reduce Spiritnons Diqnors. Water to be added to 100 volumes of spirit when of following strength: Strength Required. 90 85 80 75 70 65 60 55 50 Per cent. Per cent. Per cent. Per cent. Per cent. Per cent. Per cent. Per cent. Per cent. Per cent. 85 5-9 — — — — — 80 12.5 6-3 — — — 75 20 13-3 6-7 — — — 70 28.6 21.4 14-3 7- 1 — 65 38.5 30.8 23.1 i5-4 7-7 — 60 50 41.7 33-3 25 16.7 8-3 — 55 63.6 54-5 45-5 36.4 27.4 18.2 9.1 — 50 80 7° 60 50 40 30 20 10 40 125 112.5 100 87-5 75 62.5 50 37-5 25 30 • 200 183-3 166.7 150 133-3 116.7 loo 83*3 66.7 Illustration.— 100 volumes of spirituous liquor having 00 per cent, of spirit con- tains: alcohol 90, water 10, — 100. To reduce it to 30 per cent, there is required 200 volumes of water. Hence 200 -f- 10 5= 210, and — = — = 30 spirit ’ or 3Q per cent 210 70 70 water, 0 * Proportion of Alcohol Per Cent. In 100 Parts of Spirit , by Weight or Volume , at 6 o°. Alcohol. Specific Gravity. Alcohol. Specific G ravity . Alcohol. Specific Gravity. Alcohol. Specific Gravity. 0 I 20 .972 50 .918 80 .848 5 .991 30 •958 60 .896 90 .823 10 .984 40 •94 70 .872 100 •794 In 100 Parts of Alcohol and Water , by Weight, at 6 o°. Alcohol. Specific Gravity. Alcohol. Specific Gravity. Alcohol. Specific Gravity. Alcohol. Specific Gravity. 0 1 1.99 •99 6 5.01 .991 7-99 .987 •53 •999 3.02 •994 6.02 •99 9-°5 .985 1.02 .998 4.02 •993 7.02 .988 10. 07 .984 Tid.es of Atlaiatic and IPacific Oceans at Isthmus of Panama. (Totten.) Atlantic , Navy Bay.— Highest tide 1.5 feet; lowest . 63 feet. Pacific , Panama Bay . — Highest tide 17.72 to 21.3 feet ; lowest 9. 7 feet. State. JSq. Areas of TJ. S. Coal Fields. . Miles. Illinois Virginia Pennsylvania* . . . Kentucky 44000 21 000 I 5 437 13500 State. Ohio Indiana. . . Missourif. Michigan!. * Bituminous and Anthracite. Sq. Miles. 11 900 7 700 6000 5000 Tennessee . . Alabama Maryland. . . Georgia t Anthracite. England France 106.5' Holland ) 0 102 ° Belgium ' Denmark 1 Sweden [ . . 99. 5 0 Norway ) Russia Egypt. . Africa. . Asia . . . Suez. . . England. . .— 5 0 ] Denmark ) Holland ) 0 Sweden [ Belgium j | Norway ) Sq. Miles. 4300 3400 550 150 Extremes of Heat in Various Countries. Greece...... 105° Italy 1040 Spain 102 0 Tunis 112.5' Germany 103 o | Manilla u 3 . 5 o Extremes of temperature upon the Earth 240°. Extremes of Cold in. Various Countries. -67 c I France —24° Russia . . . . — 46° I Germany. .— 32 0 . 116. i° • 133 - 4 ° . 120 ° . 1 26. 5° Italy —10 o Fort Reliance, N. A. .—70° Semipalatinsk, “ ..—76° 192 MISCELLANEOUS ELEMENTS. NT e aii. Temperatures of Various Localities. Polar Regions. . 36° Globe 50 0 London 5*° Edinburgh 41 0 I Rome . . . 60° I I Poles — 13 0 I 1 Equator ... 82° | | Torrid Zone. 75°! Line of Perpetual Congelation, or Snow Line. Latitude. 15 20 25 Height. Feet. 14764 14 760 13 47 8 12 557 Latitude. 30 35 40 45 Height. Feet. 11484 10287 9000 7670 Latitude. 50 55 60 65 Height. Feet. 6334 5020 3818 2230 70 75 80 85 Height. Feet. 1278 1016 451 327 and in Iceland 3084 feet. At the Equator it is 15260 feet; at the Alps 8120 feet; At Polar Regions ice is constant at surface of the Earth. Limits of Vegetation in Temperate Zone. The Vine ceases to grow at about 2300 feet above level of the sea, Indian Corn at 2800, Oak at 3350, Walnut at 3600, Ash at 4800, Yellow Pine at 6200, and Fir at 6700. Periods of Gres tat ion and Number of Young. — ’ ' Weeks. 9 Weeks, Elephant. 100 Horse Camel Ass No. 1 Cow Weeks. No. 1 Sheep . . Weeks. . 21 No. 2 Buffalo . . 40 1 Goat . . . , . 22 2 1 Stag. . . . • 36 1 Beaver , .. 17 3 1 Bear . . . • 30 2 Pig .. 17 12 1 Deer . . . . 24 | 2 Wolf . . . . 10 5 Rabbit. . • 4 6 Guinea Pig. 3 3 No. 6 5 6 8 6 Periods of Incubation of Birds. Swau 42 days- Parrot, 40 days; Goose and Pheasant, 35 days; Duck Turkey, and Peafowl, 28 days; Hens of all gallinaceous birds, 21 days; Pigeon and Canaiy, 14 days. Temperature of incubation is 104 0 . Ages of Animals, etc. Bear 20; Cow, 20; Deer, 20; Rhinoceros, 20; Swine, 20; Wolf, 20, Cat, 15, Fox, 15, Dog, 15; Sheep, 10; Hare, Rabbit, and Squirrel, 7. Relative Weights of Brain. Man, 154.33; Mammifers, 29.88; Birds, 26.22; Reptiles, 4.2; Fish, 1. Buoyancy of Casks. Buoyancy of a c ask in fresh water in lbs. = »-97 «mes volume of it in U. S. gal- Ions and 10 times in Imperial gallons, less weight of cask. Transportation of Horses and Cattle. Space re 9di r ®d dfei»°is Hbr S Hors° s^Hay^ 15 lte!; ■ Beeves, Haj^ 18 lbs. ; Water, 6 gallons. Rock and Earth Excavation and Embankment. Number of Cube Feet of various Earths in a Ton. ^ n nA . cHv 18.6 1 Clay with Gravel 14.4^ Loose Earth 24 | uay ' - ~ 1 * Coarse Sand 18.6 | Earth with Gravel. . . 17.8 | Common Soil. ....... 15.6 The volume of Earth and Sand in embankment exceeds that in a primary ex- cavation in following proportions: Clay and Earth will subside about .12. MISCELLANEOUS ELEMENTS. 193 Hills or Plants in an Area of One Acre. From 1 to 40 feet apart from centres . Feet apart. No. Feet apart. No. Feet apart. No. Feet apart. | No. 1 43 560 5 1742 9 538 16 171 *•5 19360 55 1440 9-5 482 17 151 2 10890 6 1210 10 435 18 135 2-5 6969 6.5 1031 10.5 361 20 108 3 4 840 7 889 12 302 25 69 3-5 3 556 7-5 775 13 258 30 48 4 2 722 8 680 14 223 35 35 4-5 2151 8-5 692 i 5 193 40 27 Number of several Seeds in a 33ushel, and Number j>er Square Foot per Acre. Timothy. Clover . . . No. Sq. Foot. No. Sq. Foot. 41 823 360 16 400 960 960 Rye 888 390 556 290 20.4 12.8 376 Wheat "V olumes. Permanent gases, as air, etc., are diminished in their volume in a ratio direct with that of pressure applied to them. With vapor, as steam, etc. this rule is varied in consequence of presence of the temperature of vaporization Minerals. Talc 1 Gypsum 2 Mica 2.5 Carbonate of lime. 3 Carats. Mattam 367 Grand Mogul* 270.0 Orlotf. 194.25 Florentine, brilliant . 139.5 Crown of Portugal. . .138.5 * Rough 900. re Hardness of some Minei 1 Barytes • 3-5 Opal 6 Fluor-spar. . . , • 4 Quartz 7 | Feldspar . 6 Tourmalin 7 | Lapis Lazuli . 6 Garnet 7.5 Emerald 8 Topaz 8 Ruby g Diamond 10 “Weight of Diamonds. Carats. Regent or Pitt 136.75 I Star of the Southf. . 125 Koh-i-NoorJ 106.06 Piggott 82.25 I Napac 78.625! t Rough 254.5. • Carats. Dresden 76.5 Sancy 53 . 5 Eugenie, brilliant . 51 Hope (blue) 48.5 Polar Star 40.25 % Originally 793. Heat of the Sun. artowNgrton 3138740°! Waterston ,6000000° Capt. John Ericsson 4 909 86o° J Soret 10443223 0 Sundry others ranging from 2520 0 to 183600°. a j Moon. — Distance of Moon from Earth 237000 miles. Frigorific Mixture. Lowest temperature yet procured. Faraday obtained 166° by evaporation of a mixture of solid carbonic acid and sulphuric ether. Current of Rivers. A fall of . 1 of an inch in a mile will produce a current in rivers. Sandstones. condi tion 16S 8ands * one erec ^ England in 12th century are yet in good Canal Transportation. . Erie and Hudson River.— From Buffalo to New York 405 miles cost of transportation 2.46 mills per ton (inclusive of tolls) per mile. Transportation of vheat costs when it reaches New York 4.72 cents per bushel, and .61 cents ner bushel for elevating and trimming. 5 uu .01 cents per ♦ ~r Er ) e . Canal. —Four mules will tow 230 tons of freight down and IO o of 6 9 q macs 1 V ° V mg a pen ° d ° f 30 days ’ at a cost of 8 cents P cr miIe for a course R 194 MISCELLANEOUS ELEMENTS. Matter. Unit of the Physicist is a molecule, and a mass of matter is composed of them, having same physical properties as parent mass. . .. It exists in three forms, known as solid, liquid, and gaseous. Solids have mdi- vidualitv of form, and they press downward alone. Liquids have not individuality of fornAxcept in spherical form of a drop, and they press downward and sideward. Gases are wholly deficient in form, expanding in all directions, and consequently thev uress upward, downward, and sideward. Liquids are compressible to a very moderate degree. Water has been forced through pores of silver, and it may be compressed by a pressure of one pound pei square inch to the 3300000th part of its volume. Gases may be liquefied by pressure or by reduction of their temperature. Combustible matter (as coal) may be burned, a structure (as a - bouse) may be destroyed as such, and the fluid (of an ink) may be evaporated, yet the matter of which coal and house were composed, although dissipated, exists, and the water and coloring matter of the ink are yet in existence. Spaces between the particles of a body are termed pores. . All matter is porous. Polished marble will absorb moisture, as evidenced in its discoloration by presence of a colored fluid, as ink, etc. Silica is the base of the mineral world, and Carbon of the organized. M.iivu.teness of NIatter. A piece of metal, stone, or earth, divided to a powder, a particle of it, however minute, is yet a piece of the original material from which it was separated, retain- ing its identity, and is termed a molecule. , It is estimated there are 120000000 corpuscles in a drop of blood of the musk-deer Thread of a spider’s web is of a cable form, is but one sixth diameter of a fibre of silk, and 4 miles of it is estimated to have a weight of but 1 grain. . One imperial gallon (277.24 cube ins.) of w T ater will be colored by mixture therein of a grain of carmine or indigo. A grain of platinum can be drawn out the length of a mile. Film of a soap- and- water bubble is estimated to be but the 300000th part ot an inch in thickness. , . . - T . It is computed that it would require 12000 of the insect know T n as the twilignt monad to fill up a line one inch in length. . A drop of water, or a minute volume of gas, however much expanded— even to the volume of the Earth— would present distinct molecules. Gold leaf is the 280000th part of an inch in thickness. A thread of silk is 2500th of an inch in diameter. , .. A cube inch of chalk in some places in vicinity of Paris contains 100000 ol shells of the foraminifera. „ _ . There are animalcules so small that it requires 75 000 000 of them to weign a gram. "Velocity, Weight, and. Volume of Molecules. Velocity.— Collisions among the particles of Hydrogen are estimated to occur at the rate of 17 million-million-million per second, and in Oxygen less than half this number. Weight— A million-million-million-million molecules of Hydrogen are estimated to weigh but 60 grains. Volume. — 19 million-million-million molecules of Hydrogen have a volume of .061 cube ins. Diameter.— Five millions in a line would measure but .1 inch. i Cliarcoal, .AJLcoliol. Charcoal as yet has not been liquefied, nor has Alcohol been solidified. Metals. Metals have five degrees of lustre — splendent , shining , glistening , glimmering , and dull. . . All metals can be vaporized, or exist as a gas, by application to them of their ap- propriate temperature of conversion. Repeated hammering of a metal renders it brittle ; reheating it restores its tenacity. Repeated melting of iron renders it harder, and up to twelfth time it becomes stronger. Platinum is the most ductile of all metals. MISCELLANEOUS ELEMENTS. 195 Impenetrability. Impenetrability expresses the inability of two or more bodies to occupy same space at same time. A mixture of two or more fluids may compose a less volume than that due to sum of their original volume, in consequence of a denser or closer occupation of their molecules. This is evident in the mixture of alcohol and water in the proportion of 16.5 volumes of former to 25 of latter, when there is a loss of one volume. Elasticity. Elasticity is the term for the capacity of a body to recover its former volume after being subjected to compression by percussion or deflection. Glass, ivory, and steel are the most elastic of all bodies, and clay and putty are illustrations of bodies almost devoid of elasticity. Caoutchouc (India rubber) is but moderately elastic^ it possesses contractility, however, in a great degree. Momentum. Momentum is quantity of motion, and is product of mass and its velocity. Thus, the momentum of a cannon-ball is product of its velocity in feet per second and its weight, and is denominated foot-pounds. A foot-pound is the power that will raise one pound one foot. Sot* nd. Velocity of sound is proportionate to its volume; thus, report of a blast with 2000 lbs. of powder passed 967 feet in one second, and one of 1200 lbs. 1210 feet. It passes in water with a velocity of 4708 feet per second. Conversation in a low tone has been maintained through cast-iron water pipes for a distance of 3120 feet, and its velocity is from 4 to 16 times greater in metals and wood than air. Eight. Sun’s rays have a velocity of 185 000 miles per second, equal to 7 . k times around the Earth. ' _ Color Blindness Is absence of elementary sensation corresponding to red. Luminous Point. To produce a visual circle, a luminous point must have a velocity of 10 feet in a second, the diameter not exceeding 15 ins. All solid bodies become luminous at 800 degrees of heat. AX i rage. When air near to surface of Earth becomes so highly heated, as upon a sandy plain, that its density within a defined distance from it increases upwards, a line of vision directed obliquely downwards will be' rendered by refraction, gradually increasing, more and more nearly horizontal as it advances, until its direction is so great as to produce a total reflection, and the reflected ray then, by successive re- fractions, is gradually elevated until it meets the eye of the observer. Looming is inverted mirage, frequently seen over calm water, and is effect of lower or surface stratum of air being colder than that above it. Snow Flakes. 96 forms of snow flakes have been observed. AX el ted Snow Produces from .25 to .125 of its bulk in water. Strength, of Ice. Two inches thick will support men in single file on planks 6 feet apart; 4 inches will support cavalry, light guns, and carts; and 6 inches wagons drawn by horses. Temperature. Sulphuric acid and water produce a much greater proportionate contraction than alcohol and w r ater. Both of these mixtures, however low their temperature pro- duce heat which is in a direct proportion to their diminution in volume. ’ year ^ deptl1 of 45 feet > the tem P erature of the Earth is uniform throughout the Temperature of Earth increases about 1° for every 50 to 60 feet of depth, and its crust is estimated at 30 miles. A body at Equator weighs two hundred and eighty-nine parts less than at the Poles. 196 MISCELLANEOUS ELEMENTS, Colors for Drawings. Material. Colors. Mateiial. Colors. Brass .... Brick Cast Iron. P|qV Gamboge. Carmine. Neutral tint. Burned umber. “ “ light. Sepia with dark spots. Lake and Bur’d Sienna. Granite Lead Indian Ink, light. “ “ and Prussian blue. Light blue and Lake. Cobalt or Verdigris. ( Burned Sienna, deep and light, \ for dark and light wood. Prussian Blue, light. Steel Water y Earth Concrete . Copper. . . Woods Wr’ght Iron. Dird.s and. Insects, — (J/. De Lacy.) Elements of Flight— Resistance of air to a body in motion is in ratio of surface of body and as square of its velocity. Wing Surface— Extent or area of winged surface is in an inverse ratio to weight of bird or insect. A Stag-beetle weighs 460 times more than a Gnat, and has but one fourteenth of its wing surface; 150 times more than a Lady Bird (bug), and has but one fifth. An Australian Crane weighs 339 times more than a sparrow, and has but one sev- enth- 3000000 times more than a Gnat, and has but one hundred and fortieth. A Stork weighs eight times more than a Pigeon, and has but one half. A Pigeon weighs ten times more than a Sparrow, and has but one half; 97 000 times more than a Gnat, and has but one fortieth. A resisting surface of 30 sq. yards will enable a man of ordinary weight to descend safely from a great elevation. Strength of Insects. —Insects are relatively strongest of all animals. A Cricket can leap 80 times its length, and a Flea 200 times. Application for Stings and Burns. Sting of Insects. —Ammonia, or Soda moistened with water, and applied as a paste. Burns . — Hot alcohol or turpentine, and afterwards bathed with lime water and sweet oii. Cold water not to be applied. To Preserve jVIeat. Meat of any kind may be preserved in a temperature of from 8o° to ioo°, for a period of ten days, after it has been soaked in a solution of 1 pint of salt dissolved in 4 gallons of cold water and .5 gallon of a solution of bisulphate of calcium. By repeating this process, preservation may be extended by addition of a solution of gelatin or white of an egg to the salt and water. To Detect Starch, in [Milk. Add a few drops of acetic acid to a small quantity of milk ; boil it, and after it has cooled filter the whey. If starch is present, a drop of iodine j solution will produce a blue tint. This process is so delicate that it will show the presence of a milligram of starch in a cube centimeter of whey (1 grain of starch in 2.16 fluid-ounces). Detaining "Walls of Iron Diles. Sheet Piles . — 7 feet from centres, 18 ins. in width and 2 ins. in thickness, strength- j ened with 2 ribs 8 ins. in depth. { Plates .— 7 feet in length by 5 feet in width and 1 inch in thickness, with one diagonal feather 1 by 6 ins. Tie-rods 2 ins. in diameter Stone Sawing. Diamond Stone Sawing.— {Emerson.) Alabama marble 6 feet X 2.5 feet in 22 min- utes — 41 sq. feet per hour. "Wood. Sawing. 7722 feet of poplar, board measure, from 9 round logs in 1 hour. Engine 12 ins diameter by 24 ins. stroke. MISCELLANEOUS ELEMENTS. 197 Cost of Dredging. Actual cost , if on an extended ivorJc, inclusive of Delivery , if dredging into or on a vessel alongside of dredger. — ( Trautwine . ) Labor at $ 1 per day and Repairs of Plant included. Depth. Cents. Depth. Cents. Depth. Cents. Depth. Cents. Feet. Cube Yards. Feet. Cube Yards. Feet. Cube Yards. Feet. Cube Yards. 10 6 20 8 25 10 35 18 15 7 22 . 9 30 13 40 25 Discharge of Scows or Camels.-^ Towing .25 mile 4 cents per cube yard, .5 mile 6 cents, .75 mile 8 cents, and 1 mile 10 cents. Note. — A Scow is a flat-bottomed vessel or boat. A Camel is a shallow, flat- bottomed and decked vessel, designed for the transportation of heavy freight or the sustaining of attached bodies, as a vessel, by its buoyancy. Dredging. A steam dredge will raise 6 cube yards, or 8.5 tons, per hour per IP. jVIetal Boring and Turning. Boring. — Cast iron. — Divide 25 by the diameter of the cylinder in inches for the revolutions per minute. Wrought iron. — The speed is one fourth to one fifth greater than for cast iron. Brass.— The speed is about twice that for cast iron. Turning. — Cast iron.— The speed is twice that of boring. Wrought iron.— The speed is one fourth to one fifth greater than that for cast iron. Brass.— The speed is twice that of boring. Vertical boring.— The speed may be twice that of horizontal boring. The feed depends upon the stability of the machine and depth of the cut. "Well Boring. At Coventry, Eng., 750000 galls, of water per day are obtained by two borings of 6 and 8 ins., at depths of 200 and 300 feet. At Liverpool, Eng., 3000000 galls, of water per day are obtained by a bore 6 ins in diameter and 161 feet in depth. This large yield is ascribed to the existence of a fault near to it, and extending t® a depth of 484 feet. ° At Kentish Town, Eng. , a well is bored to the depth of 1302 feet. At Passy, France, a well with a bore of 1 meter in diameter is sunk to a depth of 1804 feet, and for a diameter of 2 feet 4 ins. it is further sunk to a depth of 100 feet 10 ins., or 1903 feet 10 ins., from which a yield of 5 582 000 galls, of water are obtained per day. Tempering Boring Instruments. Heat the tool to a blood-red heat; hammer it until it is nearly cold- reheat it to a blood-red heat, and plunge it into a mixture of 2 oz. each of vitriol, soda sal-am- moniac, and spirits of nitre, 1 oz. of oil of vitriol, .5 oz. of saltpetre, and -2 sails of water, retaining it there until it is cool. J Circular Saws. Revolutions per Minute.— % ins. 4500, 10 ins. 3600, and 36 ins. 1000. Masonry. Concrete or Beton should be thrown, or let fall from a height of at least 10 feet or well beaten down. 1 The average weight of brickwork in mortar is about 102 lbs. per cube foot. Blastering. ,? la ? t S rers ’ work openings, as doors, windows, etc., are com- at ? n ,f- ha !i of their areas > and cornices are measured upon their extreme edges, including that cut off by mitring. Gr la zing. In Glaziers’ work, oval and round windows are measured as squares R* 1 98 MISCELLANEOUS ELEMENTS. Corn Measure. Two cube feet of corn in ear will make a bushel of corn when shelled. Tenacity of Iron Bolts in Woods. Diameter 1.125 ins. and 12 ins. in length required for Hemlock 8 tons, and for Pine 6 tons to withdraw them. Length of Gun Barrels. (C. T. Coathupe.) The length of the barrel of a gun, to shoot well, measured from vent-hole, should not be less than 44 times diameter of its bore, nor more than 47. Hay and. Straw. Hay, loose. 5 lbs. per cube foot. Ordinarily pressed, as in a stack or mow, 8 lbs. Close pressed, as in a bale, 12 to 14 lbs. , Ordinarily pressed, as in a wagon load, 450 to 500 cube feet will weigh a ton. Straw in a bale 10 to 12 lbs. per cube foot. jS’atixral Bowers. $ Mn The power or work performed by the Sun’s evaporation is estimated at Niaaaj-a Volume of water discharged over the falls is estimated at 33000000 tons per hour, and the entire fall from Lake Erie at Buffalo to Lake Ontario is 323.35 feet. "Velocity of* Stars. According to computation of Mr. Trautwine a Star passes a range in 3' 55-91" less time each day. Service Train of a Quartermaster. Quartermaster’s train of an army averages i wagon to every 24 me “: an ? ® r "' eI1 ’ equipped army in the field, with artillery, cavalry, and trains, requires 1 horse or mule, upon the average, to every 2 men. Tides. The difference in time between high water averages about 49 minutes each day. Atlantic and Pacific Oceans.— Rise and fall of tide in Atlantic at Aspmwall 2 feet, in Pacific at Panama 24 feet. Dimensions of Drawings for Batents. United States, 8.5 X 12 inches. Eatitxxde. One minute of latitude, mean level of Sea, nearly 6076 feet = 1.1508 Statute miles. Artesian "Well. White Plains, Nev., Depth 2500 feet. Foirndation Biles. A pile if driven to a fair refusal by a ram of 1 ton, falling 30 feet, will bear 1 ton vreight for each sq. foot of its external or frictional surface, or a safe load of 750 lbs. per sq. foot of surface. N Earth.. Density of its mass 5.67. Tripolith. A new building material, compounded of Coke, Sulphate of Lime, and Oxide of Iron. It has increased tensile strength after exposure to the air, being much in excess of that of lime and cement. Gras and Electric Eight. Gas light of 16 candle power costs 5 cent per hour; Electric, 4.15 cents. Niagara. Discovered, 1678. Falls have receded 76 feet in 175 years. Height, American Falls, 164 feet; Horseshoe, 158 feet. BRIDGES. — U. S. ENSIGNS, PENNANTS, AND FLAGS. 199 Suspension. Bridges. Lengths of Spans in Feet. You-Mau, China 330 Schuylkill (Phila.) 342 Hammersmith, Eng. 422 Pesth (Danube) 660 New York and Brooklyn, 930, 1595.5, and 930; clear height of Bridge above high water at qo°, 135 feet. Niagara Lewistown and Queenstown . Cincinnati Niagara Falls. 822 1040 1057 1280 U. S. ENSIGN, PENNANTS, AND FLAGS. Ensign. — Head (Depth, or Hoist). — Ten nineteenths of its length. Field— Thirteen horizontal stripes of equal breadth, alternately red and white, beginning with red. Union. — A blue field in upper quarter, next the head, .4 of length of field, and seven stripes in depth, with white stars ranged in equidistant, horizon- tal, and vertical lines, equal in number to number of States of the Union. Pennants (Narrow). — Head. — 6.24 ins. to a length of 70 feet; 5.04 ins. to a length of 40 feet; 4.2 ins. to a length of 35 feet. Night , 3.6 ins. to a length of 20 feet, and 3 ins. to a length of 9 feet.— Boat, 2.52 ins. to a length of 6 feet. Union.— A. blue field at head, one fourth the length, with 13 white stars in a hori- zontal line. Field . — A red and white stripe uniformly tapered to a point, red up- permost. Night and Boat Pennants . — Union to have but 7 stars. Union Jack. — Alike to the Union of an Ensign in dimensions and stars. Flags. — President. — Rectangle, with arms of the U. S. in centre flf a blue field. Secretary- of Navy. — Rectangle, with a vertical white foul anchor in centre of a blue field. Admiral. — Rectangle, with 4 white stars in centre of a blue field, set as a square. Vice-Admiral. — Same as Admiral’s, with 3 white stars set as an equi- lateral triangle. Rear-Admiral. — Same as Admiral’s, with two white stars set vertical. If two or more rear-admirals in command afloat should meet, their seniority is to be indicated respectively by a Blue flag, a Red with White stars, and a White with Blue stars, and another or all others, a White flag with Blue stars. Commodore’s ( Broad Pennant). — Blue, Red, or White, according to rank, with one star in centre of field, being white in blue and red pennants, and blue in white. Swallow-tailed, angle at tail, bisected by a line drawn at a right angle from centre of depth or hoist, and at a distance from head of three fifths of length of pennant; the lower side rectangular with head or hoist; upper side tapered, running the width of pennant at the tails .1 the hoist. Head. — .6 length. Fly 1.66 -hoist. Divisional Marks. — Triangle, 1st Blue, 2d Red, 3d White. Blue vertical. Reserve Division. — Yellow Red vertical. Division mark is worn by Commander of a division of a squadron at mizzen, when not authorized to wear Broad Pennant of a Commander or Flag of an Admiral. Fly .8 hoist. Signal Numbers. — Fly i. 25 hoist. Signal Pennants , Fly 4.6 hoist . Repeaters 1.89 hoist. International , Signal Number , Square , Signal Pennants. Fly .3 hoist . 200 ANIMAL FOOD, Alimentary Principles. Primary division of Food is into Organic and Inorganic. Organic is subdivided into Nitrogenous and Non-Nitrogenous ; Inorganic is composed of water and various saline principles. The former elements are destined for growth and maintenance of the body, and are termed “ plas- tic elements of nutrition.” The latter are designed for undergoing oxidation, and thus become source of heat, and are termed “ elements ot respiration, or “Calorificient.” , Although Fat is non-nitrogenous, it is so mixed with nitrogenous matter that it becomes a nutrient as well as a calorificient. Alimentary Principles. — i. Water; 2. Sugar; 3. Gum; 4. Starch; 5- P^tine, 6. Acetic Acid; 7. Alcohol; 8. Oil or Fat. Vegetable and Animal— 9. Albumen, 10. Fibrine; 11. Caseine; 12. Gluten; 13. Gelatine; 14. Chloride of Sodium. These alimentary principles, by their mixture or union, form our ordinary foods, which bv way of distinction, may be denominated compound aliments ; thus meat is composed of fibrine, albumen, gelatine, fat, etc. ; wheat consists of starch, gluten, sugar, gum, etc. Analysis of IVIeats, Pish, ‘Vegetable s, etc. Ash, etc. Food. Water. Nitro- I genous Matter, j Fat. | Sal'.ne ^ Matter. Non-Nitro- genous Matter. Sugar. | Cellu- A lose. Al Arrowroot 18 — >- I — 82 — — Barley Meal 15 6-3 2.4 2 •69.4 4.9 — Beans, White 9.9 255 2.8 55-7 ' 2.9 Beef, roast 54 27.6 15-45 2 95 — fat 5i 14.8 29.8 44 — . lean 72 *9-3 3- 6 5‘i — salt 49.1 29.6 .2 21. 1 — — Beer and Porter. . . . 9 1 .1 — .2 — Buckwheat 13 13- 1 3 •4 64-5 3-5 Blitter and Fats — 15 — 83 2 — — Cabbage 9 1 2 •5 •7 5-8 Carrots 83 i-3 .2 1 7-4 0. 1 Cheese 36.8 33-5 24-3 5-4 — — Corn Meal 14 11. 1 # 8. 1 *•7 57*6 •4 5-9 Cream 66 2.7 26.7 1.8 — 2.8 — Egg 74 14 10.5 i-5 — yolk 52 16 3°-7 *•3 — Fish, white flesh. . . 78 18.1 2.9 1 — 1 — Eels 75 9.9 13.8 i-3 — Lobster, flesh. 76.6 19.17 1.17 1.8 1.26 — • Oysters 80.39 14.01 1.52 2.7 1.38 Liver, Calf’s 72-33 20.55 5-58 i-54 — Milk, Cow’s 86 4- 1 3-9 .8 — 5-2 Mutton, fat 53 12.4 3I i 3-5 — Oatmeal i5 12.6 5-6 3 58.4 5-4 ’ Oats 21 14.4 5-5 — 48.2 — 7.6 Parsnips 82 1. 1 •5 1 9.6 5-8 Peas i5 23 2. 1 2.5 50.2 2 3- 1 Pork, fat 39 9.8 48.9 23 — Bacon, dry. . . 15 8.8 73-3 2.9 — Potatoes 75 2.1 .2 •7 16.8 3-2 1 Poultry 74 21 3-8 1.2 — Rice i3 6-3 •7 •5 78. 1 •4 Rye Meal i5 8 2 1.8 69-5 3-7 Sugar 5 — — — 95 Tripe 68 13.2 16.4 2.4 Turnips 9 1 1.2 — .6 4-3 2. 1 Veal 63 16.5 15.8 4-7 — ■ Wheat Flour 15 10.8 2 i-7 61. 1 4.2 3.5 Bread* . 37 8.1 1.6 2-3 45-4 3- 6 - | Bran ■1 i3 18 6 — 60 — 3-2 2.5 3-3 x *7 — 2 ing most. 100 lbs. flour yield 130 lbs. bread. ANIMAL FOOD. 201 Analysis of Different Foods In their Natural Condition. Ni- trates. 'Carbon- ates. Phos- phates. Water. Ni- trates. Apples 5 10 ! 84 Milk of cow. . 5 Barley 1 7 6o. c 3. ^ Mutton . 12.5 Beans 2 A. y 0 C7. 7 3. ^ 14. 8 Oats Beef T I C J/ / j D c eo Parsnips x 7 Buckwheat . . 8.6 75-4 J 1.8 14.2 Pork 9.2 10 Cabbage 4 5 1 9° Potatoes 2.4 Chicken 19 3-5 4-5 73 “ sweet 1.5 Corn,North’n 12 73 1 14 Rice 6.5 “ South’n 35 48 3 !4 Turnips 5 Cucumbers. . . *•5 1 .5 97 Veal 16 Lamb 11 35-5 3-5 50 Wheat i5 8 40 66.4 7 50 22.5 28.4 79-5 4 16.5 69.2 4-5 3 1 *•5 •9 2.6 • 5 •5 4-5 1.6 86 43 13.6 82.8 38.5 74.2 67-5 i3-5 9°-5 63 14.2 Nitrates — Are that class which supplies waste of muscle. Carbonates — Are that class which supplies lungs with fuel, and thus furnishes heat to the system, and supplies fat or adipose substances. Phosphates — Are that class which supplies bones, brains, and nerves and gives vital power, both muscular and mental. From above it appears, that Southern corn produces most muscle and least fat and contains enough of phosphate's to give vital power to brain, and make bones strong. Mutton is the meat which should be eaten with Southern corn. The nitrates in all the fine bread which a man can eat will not sustain life beyond fifty da) r s ; but others, fed on unbolted flour bread, would continue to thrive for an indefinite period. It is immaterial whether the general quantity of food be reduced too low, or whether either of the muscle-making or heat-producing principles be withdrawn while the other is fully supplied. In either case the effect will be the same. A man will become weak, dwindle away and die, sooner or later according to the deficiency; and if food is eaten which is deficient in either principle, the appe- tite will demand it in quantity till the deficient element is supplied. AH food be- yond the amount necessary to supply the principle that is not deficient, is not only wasted, but burdens the system with efforts to dispose of it. Analysis of Fruits. Apple, white. Apricot, average Blackberry Cherry, red sour black Currant, red Gooseberry, red yellow Grape, white Peach, Dutch Pear, red Plum, yellow gage. . . . large “ black blue “ red Italian, sweet. . . Raspberry, wild Strawberry, “ Banana. Water. | Sugar. Acid. Albumi- nous sub- stances. Insoluble matter. Pec to us Sub- stances. 85 7.6 1 .22 1.83 3.88 83-5 1.8 1. 1 •51 4-7 7-55 86.4 4.44 1. 19 •51 5.26 1.72 75-4 13 1 •35 •9 5-83 3-73 80.5 8.77 1.28 .83 5-9 1 2.07 79*7 10.7 •56 I 6. 04 1-33 85-4 5-6 i-7 •36 3-74 2.4 85.6 8 i-35 •44 2.92 1.26 85-4 7 1.2 .46 3-17 2.4 80 13-78 1 •83 2.48 1.44 85 1.58 .61 .46 5-49 6.4 83-5 7-5 .07 •25 3-54 4.8 80.8 2.96 .96 .48 3-9 8 10.48 79-7 3-4 .87 •4 3-9i IJ *3 88. 7 2 1.27 •4 6.86 •23 8s 3 2.25 I -33 •43 4-23 5-85 81.3 6-73 .84 .83 4.01 5-63 83-9 3-6 2 •55 8-37 1.28 87 4 i-5 .6. 5-5 •4 73-9 1 Sugar, Pectin, Salt, Acid , etc., 26.1. Sugar and. Water in Various Products not Included in tire Tat>le. ( Per Cent.) Water. „ Sugar. Sugar, crude 95 Molasses 77 Buttermilk 6.4 Molasses. . 23 Lean beef. 72 Buttermilk 88 Water. Cabbage g 1 Ale and Beer Coffee and Tea 100 202 ANIMAL FOOD, Relative Values of Foods or Assimilating Quality to make an Equal Quantity of Rlesh. in Cattle or Sheep. [Ewart.) Turnips Carrots Beets Parsnips and Swedes . . . Meadow grass in bloom. Vetches, pods open Potatoes at maturity Oat straw, cut green . . . . Bean or Vetch straw Meadow hay Vetch “ Linseed cake Cattle. Sheep. Article. Cattle. Sheep. 800 400 Wheat bran 45 105 630 — Corn and Barley meal . . 35 600 300 Oatmeal 34 — 600 200 Beanmeal 33 — 400 — Peameal 32 — 360 90 Cabbage 500 280 200 Pea straw — 200 125 — Rye bran . — 109 200 Oats — 70 100 100 Buckwheat — . 65 9 ° — Barley — 60 50 — Pease or Beans — 54 Note. — When these values express weight in lbs., then such food will produce about 4 to 5 lbs. beef or mutton. Nutritive Constituents and Values of Rood, in G-rains per Round. Food. Bakers’ Bread. . . . . Barley Meal Beef. Beer and Porter. . . Bullock’s Liver . . . Buttermilk Carrots Cheddar Cheese . . . Cocoa Dry Bacon Fat Pork Flour, Seconds Fresh Butter Green Bacon Green Vegetables. . Indian Meal Lard Molasses Carbon. Nitrogen. 1975 2563 1854 274 934 387 508 3344 3934 59S7 4ii3 2700 6456 5426 420 3016 4819 2395 68 184 1 204 44 14 306 140 , 9 J 116 76 120 Food. Carbon. ! Mutton Now Milk 1900 cqq Oatmeal 2831 USA Pfi r«n i ps Pearl Barley JJT 2660 Potatoes 769 T AlC Red Herrings Rino RyD Meal 2693 4585 1947 438 2698 Salt, Rutter Skim Cheese Skimmed Milk Split Pease Suet, Sugar 4710 2Qg: C Turnips *yoo 263 I KA. Whev Whitefish 871 1 189 44 136 217 68 86 483 43 248 13 13 *95 The Full Daily Diet of a man is held to be 12 oz. bread, 8 oz. potatoes, 6 oz. meat, 4 oz. boiled rice with milk, .375 pint of broth or pea soup, 1 pint milk, and 1 pint of beer. Nutritive Values and Constituents of jVIilh. — ( Payen .) Animal. Nitrogenous Matter and insoluble Salts. Butter. Lactic and soluble Salts. Water. Animal. Nitrogenous Matter and. insoluble Salts. Butter. Lactic and soluble Salts. Water. Goat. . . . 4-5 4 1 5-8 85.6 Ass *•7 1.4 6.4 9°-5 Cow 4-55 3-7 5-35 86.4 Mare . . . 1.62 .2 8-75 89-43 Woman. 3-35 3-34 3-77 89-54 Ewe 4.68 4.2 5-5 85.62 Weigh. t of some Different Roods required to furnish 1220 Grains of Nitrogenous Nlatter. Lbs. Barley Meal.. 2.9 Milk 42 Potatoes 8.3 Parsnips 15-9 Cheese Lbs. Meat, fat Lbs. i-3 Bacon, fat. Lbs. .... 1.8 Pease Oatmeal i-5 Bread Meat, lean • -9 Corn Meal 1.6 Rye Meal. , Fish, White . . . 1 Wheat Flour. . i-7 Rice .... 2.8 Turnips, 15.9 lbs. ; Beer or Porter, 158.6 lbs. ANIMAL FOOD, 203 [Proportion of Sugar and Acid in Various Fruits. (Fresenius.) Fruit. Sugar. Acid. Fruit. Sugar. Per Cent. Per Cent. Per Cent. Apple 8.4 .8 Plum. . Apricot 1.8 1. 1 Prune. . . . 6-3 Blackberry 4.4 1.2 Raspberry. Currants 6.1 2 Red Pear 4 Gooseberry 7. 2 I.5 Sour Cherry 7*5 0 0 Grape 14. 9 .7 Strawberry 0. 0 Mulberry Q. 2 / I. 0 Sweet, Cherry 5-7 T -. p Peach 1.6 •7 Whortleberry 10. 0 5.8 Acid. Per Cent. i-3 •9 i-5 .1 i-3 i-3 .6 x -3 Proportion of Oil in Various Air-dry Seeds. ( Berjot .) Beechnut Hemp Watermelon Mustard . Flax Peanut . Almond.. 40 Colza 1 4° (45 Orange 40 Poppy 1 4o Analysis of different Articles of Food, with. Reference only to their Properties for giving Heat and Strength. ( Payen .) In 100 Parts. Alcohol Barley Beans Beef, meat Beer, strong. . Bread, stale. . . Buckwheat. . . Butter Carrots Caviare Cheese, Chest’r Chocolate Cod-fish, salt’d Car- bon. Nitro- gen. Substances. Car- bon. Nitro- gen. Substances. Car- bon. Nitro- gen. 52 40 42 1 9 4-5 Coffee Corn Eels 9 44 30-05 x 3-5 34 1. 1 x -7 2 Oil, Olive Oysters Pen se 98 7.18 2.13 3-66 11 3 Eggs I.Q Potatoes 44 4-5 .08 Figs, dried •i.y .92 Rice 41 •33 1.8 28 42-5 1.07 2.2 Herring, salt- ed 23 15.68 10. 06 3. 1 1 Rye Flour Salmon 4i 16 2 9 x -75 83 5-5 .64 •3 1 Liver, Calf’s. . Lobster 3-93 2.93 3-74 .66 Sardines Tea 2.09 6 27.41 4.49 4- x 3 Mackerel 19.26 8 Truffes. 2.1 .2 41.04 Milk, Cow’s. . . Wheat 9-45 4 1 x -35 3 58 16 1-52 5-02 Nuts. Oatmeal 10.65 44 x -4 i-95 “ Flour.. Wine 38.5 4 1.64 .015 nitrogenous matter is obtained. 5.5, and equivalent amount of Human and Animal Sustenance. Least Quantity of Food required to Sustain Life. (E. Smith , M.D.) Carbon. Hydrogen. Grs. , 3900 j ’ 180} Mean > I 9°- An adult man, for his daily sustenance, requires about 1220 grs. nitrog- enous matter or 200 of nitrogen, and bread contains 8.1 per cent, of it. 1220 Grs. Adult Man, 4300) , r Adult Woman, 3900) Mean » Hence, — .15 062 grains which -4- 7000 in alb. —2 lbs. 2.43 oz. of bread. These quantities and proportions are also contained in about 16 lbs. of turnips. i3 T of U nitrogen le of nutritive values > P a S e 202, turnips have 263 grains of carbon and HenCe ’^ and ~ = i6 -35 lbs. for the necessary carbon and 15.4 lbs. for the nitrogen. [Relative Value of Foods compared with IOO lbs. of Grood Flay. m Lb9 - Clover, green. . 400 Corn, green ... 275 Wheat straw . . 374 Lbs. Corn 59 Linseed cake . . 69 Wheat bran 105 ANIMAL FOOD. 204 WeigKt of Articles of Food, required to "be consumed in tbe Tinman system to develop a power equal to rais- ing 140 lbs. to a height of IO OOO feet. ( Frankland .) Weight. Substances. [Weight, j Substances. Weight. ] Substances. Lbs. I •553 •555 1 .67 | .693 • 797 ' •97 1.156 1.281 1.287 1 3 11 I "Pin** . . f Lbs. I - 34 I 1-379 i- 5°5 Salt Beef Cod-liver oil.. T) oof* fo t Tsin glass Veal, lean Sugar, lump Porter T) n f fnr Cream 2.062 2. 209 2-345 2.826 3.001 Potatoes Egg boiled Fish TT'n f of Povlr Rrpari Apples X uX 01 r 01 Iv ...... • Salt Pork Milk Ham, lean, boiled. . Egg, white of. Mackerel 3.124 3.461 Carrots Arrowroot. ....... Wheat flour Ale, bottled Cabbage Lbs. 3-65 4*3 4.615 5.068 6.316 7- 8i5 8.021 8- 745 9.685 12.02 Relative Value of Various Foods as Productive of Force when. Oxidized in the Body. Cabbage. Carrots.. White of Egg.. Milk Apples Ale Fisli Potatoes 1 Porter 2.6 Egg, hard boil’d 5-4 1.2 Veal, lean..... 2.8 Cream 5-9 1.2 Salt Beef 3-3 Egg, yolk 7- 9 1.4 Poultry 3-3 Sugar 8 1.5 Lean Beef. 3-4 Isinglass 8.7 1.5 Mackerel 3.8 Rice 8.9 1.8 Ham, lean 4 Pea Meal 9- 1.9 Salt Pork 4-3 Wheat Flour .. 9.1 2.4 Bread, crumb. . 5-i Arrowroot 9-3 Oatmeal 9.3 Cheese 10.4 Fat of Pork. 12.4 Cocoa 16.3 Pemmican . . 16.9 Butter 17.3 Bacon 17.94 Fat of Beef. . 21.6 Cod-liver Oil. 21.7 Nutritions Properties of different Vegetables and Oil- cake, compared with each other in Quantities. Rye.. Bran, wheat Corn . . . Barley . 2.5 ir s 3 3 Clover hay — Cabbage 18 Hay Wheat straw.. , 26 Potatoes Barley “ 26 “ old.. . . 20 Oat “ 27-5 Carrots i7-5 Turnips ■ 30 ; Oil-cake 1 Pease and Beans 1.5 Wheat, flour. . . 2 “ grain.. 2.5 Oats 2.5 Illustration.— 1 lb. of oil-cake is equal to 18 lbs. of cabbage. Volume of Oxygen required to Oxidize 100 parts of following Foods as con- sumed in the Body. Grape Sugar.. 106 | Starch i 2 o | Albumen 150 | Fat 2 93 Hence assuming capacity for oxidation as a measure, albumen has half value of fat asTfeXroducinl element, and a greater value than either starch or sugar. Proportion of Alcohol in IOO Parts of following Liquors. (Brande. ) Small Beer. .. 1 and 1.08 Porter 3-5 and. 5.26 Cider 5 2 and 9.8 Brown Stout. 5.5 and 6.8 Ale 6.87 and 10 Hermitage, red 12.32 Champagne 12.61 Amontillado 12.63 Frontignac 12.89 Barsac 13-86 Lisbon 18.94 Lachryraa 19.7 Teneriffe 19-79 Currant Wine 20.55 Madeira 22.27 < Port 23 Rhenish 7-58 Moselle 8.7 Johannisberger 8.71 Elder Wine 8.79 Claret ordinaire 8.99 Champagne Burg’dy, 14.57 White Port 15 Bordeaux i5- 1 Malmsey 16.4 Sherry I 7* 1 7 Sherry, old 23.86 ' Marsala 25.09 Raisin Wine 25.12 Madeira, Sercial 27.4 Cape Madeira 29.51 Tokay 9-33 AT 0 1 0 rro 17. 2 Gin Si- 6 Rudesheimer 10.72 Marcobrunner n-6 Gooseberry Wine ... 11.84 Hockheimer 12.03 Vin de Grave 12.08 Alba Flora 17-26 Hermitage, white . . . 17.43 Cape Muscat 18.25 Constantia, red 18.92 Brandy 53-39 Rum 53-68 Irish Whiskey 53-9 j Scotch Whiskey 54.32 ANIMAL FOOD. 205 Proportion of Food Appropriated and Expended "by following Animals. Oxen. Sheep. Swine. Proportion appropriated “ in manure 8 • 17.6 3i-9 16.9 “ respired 60. 1 65-5 100 100 100 Specific (Gravity of Alifk and Percentage of Cream, etc. Milk. Specific Gravity. Volume of Cream. Volume of Curd. Specific Gravity when skimmed. Milk, pure* 1030 12 6-3 1032 “ xo per cent, water. 1027 10.5 5-6 1029 “ 20 “ “ “ 1024 8.5 4.9 1026 “ 30 “ “ “ 1021 6 4.2 1023 * For a method of testing the purity of milk, see Pavv on Food (Philadelphia, 1874), page 196. Note. — T he average proportion of cream is 10, or 10 per cent. Proportion Per cent, of Starch, in sundry ‘Vegetables. Arrowroot. . . , . 82 I Wheat flour. . . 66.3 I I Oatmeal . . . . Potatoes. . . .. 18.8 Rice , . 79. 1 1 Corn meal . . . . 64.7 1 Pease ■••• 55-4 1 Turnips .. 5-1 Composition of Cheese of Different Conn tries. — ( Payen .) Fat. Nitrogen. Salt. Water. Fat. Nitrogen. Salt. Water. Neufchatel. . 18.74 2.28 4-25 61.87 Chester 25.41 5-56 4.78 30-39 Parmesan . . 21.68 5-48 7.09 30 - 3 I Gruydres . . . 28.4 5-4 4.29 32-05 Brie 24.83 2-39 5-63 53-99 Marolles 28.73 3-73 5-93 40.07 Holland .... 25.06 4.1 6.21 41.41 Roquefort. . . 32.31 5.07 4-45 26.53 Nntritive Equivalents. Computed from Amount of Ni- trogen in Snbstances -when Dried. Human AI ilk at 1 . Rice Potatoes.. Corn Rye Wheat . . . Oats. 00 00 Bread, White. Milk, Cows’ . . 1.42 2. 37 Cheese Eel 3-31 a. n Lamb Egg, White. . . Lobster. ..... Ppa.se 2. Mussel 5.28 5-7 7-56 7. 79 1.06 Lentils jy 2.76 3- °5 3-05 3-2 Liver, Ox ... . Yeal T. JO Egg, Yolk Oysters Pigeon Beef 1. x y Mutton Pork ! 1.38 Beans Salmon 7.76 Ham Herring, 9.14. 8-33 8-45 8-59 8-73 8.8 8-93 9.1 Thermometric Power and JVCechanical Energy of IO Grains of Various Snbstances in their N atnral Con- dition, when Oxidized in the Animal Body into Car- bonic -A.cid, Water, and Urea. — ( Franktand .) Substance. Water raised i°. Lifted 1 foot high. Substance. Water raised i°. Lifted 1 foot high. Substance. Water raised i°. Lifted 1 foot high. Ale, Bass’s . . Apples Arrowroot. . . Lbs. 1.99 Lbs. i -54 Cheese Lbs. 11. 2 Lbs. 8.65 Mackerel — Lbs. 4.14 Lbs. 3-2 1.48 10. 06 1.29 7.7 7 Cocoa-nibs . . Cod-liver oil. 17 11 7-3 18. 12 Milk Oatmeal 1.64 10. 1 1.25 7.8 Beef, lean . . . 3.66 2.83 Egg, h’d boil. 5-86 4-53 Pea meal .... 9-57 7-49 Bread 5-52 4.26 “ yolk 8.5 6.56 Potatoes .... 2.56 1.99 Butter 18.68 14.42 “ white... 1.48 1. 14 Porter 2.77 2. 19 Cabbage 1.08 .83 Flour, wheat. 9.87 7.62 Rice, ground. 9-52 7-45 Carrots i -33 1.03 Ham, boiled . 4-3 3-32 Sugar, grape. 8. 42 6.51 s 206 ANIMAL FOOD. Digestion. Time required, for Digestion of several Articles of Food. • (Beaumont, M.D.) Apple, sweet and mellow .... sour and mellow sour and hard Barley, boiled Bean, boiled Bean and Green Corn, boiled . Beef, roasted rare roasted dry Steak, broiled boiled boiled, with mustard, etc. Tendon, boiled 44 fried old salted, boiled Beet, boiled Bread, Corn, baked Wheat, baked, fresh . . . Butter, melted Cabbage, crude crude, vinegar crude, vin’r, boiled Carrot, boiled Cartilage, boiled Cheese, old and strong. Chicken, fricasseed. . . Custard, baked Duck, roasted Dumpling, Apple, boiled. Egg- whipped boiled hard 44 soft fried Fish, Cod or Flounder, fried . . Cod, cured, boiled. ..... Salmon, salt’d and boil’d Trout, boiled or fried. . . Fowl, boiled or roasted Goose, roasted Gelatine, boiled h. m. 1 50 2 2 50 2 2 30 3 45 3 3 30 3 2 45 3 30 5 30 4 4 15 3 45 3 i5 3 30 3 30 2 30 2 4 4 30 3 15 4 15 3 30 2 45 2 45 4 4 30 3 2 1 30 3 30 3 3 30 3 30 1 30 4 3 2 30 Heart, Animal, fried Lamb, boiled Liver, Beef’s, boiled Meat and Vegetables, hashed . Milk, boiled or fresh j Mutton, roasted broiled or boiled .... Oyster roasted stewed Parsnip, boiled Pig, sucking, roasted Feet, soured, boiled Pork, fat and lean, roasted . . . recently salted, boiled . . “ u fried . . . 44 “ broiled . 44 44 raw. . . . Potato, boiled baked roasted Rice, boiled Sago, boiled Sausage, Pork, broiled Soup, Barley Beef and Vegetables . . . Chicken - Mutton or Oyster Sponge-cake, baked Suet, Beef, boiled Mutton, boiled Tapioca, boiled Tripe, soured Turkey, roasted | Domestic . boiled Turnip, boiled Veal, roasted fried Brain, boiled Venison Steak, broiled 2 30 2 30 i5 i5 55 i5 30 30 30 5 15 4 30 4 15 3 15 3 3 30 3 20 2 30 1 1 45 3 20 1 30 4 3 3 30 2 30 5 30 4 30 2 18 2 30 2 25 3 30 4 4 50 1 45 1 35 General Notes. The per-centage of loss in the cooking of meats is as follows: Boiling 23; Baking 31 ; Roasting 34. Potatoes possess anti-scorbutic power in a greater degree than any other of the succulent vegetables. The average yearly consumption of wheat and wheat flour in Great Britain is 5.5 bushels per capita of its population. The daily ration of an Esquimaux is so lbs. of flesh and blubber.-(Sir John Rost.) ANIMAL FOOD. 207 An adult healthy man, according to Dr. Edward Smith, requires daily of Phosphoric acid from . . 32 to 79 grains. Potash 27 to 107 grains. ( Chlorine 51 *75 “ Soda 80 “ 171 “ (Or of common salt 85 u 291 “ Lime 2.3 “ 6.3 “ and of Magnesia 2.5 to 3 grains. A common fowl's egg contains 120 grains of Carbon and 17.75 of Nitrogen. An ordinary working-man requires for his daily sustenance Starch 66 Salts 04 4-535 Oxygen i -47 Albuminous matter 305 Fat = 7.23 lbs. avoirdupois. Milk.— If the milk of an animal is taken at three immediately successive periods, that which is first received will not be as rich in milk-fat as the last. In a Devon cow, milked in this manner, the first milk gave but 1.166 per cent, of fat, and the last, or that known as strippings, 5 ’ 5.81 per cent. Relative Rieliiiess of Alillc of Several ^Aiaiinals. Human Millc=. 1. Cow. Milk-fat. 1.66 Casein. 1.38 Sugar. .69 Ass Milk-fat. 5 Casein. •33 Sugar. •94 Mare I-I9 •75 •94 Sheep. . . . 2.1 .72 Goat 1.04 .69 Camel i -4 — .96 The condensation of milk reduces it to about one third of its original volume. A Farm of second-rate quality, properly cultivated, will sustain 100 head of cattle per 100 acres, besides laboring-stock (employed in cultivation of farm), and swine. — (Ewart.) Thus, calves 25 ; do. 1 year 25 ; do. 2 years 25 ; cows 25. Cane Sugar (Saccharose) — Is insoluble in absolute alcohol, and in diluted alcohol it is soluble only in proportion to its weakness. Loaf sugar, as a rule, is chemically pure. Beet Root Sugar — Contains 85 to 96 per cent, of cane sugar, 1.6 to 5.1 of organic matter, and 2 to 4. 3 of water. Honey — Contains 32 per cent, of sugar (levulose), 25.5 of water, 27.9 of dextrine, and 14.6 of other matter, as mannite, wax, pollen, and insoluble matter. Molasses — Contains 47 per cent, of cane sugar, 20. 4 of fruit sugar, 2.6 of salts 2.7 extractive and coloring matter, and 27. 3 of water. Flour . — Tests of flour, see A. W. Blyth, London, 1882, page 152. Bread. — Wheat loses of water after 1 day 7.71 per cent., 3 days 8.86, and 7 days 14.05 per cent. Sago. — 2.5 lbs. per day will support a healthy man. Fig Contains nearly as much gluten as wheat bread (as 6 to 7), and in starch and sugar it is 16 per cent, richer. Gooseberry (dry)— Is as nutritious as wheat bread. Watermelon , Vegetable marrow , and Cucumber — Contain 94, 95, and 97 per cent, of water respectively. * Onion (dry)— Contains 25 to 30 per cent, of gluten. Potato containing but 5. Cabbage, Cauliflower , Broccoli, and Leaves are generally rich in gluten, while the potato is poor. Ratio of Flesh-formers of Tubers. Per Cent. Tubers. Flesh- formers. Starch, etc. Ratio to Heat-giv’rs. Tubers. Flesh- formers. Starch, etc. Ratio to Heat-giv’rs. Beet root Turnip*. Carrot Potato •4 •5 •5 1.2 13-4 4 5 18 1:30 1:8 1 : 10 1 : 16 Parsnip Onion Sweet Potato. Yam 1.2 i -5 i -5 2.2 8.7 4.8 20.2 16.3 1 : 10 3-5 1:13 7-5 208 gravity of bodies.— geavity and weight. GRAVITY OF BODIES. Gravity acts equally on all bodies at equal distances from Earth’s centre ; its force diminishes as distance increases, and increases as dis- tance diminishes. Gravitating forces of bodies are to each other, 1. Directly as their masses. 2. Inversely as squares of their distances. Gravity of a body, or its weight above Earth’s surface, decreases as square of its distance from Earth’s centre in semi-diameters of Earth. Illustration i —If a body weighs 900 lbs. at surface of the Earth, what will it weigh 2000 miles above surface ?-Earth’s semi-diameter is 3963 miles (say 4000). Then 2000 -f 4000 = 6000 = 1.5 semi-diam's , and 900 -^-1.5— — — 4°° L0S ' Inversely , If a body weighs 400 lbs. at 2000 miles above Earth’s surface, what will it weigh at surface? 400 X 1.5 =90° lbs. 2. — A body at Earth’s surface weighs 360 lbs. ; how high must it be elevated to weigh 40 lbs.? 352 = 9 semi-diameters , if gravity acted directly; but as it is inversely as square of the distance, then -fq = 3 semi-diameters = 3 X 4000 = 12000 miles. 3 . _To what height must a body be raised to lose half its weight? As ,/i : y/ 2 :: 4000 : 5656 = as square root of one semi-diameter is to square root of two semi- diameters, so is one semi-diameter to distance required. Hence 5656 — 4000 = 1656 = distance from Earth's surface. Diameters of two Globes being equal , and their densities different , weight of a body on their surfaces will be as their densities. Their densities being equal and their diameters different , weight of them will be as their diameters. Diameters and densities being different , weight will be as their product. Illustration.— I f a body weighs 10 lbs. at surface of Earth, what will it -weigh at surface of Sun, densities being 392 and 100, and diameters 8000 and 883000 miles . 883000 X 100 -4- 8000 X 392 = 28. 157 = quotient of product of diameter of Sun and its density , and product of diameter of Earth and its density. Then 28.157 X 10 = 281.57 lbs. Note.— Gravity of a body is .00346 less at Equator than at Poles. SPECIFIC GRAVITY AND WEIGHT. Specific Gravity or Weight of a body is the proportion it bears to the weight of another body of known density or of equal volume, and which is adopted as a standard. . , , , If a body float on a fluid, the part immersed is to whole body as specmc gravity of body is to specific gravity of fluid. When a body is immersed in a fluid, it loses such a portion of its own weight as is equal to that of the fluid it displaces. 1 An immersed body, ascending or descending in a fluid, h^ a force equa to difference between its own weight and weight of its bulk of the fluid, less resistance of the fluid to its passage. . , « . ..p Water is well adapted for standard of gravity ; and as a cube foot of it at 62° F. weighs 997.68 ounces avoirdupois, its weight is taicen as the unit, or approximately 1000. SPECIFIC GRAVITY AND WEIGHT. 209 French standard temperature for comparison of density of solid bodies and determination of their specific gravities, is that of maximum density of water, at 4 0 C. or 39. i° F., and for gases and vapors under one atmosphere or .76 centimeters of mercury is 32 0 F. or o° C., and specific gravity of a body is expressed by weight in kilogrammes of a cube decimeter of that body. Densities of metals vary greatly. Potassium, Sodium, Barium, and Lithium are lighter than water. Mercury is heaviest liquid and Platinum heaviest metal. Volcanic scoriae is lighter than water. Pomegranate and Lignum-vitae are heaviest of woods. Pearl is heaviest of animal substances, and Flax and Cotton are heaviest of vegetable sub- stances, former weighing nearly twice as much as water. Zircon is heaviest of precious stones, being 4.5 times heavier than water. Garnet is 4 times heavier, Diamond 3.5 times, and Opal, lightest of all, is but twice as heavy as water. To Ascertain Specific (Gravity of a Solid. Body- Heavier tli an. W ater. Rule.— W eigh it both in and out of water, and note difference ; then, as weight lost in water is to whole weight, so is 1000 to specific gravity of body. ^ w x 1000 ~ TTT 7 Dr, — ^ = G, W and w representing weights out and m icater , and G specific gravity. Example. — What is specific gravity of a stone which weighs in air 1=; lbs in water 10 lbs.? ’ 15 — 10 = 5; then 5 : 15 1000 : 3000 Spec. Grav. To Ascertain Specific Gravity of a Body lig-liter tlian Water. Rule.— A nnex to lighter body one that is heavier than water, or fluid used ; 'weigh piece added and compound mass separately, both in and out of water, or fluid ; ascertain how much each loses, by subtracting its weight from its weight in air, and subtract less of these differences from greater. Then, as last remainder is to weight of light body in air, so is 1000 to specific gravity of body. Example.— What is specific gravity of a piece of wood that weighs 20 lbs. in air- annexed to it is a piece of metal that weighs 24 lbs. in air and 21 lbs. in water, and the two pieces in water weigh 8 lbs.? 20 -f- 24 — 8 = 44 — 8 = 36 — loss of compound mass in water ; 24 — 21 = 3 = loss of heavy body in water. 33 : 20 : *. 1000 : 606 = 24 Spec. Grav. To Ascertain Specific (Gravity of a- Blnid. Rule.— T ake a body of known specific gravity, weigh it in and out of the fluid ; then, as weight of body is to loss of weight, so is specific gravity of body to that of fluid. Example. — What is specific gravity of a fluid in which a piece of copper (spec, grav. =9000) weighs 70 lbs. in, and 80 lbs. out of it ? 80 : 80 — 70 = 10 : ; 9000 : 1125 Spec. Grav. To .Ascertain Specific (Gravity of a Solid. Body 'wliich. is soluble in Water. Rule.— W eigh it in a liquid in which it is not soluble, divide its weight out of the liquid by loss of its weight in the liquid, and multiply quotient by specific gravity of liquid ; the product is specific gravity. Example.— W hat is specific gravity of a piece of clay, which weighs 15 lbs. in air and 5 lbs. in a liquid of a specific gravity of 1500, in which it is insoluble ? 15-i-zoX 1500 = 2250 Spec. Grav s* 210 SPECIFIC GRAVITY AND WEIGHT. SOLIDS. Substances. ( Specific ( Gravity. Weight )f a Cube Inch. Metals. 2560 2670 7700 6 712 5763 470 Lb. .0926 .0906 .2785 .2428 “ wrought.... “ Bronze .2084 .017 9823 2000 •3553 .0723 Brass. Sheet, cop. 75, zinc 25. Yellow 66, “ 34. Muntz “ 60, “ 40. 8450 8300 8 200 8 380 .3056 .2997 .2966 .3026 8 100 .2930 8 214 .2972 3000 8750 8 217 8832 8 700 8060 7 39° 8 650 .1085 .3i 6 5 .2972 •3i94 .2929 .291 .2668 3129 “ ’ ordinary mean . “ cop. 84, tin 16 . . “ “ 81, “ 19 . . “ small bells, cop. 35, tin 65.... “ cop. 21, tin 74 . . Cadmium Calcium 1 580 5900 8 098 8 600 •057 Chromium. ............ •2134 Cinnabar .2929 Cobalt • 3 111 Columbium ............ 6000 .217 Copper, cast. ........... 8788 8698 8880 8880 19258 19361 17486 i5 7°9 18680 3179 u plates •3 I 4 6 “ wire and bolts.. “ ordinary mean. .32x2 .3212 .6965 .7003 Gold pure , r it Jjap'impred “ 22 carats fine “ 20 “ u Iridium •6325 .5682 .6756 it hammered 23000 7308 6900 7500 7207 7217 7065 7 218 , 7788 ■ 7 774 3 7704 , 7 698 • 7 54° , 7 808 . 8 140 • 7 744 • 11 352 .8319 .264 .2491 .2707 Iron, Cast, gun metal. . . 44 minimum u maximum “ ordinary mean * * mean F-ug .2607 .2609 •2555 .2611 .2817 .2811 “ cast, hot blast “ cold ‘‘ “ Wrought bars <( “ wire “ “ rolled plates “ “ average — “ “ Eng. rails . . “ “ Lowmoor. . , n “ pure .2787 .2779 .2722 .2819 .2938 .2801 .4106 “ ordinary mean — Lead cast tt yollftd . 11388 .4119 Lithium . 59° 1 .0213 Magnesium • 1 75° 1 .0633 Manganese . 800c 1 .2894 Mercury o . is6a2 : -S66i “ +32 0 . 13 59 s 1 *49 lS Substances. ( Specific , gravity. Weight of a Cube Inch. Metals. Mercury 6o° “ 212° . ••• 13 569 Lb. .4908 13 370 .4836 Molybdenum 8 600 • 3m Nickel 8 800 .3i 8 3 “ cast 8279 .2994 Osmium 10000 .3613 Palladium 11 350 .4105 Platinum, hammered . . . 20337 •7356 “ native 16 000 •57 s 7 “ rolled 22 069 .7982 Potassium, 59 0 865 .0313 Red lead 8940 •3 2 4 Rhodium 10650 .3852 Rubidium 1 520 .055 Ruthenium 8 600 .3111 Selenium 4 5oo .1627 Silver, pure, cast 10474 .3788 “ “ hammered. 10511 .3802 Sodium 970 .0351 Steel, minimum 7 7oo .2785 “ maximum 7900 .2857 “ plates, mean 7 806 .2823 “ soft 7 s 33 .2833 “ temper’d andhard- ened 7 818 .2828 “ wire 7 8 47 .2838 “ blistered 7823 .283 u crucible 7842 .2836 “ cast 7 M .2839 “ Bessemer 7 s 52 .284 “ ordinary mean 7 s 34 .2916 Strontium 2540 .0918 Tellurium 6 no .221 Thalium 1 1 850 .4286 Tin, Cornish, hammered. 7 39° .2673 “ “ pure 7291 .2637 Titanium 5 300 .1917 Tungsten Uranium Wolfram Zinc, cast “ rolled Woods ( Dry ) Alder Apple Ash Bamboo Bay tree Beech Birch Blackwood, India . . Boxwood, Brazil — “ France... “ Holland. . Bullet-wood Butternut 17 000 18330 7 ll 9 6 861 7191 .0149 .6629 • 2575 .2482 .26 800 793 845 690 400 Cube Foot. 50 49.562 52.812 43- 125 25 822 51-375 852 53-25 690 43- I2 5 567 35-437 720 45 898 56.125 1 031 04-437 1 328 s 3 912 57 928 5 s 376 23-5 SPECIFIC GRAVITY AND WEiGHT. 21 1 Substances. Woods {Dry). Campeachy Cedar “ Indian Charcoal, pine “ fresh burned. “ oak “ • soft wood . . . “ triturated... Cherry Chestnut, sweet Citron Cocoa Cork Cypress, Spanish Dog- wood Ebony, American u Indian Elder Elm \ “ rock Erroul, India Filbert Fir, Norway Spruce. “ Dantzic Fustic Greenlieart or Sipiri. Gum, blue “ water Hackmatack Hawthorn Hazel Hemlock Hickory, pig-nut “ shell-bark. . Holly Iron -wood. Jasmine Juniper Khair, India Lancewood, mean. . . Larch Lemon Lignum-vitse Lime Linden Locust Logwood Mahogany “ Honduras. “ Spanish... Maple “ bird’s-eye Mastic Mulberry Oak, African 11 Canadian il Dantzic. Weight Specific 0 f a Cube Gravity. Foot. 9 X 3 56 i i3 x 5 441 380 i573 280 1380 7 X 5 610 726 1040 240 644 756 i33i 1209 695 570 671 800 1014 600 512 582 970 io55 843 1000 59 2 910 860 368 792 690 760 990 770 566 1171 720 544 560 703 650 x 333 804 604 728 9*3 720 1063 560 852 750 576 849 561 897 823 872 759 * U. 57-062 35.062 82.157 27.562 2 3-75 98.312 *7-5 86. 25 44.687 38.125 45-375 6 5 *5 40.25 47- 2 5 83.187 75 562 43-437 35 625 4i 937 5o 63-375 37-5 32 36.375 60.625 65-95 52.687 62.5 37 56.875 53-75 2 3 49*5 43-125 47-5 61.875 48.125 35-375 73-187 45 34 35 43-937 40.625 83-312 50.25 37-75 45-5 57 062 45 66.437 35 53- 25 46.875 36 53.062 35.062 56.062 51-437 54- 5 47-437 Woods {Dry). Oak, English. , * ‘ green “ heart, 60 years “ live, green “ “ seasoned “ white Olive Orange Pear Persimmon Plum Pine, pitch “ red “ white “ yellow “• Norway.. Pomegranate Poon Poplar “ white Quince Rosewood Sassafras Satinwood Spruce Sycamore Tamarack . Weight Specific of a Cube Gravity. Foot. Teak (African oak) Walnut. black. Willow . Yew, Dutch. . . Spanish. ( Well Seasoned.*) Ash Beech Cherry Cypress Hickory, red Mahogany, St. Domingo. Pine, white . yellow Poplar White Oak, upland “ u James River Stones, Earths, etc. Alabaster, white “ yellow Alum Amber Ambergris Asbestos, starry Asphalte Barytes, sulphate . . . . j Beton, N. Y. St.Con’g Co. 858 932 1146 1170 1260 1068 860 680 705 661 710 785 660 590 554 461 740 *354 580 383 529 7°5 728 482 885 500 623 383 657 980 671 500 486 585 788 807 722 624 606 441 838 720 473 54i 587 687 759 2730 2699 i7 x 4 1078 866 3073 2250 4000 4865 2305 Lbs. 53-625 58.25 71.625 73-125 78.75 66.75 53-75 42.5 44.062 4i-3 x 2 44- 375 49.062 41.25 36.875 34-625 28.812 46.25 84.625 36.25 23-937 33.062 44.062 45- 5 30.125 55-312 31-25 38.937 23-937 41.062 61.25 4 X 937 3 x -25 30-375 36.562 49-25 50.437 45-125 39 n 37-875 27.562 52.375 45 29.562 33-8i2 36.687 42-937 42-437 170.625 168.687 107. 125 67-375 192.062 140.625 250 304.062 144. 06 212 SPECIFIC GRAVITY AND WEIGHT. Specific Gravity. Stones, Eartlis, etc. Basalt | Bitumen, red “ brown Borax Brick | “ pressed “ fire “ work in cement. . . “ “ u mortar Carbon Cement, Portland . “ Roman... Chalk Clay •{ with gravel. . Coal, Anthracite. , Borneo Cannel Caking Cherry Chili Derbyshire . . Lancaster Maryland Newcastle . . . Rive de Gier. Scotch Weight of a Cube Foot. “ Splint “ Wales, mean. . . Coke “ Nat’l, Va Concrete, in cement. “ mean Earth * common soil, dry “ loose “ moist sand “ mold, fresh “ rammed “ rough sand “ with gravel ... . “ Potters’ “ light vegetable. . Emery Feldspar Flint, black “ white Fluorine Fuel, Warlich’s “ Lignite Glass, bottle - “ Crown “ flint.. 2740 2864 1160 830 1714 1367 1900 2400 2201 1800 1600 2000 35°° 130° 1560 1520 2784 1930 2480 x35o 1436 1640 1290 1238 1318 1277 1276 1290 1292 1273 1355 1270 1300 1259 1300 1302 1315 1000 746 2200 2000 1216 1500 2050 2050 1600 1920 2020 1900 1400 4000 2600 2582 2594 1320 1150 130° 2732 2487 2933 3200 Substances. Specific Gravity. 171.25 179 72-3 51-7 107. 125 85-437 118.75 50 137.562 12.5 100 125 218.75 81.25 97-25 95 174 . 120.625 i55 84-375 89-75 102.5 80.625 77-375 82.375 79.812 79-75 80.625 80.75 79.562 84.687 79-375 81.25 78.687 81.25 8i-375 82. 187 62.5 46.64 137-5 125 76 93-75 128.125 28.125 100 120 126.25 i8.75 87-5 250 162.5 161.375 162.125 82.5 7 i - 8 75 81.25 170-75 155-437 183.312 196 Weight of a Cube Foot. Stones, Eartlis, etc. Glass, green optical white •• window soluble Gniess, common Granite, Egyptian red. . 14 Patapsco “ Quincy u Scotch “ Susquehanna “ “ gray Graphite Gravel, common Grindstone Gypsum, opaque . . . Hone, white, razor . Hornblende Iodine Lava, Vesuvius Lias Lime, quick “ hydraulic Limestone, white . . . “ green... Magnesia, carbonate Magnetic ore Marble, Adelaide . . . “ African “ Biscayan, black. “ Carrara “ - common... “ Egyptian. . . “ French “ Italian, white. . “ Parian “ Vermont, white. “ Silesian Marl, mean “ tough Masonry, rubble Granite. . . Limestone. . . Sandstone. Brick “ rough work Mica Millstone . Quartz. Mortar . Mud. wet and fluid u u u pressed, Nitre Oyster-shell Paving-stone Peat, Irish, light .... “ “ dense.... “ very “ .... 2642 3450 2892 2642 1250 270 2654 2640 2652 2625 2704 2800 2200 1749 2143 168 2876 3540 4940 1710 2810 1350 804 2745 3*56 3180 2400 5094 2715 2708 2695 27x6 2686 2668 2649 2708 2838 2650 2730 *75° 2340 2050 2640 2640 2160 2240 1600 2800 2484 1260 1384 1750 1630 1782 1920 1900 2092 2416 278 562 675 165.125 215.625 180.75 165.125 78.125 5-875 65-875 65 65-75 64.062 69 i75 37-5 109. 312 133-937 35-5 179-75 221.25 06.875 75-625 46-875 50.25 171.562 197-25 98.75 150 . 3 W -6 69.687 69.25 168.437 169.75 167.875 166.75 65.562 169.25 177-375 i65-57 170.625 109.375 146.25 128.125 165 165 135 140 100 175 155-25 78-75 86.5 109.375 101.875 112 120 118.75 i3°-75 151 17-375 35-125 42.187 * Specific gravity of earth is estimated at from 1520 to 2200. SPECIFIC GRAVITY AND WEIGHT. 213 SUBSTANCES. Specific Gravity. Weight of a Cube Foot. Stones, Eartlis, etc. Peat, black j 1058 1329 Lbs. 66. 125 83.062 110.625 73-5 212.5 87-5 Plaster of Paris j “ “ “ dry Plumbago 1176 3400 1400 2100 Porcelain, China 14^7^ 2765 66 5 172 8l2 66 187 Quartz 8940 558.75 68.062 1 70. 937 123.812 Too TOC Resin Rock, crystal 070 C Rotten-stone IQ8i Salt, common “ rock 2200 137-5 130.625 112.5 io 4-375 87 97-5 88.75 103.66 107.25 106.33 137-5 139.81 198.125 51-875 140.625 162.5 167 181.25 *74 Saltpetre 2090 1800 1670 1392 1560 Sand, coarse “ common “ damp and loose. . . li dried u “ ... “ dry “ mortar, Ft. Rich m’d “ “ Brooklyn.. “ silicious 1659 1716 1701 Sandstone, mean “ Sydney Schorl 2237 Scoria, volcanic 3*7° 830 Sewer pipe, mean Shale 2600 2672 2900 2784 Slate j “ purple Smalt Soapstone 2440 2730 979 c I 5 2 - 5 170.625 170.937 168.312 169 212. 5 Spar, calcareous “ Feld, blue z / DJ 2693 2704 “ “ green ‘‘ Fluor Specular ore 3400 c OC T Stalactite 320. IO7 150-937 122.562 165 164.062 169 ] 156.875 j I2 9-75 157-5 144-75 165.687 172 144 148 186 168 127.062 Stone, Bath, Engl 2415 1961 2640 2625 2704 2510 2076 “ Blue Hill “ Bluestone (basalt) “ Breakneck, N.Y. . “ Bristol, Engl “ Caen, Normandy. “ common “ Craigleith, Scotl. . “ Kentish rag, . “ Kip’s Bay, N.Y. . 11 Norfolk (Parlia- ment House). . . “ Portland, Engl... “ Staten Isl’d, N.Y. “ Sullivan Co., “ Sulphur, native 2316 2651 2759 2304 2368 2976 2688 2033 1952 1815 2720 Terra Cotta Tile ir 3-437 170 1 Trap Substances. Grranite. (Gen' l Gill more, U. S. A.) Duluth, Minn., dark Fall River, Mass., gray. . Garrison’s, N. Y. “ .. Jersey City, N. J., soap.. Keene, N. H., bluish gray Maine Millstone Ft., Conn New London, “ Quincy, Mass., light Richmond, Va “ “ gray Staten Island, N. Y Westchester Co., N. Y.. Westerly, R. I., gray Limestone. (Gen' l Gillmore , U. S. A . ) Bardstown, Ky. , dark . . Caen, France Canajoharie, N. Y Cooper Co. , Mo. , d’k drab Erie Co., N. Y , blue.... Garrison’s, N. Y Glens’ Falls, “ Joliet, 111., white Kingston, N. Y. Lake Champlain, N. Y. . Lime Island, Mich., drab Marblehead, Ohio, white Marquette, Mich., drab . Sturgeon Bay, Wis., blu- i ish drab ZVTar'ble. 1 (Gen' l Gillmore , U. S. A . ) Dorset, Vt East Chester, N. Y Italian, common Mill Creek, 111., drab.... North Bay, Wis., “ Sandstone. (Gen' l Gillmore , U. S. A.) Albion, N.Y., brown Belleville, N. J., gray. . . Berea. Ohio, drab Cleveland, u olive green Edinb’h,Sc’tl.,CraigIeith Fond du Lac, Wis., purple Fontenac, Minn., l’g’t buff Haverstraw, N. Y., red. . Kasota, Minn., pink Little Falls, N. Y., brown Marquette, Mich., purple Masillon, 0., yellow drab Medina, N. Y., pink Middletown, Ct., brown. Seneca, Ohio, red “ Vermillion, Ohio, drab. . Warrensburgh, Mo Specific Gravity. Weight of a Cuba Foot. Lbs. 2780 173-7 2635 164.7 2580 161.2 3 ° 3 ° 189.3 2656 166 2635 164.7 2706 169. 1 2660 166.25 2695 168.5 2727 170.5 2630 164.4 2861 ; 178.8 2655 165.9 2670 166.9 2670 166.9 19°° xi8.8 2685 167.8 2320 i 4 i -3 2640 165 2635 164.7 2700 168.7 2540 158.7 2690 168. 1 2750 171.9 2500 156.3 2400 150 2340 146.25 O 00 N I 73-7 2635 164.7 2875 179.7 269O 168. 1 2570 171.9 2800 W 5 2420 151-25 2259 141.2 2110 131-9 224O 140 2260 141.25 2220 138.7 2325 i 45 - 3 i 214 SPECIFIC GRAVITY AND WEIGHT. Spec. Grav. Agate 2590 Amethyst 39 20 Carnelian 2613 Chrysolite 2782 Diamond, Oriental. . . 3521 “ Brazilian.. 3444 “ pure 3520 Emerald 395° Precious Stones. Spec. Grav. Emerald, aqua ma- rine Garnet “ black... 3750 Jasper Jet Lapis lazuli. . . . Malachite Spec. Gray. Onyx 2700 Opal 2090 Pearl, Oriental 2650 Ruby 3980 Sapphire 3994 Topa^ 3 s 00 Tourmaline 3070 Turquoise 2750 Substances. ( Specific L Gravity. Weight if a Cube Foot. IVI is cell an eons. 1090 ,001292 . 965 1900 942 988 Lbs. 68.125 Atmospheric Air ,080728 60.312 Beeswax Bone 118.75 Butter 58-875 Camphor 61.75 Caoutchouc 930 95° 1650 58.125 Cotton 59-375 Dynamite 103. 125 ]£gg 1090 9 2 3 936 ; 923 1790 1 1222 — Fat of Beef | “ Hogs “ Mutton jrjax 57-687 58.5 57.687 111.875 Gamboge Gtycerin* 1 1261 78.752 Grain Barley 59° 750 500 36.875 “ ’ Wheat 46.875 “ Oats 31-25 Gum Arabic., _ . 1452 900 1000 i55o 1800 980 90.75 Gunpowder loosft ...... 56.25 4 ‘ shaken . . . . “ solid | Gutta-pfM'uhn, 62.5 96.875 112.5 61.25 Hay old compact 128.8 8.05 Horn 1689 105.562 Substances. Human body Ice, at 32 0 Indigo Isinglass Ivory Lard Leather Mastic Myrrh Nitro-Glycerine. Opium Potash Resin Snow Soap, Castile Spermaceti Starch Sugar “ .66 Tallow Wax 1070 922 1009 mi 1825 947 960 1074 1360 1600 1336 2100 1089 •0833 1071 943 950 1606 972 1326 941 964 970 57-5 63.062 69.437 114.062 59* l8 7 60 67.125 85 100 83-5 131.25 68.062 5.2 56.937 58.937 59-375 100.375 60.25 82.875 58.812 60.25 60.625 Liquids. Acid, Acetic “ Benzoic “ Citric “ Concentrated.. “ Fluoric “ Muriatic “ Nitric “ Nitrous “ Phosphoric “ “ solid.. “ Sulphuric Alcohol, pure, 6o° “ 95 per cent. . . . “ 80 “ “ 50 “ “ 40 “ ...■ “ 25 u “ 10 .... “ 5 “ “ proof spirit,* 5c per cent., 6o a “ proof spirit, 50 per cent., 8o° Ammonia, 27.9 per cent. Aquafortis, double “ single Beer Benzine Bitumen, liquid Blood (human) Brandy, .83 or .5 of spirit Bromine Cider Ether, Acetic “ Muriatic..... “ Nitric “ Sulphuric.... Honey Milk Oil, Anise-seed “ Codfish “ Whale “ Linseed., “ Naphtha “ Olive “ Palm “ Petroleum Rape Sunflower Turpentine . . . 1062 667 1034 1521 1500 1200 1217 1550 1558 2800 1849 794 816 863 934 95i 970 986 992 } 934 ' 875 1200 1034 850 io54 924 2966 1018 866 845 mo 7i5 1450 1032 986 9 2 3 923 940 850 9 a 5 969 880 9*4 926 870 66.375 41.687 64.625 95.062 93-75 75 „ 76.062 96.875 97-375 175 a 115.562 49.622 5i 53-937 58- 375 59- 437 60.625 61.625 62 58-375 54.687 55-687 81.25 75 , 64.625 53- 125 53 „ 65-875 57-75 85-375 63.625 54- 12 5 52.812 69-375 44.687 90.625 64-5 61.625 57.687 57- 687 58- 75 53- I2 5 57- l8 7 60. 562 55 57-125 57-875 54- 375 * Specific gravity y of proof spirit according to Ure’s Table for Sykes's Hydrometer, < SPECIFIC GRAVITY AND WEIGHT. 215 Substances. Liquids. Spirit, rectified Steam, at 212 0 Tar Vinegar Water, at 32 0 “ “ 39- l0 “ “ 6 2 °t “ “ 212° “ distilled, at 39 0 * .038 18. Specific Gravity. Weight of a Cube Foot. Substances. Specific Gravity. Weight of a Cube Foot. Lbs. Liquids. Lbs. 824 51-5 Water, Dead Sea 1240 77-5 .00061 .038* “ Mediterranean... 1029 64.312 1015 63-437 ‘ ‘ sea 1029 64.312 1080 67-5 “ Black Sea 1016 63-5 998.7 62.418 “ rain 1000 62.5 998.8 62.425 Wine, Burgundy 992 62 997-7 62.355 “ Champagne 997 64-375 956.4 59-64 “ Madeira 1038 62.312 998 62.379 “ Port 997 62.312 t x cube inch at standard temperature = 252.5954 grains. Compression of following fluids under a pressure of 15 lbs. per square inch: Alcohol.. .0000216 | Mercury.. .00000265 I Water.. .00004663 | Ether.. .00006158 Elastic Eliaids. 1 Cube Foot of Atmospheric Air at 32 0 weighs .080728 lbs. Avoirdupois = 565.096 grains , and at 62° 532. 679 grains. Its assumed Gravity of 1 is Unit for Elastic Fluids. Sp Spec. Gray. Acetic Ether 3.04 Ammonia 589 Atmos, air, at 32 0 . . 1 Azote 976 Carbonic acid 1.53 “ oxide 972 Carburet’d Hydrog. .559 Chlorine 2.421 Chloro-carbonic ... 3. 389 Chloroform 5.3 Cyanogen 1.815 Gas, coal ( ' 43 s ( - 75 2 Hydrochloric acid . 1. 278 Hydrocyanic “ . .942 Hydrogen 0692 Muriatic acid 1-247 t Weight of a cube foot 267.26 Nitric acid. “ oxide Nitrogen Nitrous acid Nitrous oxide .... Olefiant gas Oxygen ...... Phosphurett’d Hy- drogen Sulphuretted Hy- drogen 1 Sulphurous acid. . 2 Steam, J at 212 0 . . . Smoke. Bitum. Coal. . . . Coke Wood grains, and compared with 5. Grav. .217 1.094 •974 2.638 1-527 .9672 1. 106 1.77 •47295 .102 .105 •°9 water at Spec. Grav. Vapor. Alcohol 1.613 Bisulphuret of Carbon 2.64 Bromine 5.4 Chloric Ether 3.44 Chloroform 4.2 Ether 2.586 Hydrochlor. Ether 2.255 Iodine 8.716 Nitric acid 3.75 Spirits of Turpen- tine 5-013 Sulphuric acid .. . 2.7 “ Ether.. 2.586 Sulphur 2.214 Water 623 62° specific gravity = .000 612 3. Weight of a Cube Foot of Gases at 32 0 F., and under Pressure of one Atmos- phere, or 2116.4 lbs. per Square Foot. Lbs. Air, at 32 0 080728 u “ 62° 076 097 Alcohol 1302 Carbonic acid 12344 Carburet. Hydrog. . 044 62 Lbs. Chlorine 197 Chloroform 428 Coal gas 03536 Ether, Sulphuric. . .2093 Gaseous steam 05022 Sulphurous acid 1814 lbs. , Lbs. Hydrogen 005 594 Nitrogen 078 596 Olefiant gas 0795 Oxygen 089256 Steam 05022 To Compute Weight of a Body or Substance when Specific Gravity is given. Rule.— M ultiply specific gravity by unit or standard of body or sub- stance, and product is the weight. Or, Divide specific gravity of body or substance by 16, and quotient will give weight of a cube foot of it in lbs. Example.— S pecific gravity is 2250; what is weight of a cube foot of it? 2250 X 62.5 = 140.625 lbs. WEIGHTS OF VARIOUS SUBSTANCES. 2l6 Weights and. Volumes of various Substances v “ Ordinary Use. Substances. IMetals. „ ( copper 67 \ Brass . . j ziuc 33 J “ gun metal “ sheets..., “ wire Copper, cast “ plates Iron, cast “ gun metal “ heavy forging. “ plates “ wrought bars. . . Lead, cast. . . “ rolled. Mercury, 6o° Steel, plates. “ soft... Tin Zinc, cast . . . “ rolled.. Cube Foot. Cube Inch. Lbs. 488.75 543-75 5i3- 6 524.16 547- 2 5 543-625 450-437 466.5 479-5 481.5 486.75 7°9-5 7 IX -75 848.7487 4 8 7-75 489. 562 455-687 428.812 449-437 "Woods. Ash Bay Blue Gum Cork Cedar Chestnut Hickory, pig nut. “ shell-bark. Lignum-vitee Logwood Mahoga^Hondur’s Oak, Canadian “ English “ live, seasoned. . “ white, dry “ “ upland.. Pine, pitch “ red “ white “ well-seasoned.. Pine, yellow 52.812 51-375 64-3 i5 35.062 38.125 49-5 43- x2 5 83.312 57.062 35 66.437 54-5 58-25 66.75 53-75 42-937 4 x - 2 5 36.875 34.625 29. 562 33- 812 .2829 •3147 .297 •3033 •3179 .3167 .2607 .27 •2775 .2787 .2816 .4106 .4119 .491174 .2823 •2833 .2637 .2482 .2601 Cube Feet in a Ton. 42.414 43.601 34-S37 149-333 63.886 58-754 45.252 51.942 26.886 39-255 64 33-7 x 4 41.101 38.455 33-558 41.674 52.169 54-303 60.745 64.693 75-773 66.248 _ . r- . I Cube Feet Cube Foot. inaTon . Woods. Spruce Walnut, black, dry... Willow dry IVIiscellan eous, Air Basalt, mean Brick, fire mean Coal, anthracite — j bitumin., mean Cannel Cumberland. . . Welsh, mean.. Coke Cotton, bale, mean . . ‘ “ pressed Earth, clay common soil. “ gravel dry, sand. loose moist, sand.. 1 mold.- mud £ with gravel. Granite, Quincy “ Susquehanna Gypsum Hay, bale “ hard pressed. . Ice, at 32° India rubber “ u vulcanized Limestone Marble, mean Mortar, dry, mean Plaster of Paris. . . Water, rain “ salt “ at 62° Lbs. 3 x - 2 5 3 x - 2 5 36.562 30-375 075291 75 a 37.562 102 •75 102.5 80 94-875 84.687 81.25 62.5 14-5 20 25 a 120.025 137. i2 5 109. 312 120 93-75 128.125 128. 125 101.875 126.25 71.68 71.68 61.265 73-744 16.284 21.961 24.958 21.854 28 23.609 26.451 27.569 35- 8 4 154.48 114 . 89.6 18.569 16.335 20.49 18.667 23.893 17.482 17.482 21.987 17-742 165-75 i3-5 x 4 169 13-254 135-5 i 6 . 53 x 12 186.66 * 25 89.6 57-5 38-95 56-437 39-69 — ; I97-25 1 I -355 r 167.875 x 3-343 97.98 22.862 73-5 3°-47 6 62.5 35-84 64.312 34-83 62.355 35-955 To Compute Proportions of Two Ingredients in a Com-: po“»d, or to ^Discover Adulteration m Metals. • Rui.e. — T ake differences of each specific gravity of ingri gravity of body to proportions of the ingredients. Examfle.-A compound of gold (spec. grav. = .8.888) and silver (spec. 0>™.= xo 535) has a specific gravity of 14 ; what is proportion of each metal ? 18.888 — 14=4. 888X 10.535=51.495* 14— i »S3S=3-465Xi8.M8_6|^^ 6s-447+5i-49S:6s-447:--*4:7-835fl»«. 65-447+SM95- 5i-495..«4- 0 WEIGHTS OF VARIOUS SUBSTANCES IN BULK. 21/ Weiglits of Vario-as Safbstances per Cube Foot in 13 vxlL; Lbs. Lead, in pigs 567 Iron, “ 360 Marble, in blocks) Limestone, “ ) ' * 17 Trap 170 Granite, in blocks 164 Sandstone 141 Lbs. Potters’ clay Loam Gravel Sand Bricks, common. . • • 93 Ice, at 32 0 . . . . -. «• 57-5 Oak, seasoned .. 52 Lbs. Coal, caking 50 Wheat 48 Barley 38 Fruit and vegetables . . 22 Cotton seeds 12 Cotton 10 Hay, old 8 Ash, dry, 100 feet BM 175 ton. “ white, “ “ 141 “ Cement, struck bushel and packed* 100 lbs. Cement, Portland, bushel. no lbs. Cherry, dry, 100 BM 156 ton. Chestnut, dry, 100 BM .. . .153 “ Coal, anthracite, 1 cub. yd. broken and loose ... 1.75 yds. “ “ “ 1 ton.. 41.5 cub. feet. Coke, ton = 80 to 97 cub. feet. Earth, common soil 137-125 lbs. Earth, loose 93-75 lbs. Elm, dry, 100 feet BM 13 ton. Gypsum, ground, str. bush. 70 lbs. “ “ well shaken 80 “ Hemlock, dry, 100 feet BM. .093 ton. Hickory, “ ‘‘ “ . .197 “ Masonry, Granite, dressed. . 165 lbs. “ “ rough. . . 126 “ “ Limestone, dres’d 165 “ “ Sandstone 135 “ “ Brick, pressed .. . 140 “ “ “ com’n, rough. 100 u * One packed bushel = 1.43 loose. Comparative W eiglit of Green and. Seasoned Timber. Timber. Weight of a Cube Foot. Green. 1 Seasoned. Timber. Weight of a Green. , Cube Foot. Seasoned. American Pine Ash Beech Lbs. 44-75 58.18 60 Lbs. 30-7 50 53-37 Cedar English Oak Riga Fir Lbs. 32 71.6 4^-75 Lbs. 28.25 43-5 35-5 ^Application of tlie Ta"bles. When Weight of a Solid or Liquid Substance is required. Rule. — Ascer- tain volume of substance in cube feet ; multiply it by unit in second column of tables (its specific gravity), and divide product by 16 ; quotient will give weight in lbs. When Volume is given or ascertained in Inches. Rule. — Multiply it by unit in third column of tables (weight of a cube inch), and product will give weight in lbs. Example.— What is weight of a cube of Italian marble, sides being 3 feet? 33 x 2708 = 73 1 16 02., which - 4 - 16 = 4569.75 lbs. Or of a sphere of cast iron 2 inches in diameter? 23 x .5236 X .2607 weight of a cube inch= 1.092 lbs. When Weight of an Elastic Fluid is required. Rule. — Multiply specific gravity of fluid by 532.679 (weight of a cube foot of air at 62° in grains), divide product by 7000 (grains in a lb. Avoirdupois), and quotient will give weight of a cube foot in lbs. Example. — What is weight of a cube foot of hydrogen ? Specific gravity of hydrogen .0692. 532.679 X .0692 - 4 - 7000 := .005 265 9 lbs. To Compute W eiglit of Cast iVCetal by W eiglit of Pattern. When Pattern is of White Pine. Rule. — Multiply weight of pattern in lbs. by following multipliers, and product will give weight of casting : Iron, 14 ; Brass, 15 ; Lead, 22 ; Tin, 14 ; Zinc, 13.5. When there are Circular Cores or Prints. Multiply square of diameter of core or print by its length in inches, the product by .0175, and result is weight of pattern of core or print to be deducted from weight of pattern. T 218 balloons, shrinkage of castings, etc. To Compute Weights of Ingredients, tLat of Compound "being given. Rule. — As specific gravity of compound is to weight of compound, so are each of the proportions to weight of its material. Example.— Weight, as above, being 28 lbs., what are weights (Of the ingredients? TA " 28 * * I 7 ' 835 ! I5 - 67 W 1 *' 14 . 2« .. j6 5 . I2 33 Sl i ver , Note. —Specific gravity of alloys does not usually follow ratio of their compo- nents, it being sometimes greater and sometimes less than their mean. To Compute Capacity of a Balloon. Rule. — From specific gravity of air in grains per cube foot, subtract that of the gas with which it is inflated ; multiply remainder by volume of bal- loon in cube feet ; divide product by 7000, and from quotient subtract weight of balloon and its attachments Example.— Diameter of a balloon is 26.6 feet, its weight is 100 lbs., and specific gravity of the gas with w T hich it is inflated is .07 (air being assumed at 1); what is its capacity, specific gravity of air assumed at 527.04 grains. - 100 = 590.04 lbs. 527.04 — (527-04 X .07) 3 6 - s 9 X z6 - 63 X -5 2 3 6 . 7000 To Compute Diameter of a Balloon. Weight to be raised being given.— By inversion of preceding rule. 3 /VV X 7000 ~t~ 0 ~~~ * ' s an( j s' representing weight of air and gas .^5236 in grains per cube foot, W r weight to be raised in lbs ., and d diameter of bal- loon in feet. Illustration.— Given elements in preceding case. Then '590.04 -b 100 X 7000^- 527- °4 ~ 36-89 — 3/2 .5236 V -5236 Proof of Spirit moms Liquors. /Q854.69 _ ; 26 .6 feet. A cube inch of Proof Spirits weighs 234 grains ; then, if an immersed cube inch of any heavy body weighs 234 grains less in spirits than air, it shows that the spirit in "which it was weighed is Proof. If it lose less of its weight, the spirit is above proof ; and if it lose more, it is below proof. Illustration.— A cube inch of glass weighing 700 grains weighs 500 grains when weighed in a certain spirit; what is the proof of it? 700 — 500 == 200 == grains = weight lost in spiidt. : 234 ; : 1 : 1. 17 = ratio of proof of spirits compared to proof spirits , or Then 200 : 1 = .17 above proof Note.— For Hydrometers and Rules for ascertaining Proof of Spirits, see page 67; and for a very full treatise on Specific Gravities and on Floatation, see Jamie- son’s Mechanics of Fluids. Lond., 1837. SLriifkage of Castings. i It is customary, in making of patterns for castings, to allow for shrinkage . per lineal foot of pattern as follows : Iron, small cylinders in. per ft. u Pipes =K. “ Girders, beams, etc. = % in 15 ins. “ Large cylinders, the contraction > —Xh complete scale. w /m f ’’ B re P reseilti »g ten feet; A to E, E to G, etc., will measure one 1001. a to a, c to 1, 1 to 2, etc., will measure x-ioth of a foot. The several lines 4easUre’„nnn;T'/‘ ? ea ^ r ? . U P°“ linos- k, l, etc., wooth of a foot; and op will measure upon 7 c, 7, etc., divisions of i-ioth of a foot. Lines. To Draw a Perpendicular to a Light Line, 3. as or > 3, c A, Dig. 4 , or from a Point external to it, as A, Dig. £>, and. from any two Points, as c d, Dig. 0. With any radius as r A, r B, cut line at A and B ; then with a longer radius, as A 0, B 0 , describe arcs cutting each ~l other at 0 , and connect 0 r. ( Fig. 3. ) B Or, from A, set off A B equal to 3 B parts by scale; from A B, with radii c of 4 and 5 parts, describe arcs cut- ting at c, and connect c A. (Fig. 4. ) Note. — This method is useful where straight edges are inappli- cable. Any multiples of numbers 3, 4, 5 may be taken with same ef- foct, as 6, 8, 10, or g, 12, 15. ^ From A, with a sufficient radius, c cut linq at o c, and from them de- scribe arcs cuttiug at r, and connect Ar. (Fig. 5.) From any two points, as c d, at a proper distance anart dcsrriho tn-ro » t. 222 GEOMETRY. To Bisect a Right Line or an Arc of a Circle, and. to Draw a Perpendicu- lar to a Circular or Right Line, or a Radial Arc.-F ig* *7 . From A B as centres describe arcs cutting each other at c and d, connect c d, and line and arc are bisected at e and o. Line c d is also perpendicular to a right line as A B, and radial to a circular arc as A o B. To Draw a Line Bar all el to a Given Right Line, as c d, Rig. S. From A B describe arcs Ac, A d, and draw a line par- allel thereto, touching arcs c and d. Angles. To Describe Angles of ^30° 60°, Rig 9, and 45°, From A, with any radius, A o, de- scribe o r, and from o with a like ra- dius cut it at r, let fall perpendicular rs; then o Ar = 6o°, and Ars = 3o°. (Fig. 9 ) Set off any distance, as A B, erect perpendicular Ao = AB, and connect o B. (Fig. 10.) To Bisect Inclination of Two Lines, when Point of Intersection is Inac- cessible.— Rig- D« Upon given lines, A B, C D, at any points draw perpen- diculars e o, s r, of equal lengths, and from ojnd sdraw parallels to their respective lines, cutting at n, bisect angle o n s, connect n m, and line will bisect lines as re- quired. Rectilineal Figures. To Describe an Octagon upon a Line, as A B.-Ri . is. JClUgOXi -V — 3 — - - From points A B erect indefinite perpendiculars A/, Be; produce A B to m and n, and bisect angles mAo and nap with A u and B r. Make A u and B r equal to A B, and draw u z, r v parallel to A/, and equal to A B. From z and v, as centres, with a radius equal to AB, de- . scribe arcs cutting A/, B e, in / and 6. Connect zffe, and e v. To Inscribe any Regular Polygon in a Circle, or to Divide Circumference into a given Number of Eqtial Parts.— b i„- 13. If Circle is to contain a Heptagon. - Draw angle A . B at centre o for 360° -7 = 5 .° 4=' 5i"+.. or 5'f-, the f “* ofl upon circumference distance A B or remaining angles Ao B. GEOMETRY. 223 To Inscribe a Hexagon in. a Circie.— Fig-. 14 . Draw a diam- eter, A oB. From A and B as cen- tres, with A 0 and B o, cut circle at cm and en y and connect. m>' --'n To Describe a Hexagon about a Circle.— Fig. IS. Draw a diam- eter as a 0 b ; and with ao cut circle ate; join ac,and bisect it with ra- dius o r, through r draw e r paral- lel to c a, cutting diameter at m ; then with radius 0 m describe circle, within which describe a hexagon as above. C r — To Inscribe a Pentagon in a Circle.— Fig. 10. Draw diameters A c and m n, at right angles to each other; bisect 0 n in r, and with r A describe As; from A with A 5 describe 5 B. Connect AB, and distance is equal to one side of a pentagon. To Describe a Pentagon upon a Dine, as A B.— Fig. Draw B m per- pendicular to A B, and equal to one half of it; extend A m until m n is equal to B m. From A and B, with radius Bn, de- jp-w scribe arcs cutting " 7 each other in o; then from 0, with radius o B, describe circle A C B, and line A B is equal to one side of a pentagon upon circle described. To Describe a Regular Polygon of any required Number of Sides.— Fig. 18 . From point o, with distance 0 B, describe semicircle B b A, which divide into as many equal parts, Aa,ab,bc, etc., as the polygon is to have sides. Thus, let a Hexagon be required: From 0 to second point b of six divisions draw 0 b , and through other points, c, d, and e, draw oC,oD, etc. L VS Apply distance 0 B, from B to E, from E to D, from D to A o "" -'B C, etc. Join these points, as b C, C D, etc. To Construct a Hexagon To Construct a Square or a Rectangle on a given Dine.— Fig. 19 . 19 . m;< Xn A B n, and join 0 r. On A B as cen- tres, with A B as radius, describe arcs cutting at ■'IF c; on c describe arcs cutting at o r; and on o r describe others, cutting at ran; draw A m and a given Dine.— Fig From ends of line, A B, describe arcs cutting each other at o, and from o as a centre, with radius o A, describe a cir- cle, and with same radius set off A c, cd, Bffe , and con- nect them. Inscribe an Octagon in a Circle.— Fig. 21 Draw diameters, A C, B D, at right angles, bisect arcs, A B, B C, etc. , at s, r, o, e, and join Ao, 0 B, etc. (Fig. 21.) To Describe an Octagon, about a Circle.— Fig. 22 . Describe a square about circle A B, draw diagonals c/, e d , draw 0 i, etc., perpendicular to diagonals and touch- insr circle. fFig. 22.) 224 GEOMETRY. To Inscribe a Square in a Circle.— Fig. S 3 . Draw line A B through centre of circle ; c 9 take any radius, as A e, and describe the arcs Aee, Bee; connect ee, continuing line to C and D ; join AC, AD, etc. (Fig. 23.) To Describe a Square about A - a Circle.— Fig. 24 . Draw line A B through centre of circle. Take any radius, as A e; describe arcs Aee, Bee; connect ee, continuing line to C D. Describe B r and D r ; draw and extend B r and D r, and sides A and C parallel to them. (Fig. 24.) (lig. 28.) Tq describe a Circular Segment tliat A\ * ^ /E will Lotli fill tlie angle between two diverging lines and. touch tnem. Fig. 29 . Bisect inclined lines, A B, D E. by line e f and perpendicular thereto, B D, to define nda ry ° f seg- ment to be described. Bisect angles at B and D by lines cutting at 0, and from 0, with radius 0 ^describe arc •gVl- _>q) men. To Draw a Series of Circles between Two Incnned Lines, touching them and each othei . * qa Bisect given lines A B, C D, by line oc. * - — B From a point r in this line erect f $ perpen- dicular to A B. and on r describe circle sm, cutting centre line at n; from u erect u n perpendicular to centre line, cutting A B at n and from n describe an arc n u v. cutting A B at i\ erect x v parallel to r s , making x -D centre of next circle to be described, with roHinci nr. 11 HBfl SO Oil. Note.— Largest circle may be described first. GEOMETRY. 225 To Describe a Circle that shall pass through any three given Points, as A B C.— Digs. 31 and 32 . Upon points A and B, with any opening of a dividers, describe arcs cutting each other at ee. On points B C describe two more cutting each other in points c c. Draw lines ee and cc, and intersection of these lines, o , is centre of circle ABC. (Fig. 31.) _ £ C A tre not attainable. — From A B as centres, describe arcs A <7 B ft- B C " 6 and B c into an y num ker of equal parts, also eg and B h into a like number. Draw A 1, 2, 3, etc., and B 1, 2, etc and intersec tion of these lines as at o are points in the circle required. (Fig. 32./ Or, let A B C be given points, connect A B, AC, C B, and draw e c parallel to A B. Divide C A into a number of equal parts' as at 1, 2, and 3, and from C describe arcs through these points to meet right lines from C to points 1, 2, and 3, or A e, and before directed. (Fig. 33.) these are points in a circle, to be drawn as ■ Dl : aw a Tangent to a To Draw Tangents to a Circle from a given Point Circle from a Point with- in Circumference. - Fig. ont it.— Dig. 35. 34 . e 34 , £ ^ > 35 . , , . . . . .. , , From A draw A o, and bisect it at s: through point A draw radial line A o, describe arc through o, cutting circle at and erect perpendicular ef (Fig. 34 .) m n\ join A m or A n. To Draw from or to Circumference of a Circle, Lines leading to an Inaccessible Centre.— Fig. 36. •>/» /r v, ^ Divide whole or any given portion of % { T Jfr circumference into desired number of ! I i parts; then, with any radius less than distance of two divisions, describe arcs /' N cutting each other, as A r, b r, c r, d r, etc.; draw lines b r, c r, etc., and they will lead to centre. To draw end lines , as A r, F r. From b describe arc o, and with radius b i from A or F as centres, cut arcs A r, etc., and lines A r, F r, will lead to centre. ’ To Describe an Arc, or Segment of a Circle, of a large a7 " —■ Radius.— Fig. 3 * 7 . Draw chord AcB; also line h D i parallel with chord, and at a distance equal to height of segment; bisect chord in c, and erect perpendicular fnm anrmimh 0 A f D ’ B ?’ T Ct als ° P e TendicSai°A n, B n ^di vhhT A B and h i ? a .^ r \ ut ^ f ber e( l ual parts; draw lines i i, 2 2, etc., and divide lines A n, B n n°^ f r^ er ° f , equ A P arts in A B ; draw lines to D from each division in lines A w, B n, and at points of intersection with former lines describe arc or segment. 226 GEOMETRY. Ellipse. To Describe an Ellipse to any Length and Breadth given.— Eig. 38. 38 . E Let longest diameter be C D, and shortest EF. Take distance C o or o D, and with it, from points E and F, describe arcs h and / upon diameter C D. nl — TT] /.Jn Insert pins at h and at / and loop a string around '' ' ~ 1 * 1 them of such a length that when a pencil is introduced within it it will just reach to E or F. Bear upon string, sweep it around centre o, and it will describe ellipse. Note. — I t is a property of Ellipse that sum of two lines drawn from foci to meet in any point in curve is equal to transverse diameter. Bisect transverse axis A B at o, and on centre o erect perpendicular C D, making o D and o C each equal to half conjugate axis. From C or D, with radius A o, cut transverse axis at s s for foci. Divide A o into any number of equal parts, as i, 2, 3, etc. With radii A 1, B 1, on s and s as centres, describe arcs, and repeat this operation for all other divis- ions 1, 2, 3, etc., and these points of intersection will give line of curve. and. Two Diameters of an Ellipse. —Eig. 40. Let A B e wbe diameters of an Ellipse. Draw at pleasure two lines, q q, 0 ?n, parallel to each other, and equidistant from A and B; bisect them in points h «, and draw line ur ; bisect it ' in s, and upon s, as a centre, describe a circle at pleasure, as fl v, cutting figure in points/ v. Draw right line fv; bisect it in », and through points i s draw greatest diameter A B, and through centre, s, draw least diameter c w, parallel to fv. To Describe an Ellipse approximately DyCireular Arcs. Set off differences of axes from centre 0 to a and c or 0 A and 0 C ; draw a c and bisect it, and set ofT its half to r : draw r s parallel to a c, set off 0 n equal to 0 r, connect n s , and draw parallels r m, • nm: from m, with radii m s and s m, describe arcs through C and D, and from n and r describe arcs through A and B. Note —This method is not satisfactory when con- jugate axis is less than two thirds of transverse axis. "Witli Arcs of Three Radii-— Eig. / ry\ \ T n\ 6 1 a /r J \S/ 42 . 1 42. On transverse axis A B draw rectangle A B c d, on height 0 e\ to diagonal A e draw perpendicular dh 0 • set off 0 r equal to 0 e, describe a semicircle on A V, and produce O e to l ; set off 0 m equal to el and on 0 describe an arc with radius Owi; on A ’with radius 0 1 , cut this arc at a. Thus the five centres, 0, a, a', fc, h\ are found, from which arcs are described to form ellipse. Note —This process answers for nearly all pro- portions of ellipses. It is used in striking vaults, stone bridges, etc. GEOMETRY. 227 To Construct an Ellipse from Two Circles.— Eig. 43 . Describe two semicircles, as A B, C D, diameters of which are respectively lengths of major and minor axes. The intersection of the horizontal and vertical lines drawn from any radial line will give a point in the curve G D. To Construct an Ellipse, wTieii Two Diameters are Given.-Pig. 44 . Make c o and A v equal to each other, but less than half breadth. Draw vo, and from its centre i draw and extend perpendicular at i to d, draw d v m, make B?« = Av, draw du r, from u and v describe B r and A m, from d describe m c r, extend c z to s, and it will be centre for other half of figure. To Construct an Ellipse by Ordinates.— Fig. 45 . Divide semi-transverse axis, as A 6, into 8 or 10 divisions, as may be convenient, and erect ordi- nates, the lengths of which are equal to semi-con- jugate, multiplied by the units for each division as follows: 1 — .484 12 2 — .661 44 3 — .78063 4 — . 866 03 5 — 6 — •92 7 03 .968 24 .992 16 Divisions. 1 — .45389 2 — .6 3 — .714 14 4- . 8 Tenths. 5 — .86602 9 — *99499 6 — .91651 10 — 1 7 — -993 94 8 — .999 79 To Construct an Ellipse when Diameters do not Inter- sect at Right Angles.- Eig. 46 . Let A B and C D be given diameters. Draw boundary lines parallel to diameters, divide longest diameter into any number of equal parts, and divide shortest boundary lines into same number of equal parts. From one end of shortest diameter, D, draw radial lines through divisions of longest diame- ter, and from opposite end, C, draw radial lines to divisions on shortest boundary lines ; the intersection of these lines will give points in the curve. Arcs. To Describe a, Grothic Arc.- Eig. 4 I 7. Take line A B. At points A and B draw arcs B a and A c, and it will describe arc required. To Describe an Elliptic Arc, Chord, and Height being :\\\ given.— Eig. 48 . Bisect A B. at c ; erect perpendicular A q, and draw line q D equal and parallel to A c. Bisect A c and A q in r and n; make c 1 equal to c D, and draw line l r q ; draw also line n s D ; bisect B 5 D with a line at right angles, and cutting line c D at 0; draw line o q\ make cp equal to c lc, and draw line op i. Then, from 0 as a centre, with radius 0 D, describe arc sDi; and from lc and p as centres, with radius A lc, describe arcs A s and B i. 228 GEOMETRY. To Describe a, Grotliic Arc.— Figs. 4=0 and. SO. Divide line A B into three equal parts, e c ; from points A and B let fall perpendiculars A o and B r, equal in length to two of divisions of line A B ; s 50 . draw lines o h and r g from points . e, c; with length of cB, describe arcs /C *\ Ag and B h, and from points o and r f \ / \ describe arcs g i and i h. (Fig. 49.) j^L -} r ic 13 Or, divide line A B into three a \ / equal parts at a and &, and on points V A, a, &, and B, with distance of two / \ divisions, make four arcs intersect- ^ V 6 r ing at c and 0. c> Through points c, 0, and divisions a, 6, draw lines cf and 0 e , on points a and b describe arcs A e and B^, and on points c 0 arcs f s and e s. (I ig. 5°-) Cycloid, and Epicycloid. To Describe a Cycloid..— Fig. SI. When a circle, as a wheel, rolls over a straight right line, beginning as at A and ending at B, it completes one revolution, and measures a straight line, A B, exactly equal to circumference of circle c e r, which is termed the generating circle , and a point or pencil fixed at point r in circumference traces out a curvilinear path, ArB, termed a cycloid. A B is its base and c r its axis. Place generating circle in middle of Cy- ■p cloid, as in figure; draw a line, m n, paral- lei to base, cutting circle at e; and tangent The following are some of properties of Cycloid : Or, whole arc of Cycloid ArB = four times axis c r. Area of Cycloid A r B A = three times area of generating circle r c. Tangent n i is parallel to chord e r. n i to curve at point n. Horizontal line e n = arc of circle e r. Half-base A c=half-circumference cer. Arc of Cycloid rn = twice chord r e. Half arc of cycloid Ar=twice diameter of circle r c. To Describe Curve of a Cycloid.— IFi 52. On an indefinite line, A B, set olf co== circumference of generating circle, di- vide this line into any number of equal parts (8 in figure), and at points of divis- ion erect perpendiculars thereto. Upon p each of these lines describe a circle — — - x ~ ~ generating circle. On c 1 take 1 x — .2< 0 1, and with a; as a centre, with radius x c = .75 c x, describe an arc cutting circle at 1'; from 2 on next circle, with two distances of 1 1', ^measured as chords cut circle at 2' • from 3 on next circle, with three distances of 1 1 , cut circle at 3 , and proceed in like manner from each side until figure is complete. To Describe an Interior Epicycloid or Hypocycloid.— Eig. S 3 . If generating circle is rolled on inside of fundamental circle *as in Fig. 53, it forms an interior epicycloid , or % 1 hypocycloid , A c B, which becomes in this case nearly a < J* straight line. Other points of reference in figure cor- respond to those in Fig. 51. When diameter of generat- ing circle is equal to half that of fundamental circle^ epicycloid becomes a straight line, being diameter of the larger circle. * See explanation, Fig. 54. GEOMETRY. 229 To Describe an Exterior Epicycloid.— Eig. 54. An Epicycloid differs from a Cycloid in this, that it is generated by a point, o'", in one circle, o r, rolling upon circumference of another, A r s, instead of upon a right line or horizontal surface, former being generating circle and latter fundamental circle. Generating circle is shown in four positions, in which its generating point is indicated by o o' 0" o'". A o'" s is an Epicycloid. Involute. To Describe an Involute.-Fig. 55. Assume A as centre of a circle, b c 0; a cord laid partly upon its circumference, as be-, then the curve eimn, described by a tracer at end of cord, when unwound from a circle, is an involute. This curve can also be defined by a batten, x, rolling on a circle, as s u. 3?araT>ola. geometry. To Describe Curve of* a Parabola, Base and. Height "being given.— Big. 60 . Draw an isosceles triangle, as a b d, base of which shall be equal to and its height, c b , twice that of proposed parabola. Divide each side, a 6, d b, into any number of equal parts ; then draw lines i i, 2 2, 3 3, etc., and their intersection will define curve. (Fig. 6o.) To Describe a Parabola, any Ordinate to -A.xis and its Abscissa being given.— Big. 61 - Bisect ordinate, as A o in r; join m c n B r, and draw r s perpendicular to it, meeting axis continued to s. Set off - ’ Bc,Be, each equal to os; draw m c u perpendicular to B s, then m u is directrix and B e focus; through e and any number of points, i, i, i etc., in axis, draw double ordinates v i v, and on centre e, with radii e c, i c, etc., cut respective or- dinates at v v , etc., and trace curve through these points. ► Note.— L ine vev passing through focus is parameter. j's Spiral. * To Draw a Spiral about a given Point.— Big. 62 . Assume c the centre. Draw A Ji, divide it into twice number of parts that there are to be revolutions of line. Upon c de- scribe re, os, A h, and upon e describe rs,os, etc. Hyperbola. To Describe a Hyperbola, Transverse and Conjugate Diameters being given. Big. 63. Let A B represent transverse diameter, and C D C °DrawC e parallel to A B, and e r parallel to C D; draw o e, and with radius o e, with o as a centre, describe circle F e r, cutting transverse axis pro- duced in F and /; then will F and ./be foci of fig- In o B produced take any number of points, n, n, etc., and from F and /as centres, with A n and B n as radii, describe arcs cutting each other in s, s, etc. Through s, s, etc. , draw curve ssssBssss. Note.— If straight lines, as o e y and o r y, are drawn from ^ tremities e r, they will be asymptotes of hyperbola, property of which is to ap- proach continually to curve, and yet never to touch it. When Foci and Conjugate Axis are given.— Let F and /be foci, and C D conjugate a ?hroVgh P cTaw| ^parallel to F and /; then, Was a centre and oF as a radius, describe an arc cutting g C e at g and e; from these ^points ^et faU Perpen- diculars upon line connecting F and / and part intercepted between them, as A B, will be transverse axis. Catenary. To Delineate a Catenary, Span and. Versed Sine being given. — Big. 64 . (IF. Hildenbrand.) Divide half span, as A B, into any required number of equal parts, as i, 2, 3, and let fall BG and A 0, each equal to versed sine of curve ; divide A o into like number of parts, 1', 2', 3 as A B. Connect Ci',C 2', and 0 3', and points of intersec- tion of perpendiculars let fall from A B will give points through which curve is to be drawn. A Or, suspend a finely linked chain against a ver- tical plane, trace curve from it on the plane in accordance with conditions of given length and height, or of given width or length of arc. UUgLH ilUU UCIgui, v. — - v Note.— For other methods see D. R. Clark’s Manual, pp. 18, 19. AREAS OF CIRCLES. 23 I Areas of Circles, from St to 150. Diam. | Area. Diam. | Area. Diam. Area. 1 1 Diam. Area. St .OOO 192 3 7.0686 7 38.4846 14 I 53-938 v /16 | 7.3662 X 39 - 87 I 3 Y 156.7 /82 7.6699 X 41.2826 Y± I 59-485 ^L6 .003 068 % 7.9798 X 42.7184 Ys 162.296 % .012 272 X 8.2958 Y 44.1787 Y 165.13 * .027612 Yg X 8.618 8.9462 45.6636 47-1731 s 167.99 I7O.874 ¥ .049087 X 9.2807 % 48.7071 X 173.782 Y 9.62H 8 50.2656 15 176.715 /16 ■% 9.968 M 51.8487 X 179.673 K .IIO447 % IO.3206 3 ^ 53-4563 X 182.655 % .15033 % IO.679 X 55.0884 X 185.661 Si II.0447 Y 56.7451 X 188.692 X • i 9 6 35 % II.416 % 58.4264 X I9I.748 % .248505 Ys n -7933 % 60.1322 X 194.828 % 12.177 Ys 61.8625 X 197-933 78 4 12.5664 9 63.6174 16 201.062 ^6 .371 224 X 12.962 X 65.3968 X 204.216 .441 787 13-3641 Y 67.2008 X 207.395 13.772 % 69.0293 X 210.598 /16 .518487 X 14.1863 Y 70.8823 X 213.825 X .601 322 Xg 14.606 H 72.7599 X 217.077 % .690292 % Xg 15-033 15-465 K X 74.6621 76.5888 X X 220.354 223.655 I •7854 Si I 5-9043 10 78.54 *7 226.981 .8866 V m 16.349 X 80.5158 X 230.331 .99402 Ys 16.8002 X 82.5161 X 233.706 X& 1. 1075 Xg 17-257 X 84.5409 X 237.105 ✓£ 1.2272 Si 17.7206 x: 86.5903 X 240.529 ^6 I *353 % 18.19 x 88.6643 X 243-977 X 1.4849 % 18.6655 x 90.7628 X 247-45 A& 1.6229 % 19.147 % 92.8858 X 250.948 Y 1.767 1 5 I 9-635 11 95-0334 18 254-47 %& I- 9 I 75 x& 20.129 X 97-2055 X 258.016 M 2.073*5 Si 20.629 X 99.4022 X 261.587 x 2.2365 Xs 2 i-i 35 X 101.6234 X 265.183 % 2.4053 Si 21.6476 X 103.8691 X 268.803 % 2.58 Xg 22.166 X 106.1394 X 272.448 % 2.761 2 Ys 22.6907 M 1 08.4343 X 276.117 % 2.9483 Xg 23.221 X 1 10.7537 X 279.811 2 3.1416 Si 23-7583 12 113.098 19 283.529 /5l6 3-338 Xg 24.301 X 115.466 X 287.272 Si 3-5466 % 24.8505 X 117.859 X 291.04 X& 37584 Xg 25.406 X 120.277 X 294.832 Si 3.9761 % 25-9673 X 122. 719 X 298.648 %, 4.2 % 26.535 X 125.185 X 302.489 Ys 4.4301 % 27.1086 X 127.677 X 306.355 4.7066 Xg 27.688 % 130. 192 X 3 IO - 2 45 1/ Si 4.9087 6 28.2744 13 132.733 20 314.16 X& 5-1573 Ys 29.4648 X 135-297 X 318.099 § 5 - 4 ii 9 Y 30.6797 X 137.887 X 322.063 Xg £•6723 Ys 31.9191 X 140.501 X 326.051 Si 5-9396 Y 33-1831 X I 43 -I 39 X 330.064 % 6.2126 Ys 34-4717 X 145.802 X 334 - 102 % 6.491 8 % 35-7848 X 148.49 X 338.164 i % 6.7772 1 Ys 37.1224 % 1 5 1. 202 X 342.25 232 AREAS OF CIRCLES. Diam. Area. Diam. Area. Diam. Area. 21 346.361 28 6I5-754 35 962.115 % 350.497 X 621.264 X 969 % 354-657 X 626.798 X 975-909 % 358.842 % 632.357 % 982.842 X 363-05I X 637.941 X 989.8 % 367.285 % 643-549 % 996.783 % 371-543 % 649.182 % IOO379 X 375.826 X 654.84 % 1010.822 22 380.134 29 660.521 36 IOI7.878 X 384.466 X 666.228 X IO24.96 X 388.822 X 671.959 X IO32.065 X 393-203 X 677.714 % I °39- I 95 X 397.609 X 683.494 X 1046.349 % 402.038 % 689.299 % 1053.528 % 406.494 % 695.128 % 1060.732 % 410.973 X 700.982 X 1067.96 23 4I5-477 30 706.86 37 1075.213 X 420.004 X 7 i2 . 7 6 3 X 1082.49 X 424-558 X 718.69 X 1089.792 X 429-I35 % 724.642 % 1097.118 X 433-737 X 730.618 X 1104.469 % 438.364 % 736.619 % 1111.844 % 443-015 % 742.645 % 1 1 19.244 X 447.69 X 748.695 X 1126.669 24 452.39 31 754-769 38 . 1134.118 X 457*115 X 760.869 X 1141.591 X 461.864 X 766.992 X 1149.089 % 466.638 X 773-14 X 1156.612 X 471.436 X 779-3 x 3 X 1164.159 % 476.259 X 785-51 X 11 7 I -73 I % 481.107 % 79 I *73 2 % 1 1 79.327 % 485-979 X 797-979 X 1186.948 2 5 490.875 32 804.25 39 1 194.593 X 495.796 X 810.545 X 1202.263 X 500.742 X 816.865 X 1209.958 % 505-712 % 823.21 X 1217.677 X 510.706 X 829.579 X 1225.42 % 5I5-726 X 835-972 X 1233.188 % 520.769 % 842.391 X 1240.981. % 525-838 X 848.833 X 1248.798 26 530.93 33 855.301 40 1256.64 X 536.048 861.792 X 1264.506 X 54i-i9 X 868.309 X 1272.397 % 546.356 X 874.85 % 1280.312 X 551.547 X 881.415 X 1288.252 % 556.763 X 888.005 % 1296.217 % 562.003 X 894.62 % 1304.206 % 567.267 % 901.259 X 1312. 219 27 572.557 34 907.922 41 1320.257 X 577-87 X 914.611 X 1328.32 X 583.209 X 921.323 X 1336-407 % 588.571 X 928.061 X I344-5I9 X 593-959 X 934.822 X 1352655 % 599-371 X 941.609 X 1360.816 % 604.807 X 948.42 X 1369.001 X 610.268 X 955*255 X I377-2II 42 X X % X X % X 43 X X X X % 44 X X X 4S X % % X 46 X 47 48 X 138545 1393-7 1401.99 1410.3 1418.63 1426.99 1435-37 1443-77 1452.2 1460.66 1469.14 1477.64 1486.17 1494-73 15033 I5II-9I 1520.53 I 5 2 9- I 9 1537.86 1546.56 1555-29 1564.04 1572.81 1581.61 I 590-43 1599.28 1608.16 1617.05 1625.97 1634.92 1643.89 1652.89 1661.91 1670.95 1680.02 1689. 1 1 1698.23 1707.37 i7 l6 -54 I725-73 1734-95 1744.19 1753-45 1762.74 1772.06 1781.4 1790.76 1800.15 1809.56 1819 1828.46 I 837-95 1847.46 1856.99 1866.55 1876.14 AREAS OF CIRCLES. 233 Diam. Area. | Diam. Area. Diam. Area. Diam. Area. 49 1885.75 56 2463.OI 63 3 II 7.25 7 ° 3848.46 A 1895.38 y 8 2474.02 A 3129.64 A 3862.22 A 1905.04 A 2485.05 A 3142.04 A 3876 A 1914.72 % 2496. 1 1 A 3154.47 a 3889.8 A 1924.43 A* 2507.19 A 3166.93 A 3903.63 % 1934.16 A 2518.3 & 3 I 79 - 4 I A. 39 I 7-49 A 1943.9 1 % 2529-43 % 3191.91 A 3931-37 % I 953-69 A 2540.58 A 3204.44 A 3945-27 50 I 9 6 3-5 57 255 I -76 64 3217 71 3959-2 1973-33 2562.97 A 3229.58 A 3973-15 A 1983.18 A 2574.2 A 3242.18 A 3987.13 % 1993.06 Vs 2585-45 A 3254.81 4 , 4001.13 A 2002.97 a 2596.73 A 3267.46 A 4015.16 % 2012.89 A 2608.03 A 3280.14 4 4029.21 A 2022.85 A 2619.36 A 3292.84 A 4043.29 % 2032.82 % 2630.71 A 3305-56 A 4057-39 51 2042.83 58 2642.09 65 3318.31 72 4071.51 % 2052.85 H 2653-49 A 3331.09 A 4085.66 X 2062.9 A 2664.91 A 3343-89 A 4099.84 A 2072.98 % 2676.36 A 3356.71 A 4114.04 A 2083.08 A 2687.84 A 3369-56 A 4128.26 % 2093.2 4 2699.33 A 3382.44 A 4142.51 A 2103.35 % 2710.86 % 3395-33 A 4156.78 % 2113.52 % 2722.41 A 3408.26 A 4171.08 52 2123.72 59 2733.98 66 3421.2 73 4185.4 K 2133-94 A- 2745-57 X 3434-17 A 4199.74 A 2144.19 A 2757.2 A 3447.17 A 4214.11 % 2154.46 A 2768.84 A 3460.19 A 4228.51 A 2164.76 A 2780.51 A 3473-24 A 4242.93 % 2175.08 A 2792.21 A 3486.3 A 4257-37 % 2185.42 % 2803.93 A 3499-4 A 4271.84 % 2195.79 A 2815.67 A 3512.52 A 4286.33 53 2206. 19 60 2827.44 67 3525-66 74 4300.85 A 2216.61 A 2839.23 A 3538.83 X 4315-39 A 2227.05 A 2851.05 A 3552.02 A 4329.96 % 2237.52 % 2862.89 A 3565-24 A 4344-55 A 2248.01 A 2874.76 A 3578.48 A 4359-17 % 2258.53 A 2886.65 A 3591.74 A 4373 - 8 i % 2269.07 % 2898.57 A 3605.04 A 4388.47 % 2279.64 A 2910.51 A 3618.35 A 4403.16 54 2290.23 61 2922.47 68 3631.69 75 4417.87 A 2300.84 A 2934.46 A 3645.05 A 4432 . 6 i A 2311.48 A 2946.48 A 3658.44 A 4447-38 % 2322.15 A 2958.52 A 3671.86 A 4462.16 A 2332.83 A 2970.58 A 3685.29 A 4476.98 A 2343-55 A 2982.67 A 3698.76 A 4491.81 A 2354-29 % 2994.78 A 3712.24 A 4506.67 A 2365.05 A 3006.92 A 3725.75 A 4521.56 55 2375-83 62 3019.08 69 3739-29 76 4536.47 A 2386.65 A 3031.26 A 3752.85 X 455 i- 4 i A 2397.48 A 3043-47 A 3766.43 X 4566.36 A 2408.34 A 3055.71 A 3780.04 % 4581.35 A 2419.23 M 3067.97 A 3793.68 . X 4596.36 A 2 43°- I 4 % 3080.25 A 3807.34 A 4611.39 X 2441.07 % 3092.56 A 3821.02 A 4626.45 A 2452.03 A 3104.89 A 3834.73 A 464 I -53 234 AREAS OF CIRCLES. Diam. Area. Diam. Area. 77 4656.64 84 554 I -78 % 4671.77 H 5558.29 X 4686.92 x 5574.82 % 4702.I % 5591-37 A 47 I 7 - 3 I x 5607.95 % 4732.54 % 5624.56 % 4747-79 x 5641.18 % 4763.07 % 5657-84 78 4778.37 5674 - 5 I y a 4793-7 Ya 5691.22 x 4809.05 X 5707-94 % 4824.43 Ya 5724.69 X 4839- 8 3 x 5741-47 % 4855.26 % 5758.27 X 4870.71 x 5775-1 % 4886.18 % 579!-94 79 4901.68 86 5808.82 A 4917.21 5825.72 x 4932.75 X 5842.64 % 4948.33 X 5859-59 A 4963.92 x 5876.56 % 4979-55 % 5893-55 % 4995-19 x 59 io -58 % 5010.86 % 5927-62 So 5026.56 87 5944.69 % 5042.28 % 5961.79 X 5058.03 X 5978.91 % 5073-79 % 5996-05 x 5089.59 V * 6013.22 Ya 5105.41 % 6030.41 X 5121.25 x 6047.63 % 5 I 37 -I 2 % 6064.87 Si 5 i 53 -oi 88 6082.14 % 5168.93 % 6099.43 X 5184.87 x 6116.74 % 5200.83 % 6134.08 u 5216.82 X 615145 Ya 5232.84 Ya 6168.84 X 5248.88 X 6186.25 % 5264.94 ! % 6203.69 82 5281.03 ' 89 6221.15 Vs 5297.14 X 6238.64 X 53 I 3-28 x 6256.15 % 532944 % 6273.69 x 5345-63 x 6291.25 % 5361.84 % 6308 84 x 5378 .o 8 % 6326 45 % 5394-34 i % 6344.08 83 5410.62 90 6361.74 % 5426.93 X 6379.42 x 5443.26 X 6397 -I 3 % 5459-62 Ya 6414.86 x 5476.01 X 6432.62 % 5492.41 4 64504 % 5508.84 % 6468.21 % 5525-3 % 6486.04 Diam. Area. Diam. Area. 9 1 6503-9 98 7542.98 Ya 6521.78 % 7562.24 X 6539.68 X 7581.52 Ya 6557 - 6 I Ya 7600.82 X 6575-56 X 7620.15 % 6593-54 4 7639-5 % 6611.55 X 7658.88 Ya 6629.57 Ya 7678.28 92 6647.63 99 7697.71 Ys 6665.7 Ya 7717.16 X 6683.8 X 7736.63 % 67OI.93 % 7756.13 X 6720.08 X 7775.66 % 6738.25 Ya 7795.21 X, 675645 X 7814.78 Ya 6774.68 Ya 7834.38 93 6792.92 100 7854 Ya 6811.2 X 7893-32 X 6829.49 Ya 7932.74 Ya 6847.82 X 7972.25 X 6866.16 IOI 80II.87 Ya 6884.53 X 8051.58 X 6902.93 X 8091.39 Ya 6921.35 X 8131.3 94 6939.79 102 8171.3 Ya 6958.26 X 82H.4I X 6976.76 X 8251.61 Ya 6995.28 X 8291.91 X 7013.82 103 8332.31 Ya 7032-39 ¥ 8372.81 X. 7050.98 X 8413 4 Ya 7069.59 X 8454.09 95 7088.23 104 8494.89 X 7106.9 X 8535.78 X 7 I 25-59 X 8576.76 % 7 I 44 - 3 I X 8617.85 X 7 i6 3-°4 105 8659.03 Ya 7181.81 X 87OO 32 X 7200.6 X 8741.7 Ya 7219.41 X 8783 l8 96 7238.25 106 8824.75 Ya 7257.11 A 8866.43 X 7275-99 A 8908.2 % 7294.91 % 8950.07 X 73 * 3-84 107 8992.04 Ys 7332.8 A 9034.II X 7351-79 H 9076.28 Ya 7370.79 % 9118.54 97 7389-83 108 9160.91 Ya 7408.89 A 9203.37 X 7427.97 A 9245 93 Ya 7447.08 % 9288.58 X 7466.21 109 9331-34 Ya 7485-37 i ^ 9374- I 9 X 7504-55 A 94 I 7- I 4 i Ya 7523-75 1 X 9460.19 AREAS OF CIRCLES. 235 Diam. Area. Diam. Area. Diam. Area. | I Diam. Area. iio 9503-34 120 II309.76 130 13273.26 140 I 5 393-84 H 9546.59 X 11356.9 3 A I 3324-36 A 15448.87 A 9589-93 3< 1 1 404.2 A I 3375-56 A I 5 503-99 % 9633-37 % H 45 I -57 % 13426.85 A 15 559.22 in 0676.QI 121 II499.O4 131 13478.25 141 15614.54 H 9720-55 H II 546.61 A 13 529- 74 3^ 15669.96 A 9764.29 A II594.27 • A I 3 58 i -33 A I 5 725-48 % 9808.12 % II 642.03 A 13633-02 A 15781.09 112 9852.06 122 1 1 689.89 132 13684.81 142 15836.81 X Q 896.09 3^ II 737-85 A 13 736.69 A 15 892.62 A 9940.22 A II785.91 A 13 788.68 A 15948.53 % 9984.45 % II 834 06 A 13840.76 A 16004.54 113 IOO28.77 123 II 882.32 133 13892.94 143 16060.64 a IOO73.2 a II930.67 A 13945.22 A 16 116.85 A IO II7.72 A II 979.12 A 13997.6 a 16173.15 % IO 162.34 % 12 027.66 A 14050.07 A 16229.5 5 114 10 207.06 124 12076.31 134 14 102.64 144 16286.05 3 ^ 10251.88 A 12 125.05 3 ^ I 4 I 55 - 3 I 3 ^ 16342.65 'A IO296.79 A 12173-9 14 208.08 A 16399 35 A IO34I.8 A 12 222.84 % 14253.09 A 16456.14 115 IO386.9I 125 12 271.87 135 I 43 I 3 - 9 I 145 16513.03 X 10432.12 X 12 321.01 X - 14366.98 3 € 16570.02 A IO477.43 A 12370.25 A 14420.14 X 16627. 11 A IO522.84 % 12419.58 A 14473-4 A 16684.3 1 16 IO568.34 126 12469.01 136 14526.76 146 16741.59 A IO613.94 %■ 12518.54 X 14580.21 A 16798.97 A I0659 65 y * 12568.17 . A I 4633-77 , A , 16856.45 % IO705.44 % 12 618.09 A 14687.42 A 16914.03 117 10 751.34 127 12667.72 137 14 741.17 147 16971.71 A 10 797.34 A 12 717.64 A 14795.02 17029.48 A I0843.43 A 12 767.66 A 14848.97 X 17087.36 % 10889.62 A 12817.78 A 14903.01 A I 7 I 45-33 n8 IO935.9I 128 12867.99 138 14957.16 148 17203.4 A IO982.3 A 12918.31 A 15011.4 A 17261.57 A II 028.78 A 12968.72 A 15065.74 A 17319.84 % HO75.37 % 13019.23 A 15 120.18 . , A 17378.2 1 19 II I22.05 129 13069.84 139 I 5 I 74 - 7 I 149 17436.67 A II 168.83 A 13 120.55 A 15229.35 A 17495.23 A 11215.71 A I 3 I 7 I -35 A 15 284 08 A I 7 553-89 A II 262.69 % 13 222.26 A 15338.91 150 17671.5 To Compute Area of a Circle greater than, any in Table. Rule.— D ivide dimension by two, three, four, etc., if practicable to do so, until it is reduced to a diameter to be found in table. Take tabular area for this diameter, multiply it by square of divisor, and product will give area required. Example. — W hat is area for a diameter of 1.050? 1050-^-7 = 150; tab. area , 150= 17 671.5, which x 7 2 = 86.5 903. 5, area. To Compute Area of a Circle in 3Teet and. Inches, etc., by preceding Table. Rule.— -R educe dimension to inches or eighths, as the case may be, and take area in that term from table for that number. AREAS OE CIRCLES. 236 Divide this number by 64 (square of 8) if it is in eighths, and quotient will give area in inches, and divide again by 144 (square of 12) if it is in inches, and quotient will give area in feet. Example.— What is area of 1 foot 6.375 ins.? 1 foot 6.375 ins. = 18.375 ins. = 147 eighths. Area of 147 = 16971.71, which — 64 = 265.181 25 ins.; and by 144 = 1.84 125 feet. To Compute Area, of* a Circle Composed of* air Integer and. a Fraction. Rule. — Double, treble, or quadruple dimension given, until fraction is in- creased to a whole number, or to one of those in the table, as 3^, etc., provided it is practicable to do so. Take area for this diameter ; and if it is double of that for which area is required, take one fourth of it ; if treble, take one sixteenth of it, etc. Example.— Required area for a circle of 2.1875 ins. 2.1875 X 2 = 4.375, area for which = 15.0331, which -4- 4 = 3. 758 ins. When Diameter is composed of Integers and Fractions contained in Table. Rule. — Point off a decimal to a diameter from table, and add twice as many figures or ciphers to the right of the area as there are figures cut off from the diameter. Example i. — What is area of 9675 feet diameter? Area of 96.75 = 7351.79; hence, area = 73 5i7 9oo/ee£ 2.— What is area of 24 375 feet diameter? Area of 2. 437 5 = 4. 6664 ; hence, area = 4 66 640 000 feet. To Ascertain Area of* a Circle as 300, 3000, etc., not contained in Table. Rule. — Take area of 3 or 30, and add twice the excess of ciphers to the result. Example. — What is area of a circle 3000 feet in diameter? Area of 30 = 706. 86, hence area of 3000 = 7 068 600 feet. To Compute Area of* a Circle "by* Logarithms. Rule. — To twice log. of diameter add 1.895091 (log. of .7854), and sum is log. of area, for which take number. Example. — What is area of a circle 1200 feet in diameter? Log. 1200 X 2 -j- 1.895091 = 6.158362 1.895091 = 6.053453, and number for which = 1 130976 feet. Areas of* Birmingham Wire Grange. I>iam. | Area. 1 Diam. Area. Diam. Area. Diam. Area. No. Sq. Inch. No. Sq. Inch. No. Sq. Inch. No. Sq. Inch. I .070 686 IO .014 IO3 19 .OOI 385 28 .000 154 2 •063347 II .OII309 20 .OOO 962 29 .OOOI33 3 .052 685 12 .009 331 21 .OOO 804 30 .OOO 1 13 4 .044 488 13 .007 088 22 .OO0616 31 .OOOO78 5 .038 013 14 .OO54II 23 .OOO 491 32 .OOO064 6 •032 365 15 .004071 24 .OOO 38 33 .000 05 7 .025447 16 .003318 25 .OOO314 34 .OOOO38 8 .021 382 17 .002642 26 .OOO 254 35 .000 02 9 .OI 7 203 18 .001 886 27 .000 201 36 .000 013 CIRCUMFERENCES OF CIRCLES. 237 Circumferences of Circles, from to 150. Diam. ClRCUM. Diam. ClRCUM. 1 Diam. ClRCUM. | Diam. ClRCUM. tr .04909 3 9.4248 8 25.1328 15 47.124 1/ .Ye 9.6211 X 25.5255 Xs 47.5167 /32 X 9 - 8 I 75 X 25.9182 X 47.9094 Kg % 10.014 % 26.3109 % 48.3021 H .392 7 10.2102 X 26.7036 X 48.6948 /16 IO.406 . % 27.0963 X 49.0875 Kg •589 % IO.6029 % 27.489 % 49.4802 K •785 4 X. IO.799 % 27.8817 Vs 49.8729 ft/ X IO.9956 9 28.2744 16 50.2656 VlG .981 75 % II.I9I X 28.6671 Vs 50.6583 % 1.1781 % n.3883 X 29.0598 % 51.051 Kg 1.3744c; % II.584 % 29.4525 % 51.4437 % II.781 X 29.8452 X 51.8364 % 1.5708 % II.977 % 30.2379 % 52.2291 V /16 1.76715 Vs I2.I737 % 30.6306 % 52.6218 I -963 5 X 12.369 % 3 1 0233 % 53.0145 /8 4 I2.5664 10 3 i< 4 1 6 17 53.4072 Kg 2.15985 Y 12.762 K 31.8087 K 53-7999 % 2.3562 % I2.959I K 32.2014 % 54.1926 Kg 13.155 % 32.5941 % 54-5853 % 2.552 55 X 13.3518 X 32.9868 X - 54.978 % 2.7489 Kg 13-547 % 33-3795 % 55.3707 % 13.7445 % 33.7722 % 55.7634 Xg 2-945 25 X 13 94 % 34.1649 % 56.1561 1 3.1416 X 14.1372 11 34.5576 18 56.5488 Kg 3-337 9 X 14-333 X 34-9503 X 56.9415 Si 3-534 3 X 14.5299 X 35-343 X 57-3342 K 3.7306 % 14-725 % 35.7357 % 57.7269 K 3-927 % 14.9226 X 36.1284 X 58.1196 Kg 4-1233 X 15.119 % 36.5211 % 58.5123 % 4.319 7 Vs I5.3I53 X 36 . 9 1 38 % 58.905 Kg 4.516 % 15-511 % 37.3065 Ks 59-2977 Si 4.7124 5 15.708 12 37.6992 J 9 59.6904 Kg 4.9087 Vs 16.1007 X 38.0919 X 60.0831 % 5.105 1 X 16.4934 X 38.4846 X 60.4758 Kg 5-301 4 X 16.8861 % 38.8773 Ks 60.8685 % 5497 8 X 17.2788 X 39.27 X 61.2612 % 5.6941 % 17.6715 % 39.6627 Ks 61.6539 % 5.8905 % 18.0642 % 400554 % 62.0466 % 6.086 8 % 18.4569 X 40.4481 % 62.4393 2 6.283 2 6 18.8496 13 40.8408 20 62.832 Kg 6.4795 Vs 19.2423 Vs 4 I .2335 Vs 63.2247 % 6.675 9 X 19.635 X 41.6262 X 63.6174 Kg 6.872 2 % 20.0277 % 42.0189 % 64.0101 Si 7.0686 X 20.4204 X 42.4116 X 64.4028 Kg 7.2649 X 20.8131 % 42.8043 X 64-7955 % 746 i 3 % 21.2058 % 43-197 % 65.1882 K& 7.6576 % 21.5985 % 43-5897 % 65.5809 X 7.854 7 21.9912 14 43.9824 21 65-9736 Kg 8.0503 Vs 22.3839 X 44-3751 K 66.3663 % 8.246 7 X 22.7766 X 44.7678 X 66.759 Kg 8-443 % 23.1693 X 45.1605 X 67.1517 % 8.6394 X 23.562 X 45.5532 X 67-5444 % 8-835 7 % 23-9547 X 45-9459 X 67.9371 % 9.032 1 % 24.3474 % 46.3386 % 68.3298 X 9.2284 % 24.7401 X 46.7313 % 68.7225 238 CIRCUMFERENCES OF CIRCLES. Diam. ClBCUM. Diam. ClBCUM. Diam. ClBCUM. Diam. ClBCUM. 22 69.II52 2 9 91.1064 36 113.098 43 135.089 A 69.5079 91.4991 X 113-49 X 1 35 -48 1 A 69.9O06 A 91.8918 X 113.883 X * 35-874 X 70.2933 % 92.2845 X 114.276 X 136.267 A 70.686 A 92.6772 X 114.668 X 136.66 A 7I.O787 % 93.0699 X 115.061 X 137.052 % 71.4714 % 93.4626 X 1 15-454 X 137-445 x 7I.864I % 93-8553 X 115.846 X 137.838 23 -, 72.2568 30 94.248 37 116.239 44 138.23 X 72.6495 x 94.6407 X 116.632 X 138.623 A 73.0422 A 95-0334 X 117.025 X 139.016 x 73-4349 % -95.4261 X 1 1 7.41 7 X 139.408 x 73.8276 A 95.8188 X 117.81 X 139.801 x 74.2203 x 96.2115 X 118.203 X 140. 194 X 74 - 6 i 3 % 96.6042 X 118.595 X 140.587 % 75-0057 x 96.9969 X 118.988 X 140.979 24 75-3984 3 * 97.3896 38 119.381 45 141.372 A 75 - 79 H >8 97.7823 X 119.773 X 141.765 A 76.1838 X 98-175 X 120.166 X I 4 2 *i 57 % 76.5765 % 98.5677 X 120.559 X 142.55 A 76.9692 X 98.9604 X 120.952 X 142.943 % 77.3619 X 99-3531 X 121.344 X 1 43 -335 % 77-7546 % 99-7458 X 121.737 X 143.728 % 78.1473 % 100.1385 X 122.13 x 144.121 25 78.54 3 2 100.5312 39 122.522 46 144.514 A 78.9327 X 100.9239 X 122.915 X 144.906 A 79-3254 X 101.3166 X 123.308 X 145.299 % 79.7181 % 101.7093 X 123.7 X 145.692 A 80.1108 % 102. 102 X 124.093 X 146.084 % 80.5035 | X 102.4947 X 124.486 X 146.477 % 80.8962 X 102.8874 X 124.879 X 146.87 % 81.2889 % 103.2801 X 125.271 X 147.262 26 81.6816 33 103.673 40 125.664 47 147-635 A 82.0743 X 104.065 X 126.057 X 147.048 A 82.467 X 104.458 X 126.449 X 148.441 % 82.8597 X 104.851 X 126.842 X 148.833 A, 83.2524 X 105.244 X. 127.235 X 149.226 % 83.6451 % 105.636 X 127.627 X 149.619 % 84.0378 X 106.029 X 128.02 X 150.01 1 % 84-4305 X 106.422 X 128.413 X 150.404 27 84.8232 34 106.814 41 128.806 48 150.797 A 85.2159 X 107.207 129.198 X 151.189 A 85.6086 X 107.6 X 129.591 X 151.582 % 86.0013 X 107.992 X 129.984 X I 5 I -975 A 86.394 X 108.385 X 130.376 X 152.368 % 86.7867 X 108.778 X 130.769 X 152.76 % 87.1794 X 109.171 X 131.162 X I 53- I 53 A 87.5721 X 109.563 X I 3 I -554 X I 53-546 28 87.9648 35 109.956 42 I 3 I -947 49 I 53-938 A 88.3575 X 1 10.349 X 132.34 X I 54 - 33 I A 88.7502 X no. 741 X 132.733 X 154.724 % 89.1429 X in -134 X 133-125 X I55-ii6 A 89-5356 X in. 527 X i 33 - 5 i 8 X I 55-509 %. 89.9283 X 111.919 X I 33 - 9 11 If 155.902 % 90.32 1 X 112.312 X I 34-303 X 156.295 A 90.7137 X 112.705 X 134.696 X 156.687 CIRCUMFERENCES OF CIRCLES. 239 Diam. r ClRCUM. II Diam. I ClRCUM. Diam. ClRCUM. Diam. ClRCUM. 50 157.08 ' 57 1 79.071 64 201.062 71 223.054 X 157*473 X 179.464 X 201.455 X 223.446 X 157*865 K 179.857 X 201.848 X 223.839 X 158.258 % 180.249 X 202.24 X 224.232 X 158.651 X 180.642 X 202.633 X 224.624 H i 59*°43 % 181.035 X 203.026 X 225.017 X I 59-436 % 181.427 % 203.419 % 225.41 % 159.829 % 181.82 X 203.811 X 225.802 51 160.222 58 182.213 65 204.204 72 226.195 % 160.614 X 182.605 X 204.597 X 226.588 U 161.007 X 182.998 X 204.989 X 226.981 % 161.4 % 183.391 X 205.382 X 227.373 X 161.792 X 183.784 X 205.775 X 227.766 % 162.185 % 184.176 X 206.167 X 228.159 X 162.578 % 184.569 X 206.56 % 228.551 % 162.97 % 184.962 X 206.953 X 228.944 52 163.363 59 185.354 66 207.346 73 229.337 H 163.756 X 185.747 X 207.738 229.729 X 164.149 X 186.14 X 208.131 X 230.122 % 164.541 % 186.532 X 208.524 X 230.515 X 164.934 X 186.925 X 208.916 X 230.908 % 165.327 % 187.318 X 209.309 X 231.3 X 165.719 % 187.711 M 209.702 % 231.693 X 166.112 X 188.103 X 210.094 X 232.086 53 166.505 60 188.496 67 210.487 74 232.478 X 166.897 188.889 X 210.88 232.871 X 167.29 X 189.281 X 211.273 233.264 X 167.683 % 189.674 X 211.665 X 233*656 X 168.076 X 190.067 X 212.058 X 234.049 X 168.468 % 190.459 X 212.451 X 234.442 X 168.861 X 190.852 M 212.843 X 234*835 % 169.254 % 191-245 X 213.236 X 235.227 54 169.646 6t 191.638 68 213.629 75 235.62 X 170.039 X 192.03 X 2 14.02 1 X' 236.013 x 170.432 X 192.423 X 214.414 X 236.405 x 170.824 % 192.816 X 214.807 X 236.798 X 171.217 X 193.208 X 215.2 X 2 37 * I 9 I % 171.61 % 193.601 X 215.592 X 237.583 x 172.003 % I 93*994 X 215.985 X 237*976 % 172.395 X 194.386 X 216.378 X 238.369 55 172.788 62 194.779 69 216.77 76 238.762 X 173.181 X 195*172 217.163 X 239*154 X 173*573 X 195*565 X 217*556 X 239*547 % 173.966 X 195*957 X 217.948 X 239.94 X 1 74*359 X 196.35 X 218.341 X 240.332 % 174*751 X 196.743 X 218.734 X 240.725 % I 75 -I 44 % 197*135 % 219. 127 X 241.118 % 175*537 : X 197.528 X 219.519 X 241.51 56 175*93 63 197. 921 70 219.912 77 241.903 X 176.322 X 198.313 X 220.305 'X- 242.296 X 176.715 X 198.706 X 220.697 X 242.689 % 177.108 X *99-099 X 221.09 X 243.081 X 177*5 X 199.492 X 221.483 X 243.474 $ 177.893 X 199.884 X 221.875 X 243.867 % 178.286 % 200.277 % 222.268 X 244.259 % 178.678 1 X 200.67 X 222.661 X 244.652 240 CIRCUMFERENCES OF CIRCLES. Diam. Circum. ' ! Diam. Circum. Diam. | 78 245-045 ^85 267.036 9 2 t/ X 245-437 % 267.429 X X 245-83 , M 267.821 X X 246.223 % 268.214 X X 246.616 X 268.607 X X 247.008 % 268.999 X X 247.401 % 269.392 X % 247.794 % 269.785 X 79 248.186 86 27O.I78 93 1/ X 248.579 X 27O.57 X 248.972 X 270.963 X X 249.364 % 271.356 X X 249-757 X , 271.748 X. X 250.15 % 272. 141 X X 250.543 % 272.534 X X 250.935 % 272.926 X 80 251.328 8 ‘7 273-3I9 94 Vs 251. 721 ft 273-7 12 X K 252.113 274.105 X X 252.506 X 274.497 X X 252.899 X 274.89 X X 253.29 1 ft 275.283 X X 253.684 275- 6 75 X % 254.077 % 276.068 X 81 254-47 88 276.461 95 X 254.862 X 276.853 X X 255-255 X 277.246 X X 255.648 % 277.629 X X 256.04 X 278.032 X X 256.433 % 278.424 X X 256.826 % 278.817 X X 257.218 % 279.21 X 82 257.611 89 279.602 96 X 258.004 X 279-995 X X 258.397 X 280.388 X X 258.789 X 280.78 X X 259.182 X 281.173 X X 259-575 ft 281.566 $ X 259.967 281.959 X X 260.36 % 282.351 X 83 260.753 90 282.744 97 X 261.145 X 283.137 X X 261.538 X 283.529 X X 261.931 X 283.922 X X 262.324 X 284.315 x X 262.716 X 284.707 g X 263.109 X 285.1 X X 263.502 X 285.493 X 84 263.894 9 1 285.886 98 X 264.287 X 286.278 x X 264.68 X 286.671 X X 265.072 X 287.064 X . X 265.465 X 287.456 X X 265.858 % 287.849 X 266.251 % 288.242 X X 266.643 \ % 288.634 X 289.027 289.42 289.813 290.205 290.598 290.991 291.383 291.776 292.169 292.562 292.954 293-347 293 - 74 2 94 - 1 3 2 294- 525 294.918 295 - 3 1 295-703 296.096 296.488 296.881 297.274 297.667 298.059 298.452 298.845 299.237 299.63 300.023 300.415 300.808 301. 201 301-594 301.986 302.379 302.772 3 ° 3- i6 4 303 - 557 3 ° 3-95 304 - 34 2 305-521 308.27 309.84 Diam. Circum. 99 311.018 ft 311.411 X 311.804 % 312.196 ft 312.589 ft 312.982 % 313-375 % 3 x 3-767 100 314.16 X 3 1 4-945 X 315-731 X 316.516 IOI 317.302 X 318.087 X 318.872 X 3 j 9-658 102 320.443 X 321.229 X 322.014 X 322.799 io3 1/ 323-585 X 324.37 X 325-156 X 325-941 104 326.726 X 327.512 X 328.297 X 329.083 105 329.868 X 330-653 X 33 1 -439 X i 332.224 106 333-01 X 1 333-795 X 334-58 X 335-366 107 1 336.151 X 336.937 34 337.722 338.507 108 . 339-293 : X 340.078 K 340.864 l % 341.649 » io 9 . 342.434 ) X 1 343-22 : X 344-005 i- X 344 - 79 1 7 IIO 345.576 X 346.361 2 X 347-147 5 x 347.932 3 III 348.718 34 349-503 3 ft 350.288 5 % 351-074 CIRCUMFERENCES OF CIRCLES. 24 1 Diam. ClRCUM. | Diam. Circum. Diam. 1 Circum. Diam. J Circum. 1 12 351-859 121 380.134 130 408.408 *39 436.682 X 352.645 X 380.919 X 409.192 X 437-467 X 353-43 X 381.704 X 409.979 X 438.253 X 354-215 X 382.49 X 410.763 X 439-037 1 13 355 - 001 122 383.275 131 411-55 140 439.824 X 355-786 X 384.061 X 412.334 X 440.608 X 356.572 X 384.846 X 413.12 X 441-395 x 357-357 X 385.631 X 413-905 X 442.179 1 14 358.142 123 386.417 132 414.691 141 442.966 X 358.928 X 387.202 X 415.476 X 443-75 X 359 - 7 I 3 X 387.988 X 416.262 X 444-536 X 360.499 % 388.773 X 417.046 X 445-321 1 15 361.284 124 389-558 133 417-833 142 446.107 X 362.069 ¥ 390.344 X 418.617 X 446.891 X 362.855 X 39 I - I2 9 X 419.404 X 447.678 X 363-64 X 39 I - 9 I 5 X 420.188 X 448.462 116 364.426 125 392.7 134 420.974 *43 449.249 X 365.211 X 393-484 X 421.759 X 450.033 X 365-996 X 394.271 X 422.545 X 450.82 X 366.782 X 395-055 X 423-33 X 451-604 117 367.567 126 395-842 135 424.116 144 452.39 X 368.353 X 396.626 X 424.9 X 453-175 X 369.138 X 397.412 X 425.687 X 453 - 96 i X 369.923 X 398.197 X 426.471 X 454-745 118 370.709 127 398.983 136 427.258 I 45 455-532 X 371.494 X 399.768 X 428.042 X 456.316 X 372.28 X 400.554 X 428.828 X 457-103 X 373.065 X 401.338 X 429.613 146 458.674 119 373-85 128 402.125 137 430.399 X 460.244 X 374-636 X 402.909 X 431-183 147 461.815 X 375-421 X 403.696 X 431-97 X 463.386 X 376.207 X 404.48 X 432.754 148 464-957 120 376.992 129 405.266 138 433-541 X 466.528 X 377-777 X 406.051 X 434-325 149 468.098 X 378.563 X 406.837 X 435 - I 12 X 469.669 X 379-348 X 407.622 X 435-896 1 150 471.24 To Compute Circumference of a Diameter greater tban any- in preceding Table. Rule. — Divide dimension by two, three, four, etc., if practicable to do so, until it is reduced to a diameter in table. Take tabular circumference for this dimension, multiply it by divisor, according as it was divided, and product will give circumference required. Example. — What is circumference for a diameter of 1050? 1050-^-7 = 150; tab. circum 150 = 471.24, which X 7 = 3298.68, circumference. To Compute Circumference of a Diameter in Feet and Indies, etc., "by" preceding Table. Rule. — Reduce dimension to inches or eighths, as the case may be, and take circumference in that term from table for that number. Divide this number by 8 if it is in eighths, and by 12 if in inches, and quotient will give circumference in feet. X CIRCUMFERENCES OF CIRCLES. 242 Example.— Required circumference of a circle of 1 foot 6.375 ins. 1 foot 6.375 ins. = 18.375 ins. = 147 eighths. Circum. of 147 = 461.815. which -5- 8 = 57-727 ins.; and by 12 = 4.810 6 feet. • Compute Circumference for a Diameter composed of an integer and a Fraction. Rule. — Double, treble, or quadruple dimension given, until fraction is in- creased to a whole number or to one of those in the table, as etc., pro- vided it is practicable to do so. Take circumference for this diameter ; and if it is double of that for which circumference is required, take one half of it ; if treble, take one third of it ; and if quadruple, one fourth of it. Example.— Required circumference of 2.218 75 ins. 2.21875 X 2 = 4.4375, which x 2 = 8.875; circum. forf\diich = 27.8817, which -i- 4 = 6.9704 ins. When Diameter consists of Integers and Fractions contained in Table. Rule.— P oint a decimal to a diameter in table, take circumference from table, and add as many figures to the right as there are figures cut off. Example.— What is circumference of a circle 9675 feet in diameter? Circumference of 96. 75 — 303. 95 ; hence, circumference of 9675 = 30 395 To Ascertain Circumference for a Diameter, as (500, £5000, etc., not contained in Talxle. R u le. — Take circumference of 5 or 50 from table, and add the excess of ciphers to the result. Example.— What is circumference of a circle 8000 feet in diameter? Circumference of 80 == 251. 38; hence, circumference of 8000 = 25 138 feet. To Compute Circumference of a Oircle by Logarithms. Rule.— To log. of diameter add .497 01 G°»- 3- I 4 I 6), and sum is log. *j of circumference, from which take number. Example. — What is circumference of a circle 1200 feet in diameter? Log. 1200= 3-079 18 -f-. 497 01 = 3.576 19, and number for which = 3769.91 /erf. Circumferences of Birmingham Wire Gauge. Circum. I Diam. 1 Circum. Diam. Circum. Diam. Circum. j Ins. .942 48 .892 21 .81367 •747 7 .691 15 .637 74 .565 49 .51836 .46495 | No. ' IO 11 12 13 14 . 15 I I 1 18 Ins. .42097 .37699 .342 43 .298 45 .260 75 .226 19 .2042 .18221 .15394 No. 19 20 21 22 23 24 25 26 27 Ins. .13195 .IO995 .100 53 .087 96 .078 54 .069 II .062 83 .05655 .050 26 No. 28 29 30 31 32 33 34 35 36 Ins. | .04398 1 .040 84 j •037 7 ; .031 41 < .028 27 .025 13 .021 99 .015 71 .012 57 o\oi -£* oj io h vb oo- i 598 •9 •5 47 i 7 - 3 o8 7 243-474 83 .6 4729.4903 243.7882 .1 •7 4741.6876 244.1023 .2 .8 4753-9005 244.4165 •3 •9 4766.1292 244.7306 •4 78 4778.3736 245.0448 N •5 .1 4790-6337 245-359 .6 .2 4802.9095 245-6731 •7 .3 4815.201 245-9873 .8 .4 4827.5082 246.3014 •9 .5 4839.8311 246.6156 84 .6 4852.1698 246.9298 .1 .7 4864.5241 247.2439 .2 .8 4876.8942 247 - 558 i •3 •9 4889.2799 247.8722 •4 79 4901.6814 248.1864 •5 .1 4914.0986 248.5006 .6 .2 4926.5315 248.8147 •7 •3 4938-98 249.1289 .8 •4 4951.4443 249.443 ! •9 •5 4963.9243 249-7572 i 85 .6 4976.4201 250.0714 .1 .7 4988.9315 250.3855 .2 .8 5001.4586 250.6997 •3 •9 5014.0015 251.0138 •4 80 5026.56 251.328 ! -5 .1 5039 - 1 343 251.6422 •6 .2 5051.7242 251-9563 •7 .3 5064.3299 252.2705 .8 •4 5076.9513 252.5846 ■9 •5 5089.5883 252.8988 86 .6 5102.2411 253213 .1 •7 5114.9096 253-527! .2 .8 5127.5939 253-8413 •3 •9 5140.2938 254- I 554 •4 81 5153.0094 254.4696 •5 .1 5165.7407 254.7838 .6 .2 5178.4878 255-0979 •7 .3 5191.2505 255-4121 ! .8 •4 5204.0289 255.7262 •9 .5 5216.8231 256.0404 87 .6 5229.633 256.3546 .1 .7 5242.4586 256.6687 .2 .8 5255-2999 256.9829 •3 •9 5268.1569 257.297 1 * 4 82 5281.0296 257.6112 •5 .1 5 2 93 - 9 l8 257-9254 .6 .2 5306.8221 258.2395 •7 .3 5319.742 258.5537 .8 •4 5332.6775 258 8678 •9 5345.6287 5358.5957 537I-5784 5384-5767 5397-5908 5410.6206 5423.6661 5436.7273 5449.8042 5462.8968 5476.0051 5489.1292 5502.2689 5515.4244 5528.5955 5541.7824 5554-985 5568.2033 5581.4372 55946869 5607.9523 5621.2335 5 6 34-5303 5647.8428 5661.1711 5674 - 5 I 5 5687.8747 57 oi - 2 5 5714.6411 5728.0479 5741.4703 5754.9085 5768.3624 5781.8321 5795-3174 5808.8184 5822.3351 5835.8676 58494157 15862.9796 5876 . 559 1 5890.1544 5903-7654 59 I 7 - 39 21 5931.0345 5944.6926 5958.3644 5972.0559 5985.7612 5999.4821 6013.2187 6026.9711 6040.7392 6054.5229 6068.3224 259.182 2594962 259.8103 260.1245 260.4386 260.7528 261.067 261.3811 261.6953 262.0094 262.3236 262.6378 262.9519 263.2661 263.5802 263.8944 264.2086 264.5227 264.8369 265.151 265.4652 265.7794 266.0935 266.4077 266.7218 267.036 267.3502 267.6643 267.9785 268.2926 268.6068 268.921 269.2351 269.5493 269.8634 270.1776 270.4918 270.8059 271.1201 271.4342 271.7484 272.0626 272.3767 272.6909 273.005 273.3192 273 6334 273 9475 274.2617 2745758 274.89 275.2042 275-5183 275 - 83 2 5 276.1466 AREAS AND CIRCUMFERENCES OF CIRCLES. 25 Diam. Area. ClRCUM. 88 6082.1376 276.4608 .1 6095.9685 276.775 .2 6109 8151 277.089I •3 6123.6774 277.4O33 •4 6 I 37-5554 277.7174 •5 6151.4491 278.O316 .6 6165.3586 278.3458 •7 6179.2837 278.6599 .8 6193.2246 278.974I •9 ! 6207.1811 279.2882 89 6221.1534 279.6024 .1 6235.1414 279.9166 .2 6249.1451 280.2307 •3 6263.1644 280.5449 •4 6277.1995 28O.859 •5 6291.2503 28l.I732 .6 6305-3169 281.4874 •7 63 I 9 - 399 1 281.8015 .8 6333497 282.1157 •9 6347.6107 282.4298 90 6361.74 282.744 .1 6375 - 885 I 283 0582 .2 6390.0458 283.3723 •3 6404.2223 283.6865 •4 ! 6418.4144 284.OOO6 •5 6432.622 3 284.3I48 .6 6446.8459 284.629 •7 6461.0852 284.943I .8 6475-3403 285.2573 •9 6489.61 1 285.5714 9 i 6503.8974 285.8856 .1 6518.1995 286.I998 .2 6532-5174 286.5139 •3 6546.8509 286.828l •4 6561.2002 287.I422 •5 6575-5651 287.4564 .6 6589-9458 287.7706 •7 6604.3422 288.0847 .8 6618.7543 288.3989 •9 6633.1821 288.7I3 92 6647.6256 289.O272 .1 6662.0848 289.34I4 .2 6676.5598 289.6555 •3 6691.0504 289.9697 •4 6705.5567 29O.2838 •5 6720.0787 29O.598 .6 6734.6165 29O.9I2 1 •7 6749.17 291.2263 .8 6763.7391 291.5405 •9 6778.324 29I.8546 93 6792.9246 292.1688 .1 6807.5409 292.483 .2 6822.1729 292.797I •3 6836.8206 293. II I 3 •4 6851.484 | 293 4254 1 Diam. Area. ClRCUM. •5 6866.1631 293-7396 .6 6880.858 294.O538 •7 6895.5685 2943679 .8 6910.2948 294.6821 •9 6925.0367 294.9962 94 6939-7944 295.3IO4 .1 6954.5678 295.6246 .2 6969-3569 295-9387 •3 6984.1616 296.2529 -4 6998.9821 296.567 -5 7013.8183 296.8812 .6 7028.6703 297 -I 954 •7 7043.5379 297-5095 .8 7058.4212 297.8237 •9 7073.3203 298.I378 95 7088.235 298.452 .1 7103.1655 298.7662 .2 7118.II16 299.0803 •3 7133-0735 299-3945 •4 7148.0511 299.7086 •5 7163.0443 300.0228 .6 7 1 78.0533 300.337 •7 7193.078 300.6511 .8 7208.1185 3OO.9653 •9 7223.1746 301.2794 96 7238.2464 3OI.5936 .1 72533339 3OI.9O78 .2 7268.4372 302.2219 •3 7283.5561 302.5361 •4 7298.6908 302.8502 •5 7313.8411 3O3.1644 .6 7329 0072 3O3.4786 •7 7344.189 3O3.7927 .8 7359-3865 3O4.I069 •9 ' 7374-5997 304 - 42 I 97 7389.8286 304-7352 .1 7405.0732 305.0494 .2 7420.3335 305-3635 •3 7435.6096 3O5.6777 •4 7450.9013 3O5.9918 •5 7466.2087 306.306 .6 7481.5319 306.6202 •7 7496.8708 306.9343 .8 7512.2253 307.2485 •9 7527-5956 3O7.5626 98 7542.9816 307.876 8 .1 7558.3833 308.191 .2 7573-8007 308.5051 -3 7589-2338 308.8193 •4 7604.6826 309.1334 -5 7620.1471 309.4476 .6 7635.6274 309.7618 •7 7651.1233 310.0759 .8 7666.635 3103901 •9 7682.1623 | 310.7042 252 AREAS AND CIRCUMFERENCES OF CIRCLES. Diam. Area. ClRCUM. Diam. 99 7697.7054 3II.O184 •5 .1 77 I 3 * 26 4 2 311.3326 .6 .2 7728.8337 3II.6467 •7 •3 7744.4288 3II.9609 .8 •4 7760.0347 3 I 2.275 •9 Area. 7775*6563 779 I,2 937 7806.9467 7822.6154 7838.2999 712.5892 312.9034 3 I 3* 2I 75 3i3*53i7 3I3-8458 To Compute Area or Circumference of a Diameter greater than any in. preceding Table. See Rules, pages 235-6 and 241-2. Or, If Diameter exceeds 100 and is less than 1001. Put a decimal point, and take out area or circumference as for a Whole Number by removing decimal point, if for an area, two places to right , and if for a circumference, one place. Example.— What is area and what circumference of a circle 967 feet in diame- tG Area of 96.7 is 7344.189; hence, for 967 it is 734 418.9; and circumference of 96.7 is 303.7927, and for 967 it is 3037.927. To Compute Area and Circumference of a Circle by Log- arithms. See Rules, pages 236, 242. Areas and. Circumferences of Circles. From i to 50 Feet {advancing by an Inch). Or, From i to 50 Inches {advancing by a Twelfth). Diam. Area. ClRCUM. Diam. Area. ClRCUM. I ft ' 1 2 Feet. .7854 • 9 2I 7 1.069 Feet. 3 * I 4 l6 3*4034 3.6652 3 A 1 2 Feet. 7.0686 7.4668 7.8758 Feet. 9.4248 9,6866 9.9484 3 1.2272 3 * 9 2 7 3 8.2958 10.2102 A *•3963 4.1888 4 8.7267 IO.472 T- 5 6 i *5763 1.7671 4.4506 4 * 7 I2 4 5 6 9.1685 9.6211 10.7338 IO.9956 7 1.969 4 * 974 2 7 IO.0848 II.2574 8 2.1817 5* 2 36 8 10.5593 II.5192 9 2.4053 5*4978 9 II.O447 II.781 10 2.6398 5*7596 10 11*541 I2.O428 11 2.8853 6.0214 11 12.0483 12.3046 2 ft . 3 * I 4 l6 6.2832 4 /*• 12.5664 12.5664 1 3.4088 6-545 1 130955 12.8282 2 3.687 6.8068 2 I 3-6354 I3.O9 3 4 3-9761 4.2761 7.0686 7*3304 3 4 14.1863 14.7481 I 3 - 35 I 8 13.6136 5 4.5869 7.5922 5 15.3208 I 3-8754 6 4.9087 7*854 6 I 5-9043 14.1372 7 8 5 * 2 4 i 5 5.5852 8.1158 8.3776 7 8 16.4989 17.1043 I4.499 14.6608 9 10 11 5-9396 6.305 6.6814 8.6394 8.9012 9 i6 3 9 10 1 11 17.7206 18.3478 18.9859 14.9226 15.1844 15 4462 AREAS AND CIRCUMFERENCES OF CIRCLES. 253 Diam Area. ClRCUM. Feet. Feet. 5 A I 9-635 15.708 1 20.2949 15.9698 2 20.9658 16.2316 3 21.6476 16.4934 4 22.3403 16.7552 5 23 0439 I7.OI7 6 23-7583 17.2788 7 24.4837 17.5406 8 25.22 17.8024 9 25-9673 18.0642 10 26.7254 18.326 11 27.4944 18.5878 6 ft, 28.2744 18.8496 I 29.0653 I9.III4 2 29.867 I 9-3732 3 30.6797 I 9-635 4 3 I -5033 19.8968 5 32.3378 20.1586 6 33 -I 83 I 20.4204 7 34.0394 20.6822 8 34.9067 20.944 9 35-7848 21.2058 10 36.6738 21.4676 11 37-5738 21.7294 7 /<- 38.4846 21.9912 1 39.4064 22.253 2 40.339 22.5148 3 41.2826 22.7766 4 42.2371 23.0384 5 43.2025 23.3002 6 44.1787 23.562 7 45 -I 659 23.8238 8 46.164.I 24.0856 9 47 .I 73 I 24-3474 10 48.193 24.6092 11 49.2238 24.871 8 ft. 50.2656 25.1328 1 5 I- 3 I 83 25-3946 2 52.3818 25.6564 3 53.4563 25.9182 4 54 - 54 I 7 26.18 5 55.638 26.4418 6 56.7451 26.7036 7 57.8632 26.9654 8 58.9923 27.2272 9 60.1322 27.489 10 61.283 27.7508 11 62.4448 28.0126 9 ft' 63.6174 28.2744 1 64.801 28.5362 2 65.9954 28.798 3 67.2008 29.0598 4 68.417 29.3216 5 69.6442 29-5834 Diam. Area. ClRCUM. 6 Feet. 70.8823 Feet. 29.8452 7 7 2 -i 3 I 4 3O.IO7 8 73.3913 30.3688 9 74.6621 30.6306 10 75-9439 30.8924 11 77-2365 31.1542 10 ft. 78.54 31.416 1 79-8545 31.6778 2 81.1798 3 I -9396 3 82.5161 32.2014 4 83-8633 32.4632 5 85.2214 32.725 6 86.5903 32.9868 7 87.9703 33.2486 8 89.3611 33-5104 9 90.7628 33.7722 10 92.1754 34-034 11 93-599 34-2958 11ft. 95-0334 34-5576 1 96.4787 34.8194 2 97-935 35.0812 3 99.4022 35-343 4 100.8803 35.6048 5 102.3693 35.8666 6 103.8691 36.1284 7 105.38 36.3902 8 106.9017 36.652 9 108.4343 36.9138 10 109.9778 37 -I 756 11 in -5323 37-4374 12 ft. 113.0976 37.6992 1 114.6739 37.961 2 116.261 38.2228 3 117.8591 38.4846 4 119.468 38.7464 5 121.088 39.0082 6 122.7187 39-27 7 124.3605 39 - 53 I 8 8 126.0131 39-7936 9 127.6766 40.0554 10 129.351 4 °- 3 I 7 2 11 131.0366 40.579 13 A 132.7326 40.8408 1 134.4398 41.1026 2 136.1578 41.3644 3 137.8868 41.6262 4 139.6267 41.888 5 I 4 I -3774 42.1498 6 I 43 - I 39 I 42.4116 7 I 44 - 9 II 7 42.6734 8 146.6953 42.9352 9 148.4897 43-197 10 150.295 43-4588 11 152.1113 43.7206 Y \ J >1 AM. I ft- I 2 3 4 5 6 7 8 9 io ii 5 ft* i 2 3 4 5 6 7 8 9 io ii 6/*. i 2 3 4 5 6 7 8 9 io ii iT ft. i 2 3 4 5 6 7 8 9 io ii i8 ft. i 2 3 4 5 IE AS AND CIRCUMFERENCES OF CIRCLE! Feet. 1539384 I55-7764 157.6254 I59-4853 161.3561 163.2378 165.1303 167.0338 168.9483 170.8736 172.8098 I 74-7569 i 7 6 - 7 i 5 178.684 180.6638 182.6546 184.6563 186.6689 188.6924 190.7267 192. 7721 194.8283 196.8954 198.9734 201.0624 203.1622 205.273 207.3947 209.5273 211.6707 213.8252 215.9904 218.1667 220.3538 222.5518 224.7607 226.9806 229.2113 23 I -453 233-7056 235.9691 238.2434 240.5287 242.8249 245. 132 1 247.4501 249-779 252.1188 254.4696 256.8312 259.2038 261.5873 263.9817 266.3869 ClRCUM. Diam. Area. Feet. Feet. 43.9824 6 268.8031 44.2442 7 271.2302 44.506 8 273.6683 44.7678 9 276.II72 45.0296 IO 278.577 45.2914 11 281.0477 45-5532 19 .A 283.5294 45-815 1 286.0219 46.0768 2 288.5255 46.3386 3 291.0398 46.6004 4 293.5651 46.8622 5 296.IOI2 47.124 6 298.6483 47-3858 7 3OI.2064 47-6476 8 303-7753 47.9094 9 306.3551 48.1712 IO 308.9458 48.433 11 3 H -5475 48.6948 20 j 65 . 314.16 48.9566 1 316.7834 49.2184 2 319.4178 49.4802 3 322.0631 49.742 4 324 - 7 I 93 50.0038 5 327.3864 50.2656 6 330.0643 50-5274 7 33 2 . 753 2 50.7892 8 335-4531 51-051 9 338.1638 51.3128 IO 340.8854 5 I -5746 11 343.618 51.8364 21 ft. 346.3614 52.0982 1 349 -H 57 52-36 2 351.881 52.6218 3 354-6572 52.8836 4 357-4442 53- I 454 5 360.2422 53 - 407 2 6 363.0511 53-669 7 365.8709 53 - 93 o 8 8 368.7017 54.1926 9 371-5433 54-4544 IO 374-3958 54.7162 i - 11 377-2592 54-978 ! 22 ft. 380.1336 55-2398 1 383.0188 ! 55 - 5 oi 6 2 385 - 9 I 5 1 55-7634 3 388.8221 56.0252 4 391-74 56.287 5 394.6689 I 56.5488 6 397.6087 56.8106 7 400.5594 57.0724 8 403.521 1 | 57-3342 9 406.4936 ! 57 596 IO 409.477 l 57-8578 11 412.4713 AREAS AND CIRCUMFERENCES OF CIRCLES. 255 Diam. Area. ClRCUM. Feet. Feet. 2 3 A 415.4766 72.2568 I 418.4927 72.5186 2 421.5198 72.7804 3 424-5578 73.0422 4 427.6067 73-304 5 4.IO.6664 73-5658 6 433-7371 73.8276 7 436.8187 74.0894 8 439-91 74-3512 9 443.0147 74 - 6 i 3 IO 446.129 74.8748 ii 449.2542 75.1366 24 A 452.3904 75-3984 I 455-5374 75.6602 2 458.6954 75.922 3 461.8643 76.1838 4 465.044 76.4456 5 468.2347 76.7074 6 47 I -4363 76.9692 7 474.6488 77.231 8 477-872 3 77.4928 9 481.1066 77-7546 IO 484.3518 78.0164 ii 487.6076 78.2782 25 A 490.875 78.54 1 494.1529 78.8018 2 497.4418 79.0636 3 500.7416 793254 4 504.0523 79.5872 5 507-3738 79.849 6 510.7063 80.1108 7 514.0485 80.3726 8 517.404 80.6344 9 520.7693 80.8962 IO 524.1454 81.158 ii 527.5324 81.4198 26 ft. 530.9304 81.6816 1 534-3397 81.9434 2 537-759 82.2052 3 541.1897 82.467 4 544 - 63 I 3 82.7288 5 548.0837 82.9906 6 55 I- 547 I 83.2524 7 555-0214 83.5142 8 558.5066 83.776 9 562.0028 84.0378 IO 565.5098 84.2996 11 569.0277 84.5614 27 A 572.5566 84.8232 1 576.0963 85.085 2 579.6467 85.3468 3 583.2086 85.6086 4 586.781 85.8704 5 590.3644 86.1322 Diam. Area. ClRCUM. Feet. Feet. 6 593-9587 86.394 7 597-5639 86.6558 8 60I.18 86.9176 9 604.8071 87.1794 10 608.445 87.4412 11 612.0938 87.703 to 00 > 6 I 5-7536 87.9648 1 619.4242 88.2266 2 623.IO58 88.4884 3 626.7983 88.7502 4 630.5016 89.OI2 5 634.2159 89.2738 6 637.94II 89-5356 7 641.6772 89.7974 8 645.4243 90.0592 9 649.1822 90.321 10 652.951 90.5828 11 656.7307 90.8446 29 A 660.5214 91.1064 1 664.3229 91.3682 2 668.1354 9 t - 6 3 3 671.9588 91.8918 4 675 - 793 I 92.1536 5 679.6382 92.4154 6 683.4943 92.6772 7 687.3613 92.939 8 691.2393 93.2008 9 695.1281 93.4626 10 699.0278 93-7244 11 702.9384 93.9862 2>o ft- 706.86 94.248 1 710.7924 94.5098 2 7 I 4-7358 94.7716 3 718.6901 95-0334 4 722.6553 95.2952 5 726.6313 95-557 6 730.6183 95.8188 7 734.6162 96.0806 8 738.6251 96.3424 9 742.6448 96.6042 10 746.6754 96.866 11 750.7164 97.1278 3 l/^ 754.7694 97.3896 1 758.8327 97.6514 2 762.907 97.9132 3 766.9922 98.175 4 771.0883 98.4368. 5 775-1952 98.6986 6 779 - 3 I 3 I 98.9604 7 783.4419 99.2222 8 787.5817 99.484 9 79 I -7323 99-7458 10 795.8938 100.0076 11 800.0662 100.2694 256 AREAS AND CIRCUMFERENCES OF CIRCLES. Diam. Area. ClRCUM. Diam. 32 ft. Feet. 804.2496 Feet. IOO.5312 6 I 808.4439 IOO.793 7 2 812.649 IOI.0548 8 3 816.8651 IOI.3166 9 4 ’ 821.092 IOI.5784 10 5 825.3299 IOI.8402 11 6 829.5787 102. 102 37 A 7 833-8384 IO2.3638 1 8 838.IO9I IO2.6256 2 9 842.3906 IO2.8874 3 10 846.683 I 03 -I 49 2 4 11 850.9863 IO3.4I I 5 33 ft: 855-3006 IO3.6728 6 I 859.6257 I 03 - 934 6 7 2 863.9618 104.1964 8 3 868.3088 104.4582 9 4 872.6667 104.72 10 5 877-0354 104.9818 11 6 881.4151 105.2436 38 A 7 885.8057 105-5054 1 8 890.2073 105.7672 2 9 894.6197 106.029 3 10 899.043 106.2908 4 11 903.4772 106.5526 5 34 A 907.9224 106.8144 6 1 912.3784 107.0762 7 2 916.8454 107.338 8 3 9 2I - 3 2 33 107.5998 9 4 925.812 107.8616 10 5 930.31 1 7 108.1234 11 6 934.8223 108.3852 39 A 7 939-3439 108.647 1 8 943.8763 108.9088 2 9 948.4196 109.1706 3 10 952.9738 109.4324 4 11 957 - 539 2 109.6942 5 35 A 962.115 109.956 6 1 966.7019 110.2178 7 2 971.2998 110.4796 8 3 975.9086 1 10.7414 9 4 980.5287 hi .0032 10 5 985.1588 111.265 11 6 989.8005 111.5268 40 A- 7 994-4527 111.7886 1 8 999.116 1 1 2.0504 2 9 1003.7903 112.3122 3 . 10 1008.4754 112.574 4 11 1013.1714 112.8358 5 3 6 A 1017.8784 113.0976 6 1 1022.5962 1 13-3594 7 2 1027.325 113.6212 8 3 1032.0647 113.883 9 4 1036.8153 114.1448 10 5 1041.5767 114.4066 11 ClRCUM. Feet. IO46.3491 IO51.1324 IO55.9266 1060.7318 1065.5478 IO7O.3747 1075.2126 1080.0613 1084.921 1089.7916 IO94.673I 1099.5654 I IO4.4687 IIO9.3829 III4.308 III9.244I II24.I9I II29.I489 1 134. H 76 1139.0972 II44.0878 II49.0893 II54.IOI7 H59* I2 49 1164.1591 1169.2042 1174.2603 n 79.3272 1184.405 1189.4937 H94-5934 1199.7039 1204.8254 1209.9578 1215.101 1220.2552 1225.4203 1230.5963 i235-7 8 33 1240.9811 1246.1898 1251.4094 1256.64 1261.8814 1267.1338 1272.3971 1277.6712 1282.9563 1288.2523 I293-559 2 1298.877 1304.2058 i3°9-5454 i 3 i 4-8959 Feet. II4.6684 1 14.9302 II5.I92 II 5-4538 II 5 - 7 I 56 H 5-9774 116.2392 116.501 116.7628 117.0246 117.2864 117.5482 117.81 118.0718 118.3336 118.5954 118.8572 119.119 119.3808 119.6426 1 19.9044 120.1662 120.428 120.6898 120.9516 121.2134 121.4758 121.737 121.9988 122.2606 122.5224 122.7848 123.046 123.3078 123.5696 123.8314 124.0932 124-355 124.6168 124.8786 125. 1404 125.4022 125.664 125.9258 126.1876 126.4494 126.7112 126.973 127.2348 127.4966 127.7584 128.0202 128.282 128.5438 A i 3 4 5 6 7 8 9 io ii A i 2 3 4 5 6 7 8 9 io ii A i 2 3 4 5 6 7 8 9 io ii A i 2 3 4 5 6 7 8 9 io ii A i 2 3 4 5 AND CIRCUMFERENCES OF CIRCLES. 257 Area. Feet. 132 O .2574 1325.6297 I 33 I - OI 3 I 33^*4°7 2 1341.8123 1347.2282 1352.6551 1358.0929 1363-5416 1369.0013 i374-47i8 1379-9532 I385-4456 1390.9488 1396.463 1401.9881 1407.5241 1413.0709 1418.6287 1424.1974 1429.777 1435-3676 1440.969 1446.5813 1452.2046 1457.8387 1463-4838 1469.1398 1474.8066 1480.4844 1486.1731 1491.8717 1497.5833 1503.3047 1509037 1514.7802 1520.5344 1526.2994 1532.0754 1537.8623 1543.66 1549.4687 1555-2883 1561.1188 1566.9603 1572.8126 15786756 1584.5499 1590.435 1596.3309 1602.2378 1608.1556 1614.0843 1620.0238 ClRCUM. Feet. 128.8056 129.0674 I 29.3292 129.591 129.852 8 130.1146 130.3764 130.6382 130.9 131.1618 131.4236 131.6854 I 3 I -9472 132.209 132.4708 132.7326 132.9944 133.2562 i33-5i8 1 33 - 7798 1 34 - ° 4 I 6 134- 3034 134.5652 134.827 135.0888 i V 35-35 o 6 135.6124 135- 8742 136.136 136.3978 136.6596 136.9214 137.1832 137 - 445 137.7068 137.9686 138.2304 138.4922 138- 754 139.0158 139.2776 139 - 5394 139.8012 140.063 140.3248 140.5866 140.8484 141.1102 I4I-372 141.6338 141.8956 142.1574 142.4192 142.681 Diam. Area. ClRCUM. Feet. Feet. 6 1625.9743 142.9428 7 1631.9357 143.2046 8 1637.9081 143.4664 9 1643.8913 143.7282 10 1649.8854 143-99 11 1655.8904 144.2518 46 A 1661.9064 144.5136 1 1667.9332 144-7754 2 1673.971 145.0372 3 1680.0197 145.299 4 1686.0792 145.5608 5 1692.1497 145.8226 6 1698.2311 146.0844 7 1704.3195 146.3462 8 I 7 IO .4267 146.608 9 1716.5408 146.8698 10 1722.6658 147.1316 11 1728.8017 147-3934 47 A- 1734.9486 147.6552 1 1741.1063 147 917 2 1747.275 148.1788 3 I 753-4546 148 . 44.06 4 1 759 . 645 1 148.7024 5 1765.8464 148.9642 6 1772.0587 149.226 7 1778.2819 149.4878 8 1784.516 149.7496 9 1790.7611 1 50.01 14 10 I 797 -OI 7 150.2732 11 1803.2838 150.535 48 A- 1809.5616 150.7968 1 1815.8502 151.0586 2 1822.1498 1 5 1. 3204 3 1828.4603 151.5822 4 1834.7817 151.844 5 1841.1139 152.1058 - 6 1847.4571 152.3676 7 1853.8112 152.6294 8 1860.1763 152.8912 9 1866.5522 I 53 -I 53 IO 1872.939 153.4148 11 1879.3367 153 6766 49 A 1885.7454 153-9384 1 1892.1649 154.2002 2 1898.5954 154.462 3 1905.0368 154.7238 4 1911.4897 154.9856 5 1917.9522 155 . 2474 . 6 1924.4263 155-5092 7 I 93 °- 9 II 3 I 55 - 77 I 8 I 937-4073 156.0328 9 I 943 - 9 I 4 2 156.2946 IO 1950.4318 156.5564 11 1956.9604 156.8182 50 A I963-5 i 57 -o 8 258 SIDES OF SQUARES EQUAL TO AREAS. Sides of Squares— equal in. Area to Diameter from 1 to 100 . Side of Sq. )iam. Side of Sq. Diam. S I .8862 14 ] X I.IO78 X 3 X I-3293 X ] % 1.5509 % 2 I.7724 15 X I.994 X X 2.2156 X % 2.4371 % 3 2.6587 16 2.8802 X 3.IO18 X X 3-3233 % 4 3-5449 17 X 3-7665 X X 3.988 X % 4.2096 % 5 4-43 11 18 X 4.6527 X X 4.8742 X % 5-0958 % 6 5-3 T 74 X 5.5389 X X 5-7605 X % 5.982 % 7 6.2036 20 X 6.4251 X X 6.6467 X % 6.8683 % 8 7.0898 21 X 7-3 TI 4 X X 7-5329 X X 7-7545 % 9 7.976 22 X 8.1976 X X 8.4192 X % 8.6407 % 10 8.8623 23 X 9.0838 X X 9-3054 X % 9.5269 H II 9-7485 24 X 9-97 X X 10.1916 X % 10.4132 X 12 10.6347 25 X 10.8563 X X 11.0778 X % 11.2994 % 13 11.5209 26 X 11.7425 X X 11.9641 X % 12.1856 % 14.4012 14.6227 14.8443 15.0659 15.2874 I5-509 I5-7305 15-9521 16.1736 16.3952 16.6168 16.8383 17.0599 17.2814 I7-503 17.7245 17.9461 18.1677 18.3892 18.6108 18.8323 19-0539 19.2754 19.497 19.7185 19.94OI 20.1617 20.3832 20.6048 20.8263 21.0479 21.2694 21.491 21.7126 21.9341 22.1557 22.3772 22.5988 22.8203 23.0419 23.2634 23-485 23.7066 27 28 29 X 30 X X % I ¥ 32 3^ 33^ 34 X 35 36 X 37 38 3^ X % 39 x % 23.9281 24.1497 24.37 12 24.5928 24.8144 25-0359 25-2575 25-479 25.7006 25.9221 26.1437 26.3653 26.5868 26.8084 27.0299 27-2515 27-473 27.6946 27.9161 28.1377 28.3593 28.5808 28.8024 29.0239 29- 2455 29.467 29.6886 29.9102 30.1317 30- 3533 30.5748 30.7964 31.0179 31- 2395 31.4611 31.6826 31.9042 32.1257 32- 3473 32.5688 32.7904 I 33 0112 33- 2335 33- 4551 33.6766 33.8982 34- H97 34- 3413 34.5628 34.7884 35.006 35- 2275 a Circle. Diam. Side of Sq. 40 35-449 1 X 35.6706 X 35.8922 % 36-1137 41 36.3353 X 36.5569 X 36.7784 42 3€ 43 % 44 X X % 45 46 X % 47 X 49 X X % X X % 50 X 51 X X % 52 X 37 37-2215 37-4431 37.6646 37.8862 38.1078 38.3293 385509 38.7724 38.994 39-2155 39-4371 39- 6 587 39.8802 40.1018 40.3233 40.5449 40.7664 40.988 41.2096 4 I -43 11 41.6527 41.8742 42.0958 42.3173 42.539 42.7604 42.982 43.2036 43- 425I 43.6467 43.8682 44.0898 44- 3H3 44-5329 44-7545 44.976 45.1976 45.4191 45.6407 45.8622 46.0838 46-3054 46.5269 46-7485 SIDES OF SQUARES EQUAL TO AREAS. Diam. Side of Sq. 1 Diam. Side of Sq. Diam. Side of Sq. I Diam 53 w 46.97 65 57.6047 77 68.2395 89 34 A 47.1916 A 57.8263 X 68.461 34 47 - 4 I 3 I 34 58.0479 A 68.6826 3^ 54 47-6347 54 58.2694 % 68.9041 54 54 47.8562 66 58.491 78 69.1257 90 Va a 48.0778 X 58.7125 34 69.3473 a 48.2994 A 58.9341 A 69.5688 7± 1/ % 48.5209 % 59-1556 54 69.7904 72 55 48.7425 67 59-3772 79 7O.OII9 /4 a 48.964 X 59-5988 34 70.2335 91 a 49 - 1 856 A 59.8203 X 70.455 A 54 49.4071 54 60.0419 % 70.6766 Ai 56 49.6287 68 60.2634 80 70.8981 54 3€ 49-8503 % 60.485 34 7 I - II 97 92 3? 50.0718 A 60.7065 34 7 I - 34 I 3 3i % 50.2934 % 60.9281 Si 71.5628 34 57 w 50 . 5 H 9 69 61.1497 81 7 x - 7844 54 A 50.7365 * 61.3712 A 72.0059 A 50958 61.5928 A 72.2275 34 54 51.1796 h 61.8143 % 72.4491 34 58 51.4012 70 62.0359 82 72.6706 54 A 51.6227 1 « 62.2574 34 72.8921 94 3? 5 I -8443 i x 62.479 X 73- II 37 % 52.0658 i 54 62.7006 Si 73-3353 A \/ 59 52.2874 71 62.9221 83 73-5568 A 3? A 52.5089 34 63 -I 437 34 73.7784 3 1 52.7305 x 63-3652 X 73-9999 95 54 52.9521 , % 63.5868 Si 74.2215 A 60 53 -I 736 72 63.8083 84 74-4431 A A 53-3952 A 64.0299 34 74.6647 % 3? 53-6167 A 64.2514 A * 74.8862 96 5i 53-8383 54 64.4730 % 75.1077 X 61 54-0598 73 64.6946 85 75-3293 34 a 54.2814 ! 34 64.9161 A 75 . 55 o 8 54 3 ? % 54-503 1 54-7245 X 54 65.1377 65.3592 A % 75.7724 75-9934 97 1 / 62 54.9461 74 65.5808 86 76.2155 /4r 34 a 55.1676 34 65.8023 A 76.4371 54 a 55-3892 A 66.0239 A 76.6586 % 55 6107 | % 66.2455 % 76.8802 98 63 55.8323 75 66.467 87 77.1017 A 1/ a a 56.0538 56.2754 34 34 66 6886 66.9104 A A 77-3233 77-5449 A 54 54 56.497 Si 67.1317 % 77.7664 99 64 56.7185 76 67-3532 88 77.988 A A 56.9401 A 67.5748 34 78.2095 A 3? 57.1616 A | 67.7964 A 78.4316 I 54 54 57-3832 i! Si 1 68.0179 % 78.6526 j 100 Application of Table. To Ascertain, a Square tliat Las same Area a Circle. Il l us.— I f side of a square that has same area as a circle of 7-3 2s i By Table of Areas, page 233, opposite to 73.25 is 4214.11; and i 04.9161, which is side of a square having same area as a circle of tha 259 Side of Sq. 78.8742 79-0957 79-3173 79-5389 79.7604 79.982 80.2035 80.4251 80.6467 80.8682 81.0898 81.3113 81.5329 81.7544 81.976 82.1975 82.4191 82.6407 82.8622 83.0838 83-3053 83.5269 83.7484 83-97 84.1916 84.4131 84.6347 84.8562 85.0778 85-2993 85.5209 85-7425 85.9646 86.185 86.4071 86.6289 86.8502 87.0718 87-2933 87-5449 : 87.7364 ! 87.958 I 88 1796 | 88.4011 I 88.6227 a Griven i. is required, this table is diameter. 26 o LENGTHS OF CIRCULAR ARCS. Lengths of Circular Arcs, up to a Seiuicircle. Diameter of a Circle — i, and divided into 1000 equal Parts. H’ght. Length. H’ght. Length. | I rght.; Length. 1 1 H’ght. Length. I I’ght.; I .1 I.O2645 •15 I.O58Q6 .2 1 : [.IO348 -25 I - 159 12 - 3 1 .IOI I.O2698 •151 I.05973! .201 : 1. 104 47 .251 I.16033 . 3 QI 1 .102 I.O27 52 .152 I.06051 I .202 1.105 48 .252 I.16157 . ,302 1 .103 I.O2806 •153 I.061 3 .203 I.I065 -253 I.16279 . •303 3 .IO4 1.0286 •154 1 .062 09 .204 1.107 52 •254 I.16402 ■304 3 .105 I.O29 14 .155 1.062 88 .205 I.IO855 -255 I.16526 .305 3 .106 I.O297 .156 1 .063 68 .206 I.IO958 .256 I.16649 .306 3 .107 1 .030 26 .157 1.06449 .207 I.II062 -257 I.167 74 •307 ] .IO8 1 .030 82 .158 1.0653 .208 I. II165 .258 I.16899 .308 ] .IO9 1.03139 -159 1.066 11 .209 I.II269 -259 I.17O 24 .309 : .11 I.03196 .16 1.06693 .21 I.II 374 .26 I-I 7 I 5 • 3 1 .III I.O32 54 .l6l 1.067 75 .211 I.II 479 .261 1. 17275 • 3 33 .112 I.O33 12 .162 1 .068 58 .212 I.II584 .262 1.17401 • 3 12 • II 3 1-033 7 1 .163 1.06941 -213 ! I.I1692 .263 1.17527 • 3 3 3 .114 1.0343 .164 1.07025 .214 I.II796 .264 1.17655 • 3 3 4 .115 1.0349 .165 1. 07 1 09 .215 I.II904 .265 I.17784 • 3 3 5 .Il6 1-03551 .166 1. 07 1 94 .216 1 .120 1 1 .266 I.I 79 I 2 • 3 l6 -II 7 1.03611 .167 1.072 79 .217 I.I 2 I l8 .267 I.1804 • 3 3 7 .118 1 .036 72 .168 1-073 65 .218 1 .122 25 .268 I.18162 .318 .119 1-037 34 .169 1.074 51 .219 ! 1.123 34 .269 I.182 94 • 3 3 9 .12 1.03797 •17 1-075 37 .22 1.12445 .27 I.18428 •32 .121 1.0386 .171 1.076 24 .221 1. 125 5 6 .271 I.18557 .321 .122 1.03923 .172 1.077 11 .222 1.12663 .272 1.186 88 .322 .123 1.03987 •173 1.07799 .223 1. 12774 •273 1. 188 19 •323 .124 1.040 51 .174 1.07888 .224 1.12885 •274 1.18969 •324 .125 1. 041 16 • 175 1.07977 .225 1. 129 97 .275 1.19082 .325 .126 1.04 1 81 .176 1.08066 .226 1.13108 .276 1. 192 14 .326 .127 1.04247 .177 1. 08 1 56 .227 1. 132 19 •277 I-I 9345 •327 .128 1-04313 .178 1.082 46 .228 II 333 I .278 1.19477 .328 .129 1.0438 .179 1-08337 .229 1.13444 .279 1.196 1 .329 .13 1.04447 .18 1 .084 28 •23 1 . 13557 .28 1 . 197 43 •33 •I 3 1 1-045 15 .l8l 1.085 19 .231 1.1367 1 .281 1.19887 • 33 1 .132 1.045 84 .182 1.086 11 .232 1.13786 .282 1.200 11 .332 • 1.33 1.04652 .183 1.08704 •233 I-I 3903 .283 1. 201 46 •333 • I 34 1.04722 .184 1.08797 •234 1. 140 2 .284 1.202 82 •334 • I 35 1 .047 92 .185 1 .088 9 •235 1.14 1 36 .285 1.204 19 -335 .136 1.048 62 .186 1.089 84 .236 1. 142 47 .286 1.205 58 •336 .137 1.04932 .187 1.090 79 •237 i-i 43 6 3 .287 1.20696 •337 .138 1.05003 .188 1. 091 74 .238 1.1448 .288 1.208 28 •338 .139 1-05075 .189 1.09269 •239 I-I 4597 .289 1.20967 •339 .14 1. 05 1 47 .19 1.09365 .24 I-I 47 I 4 .29 1. 21202 •34 .141 1 .052 2 .191 1.09461 .241 1.14831 .291 1.21239 • 34 1 .142 1.05293 , .I 9 2 1-095 57 .242 1. 149 49 ► .292 1.21381 ■342 • i 43 1.05367 •193 1.09654 •243 1.15067 ’ -293 i 1.2152 •343 .144 1.054 41 .194 1-09752 .244 1.15186 > .294 . 1.21658 •344 .145 1.05516 * -195 1.0985 -245 I-I 53 > .295 : 1.21794 *345 .146 • 1-05591 .196 1 1.09949 1 .246 • I-I 542 S ) .296 ) 1.21926 •346 .147 ' 1.05667 ' I -197 1.10048 1 - 2 47 I-I 554 S ) .297 1.220 01 •347 .148 1 1-05745 ! -198 ; 1.10147 .248 ■ 1.1567 .298 1.22203 •348 .145 ) 1.0581c > .195 ) 1. 102 47 - .245 ► 1 - I 579 ] [ .29911.22347 •349 1.24946 1.25095 1.25243 1-25391 1-25539 1.256 86 1.25836 1.25987 1.26137 1.262 86 1.26437 1.265 88 1.2674 1.268 92 1.27044 1. 271 96 1.27349 1.27502 | 1.27656 1.2781 ! 1.27964 1.281 18 1.282 73 1.284 28 1.285 83 1.28739 1.288 95 1.29052 1.29209 1.29366 1.295 2 3 1.29681 1.29839 LENGTHS OF CIRCULAR ARCS. 26l H’ght. | Length. H’ght Length. H’ght Length. H’ght. Length. H’ght. Length. •35 I.29997 .38 1.34899 • 4 i I.4OO77 •44 1-455 12 •47 I.5H85 • 35 i I.301 56 .381 1.35068 .411 I.402 54 .441 1.45697 .471 I. 5 I 3 78 •352 I- 303 I 5 .382 1-35237 .412 I.40432 •442 1.45883 •472 i. 5 x 5 7 i •353 I.30474 .383 1.35406 • 4 i 3 I.406 I •443 1.46069 •473 I - 5 I 7 64 •354 I.30634 •384 i -355 75 • 4 r 4 I.40788 •444 1.462 55 •474 1.51958 •355 I.30794 •385 1-35744 • 4 i 5 I.40966 •445 1.46441 •475 1.521 52 •356 1-30954 •386 I- 359 I 4 .416 I.41145 •446 r. 466 28 •476 1.52346 •357 I- 3 III 5 •387 1.36084 .417 I.41324 •447 1.46815 •477 1 .525 41 •358 I.31276 .388 1.36254 .418 I- 4 I 503 •448 1.47002 .478 1.52736 •359 I- 3 I 437 •389 1-36425 .419 I.41682 •449 1.47189 •479 1.52931 •36 I- 3 I 599 •39 1.36596 .42 I.41861 •45 1 -473 77 .48 I -53 I 26 .361 i - 3 i 7 6 i • 39 i 1.36767 .421 I.42041 • 45 i 1 -475 65 .481 I -533 22 .362 1-31923 •392 I -36939 .422 1.422 22 •452 1 -477 53 .482 i .535 i 8 •363 1.32086 •393 1.37111! •423 I.424O2 •453 1.47942 •483 I -537 14 •364 1.32249 •394 1.37283 •424 I.425 83 •454 1.48131 •484 1-539 1 •365 1-32413 •395 1 - 37455 ; •425 I.42764 •455 1.4832 •485 1. 541 06 .366 I -325 77 •396 1.37628: .426 I.42945 •456 1.48509 .486 1.54302 •367 1-32741 •397 1.37801 .427 1. 431 27 •457 1.48699 .487 1-544 99 .368 1.32905 •398 1-379 74 .428 I -43309 •458 1.488 89 .488 1.54696 •369 1.33069 •399 1.38148 •429 I.434 91 •459 1-490 79 .489 •49 1.54893 I - 55°9 •37 I -332 34 •4 1.38322 •43 i -436 73 .46 1.49269 • 49 1 1.552 88 • 37 i 1-33399 .401 1.38496 • 43 i 1.43856 .461 1.4946 •492 1.554 86 •372 I -33564 .402 1.38671 •432 1.44039 .462 1.49651 •493 i -556 85 •373 1-3373 •403 1.388 46 •433 1.442 22 •463 1 .498 42 •494 I .558 54 •374 1-33896 ; •404 1.39021 •434 1.44405 •464 1 -5oo 33 •495 1.560 83 •375 1-34063 •405 I - 39 1 96 •435 1.445 89 •465 1.502 24 •496 1.562 82 •376 1.34229 .406 1-393 72 •436 1-447 73 .466 1.50416 •497 1.56481 •377 1 -343 96 ! •407 1-39548 •437 : 1-44957 •467 1 .506 08 •498 1.5668 •378 1-34563 ! .408 1.39724 •438 I- 45 I 42 .468 1.508 •499 1.568 79 •379 I *347 3 1 ; •409 !-399 •439 i i- 45327 ll •469 1.50992 •5 1 1-570 79 To ^Ascertain Length, of an TAro of 1 a Circle by pre- ceding Table. Rule.— D ivide height by base, find quotient in column of heights, take length for that height opposite to it in next column on the right hand. Multiply length thus obtained by base of arc, and product will give length. Example.— What is length of an arc of a circle, base or span of it being 100 feet and height 25? ’ 25 100 = .25; and .25, per table, = 1. 159 12, length of base, which , multiplied by 100 = 115.912 feet. ’ * J When , in division of a height by base , the quotient has a remainder after third place of decimals , and great accuracy is required . Rule.— T ake length for first three figures, subtract it from next following length ; multiply remainder by this fractional remainder, add product to first length, and sum will give length for whole quotient. Example.— What is length of an arc of a circle, base of which is « feet and height or versed sine 8 feet? * ’ ,Jtj 5= "‘ 22 * 5714 '’ ^butor length for .228 = 1.13331, and for .229 = !. 13444 the difference between which is .001 13. Then .5714 x .001 13 = .000645682. Hence .228 = 1. 13331", ■ and .0005714= .000645682 „„„ ^ 7 . , , 1-133955682, the sum by which base of arc is to be multiplied ; and 1. 133 955 682 X 35 = 39.688 45 feet . J frees. | I 2 3 4 5 6 7 8 9 io ii 12 13 14 i5 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3° 3i 32 33 34 35 36 37 38 39 40 41 42 43 44 45 r r« Ru )ly it Exj OF CIRCULAR ARCS. ►F Circular .Arcs from 1° to 1£ (Radius = 1 .) sgrees. Length. | j Degrees. Length. Degrees. \ 46 .8028 i 9 1 I .5882 136 47 .8203 92 I .6057 137 48 •8377 93 I. 623 I 138 49 •8552 94 I .6406 139 50 51 52 53 54 .8727 .8901 1 .9076 •925 .9424 ! 95 96 97 98 99 I .6581 1.6755 I .693 I. 7 IO 4 I .7279 140 141 142 143 144 55 •9599 ! 100 1*7453 145 56 •9774 : : IOI I .7628 146 57 .9948 j 102 I .7802 147 58 1. 0123 j 103 1.7977 148 59 1.0297 ! 104 I .8151 149 60 1.0472 j 105 106 I .8326 I .85 150 61 62 1 .0646 1.0821 10 7 108 I .8675 I .8849 1 5 1 152 63 64 1.0995 1.117 109 I .9024 1 53 154 65 1 -1345 j no I -9 I 99 155 66 1. 1519 ! ill 1-9373 156 67 1.1694 I 1 12 1.9548 157 68 1. 1868 113 1.9722 158 69 1.2043 114 1.9897 159 1. 2217 115 2.0071 160 70 Il6 2.0246 161 71 72 1.2392 k 1.2566 117 Il8 2.042 2.0595 162 16-2 73 74 1.2741 1.2915 119 2.0769 164 75 1.309 120 2.0944 , 165 76 1.3264 121 2.1118 166 77 1-3439 122 2.1293 1 167 78 i- 3 6i 3 123 2.1467 168 79 1.3788 124 2.1642 169 80 81 82 83 84 i-39 6 3 I-4I37 I-43I2 1.4486 1 .4661 125 126 127 128 129 2.1817 2. 1991 2.2166 2.2304 2.2515 j 170 171 ! J 72 173 174 85 1-4835 130 2 . 26 S 9 | 175 86 1 . 5 01 131 2.2864 ; 176 87 1.5184 132 2.3038 ; i77 88 1-5359 133 2.3213 1 89 1-5533 134 2.3387 179 90 1.5708 135 2.3562 180 .11. Length, of a Circular Arc by )lumn opposite to degrees of arc, take length, circle. ;r of degrees in an arc are 45 0 , and diameter of cir< igthx 54-2 = !. 9635/^. LENGTHS OF ELLIPTIC ARCS. 263 Lengths of Elliptic Arcs. Up to a Semi-ellipse. Transverse Diameter = 1, and divided into 1000 equal Parts. H’glit. 1 Length. 'H’ght Length. H’ght Length. H’ght Length. H’ght .1 Length. .1 1. 04 1 62 •15 1-0933 .2 1. 150 14 •25 1. 21 1 36 •3 I.27669 .IOI I.O42 62 •151 1.09448 I .201 I-I 5 I 3 I .251 I.21263 .301 I.27803 .102 I.O4362 •152 1.09558 .202 I.15248 .252 1. 2139 .302 I - 2 7937 .103 1 .044 62 •153 1.09669 I -203 I.15366 •253 I.21517 •303 1.280 71 .104 I.O45 62 1 -154 1.0978 .204 I.15484 •254 I.21644 •304 1.28205 .105 I.O4662 •155 1.09891 .205 I.15602 •255 1. 217 72 •305 1.28339 .106 1 .047 62 .156 1. 100 02 .206 1 . 1572 .256 1. 219 .306 1.284 74 .107 I.O4862 •157 1. 101 13 .207 1-15838 •257 1.220 28 •307 1.286 09 .108 1.049 62 .158 1. 102 24 .208 I - I 5957 .258 1. 221 56 .308 1.28744 .IO9 1.05063 •159 iio 335 .209 1. 160 76 •259 I.22284 •309 1.288 79 .11 1.05164 .16 1. 104 47 .21 1.161 96 .26 1.224 12 • 3 i 1.290 14 .III 1.05265 .161 1.1056 .211 1.163 *6 .261 1. 2254I • 3 IX 1. 291 49 .112 1.05366 .162 1.10672 .212 1.16436 .262 1.226 7 .312 1.292 85 •113 1.05467 .163 1. 107 84 •213 1-16557 .263 I.22799 • 3 X 3 1.294 21 .114 1.05568 .164 1.10896 .214 1. 166 78 .264 I.22928 • 3 X 4 1-295 57 •115 1.05669 .165 1.11008 •215 1.16799 .265 I.23057 • 3 X 5 1.29603 . Il 6 1-057 7 .166 1. hi 2 . 2 l 6 1.1692 .266 I.23186 • 3 l6 1.298 29 .117 I.O5872 .167 1.11232 .217 1. 1 70 41 .267 1 - 2 33 15 • 3 i 7 1.29965 .118 1.05974 .168 I .H 344 . 2 l 8 I - I 7 1 63 .268 x - 234 45 .318 1. 301 02 .119 1 .060 76 .169 1.11456 .219 1.17285 .269 !-235 75 • 3 X 9 1.30239 .12 ! 1. 061 78 •17 1.11569 .22 1. 17407 .27 1 - 2 37 05 •32 1 -303 76 .121 1.0628 .171 1.11682 .221 I - I 75 29 .271 1 -238 35 •321 1 -305 13 .122 1.06382 .172 I - II 795 .222 1.17651 .272 1.23966 .322 1-3065 .123 1 1.06484 •173 1.11908 .223 I - I 77 74 •273 1.24097 •323 1.30787 .124 1.06586 .174 1.12021 .224 1.17897 •274 1.242 28 •324 1.30924 .125 1.06689 •175 1.12134 .225 1.1802 •275 1 - 2 43 59 •325 1.31061 .126 1.067 92 .176 1. 12247 .226 1.18143 .276 1.2448 .326 1.3H98 .127 1.06895 .177 1.1236 .227 1.18266 •277 1.24612 •327 r - 3 T 335 .128 ; 1.06998 .178 1. 124 73 .228 1.1839 .278 1.24744 .328 I - 3 I 4 72 .I29 1. 070 01 .179 1.12586 .229 1.18514 •279 1.248 76 •329 1.316 1 • X 3 1.07204 .18 1.12699 •23 1.18638 .28 1.250 1 •33 i - 3 i 748 •131 1.07308 .181 1.128 13 .231 1.18762 .281 1.25142 • 33 i 1.31886 .132 1.074 12! .182 1. 12927 .232 1. 188 86 .282 1.25274 •332 1.320 24 • x 33 1.075 16 1 .183 1.13041 •233 1. 190 1 .283 1.254 06 •333 1.321 62 •134 1.07621 .184 I-I 3 I 55 •234 I - I 9 1 34 .284 1-25538 •334 1-32 3 • x 35 1.07726 .185 1.13269 •235 1.19258 .285 1.2567 •335 1.32438 .136 1.07831 .186 I - I 33 83 .236 1.19382 .286 1.25803 •336 1.325 76 • x 37 1.07937 .187 I-I 3497 •237 1.19506 .287 1.25936 •337 1-327 x 5 .138 1 .080 43 .188 1.136 11 .238 1.1963 .288 1.26069 •338 1.328 54 -39: 1. 08 1 49 .189 1.13726 •239 I-I 9755 .289 1.262 02 •339 1 -329 93 .14 1.082 55 .19 1.13841 •24 1.1988 .29 1-26335 •34 I - 33 1 32 .141 1.083 6 2 .191 I - I 39 56 .241 1.20005 .291 1.26468 • 34 x 1.332 72 .142 1.08469 .192 1. 140 71 .242 1. 201 3 .292 1.26601 .342 1 33412 .143 1.085 76 • I 93 1.141 86 •243 1.202 55 •293 1.26734 ■343 1 1 335 52 • x 44 1.08684 .194 1. 143 01 .244 1.2038 •294 1.26867 •344 | 1 33692 • x 45 1.08792 •195 1. 144 16 •245 1.205 °6 •295 1.27 •345 x -33833 .146 1.08901 .196 i-i 453 i .246 1.20632 .296 1.27133 •346 x -339 74 • x 47 1.090 1 .197 1.146 46 .247 1.20758 .297 1.27267 •347 I - 34 1 15 .148 1.091 19 .198 1.14762 .248 1.208 84 .298 1. 27401 •348 1.342 56 .149 1.092 28; .199 1.14888 .249 1. 210 1 .299 i,2 7535 •349 1-34397 264 LENGTHS OF ELLIPTIC ARCS. H’ght. Length. H’ght •35 •35i •352 •353 •354 •355 •356 •357 .358 1-345 39 1.34681 1.34823 i-349 6 5 1.35108 1.352 51 1.35394 1-355 37 _ 1.3568 •359 I 1 *358 23 .36 I 1.35967 .361 I 1.361 11 .362 | 1.36255 .363 | I.36399 •3 6 4i 1.36543 .36s 1.36688 .366 1 1.36833 .367 ! 1-36978 .368 1-371 23 .369 1-37268 1-374 14 1.37662 1.37708 1.378 54 .^,1.38 •375 j 1-38146 .376 1 1.38292 •377 | 1.38439 .378 i 1-38585 •379 i 1 *387 32 .38 U.38879 •37 •37i .372 •373 •374 Length. 1 1 H’ght. I Length. I j H’ght. I Length. | H’ght.j Length. .381 .382 .383 •384 .385 .386 •387 .388 •389 •39 1.39024 I-39 1 69 I-393I4 1-394 59 1.39605 1.397 5i 1.39897 1.40043 1 .401 89 w 1.40335 .391 j 1.40481 .392 ! 1.40627 •393 I -4°7 73 .394; 1.409 19 •395 ! i. 4 io6 5 1.412 11 i*4 I 3 57 1.41504 1.4:1651 1.41798 1.41945 1.420 92 1.42239 1.42386 •396 •397 •398 •399 •4 .401 .402 •403 •404 .405 .406 .407 .408 .409 .41 .411 .412 •4i3 .414 •4i5 .416 .417 .418 .419 .42 .421 .422 .423 .424 .425 .426 •427 .428 .429 •43 •43 1 •432 •433 •434 •435 •436 ■437 •438 ■439 .44 ,441 ,442 ■443 ■444 ■445 .446 ■447 .448 .449 •45 .451 .452 •453 •454 •455 •456 •457 .458 •459 1.425 33 j 1.42681 J 1 1.42829 | 1.42977, I -43 I 25 II 1.432 73 1.434 21 1.435 69 1.43718 1.43867 1.440 16 1.44165 1.443 H 1.44463 1.44613 I.44763 1.449 x 3 1.45064 1.452 14 1.45364 1-455 J 5 1.45665 1.45815 1.45966 1.461 67 1.46268 1 .464 19 1-4657 1.46721 1.468 72 1.47023 1. 471 74 1.47326 1.474 78 1-4763 1.47782 1-47934 1 .480 86 1.482 38 1.48391 1.48544 1.486 97 1.4885 1.49003 I-49 1 57 1.493 1 1 1.49465 1.496 18 1.497 71 1-499 24 1.50077 1.5023 1.50383 1.50536 1.50689 .46 .461 .462 •463 •464 •465 .466 •467 .468 .469 •47 .471 .472 •473 •474 •475 .476 •477 .478 ■479 .48 .481 .482 ■483 •484 ■485 .486 .487 .488 .489 •49 •49 1 .492 •493 •494 •495 •496 ■497 •498 ■499 •5 .501 .502 .503 .504 .505 .506 .507 .508 •509 .51 •5” .512 •5i3 •5i4 .515 i.594o8 .516 1.59564 .517 1-597 2 .518 i.59 8 76 .519 1.60032 .52 1. 601 88 .521 1.60344 .522 1.605 .523 1.60656 .524 1.60812 .525 1.60968 .526 1.61124 .527 1.6128 .528 1.61436 .529 1.61592 .53 1.61748 .531 1.61904 .532 1.6206 .533 1.62216 •534 1.62372 •535 1.62528 .536 1.62684 .537 1.6284 .538 1.62996 •539 1-63152 •54 1.63309 .541 1.63465 ,542 1.63623 •543 1-6378 .544 1.63937 .545 1.64094 .546 1.642 51 .547 1.64408 .548 1.64565 .549 1.64722 .55 1.64879 .551 1.65036 .552 1.65193 •553 1-653 5 •554 1 -655 07 -555 1-65665 .556 1.65823 •557 1.65981 .558 | 1. 661 39 .559 1.66297 .56 1.66455 .561 1.66613 .562 1.66771 .563 1.66929 .564 1.67087 .565 1-67245 1.58784 1 .566! 1.67403 1.5894 -567; i-6756i ( 1.59096 .568 1.67719 1.59252: .569 ; i - 6 78 77 I.508 42 ; 1.50996 I-5 11 5 1-51304! 1.51458 I.516 12 ; I.51766; I.5I92 1.520 74 1.52229 1.523 84 1.525 39 1.52691 1.52849 1.530 04 I-53 1 59 1-533 *4 j 1.53469: I-53625 i .537 8i | 1-539371 i.540 93> 1.542 49 I.54405 i .545 6i 1.547 18 1.548 75 1.55032 1.55189 1.55346 1.55503' I.5566 I i.558i7i 1-559 74 1.561 3 1 1.56289 1.56447 1.56605 1.56763 1.56921 1.57089 1.572 34 1.57389 1-57544 1.57699 1.57854 1.58009 1.58164 1 I.583 x 9 1.584 74 1.58629 .57 1 1.68036 j 57 1 1 1.68195 .572 j 1.68354 •573 i -685 13 •574 1-68672 •575 1.68831 .576 | 1.6899 •577 1-69149 .578 1.69308 •579 i 1.69467 .58 .581 .582 .583 .584 .585 .586 •587 .588 .589 •59 •59i •592 593 •594 ■595 ■596 •597 •598 •599 .6 .601 .602 .603 .604 .605 .606 .607 .608 .609 .61 .611 .612 .613 .614 .615 .616 .617 .618 .619 .620 .621 .622 .623 .624 1.69626 1.69785 1.69945 I. 701 05 1.702 64 1.70424 1.705 84 1.70745 1.70905 I I. 71065 1. 71225 1. 712 86 .71546 i*7 1 7 °7 1.71868 1.720 29 1. 721 9 1-7235 1.725 11 1.72672 1.72833 1.72994 1.73155 1-733 16 1.73638 1-73799 : I-739 6 1. 741 21 1.742 83 1.74444 1.74605 1.74767 , 1.74929 ' I.7509 1 : 1.752 52 < 1-754 J 4 ; 1.755 76 , 1.75738 1-759 1.760 62 1.762 24 1.76386 1.765 48 1.7671 LENGTHS OF ELLIPTIC ARCS. H’ght. Length. | H’ght . Length. H’ght. .625 I.768 72 .68 I.858 74 •735 .626 I.77034 .681 I.86039 •736 .627 1. 771 97 .682 I.862 05 •737 .628 I -773 59 .683 I.8637 •738 .629 1 1.775 2i .684 1-865 35 •739 •63 1.77684 .685 I.867 •74 .631 ; J-77847 .686 x. 868 66 •74i .63 2 1.780 09 .687 1.87031 •742 •633 1.781 72 .688 1.871 96 •743 •634 I.78335 .689 1.87362 •744 .635 1.784 98 .69 1.87527 •745 .636 1.7866 .691 1.87693 .746 •637 1.788 23 .692 1.87859 •747 .638 1.789 86 •693 1.880 24 •748 •639 1. 791 49 .694 1. 881 9 •749 .64 1.793 12 •695 1.88356 •75 .641 1-794 75 .696 1.885 22 •75* .642 1.70038 .697 1.88688 •752 •643 1.79801 .698 1.888 54 •753 .644 1.79964 .699 1.8902 •754 •645 1. 801 27 •7 1.891 86 •755 .646 1.8029 .701 1.89352 •756 .647 1.80454 .702 1.895 19 •757 .648 1.806 17 : •703 1.896 85 •758 .649 1.8078 •704 1.89851 •759 .65 1.80943! •705 1.900 17 .76 .651 1.811 07 .706 1. 901 84 .761 .652 1.812 71 1 •707 1-9035 .762 •653 i-8i4 35 .708 1-90517 .763 .654 1.81599! •709 x. 906 84 .764 •655 1.81763! •7i 1.90852 •765 .656 1.81928 .7x1 1.910 19 .766 •657 1.82091 .712 1.911 87 .767 .658 1.82255 •7i3 I -9 I 3 55 .768 •659 1.824 19 .714 I -9 I 5 23 .769 .66 1.825 83 .715 1.91691 •77 .661 1.82747 .716 1.91859 •77* .662 1.829 11 .717 1.92027 •772 .663 1-830 75 .718 1-92195 •773 .664 1.8324 .719 1.92363 •774 .665 1.83404 .72 1-925 3* •775 .666 1.835 68 .721 1.927 •776 .667 1-837 33 .722 1.92868. •777 .668 1.83897 •7 2 3 x. 93^36 .778 .669 1.84061 .724 1 -933 04 •779 .67 1.842 26 •725 1-93373 .78 .671 1.84391 .726 *•935 4* .781 .672 1.845 56 ‘73? *•937 * .782 •67 3 1.847 2 .728, 1.93878 •783 .674 1.84885 ,7291 1.94046 •784 •675 1-8505 •73 1.942 15 •785 .676 I-853-JL5 I •73i *•94383 .786 .677 1-853 79 : •732 I-945 5 2 .787 .678 1-85544 •733 1.94721 .788 .679 1.85709:! •734 1.9489 | •789 Length. H’ght. Length. H’ght. 1 -950 59 1.952 28 1 -953 97 1.95566 *•95735 1-95994 1 .960 74 x .962 44 1.964 14 1.96583 I-967 53 1.96923 1.97093 1.972 62 1.97432 1.97602 1.97772 1 -979 43 1.981 13 1.98283 I-984 53 1.986 23 1.98794 1.98964 I -99 1 34 1- 99305 1.99476 1.99647 1.998 18 1.99989 2.001 6 2.00331 2.005 02 2 .006 73 2.008 44 2 .010x6 1 2.01187 I 2.0x359 ! 2.01531 2.01702 2.0x874 2.020 45 2.022 17 2.023 89 2.02561 2.02733 2.02907 2.0308 2.03252 2.03425 2 - 035 98 2.037 71 2.03944 2.041 17 2.0429 Z •79 .791 .792 •793 •794 •795 .796 •797 .798 •799 .8 .801 .802 .803 .804 •805 .806 .807 .808 .809 .81 .811 .812 •813 .814 .8x5 .816 .817 .818 .8x9 .82 .821 .822 .823 .824 .825 .826 .827 .828 .829 •83 .831 .832 •833 •834 •835 .836 •837 .838 •839 .84 .841 .842 •843 -844 2.04462 2.04635 2.04809 2.04983 2.05157 2.05331 2-05505 2.056 79 2.05853 2.060 27 2.062 02 2.06377 2.065 52 2.067 27 2.06901 2.070 76 2.07251 2.07427 2.07602 2.07777 2.07953 2.081 28 2.08304 2.0848 2.086 56 2.08832 2.09008 2.091 98 2.0936 2.095 36 2.09712 2.098 88 2.10065 2.102 42 2.104 19 2.105 96 2.107 73 2.1095 2. in 27 2.11304 2.11481 2.11659 2.11837 2.120 15 2.121 97 2.123 71 2.125 49 2.12727 2.12905 2.13083 2.13261 2.13439 2.13618,. 2.13797 2-I3976 J •845 .846 •847 .848 •849 .85 .851 .852 .853 •854 .855 .856 .857 858 •859 ,86 ,861 ,862 ,863 .864 .865 .866 .867 .868 .869 .87 .871 .872 .873 •874 •875 .876 •877 .878 •879 .88 .881 .882 .883 .884 .885 .886 .887 .888 .889 .89 j .891 .892 •893 -S94 .895 .896 -89? .898, •8 99 ' 265 Length. 2.14155 2.143 34 2.145 13 2.146 92 2.148 71 2.I505 2.15229 2.15409 2.15589 2-1577 2-1595 2.l6l 3 2.16309 2.16489 2.16668 2.168 48 2.17028 2.17209 2.17389 2-175 7 2.177 51 2-I7932 2.l8l 13 2.18294 2.184 75 2.186 56 2.18837 2.190 18 2.192 2.19382 2.19564 2.19746 2.19928 2.201 1 2.202 92 2.204 74 2.20656 2.20839 2.2x022 2.21205 2.21388 221571 2.21754 2.21937 2.221 2 2.22303 2.22486- 2.226 7 2.228 54 2.23038. 2.232 22: 2.234 q6j 2-2359; 2-23774; 2 • 239.58 266 LENGTHS OF ELLIPTIC ARCS. H’ght. | •9 ! .901 1 • 9° 2 •903 • 9°4 ’ 9 °§ I .906 1 .907 1 .908 .909 Length. 2.241 42 2.243 2 5 2 24508 2.24691 2.248 74 2.25057 2.2524 2.25423 2.256 06 2.25789 H’ght. | .92 .921 .922 • 9 2 3 • 9 2 4 • 9 2 5 .926 .927 .928 .929 Length. H’ght . 1 2.27803 2 . 27987 ] 2 . 28 l 7 I 2.283 54 2.28537 2.287 2 2.289 03 2.29086 2.292 7 2.294 53 •94 1 .941 .942 •943 •944 •945 .946 •947 .948 •949 .91 2.25972 .911 j 2.26155 .912 j 2.26338 .913 | 2.26521 .914 2.26704 .915 ] 2.26888 .916 2.27071 .917 ; 2.27254 .918 2.27437 .919 2.2762 •93 • 93 i •932 •933 •934 •935 •936 •937 •938 •939 2.29636 •95 2.2982 • 95 1 2.30004 •952 2.30188 •953 2.303 73 •954 2.305 57 •955 2.30741 •956 2.30926 •957 2.311 11 •958 2.31295 •959 Length. | H’ght. Length, j H’ght. 1 2 - 3 T 4 79 j •96 2-352 41 .98 2.31666 .961 2.35431 .981 2 . 31852 ; .962 2.35621 .982 2.32038 •963 2.358 I .983 2.32224 .964 2.36 .984 2.324 1 1 .965 2.361 91 .985 2.32598 .966 2.36381 .986 2.32785 .967 2.36571 .987 2.32972 .968 2.367 62 .988 2.3316 .969 2.36952 .989 •99 2.33348 •97 2 . 371 43 • 99 1 2-335 37 .971 2-37334 .992 2.337 26 .972 2.37525 •993 2.33915 •973 2.377 16 •994 2.341 °4 •974 2.37908 •995 2.342 93 •975 2.381 .996 2.34483 .976 2.38291 •997 2346 73 •977 2.384 82 •998 2.348 62 .978 2.386 73 •999 1 2.35051 1-979 2.388 64 1 . Length. 2.390 55 2.39247 2-39439 2.3963 1 2.39823 2.40016 2.402 08 2.404 2.405 92 2.407 84 2.40976 2.4II 69 2.41362 2.41556 2.41749 2.41943 2.42136 2.42329 2.425 22 2.427 15 2.42908 To Ascei'tsain. Eeragtli of* an Elliptic Arc (riglit Semi- Ellipse) by preceding Talole. Rule. — Divide height by base, find quotient in column of heights, and take length for that height from next right-hand column. Multiply length thus obtained by base of arc, and product will give length. Example.— What is length of arc of a semi-ellipse, base being 70 feet, and height • 30. 10 feet ? 30. io-j- 70 — .43 ; and 43 , .-per table, = 1.46268. Then 1.46268 X 70 = 102.387 6 feet. When Carve is not that of a right Semi-Ellipse , Height being half of Trans- verse Diameter. Rule. — Divide half base by twice height, then proceed as in preceding example ; multiply tabular length by twice height, and product will give , length. Example. — What is length of arc of a semi-ellipse, height being 35 feet, and base ] 60 feet? 60 -r- 2 = 30, and 30-7-35X2 = . 428, tabular length of which = 1.459 66. Then 1.45966 X 35 X 21= 102.1762 feet. When , in Division of a Height by Base , Quotient has a Remainder after third Place of Decimals, and great Accuracy is required, Rule. — Take length for first three figures, subtract it from next following j length; multiply remainder by this fractional remainder, add product to j first length, and sum will give length for whole quotient. Example. — What is length of an arc of a semi-ellipse, base being 171.3 feet and < height 125 feet? 171.3 -4- 2 = 85.65. and 125 X 2 = 250. 171. 3 -r- 250 = .3426 ; tabular length for } .342 = 1.334 I2 > aD( l -343 =■ i-335 52, the difference between which is .0014. Then .6 X .0014 = .0084. Hence, .342 = 1.334 12 .0006= .0084 1.342 52, the sum by ivhicli base of arc is to be multiplied ; and 1.34252 X 171. 3 = 229.973 676/e^. AREAS OF SEGMENTS OF A CIRCLE. 267 Areas of Segments of a Circle. Th e Diameter of a Circle = 1, and divided into 1000 equal Parts. Verse* Sine. Seg. Area. .OOI .OOOO4 .002 .OOO 12 .OO3 .000 22 .OO4 .OOO34 .005 .OOO47 .006 .000 62 .007 .OOO78 .008 .OOO95 .009 .OOI 13 .01 •OOI 33 .Oil •OOI53 .012 •001 75 .013 .OOI 97 .014 .002 2 .015 .OO244 .016 .002 68 .017 .002 94 .018 ; -0032 .019 •OO347 .02 •OO375 .021 .OO403 .022 .OO432 .023 .OO462 .024 ; .OO492 .025 .00523 .026 ; -00555 .027 .00587 .028 .00619 .029 ! •00653 •03 I .00686 •031 j .00721 .032 .00756 •033 .00791 •034 .008 27 •035 .00864 .036 .00901 •037 .00938 •038 ; .009 76 •039 .01015 *°4 ; .01054 ! -041 .01093 f .O42 •01133 ! -°43 | .on 73 •044 .012 14 •045 •OI255 .046 .OI297 •047 •01339 .048 .OI382 .049 •OI425 •05 .OI468 .051 .015 12 Versed Sine. Seg. Area. Versed Sine. Seg. Area. 1 1 Versed J Sine. Seg. Area. V ersed Sine. Seg. Area. .052 .015 56 .103 .042 69 •154 •076 75 .205 .11584 •053 .Ol6oi .IO4 .0431 j - x 55 •07747 .206 .11665 •054 .01646 .105 .04391 1 .156 .0782 .207 .11746 •055 .01691 .106 .044 52 , • I 57 .07892 .208 .11827 .056 •01737 .107 •045 14 : .158 •07965 .209 .11908 •057 .OI783 .108 •045 75 • z 59 .080 38 .21 .119 9 .058 .0183 .109 .04638 j .16 .081 11 .211 .120 71 •059 .Ol8 77 .11 •047 .161 .08 1 85 .212 •121 53 .06 .OI924 .III .04763 .162 .082 58 •213 •12235 .061 .OI972 .112 .048 26 .163 .08332 .214 .123 17 .062 .020 2 •113 .048 89 .164 .084 06 .215 •12399 .063 .020 68 .114 •04953 ‘ .165 .084 8 .216 .12481 .064 .021 17 •115 .050 16 .166 •08554 .217 •12563 .065 .021 65 • Il6 .0508 .167 .086 29 .218 .126 46 .066 .022 15 .117 •05145 .168 .08704 .219 .127 28 .067 .O2265 .Il8 .05209 .169 •087 79 .22 .128 II .068 •023 15 •I I 9 •052 74 • x 7 .088 53 .221 .12894 .069 .02336 .12 •05338 .171 .089 29 .222 .12977 •07 .O24 1 7 .121 .05404 .172 .09004 •223 .1306 .071 .024 68 .122 .05469 • x 73 .0908 .224 •13144 .072 .025 19 .123 •055 34 • I 74 •09 x 55 .225 .13227 •073 .02571 .124 .056 •175 .09231 .226 •133 II .074 .026 24 .125 .05666 .176 •09307 .227 •133 94 .075 .026 76 .126 •057 33 • J 77 •09384 .228 •134 78 .076 .027 29 .127 •05799 .178 .0946 .229 •135 62 •077 .027 82 ^.128 .05866 • x 79 •095 37 •23 .136 46 .078 .02835 *129 •05933 .18 .09613 .231 • I 373 I .079 .02889 •13 .06 .181 .0969 .232 •13815 .08 .02943 •131 .060 67 .182 .097 67 •233 •139 .081 .02997 .132 •06135 .183 .09845 •234 •139 84 .082 .03052 •133 .06203 .184 .09922 •235 .14069 .083 .031 07 •134 .062 71 .185 .1 .236 •141 54 .084 .031 62 • x 35 •06339 .186 .100 77 •237 •142 39 .085 .032 18 .136 .064 07 .187 •ioi 55 •238 •143 24 .086 .032 74 • x 37 .064 76 .188 .10233 •239 .14409 .087 .088 .089 •0333 •03387 .138 • T 39 •065 45 .066 14 .189 .19 .10312 .1039 .24 .24I •14494 .145 8 •034 44 .14 .066 83 .191 .104 68 .242 .14665 .09 •03501 .141 •06753 .192 •10547 •243 •I 475 I .091 •035 58 .142 .068 22 • T 93 .10626 .244 •14837 .092 .036 16 •i 43 .068 92 .194 .10705 •245 .149 23 •093 .036 74 .144 .069 62 • I 95 .107 84 .246 .15009 •094 •03732 •i 45 •07033 .196 .10864 •247 •15095 •095 •0379 .146 .071 03 .197 .10943 .248 .151 82 .096 •03849 .147 .07174 .198 .11023 .249 .15268 .097 .03908 .148 •07245 • I 99 .111 02 .25 •i 53 55 .098 .03968 .149 •073 16 .2 .111 82 .251 •I 544 1 .099 .040 27 • T 5 •07387 .201 .11262 .252 • 15528 .1 .040 87 • x 5i •07459 .202 •ii343 •253 •15615 .IOI .041 48 .152 •07531 .203 .11423 •254 .15702 .102 .04208 | •153 .07603 .204 •11503 •255 •15789 268 AREAS OF SEGMENTS OF A CIRCLE. Seg. Area. .256 .257 .258 •259 .26 .261 .262 .263 .264 .265 .266 .267 .268 .269 .27 . 27 I .272 •273 .274 •275 .276 .277 .278 .279 .28 .281 .282 .283 .284 .285 .286 .287 .288 .289 .29 .291 .292 •293 .294 •295 .296 .297 .298 •299 •3 • 3 01 .302 •303 •304 .158 76 • i 59 6 4 .16051 .16139 .162 26 .16314 | Versed Sine. •305 .306 •307 .308 •309 •31 Seg. Area. .16402 .311 .1649 • 3 12 .16578 • 3 i 3 .16666 .314 .16755 •315 .168 44 • 3 l6 .16931 • 3 i 7 .1702 .318 .17109 • 3 I 9 • I 7 1 97 .32 .17287 .321 .173 76 .322 .17465 .323 .175 54 •324 .17643 .325 .17733 .326 .17822 .327 .179 12 .328 .18002 •329 .18092 •33 .181 82 . 33 i .182 72 •332 .18361 •333 .18452 •334 .185 42 •335 .18633 •336 .18723 •337 .188 14 .338 .18905 •339 .18995 •34 .19086 • 34 i • I 9 I 77 .342 .192 68 •343 .1936 •344 .19451 •345 .19542 .346 .19634 •347 .197 25 •348 .19817 •349 .19908 •35 .2 • 35 i .20092 .352 .201 84 •353 V ersed Sine. Seg. Area. ,302 76 .20368 .2046 •205 53 .20645 .207 38 .20923 .210 15 .211 08 .21201 .212 94 .21387 .2148 .215 73 .21667 .2176 •21853 .21947 .220 4 .221 34 .222 28 .22321 .22415 .22509 .22603 .226 97 .22791 .228 86 .2298 .230 74 .231 69 .23263 .23358 .234 53 .23547 .23642 .23737 .23832 .23927 .24022 .241 17 .242 12 .24307 .24403 .244 98 •24593 .246 89 •354 •355 .356 •357 .358 •359 .36 .361 .362 •363 .364 •365 .366 .367 .368 .369 •37 . 37 i •372 •373 •374 •375 .376 •377 •378 -379 .38 .381 .382 .383 •384 .385 .386 .387 .388 .389 •39 • 39 1 .392 •393 •394 •395 .396 •397 •398 •399 •4 .401 l| .402 Versed ] Sine. .2488 .24976 .25071 .25167 .25263 .25359 -25455 .255 51 .25647 -25743 .258 39 . 2593 6 .260 32 .261 28 .262 25 ,26321 .264 18 .265 14 .266 II .267 08 .268 04 . 269 OI .26998 .27095 .271 92 .27289 .27386 .27483 .27580 .27677 -277 75 .278 72 .27969 .28067 .281 64 .282 62 .28359 .28457 .285 54 .28652 .2875 .28848 .28945 .29043 .291 41 .29239 .29337 .29435 .29533 •403 .404 .405 ,406 .407 .408 .409 .41 .411 412 413 .414 415 .416 .417 .418 419 .42 ,421 422 .423 .424 425 426 427 •43 •44 l Seg. Area. f ersed 1 Sine. Seg. Area. .29631 .452 - -344 77 .297 29 -453 ■345 77 .298 27 •454 .34676 .29925 •455 •347 76 .30024 .456 .348 75 .3OI 22 •457 •349 75 .302 2 .458 •350 75 .30319 •459 . 35 i 74 •30417 .46 •352 74 .305 15 .461 •35374 .30614 .462 •354 74 •3°7 I 2 .463 •355 73 .308H .464 .35673 .30909 .465 •357 73 .31008 .466 .35872 .3HO7 .467 .359 72 • 3 I2 o 5 .468 .360 72 .31304 .469 .361 72 • 3 I 4°3 •47 .362 72 .31502 .471 •36371 .316 .472 .36471 • 3 i6 99 •473 .36571 * 3 I 7 98 •474 .36671 > - 3 i8 97 •475 .367 7 i .31996 .476 .36871 1 .32095 •477 .36971 > .32194 .478 •37071 .32293 •479 • 37 1 7 ; .32391 .48 •372 7 5 .3249 .481 •373 7 $ .325 9 .482 •374 7 [ .32689 •483 •375 7 5 .32788 .484 •3767 5 .32887 .485 •377 7 7 .32987 .486 .378 7 3 .33086 •487 •3797 ? . 331 85 .488 .3807 .332 84 •489 .3817 1 .33384 •49 .3827 2 .33483 • 49 1 •383 7 3 -33582 .492 .3847 4 .33682 •493 •385 7 5 .337 81 •494 .3867 6 .3388 •495 •387 7 7 - 339 8 .496 .3887 8 .340 79 , .497 •389 7 9 - 34 i 79 1 .498 .390 7 ; .342 78 i -499 .391 7 ;i -34378 1 1 *5 .392 7 To Compute Area of a Segment of a Circle preceding Tat>le. Rule —Divide height or versed sine by diameter of circle ; find quotient in column of versed sines. Take area for versed sine opposite to it m next col- umn on right hand, multiply it by square of diameter, and it will gi\e area. AREAS OF ZONES OF A CIRCLE. 269 Example —Required area of a segment of a circle, its height being 10 feet and diameter of circle 50. 6 10 -f- 50 = . 2, and . 2, per table , = . 1 1 1 82 ; then . 1 1 1 82 X 50 2 = 279. 55 feet. When, in Division of a Height by Base , Quotient has a Remainder after I hird Place of Decimals , and great Accuracy is required. Rule.— T ake area for first three figures, subtract it from next following- area, multiply remainder by said fraction, add product to first area and sum will give area for whole quotient. lieVh^nt ' ? 1 iS area ° f a segraent of a circle > diameter of which is 10 feet, and i. 575 r i°~i 575 ; tabular area for .157 = .07892. and for .158 = .070 6q the dif- ference between which is .00073. y 5 79 aij Then .5 x .00073 = .000365. H ence ! .157 =.07892 .0005 = .006 365 of circle is to be multiplied ; and .079 285 = Square °f diameter Areas of Zones of a Circle. The Diameter of a Circle — 1, and divided into 1000 equal Parts. H’ght j Area. |[ Area. | H’ght Area. H’ght. Area. H’ght. Area. .OOI .002 .003 .OO4 .005 .006 .007 .008 .OO9 .OI .Oil .012 .013 .OI4 .015 .Ol6 .OI7 .018 .OI9 .02 .021 .022 .023 .024 .025 .026 .027 .028 .O29 •03 .031 .O32 •033 •034 .OOI .002 .003 .004 1 .005 .006 .007 .008 .009 .OI .on .012 .013 .014 .015 .016 .017 .018 .019 .02 .021 .022 .023 .024 .025 .02599 .02699 .02799 .028 98 .02998 1 .03098 .03198 .03298 •03397 1 •035 .036 •037 .038 •039 .04 .041 .042 •043 .044 •045 .046 .O47 .048 .049 •05 .051 .O52 •053 •054 •055 .056 •057 .058 •059 .06 .061 .062 .063 .064 .065 .066 .067 .068 •03497 •03597 .03697 .03796 .03896 .03996 •040 95 .041 95 .04295 .04394 .04494 •04593 .04693 •04793 .048 92 .04992 .05091 .0519 .0529 •05389 .05489 .055 88 .056 88 •05787 .05886 .059 86 .060 85 .061 84 .062 83 .063 82 .06482 .0658 .0668 -0678 .069 •07 .071 .072 •073 .074 •075 .076 •077 .078 •079 .08 .081 .082 .083 .084 .085 .086 .087 .088 .089 .09 .091 .O92 •093 •°94 •095 .096 •097 .098 •°99 .1 .IOI .102 2 .068 78 .06977 .070 76 •07175 .072 74 •°73 73 •074 72 •075 5 .07669 .07768 .07867 .07966 .080 64 .081 63 .082 62 .0836 .08459 •085 57 .086 56 .08754 .08853 .08951 .0905 .091 48 .09246 •09344 .09443 .0954 .09639 •09737 •09835 •09933 .10031 .101 29 :* .103 .IO4 .105 .106 .107 .108 .IO9 .11 .III .112 •113 .114 •115 .Il6 .117 .118 •II 9 .12 .121 .122 .123 .I24 .125 .126 .127 .128 .I29 •L 3 .131 .132 •133 •134 •135 •136 .102 27 •IO325 . IO4 22 .105 2 .I0618 .IO715 .IO813 .IO911 .11008 .III06 .11203 •113 .II398 •II 495 .II592 .1169 .II787 .11884 .II981 .120 78 • 121 75 .122 72 .12369 .12469 .125 62 .12659 • I2 7 55 .12852 .12949 •13045 .13141 •13238 •13334 •1343 •137 .138 •139 .14 .141 .142 •143 .144 •145 .146 .147 .148 .149 •15 •151 •152 •153 •154 •155 .156 •157 .158 •159 .16 .l6l .162 .163 .164 .165 .166 .167 .168 .169 •*7 •I 35 2 7 • t 36 2 3 •137 19 •13815 .13911 .14007 .14103 .14198 .14294 • T 439 .14485 .14581 • i 4 6 7 7 .14772 .14867 .149 62 .15058 •15153 .15248 • I 53 43 •15438 •155 33 .15628 •i 57 2 3 .15817 •15912 .16006 .16101 .16195 .1629 .16384 .16478 .165 72 .16667 270 AREAS 0E ZONES OF A CIRCLE, H’ght. .171 .172 .173 .174 •175 .176 .177 .178 .179 .18 .181 .182 .183 .184 .185 .186 .187 .188 .189 .19 .191 .192 .193 .194 .195 .196 .197 -198 .199 .2 .201 .202 .203 .204 .205 .206 Area. H’ght. Area. H’ght .21805 .21894 .21983 .220 72 .221 6 l .222 5 .22335 .233 I .22427 .234 .225 15 .235 I .22604 .236 I .22692 .237 .2278 .2^8 .22868 .239 .207 .208 .209 .21 .211 .212 .213 .214 .215 .216 .217 .218 .219 .22 .221 .222 .223 .224 .225 ,16761' ,16855 .16948 .170 42 .17136 .1723 .173 23 .17417 .1751 .17603 .17697 .1779 .17883 .179 76 .18069 .181 62 .182 54 .i 8 347 .1844 .185 32 .18625 .18717 .18809 .18902 .18994 .19086 .191 78 .1927 .193 61 .194 53 .19545 .19636 .197 28 .198 19 .1991 .20001 .20092 .201 83 .202 74 .20365 .204 56 .205 46 .20637 .207 27 .208 18 .20908 .20998 .21088 .211 78 .21268 .21358 .21447 -21537 .21626 .217 16 .226 .227 .228 .229 .23 .231 .232 .24 .241 .242 •243 .244 •245 .246 .247 .248 .249 .25 .251 .252 .253 .254 .255 .256 •257 .258 .259 .26 .261 .262 .263 .264 .265 .266 .267 .268 .269 .27 <27 1 .272 •273 .274 .275 .276 •277 .278 •279 .28 .229 56 .23044 .231 3i .232 19 .23306 .23394 .234 81 .235 68 .236 55 .237 42 .238 29 .239 1 5 .24002 .240 89 .241 75 .242 61 •243 47 .244 33 .245 19 .246 04 .2469 .247 75 .248 61 .249 46 .25021 .251 16 .25201 .25285 •253 7 .254 55 .25539 .25623 .25707 .25791 .25875 .25959 .26043 .261 26 .26209 .26293 ^263 76 .26459 .281 .282 .283 .284 .285 .286 .287 .288 .289 .29 .291 .292 .293 .294 .295 .296 .297 .298 •299 •3 • 3 01 .302 •303 I .304 I .305 .306 .307 • 3° 8 •309 • 3 1 • 3 11 • 3 12 .313 .314 .315 .316 317 .318 ■3i9 •32 .321 .322 .323 .324 .325 .326 .327 .328 •329 •33 .331 .332 •333 •334 •335 ,265 41 .266 24 ,267 06 .267 89 .268 71 .26953 ,27035 .271 17 .27199 .272 8 .273 62 .27443 .27524 .27605 .276 86 .27766 .27847 .27927 .280 07 .28088 .281 67 .28247 I .28327 .28406 I .28486 I .28565 .28644 .28723 .28801 .2888 .28958 .29036 .291 15 .291 92 .292 7 .29348 .294 25 .295 02 .2958 .296 56 .297 33 .298 1 .298 86 .299 62 •30039 .301 14 .3019 .30266 .30341 •3°4 16 .304 9 1 .305 66 .30641 .307 15 .336 •337 •33 8 i *339 •34 .341 .342 •343 •344 •345 •346 •347 .348 •349 •35 I *351 •352 •353 •354 •355 •356 357 •35 8 •359 •36 .361 .362 .363 .364 .365 .366 •367 .368 •369 •37 •37i .372 •373 •374 •375 .376 •377 •37 8 •379 •3 8 .381 •3 82 .3 8 3 .384 .385 .386 .387 .388 -389 .3079 II -39 .308 64 30938 .310 12 •3 io8 5 •3 11 59 •3 I2 3 2 31305 - 3 * 3 1 % •3145 •31523 ■31595 - 3 l66 7 •3 X 7 39 •3 l8 11 .31882 •3*954 ,320 25 .320 96 .321 67 32237 .32307 323 77 •32447 .325 17 .32587 .32656 •32725 .327 94 .328 62 .329 3 1 .32999 •33067 •33 1 35 •33203 .332 7 33337 •33404 334 7 1 •335 37 33604 •336 7 •337 35 .33801 .33866 .33931 .33996 .34061 .34125 .3419 .342 53 .34317 •3438 •34444 .34507 .34569 H’ght. I Area. .391 .34632 •392 | .34694 •393 1 •347 56 •394 .348 18 •395 .348 79 .396 •3494 •397 .35001 .398 •35062 •399 •35 1 22 •4 .35182 .401 .35242 .402 •35302 .403 404 .405 .406 .407 .408 .409 .41 .411 ,412 •4i3 .414 ■415 .416 .417 .418 .419 .42 .421 .422 •423 .424 .425 .426 •427 .428 •429 •43 .431 .432 •433 •434 •435 .436 •437 .438 •439 •44 .441 .442 •443 .444 •445 .35361 ■3542 •35479 •35538 •35596 •35654 •357II .35769 .35826 .35883 •35939 •35995 .36051 .361 07 .361 62 .362 17 ,362 72 .36326 •3638 .36434 .36488 .36541 .36594 .36646 .36698 .3675 .368 02 .36853 .36904 .36954 .37005 .370 54 •37 1 °4 •37153 .37202 •372 5 .37298 .37346 •37393 •3744 .37487 •375 33 •375 79 AREAS OF ZONES OF A CIRCLE. 271 H’ght. Area. | H’ght. Area. H’ght. Area. H’ght. Area. H’ght. Area. .446 .37624 •457 .38096 .468 •385 14 •479 .38867 •49 • 39 1 37 •447 .37669 •458 •381 37 .469 •385 49 .48 •38895 • 49 1 • 39 i 56 .448 ! -37714 •459 •381 77 •47 •385 83 .481 •389 23 .492 • 39 i 75 •449 •377 58 .46 .382 16 .471 .38617 .482 •3895 •493 .391 92 •45 .37802 .461 •382 55 .472 .3865 •483 .389 76 •494 .39208 • 45 i I •37845 .462 .382 94 •473 .38683 •484 .390 01 •495 •39223 .452 : .37888 •463 •38332 •474 •387 15 •485 .390 26 .496 •392 36 ♦453 •37931 .464 •38369 •475 •38747 .486 •390 5 •497 •39248 •454 ! •379 73 •465 .38406 .476 •387 78 •487 •390 73 .498 •392 58 •455 .380 14 .466 •384 43 •477 .38808 .488 •390 95 •499 .392 66 • 456 : .38056 .467 •384 79 1 •478 .38838 •489 •39117 •5 •392 7 This Table is computed only for Zones, lonyest Chord of which is Diam- eter. To Compute Area of a Zone "by preceding Table. When Zone is Less than a Semicircle . Rule— D ivide height by diameter, find quotient in column of heights. Take area for height opposite to it in next column on right hand, multiply it by square of longest chord, and product will give area of zone. Example.— Required area of a Zone, diameter of which is 50, and its height 15. 15 4- 50 = .3; and .3, as per table, — . 280 88. Hence . 280 88 X 50 2 = 702.2 area. When Zone is Greater than a Semicircle. Rule.— T ake height on each side of diameter of circle, and ascertain, by preceding Rule, their respective areas ; add areas of these two portions to- gether, and sum will give area. Example. — Required area of a zone, diameter of circle being 50, and heights of zone on each side of diameter of circle 20 and 15. 20-h- 50 = . 4; .4, as per table, ==.351 82; and .351 82 X so 2 = 879.55. i£-^5o = -3; .3, as per table, =.28088; and .28088 X so 2 = 702.2. Hence 879.554-702.2 = 1581.75 area. When, in Division of a Height by Chord, Quotient has a Remainder after Third Place of Decimals, and great Accuracy is required. Rule.— T ake area for first three figures, subtract it from the next follow- ing area, multiply remainder by said fraction, and add product to first area ; sum will give area for whole quotient. Example. — What is area of a zone of a circle, greater chord being 100 feet, and breadth of it 14 feet 3 ins.? 5 14 feet 3 ins. — 14-25, and 14. 25 -=-100=3.1425; tabular length for and for . 143 — . 141 03, difference between which is .00096. Then . 5 x .000 96 = .000 48. Hence .142 = . 140 07 .0005 = .000 48 .142 = .140 07, chord is to be multiplied ; . 1 40 55 , mm by which square of greater and . 140 55 X 100 2 ■= 1405.5 feet. [ lsh.it. I 2 3 4 5 6 7 8 9 io ii 12 13 14 i 5 16 17 18 19 20 21 22 23 24 2 5 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 SQUARES, CUBES, AND ROOTS. es. Cubes, and Square and Cube From 1 to 1600. Square. I 4 9 16 25 36 49 64 81 1 00 1 21 144 1 69 1 96 225 256 2 89 324 3 61 400 441 484 529 5 76 625 6 76 729 784 841 900 961 1024 1089 11 56 1225 1296 1369 1444 1521 1600 1681 1764 1849 1936 2025 21 16 2209 2304 2401 2500 2601 2704 2809 29 16 Cube. Square Root. I 8 I I.4142136 27 I-73 2 0508 64 2 125 2.236068 216 2-449 489 7 343 2.645 751 3 512 2.828 427 1 729 3 1 000 3.1622777 I33 1 3.316624 8 1 728 3.464 101 6 2197 3 605 5513 2 744 3.741 6574 3 375 3.8729833 4096 4 A 4 9*3 4.123 1056 5832 4.242 640 7 6859 4*358 598 9 8000 4*472 136 9261 4.582 575 7 10648 4.6904158 12 167 4*795 831 5 13 824 4.898 9795 15625 5 17 576 5.0990195 19683 5.1961524 21952 5.291 5026 24 389 5.385 l6 4 8 27000 5.4772256 29 79 1 5.5677644 32 768 5.6568542 35 937 5.744 5626 39 3°4 5.8309519 42 875 5.9160798 46656 6 50653 6.082 762 5 54872 6.164414 59319 6.244 998 64000 6.324 555 3 68 921 6.4031242 74 088 6.480 740 7 79 507 6.557 438 5 85184 6.633 2496 9 1 I2 5 6.708 2039 97 336 6.782 33 103 823 6.8556546 1 10 592 6.928 203 2 117649 7 . 0 125 000 7.071 067 8 132 651 7.141 4284 140 608 7.21 1 1026 148877 7.280 1099 157464 7.3484692 Numbe 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 7 i 7 2 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 9 i 92 93 94 95 96 97 - 98 . 99 100 101 102 103 104 105 SQUARES, CUBES, AND ROOTS. 273 Square. f Cube. 30 25 166375 3136 175 616 32 49 185 193 3364 195 1 12 34 81 205 379 36 OO 216000 3721 226 981 3844 238 328 3969 250 047 40 96 262 144 42 25 274 625 43 56 287 496 44 89 300 763 46 24 314 432 4761 328 509 4900 343000 5041 357 9 11 5184 373 248 53 29 389017 54 76 4°5 224 5625 421 875 57 76 438 976 59 29 456533 60 84 474 552 6241 493 039 64 00 512000 6561 53 i 44 i 6724 551368 68 89 57i 787 7056 592 704 72 25 614 125 7396 636056 7569 658 503 77 44 681 472 7921 704 969 81 00 729000 82 81 753 571 8464 778688 86 49 804 357 88 36 830 584 9025 857 375 92 16 884 736 9409 912673 96 04 941 192 9801 970 299 1 0000 1 000000 1 0201 1 030 301 1 0404 1 061 208 1 0609 1 092 727 1 08 16 1 124864 1 1025 1 157625 1 1236 1 191 016 1 1449 1 225 043 1 1664 1 259712 1 1881 1 295 029 1 21 00 1 331 000 Square Root. Cube Root. 7.416 198 5 3.802 952 5 7 - 4833 I 4 8 3.825 862 4 7-549834 4 3.848 501 I 7 - 6 i 5 773 1 3.8 708766 7.681 145 7 3.8929965 7-745 9667 3.9148676 7.810249 7 3.936 497 2 7.874 0079 3-957 89 I 5 7-937 253 9 3 - 979057 I 8.062 257 7 4 4.020 725 6 8.1240384 4.041 240 1 8-185 352 8 4.061 548 8.246 21 1 3 4.081 655 1 8.306 623 9 4.101 566 1 8.3666003 4.121 2853 8.426 149 8 4.140817 8 8.485 281 4 4.160 167 6 8-544 003 7 4-179 339 8.602 3253 4.1983364 8.660 254 4.217 1633 8.7177979 4-235 823 6 8.774 9644 4.254321 8.831 7609 4.272 6586 8.888 194 4 4.290 840 4 8.9442719 4.308 869 5 9 4.326 748 7 9.0553851 4.344 4815 9-1104336 4.362 070 7 9.165 1514 4-379 5 I 9 I 9.219 544 5 4.396 829 6 9.2736185 4.414 004 9 9.327 379 1 4.4310476 9.380831 5 4.447 960 2 9-433 98 I 1 4.464 745 1 9.486833 4.481 404 7 9-539392 4-497 94 i 4 9.591 663 4 . 5 H 357 4 9-643 650 8 4-530 654 9 9-695 359 7 4-5468359 9.746 794 3 4.562 902 6 9-797 959 4-578 857 9.848 8578 4-5947009 9.899 494 9 4-6io 436 3 9.9498744 4.626065 10 4-641 588 8 10.049 875 6 4-6570095 I0 -099 504 9 4.672 3287 10.148 891 6 4.687 548 2 10.198039 4.702 6694 10.246 950 8 4.717694 10.295 630 1 4.732 6235 10.344 °8o 4 4-747 4594 IO - 39 2 304 8 4.762 203 2 10.440 306 5 4.776 8562 10.488 088 5 4 - 79 1 4199 JMBEK. Ill 1 12 113 114 115 Il6 117 Il8 IT 9 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 SQUARES, CUBES, AND ROOTS. Square. CUEE. I 23 21 I 367 63I 125 44 I 4O4 928 I 2769 I 442 897 I 2996 i 481 544 1 32 25 1 520 875 i 34 56 1 560 896 1 3689 1 601 613 139 24 1 643 032 1 41 61 1 685 159 1 4400 1 728000 1 4641 1 77 1 5 61 1 48 84 1815848 1 51 29 1 860 867 1 53 76 1 906 624 1 5 6 25 1 953 125 1 58 76 2000376 1 61 29 2 048 383 1 6384 2097 152 1 6641 2 146689 1 6900 2 197000 17161 2 248 091 1 74 24 2 299 968 1 7689 2 352 637 1 79 56 2 406 104 1 82 25 2 460 375 1 8496 2 5 t 5 456 1 87 69 2 57 1 353 19044 2 628 072 1 93 21 2 685 619 1 9600 2 744000 1 9881 2803221 201 64 2 863 288 20449 2 924 207 20736 2 985 984 2 1025 3 048 625 2 13 16 3 1 12 136 2 1609 3 1 7 6 523 2 19 04 3241 792 2 2201 3 307 949 2 2500 3375000 2 2801 3 442 951 23104 3511008 23409 3 581577 23716 3652 264 24025 3 723 875 2 43 3 6 3796416 24649 3 869 893 24964 3944312 2 52 81 4019679 2 5600 4096000 25921 4 173 281 262 44 4251 528 26569 4 330 747 26896 4410944 2 72 25 4492 125 2 75 56 4574296 Square Root. 10.535 6 53 8 IO.583OO52 10.630 145 8 10.6770783 10.723 805 3 10.770 3296 10.8166538 10.862 7805 10.908 7121 10.954 4512 11 11.045 361 11.090 5365 11. 135 5287 11.1803399 11.224972 2 11.269427 7 i I *3 I 3 708 5 11.3578167 1 1. 401 7543 11.445 5231 11.489 125 3 II-53 2 5626 n-575 8369 11.61895 1 1. 661 9038 11.7046999 n-747 340 1 11.789 826 1 11.832 1596 11.874 342 1 11.9163753 11.958 260 7 12 12.041 5946 12.083046 12:124355 7 12.165 5251 ,12.2065556 12.247448 7 12.288 205 7 12.328 828 12.3693169 12.4096736 12.449 8996 12.489996 12.5299641 12.569 805 1 12.6095202 12.649 1106 12.6885775 12.727 922 1 12.767 1453 12.806 248 5 12.845 232 6 12.884098 7 SQUARES, CUBES, AND ROOTS. 275 Number. | Square. | Cube. 167 168 t 169 1 70 1 7 1 172 173 174 175 176 2 78 89 2 82 24 28561 2 89 OO 29241 295 84 29929 302 76 30625 30976 177 178 179 180 181 182 183 184 185 186 187 1 88 189 190 191 192 193 1 94 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 31329 3 1684 32041 32400 32761 33124 3 34 89 3 38 56 3 42 25 3450 3 4969 3 53 44 3 57 21 361 00 36481 36864 3 72 49 37636 380 25 38416 38809 3 92 04 39601 40000 40401 40804 4 12 09 4 16 16 420 25 42436 42849 43264 4 36 8i 441 00 4 45 21 4 4944 4 53 69 4 57 96 462 25 46656 4 70 89 4 75 24 4 796i 4 8400 48841 4 92 84 4657463 4 74i 632 4 826 809 4913000 5 000 21 1 5 088 448 5 177 717 5 268 024 5 359375 5 45i 776 5 545 233 5639 752 5 735 339 5 832 000 5 929 74i 6 028 568 6 128487 6 229 504 6331625 6434 856 6 539 203 6 644 672 6751 269 6 859 000 6967 871 7077888 7189057 7301384 7 4i4 875 7 529 536 7 645 373 7762 392 7 880 599 8 000 000 8 120601 8 242 408 8365 427 8489664 8 615 125 8 741 816 8 869 743 8998912 9 129329 9 261 000 9393 931 9528128 9663597 9800344 9938 375 10077696 10218313 10 360 232 10 503 459 10 648 000 10 793 861 10 941 048 Square Root. 12.922 848 12.961 481 4 13 13.038 404 8 i 3*°76 6968 13.114877 13-1529464 13.190906 13.228 7566 13.2664992 I 3*3°4 134 7 1 3-34 I 664 1 I 3-379 °88 2 13.4164079 *3-453 624 I 3-49° 737 6 i 3-5 2 7 7493 *3-564 66 13.601 4705 13.638 181 7 *3-674 794 3 I 3-7 11 3°9 2 I 3-747 727 1 13.784 0488 13.820275 13-8564065 i 3-892 44 13- 9283883 13.96424 14 *4-°35 668 8 14.0712473 14.106 736 1 4- I 4 2 1356 I 4- I 77 4469 14.212 6704 14.247 806 8 14.282 856 9 14.317821 1 I 4-35 2 700 1 14.387 494 6 14.422 205 1 i 4-456 832 3 I 4-49 I 376 7 I4-525 839 14.560219 8 *4-594 5*9 5 14.628 738 8 14.662 878 3 14.6969385 *4-730 919 9 14.764823 1 *4-798 648 6 *4-832 397 14.866 068 7 148996644 i Cube Root. 5-5068784 5.5178484 5.5287748 5-539658 3 5.550 4991 5.561 2978 5-572 0546 5.582 7702 5-593 444 7 5.604078 7 5.6146724 5.625 2263 5.635 7408 5.6462162 5.6566528 5.667 051 1 5-6774114 5.687 734 5.6980192 5.708 2675 5-7*8479* 5.728 6543 5.738 793 6 5.748 897 1 5.758 9652 5.768 9982 5.778 9966 5.788 9604 5.798 89 5.808 785 7 5.8186479 5.8284767 5.8382725 5-8480355 5.857 766 5.8674643 5.877 1307 5.886 765 3 5.896 3685 5.9059406 5.9*54817 5.924 9921 5.934472 1 5.943 922 5- 953 34*8 5.962 732 5.972 092 6 5.981 424 59907264 6 6.009 245 6.018 461 7 6.027 650 2 6.036 810 7 6 - 045 943 5 6 055 048 9 UMBEE. 223 224 225 226 227 228 229 23° 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 2^6 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 SQUARES, CUBES, AND ROOTS. Square. Cube. 4 97 29 II089567 501 76 1 1 239 424 50625 II390625 5 IO 76 II543176 51529 11697083 5 1984 II 852352 52441 12 008 989 52900 12 1670OO 5 33 6i 12326391 5 38 24 12 487 l68 5 42 89 12649337 54756 12812904 5 52 25 I2977875 5560 13 144 256 56169 13312 053 56644 I3481 272 5 7 1 21 I3651919 5 7600 I3824 OOO 5 8081 13 997 521 585 64 14 172 488 5 90 49 14348907 59536 14 526 784 60025 14 706 125 605 16 14886936 6 1009 15069223 6 1504 15252992 6 2001 15 438 249 6 25 00 15 625000 63001 15813251 63504 16 003 008 64009 16194277 645 16 16 387 064 65025 16581375 65536 16 777 216 66049 16974 593 66564 I7I735I2 6 7081 17 373 979 6 7600 17 576000 681 21 17 779581 68644 17984728 691 69 18 191 447 69696 18 399 744 702 25 18 609 625 7°7 56 18821 096 7 12 89 19034163 7 18 24 19 248 832 7 2361 19 465 109 7 2900 19 683 OOO 7 34 4 1 19902 511 7 39 84 20 123648 7 45 29 20346417 7 50 76 20 570 824 75625 20 796875 7 61 76 21 024 576 767 29 21 253933 7 72 84 21 484 952 Square Root. 14.9331845 14.966 629 5 *5 . 15-033 2964 15.0665192 15.0996689 i5-*32 746 i 5 -i 6 5 7509 15.198 6842 15-231 546 2 15264 337 5 15.297 0585 I5-329 7°9 7 15.362 291 5 15-394 804 3 15.4272486 15.459 6248 I 5-49 I 933 4 15.5241747 I5-556349 2 I5-588 457 3 15.6204994 15.6524758 15.684 3871 15-7162336 15.748015 7 15-779 733 8 15.811 3883 15.842 9795 I5-874 507 9 15-9059737 15-937 377 5 15.9687194 16 16.031 2195 16.062 378 4 16.0934769 16.1245155 16.155 4944 16.186414 1 16.217 274 7 16.248 0768 16.278 8206 16.309 5064 16.3401346 16.370 705 5 16.401 219 5 # 16.431 676 7 16.462 077 6 16.492 422 5 16.522 71 1 6 16.5529454 16.583124 16.613247 7 16.643317 16.678 332 N UMBE 2 79 280 28l 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 31 1 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 SQUARES, CUBES, AND ROOTS. 277 Square. Cube. Square Root. I Cube Root. 77841 21 7I7 6 39 16.703 293 I 6.534 335 I 7 8400 21 952OOO 16.733 200 5 6.542 132 6 7 8961 22 188041 16.7630546 6.549911 6 7 95 24 22 425 768 16.792 8556 6.5576722 80089 22 665 187 16.822 603 8 6.5654144 806 56 22 906 304 16.8522995 6.573 138 5 8 12 25 23 I49 125 16.881 943 6.580 8443 8 17 96 23 393 656 16.9115345 6.588 532 3 823 69 23 639 903 16.9410743 6.596 202 3 82944 23 887 872 16.970 562 7 6.603 854 5 83521 24 137 569 17 6.61 1 489 841 00 24 389 OOO 17.029 3864 6.619 106 84681 24642 1 71 17.058 722 1 6.626 705 4 85264 24 897 088 17.088 007 5 6.634 287 4 85849 25 153 757 17.117 242 8 6.641 852 2 86436 25412 184 17.1464282 6.649 399 8 870 25 25 672 375 I7-I75 564 6.656 930 2 8 76 16 25 934 336 17.2046505 6.664 443 7 8 82 09 26 198 073 17.233 6879 6.6719403 8 88 04 26 463 592 17.262 6765 6.679 42 8 9401 26 730899 17.291 6165 6.686 883 1 90000 27000000 17.320 508 1 6.6943295 90601 27 270 901 I7*34935i6 6.701 7593 9 1204 27543608 17.378 1472 6.709 1729 9 1809 27 818 127 17.406 895 2 6.71657 924 16 28 094 464 17*435 595 8 6.723 9508 930 25 28372 625 17.4642492 6.731 3i5 5 93636 28 652616 I7-492 8557 6.738 664 1 942 49 28934443 17.521 415 5 6.745 9967 948 64 29218 112 17.5499288 6.753 3134 9 54 8i 29 503629 I7-578395 8 6.7606143 961 00 29 791 OOO 17.6068169 6.767 8995 967 21 30080 231 17*635 192 1 6.775 169 9 7344 30371328 17.663521 7 6.782 4229 97969 30664297 17.691 806 6.789 661 3 98596 30959 144 17.720045 1 6.796 8844 992 25 31 255 875 17.748 2393 6.804 °9 2 1 998 56 31 554496 17.7763888 6.811 284 7 10 04 89 31855013 17.8044938 6.818462 10 11 24 32157432 17*832 554 5 6.825 624 2 10 1761 32 461 759 17.860571 1 6.832 771 4 102400 32 768 OOO 17.8885438 6.839 9°3 7 10 3° 4 1 33076161 17.916472 9 6.847 02 1 3 10 36 84 33386248 . I7-944 3584 6.854 124 1043 29 33 698 267 17.972 2008 6.86l 212 1049 76 34012224 18 6.868 285 5 10 56 25 34328125 18.027 7564 6.875 344 3 10 62 76 34645976 18.055 47° 1 6.882 388 8 10 69 29 34 965 783 18.083 Hi 3 6.889418 8 10 75 84 35 287552 18. no 770 3 6.896 434 5 108241 35611 289 18.1383571 6.903435 9 108900 35 937 000 18.165 902 1 6.910423 2 10 95 61 36 264 691 18.1934054 6.9173964 1 1 02 24 36 594 368 18.220 867 2 6.924 3556 1 1 08 89 36926037 18.248 287 6 6.9313088 n 1556 37 259 704 A A 18.275 6669 6.938 232 1 337 33$ 339 340 34i 342 343 344 345 34^ 347 34§ 349 350 35i 352 353 354 355 35^ 357 358 359 360 361 362 3^3 364 365 366 367 368 3^9 37c 37i 372 37: 374 37! 37; 37* 37< 38c 38: 38: 38. 38. 38 38 38 38 SQUARES, CUBES, AND ROOTS. Square. II 22 25 1 1 28 96 1135 69 1 1 42 44 II 49 21 1 1 56 OO II 62 8 l 1 1 69 64 II 7649 H 83 36 119025 11 97 16 1204 09 12 11 04 12 1801 12 2500 123201 123904 124609 1253 16 126025 12 67 36 127449 12 81 64 128881 129600 J 13 03 21 j 13 1044 13 I 7 69 132496 1332 25 13 39 56 134689 13 54 24 1361 61 136900 137641 1383 84 13 9 1 29 1398 76 1406 25 14 J 3 76 1421 29 14 28 84 143641 144400 i 4 5 l61 14 59 24 14 66 89 147456 148225 j 14 89 96 14 97 69 15 05 44 15 13 21 15 21 00 I Cube. 37 595 375 37 933056 38 272 753 38614472 38958219 39 304000 39 651821 40 001 688 40 353607 40 707 584 41 063 625 41 421 736 41 781 923 42 144 192 42 508 549 42 875 000 43 243 551 43614208 43 986977 44361 864 44 738 875 45 118016 45 499 293 45 882712 46 268 279 46 656 000 47 °45 83 1 47 437 928 47 832 147 48 228 544 48 627 125 49027 896 49 430 863 49836032 50 243 409 50653000 51 064 81 1 51478 848 51895117 52 3 T 3 624 52 734 375 53 157 376 53 582 633 54010152 j 54 439939 ! KA 872 OOO 55 306 341 | 55 742 968 ; 56 181 887 56 623 104 57066625 57512456 57960603 58411072 58 863 869 593 i 900° Square Root. 18.3030052 18.330 302 8 18.357 559 8 18.3847763 18.4H 952 6 18.439 088 9 18.466 185 3 18.493 242 18.5202592 18.547 237 18.574175 6 18.601 O75 2 18.627936 18.6547581 18.681 541 7 18.708 2869 18.734994 18.761 663 18.788 2942 18.814887 7 18.841 4437 18.867 962 3 18.894 4436 18.920 887 9 18.947 2953 18.973 666 19 19.026 297 6 19-0525589 19.078 784 19.1049732 19.131 1265 19.157 244 1 19.183 3261 19.209 372 7 19-235 384 1 19.2613603 19.287 301 5 | i9-3 I 3 207 9 I9-3390796 j 19.364 9 l6 7 | 19.3907194 19.4164878 19.442 222 1 19.467 922 3 19-493 588 7 X 9-5 I 9 221 3 19.544 820 3 19-57° 3858 19-595 9*7 9 19.621 4169 19.646 882 7 19.6723156 19.697 7156 19.723 0829 19.7484177 I SQUARES, CUBES, AND ROOTS. 279 Number Square. Cube. Square Root. 1 Cube Root. 391 152881 59776471 I 9-773 7 I 9 9 7.3123828 392 15 3664 60 236 288 19.798 9899 7.3186114 393 15 44 49 60698 457 19.824 227 6 7.324 8295 394 15 52 36 6l 162 984 i 9-849433 2 7 * 33 1 0369 395 156025 6l 629 875 19.874 6069 7*337 233 9 396 15 68 16 62 099 136 19.899 748 7 7.343 420 5 397 15 7609 62 570 773 19.924 858 8 7.349 596 6 398 158404 63 044 792 J 9-949 937 3 7*355 762 4 399 15 92 01 63 521 199 I 9-974 984 4 7.3619178 400 16 0000 64 000 000 20 7.368 063 401 16 08 01 64481 201 20.024 984 4 7.374 1979 402 16 1604 64 964 808 20.049937 7 7.380322 7 403 ! 162409 65 450 827 20.074 859 9 7.386 437 3 404 1632 16 65 939 264 20.099 75 1 2 7.392 541 8 405 1640 25 66 430125 20.124611 8 7.3986363 406 1648 36 66 923 416 20.149441 7 7.404 7206 407 1656 49 67419 i 43 20.174 241 7.410795 408 1664 64 67917312 20.1990099 7.4168595 409 16 72 81 68417929 20.223 748 4 7.422 914 2 410 16 8l OO 68 921 000 20.248 456 7 7.428 958 9 4 ir 168921 69426 531 20.273 134 9 7.434 993 8 412 16 97 44 69934528 20.297 783 1 7.441 018 9 4 r 3 1705 69 70 444 997 20.322 401 4 7*447 °34 2 414 171396 70 957 944 20.346 989 9 7-453 0399 4 r 5 17 22 25 7 M 73 375 20-371 548 8 7.459 035 9 416 1730 56 71 991 296 20.396 078 1 7.465 022 3 417 1738 89 72511713 20.4205779 7.470999 1 418 174724 73 034 632 20.445 048 3 7.476 966 4 419 i 7 55 6i 73 560059 20.469 489 5 7.482 924 2 420 1 7 64 00 74 088 000 20.493 901 5 7.488 872 4 421 17 72 41 74618461 20.518 2845 7.494811 3 422 j 17 80 84 75 151 448 20.542 638 6 7.500 740 6 423 ; 1 7 89 29 75 686 967 20.566 963 8 7.506660 7 424 1797 76 76 225 024 20.591 260 3 7.512 571 5 425 18 06 25 76 765 625 20.615 528 1 7.518473 426 18 14 76 77308776 20.639 767 4 7.524 365 2 427 18 23 29 77854483 20.663 978 3 7.530 248 2 428 | 183184 78 402 752 20.688 1609 7.536 122 1 429 18 4041 78 953 589 20.712 315 2 7.541 986 7 430 18 4900 79 507 000 20.736441 4 7.547 842 3 43 i 185761 80 062 991 20.7605395 7*553 688 8 432 18 66 24 80621 568 20.784609 7 7.5595263 433 18 74 89 81 182 737 20.808 652 7*565 354 8 434 18 83 56 81 746504 20.832 666 7 7.571 1743 435 18 92 25 82312875 20.856 653 6 7.576 9849 436 190096 82 88 1 856 20.880613 7.582 786 5 437 190969 8 3 453 453 20.904545 7*5885793 43 8 19 18 44 84 027 672 20.928 449 5 7.594 363 3 439 19 27 21 84 604519 20.952 326 8 7.600 1385 440 193600 85 184000 20.976 177 7.605 904 9 441 194481 85 766 121 21 7.611 662 6 442 195364 86350888 21.023 796 7.617 411 6 443 1962 49 86938 307 21.0475652 7.623 151 Q 444 19 71 36 87528384 21.071307 5 7.628 883 7 445 19 80 25 88 121 125 21.095 023 1 7.634 606 7 446 1 198916 88 716 536 21.1187121 7.6403213 28 o SQUARES, CUBES, AND ROOTS. Number. Square. 447 19 98 09 448 20 Q7 04 449 20 l6oi 450 20 25 OO 45 i 20 34 OI 452 20 43 04 453 20 52 09 454 20 61 l6 455 20 70 25 456 20 79 36 457 20 88 49 458 20 97 64 459 21 0681 460 21 1600 461 21 25 21 462 21 3444 463 21 43 69 464 21 52 96 465 21 62 25 466 21 71 56 467 21 80 89 468 21 90 24 469 21 9961 470 22 0900 47 i 22 18 41 472 22 27 84 473 22 37 29 474 22 46 76 475 22 56 25 476 22 65 76 477 22 75 29 478 22 84 84 479 22 94 41 480 230400 481 23 1361 482 23 23 24 483 23 32 89 484 2 3 42 56 485 23 52 25 486 23 61 96 487 23 7 1 69 488 23 81 44 489 2391 21 490 2401 00 49 1 24 1081 492 24 20 64 493 243049 494 24 40 36 495 24 50 25 496 24 60 16 497 24 70 09 498 24 80 04 499 24 90 01 5 °o 25 00 00 501 25 1001 502 25 20 04 Cube. I Square Root. 89 314 623 89915 39 2 90518849 91 125 000 91 733 8 5 i 9 2 345 408 9 2 959 6 77 93576664 94 ^375 94818816 95 443993 96071 912 96 702 579 97 336000 97972 181 98611 128 99252847 99897 344 100 544 625 101 194696 101 847 563 102 503 232 103 16 1 709 103 823 000 104487 hi 105 154048 105 823817 106 496 424 107 171 875 107 850 176 108 531333 109 215 352 109 902 239 no 592 OOO III 284 641 III 980 168 112678 587 113379904 114084 125 1 14 791 256 115501303 116214272 116930 169 1 1 7 649 000 118370771 1 19 095 488 119823157 120553 784 121 287375 122023936 122 763 473 123 505 99 2 124 251499 125 000000 125 75 1 501 126 506 008 21.1423745 21.166010 5 21.189 620 1 21.2132034 21.236 7606 21.260 291 6 21.283 7967 21.307 275 8 21.330 729 21.3541565 21.377 558 3 21.4009346 21.424 285 3 21.447 6106 21.4709106 21.494 1853 2 i« 5 I 7 434 8 21.5406592 21.5638587 21.587 033 1 21.610 182 8 21.633307 7 21.656407 8 21.679 4834 21.702 5344 2 i. 7 2 5 561 21.748 5632 21.771 541 1 21-794 494 7 21.8174242 21.840 3297 21.863211 1 21.8860686 21.908 9023 21.931 7122 21.954498 4 21.977 261 22 22.022 715 5 22.045 407 7 22.0680765 22.090 722 22.1133444 22.135 9436 22.1585198 22.181 073 22.2036033 22.226 no 8 22.248 595 5 22.271 057 5 22.2934968 22 StSW 6 22.3383079 22.3606798 22.383 0293 22.405 356 5 Cube Root. 7.646 027 2 7.651 724 7 7.6574138 7.6630943 7.668 766 5 7.6744303 7.680085 7 7.685 732 8 j.691 371 7 7.697 002 3 7.702 6246 7.708 238 8 7.7138448 7.7194426 7.7250325 7.7306141 7.736 187 7 7-741 7532 7-747 310 9 7.752 8606 7.7584023 7-763 936 i 7.769 462 7.7749801 7.780 4904 7.7859928 7 - 79 1 487 5 7.7969745 7.802 453 8 7.807 9254 7.8133892 7.818 845 6 7.824 294 2 7-829 735 3 7.835 1688 7.840 594 9 7.8460134 7.8514244 7.856 828 1 7.862 224 2 7.867 613 7.8729944 7.878 3684 7-883 735 2 7.8890946 7.894 446 8 7.899 791 7 7.905 1294 7 - 9 io 4599 7 - 9 I 5 7832 7.921 0994 7.926 408 5 7 - 93 1 7 i° 4 7-937 005 3 7.942 293 1 7-947 5739 Ncmbe 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 54 i 542 543 544 545 546 547 548 549 550 55 i 552 553 554 555 556 557 558 SQUARES, CUBES, AND ROOTS. 28l Square. Cube. 25 3009 127263527 25 40 16 128 024 064 25 50 25 128 787 625 25 60 36 129 554216 25 70 49 130 323 843 25 80 64 131 096512 259081 I31 872 229 26 01 00 132651 OOO 26 11 21 I 33 432831 2621 44 134 217 728 2631 69 135 005 697 2641 96 *35 796 744 26 52 25 136590875 26 62 56 J 37 388 096 26 72 89 138188413 26 83 24 138991 832 269361 x 39 798 359 270400 140 608000 27 1441 141 420 761 27 24 84 142 236 648 27 35 29 *43 055667 2745 76 I 43 877 824 2756 25 144 703 125 27 66 76 *45 53 i 576 27 7729 146363 183 27 87 84 r 47 197 952 2798 41 148 035 889 28 09 00 148 877 OOO 28 1961 149 721 291 28 30 24 150 568 768 28 40 89 I 5 I 4 X 9 437 285156 I 5 2 273 304 28 62 25 *53 I 3°375 28 72 96 ^3990656 28 83 69 J 54 854153 289444 155 720872 29 05 21 156 590819 29 16 00 157464000 29 2681 J 5 8 340421 29 37 64 159 220 088 294849 160 103007 29 5936 160989 184 29 70 25 161 878 625 2981 16 l6 2 771 336 29 92 09 163667323 30 03 04 164566 592 30 1401 165 469 I49 30 2500 166 375 OOO 303601 167 284151 304704 168 196 608 36 58 09 169112377 3069 16 170031464 30 80 25 1 70 953 875 309136 171 879616 3102 49 172 808693 3 i 1364 173 74 i 1 12 A A* Square Root. 22.427 661 5 22.4499443 22.472 205 I 22.494 443 8 22.5166605 22 - 538 855 3 22.561 0283 22.583 1796 22.605 309 1 22.627 417 22.649 503 3 22.671 568 1 22.693 61 1 4 22.715633 4 22.737634 22.7596134 22.7815715 22 8035085 22.825 4244 22 8473193 22.8691933 22 891 0463 22.912 8785 22.934 6899 22.956 4806 22.978 2506 23 . 23.021 7289 23- 043 437 2 23.065 125 2 23.086 792 8 23.10844 23.130067 23.1516738 23.173 2605 23.194 827 23.2163735 ^3-237 900 1 23.2594067 23.280 8935 23-302 3604 23-323 8076 23-345 235 1 23.366 6429 23.388031 1 23-409 399 8 23-430 749 23-452 078 8 23-473 389 2 23.494 680 2 23 - 5 I 5 952 23-5372046 23 558 438 23-5796522 23.600 847 4 23.622 023 6 I j Cube Root. 7.952 847 7 7.9581144 7-963 374 3 7.968 627 I 7-973 873 I 7- 979 112 2 7.9843444 7.989 569 7 7.994 7883 8 8.005 204 9 8.010 403 2 80155946 8.020 7794 8.025 957 4 8.031 128 7 8.036 293 5 8041 451 5 8.046603 8- 051 747 9 8.056 8862 8.062 018 8.067 14 3 2 8.072 262 8.077 374 3 8.082 48 8.0875794 8.092 672 3 8 097 7589 8.102 839 8.107 912 8 8.112 9803 8.118041 4 8.123 0962 8.128 144 7 8.133187 8.138 223 8.1432529 8.148 276 5 8-153 2939 8.158 305 1 8.1633102 8.168 3092 8.173 302 8.1782888 8.1832695 8.188 244 1 8.193212 7 8-1981753 8.203 131 9 8.208 082 5 8.213027 1 8.2179657 8.222 898 5 8.227 8254 8.232 7463 2! 282 SQUARES, CUBES, AND ROOTS. jfu Number. I Square. 559 560 561 562 563 564 565 566 567 568 569 57° 571 57 2 573 574 575 57 6 577 578 579 580 581 582 583 584 585 586 587 588 589 59° 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 61 1 612 613 614 31 24 81 31 3600 3 1 47 21 3 1 58 44 31 6969 31 80 96 31 92 25 32 03 5 6 32 14 89 32 26 24 32 37 61 32490° 32 60 41 32 71 84 32 83 29 32 94 76 33 06 25 33 17 7 6 33 29 29 334084 33 52 4i 336400 33 75 61 338724 33 98 89 34 i° 56 3422 25 34 33 96 3445 69 34 57 44 346921 3481 00 34 92 81 35 04 64 35 i 6 49 35 28 36 35 40 25 35 52 16 35 6 4°9 35 7604 35 8801 36 00 00 36 12 01 36 24 04 363609 36 48 16 36 60 25 36 72 36 368449 36 96 64 37 08 81 47 21 00 37 33 21 37 45 44 37 57 69 376996 Square Root. Cube Root. 174 676879 175616000 176558 481 177 504 328 178453 547 179406 144 180 362 125 181 321 496 182 284 263 183 250 432 184220009 185 193000 186 169411 187 149 248 188 132517 189 1 19 224 190 109 375 191 102 976 1 192 100 033 193 100 552 194 104 539 195 112000 196 122 941 197 *37 368 198155287 199 176 704 200 201 625 201 230 056 202 262 003 203 297 472 204 336 469 205 37900 ° 206425 071 207 474 688 208 527 857 209 584 584 210644 875 21 1 708 736 212 776 173 213847 I 9 2 214921 799 216000000 217 081 801 218 167 208 219256 227 220 348 864 221 445 I2 5 222 545 016 223 648 543 224 755 7 12 225 866 529 226981 000 228099 131 229 220 928 230 346 397 231 475 544 23.643 180 8 23.664 319 1 23.685 4386 23.706 539 2 23.727 621 23.748 6842 23.769 728 6 23-79° 754 5 23.811 761 8 23.832 750 6 23.853 7209 23.874672 8 23.8956063 23.916521 5 23.937 4 1 8 4 23.958 2971 23-979 J 57 6 24 24.020 824 3 24.041 630 6 24.062 418 8 24.083 189 1 24.103 94 1 6 24.1246762 24.1453929 24.166091 9 24.1867732 24.207 436 9 24.228 082 9 24.2487113 24.269 322 2 24.289915 6 24.310491 6 24 33 1 °5° 1 24-35 1 59 1 3 24-372 n5 2 24-392 621 8 24 413 in 2 24-433 583 4 24.454 038 5 24.474 476 5 24.494 897 4 24-5 1 5 3° 1 3 24-535 688 3 24.5560583 24.576 41 1 5 24- 59 6 747 8 24.617067 3 24-637 37 24.657 656 24.677 925 4 24.698 178 1 24.7184142 24.738 6338 24.758 8368 24.779 0234 8.237 661 4 8.2425706 8.247 474 8.252 3715 8.257 263 3 8.262 1492 8.267 0294 8.2719039 8.2767726 8.281 625 5 8.2864928 8.291 344 4 8.2961903 8.301 0304 8.305 865 1 8.310 694 1 8.315 517 5 8.320 335 3 3.325 147 5 8.329954 2 8.334 755 3 8.339 550 9 8-344 34i 8-349 I2 5 6 8.353 904 7 8.3586784 8.3634466 8.368 209 5 8.372 9668 8-377 7 l8 8 8.382 465 3 8.387 2065 8.391 942 3 8.3966729 8.401 398 1 8.406 1 18 8.4108326 | 8.415 54i 9 8.420246 8.424 944 8 8.429 6383 8.434 3 2 6 7 8.439009 8 8.4436877 8.448 360 5 8.4530281 8.4576906 8.462 347 9 8.467 8.471 647 1 8.476 289 2 8.480926 1 8.4855579 8.490 1848 8.494 806 5 8.4994233 i !> UMJ 6ij 6i< 6i' 6i< 6i< 62( 62] 62: 62^ 624 62= 62^ 62 y 628 62g 63O 631 632 633 634 635 636 637 638 639 64O 64I 642 643 644 645 646 647 648 949 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 SQUARES, CUBES, AND ROOTS. 37 82 25 37 94 56 380689 38 19 24 383161 384400 38 5641 38 68 84 38 81 29 3893 76 390625 39 18 76 393129 3943 84 395641 39 69 00 3981 61 39 94 24 40 06 89 40 19 56 40 32 25 40 44 96 405769 40 70 44 4083 21 409600 41 08 81 41 21 64 413449 41 47 36 416025 41 73 16 41 86 09 ; 419904 42 1201 | 42 25 00 42 38 01 425104 42 6409 42 77 16 429025 430336 43 16 49 432964 43 4281 43 56oo 43 6921 43 82 44 43 95 69 4408 96 44 22 25 4435 56 4448 89 4462 24 44 75 61 448900 232 608 375 233 744 896 234885 1 13 236 029 032 237176659 238 328 000 239 483 061 240 641 848 241 804 367 242970 624 244 140625 245 134 376 246 491 883 247 673 152 248 858 189 250 047 000 25 1 239 591 2 52 435 968 253 636 137 254 840 104 256047875 257 259456 258474853 259 694 072 260917 119 262 144 000 263374 721 264 609 288 265 847 707 267 089 984 268 336 125 269585 136 270 840023 272 097 792 273 359 549 274 625000 275 894 451 277 167 808 278445077 279 726 264 281 on 375 282 300416 283 593393 284 890312 286 191 179 287 496000 288 804 781 290117528 291 434 247 292 754 944 294079625 295 408 296 296740963 298 077 632 299418309 300 763 000 24.799 19 3 5 24.8193473 24.8394847 24.859 605 8 24.879 7106 24.899 799 2 24.919871 6 24- 939927 8 24.959 9679 24.979992 25 25.019992 25.039 968 1 25.0599282 25.0798724 25.099 8008 25.1197134 25.1396102 25- I5949I3 25-I79356 6 25.199 2063 25.2190404 25-238 858 9 25.258 661 9 25.2784493 25.298 221 3 25-3i7 977 8 25-337 7*8 9 25-357 444 7 25-377 *55 * 25-396 8502 25-4*65301 25.436 194 7 25-455 8441 25475 478 4 25-495 097 6 25.514701 6 25-534 2907 25 553 8647 25-573 423 7 25.592 9678 25.6124969 25.632011 2 25-6515107 25.6709953 25.6904652 25-7099203 25.729 3607 25-748 7864 25.768 1975 25-787 593 9 25.8069758 25.8263431 25-845 696 25-8650343 i 25-8843582 | 283 Cube Root. 8.504 035 8.508641 7 8.513 243 5 8.5178403 8.522 432 I 8.5270189 8.5316009 8-536I78 8.540 750 I 8-545 3*7 3 8.5498797 8-554 437 2 8.5589899 8.563 537 7 8.568 080 7 8.572 6189 8-577I523 8.581 6809 8.586 204 7 8-590 723 8 8-595 238 8.599 747 6 8.604 252 5 8.608 752 6 8.613 248 8.617 7388 8.622 224 8 8.626 706 3 8.631 183 8.635 655 1 8.640 122 6 8.644 585 5 8.6490437 8.6534974 8.6579465 8.662 391 1 8.666 831 8.671 2665 8.675 6974 8.680 123 7 8.684 545 6 8.688 963 8-6933759 8.697 7843 8.702 1882 8.7065877 8.710982 7 8-7*5 3734 8.7197596 8.724 141 4 8.7285187 8.732 891 8 8.7372604 8.741 6246 8.745 9846 8.7503401 284 SQUARES, CUBES, AND ROOTS. Number. Square. Cube. Square Root. 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 • 688. 689 690 691 692 693 694 695 696 697 698 699 700 701 702 7°3 7°4 7°5 706 707 708 7°9 710 711 712 713 714 715 716 717 718 719 720 721 722 7 2 3 724 7 2 5 726 45 02 41 45 15 84 45 29 29 45 42 76 45 56 25 45 69 76 45 83 29 45 9684 46 1041 46 24 00 463761 465124 46 64 89 46 78 56 4692 25 47 °5 96 47 x 9 6 9 47 3344 47 47 2i 47 61 00 47 748 i 47 88 64 48 02 49 48 16 36 483025 48 44 16 48 5809 48 72 04 48 8601 490000 49 1401 49 28 04 4942 09 49 56 16 49 7° 25 49 8436 4998 49 50 12 64 50 26 81 5041 00 50 55 21 50 69 44 50 83 69 509796 51 1225 51 26 56 51 4089 5 X 55 24 51 6961 51 8400 5 1 98 4 1 52 1284 52 27 29 52 4 1 76 52 5625 52 7° 76 302 III 711 303 464 448 304821 217 306 182 024 307 546 875 308 915 776 310288 733 311665 752 313046839 314 432 OOO 315821241 317214568 318611 987 320013504 321 419 125 322 828 856 324 242 703 325 660 672 327 082 769 328 509 OOO 329939371 331 373 888 332812557 334255 3 8 4 335 7° 2 375 337153530 338608873 340 068 392 34 1 S3 2 099 343 000 000 344472101 345 948 408 347428927 348913664 350 402 625 351895816 353393 243 354 894 9 1 2 356 400 829 3C7 QII 000 359425 43 x 360944 128 362 467 097 363 994 344 365 525 875 367 061 696 368601 813 370 146 232 37 1 694 959 373 248 000 374 805 361 376 367 048 377 933067 379 503 424 381 078 125 1 382657176 25.903 667 7 25.922 962 8 25.942 243 5 25.961 51 25.980 762 1 26 26.0192237 26.038 433 1 26.057 628 4 26.0768096 26.095 976 7 26.115 1297 26.134 268 7 26.153 393 7 26.172 5047 26.191 601 7 26.2106848 26.229 754 1 26.248 809 5 26.267 851 1 26.2868789 26.305 892 9 26.324 893 2 26.3438797 26.362 852 7 26.381 81 1 9 26.400 757 6 26.419 689 6 26.438 608 1 26.457 5 X 3 X 26.476 404 6 26.495 282 6 26.5141472 26.532 998 3 26.5518361 26.5706605 26.589 47 16 26.608 269 4 26.6270539 26.645 825 2 26.664 583 3 26.683 328 1 26.702 0598 26.720 7784 26.739 4839 26.7581763 26.776 855 7 26.795 522 26.8i4 x 75 4 26.8328157 26.851 443 2 26.8700577 26.888 659 3 26.907 248 1 26.925 824 26.944 387 2 Cube Root. 8.754 69 1 3 8.759 °3 8 3 8.763 3809 8.7677192 8.7720532 8.776 383 8.780 7084 8.785 0296 8.789 3466 8.793 6 59 3 8.797 967 9 8.802 272 1 8.8065722 8.810868 1 8.8151598 8.819447 4 8.823 730 7 8.8280099 8.832 285 8.8365559 8.840 822 7 8.845 085 4 8.849 344 8.853 598 5 8.8578489 8.862 095 2 8.8663375 8.8705757 8.874 809 9 8.87904 8.883 266 1 8.887 488 2 8.891 7063 8.895 9204 8.9001304 8.904 33 6 6 8.908 538 7 8.9127369 8.916931 1 8.921 1214 8.925 3°7 8 8.929 49° 2 8.9336687 8.937 8433 8.942014 8.946 1809 8.950 343 8 8-954 5029 8.958 658 1 8.962 809 5 8.966 957 8.971 1007 8.975 240 6 8.979 3766 8.983 5089 8.9876373 i i 1 j i 1 I UMBI 727 728 729 730 731 732 733 734 735 73 6 737 738 739 740 74i 742 743 744 745 746 747 748 749 750 75i 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 SQUARES, CUBES, AND ROOTS. Square. Cube. Square Root. 52 85 29 52 99 84 53 i4 4i 53 2900 53 436i 53 58 24 53 72 89 53 87 56 5402 25 54 16 96 54 3 1 69 54 46 44 5461 21 54 76 00 549081 550564 55 2049 55 35 36 55 50 25 55 65 16 55 80 09 55 95 04 56 1001 56 2500 56 4001 565504 56 70 09 56 85 16 570025 5715 36 57 30 49 57 45 64 576081 57 7600 5791 21 580644 58 21 69 583696 58 52 25 586756 58 82 89 58 98 24 59 1361 59 29 00 5944 41 59 5984 59 75 29 5990 76 6006 25 60 21 76 603729 6052 84 6068 41 60 84 00 609961 61 1524 384 240583 385 828 352 387420489 389017000 390617891 392 223 168 393832 837 395 446904 397065375 398 688 256 400315553 40 1 947 272 403 583 419 405 224 000 406 869 021 408518488 410 172 407 41 1 830 784 413493625 415 160936 416832 723 418508 992 420489 749 421 875000 423 564751 425 259 008 426957777 428 661 064 430368875 432 081 216 433 798093 435 5i9 512 437 245 479 438 976 000 440 71 1 081 442 450 728 444 194 947 445 943 744 447697 125 449 455 096 45 1 217663 452 984 832 454 756 609 456 533ooo 458314011 460 099 648 461 889917 463 684 824 465 484 375 467 288 576 469097433 470910952 472 729 139 474 552 000 476379 54i 478 21 1 768 26.962 937 z 26.981 475 1 27 27.018 512 2 27.037011 7 27.055 498 5 27-073 972 7 27.092 434 4 27.1108834 27.1293199 27-I47 743 9 27.1661554 27-184 5544 27.202 941 27.2213152 27.239 676 9 27.258 026 3 27.2763634 27.294688 1 27.3130006 27-33I 300 7 27*3495887 27.367 8644 27.3861279 27.4043792 27.422 6184 27.440 845 5 27-4590604 27.4772633 27.495 454 2 27-513633 27-531 7998 27-549 954 6 27.568 097 5 27.5862284 27.604 347 5 27.622 4546 27-6405499 27.6586334 27.676 705 27.694 764 8 27.7128129 27.7308492 27.7488739 27.7668868 27.784 888 27.802 877 5 27.820 8555 27.838 821 8 27.856 7766 27.8747197 27.8926514 27-9105715 27.928480 1 27-9463772 27.964 262 9 285 I Cube Root. 8.991 762 8.995 882 9 9 9.0041134 9.008 222 9 9.012 3288 9.0164309 9.0205293 9.024 623 9 9.028 7149 9.032 802 1 9.036 885 7 9.040 965 5 9.045 041 7 9.049 1142 9-053 183 1 9.0572482 9.061 309 8 9.065 367 7 9.069 422 9.0734726 9-077 5I9 7 9.081 563 1 9.085 603 9.089 639 2 9.0936719 9.097 701 9.101 7265 9-io5 7485 9.109 7669 9.113 7818 9.1177931 9.121 801 9-125 8053 9.129 806 1 9-133 8034 9-I37 797I 9-141 7874 9- I 45 774 2 9.149 757 6 9-153 737 5 9* I 57 7U39 9.161 6869 9.165 6565 9.169622 5 9-I73 585 2 9-177 544 5 9.181 5003 9 -i 85 4527 9.189401 8 9-193 3474 9-197 2897 9.201 2286 9.205 164 1 9.2090962 9.213025 286 SQUARES, CUBES, AND ROOTS. 7 8 3 784 785 786 787 788 789 790 79 1 792 793 794 795 79 6 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 ..L Square Root. 6l 30 89 6l 46 56 6l 62 25 61 77 96 61 93 69 62 09 44 62 25 21 62 41 00 62 5681 62 72 64 62 88 49 630436 63 20 25 63 36 16 63 52 09 63 68 04 63 84 01 64 00 00 64 1601 64 32 04 64 48 09 64 64 16 64 80 25 64 96 36 65 1249 65 28 64 65 4481 65 61 00 65 77 21 65 9344 660969 66 25 96 66 42 25 66 58 56 66 74 89 66 91 24 67 07 61 67 24 00 67 40 41 67 56 84 67 73 2 9 67 89 76 68 06 25 68 22 76 6839 29 685584 68 72 41 68 89 00 6905 61 69 22 24 69 38 89 695556 69 72 25 69 88 96 70 05 69 70 22 44 480048 687 481 890 304 483 736625 485 587 656 487 443 403 489303872 491 169 069 493039 ooo 494 9 i 3 6 7 i 496 793 °88 498 677257 500 566 184 502459875 504 358336 506 261 573 508 169 592 510082 399 512 000000 513 922 401 515 849 608 517 781627 519718464 521 660 125 523 606 616 525 557 943 5 2 7 5*4 112 529475 I2 9 53 1 44 1 000 533 4« 73J 535 387 328 537 367 797 539353 x 44 541 343 375 543 338490 545 338 513 547 343432 549353 259 K C I 368 OOO 553387661 555 4 12 248 557 44 1 767 559476 224 561515625 563 559976 565 609 283 567 663 552 569 722 789 571 787 ooo 573856 I9 1 575 93 0 368 578009537 580093 704 582 182 875 584 277 056 586 376 253 588 480 472 27.982 137 2 28 28.017 851 5 28.035 691 5 28.0535203 28.071 337 7 28.089 1438 28.1069386 28.124 722 2 28.1424946 28.160255 7 28.1780056 28.195 7444 28.213472 28.231 1884 28.248 893 8 28.266 588 1 28.284271 2 28.301 943 4 28.3196045 28.337 2546 28.354 8938 28.372 521 9 28.390 139 1 28.407 745 4 28.425 340 8 28.442 925 3 28.460 498 9 28.478061 7 28.495 613 7 28.5131549 28.530 68 5 2 28.548 2048 28.565 713 7 28.5832119 28.600 699 3 28.618 176 28.635 642 1 28.653 0976 28.670 542 4 28.687 976 6 28.7054002 28.7228132 28.740215 7 28.757 607 7 28.774 989 1 28.792 360 1 28.809 7206 28.8270706 28.844 4 10 2 28.861 739 4 28.879 0582 28.896 366 6 28.9136646 28.9309523 28.948 229 7 9.2169505 9.2208726 9.224 791 4 9.228 706 8 9.2326189 9.236527 7 9.2404333 9-244 335 5 9.248 234 4 9-252 13 9.256 022 4 9.259 91 1 4 9.263 797 3 9.267 679 8 9-271 5592 9-275 435 2 9.279 308 1 9.2831777 9.287 044 9.290907 2 9.294 767 1 9.298 623 9 9.302 477 5 9.3063278 9.310175 9 - 3 I 4 OI 9 9 - 3 I 7 8599 9.3216975 9-325 532 9-329 3 6 3 4 9 - 333 I 9 I 6 9-337 016 7 9 - 34 ° 838 6 9344 657 5 9.3484731 9.352 285 7 9-3560952 9-359 9 01 6 9-363 704 9 9-367 505 1 9 - 37 1 3°2 2 9-375 096 3 9.378 8873 9.382 675 2 9.38646 9.390 241 9 9.3940206 9.397 7964 9.401 569 1 9-405 338 7 9.409 105 4 9.412 869 9.4166297 9.420 387 3 9.424 142 9.4278930 Numbe 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 SQUARES, CUBES, AND ROOTS. Square. Cube. Square Root. 70 3921 70 5600 70 72 8l 70 89 64 71 0649 7 1 2 336 714025 715716 71 7409 71 91 04 72 08 01 72 25 00 72 4201 72 5904 72 7609 72 93 16 73 10 25 732736 73 44 49 7361 64 73 78 81 739600 74 13 21 74 30 44 744769 746496 74 82 25 74 99 56 75 1689 75 34 24 75 5i6i 75 69 00 75 8641 760384 76 21 29 763876 76 56 2 5 76 73 76 7691 29 7708 84 77 2641 774400 7761 61 77 79 2 4 77 96 89 78 14 56 78 32 25 784996 78 67 69 78 85 44 790321 79 21 00 7938 8i 79 5664 79 74 49 7992 36 590589719 592 704 000 594 823 321 596 947 688 599077107 601 21 1 584 603351 125 605 495 736 607 645 423 609800 192 61 1 960049 614 125 000 616295 051 618470 208 620 650477 622 835 864 625 026375 627 222 016 629 422 793 631 628 712 633 839 779 636 056 000 638277381 640 503 928 642 735 647 644 972 544 647214625 649 461 896 651714363 653 972 032 656 234 909 658 503 000 660 776311 663 054 848 665 338617 667627 624 669921 875 672 221 376 674 526 133 676 836 152 679 151 439 681 472 000 683 797 841 686 128 968 688 465 387 690 807 104 693 i 54 125 695 506456 697 864 103 700 227072 702 595 369 704 969 000 707 347 97i 709 732 288 712 121 957 714516984 28.965 496 7 28.982 753 5 29 29.0172363 29.034 462 3 29.051 678 1 29^68 883 7 29.086 079 1 29.103 2644 29.120 4396 29.1376046 29-I54 759 5 29.1719043 29.189039 29.206 163 7 29.223 278 4 29.246383 29-257 477 7 29.274 562 3 29.291 637 29.308 701 8 29-325 7566 29.342 801 5 29-3598365 29.376 861 6 29-393 876 9 29.410882 3 29.427 877 9 29.444 863 7 29.461 839 7 29.478 805 9 29.495 762 4 29.512 709 1 29.529 646 1 29-546573 4 29-563491 29-5803989 29-597 297 2 29.614 185 8 29.631 064 8 29.647 934 2 29.664 793 9 29.681 644 2 29.698 484 8 29-7I5 3I5 9 29-732 1375 29.7489496 29*765 752 1 29.782 545 2 29.7993289 29.816 103 29.832 867 8 29.849 623 1 29.866 369 29.883 105 6 29.899 832 8 287 Cube Root. 9.431 642 3 9-435 38 9.439 130 7 9.4428704 9.446 607 2 9-450 34i 9.4540719 9 457 799 9 9.461 5249 9.465 247 9.468 966 1 9.472 682 4 9476395 7 9.480 106 1 9.483 813 6 9.4875182 9.491 22 9.4949188 9.498 614 7 9.502 307 8 9-505 998 9.5096854 9-5133699 9-5170515 9.520 7303 9.5244063 9.5280794 9-53i 749 7 9-535 4I7 2 9-539 081 8 9-542 743 7 9.546 402 7 9-5500589 9-553 7I2 3 9-557 363 9.561 0108 9-5646559 9.568 298 2 9-57r937 7 9-575 574 5 9-579 2085 9.582 839 7 9.586 468 2 9-590 093 7 9-593 7i69 9-597 337 3 9.600 954 8 9.604 569 6 9.608 181 7 9.611 791 1 9-6i5 397 7 9.619001 7 9.622 603 9.626 201 6 9.629 797 5 9-633 390 7 SQUARES, CUBES, AND ROOTS. 2 Ni 288 Square. 895 896 897 898 899 900 901 902 9°3 9°4 9°5 906 9°7 908 9°9 910 9 11 912 9 T 3 9*4 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 93 ° 93 1 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 80 10 25 80 28 16 804609 80 64 04 80 82 01 81 0000 81 1801 81 3604 81 5409 81 72 16 81 9025 820836 82 26 49 82 44 64 82 62 81 82 81 00 82 9921 831744 833569 835396 83 72 25 839056 840889 84 27 24 84 45 61 84 6400 84 82 41 85 00 84 85 19 29 85 37 76 85 56 25 85 74 76 85 93 29 86 1 1 84 86 30 41 86 4900 86 67 61 86 86 24 87 04 89 87 23 56 87 42 25 87 6096 87 7969 879844 88 1721 88 36 00 88 5481 88 73 64 88 92 49 89 11 36 893025 89 49 16 89 6809 89 87 04 90 06 01 902500 Square Root. 716917375 7 i 93 2 3 * 3 6 721 734 273 724150 792 726 572 699 729000000 I 73 1 432 701 733 870 808 73 ^ 3 X 4 3 2 7 738 763 264 741 217625 743 677416 746 142 643 748613312 751089 429 753571000 756058031 758550528 761 048 497 7 6 3 55 i 944 766 060 875 768 575 296 771 095213 773 620 632 776 15 1 559 778 688 000 781 229961 783 777 448 786330467 788 889 024 79 1 453 I2 5 794 022 776 796 597 983 799178 752 801 765 089 804357000 806 954 491 809557568 812 166237 814780 504 817400375 820 025 856 822 656 953 825 293 672 827936019 830 584 000 833237621 835 896 888 838 561 807 841 232 384 843 908 625 846590536 849278 123 851 97i 392 854670349 857 375 000 29.916 5506 29-933 259 1 29.9499583 29.966 648 1 29.983 3287 30 30.016 662 30.0333148 30.0499584 30.066 592 8 30.0832179 30.099 8339 30.1164407 30-1330383 30.149 6269 30.1662063 30.1827765 30.199 337 7 30.215 8899 30.232 432 9 30.248 966 9 30.265 491 9 30.282 007 9 30.298 514 8 30.3150128 30 - 33 1 501 8 30.3479818 30.364 452 9 30.3809151 30-397 368 3 30.4138127 30.430 248 1 30.446 674 7 30.463 092 4 30.479 5 oi 3 30.495 9 ° 1 4 30.512 292 6 30.528 675 30.545 048 7 30.5614136 30.577 769 7 30.5941171 30.6104557 30.626 785 7 30.643 1069 30.6594194 30.675 723 3 30.692 018 5 30.708 305 1 30.724 583 30.7408523 30 . 757 JI 3 30.773 365 l 30.789 608 6 30.805 843 6 30.822 07 Cube Root. 9.636 981 2 9.640 569 9.6441542 9.647 736 7 9.651 3166 9.654 893 8 9.658 468 4 9.662 040 3 9.6656096 9.669 1 76 2 9.672 74O 3 9.676 3OI 7 9.679 860 4 9.6834166 9.686 97O I 9.690 521 I 9.694 069 4 9.6976151 9.7OI 1583 9.704 6989 9.708 2369 9.7H 7723 9 - 7 I 5 305 I 9-7 l8 835 4 9.722 363 1 9.725 888 3 9.7294109 9'732 930 9 9.7364484 9-739963 4 9-743 475 8 9-746985 7 . 9-750 493 9-753 997 9 9-757 500 2 ; 9.7610001 ' 9.7644974 * 9.7679922 j 9 - 77 1 484 5 | 9-774 9743 j 9.7784616 - 9.7819466 j 9.7854288 9.7889087 1 9.7923861 9.795861 1 J 9-7993330 9.8028036 9.806271 1 9.8097362 9.813 1989 9.816659 1 9.820 1169 9-8235723 9.827 025 2 I 9- 8 30 475 7 UMBER 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 SQUARES, CUBES, AND ROOTS. 289 Square. I Cube. ' Square Root. | Cube Root. 9044OI 860085 351 30.838 287 9 9.8339238 906304 862 8oi 408 30.854 4972 9-837 369 5 9082 09 865 523177 30.870 698 I 9.840812 7 91 OI 16 868 250 664 30.886 890 4 9.8442536 91 20 25 870983875 30.903 074 3 9.847 692 9!3936 873 722 816 30.919 247 7 9.851 128 9 1 5849 876467 493 3 0 .935 4l6 6 9.854 561 7 91 7764 879217912 30.951 575 1 9.8579929 91 9681 881 974079 30.967 725 1 9.861 421 8 92 1600 884 736 000 30.983 866 8 9.864 848 3 92 3521 887 503 681 3i 9.868 272 4 92 54 44 890 277 128 31.016 1248 9.871 694 1 92 7369 893056 347 31.032 2413 9.875 1135 Q2 Q2 Q6 895 841 344 31.048 3494 9-878 530 5 93 12 25 898632 125 31.0644491 9.881 945 1 93 3i 56 901 428 696 31.0805405 9-885 357 4 93 5089 904 231 063 31.096 6236 9.888 767 3 93 70 24 907 039 232 31.112 698 4 9.8921749 93 89 61 909853 209 31.128 7648 9.895 580 1 940900 912 673000 31.144 823 9.898 983 942841 915498611 31.160 872 9 9.902 383 5 94 47 84 918330048 31.1769145 9.905 781 7 94 67 29 921 167317 31.1929479 9.909 177 6 94 86 76 924010424 31.208973 1 9.912 571 2 950625 926 859375 31.224 99 9.9159624 95 25 76 929 714 176 31.2409987 9-9I935I3 95 45 29 932 574 833 31.2569992 9.922 7379 95 64 84 935 441352 31.2729915 9.926 122 2 95 8441 938 3 1 3 739 31.2889757 9.929 5042 960400 941 192 000 31.304951 7 9.932 8839 96 23 61 944076 141 31.3209195 9.936 261 3 9643 24 946 966 168 31.3368792 9.9396363 96 62 89 949 862 087 31.3528308 9.943 009 2 96 82 56 952 763 904 31.3687743 9-946379 7 97 02 25 955671625 31.384 7097 9.949 7479 97 21 96 958 585 256 31.4006369 9.9531138 974169 961 504 803 31.4165561 9956477 5 9761 44 964 430 272 31432 4673 9-9598389 97 81 21 967 361 669 31.4483704 9.963 198 1 9801 00 970 299 000 31.4642654 9.9665549 98 2081 97 3 242271 31.4801525 9.9699095 98 40 64 976 191 488 31.4960315 9.973 261 9 98 6049 979 146657 31.5119025 9.976612 98 80 36 982 107 784 31.5277655 9-9799599 990025 985 074 875 31.5436206 9.983 305 5 99 20 16 988 047 936 31.5594677 9.986 648 8 994009 991 026 973 3i-575 3o68 9.98999 996004 994 01 1 992 31.591 138 9.993 3289 998001 997 002999 31.606961 3 9.996665 6 1 000000 1 000000000 31.622 7766 10 1 000201 1 003003001 31.638 584 10.003 322 2 1 00 40 04 1 006012008 31.6543836 10.006662 2 1 006009 1 009027027 31.6701752 10.009 989 9 1 00 80 16 x 012 048 064 31.685 959 10.013 315 5 1 01 00 25 1 015075 125 31.701 7349 10.016 638 9 1 01 20 36 1 018 108216 B B 3i-7i7 5o 3 10.019960 1 290 SQUARES, CUBES, AND ROOTS, Number. Square. Cube. Square Root. | Cube Root. IOO7 I OI 40 49 I 021 147 343 31.733 2633 IO.023 279 I 1008 I OI 60 64 I O24 192 512 31.7490157 10.026 595 8 IOO9 I 01 8081 I 027 243 729 31.764 7603 IO.029 910 4 IOIO I 02 OI OO I O30 30I OOO 31.7804972 10.033 222 8 IOII I 02 21 21 IO33 364 331 31.796 2262 IO.O36 533 IOI2 I 02 41 44 IO36433 728 31.8119474 IO.O39 841 1013 I 02 6l 69 I O39 509 197 31.8276609 IO.O43 ^69 IOI4 I 02 8l 96 I 042 590 744 31.8433666 10.046 450 6 1015 I 03 02 25 I 045 678 375 31.8590646 10.049 752 1 IOl6 I 03 22 56 I 048 772 096 31.874 7549 10.053 0514 1017 I 03 42 89 I 051 871 913 31.8904374 10.056 348 5 IOl8 I 03 63 24 IO54977832 31.9061123 10.0596435 1019 I 03 83 6l I 058 089 859 31.921 7794 10.062 936 4 1020 I 04 04 OO I 061 208 OOO 3 I -937 438 8 10.066 227 1 1021 I 04244I I 064 332 261 31.9530906 10.0695156 1022 I 04 44 84 I 067 462 648 31.968 7347 10.072 802 1023 I 04 65 29 I 070 599 167 31.9843712 10.076 0863 1024 I 04 85 76 I 073 74I 824 32 10.079 368 4 1025 105 06 25 I 076 890 625 32.OI5 621 2 10.082 648 4 1026 I 05 26 76 1080045576 32.O31 2348 10.085 926 2 1027 I 05 47 29 I 083 206 683 32.046 840 7 10.089 201 9 1028 I 05 67 84 I 086 373 952 32.062 439 I 10.092 475 5 1029 I 05 884I I 089 547 389 32.078 0298 10.095 7469 1030 I 06 09 OO I 092 727 OOO 32.0936131 10.0990163 1031 I 06 2961 1 095 912 791 32.IO9 188 7 10.102 2835 1032 I 06 50 24 1 099 104 768 32.124 7568 10.105 5487 1033 I 06 70 89 I 102 302 937 32.1403173 10.108 81 1 7 1034 I 06 91 56 I IQ 5 507 304 32.1558704 10.1120726 1035 I 07 12 25 I 108 717875 32.1714159 10.1155314 IO36 I 07 32 96 I III934656 32.1869539 10.118 5882 IO37 107 53 69 I II5157653 32.202 4844 10.121 842 8 IO38 107 7444 I H8386872 32.2180074 10.126 095 3 1039 1 07 95 21 I 12 1 622 319 32.2335229 10.128 3457 IO4O 1 08 16 00 I I24864OOO 32.24903I 10.131 5941 IO41 1 08 36 81 I I28 III 921 32.264 5316 10.134 8403 IO42 1085764 I 13 1 366088 32.280 0248 10.138 0845 1043 1 08 78 49 I I34 626 507 32.2955105 10.141 3266 IO44 1 08 99 36 I I37 893 184 32.310988 8 10.144 5667 1045 1 09 20 25 I I4I l66 125 32.326 4598 10.147 804 7 IO46 1 0941 16 I 144445 336 32.341 92 3 3 10. 15 1 0406 IO47 1 09 62 09 I I47 73O 823 32.357 3794 10.1542744 IO48 1 09 83 04 I 15 1 022 592 32.372 828 1 10.157 5062 IO49 1 10 04 01 I 154320649 32.388 2695 10.160 7359 1050 1 102500 I 157 625 OOO 32.403 7035 10.163 9636 1051 1 104601 1 160935 651 32.419 130 1 10.167 1893 1052 1 106704 1 164 252 698 32.434 5495 10.170 412 9 1053 1 10 88 09 1 167 575 877 32.4499615 10.1736344 1054 1 1 1 09 16 1 170905464 32.465 3662 10.1768539 1055 1 11 3025 1 174 241 375 32.480 763 5 10.180071 4 IO56 1 11 5136 1 177583616 32.4961536 10.1832868 *057 1 11 7249 1 180 932 193 32.5115364 1 10.1865002 1058 1 119364 1 184287112 32.5269119 10.1897116 1059 1 12 1481 1 187 648 379 32.542 2802 10.1929209 ,1060 1 12 3600 1 191 016000 32.5576412 10.196 1283 1061 1 12 5721 1 1943899S1 32.572 9949 10.1993336 1062 1 12 7844 1 197 770328 32.5883415 10.202 5369 SQUARES, CUBES, AND ROOTS. 29I Number. Square, Cube. Square Root. Cube Root. 1063 I 129969 I 201 157047 32.603 680 7 10.205 738 2 1064 I 13 2096 I 204 550 144 32.6190129 10.208 937 5 1065 I 1342 25 I 207 949 625 32.6343377 10.212 I34 7 1066 i 13 6 3 56 1211355496 32.6496554 10.215 33 1067 1 138489 I 214 767 763 32.6649659 IO.2185233 1068 1 14 06 24 I 218 186 432 32.680 2693 10.221 7146 1069 1 142761 I 221 6l I 509 32.695 565 4 10.224 9039 IO7O 1 14 49 00 I 225 043 OOO 32.7108544 IO.22809I 2 IO71 1 14 70 41 I 228480911 32.726 1363 IO.23I 2766 IO72 1 1491 84 I 231 925 248 32.741 41 1 I IO.2344599 1073 1 15 *3 2 9 1235 376017 32.7566787 IO.237 641 3 IO74 1 15 34 76 I 238 833 224 32. 77 1 939 2 IO.24O 820 7 IO75 1 155625 I 242 296 875 32.787 1926 IO.243 998 I IO76 1 15 77 76 I 245 766976 32.802 4389 IO.2471735 IO77 1 159929 I 249 243 533 32.817 6782 10.250 347 IO78 1 16 20 84 I 252 726 552 32.8329103 IO.2535186 IO79 1 1642 41 I 256216039 32.8481354 10.256 688 1 1080 1 166400 I 259 712 OOO 32.8633535 10.259 8557 Io8l 1 168561 I 263 214 441 32.8785644 10.263021 3 1082 1 1707 24 I 266 723 368 32.8937684 10.266 185 1083 1 17 28 89 I 27O 238 787 32.9089653 10.269 346 7 IO84 1 175056 I 273 760 704 32.9241553 10.272 5065 1085 1 1772 25 I 277 289 125 32.9393382 10.275 6644 1086 1 179396 I 280 824 056 32.9545141 10.278 8203 1087 1 18 15 69 I 284 365 503 32.969683 10.281 9743 1088 1 18 37 44 I 287913472 32.984 845 10.285 1264 1089 1 18 5921 I 291 467 969 33 10.288 2765 IO9O 1 1881 00 1 295 029 OOO 33.015 148 10.291 424 7 IO9I 1 1902 81 1 298 596 571 33.030 289 1 10.2945709 IO92 1 19 24 64 1 302 170688 33.045 4233 10.2977153 IO93 1 194649 1305 751357 33-o6o 550 5 10.300 8577 IO94 1 19 68 36 1309338 584 33-075 670 8 10.303 998 2 1095 1 19 90 25 1312 932 375 33.090 784 2 10.307 136 8 IO96 1 20 12 16 1 316 532 736 33.105 8907 10.3102735 IO97 1203409 1 320 139 673 33.1209903 10.313 408 3 IO98 1 20 56 04 1 323 753 192 33.136083 10.316541 1 IO99 1 20 78 01 1 327 373 299 33.151 1689 10.319672 1 I IOO 1 21 0000 1 331 000000 33.1662479 10.322 801 2 IIOI 1 21 2201 1334 633301 33.181 32 10.325 9284 1102 1 21 4404 1 338 273 208 33.1963853 10.3290537 1103 1 21 6609 i 34 i 9 i 9 727 33.2114438 10.332 177 IIO4 1 21 88 16 1 345 572 864 33.2266955 10.335 2985 1105 1 22 1025 1 349 232 625 33.241 5403 10.3384181 II06 1 22 32 36 1 352 899 016 33-2565783 10.341 535 8 1107 1 225449 1356572043 33.2716095 10.344651 7 IIO8 1 22 76 64 1 360 251 7J2 33.286 633 9 10.347 765 7 IIO9 1 22 98 81 1 363 938 029 3 3-3° 1 651 6 10.350 8778 iiio 1 2321 00 1 367 631 000 33.3166625 10.353988 IIII 1 23 4321 1 37 1 330631 33-33 1 6666 10.3570964 1112 1 23 65 44 1375036928 33.346664 10.360 202 9 1113 1 23 87 69 1378 749897 33.361 654 6 10.363 307 6 1114 1 240996 1382469544 33-3766385 10.3664103 1115 1 24 32 25 1 386 195 875 33 - 39 1 615 7 10.3695113 1116 124 5456 1 389 928 896 33.406 586 2 10.372 610 3 1117 1 24 76 89 1393668613 33.4215499 10.375 707 6 1118 1 24 99 24 1 397 4*5 ° 3 2 33-436 507 10.378 803 292 SQUARES, CUBES, AND ROOTS. Number. Square. III9 I 25 21 6l 1120 I 25 44 OO 1121 I 256641 1122 i 25 88 84 1123 1 26 1 1 29 1124 1 2633 76 1125 1 26 56 25 1126 1 26 78 76 1127 1 2701 29 1128 1 27 23 84 1129 1 27 4641 1130 1 27 6900 II3I 1 27 91 61 1132 i 28 14 24 1 133 1 28 36 89 1134 1 28 59 56 1135 1 28 82 25 II36 1 29 04 96 1137 1 29 27 69 II38 1 29 5044 1139 1 29 7321 II4O 1 29 96 00 II4I 1 30 18 81 II42 1304164 1143 130 6449 1144 1308736 1145 1 31 1025 II46 1 3 1 33 16 1147 1315609 II48 1 31 7904 1149 1 32 02 01 1150 1 32 25 00 II5I 1 32 48 01 1152 1 32 71 04 1153 132 9409 1154 133 17 16 1155 1334025 II56 13363 36 1157 1338649 1158 1340964 1159 1343281 Il6o 1 34 5600 Il6l 134 7921 .El62 1 35 02 44 I163 1 35 2 5 69 I164 1 35 48 96 I165 1 35 7 2 2 5 Il66 1 35 95 56 I167 1 36 18 89 Il68 13642 24 I169 1366561 1170 1 36 89 00 II7I 1371241 1172 1 37 35 84 1173 137 59 2 9 1174 1 37 82 76 Cube. I 401 168 159 I 404 928 OOO I 408 694 561 I 412 467 848 I 416 247 867 I 420 034 624 I 423 828 125 I 427 628 376 i 43 1 435 383 I 435 249 152 i 439 069 689 1 442 897 000 1 446 731 091 1450571968 i 454 4 i 9 6 37 1 458 274 104 1462135 375 1 466 003 456 1469878353 1 473 760 072 1 477 648 619 1 481 544000 1 485 446 221 1489355 288 1 493 271 207 1497 193984 1 501 123625 1 505 060 136 1 509 603 523 1512953 792 1 516910949 1 520 875 OOO 1524845 951 I 528 823 808 I 532 808 577 1 536 800 264 1540 798875 1 544804416 1 548816893 1552 836312 1 556 862 679 1 560 896 OOO 1564936 281 1 568 983528 1573037 747 1577098944 1 581 167 125 1 585 242 296 1589324463 *593 4*3 632 1 597 509809 1 601 613 OOO I 605 723 21 1 I 609 84O 448 1613964717 I 618096024 1 Square Root. 33-451457 3 33.4664OI I 33.481 338 I 33.496 2684 33.511 1921 33.5261092 33.5410196 33-555 923 4 33.570 8206 33-585 7II2 33.600 595 2 33- 6i 5 47 2 6 33-630 3434 33.645 207 7 33.6600653 33.674 9 i6 5 33.689 761 33-704 599 1 33.7194306 33-734255 6 33-749074I 33.7638860 33.7786915 33-793 490 5 33.808 283 33.823 069 1 33-8378486 37.852 621 8 33.8673884 33.882 148 7 33.8969025 33.9116499 33.9263909 33.941 1255 33-955 853 7 33- 970 575 5 33.985 291 34 34.014 702 7 34.029399 34.044 089 34.058 772 7 34- 073 450 1 34.088 12 1 1 34.102 7858 34.1174442 34.132 0963 34.146 7422 34.161381 7 34.176015 34.190642 34.205 262 7 34.2198773 34.2344855 34.2490875 34.2636834 Cube Root. IO.381 8965 IO.384 988 2 IO.388 078 I IO.39I l66 I IO.394 252 3 10.397 336 6 IO.4OO 419 2 IO.4034999 IO.406 578 7 IO.409 655 7 IO.412 73I IO.415 8044 IO.418 876 10.421 945 8 10.425 013 8 10.428 08 10.431 1443 10.434 2069 10.437 267 7 10.4403267 10.443 3839 10.4464393 10.4494929 10.452 5448 10.455 594 8 10.458 643 1 10.4616896 10.4647343 10.4677773 to.4708185 10.4738579 10.476 895 5 ♦ 10.47993*4 10.482 965 6 10.485 998 10.489 028 6 10.4920575 10.495 084 7 10.498 no 1 10.501 133 7 10.504 1556 10.5071757 I 10.510 194 2 IO.513 2109 ; IO.5162259 j IO.519 239 I | IO.5222506 IO.525 260 4 ' IO.528 268 5 10.53 1 2749 ; 10.5342795 10.537 282 5 10.540 283 7 10.543 283 2 10.546 281 10.5492771 SQUAKES, CUBES, AND BOOTS. 293 Number. Square. Cube. Square Root. Cube Root. H 75 I 38 06 25 I 622 234 375 34.278 273 IO.552271 5 1176 I 38 29 76 I 626 379 776 34.292 856 4 10.555 264 2 1177 i 38 53 29 I 630 532 233 34-307 4336 IO.558 255 2 1178 138 76 84 I 634 691 752 34.3220046 10.561 2445 1179 I 390041 1638 858 339 34-336 5694 IO.564 232 2 Il8o 139 2400 I 643 032 OOO 34.351 1281 IO.567 2l8 I Il8l 13947 6l I 647 212 741 34.365 6805 IO.57O 202 4 1182 139 7124 I 651 4OO 568 34.380 226 8 IO.5731849 1183 139 94 89 I 655 595 487 34.394 767 10.576 165 8 1184 I 40 18 56 I 659 797 504 34.409301 1 IO.5791449 1185 I 40 42 25 I 664 006 625 34.4238289 IO.582 122 5 Il86 I 40 65 96 i 668 222 856 34.438 3507 IO.585 098 3 1187 I 40 89 69 1 672 446 203 34.452 866 3 IO.5880725 Il88 141 1344 1 676676672 34.467 375 9 IO.59I O45 H89 I 41 37 21 1 680 914 269 34.481 8793 IO.594OI58 II90 I 41 6l OO 1 685 159000 344963766 IO.596 985 II91 I 41 8481 1 689410871 34.5108678 IO.5999525 II92 I 42 08 64 1 693 669 888 34-525 353 IO.6029184 1193 I 42 32 49 1697936057 34-539 8321 10.605 882 6 II94 I 42 56 36 1 702 209 384 34554 3051 10.608 845 1 1195 I 42 80 25 1 706 489 875 34.568 772 10.61 1 806 1196 I430416 1 710777536 34.583 232 9 IO.614 765 3 1197 i 43 28 09 .1715072373 34.597 687 9 10.617 722 8 II98 1435204 1 719374 392 34.6121366 10.6206788 II99 143 7601 1723683599 34.6265794 10.623 633 1 1200 1 44 00 00 1 728 000 000 34.641 016 2 10.626 585 7 1201 1 44 24 01 1 73 2 323 601 34.6554469 10.629 536 7 1202 1 44 48 04 1 736654408 34.669871 6 10.632 486 1203 1 44 72 09 1740992 427 34.684 290 4 10.635 433 8 1204 1 44 96 16 1 745 337664 34.698 703 1 10.638 379 9 1205 1 45 20 25 1 749690 125 34.713 1099 10.641 3244 1206 1 45 44 36 1 754049816 34-7275107 10.644 267 2 1207 145 6849 1 758416743 34.741 905 5 10.647 208 5 1208 1 45 92 64 1 762 790912 34.756 2944 10.650 148 1209 1 46 1681 1 767 172 329 34.7706773 10.653 086 1210 1 46 41 00 1 771 561 000 34.785 0543 10.656 022 3 I2II 1466521 1775 956931 34.7994253 10.658957 1212 | 1 46 89 44 1 780360 128 34.8137904 10.661 890 2 1213 1 1471369 1 784 770 597 34.828 149 5 10.664821 7 1214 | 1473796 1 789 188 344 34.842 502 8 10.667 751 6 1215 1 47 62 25 1793 613 375 34.856 8501 10.6706799 I2l6 1 1 47 86 56 1 798 045 696 34.871 191 5 10.673 606 6 1217 1 48 10 89 1 802 485 313 34.885 527 1 10.676531 7 1218 ! 1483524 1 806 932 232 34.8998567 10.6794552 1219 j 1485961 1 81 1 386459 34.9141805 10.682 377 1 1220 1 1 48 84 00 1 815 848000 34.928 4984 10.685 297 3 1221 1 49 08 41 1 820316861 34.9428104 10.688 216 1222 14932 84 1 824 793 048 34.9571166 10.691 133 1 1223 1495729 1 829 276 567 34.9714169 10.694 048 6 1224 1 4981 76 1 833 767 424 34.9857114 10.696 962 5 1225 1 500625 1 838 265 625 35 10.699 874 8 1226 1 5030 76 1 842 771 176 35.0142828 10.702 785.5 1227 15055 29 1 847 284 083 35-0285598 10.7056947 1228 150 7984 1 851 804352 35.042 8309 10.708 6023 1229 1 51 0441 1 856 331 989 35-0570963 10.71 1 5083 J230 I 51 2900 1 860 867 000 35 - 07 I 355 8 10.714 412 7 B B* 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 SQUARES, CUBES, AND ROOTS. Square. 15153 61 I 51 78 24 I 52 02 89 I 52 27 56 I 52 52 25 I 527696 I 5301 69 153 2644 i 53 5i 21 1 53 7600 1 540081 1 54 25 64 154 50 49 1 54 75 36 1 55 00 25 1 55 25 16 155 5009 i 55 75 04 1 56 00 01 1 56 25 00 1 56 5001 1 56 75 04 1 570009 1 57 25 16 157 50 25 1 57 75 36 1 580049 1 58 25 64 1585081 1 58 76 00 1 5901 21 159 2644 1595169 159 7696 1 60 02 25 1 60 27 56 1 60 52 89 1 60 78 24 1 61 03 61 1 61 2900 1 61 5441 1 61 79 84 1 62 05 29 1 62 30 76 1 62 56 25 1 62 81 76 1 63 07 29 1 63 32 84 1635841 1 63 84 00 1 640961 16435 24 1 64 60 89 1 64 86 56 1 65 1225 1653796 Cube. I 865 409 391 I 869 959 168 1874516337 I 879 080 904 1883 652 875 i 888 232 256 1 892819053 1897413272 1 902 014 919 1 906 624 000 1 91 1 240521 1 915 864488 1920495 907 1 925 134 784 1 929 781 125 1934 434 936 1939096223 1 943 764 992 1 948 441 249 1953 125000 1957816251 1 962 515 008 1 967 221 277 1971935064 1976 656375 1 981 385 216 1 986 121 593 1990865512 1995616979 2 000 376 000 2005 142 581 2009916 728 2 014 698 447 2019487 744 2 024 284 625 2 029 089 096 2033901 163 2 038 720 832 2 043 548 109 2 048 383 000 2053 225 51 1 2 058 075 648 2062 933417 2 067 798 824 2 072 671 875 2077552576 2 082 440 933 2087 336952 2 092 240 639 2097 152000 2 102 071 041 2 106 997 768 2 hi 932 187 2 116874304 2 121 824 125 2 126 781 656 Square Root. 35.0856096 35.0998575 35.II40997 35.128 3361 35.142 5568 35.156 791 7 35.1710108 35.185 2242 35.1994318 35.2136337 35.2278299 35.242 0204 35.256 2051 35.270 3842 35.2845575 35.298 725 2 35 - 3 12 887 2 35.3270435 35-341 194 1 35-355 339 1 35-3694784 35.383612 35-397 74 35.4118624 35.425 9792 35.4400903 35.454 195 8 35.468 295 7 35.482 39 35.4964787 35.5105618 35.5246393 35-538 7 II 3 35-552 777 7 35.566 8385 35 5808937 35-594 943 4 35.608 9876 315.6230262 35-637 0593 35.6510869 35.665 109 35.679 125 5 35 - 693 I 366 35.707 142 1 35.721 1422 35-735 136 7 35.7491258 35-763 109 5 35.7770876 35.791 0603 35.805 0276 35.8189894 35-832 945 7 35.846 8966 35.860 842 1 Number. 1287 1288 1289 1290 1291 1292 1293 I294 1295 1296 I297 1298 1299 1300 I 3 ° i 1302 1303 1304 1305 1306 1307 1308 1309 1 310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 *333 z 334 *335 1336 I 337 1338 *339 1340 I 34 i 1342 SQUARES, CUBES, AND ROOTS. 295 Square. Cube. Square Root. Cube Root. I 65 63 69 2 13 1 746903 35.874 782 2 IO.877427 I 165 8944 2 136 719 872 35.888 7169 10.880 243 6 i 66 15 21 2 141 7OO 569 35.902 646 I 10.883 058 7 1 6641 00 2 I46 689 OOO 35.9165699 IO.885 8723 166 6681 2 151685 I7I 35.9304884 10.888 684 5 1 66 92 64 2 156689 088 35.9444OI 5 10.891 495 2 1 67 1849 2 l6l 7OO 757 35-958 309 2 10.894 304 4 167 4436 2 l66 720 184 35.972 211 5 10.897 112 3 1 67 70 25 2 I 7 I 747 375 35.9861084 10.8999186 1 67 96 16 2 176 782 336 36 10.902 723 5 1 68 22 09 2 181 825 073 36.OI38862 10.905 5269 1 68 48 04 2 186875 592 36.027 767 I 10.908 329 1 68 74 01 2 191 933 899 36.041 6426 10.911 1296 1 690000 2 197 OOO OOO 36.055 512 8 10.913928 7 1 69 26 01 2 202 O73 9OI 36.0693776 10 916 7265 1 69 52 04 2 207 I55 608 36.083 237 I 10.919 522 8 1 69 78 09 2 212 245 I27 36.097 091 3 10.922 317 7 1 7004 16 2 217 342 464 36. 1 IO 94O 2 10.925 hi 1 1 7030 25 2 222 447 625 36.124 7837 10.927 903 1 1 705636 2 227 560616 36.138 622 10.930 693 7 1 70 82 49 2 232 68l 443 36.152455 10.933 482 9 1 71 08 64 2 237 8lO 1 12 36.1662826 10.936 270 6 1 713481 2 242 946 629 36.180 105 ro.9390569 1 71 61 00 2 248 09I OOO 36.I93922I 10.941 841 8 1 71 87 21 2 2 53 2 43 23 1 36.207 734 10.9446253 1 72 13 44 2 258 403 328 36.221 5406 10.9475074 1 723969 2263571 297 36.235 3419 10.950 188 1 72 65 96 2 268 747 I44 36.2491379 10.952 967 3 1 7292 25 2 2 73 930 875 36.262 928 7 10.955 745 1 1731856 2 279 122 496 36.2767143 10.958 521 5 1 734489 2 284 322 013 36.290 4946 10.961 296 5 1 73 71 24 2289529 432 36.304 269 7 10.964 070 1 1 7397 6i 2 2 94 744 759 36.3180396 10.966 842 3 i 74 24 00 2 299 968 000 36.3318042 10.969 613 1 1 745041 2 305 199 161 36.345 563 7 10.972 382 5 1 74 7684 2310438 248 36.359317 9 10.9751505 1 75 03 29 2 315 685 267 36.373 067 10.977 917 1 1 75 29 76 2 320 940 224 36.3868108 10.980 682 3 1 75 56 25 2 3 2 6 203 125 36.4OO5494 10.983 446 2 1 75 82 76 2 33 M 73 976 36.4142829 10.986 208 6 1 76 09 29 2 336 75 2 783 36.4280II 2 10.988 969 6 17635 84 2342039552 36.441 7343 10.991 7293 1 766241 2 347 334 289 36.455 452 3 10.994 487 6 1 76 89 00 2 35 2 637 000 36.469 165 10.997 2445 1771561 2 357 947 69 i 36.482 872 7 11 1 7742 24 2 363 266 368 36.4965752 11.002 754 1 1 77 68 89 2 368 593 037 36.5IO2725 11.005 5069 1 77 95 56 2 373 9 2 7 704 36.523 9647 11.008 2583 1 78 22 25 2 379 270 375 36.537 6518 1 1. 01 1 008 2 1 78 48 96 2 384621 056 36.551333 8 11.0137569 1 78 75 69 2 389979 753 36 565 010 6 1 1. 016 504 1 1790244 2 395 346 47 2 36 5786823 11. 019 25 1 79 2921 2 400 721 219 36.592 3489 11.021 9945 1 79 5600 2 406 104000 36.6060104 11.024 7377 1 7982 81 2411 494821 36.6196668 11.0274795 1 800964 2 416 893 688 36.6333181 11.030 2199 296 Number. 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 i37i 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 i39i 1392 1393 1394 1395 1396 1397 1398 SQUARES, CUBES, AND ROOTS. Square. I 80 36 49 I 80 63 36 I 80 90 25 I 8l 17 16 I 8l 44 09 I 8l 71 04 I 8l 9801 I 82 25 OO I 82 52 OI I 82 79 04 I 830609 I 83 33 16 1836025 1838736 I 84 14 49 I 84 41 64 i 84 68 81 1 84 96 00 1 85 23 21 185 5044 1 85 77 69 1 86 04 96 186 32 25 1 86 59 56 1 86 86 89 1 87 14 24 1 87 41 61 1 87 69 00 1 87 9641 1 88 23 84 1 88 51 29 1 88 78 76 1 89 06 25 1 89 33 76 1 8961 29 1 89 88 84 1 90 1641 1 904400 1 90 71 61 1909924 1 91 26 89 1 9 1 54 56 1 91 82 25 1 92 09 96 1 92 37 69 192 6544 192 9321 1 93 2 1 00 193 4881 193 7664 1940449 1943236 1 94 60 25 1 94 88 16 1 95 1609 195 4404 Cube. 2 422 300 607 2427715584 2 433 x 38 625 2 438 569 736 2 444 008 923 2449456 192 2 454 911549 2 460 375 000 2465 846551 2 471 326 208 2476813977 2 482 309 864 2487813875 2493326016 2 498 846 293 2 504 374 712 2 509 91 1 279 2 C15 456000 2 521 008 881 2 526 569 928 2 532 139 147 2537716544 2 543 302 125 2 548 895 896 2 554 497 863 2 560 108032 2565 726409 2 57 1 353000 2 576 987 81 1 2 582 630 848 2 588 282 117 2 593 94 1 624 2599609375 2 605 285 376 2 610969633 2 616662 152 2 622 362 939 2 628 072 000 2633 789 341 2639514968 2 645 248 887 2 650 991 104 2 656 741 625 2 662 500 456 2 668 267 603 2 674 043 072 2 679 826 869 2 685 619000 2 691 419 471 2 697 228 288 2 703 045 457 2 708 870 984 2 714704875 2 720 547 136 2726397773 2 732 256 792 Square Root. 36.6469644 36.660 605 6 36.674 241 6 36.687 872 6 36.701 4986 36.715 1195 36.728 7353 36.742 346 1 36- 755 9519 3 6 -7 6 9 552 6 36.783 1483 36.796 739 36.8103246 36.823 9053 36.8374809 36.851 051 5 36.864 617 2 36.8781778 36.891 733 5 36.905 284 2 36.918 8299 36.932 3706 36.9459064 36.959 437 2 36.972 963 1 36.986 484 37 37 - oi 3 5 ii 37.027 017 2 37.0405184 37.0540146 37.067 506 37.089 9924 ,37 094 474 37- io 7 95o6 37.121 4224 37-134 8893 37.1483512 37.161 8084 37.175 2606 37.188 7079 37.202 1505 37.2155881 37.229 0209 37.242 4489 37-255 872 37.269 2903 37.282 703 7 37.2961124 37-309 5 J 6 2 37.3229152 37-3363094 37-349698 8 37.3630834 37.376 4632 37 - 3 8 9 8 382 Number 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 14H 1412 1413 1414 I 4 I 5 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 i43 6 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 i45i 1452 1453 1454 SQUARES, CUBES, AND ROOTS. 297 Square. 1 95 72 OI I 96 OOOO I 96 28 OI 1 96 56 04 I 96 8409 i 97 12 16 1 97 40 25 1 97 68 36 197 9649 1 98 24 64 1 98 52 81 1 98 81 00 1 9909 21 19937 44 19965 69 1999396 2 00 22 25 2 00 50 56 2 00 78 89 2 01 07 24 201 3561 2 01 64 00 2 OI 92 41 2 02 20 84 2 02 49 29 2 02 77 76 2 03 06 25 2 03 34 76 20363 29 2039184 2 04 20 41 2 04 49 OO 2 04 7761 2 05 06 24 2053489 2 05 63 56 2 05 92 25 2 06 20 96 2 06 49 69 2 06 78 44 207O72I 2 07 36 OO 2076481 2079364 2 08 22 49 2 08 51 36 2 08 80 25 2 O909 l6 2 09 38 09 2 09 67 04 209960I 2 10 25 00 2 IO 54OI 2 IO 83 04 2 II 12 09 2 II 41 l6 Square Root. 2 738 124 199 2 744 OOO OOO 2 749 884 201 2 755 77 6 808 2 761 677 827 2 767 587 264 2 773 505 125 2 779 43 1 4i 6 2 785 366 143 2791 30 9312 2 797 260 929 2 803 221 000 2809 189 531 2 815 166 528 2 821 151 997 2 827 145 944 2 833 148 375 2 839 159 296 2845 178713 2 851 206 632 2857243 059 2 863 288 000 2 869341 461 2 875 403 448 2881473967 2 887 553 024 2 893 640 625 2 899 736 776 .2 905 841 483 2911 954 75 2 2918076589 2 924 207 000 2 930 345 99i 2936 493 568 2 942 649 737 2948814 504 2954987875 2961 169856 2967360453 2 973 559672 2 979 767519 2 985 984 000 2992 209 121 2 998 442 888 3004685 307 3 010 936 384 3017 196 125 3 023 464 536 3 029 741 623 3036027392 3042 321 849 3 048 625 000 3054936851 3061 257408 3067 586677 3073924664 37.4032084 37-4165738 37.4299345 37.443 2904 37.4566416 37.469988 37-4 8 3 3296 37.4966665 37-509 99 8 7 37.523 326 1 37-5366487 37.5499667 37-563 2799 37.5765885 37.589 8922 37-603 191 3 37.6164857 37.629 775 4 37.6430604 37*656 340 7 37.6696164 37.682 887 4 37 696 1536 37.7094153 37.722 672 2 37-735 924 5 37.749 1722 37.7624152 37-775 653 5 37.788 8873 37.802 1163 37.8153408 37.828 5606 37.841 7759 37.854 9864 37.868 192 4 37 - 88 I 393 8 37.894 5906 37.907 782 8 37.9209704- 37-934 153 5 37-947 33i 9 37.960 505 8 37- 973 675 1 37.9868398 38 38.0131556 38.026 306 7 38.0394532 38.052 595 2 38.065 732 6 38.078 8655 38.0919939 38.105 1178 38.1182371 38- I3 1 35 1 9 Cube Root. II.184 225 2 II. 186 889 4 II.1895523 II. 192 2139 II.1948743 II - I 97 533 4 11.200 191 3 11.202 847 9 11.205 5032 11.208 1573 11.210810 1 11.213461 7 11.216 112 11.218 761 1 1 1. 221 408 9 11.224005 4 11.226 700 7 11.229 344 8 11. 231 987 6 11.234 629 2 11.2372696 11.239908 7 11.242 5465 11.245 183.1 11.247 818 5 11.250 4527 11.2530856 11.2557173 11.258 3478 11.260977 11.263605 11.266 231 8 11.268 8573 11. 271 481 6 11.274 io 4 7 11.276 7266 11.279 347 2 11.281 966 6 11.2845849 11.287 201 9 11.289 817 7 11.292 4323 11.295 0457 11.297 657 9 1 1 .300 268 8 11.302 878 6 11.305 4871 11.308 094 5 1 1. 310 7006 11.3133056 II -3 I 5 9°94 11.3185119 11.321 1132 II-323 7I3 4 11.3263124 11.3289102 298 Number. 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 147I 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 i486 1487 1488 1489 149O 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 SQUARES, CUBES, AND ROOTS. Square. Cube. Square Root. 2 1 1 70 25 3080271375 38.144462 2 2 II 9936 3086 626816 38.1575681 2 12 28 49 3 092 990 993 38.1706693 2 125764 3099363912 38.183 7662 2 12 86 8 l 3 105 745 579 38.1968585 2 13 1600 3 112 136000 38.2099463 21345 21 3 MS 53 s 181 38.223 029 7 2 13 74 44 3124 943128 38.236 108 5 2 14 03 69 3I3 1 359 847 38.249 182 9 2 143296 3 137 785 344 38.262 252 9 2 14 62 25 3144 219625 38.2753184 2 I 49 I 56 3 150 662 696 38.2883794 2 15 2089 3157114563 38.301 436 2 15 50 24 3 163 575 232 38.3144881 2 15 7961 3170044 709 38-3 2 7 5358 2 l 6 09 OO 3176523000 38.340579 2 16384 I 3 183010 hi 38.3536178 2 1667 84 3189 506048 38.366 6522 2 1697 29 3 196010817 38.379 682 1 2 1 7 26 76 3 202 524 424 38.392 707 6 2175625 3 209 046 875 38.405 728 7 2 1785 76 3215578176 38.418 745 4 2 l 8 15 29 3222 118333 38.431 757 7 2 1844 84 3 228 667 352 38.444 765 6 2 l 8 744 I 3 235 225 239 38.457 769 I 2 19 04 OO 3 241 792 000 38.470 768 I 2 193361 3248 367641 38.483 762 7 2 196324 3 254 952 168 38.496 753 2 I 992 89 3 261 545 587 38-509 739 2 20 22 56 3 268 147 904 38.522 7206 2 20 52 25 3274 759125 38.5356977 2 20 8 l 96 3 281 379 256 38.548 670 5 2 21 1169 3 288 008 303 38.5616389 2 21 41 44 3 294 646 272 38.574603 2 21 71 21 3 301 293 169 38.587 562 7 2 22 01 OO 3 307 949 000 38.600 518 1 2 22 3081 3314613 771 38.6134691 2 22 60 64 3321287488 38.626415 8 2 22 90 49 3327970157 38.639 358 2 2 23 20*36 3 334661 784 38.652 296 2 2 23 50 25 3341362375 38.665 229 9 2 23 80 l 6 3348071936 38.6781593 2 24 IOO 9 3 354 790 473 38.691 084 3 2 24 40 O 4 3361517992 38.704005 2 24 70 OI 3368 254499 38.7169214 2 25 OO OO 3 375 oooooo 38.7298335 2 25 30 OI 3 38i 754 501 38.742 741 2 2 25 60 04 3388518008 38.755 644 7 2 25 90 09 3 395 290527 38.768 5439 2 2620 l 6 3 402 072 064 38.7814389 2 26 50 25 3 408 862 625 38.7943294 2 26 80 36 3415662216 38.8072158 2 27 IO 49 3422 470843 38.820097 8 2274064 3429 288512 38.8329757 2 27 70 8 l 3436115229 38.845 849 1 2 28 oI OO 3442 951000 38.8587184 Number. I5II 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 T 533 1534 1535 1536 1537 1538. 1539 1540 1541 1542 1543 1544 1545 1546 1547 1548 T 549 1550 i55i 1552 1553 1554 1555 1556 1557 1558 1559 1560 1561 1562 1563 1564 1565 1566 SQUARES, CUBES, AND ROOTS. 299 Square. 22831 21 2 28 6l 44 22891 69 2 299I 96 2 29 52 25 2 29 82 56 230 12 89 2 30 43 24 2 30 73 6l 231 O4OO 2 3 1 34 41 2316484 23195 29 2 32 25 76 2 32 56 25 232 86 76 2331729 2334784 2 33 7841 2 340900 2343961 2347024 2 35 00 89 2 35 31 56 2 35 62 25 2359296 2 36 23 69 236 54 44 236 8521 237 1600 2 37 46 81 2 37 77 64 2380849 238 3936 2 38 70 25 2 39 01 16 2393209 2396304 2399401 2 40 25 00 2405601 2 40 87 04 241 1809 241 49 16 2 41 80 25 242 11 36 2 42 42 49 242 7364 2430481 243 3600 2436721 2439844 2442969 2 44 60 96 24492 25 2 45 23 56 Cube. 3 449 795 831 3456649 728 3463512697 3 470 384 744 3 477 265 875 3484156096 3 49i 055 4i3 3497963832 3 504 881 359 3511 808000 3518743 761 3 525 688 648 3 532 642 667 3 539 6 °5 824 3546578 125 3 553 559 576 3560558 183 3 567 549 952 3 574 558889 3581 577000 3 588 604 291 3 595 640 768 3602 686437 3609741 304 3616805 375 3 623 878 656 3630961 153 3 638 052 872 3645153819 3 652 264 000 3659383 421 3666512 088 3673 650 007 3680797 184 3687953 625 3695119336 3 702 294 323 3 709478 592 3716672 149 3 723 875 000 3731087 151 3 738 308 608 3 745 539 377 3 752 779464 3 760 028 875 3 767 287 616 3 774 555 693 3781833 1 12 3789119879 3 796 416 000 3803721481 3811036 328 3818360 547 3 825 641 144 3833037 125 3 840 389 496 Square Root. 38.8715834 38.884 444 2 38.8973OO6 38.9IO 1529 38.923OOO9 38.935 844 7 38.948 684 I 38.961 5194 38.974 350 5 38.9871774 39 39.012 8184 39.025 632 6 39.038 442 6 39.051 2483 39.0640499 39.076 8473 39.089 640 6 39.102 4296 39.115 2144 39- I2 7 995 1 39.140771 6 39- x 53 5439 39.166312 39.179076 39- I 9 I 835 9 39.204 591 5 39.2 1 7 343 1 39.2300905 39.242 833 7 39-255 572 8 39.268 307 8 39.2810387 39.2937654 39.306 488 39.319 206 5 39.331 920 8 39.344631 1 39-357 337 3 39.3700394 39.382 737 3 39-395*431 2 39.408 121 39.420 806 7 39-433 4883 39.446 165 8 39.4588393 39.471 508 7 39.484 174 39.4968353 39.5094925 39.522 145 7 39-534 794 8 39-547 4399 39.5600809 39.5727179 Cube Root. JI -475 0562 11.4775871 11.480 1169 11.4826455 11.485 1731 11.4876995 11.4902249 11.492 7491 11.4952722 H.497 7942 11.500 315 1 11.502 8348 11.505 353 5 11.507871 1 11.5103876 11.512903 11.5154173 n.5 I 7 9305 11.5204425 11.522 9535 11.525 4634 i 1.527 972 2 11.5304799 11.532 9865 11.535 492 11.5379965 11.5404998 11.543002 1 11.545 503 3 11.548 0034 11.550 5025 11.5530004 n-555 497 3 11 -557 993 1 11.560487 8 11.562 981 5 11.565 474 11.567 9655 11.570 455 9 11.572 9453 n-575 433 6 11.5779208 11.5804069 11.582 891 9 11.585 375 9 11.5878588 11.590 3407 11.592 821 5 11.595 3013 n-597 7799 11.600 257 6 11.602 7342 11.605 209 7 11.607 684 1 11.6101575 11.6126299 300 SQUARES, CUBES, AND ROOTS. Number. Square. Cube. Square Root. Cube Root. 1567 2 45 54 89 3 847 75 i 263 39-585 350 8 II.615 IOI 2 1568 2 45 86 24 3855 123 432 39-597 979 7 II.617571 5 1569 2 46 17 61 3 862 503 009 39.6106046 1 1 .620 040 7 1570 2 46 49 00 3 869 893 000 39.623 225 5 11.622 508 8 I 57 1 2 46 8041 3877292 41 1 39.635 842 4 11.624 9759 I 57 2 2 47 1 1 84 3 884 701 248 39.648 455 2 11.627 442 1573 2 47 43 2 9 3892 119517 39.661 064 11.629 907 1574 2 47 74 76 3 899 547 224 39.673 668 8 11.632 371 1575 2 48 06 25 3 906 984 375 39.686 269 6 11.634 8339 1576 2 48 37 76 3914430 976 39.698 866 5 11.637 2957 1577 2 48 69 29 3921 887033 39.7114593 11.639 7566 1578 2 49 00 84 3 9 2 9 35 2 55 2 39 - 7 2 4 048 1 11.642 216 4 1579 2 49 32 41 3 936 827 539 39-736 632 9 11.644675 1 1580 2 49 64 00 3 944312000 39.7492138 11.647 1329 1581 249 9561 3 951 805 941 39.761 7907 n.649 5895 1582 2 50 27 24 3 959 309 368 39-774 363 6 11.652045 2 1583 2 50 58 89 3 966 822 287 39.7869325 11.6544998 1584 2 50 90 56 3 974 344 7°4 39.799 497 5 11.656 9534 1585 2 51 22 25 3 981 876 625 39.8120585 11.659 4059 1586 2 5 1 53 96 3 989 418 056 39.8246155 1 1. 661 857 4 1587 251 8569 3 996 969 003 39.8371686 11.6643079 1588 2 52 1744 4 004 529 472 39.8497177 11.666 757 4 1589 2524921 4 012 099 469 39.862 262 8 1 1 .669 205 8 159 ° 2 52 81 00 4 019 679000 39.874 804 11.671 6532 159 1 2 53 12 81 4 027 268 071 39.887 341 3 11.674 099 6 i 59 2 2 53 44 64 4 034 866 688 39.899 874 7 11.676 5449 1593 2 53 76 49 4 042 474 857 39.912 404 1 11.678 9892 1594 2 54 08 36 4 050 092 584 39.924 929 5 1 1. 681 432 5 1595 2 54 40 25 4057 719875 39-937 451 1 11.683 874 8 1596 2 54 72 16 4065 356 736 39.9499687 11.686 316 i 1597 2 55 04 09 4073003 173 39.962 482 4 11.688 7563 1598 2 55 36 04 4 080 659 192 39.974 992 2 11.691 1955 1599 255 6801 4 088 324 799 39.987 498 11.6936337 1600 2 560000 4 096 000 000 40 11.6960709 Uses of preceding table may be greatly extended by aid of following Rides : To Compute Square or Cube of a liiglier iNTnm'ber than, is contained, in Table. When Number is divisible by a Number without leaving a Remainder . Rule.— I f number exceed by 2, 3, or any other number of times, any number contained in table, multiply square or cube of that number in table by square of 2, 3, etc., and product will give result. Example. — Required square of 1700. 1700 is 10 times 170, and square of 170 is 2 8900. Then, 2 89 00 X 10 2 = 2 89 00 00. 2. — What is cube of 2400? 2400 is twice 1200, and cube of 1200 is 1 728000000. Then 1 728 000 000 X 2 3 = 13 824 000 000. When Number is an Odd Number. Rule.— Take the two numbers nearest to each other, which, added together, make that sum; then from sum of squares or cubes of these tw r o numbers, multi- plied by 2, subtract 1, and remainder will give result. SQUARES, CUBES, AND ROOTS, 301 Example. — What is square of 1745? Two nearest numbers are j g 73 } — 1745. Then, per table, fffig ggg I 52 25 13 x 2 = 3 045 026 — I = 3 04 50 25. To Compute Square or Cube Root of a, liiglier NTumber tban is contained, in Table. When Number is divisible by 4 or 8 without leaving a Remainder. Rule. — D ivide number by 4 or 8 respectively, as square or cube root is required; take root of quotient in table, multiply it by 2, and product will give root required. Example. — What are square and cube roots of 3200? 3200 -f- 4 = 800, and 3200 -7- 8 — 400. Then, square root for 800, per table, is 28.28 42 71 2, which, being X 2 = 56.56 85 424 root Cube root for 40x3, per table, is 7.368 063, -which, being X 2 = 14.736 126 root. When the Root (which is taken as Number ) does not exceed 1600. The Numbers in table are roots of squares or cubes, which are to be taken as numbers. Illustration. —Square root of 6400 is 80, and cube root of 512000 is 80. When a Number has Three or more Ciphers at its right hand. Rule. — Point off number into periods of two or three figures each, according as square or cube root is required, until remaining figures come within limits of table; then take root for these figures, and remove decimal point one figure for every pe- riod pointed otf. Example. — What are square or cube roots of 1 500000? 1 500 000 = 150, remaining figure, square root of which= 12. 247 45 ; hence 1224. 745, square root 1 500000 = 1500, remaining figures, cube root of which = 11.447 14 ; hence 1 14. 47 14, cube root To Ascertain. Cxi "be Root of any Number over 1GOO. Rule.— F ind by table nearest cube to number given, and term it assumed cube; multiply it and given number respectively by 2 ; to product of assumed cube add given number, and to product of given number add assumed cube. Then, as sum of assumed cube is to sum of given number, so is root of assumed cube to root of given number. Example. — What is cube root of 224 809? By table, nearest cube is 216000, and its root is 60. 216 000 X 2 + 224 809 = 656 809, And 224809 X 24-216000 = 665618. Then 656809 : 665618 60 : 60.804-j-, root To Ascertain Square or Cube Root of* a NTumber con- sisting of Integers and. Decimals. Rule. — Multiply difference between root of integer part and root of next higher integer by decimal, and add product to root of integer given; the sum will give root of number required. This is correct for Square root to three places of decimals, and for Cube root to seven. C c 202 SQUARES, CUBES, AND ROOTS. Example.— What is square root of 53.75, and cube root of 843.75? V54 =7.3484 ^844 = 9.4503 V 53 =7.2801 V 8 43 =9.4466 .0683 .0037 •75 •75 .051 225 .002775 a/53 =7-2801 ^843 =9.4466 V 53.75 = 7 . 33 i 325 ^843.75 = 9.449 375 When the Square or Cube Root is required for Numbers not exceeding Roots given in Table. Numbers in table are squares and cubes of roots. Rule —Find by table, in column of numbers that number representing figures of integer and ’decimals for which root is required, and point it off decimally by Diaces of 2 or 3 figures as square or cube root is required; and opposite to it, in column of roots, take root and point off 1 or 2 additional places of decimals to those in root, as square or cube root is required, and result is root required. Example i.— What are square roots of .15, 1.50, and 15.00? In table, 15 has for its root 3.87 29 8 ; hence .38 72 98 = square root for . 15. 1 ko has for its root 12. 24 74 5 ; hence 1. 22 47 45 = square root for 1.50. 1500 has for its root 38.72 98; hence 3.87 29 8 = square root for 15. 2 ._What are cube roots of . 15, 1. 50, and 15.00 ? Add a cipher to each, to give the numbers three places of figures, as .150, 1.500, and 15.000. In table 150 has for its root 5.3133; hence .531 33 = cube root of. 15. r coo has for its root 11.447; hence 1.1447 = culje root °f x *5°* . _ , 15 has for its root 2.4662; and 15.000, by addition of 3 places of figures, has 24.662 ; hence 2.4662 = cube root of 15.00. To Ascertain Square or Cu/be Roots of Decimals alone. Rule.— Point off number from decimal point into periods of two or three figures each, as square or cube root is required. Ascertain from table or , b ^ c ^ c ^^ 10n root of number corresponding to decimal given, the same being read off by ^remo^ ’ng the decimal point one place to left for every period of 2 figures if square root is required, and one place for every period of 3 figures if cube root is required. Example. — What are square and cube roots of .810, .081, and .0081 ? .810, when pointed off = . 81, and-v/.8i — -9* >0 gi 5 “ “ =.081, “ a/* 0 ^ 1 =*2846. .0081, “ “ “ =.0081, “ V- 0081 = .09. .810, when pointed off = .810, and a/. 810 = 932 17. .081, “ “ “ = .081, “ ^.081 =.43267. .0081, “ “ “ = .0081, “ ty.QoZl =.20083. To Compute Root of* a, Number. Rule.— Take square root of its square root. Example. — What is the -fy of 1600? ^1600 = 40, and V4° = 6.32 45 55 3. To Compute 6tli Root of* a Number. Rule.— Take cube root of its square root. Example. — What is the of 441 ? V44X = at, and ^21 = 2. 7 589 243- FOURTH AND FIFTH POWERS OF NUMBERS. 303 4 tli and. 5 th. Powers of Numbers. From 1 to 150. Number. 4th Power. 5th Power. Number. 4th Power. 5 th Power. ■I 1 1 64 16 777 216 1 073 741 824 2 16 32 65 17 850 625 1 160 290 625 3 81 243 66 18974736 1 252 332 576 4 256 1 024 $7 20 151 121 1 350125 107 5 625 3125 68 21 381 376 1453 933 568 6 I 296 7 776 69 22 667 121 1 564 031 349 7 2 401 16 807 70 24 010000 1 680 700 000 8 4096 32768 7 1 25 411 681 1 804 229351 9 6561 59 °49 72 26 873 856 1 934917632 10 10000 100 000 73 28 398 241 2073071 593 11 14 641 161 051 74 29 986 576 2219006624 12 20736 248 832 75 31 640 625 2373046 875 13 28 561 37 i 293 76 33362 176 2 535 525 376 14 38416 537824 77 35153041 2706 784157 i 5 50625 759 375 78 37015056 2887174368 16 65 536 1 048 576 79 38950081 3077056 399 17 83 521 1419857 80 40 960 000 3 276 800000 18 104976 1 889 568 81 43046721 3 486 784 401 J 9 130 321 2 476 099 82 45 212 176 3 707 398 432 20 160000 3 200 000 83 47 458 321 3 939 ° 4 ° 643 21 194 481 4084 IOI 84 49787136 4 182 119 424 22 234256 5153632 85 52 200625 4437053125 23 279841 6436 343 86 54708016 4704270 176 24 331 776 7 962 624 87 57 289761 4 984 209 207 25 390625 9 765 625 88 59969536 5277319168 26 456976 11 881 376 89 62 742 241 5 584 059 449 27 53 i 441 14348907 90 65 610000 5 904 900 000 28 614 656 17 210368 9 1 68 574 961 6 240 321 451 29 707 281 20 51 1 149 92 71 639 296 6590815 232 30 810000 24 300 000 93 74 805 201 6956883693 31 923 52 i 28 629 151 94 78 074 896 7 339 040 224 32 1 048 576 33 554 432 95 81 450625 7 737 809375 33 1 185 921 39 393 96 84 034656 8153726 976 34 1 336 336 45 435 424 97 88 529 281 8587 340257 35 1 500625 52521875 98 92 236 816 9 039 207 968 36 1 679616 60 466 176 99 96 059 601 9 509 900 499 37 1 874 161 69 343 957 100 100 000 000 10 000 000 000 38 2 085 136 79235168 IOI 104 060 401 10 510 100 501 39 2313441 90224 199 102 108 243 216 11 040 808 032 40 2 560 000 102 400000 103 112 550881 11 592 740743 4 i 2 825 761 1 15 856 201 104 116 985 856 12 166 529024 42 3 1 1 1 696 130 691 232 105 121 550625 12 762 815 625 43 3 418 801 147 008 443 106 126 247 696 13382 255776 44 3748096 164916224 107 131 079601 14025517307 45 4 100625 184 528125 108 136 048 896 14693 280768 46 4 477 456 205 962 976 109 141 158 161 15386 239549 47 4 879681 229345007 no 146 410 000 16 105 100000 48 5 308 416 254 803 968 III 151 807 041 16850581 551 49 5 764 801 282 475 249 112 I 57 35 i 936 17 623 416 832 5 o 6 250000 312 500000 H 3 163 047 361 18424351793 5 i 6 765 201 345025251 114 168 896016 19 254 *45 824 52 7311616 380 204 032 H 5 174900625 20113581875 53 7890 481 418195493 Il6 181 063 936 21 003 416 576 54 8 503 056 459165 024 117 187 388 721 21924480357 55 9 i 5o625 503 284 375 Il8 *93 877 776 22 877577568 56 9 834 496 550 731776 119 200 533 9 21 23 863 536 599 57 10 556 001 601 692 057 120 207 360 000 24 883 200 000 58 11 316496 656356768 121 214358881 25 937 424 601 59 12 117 361 714924299 122 22 i 533 456 27027 081 632 60 12960000 777 600 000 123 228 886 641 28153056 843 61 13 845 841 844 596 3 GI 124 236 421 376 29 316 250 624 62 14776336 916 132 832 125 244 140625 30517578125 63 15752 961 992 436 543 126 252047 376 31 757 969 376 POWERS OF NUMBERS. — RECIPROCALS. 4th Power. 5th Power, ] Number. | 4th Power. 5th Power. 260 144 641 268 435 456 276 922 881 285610000 294 499 921 3°3 595 77 6 312900721 322417936 332150625 342 102016 352 275 361 362673936 33 ° 3 8 369 4°7 34 359 738 368 35723051649 37 129 300000 38579 489 651 40074642432 41615795893 43 204 003 424 44 840 334 375 46 525 874 W 6 48261724457 50049003168 I 139 140 1 Hi 142 143 144 145 146 147 148 149 1 150 373 3 CI 641 384 160000 395254161 406 586 896 418 161 601 429 981 696 442 050 6?5 454 37 i 856 466 948 881 479785216 492 884 401 506250000 51 888 844 699 53 782 400000 55 730 836 701 57 735 339232 59 797 108943 61917364224 64 097 340 625 66 338 290976 68 641 485 507 71 008 211 968 73 439 775 749 75 937 500000 greater than is 304 Number. 127 128 129 130 132 133 134 135 136 137 138 To Compute 4:th Power of a Number contained, in Table. Rule.— Ascertain square of number by preceding table or by calculation, and square it; product is power required. Example.— What is 4th power of 1500? 1500 2 = 2 250 000, and 2 250 ooo 2 = 5 062 500 000 000. To Compute 5 th. Power of a Number greater than is contained in Table. Rule.— Ascertain cube of number by preceding table or by calculation, and mul- tiply it by its square; product is power required. To Compute 4th and 5th Powers by another NLethod. rule —Reduce number by 2 until it is one contained within table. Take power which is required of that number, and multiply it by 16, 16 2 , or i63 respectively for each Son, by 2 for 4 th poier, and by 32, 32=, or 32 3 respectively for each division by 2 for 5th power. Example.— What are the 4th and 5th powers of 600? 600 -r- 2 = 300, and 300 -T- 2 == 150. The 4 th power of 150, per table, = 506 250000, which x 16 2 , multiplier for a second division 256 = 129 600000 000, 4 th power. Again, the 5 th power of 150 = 75 937 500000, which X 3 2 J > multiplier for a second division 1024 — 77 760000000000 — power. To Compute €>th Power of* a Number. Rule.— Square its cube. Example.— What is the 6th power of 2? 2 23 = 64. To Compute 4tli or 5th Root of a Number per Table. Rule— Find in column of 4 th and 5 th powers number given, and number from which that power is derived will give root required. Example.— What is the 5th root of 3 200000? 3 200000 in table is 5th power of 20; hence 20 is root required. RECIPROCALS. Reciprocal of a number is quotient arising from dividing 1 by number; thus, re- ciprocal of 2 is 1 -7- 2 = • 5 Product of a number and its reciprocal is always equal to 1 ; thus, 2 X .5 = *• Reciprocal of a vulgar fraction is denominator divided by numerator ; thus, - = • 5 - LOGARITHMS. 305 LOGARITHMS. Logarithms of Numbers. Logarithms are a series of numbers adapted to facilitate the operation of numerical computation, Addition being substituted for Multiplication, Subtraction for Division, Multiplication for Involution, and Division for Evolution. The Logarithm of a number is the exponent of a power to which 10 must be raised to give that number. It is not necessary, however, that the base should be 10, it may be any other num- ber; but Tables of Logarithms, in common use, are computed with 10 as the base. Thus, Number 100 Log. = 2, as io 2 base and exponent = 100. “ 10000 u = 4, u io 4 u “ u = 10000. The Unit or Integral part of a Logarithm is termed the Index , and the Decimal part the Mantissa ; the sum of the index and mantissa is the Logarithm. The Index of the Logarithm of any number, Integral or Mixed , when the base is 10, is equal to the number of digits to the left of the decimal point less 1. From o to 9, it is o; from 10 to 99, it is 1, and from 100 to 999, it is 2, etc. Thus, logarithm of 3304 = 3.51904, 3 being the index and .51904 the mantissa. The Index of the Logarithm of a Decimal Fraction is a negative number, and is equal to the number of places which the first significant figure of the decimal is re- moved from the place of units. Thus, index of logarithm .005 is 3 or — 3, the first significant figure, 5, being re- moved three places from that of units. The bar or minus sign is placed over an index to indicate that this alone is negative, while the decimal part is positive. The Difference is the tabular difference between the two nearest logarithms. The Proportional Part is the difference between the given and the nearest less tabular logarithm. The Arithmetical Complement of a number is the remainder after subtracting it from a number consisting of 1, with as many ciphers annexed as the number has integers. When the index of a logarithm is less than 10, its complement is ascer- tained by subtracting it from 10. Number. 4743 474 - 3 - •• 47 - 43 - 4 - 743 ' Illustrations. Logarithm. 3.676053 2.676053 1.676053 .676053 I Number. •4743 •• •°47 43 • .004 743 Logarithm. ^.676053 2.676053 3-676053 Computation of Negative Indices. To add two Negative Indices. Add them and put the sum negative. As 5 -j- 3 = 8*. To add a Positive and Negative Index. Subtract the less from the greater and to remainder give the positive or negative sign, according as the positive or nega- tive index is the greater. As 6 + 2"— 4, and 6 + 2 = 4. Illustration. — Add 6.387 57 and 2.924 59. 6.387 57 2 - 9 2 4 59 5.31216 Here the excess of 1 from 13 in the first decimal place, being positive, is carried to the positive 6, which makes 7, and 7 — 2 = 5. To Subtract a Negative Index. Change its sign to plus or positive, and then add it as in addition.^ As 3 from 2, = 3 + 2 = 5. And 5 from 2, = 5 + 2 = 3 ; also 3 from 5, =3 + 5 = 2. Illustration.— Subtract 5.765 52 from 2.346 74. 2.346 74 5-76552 2.581 22 Here, excess of 1 in the first decimal place used with the .3 in subtracting the .8 from the 1.3 is to be subtracted from the upper number 2, which makes it 3; then logarithms. 306 To Subtract a Positive Index ._ Change its sign to negative, and then add as in addition. As 2 — 2 = 2 -J- 2 = 4. To Multiply a Negative Index. Multiply the fractional parts by the ordinary rule, then multiply the negative index, which will give a negative product, and when an excess over io is to be carried, subtract the less index lrom the greater, and the re- mainder gives the positive or negative index, according as the positive or negative index is the greater. As 2 X 5 = 10, and 1 to be carried = 9. Illustration. — Multiply 2.3681 by 2, and 3.7856 by 6. 2.3681 3.7856 6 4.7362 H- 7 I 3 6 Here 2X2 = 4, also 3X6 = 18, with a positive excess of 4 = 14. To Divide a Negative Index. If index is divisible by divisor, der, put quotient with a negative sign. If negative exponent .is not ^visible by divisor add such a negative number to it as will make it divisible, and prefix an equal positive integer to fractional part of logarithm; then d 1 v ; de mcreased n ega- tive exponent and the other part of logarithm separately by ordinary nd^and for- mor quotient, taken negatively, will be index to fractional part of quotient. As 6_^ 3 == 2. ^ = 3 requires 2 to be addedor 2 to be subtracted, to make it divisible without a remainder, then 10 -f 2 = 12, 12-4-3 = 4) and 2 ( tlie sum subtracted) -r- 3 — .66, the quotient therefore is 4.66. Illustration i. — Divide 6.324282 by 3. 6. 324 282 -4-3 = 2. 108 094. 2.— Divide 14.326745 by 9. 14 . 326 745 - 4 - 9 = 18 + 4-326 745 - 4 - 9 = 2. 480 749+. Here 4 is added toTZ, that the sum 18 may be divided by 9, and as 4 is added, 4 must be 4 prefixed to the fractional part of the^ logarithm, and thus the value of the logarithm is unchanged, for there is added 4, and 4 = 0, or 4 is subtracted and 4 added. To Ascertain Logaritlirn. of’ a Number Toy Tatole. When the Number is less than tot. Look into first page of table, and opposite to number is its logarithm with its index prefixed. Illustration.— Opposite 7 is .845098, its logarithm; hence 70=1.84509 , .7 — I.845098, and .07 = 2.845098. When the Number is between 100 and 1000. r» mP , Find the aiven number in left-hand column of table headed No., and un- der o fn^eiTcolumTTs dectaal part of its logarithm, to which is to be prefixed a whofe Sumber for an index, of i or *, according as the number consists of a or 3 figures. Example.— What is logarithm of 450, and what of .4^? Log. 450 = 2.653 213, and of .45 = 1-653 213. When the Number is between 1000 and 10000. index. Example.— What is logarithm of 4505, and what of .04505 ? Log. 4505 = 3.653 695, and of .045 05 = 2.653 695. LOGARITHMS. 307 When the Number consists of Five Figures. Rule.— Find the logarithm of the number composed of the first four figures as preceding, then take the tabular difference from the right-hand column under D and multiply it by the fifth figure; reject the right-hand figure of the product and add the other figures, which are, and are termed, a proportional part to the logarithm found as above, observing that the right-hand figure of the proportional part is to be added to that of the logarithm, and the rest in order. Example. — Required logarithm of 83 407 ? Note.— When the number consists of less than 4 figures conceive a cipher an- nexed to make it four. Log. of 8340 (83 407) == 4.921 1 66 Tabular difference 52, which x 7 (5th figure) = 364 = 364 4. 92 1 202 4 logarithm. The difference of the numbers is nearly proportionate to the difference of their logarithms. Thus, difference between the numbers 8340 and 8341, the next in order, is 1, and the difference between their logarithms or tabular difference is 52. The log. of this 1 in the 4th place is therefore 52. The correction then, for the 7 nf the 5th place, which is .7 of 1 in the 4th place, is ascertained by the proportion 1 : 52 :: .7 : 36.4. The correction is obtained by multiplying the tabular difference by 7, rejecting the right hand figure of the product, if the log. is to be confined to six decimal places. When the Number consists of any Number over Four Figures. Rule. — Proceed as for four figures for the first four, multiplying the tabular dif- ference by the excess of figures over 4 and rejecting one right-hand figure of the product for a number of five figures, and two for one of six, and so on. Example i.— Required logarithm of 834079? Log. of 8340 (834079)= 5.921 166 Tabular difference 52, which x 79 = 4108 5.92120708 logarithm. 2. — Required logarithm of 8340794? Log. of 8340 (8 340 794) = 6.921 166 Tab. diff. 52, which x 794 (5th, 6th, and 7th figures) = 41 288 Or, Log. of 8340 = “ “ 7 (5th figure) X 52 tab. dif. “ “ 9 (6th “ ) X 5 2 “ “ “ “ 4 (7th “ ) X 52 “ Log. with index for 7 figures .... 6.921 207 288 logarithm. .921 166 208 6.921 207 288 To Ascertain. Logarithms of a Affixed Number. Rule. — Take out logarithm of the number as if it were an integer or whole num- ber, to w r hich prefix the index of the integral part of the number. Example.— What is logarithm of 834.0794? Mantissa of log. of 8 340 794 = 9 212 073 ; hence log. of 834.0794 = 2.921 207 3. To Ascertain Logarithm of a Decimal If raction. Rule. — Take logarithm from table as if the figures were all integers, and prefix index as by previous rules. Example.— L ogarithm of .1234=7.091 305. To Ascertain Logarithm of a ‘V'nlgar Fraction. Rule. — R educe the fraction to a decimal, and proceed as by preceding rule. Or, subtract logarithm of denominator from that of numerator, and the difference will give logarithm required. Example.— Logarithm of ^ = .1875. Log. .1875 = 1.273 001 logarithm. Or, Log. 3 = .477 121 u 16 = 1.20412 1.273001 logarithm. logarithms. 308 To _A.scertai.il. tHe Number Corresponding to a Given Logaritlim. When the given or exact Logarithm is in the Table. Operation.— Opposite to first two figures of logarithm, neglecting the index in column o look for the remaining figures of the log. in that column or in any of the n^ne at the' right thereof; the first three figures of the number will be found at the fpft in column under No., and the fourth at top directly over the log The number is to be made to correspond to index of logarithm, by pointing off decimals or prefixing ciphers. Illustration.— What is number corresponding to log. 3.963977 ? Opposite to 963977, in page 329, is 920, and at top of column is 4; hence, num- ber = 9204. When the given or exact Logarithm is not in the Table. Operation -Take the number for the next less logarithm from table, which will giV T e o fl a^ logarithm in table from the given Inlrithm add ciphers and divide by the difference in column D opposite the KlthS Annex quotient to the four figures already ascertained, and place deci- mal point. Illustration i.— What is number corresponding to log. 5.921 207 . Given log. = 5.921 207 8340 Next less in table 5.921 166 D = 52) 4100 (78-f- 78 364 834 078 Hence, number = 834 078. 460 416 44 2. —What is number corresponding to log. 3.922853? Given log. = 3.922853 8372 Next less in table 3.922 829 D = 52) 2400 (46 -j- 46 208 837 246 Hence, number = 8372.46. 320 3 12 8 Multiplication. Rule.— Add together the logarithms of the numbers and the sum will give the logarithm of the product. Example 1.— Multiply 345.7 by 2.581. Log- 345*7 =2.538699 u 2.581= .411788 2.950487 log. of product. Number = 892. 251. 2.— Multiply .03902, 59.71, and .003 147. Log. .03902 =2.591287 u 59- 71 =1.776047 “ .003147 = 3-497 897 3.865 231 log. of product. Number = .007 332 15. Division. Rule. -From logarithm of dividend subtract that of divisor, and remainder wiU give logarithm of the quotient. Example.— Divide 371.4 by 5 2 - 37- Log. 371.4 =2.569842 “ 52-37 = i-7i9° 8 3 • 850759 log. of quotient. Number = 7.091 8 0 . LOGARITHMS. 309 Rule of Three, or Proportion. Rule.— Add together the logarithms of the second and third terms, from their sum subtract logarithm of the first, and the remainder will give logarithm of the fourth term. Or, instead of subtracting logarithm of first term, add its Arithmetical Comple- ment , and subtract 10 from its index. Example 1.— What is fourth proportional to 723.4, .025 19, and 3574? As 723.4 log. — 2.859379 Is to .02519 li ==2.401228 So is 3574 “ = 3-553 155 1-954 3 8 3 First term “ 2.859379 1.095 004 log. of 4 th term. Number = . 124 453. By Arithmetical Complement. Illustration.— As 723.4 log. = 2.859 379> Ar - com. = 7- 140621 Is to .02519“ = 7.40x228 So is 3574 “ = 3-553 *55 , T 1. 095 004 log. of \th term. Number = .124 453. ^ 2— If an engine of 67 IP cari raise 57 600 cube feet of water in a given time, what IP is required to raise 8 575 000 cube feet in like time ? L °g- 8 575 000 = 6.933 234 67 = 1.826 075 u 8-759 3°9 57600 = 4.760422 3.998 877 log. of 4th term. Number = 9974.4 cube feet. 3. — If 14 men in 47 days excavate 5631 cube yards, what time will it require to excavate 47 280 at same rate of excavation ? 394 . 626 days. Involution. Rule.— Multiply logarithm of given number by exponent of the power to which it is to be raised, and the product will give the logarithm of the required power. Example.— What is cube of 30.71 ? Log. 30.71 = 1.487 28 3 4.461 84 log. of power. Number = 28 962.73. Evolution. Rule.— Divide logarithm of given number by exponent of the root which is to be extracted, and quotient will give logarithm of required root. Example 1.— What is cube root of 1234? Log. 1234 = 3.091 315 Divide by 3 = 1.030438 log. of root. Number = 10.72601. 2.— What is 4th root of .007654? Log. .007654 = 3.883888 Divide by 4 (here 34-1 + 1) = 1.470 972 log. of root. Number = .295 78. To Ascertain Reciprocal of a IN’uirn.ber. Rule.— Subtract decimal of logarithm of the number from .000000; add 1 to in- dex of logarithm and change its sign. The result is logarithm of the reciprocal. Example.— Required reciprocal of 230? .000000 Log. 230 = 2.361 728 3.638 272 = log. of .004 348 reciprocal. 3io LOGARITHMS. Simple Interest. Rule —Add together logarithm of principal, rate per cent., and time in years, from the sum subtract 2, and the remainder will give logarithm of the interest. Example.— What is interest on $ 500, @ 6 per cent., for 3 years? Log. 500 = 2.69897 6 = .77815* 3= *477 *21 3.954 242 2 1.954 242 log. of interest. Number = 90 dollars. Compound Interest. rule. Compute amount of $ . or £ i, etc, at the given rate of interest for one vpar for the first term, which is termed the ratio. . ^ Multiply logarithm of ratio by the time, add to product logarithm of the principal, and sum is logarithm of the amount. given Hates Her Cent. Rate. j Log. of Ratio. Rate. Log. of Ratio. Rate. Log. of Ratio. j Rate. Log. of Ratio. .004321 4 .005 395 . 006 466 .0075344 .008 600 2 .009 6633 .010723 9 . 011 781 8 .012 837 2 3-25 3-5 3- 75 4 4- 25 4-5 4- 75 5 5- 25 .013 890 1 .0149403 .015 988 1 .0170333 .018076 1 .019 116 3 . 020 1 54 .021 189 3 .022 222 I 5-5 5- 75 6 6.25 6- 5 6- 75 7 7' 2 5 7- 5 .0232525 .024 2804 *025 305 9 .026 328 9 .027 3496 .028 763 9 .029 3838 .030 3973 .031 408 5 I 7-75 8 8 - 2 5 8.5 8 - 75 9 9.25 9- 5 1 9-75 .0324373 .033 423 8 .034 4279 .035 4297 .0364293 .037 426 5 .038 421 4 .0394141 .040404 5 1.25 i-5 1- 75 2 2.25 2.5 2- 75 3 Example.— What will $364, at 6 per cent, to in 23 years? Los of ratio from above table .025 3059 23 per annum, compounded yearly, amount 364 • 5420357 2.561 IOI 3.1031367 log. of amount. Number = 1268.05 doll. Miscellaneous Illustrations. 1. What is area and circumference of a circle of 21.72 feet in diameter? 1.336 860 Log. of 21. 72 s =2-673720 “ “ .7854 = 1.895091 u « 2.568811 =37°- 54 feet area. Log. of 21.72 =2.33686 41 “ 3.1416 =_ 1 497£5 u a j.839 71 = 68.236 feet circum. 2. Sides of a triangle are 564, 373, and 747 feet; what is its area? Log. of sides 564 + 373 + 74 7 _ 2 g2$ 3I2 “ “ .5 side — a = 842 — 564 = 2.444045 “ “ .5 side — 6 = 842 — 373 — 2.671 173 “ “ .5 side — c =842 — 747 =± 9 TTJ 2 ± 2 )10.018 254 Area = Number of 5.009 127 = 1021.24 feet. 3 .-_ What is logarithm of 8 r 8 x 3 6 36 Log. ,3-6 o _ t. x log. 8 = 3.6 X .90309 = 3.251 124- Number = 1782.89. LOGARITHMS OF NUMBERS. 311 Logarithms of !NAimT>ers. From 1 to 10 000. No. j Logarithm. | No. | Logarithm. || No. Logarithm. | No. Logarithm. 1 *° 26 *•414 973 51 I.70757 76 1.880814 2 .301 03 27 I - 43 I 364 52 I.716 003 77 1.886 491 3 1 -477 121 28 1.447 158 53 I.724 276 78 1.892095 4 1 .60206 29 1.462 398 54 I- 73 2 394 79 1.897 627 5 .698 97 30 1.477 121 55 1.740363 80 1.90309 6 •778 151 31 1. 49 1 362 56 1.748 188 81 1.908 485 7 .845 098 3 2 1-505 15 57 1-755 875 82 1-913 814 8 .90309 33 1.518514 58 1.763 428 83 1.919 078 9 •954 243 34 I - 53 I 479 59 1.770 852 84 1.924279 10 1 35 1.544 068 60 1.778 151 85 1.929 419 11 1. 041 393 36 i -556 303 61 1-785 33 86 1.934 498 12 1.079 J 8i 37 1.568 202 62 1.792 392 87 1.939 519 13 I * II 3 943 38 1.579 784 63 1-799 34 i 88 1.944 483 14 1.146 128 39 1. 591 065 64 1.806 18 89 1.949 39 15 1.176 091 40 1.602 06 65 1.812 913 90 i -954 243 16 1.204 12 41 1.612 784 66 1.819544 91 1.959 041 17 1.230 449 42 1.623 249 67 1.826 075 92 1.963 788 18 1-255273 43 1.633468 68 1.832 509 93 1.968 483 19 1.278 754 44 1-643 453 69 1.838 849 94 1.973 128 20 1. 301 03 45 1.653213 70 1.845098 95 1.977 724 21 1.322 219 46 1.662 758 71 1.851 258 96 1.982 271 22 i-342 423 47 1.672 098 72 I -857 332 97 1.986 772 23 1.361 728 48 1. 681 241 73 1-863323 98 1. 991 226 24 1.380 21 1 49 1.690 196 74 1.869232 99 1.995 635 25 1 -397 94 50 1.69897 75 1.875061 100 2 No. 0 1 2 3 4 5 6 7 8 9 D 100 00- 0000 0434 0868 1301 1734 2166 2598 3029 346 i 3891 432 101 102 00- 00- 432 i 86 475 i 9026 5181 945 i 5609 9876 6038 6466 6894 7321 7748 8174 428 425 102 01- — — — 03 0724 1147 157 T 993 2415 424 103 01- 2837 3259 368 4 i 4521 494 536 5779 6197 6616 420 104 01- 7033 745 i 7868 8284 87 9116 9532 9947 — 417 104 02- — — — — — 0361 0775 416 105 _ 02- 1189 1603 2016 2428 2841 3252 3664 4075 4486 4896 412 IOO TO7 02- 02- 53 o 6 9384 5715 9789 6125 6533 6942 735 7757 8164 8571 8978 408 405 IO7 T lo 3 - — — oi 95 06 1004 1408 1812 2216 2619 3021 404 IOo IO9 ° 3 - ° 3 “ 3424 7426 3826 7825 4227 8223 4628 862 5029 9017 543 9414 583 9811 623 6629 7028 400 398 IO9 j 04- — — — — — 0207 0602 0998 397 110 04- I 393 1787 2182 2576 2969 3362 3755 4148 454 4932 393 in 112 04- 04- 5323 9218 5714 9606 6105 9993 6495 6885 7275 7664 8053 8442 883 389 388 112 113 05- 05- 3078 3463 3846 038 423 0766 4613 ii 53 4996 1538 5378 1924 576 2309 6142 2694 6524 386 383 114 1 14 05- 06- 6905 7286 7666 8046 8426 8805 9^5 9563 9942 383 2 _ — — — — 032 379 No. | 0 1 2 3 4 5 6 7 8 9 D 312 LOGARITHMS OF NUMBERS. No. 0 1 2 ■ 3 4 I 5 6 7 8 9 D 115 06- 0698 1075 1452 1829 2206 2582 2958 3333 3709 4083 376 116 06- 4458 4832 5206 558 5953 6326 6699 7071 7443 7815 373 117 06- 8186 8557 8Q28 9298 9668 j — — — — — 380 117 07- -- — — — — 0038 0407 0776 1145 1514 370 118 07- 1882 225 2617 2985 3352 3718 4085 4451 4816 5182 366 119 07- 5547 5912 6276 664 7004 7368 773 i 8094 8457 8819 363 120 07- 9181 9543 9904 362 120 08- — — 0266 0626 0987 1347 1707 2067 2426 360 121 08- 2785 3144 3503 3861 4219 4576 4934 5291 5647 6004 357 122 08- 636 6716 7071 7426 7781 8136 849 8845 9 ! 9 8 9552 355 123 08- 9905 355 123 09- — 0258 0611 0963 1315 1667 2018 237 2721 3071 353 124 09- 3422 3772 4122 4471 482 5169 5518 5866 6215 6562 349 125 09- 691 7257 7604 7951 8298 8644 899 9335 9681 — 348 125 10- — 0026 346 126 10- 0371 0715 1059 1403 1747 2091 2434 2777 3 ii 9 3462 343 127 10- 3804 4146 4487 4828 5169 55 i 5851 619I 6531 6871 34 i 128 10- 721 7549 7888 8227 8565 8903 9241 9579 99io — 338 128 11- — 0253 337 129 11- 059 0926 1263 1599 1934 227 2605 294 3275 3609 335 130 11- 3943 4277 4611 4944 5278 5611 5943 6276 6608 694 333 131 11- 7271 7603 7934 8265 8595 8926 9256 9586 9915 — 33 i I 3 I 12- — 0245 330 132 12- 0574 0903 1231 156 1888 2216 2544 2871 3198 3525 328 133 12- 3852 4 Y?8 4504 483 5156 5481 5806 6131 6456 6781 325 134 12- 7105 7429 7753 8076 8399 8722 9045 9368 969 — 323 , 134 13 - 0012 323 135 13- 0334 0655 0977 1298 1619 1939 226 258 29 3219 321 136 13- 3539 3858 4177 4496 4814 5133 5451 5769 6086 6403 318 137 13- 6721 7037 7354 7671 7987 8303 8618 8934 9249 9564 316 138 13- 9879 3 i 5 138 14- — 0194 0508 0822 1136 145 1763 2076 2389 2702 3 i 4 139 14- 3 oi 5 3327 3639 395 i 4263 4574 4885 5196 5507 5818 3 ii 140 14- 6128 6438 6748 7058 7367 7676 7985 8294 8603 89II 309 • * 4 i 14- 9219 9527 9835 — — — — 7 — — — 308 141 15- — — - — 0142 0449 0756 1063 137 1676 1982 307 142 15- 2288 2594 29 3205 35 i 3815 412 4424 4728 5032 305 143 15- 5336 564 5943 6246 6549 6852 7154 7457 7759 8o6l 303 144 15- 8362 8664 8965 9266 9567 9868 — — — — 302 144 16- — — — — — — 0168 0469 0769 1068 301 145 16- 1368 1667 1967 2266 2564 2863 3161 346 3758 4055 299 ■ 146 16- 4353 465 4947 5244 554 i 5838 6134 643 6726 7022 297 \ 147 16- 7317 7613 7908 8203 8497 8792 9086 938 9674 9968 295 j 148 17- 0262 0555 0848 1141 1434 1726 2019 2311 2603 2895 293 l 149 17- 3186 3478 3769 406 435 i 4641 4932 5222 5512 5802 291 150 17- 6091 6381 667 6959 7248 7536 7825 8113 8401 8689 289 151 17- 8977 9264 9552 9839 — — — — — — 287 } 151 18- — — — — 0126 0413 0699 0986 1272 I 53 8 287 152 18- 1844 2129 2415 27 2985 327 3555 3839 4123 4407 285 153 18- 4691 4975 5259 5542 582s 6108 6391 6674 6956 7239 283 154 18- 7521 7803 8084 8366 8647 8928 9209 949 9771 — 281 154 19- 0051 281 No. 0 1 2 3 4 5 6 7 8 9 eT LOGARITHMS OF NUMBERS. 313 No. 0 r 2' 3 4 5 6 7 8 9 155 19- 0332 0612 0892 1171 1451 173 201 2289 2567 2846 156 19- 3125 3403 3681 3959 4237 4514 4792 5069 5346 5623 157 19- 59 6176 6453 6729 7005 7281 7556 7832 8107 8382 158 19- 8657 8932 9206 9481 9755 — — — 158 20- — — — — — 0029 0303 0577 085 1124 159 20- 1397 167 1943 2216 2488 2761 3033 3305 3577 3848 160 20- 412 439i 4663 4934 5204 5475 5746 6016 6286 6556 161 20- 6826 7096 7365 7634 7904 8173 8441 871 8979 9247 162 20- 9515 9783 162 21- — — 0051 0319 0586 0853 1121 1388 1654 1921 163 21- 2188 2454 272 2986 3252 35i8 378 3 4049 4314 4579 164 21- 4844 5109 5373 5638 5902 6166 643 6694 6957 7221 165 21- 7484 7747 801 8273 8536 8798 906 9323 9585 9846 166 22- 0108 037 0631 0892 ii53 1414 1675 1936 2196 2456 167 22- 2716 2976 3236 3496 3755 4015 4274 4533 4792 5051 168 22- 5309 5568 5826 6084 6342 66 6858 7ii5 7372 763 169 22- 7887 8144 84 8657 8913 917 9426 9682 99 38 — 3:69 23- — — — — — — — — — 0193 170 23- 0449 0704 096 1215 147 1724 ! 97 9 2234 2488 2742 171 23- 2996 325 3504 3757 4011 4264 4517 477 5023 5276 172 23- 5528 578 i 6033 6285 6537 6789 7041 7292 7544 7795 173 23- 8046 8297 8548 8799 9049 9299 955 98 173 24- — 005 03 174 24- 0549 0799 1048 1297 1546 1795 2044 2293 2541 279 175 24- 3038 3286 3534 3782 403 4277 4525 4772 5019 5266 376 24- 55i3 5759 6006 6252 6499 6745 6991 7237 7482 7728 177 24- 7973 8219 8464 8709 8954 9198 9443 9687 9932 — 177 25- — 178 25- 179 25- 180 ! 25- 181 2s- 182 183 I 184 185 186 042 0664 0908 2853 3°9 6 3338 5273 5514 5755 7679 7918 8158 0071 031 0548 2451 2688 2925 4818 5054 529 7172 7406 7641 9513 9746 998 II 5 I 1395 358 3822 5996 6237 8398 8637 0787 1025 3162 3399 55 25 5761 7875 81 1 186 187 27- 1842 2074 2306 188 27- 4158 4389 462 189 ! 27- 6462 6692 6921 190 27- 8754 8982 9211 190 191 192 193 194 195 196 197 198 j 199 ! 199 No. 27- 27- 27- 2 7 ~ 27- 28- 28- 28- 28- 28- 29- 2 9 - 2 9 - 2 9 - 29- 30- 0213 0446 2538 277 485 5081 7151 738 9439 9 66 7 1033 1261 3301 3527 5557 5782 7802 8026 0035 0257 2256 2478 4466 4687 6665 6884 8853 9071 1488 1715 1942 3753 3979 4205 6007 6232 6456 8249 8473 8696 048 0702 0925 2699 292 3141 4907 5127 5347 7104 7323 7542 9289 9507 9725 1638 4064 6477 8877 1263 3636 5996 8344 1881 2125 4306 4548 6718 6958 9116 9355 1501 1739 3873 4109 6232 6467 8578 8812 2368 261 479 5031 7198 7439 9594 9833 1976 2214 4346 4582 6702 6937 9046 9279 0679 0912 3001 3233 5311 5542 7609 7838 9895 — — 0123 2169 2396 4431 4656 6681 6905 892 9143 1147 1369 3363 3584 5567 5787 7761 7979 ii44 1377 1609 3464 3696 3927 5772 6002 6232 8067 8296 8525 035 1 0578 0806 2622 2849 3075 4882 5107 5332 7i3 7354 7578 9366 9589 9812 1591 1813 2034 3804 4025 4246 6007 6226 6446 8198 8416 8635 D 279 278 276 275 274 272 271 269 268 267 266 264 262 261 259 258 257 256 255 253 252 251 250 249 248 246 246 245 243 242 241 239 238 237 235 234 234 233 232 230 229 228 228 227 226 225 223 222 221 220 219 218 218 T)~ 3I4 LOGARITHMS OF NUMBERS. No. 0 1 2 ■ 3 4 5 6 7 8 9 D 200 30- io 3 1247 1464 1681 1898 2114 2331 2547 2764 298 217 201 30- 3196 3412 3628 3844 4059 4275 4491 4706 4921 5136 216 202 30- 5351 5566 5781 5996 6211 6425 6639 6854 7068 7282 215 203 30- 7496 771 7924 8137 83s 1 8564 8778 8991 9204 9417 ! 213 204 30- 963 9843 1 213 204 31- — — 0056 0268 0481 0693 0906 1118 133 1542 212 205 3i“ 1754 1966 2177 2389 26 2812 3023 3234 3445 3656 211 206 3i“ 3867 4078 4289 4499 471 492 513 534 555i 576 | 210 207 31- 597 618 639 6599 6809 7018 7227 7436 7646 7854 : 209 208 31- 8063 8272 8481 8689 8898 9106 9314 9522 973 9938 ! 208 209 32- 0146 0354 0562 0769 0977 1184 1391 1598 1805 2012 1 207 210 32- 2219 2426 2633 2839 3046 3252 3458 3665 3871 4077 206 211 32- 4282 4488 4694 4899 5105 53i 5516 572i 5926 6131 205 212 32- 6336 6541 6745 695 7155 7359 7563 7767 7972 8176 204 213 32- 838 8583 8787 8991 9194 9398 9601 9805 — — 204 213 .33“ 0008 0211 203 214 33“ 0414 0617 0819 1022 1225 1427 163 1832 2034 2236 202 215 .33“ 2438 264 2842 3044 3246 3447 3649 385 4051 4253 202 216 33“ 4454 4 6 55 4856 5057 5257 5458 5658 5859 6059 626 201 217 33“ M 666 686 706 726 7459 7659 7858 8058 8257 200 2x8 33- 8456' 8656 8855 9054 9253 945i 965 9849 — — 200 2x8 34- — 0047 O246 199 2x9 34- 0444 0642 0841 1039 1237 1435 1632 183 2028 2225 198 220 .34“ 2423 262 2817 3014 3212 3409 3606 3802 3999 4196 I 97 221 .34“ 4392 4589 478s 4981 5178 5374 557 5766 5962 6157 196 .222 34“ 6 353 6549 6744 6939 7135 7.33 7525 772 79 T 5 8ll *95 223 .34-8305 85 8694 888 ; 9083 9278 9472 9666 986 — 194 ' 223 .224 35- .35-0248 0442 0636 0829 1023 1216 141 1603 1796 0054 1989 I 94 !93 ; 225 35- 2x83 2375 2568 2761 2954 ,3 I 47 3339 3532 3724 39 l6 I 93 • 226 35- 4io8 4301 4493 4685 4876 5068 526 5452 5643 5834 192 • 227 35-6026 6217 6408 6599 679 6981 7172 7363 7554 7744 I 9 I .228 35“ 7935 8125 8316 8506 8696 8886 9076 9266 9456 9646 19a .229 .35- 9835 189 229 36- — 0025 0215 0404 0593 0.783 0972 1161 135 1539 189 ; 280 36- .1728 I 9 I 7 2105 2294 2482 2671 2859 304s 3236 3424 188 ; 231 36- 3612 38 3988 4176 4363 4551 4739 4926 5ii3 5301 188 232 36- 5488 5675 5862 6049 6236 6423 661 6796 6983 7169 187 233 36- 7356 7542 7729 79*5 8101 8287 8473 8659 8845 903 186 234 36- 9216 9401 9587 9772 9958 — — — — — 186 235 37- “ — — — — 0143 0328 0513 0698 0883 185 235 37- xo68 1253 1437 1622 1806 1991 2175 236 2544 2728 184 i 236 37- 2912 3096 328 3464 3647 3831 4015 4198 4382 4565 184 237 37" 4748 4932 5ii5 5298 548 i 5664 5846 6029 6212 6394 183 ! -238 37- ^577 6759 6942 7124 7306 7488 767 7852 8034 8216 182 239 37" 8398 858 8761 8943 9 I2 4 9306 9487 9668 9849 — l82 , 239 3 S 003 , lSl 240 38- 0211 0392 0573 0754 0934 11x5 1296 1476 1656 1837 ; 181 ; 241 38- 2017 2197 2377 2557 2737 2917 3097 3277 3456 3636 180 242 38- 3815 3995 4174 4353 4533 4712 4891 507 5249 54 2 8 179 243 38- 5606 5785 5964 6142 6321 6499 6677 6856 7034 7212 178 244 38- 739 7568 7746 7923 8101 8279 8456 8634 88 1 1 8989 178 1 0 1 2 3 4 I 5 6 7 8 9 | D logarithms of numbers. 315 No. | 0 245 38- 9 l66 245 39- “ 39- °935 39- 2697 39" 4452 39" 6i 99 39“ 794 39- 9674 40- — 40- 1401 40- 3121 40- 4834 40- 654 40- 824 40- 9933 41- — 41- 162 41- 33 9343 952 9 6 9 8 9 S 75 246 247 248 249 250 251 251 252 253 254 255 256 257 257 258 259 260 261 262 263 263 264 265 266 267 268 269 269 270 271 1 1 12 1288 1464 2873 3048 3224 4627 4802 4977 6374 6548 6722 8114 8287 8461 9S47 — — 002 0192 1573 J 745 X 9 X 7 3292 3464 3635 5005 517 6 5346 671 6881 7°5 r 841 8579 8749 41- 4973 41- 6641 41- 8301 41- 9956 42- — 42- 1604 42- 3246 42- 4882 42- 6511 42- 8135 42- 9752 43- — 43- 1364 ; 43- 2969 272 1 43- 45^9 273 43" 6i6 3 274 j 43- 7751 27 5 43" 9333 275 ! 44" — 276 44- 0909 277 44- 248 278 44- 4045 279 44“ 5604 280 I 44“ 7158 281 44- 8706 281 45- — 282 45- 0249 283 45“ j 786 284 45“ 33i8 235 45- 4845 286 45- 6366 287 45- 7882 288 45- 9392 288 46- — 289 46- 0898 No. I 0 0102 027 1 044 1788 1956 2124 3467 3635 3803 514 5307 5474 6807 6973 7139 8467 8633 8798 0121 0286 0451 1768 1933 2097 34i 3574 3737 5045 5208 5371 6674 6836 6999 8297 8459 8621 99 r 4 — “ — 0075 0236 1525 1685 1846 3 T 3 329 345 4729 4888 5048 6322 6481 664 7909 8067 8226 9491 9648 9806 1066 1224 1381 2637 2793 295 4201 4357 45 T 3 576 59 x 5 6071 73 1 3 74 68 7623 8861 9015 917 0403 0557 0711 194 2093 2247 3471 3624 3777 4997 5*5 5302 6518 667 6821 8033 8184 8336 9543 9 6 94 9 8 45 1641 34 5152 6896 8634 0365 2089 3807 55 x 7 7221 8918 0609 2293 397 5641 7306 8964 0616 2261 39 QI 5534 7161 8783 0051 1817 3575 5326 7071 8808 0538 2261 3978 5688 739 1 9087 0777 2461 4*37 5808 7472 9 I2 9 0781 2426 4065 5697 7324 8944 0398 ; 0559 2007 I 2167 3 61 ! 377 5207 : 5367 6799 6957 8384 9964 1538 3106 4669 6226 7778 8542 0228 0405 0582 1993 2169 2345 375 1 3926 4 101 55 01 5676 585 7245 7419 7592 8981 9154 9328 0711 0883 1056 2433 2605 2777 4149 432 4492 5858 6029 6199 7561 773 x 19 01 9257 9426 9595 0946 3114 1283 2629 2796 2964 4305 4472 4639 5974 6141 -6308 7638 7804 797 9295 946 96 2 5 0945 in 1275 259 2754 2918 4228 4392 45 55 586 6023 6186 7486 7648 7811 9106 9268 9429 072 0881 1042 2328 2488 2649 393 409 4249 5526 5685 5844 7116 7275 7433 8701 8859 9017 D 177 177 176 176 175 174 9501 I x 73 — *73 1228 173 2949 I 172 0759 2521 4277 6025 7766 4663 637 807 9764 0122 1695 3263 4825 6382 7933 9324 j 9478 086s ; 1018 24 393 5454 6973 8487 9995 1048 1198 1348 1499 2553 4082 k6o6 7 I2 5 8638 0146 1649 0279 0437 0594 1852 2009 2166 34*9 357 6 3732 4981 5137 5293 6537 6692 6848 8088 8242 8397 9633 9787 9941 1172 1326 1479 2706 2859 3012 4235 4387 454 5758 591 6062 7276 7428 7579 8789 894 9091 0296 0447 0597 1799 1948 2098 171 17 1 170 169 169 169 168 167 167 166 165 165 165 164 164 163 162 162 162 161 161 160 159 159 158 158 158 157 157 156 155 i55 154 154 154 153 153 152 x 52 i5 x 151 0748 151 2248 | 150 0 “nr 1451 3 I 3 2 4806 6474 8135 979 1 1439 3082 4718 6349 7973 959 1 1203 2809 4409 6004 7592 9*75 0752 2323 3889 5449 7003 8552 0095 1633 3 l6 5 4692 6214 773 1 9242 3 1 6 LOGARITHMS OF NUMBERS. No. 0 1 2 3 4 5 6 7 8 9 D 290 46- 2398 2548 2697 2847 2997 3146 3296 3445 3594 3744 150 291 46- 3893 4042 4191 434 449 4639 4788 4936 5085 5234 149 292 40- 5383 5532 568 5829 5977 6126 6274 6423 6571 6719 149 293 46- 6868 7016 7164 7312 746 7608 7756 7904 8052 82 148 294 46- 8347 8495 8643 879 8938 9085 9233 938 9527 9675 148 295 46- 9822 9969 147 295 47- — — 0116 0263 041 0557 0704 0851 0998 1145 147 296 47- 1292 1438 1585 1732 1878 2025 2171 2318 2464 261 146 297 47- 2756 2903 3049 3195 3341 3487 3633 3779 3925 4071 146 298 47- 4216 4362 4508 4653 4799 4944 509 5235 5381 5526 146 299 47- 5671 5816 5962 6107 6252 6397 6542 6687 6832 6976 145 300 47- 7121 7266 7411 7555 77 7844 7989 8i33 8278 8422 145 30 1 47- 8566 8711 8855 8999 9H3 9287 9431 9575 9719,9863 144 302 45- 0007 0151 0294 0438 0582 0725 0869 1012 1156 1299 144 3°3 48- 1443 1586 1729 1872 2016 2159 2302 2445 2588 2731 143 304 48- 2874 3° l6 3159 3302 3445 3587 373 3872 4015 4157 143 305 48- 43 4442 4585 4727 4869 5011 5153 5295 5437 5579 142 306 48- 5721 5863 6005 6147 6289 643 6572 6714 6855 6997 143 307 48- 7138 728 742i 7563 7704 7845 7986 8127 8269 841 141 308 48- 8551 8692 8833 8974 9 IJ 4 9255 9396 9537 9677 9818 141 309 48- 9958 140 309 49- — 0099 0239 038 052 0661 0801 0941 1081 1222 140 310 49- 1362 1502 1642 1782 1922 2062 2201 2341 2481 2621 140 3ii 49- 276 29 304 3179 33i9 3458 3597 3737 3876 4015 139 312 49- 4155 4294 4433 4572 4711 485 4989 5128 5267 5406 139 3i3 49- 5544 5683 5822 596 6099 6238 6376 6515 6653 6791 139 3i4 49- 693 7068 7206 7344 7483 7621 7759 7897 8035 8173 138 315 49- 8311 8448 8586 8724 8862 8999 9 r 37 9275 9412 955 138 3 l6 49- 9687 9824 9962 . — — — 137 3 l6 50- — — — 0099 0236 0374 0511 0648 0785 0922 137 ' 3i7 50- 1059 1196 *333 147 1607 1744 188 2017 2154 2291 137 318 50- 2427 2564 27 2837 2973 3 io 9 3246 3382 3518 3655 136 3i9 50- 379i 3927 4063 4199 4335 4471 4607 4743 4878 5014 136 i 320 50- 515 5286 542i 5557 5693 5828 5964 6099 6234 637 136 321 50- 6505 664 6776 6911 7046 7181 7316 745i 7586 7721 135 322 50- 7856 7991 8126 826 8395 853 8664 8799 8934 9068 135 323 50- 9203 9337 9471 9606 974 9874 — — — — 134 323 51- — — — — — — 0009 0143 0277 0411 J 34 324 51- 0545 0079 0813 0947 1081 1215 1349 1482 1616 175 134 325 51- 1883 2017 2151 2284 2418 2551 2684 2818 2951 3084 133 ,■ 326 51- 3218 3351 3484 3617 375 3883 4016 4149 4282 4415 133 327 51- 4548 4681 4813 4946 5079 1 5211 5344 5476 5609 5741 133 \ 328 51- 5874 6006 6139 6 2 7 j 6403 : 6535 6668 68 6932 7064 132 S 329 51- 7196 7328 746 7592 7724 7855 7987 8119 8251 8382 132 \ 330 51- 8514 8646 8777 8909 904 9171 9303 9434 9566 9697 131 33i 51- 9828 9959 — — — — — — — — 131 33i 52- — — 009 0221 0353 0484 0615 0745 0876 1007 131 332 52- 1138 1269 14 153 1661 1792 1922 2053 2183 2314 131 333 52- 2444 2575 2705 2835 2966 3096 3226 3356 3486 3616 130 334 52- 3740 3870 4006 4136 4266 4396 4526 4656 4785 4915 130 No. 0 1 2 3 4 5 6 7 8 9 LOGARITHMS OF NUMBERS. 317 No. 335 336 337 338 338 339 340 341 342 343 344 345 346 346 347 348 349 350 351 352 353 354 354 355 356 357 358 359 360 361 362 363 363 364 365 366 367 368 369 370 371 371 372 373 374 375 376 377 378 379 No. 0 1 2 3 4 52- 5045 5i74 5304 5434 5563 52- 6"nq 6469 6598 6727 6856 52- 763 7759 7888 8016 8145 52- 8917 9045 9 X 74 9302 943 53 53“ 02 0328 0456 0584 0712 53- *479 1607 1734 1862 199 53- 2754 2882 3°°9 3136 3264 53- 4026 4153 428 4407 4534 53- 5294 542i 5547 5074 58 53" 6 55§ 6685 6811 6937 7063 53" 7819 7945 8071 8197 8322 53“ 9076 9202 9327 9452 9578 54- 54- 0329 0455 058 0705 083 54- 1579 1704 1829 1953 2078 54- 2825 295 3074 3199 3323 54- 4068 4192 4316 444 4564 54- 5307 543i 5555 5678 5802 54- 6543 6666 6789 6913 7036 54- 7775 7898 8021 8144 8267 54- 9003 9126 9249 9371 9494 55" 55- 0228 0351 0473 0595 0717 55- 145 1572 1694 1816 1938 55- 2668 279 2911 30 33 3 X 55 55- 3883 4004 4126 4247 4368 55- 5094 5215 533 6 5457 5578 55- 6303 6423 6544 6664 6785 55- 7507 7627 7748 7868 7988 55- 8709 8829 8948 9068 9188 55- 9907 56- — 0026 0146 0265 0385 56- IIOI 1221 i34 1459 1578 56- 2293 2412 2531 265 2769 56- 3481 36 37i8 3837 3955 56- 4666 4784 4903 5021 5i39 56- 5848 5966 6084 6202 632 56- 7026 7*44 7262 7379 7497 56- 8202 8319 8436 8554 8671 56- 9374 9491 9608 9725 9842 57“ — 57- 0543 066 0776 0893 IOI 57- 1709 1825 1942 2058 2174 57- 2872 2988 3104 322 3336 57- 4031 4147 4263 4379 4494 57- 5i88 5303 5419 5534 565 57" 6341 6457 6572 6687 6802 57- 7492 7607 7722 7836 795i 57- 8639 8754 8868 8983 9097 0123 56789 5693 5822 5951 6081 621 6985 71 14 7243 7372 7501 8274 8402 8531 866 8788 9559 9687 9815 9943 — — — — — 0072 084 0968 1096 1223 1351 2117 2245 2372 25 2627 3391 3518 3645 3772 3899 4661 4787 4914 5041 5167 5927 6053 618 6306 6432 7189 7315 7441 7567 7693 8448 8574 8699 8825 8951 9703 9829 9954 — — — — — 0079 0204 0955 108 1205 133 1454 2203 2327 2452 2576 2701 3447 3571 3696 382 3944 4688 4812 4936 506 5183 5925 6049 6172 6296 6419 7159 7282 7405 7529 7652 8389 8512 8635 8738 8881 9616 9739 9861 9984 — — — — — 0106 084 0962 1084 1206 1328 206 2181 2303 2425 2547 3276 3398 3519 364 3762 4489 461 4731 4852 4973 5699 582 594 6061 6182 6905 7026 7146 7267 7387 8108 8228 8349 8469 8589 9308 9428 9548 9667 9787 0504 0624 0743 0863 0982 1698 1817 1936 2035 21 74 2887 3006 3125 3244 3362 4074 4192 4311 4429 4548 5 2 57 5376 5494 5612 573 6437 6555 667 3 6791 6909 7614 7732 7849 7967 8084 8788 8905 9023 914 9257 9959 — — — — — 0076 0193 0309 0426 H26 1243 1359 1476 1592 2291 2407 2523 2639 2755 345 2 3568 3684 38 39*5 461 4726 4841 4957 5072 5765 588 5996 61 1 1 6226 6917 7032 7147 7262 7377 8066 8181 8295 841 8525 9212 9 326 9441 9555 9669 6 7 8 9 D 129 129 129 128 128 128 128 127 127 126 126 126 126 125 125 125 124 124 124 123 123 123 123 122 122 121 121 121 120 120 120 120 119 119 ng 119 118 1 18 115 117 117 117 117 11 6 116 1 id 1 IS *15 1 14 D 4 I 5 D D* Nol 0 1 2 3 4 5 6 7 8 9 D 380 080 57 - 58- 9784 9898 0012 0126 0241 0355 0469 0583 0697 0811 114 114 381 58 - 0925 1039 ii 53 1267 1381 1495 1608 1722 1836 195 114 382 58 - 2063 2177 2291 2404 2518 2631 2745 2858 2972 3085 114 383 58 - 3 X 99 33 12 3426 3539 3652 3765 3879 3992 4105 4218 ZI 3 384 58 - 433 1 4444 4557 467 4783 4896 5009 5122 5235 5348 **3 385 58 - 5461 5574 5686 5799 59 X2 6024 6137 625 6362 6475 i *3 386 58 - 6587 67 6812 6925 7037 7149 7262 7374 7486 7599 112 387 58 - 77H 7823 7935 8047 816 8272 8384 8496 8608 872 1 12 388 58 - 8832 8944 9056 9167 9279 939 1 9503 9 6i 5 9726 9838 1 12 389 389 58 - 59 - 995 0061 0173 0284 0396 0507 0619 073 0842 0953 1 12 1 12 390 59 “ 1065 1176 1287 1399 151 1621 x 732 1843 1955 2066 III 39 1 59 - 2177 2288 2 399 251 2621 2732 2843 2954 3064 3*75 III 39 2 59 - 3286 3397 3508 3 6 * 8 3729 384 395 4061 4 I 7 I 4282 III 393 59 “ 4393 4503 4614 4724 4834 4945 5055 5165 5276 5386 no 394 59 - 5496 5606 57 x 7 5827 5937 6047 6157 6267 6377 6487 no 395 59 “ 6597 6707 6817 6027 7037 7146 7256 7366 7476 7586 no 396 59 - 7695 7805 79*4 8024 8134 8243 8353 8462 8572 8681 no 397 59 - 8791 89 9009 9**9 9228 9337 9446 9556 9665 9774 109 398 -208 59 - 60- 9883 999 2 0101 021 0319 0428 0537 0646 0755 0864 109 109 399 60- 0973 1082 1191 1299 1408 i 5 x 7 1625 1734 1843 i 95 i 109 400 60- 206 2169 2277 2386 2494 2603 2711 2819 2928 3 ° 3 6 108 401 60- 3144 3 2 53 33 6 * 3469 3577 3686 3794 39 ° 2 401 4118 108 402 60- 4226 4334 4442 455 4658 4766 4874 4982 5089 5 X 97 108 4°3 60- 5305 54*3 5521 5628 5736 5844 595 x 6059 6166 6274 108 404 60- 6381 6489 6596 6704 6811 6919 7026 7 *33 7241 7348 107 405 60- 7455 7562 7669 7777 7884 799 i 8098 8205 8312 8419 107 406 60- 8526 8633 874 8847 8954 9061 9 i6 7 9274 9381 9488 107 40 7 60- 9594 9701 9808 9914 — — — — 107 AO 7 61- — 0021 0128 0234 034 x 0447 0554 107 408 61- 066 0767 0873 0979 1086 1192 1298 1405 1511 1617 106 409 61- 1723 1829 1936 2042 2148 2254 236 2466 2572 2678 106 410 61- 2784 289 2996 3 102 3207 33 x 3 34 x 9 3525 363 3736 106 4 11 61- 3842 3947 4053 4159 4264 437 4475 458 i 4686 4792 106 4.12 61- 4897 5003 5X08 5213 5319 5424 5529 5634 574 5845 105 T" 413 61- 595 6055 6l6 6265 637 6476 6581 6686 679 6895 105 4 i 4 61- 7 7105 721 73*5 742 7525 7629 7734 7839 7943 105 415 61- 8048 8 i 53 8257 8362 8466 857 x 8676 878 8884 S989 105 416 416 417 61- 62- 62- 9°93 0136 9 * 9 8 024 9302 0344 9406 0448 95 ii 0552 9 6i 5 0656 9719 076 9824 0864 9928 0968 0032 1072 *05 104 104 418 62- 1176 128 X 384 1488 1592 1695 I 799 I 9°3 2007 211 104 419 62- • 2214 2318 2421 25 2 5 2628 2732 283s 2939 3042 3 x 46 104 4*20 62- • 3 2 49 3353 3456 3559 3663 3766 3869 3973 4076 4 X 79 103 421 62- - 4282 4385 4488 459 1 4695 4798 49°! 5004 5107 521 103 422 62- ■ 53 12 54*5 55*8 5621 5724 5827 5929 6032 6 i 35 6238 103 423 62- 634 6 443 6546 6648 6751 6853 6956 7058 7161 7263 103 424 62- ■ 7366 7468 757 i 7673 7775 7878 798 8082 8185 8287 102 No. 0 1 . 2 3 4 5 6 7 8 9 D LOGARITHMS OF NUMBERS. 3 X 9 No. 0 1 2 3 4 5 6 7 8 9 D 425 62- 8389 8491 8593 8695 8797 89 9002 9 io 4 9206 9308 102 426 62- 94 i 9512 9613 9715 9817 9919 — — — 102 426 63- — — — — — 0021 0123 0224 0326 102 427 6 3 ‘ 0428 053 0631 0733 0835 0936 1038 ii 39 1241 1342 102 428 63- 1444 1545 1647 1748 1849 1951 2052 2153 2255 2356 IOI 429 63- 2 457 2 559 266 2761 2862 2963 3064 3 i6 5 3266 3367 IOI 430 63- 3468 3569 367 3771 3872 3973 4074 4 i 75 4276 4376 IOI 43 1 63- 4477 4578 4679 4779 488 4981 5081 5182 5283 5383 IOI 432 63- 5484 5584 5685 5785 5886 5986 6087 6187 6287 6388 IOO 433 63- 6488 6588 6688 6789 6889 6989 7089 7189 729 739 IOO 434 63- 749 759 769 779 789 799 809 819 829 8389 IOO 435 63- 8489 8589 8689 8789 8888 8988 9088 9188 9287 9387 IOO 436 63- 9486 9586 9686 9785 9885 9984 — 0183 — — IOO 436 64- — — — — — — 0084 0283 0382 99 437 64- 0481 0581 068 0779 0879 0978 1077 1177 1276 1375 99 438 64- 1474 1573 1672 1771 1871 197 2069 2168 2267 2366 99 439 64- 2465 2563 2662 2761 286 2959 3058 3156 3255 3354 99 440 64- 3453 355 i 365 3749 3847 3946 4044 4 i 43 4242 434 99 44 1 64- 4439 4537 4636 4734 4832 493 i 5029 5127 5226 5324 98 44 2 64- 54 22 552 i 5619 5717 5815 5913 6011 611 6208 6306 98 443 64- 6404 6502 66 6698 6796 6894 6992 7089 7 i8 7 7285 98 444 64- 7383 7481 7579 7676 7774 7872 7969 8067 8165 8262 98 445 64- 836 8458 8555 8653 875 8848 8945 9°43 9*4 9 2 37 97 446 64- 9335 9432 953 9627 9724 9821 9919 — — — 97 446 65- — — — — — — — 0016 0113 021 97 447 65- 0308 0405 0502 0599 0696 0793 089 0987 1084 1181 97 448 65- 1278 1375 1472 1569 1666 1762 1859 1956 2053 215 97 449 65 - 2246 2343 244 2536 2633 273 2826 2923 3 OI 9 3 116 97 450 65- 3 2 i 3 3309 3405 3502 3598 3695 379 1 3888 3984 408 96 45 i 65- 4177 4273 4369 4465 4562 4658 4754 485 4946 5042 96 45 2 65- 5138 5235 533 i 5427 5523 5619 5715 581 5906 6002 96 453 65- 6098 6194 629 6386 6482 6577 6673 6769 6864 696 96 454 65 - 7056 7152 7247 7343 7438 7534 7629 7725 782 7916 96 455 65- 8011 8107 8202 8298 8393 8488 8584 8679 8774 887 95 456 65- 8965 906 9 r 55 925 9346 9441 9536 9631 9726 9821 95 457 65- 99 l6 95 457 66- 0011 0106 0201 0296 ° 39 I 0486 0581 0676 0771 95 458 66- 0865 096 1055 ii 5 1245 1339 1434 1529 1623 1718 95 459 66- 1813 1907 2002 2096 2191 2286 238 2475 2569 2663 95 460 66- 2758 2852 2947 3041 3135 323 3324 34 i 8 3512 3607 94 461 66- 3701 3795 3889 3983 4078 4172 4266 436 4454 4548 94 462 66- 4642 4736 483 4924 5018 5 112 5206 5299 5393 5487 94 463 66- 558 i 5675 5769 5862 5956 605 6143 6237 6331 6424 94 464 66- 6518 6612 6705 6799 6892 6986 7079 7 i 73 7266 736 94 465 66- 7453 7546 764 7733 7826 792 8013 8106 8199 8293 93 466 66- 8386 8479 8572 8665 8759 8852 8945 9038 9 L 3 i 9224 93 467 66- 9317 941 9503 9596 9689 9782 9875 9967 — — 93 467 67- 006 0153 93 468 67- 0246 0339 0431 0524 0617 071 0802 0895 0988 108 93 469 67- ii 73 1265 1358 i 45 i 1543 1636 1728 1821 1913 2005 93 No. 0 1 2 3 4 5 6 7 8 9 D 320 LOGARITHMS OF NUMBERS. No. 0 1 2 3 4 5 6 7 8 9 j D 470 67- 2098 219 2283 2375 2467 256 2652 2744 2836 2929 92 47 1 67- 3021 3ii3 3205 3297 339 3482 3574 3666 3758 385 92 472 67- 3942 4034 4126 4218 43i 4402 4494 4586 4677 4769 92 473 67- 4861 4953 5045 5137 5228 532 5412 5503 5595 5687 92 474 67- 5778 5«7 5962 6053 61 45 6236 6328 6419 6511 6602 i 92 475 67- 6694 6785 6876 6968 7059 7i5i 7242 7333 7424 75i6 9 1 476 67- 7607 7698 7789 7881 7972 8063 8i54 8245 8336 8427 9 T 477 67- 8518 8609 87 8791 8882 8973 9064 9 I 55 9246 9337 | 9i 478 67- 9428 9519 961 97 9791 9882 9973 — — — I 9i 478 68- — 0063 oi54 0245 9 1 479 68- 0336 0426 0517 0607 0698 0789 On CO O 097 106 1151 | 9i 480 68- 1241 1332 1422 1513 1603 1693 1784 1874 1964 2055 90 481 68- 2145 2235 2326 2416 2506 2596 2686 2777 2867 2957 90 482 68- 3047 3*37 3227 3317 3407 3497 3587 3677 3767 3857 90 483 68- 3947 4037 4127 4217 4307 4396 4486 4576 4666 4756 90 484 68- 4845 4935 5025 5ii4 5264 5294 5383 5473 5563 5652 90 485 68- 5742 5831 592i 601 61 6189 6279 6368 6458 6547 89 486 68- 6636 6726 6815 6904 6994 7083 7172 7261 735i 744 89 487 68- 7529 7618 7707 7796 7886 7975 8064 8i53 8242 8331 89 488 68- 842 8509 8598 8687 8776 8865 8953 9042 9 J 3i 922 89 489 68- 9309 9398 9486 9575 9664 9753 9841 993 — — 89 489 69- 0019 0107 89 490 69- 0196 0285 0373 0462 055 0639 0728 0816 0905 0993 89 49 1 69- 1081 1 17 1258 1347 1435 1524 1612 17 1789 1877 88 49 2 69- 1965 2053 2142 223 2318 2406 2494 2583 2671 2759 88 493 69- 2847 2935 3023 3111 3i99 3287 3375 3463 355i 3639 88 494 69- 3727 3815 39°3 399i 4078 4166 4254 4342 443 4517 88 495 69- 4605 4693 4781 CO 0 00 4956 5044 5131 5219 5307 5394 88 496 69- 5482 5569 5657 5744 5832 5919 6007 6094 6182 6269 87 497 69- 6356 6444 6531 6618 6706 6793 688 6968 7055 7142 87 498 69- 7229 7317 7404 7491 7578 7665 7752 7839 7926 8014 87 499 69- 8101 8188 8275 8362 8449 853s 8622 8709 8796 8883 87 600 69- 897 9057 9 T 44 9 2 3 x 9377 9404 9491 9578 9664 975i 87 501 69- 9838 9924 87 501 70- — — 0011 0098 0184 0271 0358 0444 0531 0617 87 502 70- 0704 079 0877 0963 105 1136 1222 1309 1395 1482 86 503 70- 1568 1654 1741 1827 1913 1999 2086 2172 2258 2344 86 504 70- 2431 2517 2603 2689 2775 2861 2947 30 33 3ii9 3205 S6 505 70- 3291 3377 3463 3549 3635 3721 3807 3893 3979 4065 86 506 70- 4151 4236 4322 4408 4494 4579 4665 475i 4837 4922 86 507 70- 5008 5094 5i79 5265 535 5436 5522 5607 5693 5778 86 508 70- 5864 5949 6035 612 6206 6291 6376 6462 6547 6632 85 509 70- 6718 6803 6888 6974 7059 7144 7229 73i5 74 7485 85 510 70- 757 7655 774 7S26 7911 7996 8081 8166 8251 8336 85 5ii 70- 8421 8506 859 1 8676 8761 S846 8931 9 OI 5 9i 9185 85 512 70- 927 9355 944 9524 9609 9694 9779 9863 9948 — 85 512 71- 0033 85 5i3 71- 0117 0202 M 0 00 0 0456 054 0625 071 0794 0879 85 514 71- 0963 1048 1132 1217 1301 1385 147 1554 1639 1723 84 No. 0 1 2 3 4 5 6 7 8 9 eT LOGARITHMS OF [NUMBERS. 32 1 No. 0 2 3 4 5 6 7 8 9 D 515 71- 1807 1892 1976 206 2144 2229 2313 2397 2481 2566 84 5i6 71- 265 2734 2818 2902 2986 307 3154 3238 3323 3407 84 5i7 71- 3491 3575 3659 3742 3826 391 3994 4078 4162 4246 84 5i8 71- 433 4414 4497 458 i 4665 4749 4833 49 l6 5 5084 84 519 71- 5167 5251 5335 5418 5502 5586 5669 5753 5836 592 84 520 71- 6003 6087 617 6254 6337 6421 6504 6588 6671 6754 83 521 71- 6838 6921 7004 7088 7171 7254 7338 7421 7504 7587 83 522 71- 7671 7754 7837 792 8003 8086 8169 8253 8336 8419 83 523 71- 8502 8585 8668 8751 8834 8917 9 9083 9165 9248 83 524 71- 9331 9414 9497 958 9663 9745 9828 9911 9994 — 83 524 72- — — 0077 83 525 72- 0159 0242 0325 0407 049 0573 0655 0738 0821 0903 83 526 72- 0986 1068 1151 1233 1316 1398 1481 1563 1646 1728 82 527 72- 1811 1893 1975 2058 214 2222 2305 2387 2469 2552 82 528 72- 2634 2716 2798 2881 2963 3045 3127 3209 3291 3374 82 529 72- 3456 3538 362 3702 3784 3866 3948 403 4112 4194 82 530 72- 4276 4358 444 4522 4604 4685 4767 4849 493i 5013 82 53 i 72- 5095 5176 5258 534 5422 5503 5585 5667 5748 583 82 532 72- 5912 5993 6075 6156 6238 632 6401 6483 6564 6646 82 533 72- 6727 6809 689 6972 7053 7134 7216 7297 7379 746 81 534 72- 754i 7623 7704 7785 7866 7948 8029 811 8191 8273 81 535 72- 8354 8435 8516 8597 8678 8759 8841 8922 9°°3 9084 81 536 72- 9165 9246 9327 9408 9489 957 9651 9732 9813 9893 81 537 72- 9974 — 81 537 73- — 0055 0136 0217 0298 0378 0459 054 0621 0702 81 538 73- 0782 0863 0944 1024 1105 1186 1266 1347 1428 1508 81 539 73- 1589 1669 175 183 1911 1991 2072 2152 2233 2313 81 540 73- 2394 2474 2555 2635 2715 2796 2876 2956 3037 3ii7 80 54i 73- 3 J 97 3278 3358 3438 35i8 3598 3679 3759 3839 3919 80 542 73- 3999 4079 416 424 432 44 448 456 464 472 80 543 73- 48 488 496 504 512 52 5279 5359 5439 5519 80 544 73- 5599 5679 5759 5838 59 j 8 5998 6078 6157 6237 6317 80 545 73- 6397 6476 6556 6635 6715 6795 6874 6954 7034 7ii3 80 546 73- 7i93 7272 7352 743i 75ii 759 767 7749 7829 79 °8 79 547 73- 7987 8067 8146 8225 8305 8384 8463 8543 8622 8701 79 548 73- 8781 886 8939 9018 9097 9 I 77 9256 9335 9414 9493 79 549 73- 9572 965* 973i 981 9889 9968 — 79 549 74- — — — — — — 0047 0126 0205 0284 79 550 74- 0363 0442 0521 06 0678 0757 0836 0915 0994 1073 79 55 i 74- 1152 123 1309 1388 1467 1546 1624 1703 1782 186 79 552 74- 1939 2018 2096 2175 2254 2332 2411 2489 2568 2647 79 553 74- 2725 2804 2882 2961 3039 3118 3196 3275 3353 343 1 78 554 74" 35i 3588 3667 3745 3823 3902 398 4058 4136 4215 78 555 74- 4293 437i 4449 4528 4606 4684 4762 484 4919 4997 78 556 74- 5075 5153 5231 5309 5387 5465 5543 5621 5699 5777 78 557 74- 5855 5933 6011 6089 6167 6245 6323 6401 6479 6556 78 ! 558 ! 74- 6634 6712 679 6868 6945 7023 7101 7179 7256 7334 78 559 ; 74- 7412 7489 7567 7645 7722 78 7878 7955 8033 811 78 No. j 0 1 2 3 4 5 6 7 8 9 D 322 LOGARITHMS OF NUMBERS. No. | 0 1 2 3 4 5 6 7 8 9 j D 5 C 0 74- 8188 8266 8343 8421 8498 8576 8653 8731 8808 8885 77 561 74 - 8963 904 9118 9 X 95 9272 935 9427 9504 9582 9659 77 562 74 - 97 36 9814 9891 9968 — — — — — — 77 562 75 - — — — — 0045 0123 02 0277 0354 0431 77 563 75* 0508 0586 0663 074 0817 0894 0971 1048 1125 1202 77 564 75- 1279 1356 1433 x 5 i 1587 1664 1741 1818 1895 1972 77 505 75- 2048 2125 2202 2279 2356 2433 2509 2586 2663 274 77 566 75- 2816 2893 297 3047 3 X 23 32 3277 3353 343 3506 77 567 75 - 3583 366 3736 3813 3889 3966 4042 4119 4195 4272 77 568 75 - 4348 4425 4501 4578 4654 473 4807 4883 496 5036 76 569 75- 5112 5189 5265 534 i 54 x 7 5494 557 5646 5722 5799 76 570 75 - 5875 595 i 6027 6103 618 6256 6332 6408 6484 656 76 57 i 75- 6636 6712 6788 6864 694 7016 7092 7168 7244 732 76 572 75 " 7396 7472 7548 7624 77 7775 7851 7927 8003 8079 76 573 75 - 8155 823 8306 8382 8458 8533 8609 8685 8761 8836 76 574 75- 8912 8988 9063 9*39 9214 929 9366 9441 9517 9592 76 575 75- 9668 9743 9819 9894 997 _ __ — — — 76 575 76- — — — — — 0045 0121 0196 0272 0347 75 576 76- 0422 0498 0573 0649 0724 0799 0875 095 1025 IIOI 75 577 76- 1176 1251 1326 1402 1477 1552 1627 1702 1778 1853 75 578 76- 1928 2003 2078 2153 2228 2303 2378 2453 2529 2604 75 579 76- 2679 2754 2829 2904 2978 3053 3128 3203 3278 3353 75 580 76- 3428 3503 3578 3653 3727 3802 3877 3952 4027 4101 75 581 76- 4176 4251 4326 44 4475 455 4624 4699 4774 4848 75 582 76- 4923 4998 5072 5 X 47 5221 5296 537 5445 552 5594 75 583 76- 5669 5743 5818 5892 5966 6041 6115 619 6264 6338 74 584 76- 6413 6487 6562 6636 671 6785 6859 6933 7007 7O0 2 74 585 76- 7156 723 7304 7379 7453 7527 7601 7675 7749 7823 74 586 76- 7898 7972 8046 812 8194 8268 8342 8416 849 8564 74 587 76- 8638 8712 8786 886 8934 9008 9082 9156 923 9303 74 588 76- 9377 945 i 9525 9599 9673 9746 982 9894 9968 — 74 588 77 0042 74 589 77 " 01I 5 0189 0263 033 6 041 0484 0557 0631 0705 0778 74 590 77- 0852 0926 0999 1073 1146 122 1293 1367 144 1514 74 59 1 77 - x 5 8 7 1661 1734 1808 1881 1955 2028 2102 2175 2248 , 73 592 77" 2322 2395 2468 2542 2615 2688 2762 2835 2908 2981 ; 73 593 77 - 3055 3128 3201 3274 3348 342 i 3494 3567 364 3713 : 73 594 77 " 3786 386 3933 4006 4079 4152 4225 4298 437 1 4444 ; 73 595 77 - 45 T 7 459 4663 4736 4809 4882 4955 5028 5 1 5173 73 596 77- 5246 53 x 9 5392 5465 5538 561 5683 5756 5829 5902 73 597 77 - 5974 6047 612 6193 6265 6338 6411 6483 6556 6629 73 598 77- 6701 6774 6846 69 t 9 6992 7064 7137 7209 7282 7354 73 599 77 - 7427 7499 7572 7644 7717 7789 7862 7934 8000 8079 72 000 77 - 8151 8224 8296 8368 8441 8513 8585 8658 873 8802 72 601 77- 8874 8947 9019 9091 9 i6 3 9236 9308 938 9452 9524 72 602 77 - 9596 9669 9741 9813 9885 9957 — — — — • ! 72 602 78- — — — — — — 0029 0101 oi 73 0245 | 72 603 78- 0317 0389 0461 0533 0605 0677 0749 1 0821 0893 0965 72 604 78- 1037 1109 1 1181 1253 1324 1396 1468 154 1012 1084 72 No. 0 1 2 3 4 5 6 7 8 9 D LOGARITHMS OP NUMBERS. 323 No. 1 2 3 4 5 6 7 8 9 D 605 78- 1755 1827 1899 1971 2042 2114 2186 2258 2329 2401 72 606 78- 2473 2544 2616 2688 2759 2831 2902 2974 3046 3117 72 607 78- 3189 326 3332 3403 3475 3546 3618 3689 3761 3832 7i 608 78- 3904 3975 4046 4118 4189 4261 4332 4403 4475 4546 7i 609 78- 4617 4689 476 4831 4902 4974 5045 5116 5187 5259 7i 610 78- S33 5401 5472 5543 5615 5686 5757 5828 5899 597 7i 611 78- 6041 6112 6183 6254 6325 6396 6467 6538 6609 668 7i 612 78- 6751 6822 6893 6964 7035 7106 7177 7248 7319 739 7i 613 78- 746 753i 7602 7673 7744 7815 7885 7956 8027 8098 7i 614 78- 8l68 8239 831 8381 8451 8522 8593 8663 8734 8804 7i 615 78- 8875 8946 9016 9087 9 T 57 9228 9299 9369 944 95i 7i 616 78- 9581 9 6 5i 9722 9792 9863 9933 — — — — 70 616 79" 0004 0074 0144 0215 70 617 79- 0285 0356 0426 0496 0567 0637 0707 0778 0848 0918 70 618 79- 0988 1059 1129 1199 1269 134 141 148 155 162 70 619 79- 1691 1761 1831 1901 1971 2041 2111 2181 2252 2322 70 620 79- 2392 2462 2532 2602 2672 2742 2812 2882 2952 3022 70 621 79- 3092 3162 3231 3301 337i 344i 35ii 358 i 3651 3721 70 622 79~ 379 386 393 4 407 4139 4209 4279 4349 4418 70 623 79" 4488 4558 4627 4697 4767 4836 4906 4976 5045 5ii5 70 624 79- 5185 5254 5324 5393 5463 5532 5602 5672 574i 5811 70 625 79- 588 5949 6019 6088 6158 6227 6297 6366 6436 6505 69 626 79- 6574 6644 6713 6782 6852 6921 699 706 7129 jigS 69 627 79- 7268 7337 7406. 7475 7545 7614 7683 7752 7821 789 69 628 79- 70 8029 8098 8167 8236 8305 8374 8443 8513 8582 69 629 79" 8651 872 8789 8858 8927 8996 9065 9134 9203 9272 69 630 79- 934i 9409 9478 9547 9616 9685 9754 9823 9892 9961 69 631 80- 0029 0098 0167 0236 0305 0373 0442 0511 058 0648 69 632 80- 0717 0786 0854 0923 0992 1061 1129 1198 1266 1 335 69 633 80- 1404 1472 i54i 1609 1678 1747 1815 1884 1952 2021 69 634 80- 2089 2158 2226 2295 2363 2432 25 2568 2637 2705 69 635 80- 2774 2842 291 2979 3047 3116 3184 3252 33 21 3389 68 636 80- 3457 3525 3594 3662 373 3798 3867 3935 4003 4071 68 637 80- 4139 4208 4276 4344 4412 448 4548 4616 4685 4753 68 638 80- 4821 4889 4957 5025 5093 5161 5229 5297 5365 5433 68 639 80- 5501 5569 5637 5705 5773 5841 5908 5976 6044 6112 68 640 80- 618 6248 6316 6384 6451 6519 6587 6655 6723 679 68 641 80- 6858 6926 6994 7061 7129 7197 7264 7332 74 7467 68 642 80- 7535 7603 767 7738 7806 7873 7941 8008 8076 8i43 68 643 80- 8211 8279 8346 8414 8481 8549 8616 8684 8751 8818 67 644 80- 8886 8953 9021 9088 9156 9223 929 9358 9425 9492 67 645 80- 956 9627 9694 9762 9829 9896 9964 — — — 67 645 81- — 0031 0098 0165 67 646 81- 0233 g >3 0367 0434 0501 0569 0636 0703 077 0837 67 647 81- 0904 0971 1039 1106 11 73' 124 1307 1374 1441 1508 67 648 81- 1575 1642 1709 1776 1843 191 1977 2044 2111 2178 67 649 81- 2245 2312 2379 2445 2512 2579 2646 2713 278 2847 67 650 81- 2913 298 3047 3114 3181 3247 3314 338 i 3448 3514 67 651 81- 3581 3648 3714 3781 3848 39i4 398 i 4048 4114 4181 67 652 81- 4248 4314 4381 4447 4514 4581 4647 4714 478 4847 67 653 81- 4913 498 5046 5ii3 5i79 5246 5312 5378 5445 55ii 66 654 81- 5578 5644 57ii 5777 5843 59i 5976 6042 6109 6i75 66 No. 0 1 2 3 4 5 6 7 8 9 "IT 324 LOGARITHMS OF NUMBERS. No. | 0 12. 3 4 655 ! 8i- 6241 6308 6374 644 6506 656 ! 81- 6904 697 7036 7102 7169 657 I 81- 7565 7631 7698 7764 783 658 81- 8226 8292 8358 8424 849 659 81- 8885 8951 9017 9083 9149 660 81- 9544 961 9676 9741 9807 660 82- — — — — — 661 82- 0201 0267 0333 0399 0464 662 82- 0858 0924 0989 1055 1 12 663 82- 1514 1579 1645 171 1775 664 82- 2168 2233 2299 2364 243 665 82- 2822 2887 2952 3018 3083 666 82- 3474 3539 3605 367 3735 667 82- 4126 4191 4256 4321 4386 668 82- 4776 4841 4906 4971 5036 669 82- 5426 5491 5556 5621 5686 670 82- 6075 614 6204 6269 6334 671 82- 6723 6787 6852 6917 6981 672 82- 7369 7434 7499 7563 7 62 8 673 82- 8015 808 8144 8209 8273 674 82- 866 8724 8789 8853 8918 67 5 82- 9304 9368 9432 9497 9561 676 82- 9947 — — — — 676 83- — oori. 0075 0139 0204 677 83- 0589 0653 0717 0781 0845 678 83- 123 1294 1358 1422 i486 679 83- 187 1934 1998 2062 2126 680 83- 2509 2573 2637 27 2764 681 83- 3147 3211 3275 3338 3402 682 83- 3784 3848 3912 3975 4039 683 83- 4421 4484 4548 4611 4675 684 83- 5056 512 5183 5247 531 685 83- 5691 5754 5817 5881 5944 686 83- 6324 6387 6451 6514 6577 687 83- 6957 702 7083 7146 721 688 83- 7588 7652 7715 7778 7841 689 83- 8219 8282 8345 8408 8471 690 83- 8849 8912 8975 9038 9101 691 83- 9478 9541 9604 9667 9729 691 84- — — — — — 692 84 0106 0169 0232 0294 0357 693 84- 0733 0796 0859 0921 0984 694 84- 1359 1422 1485 1547 161 695 84- 1985 2047 21 1 2172 2235 696 84- 2609 2672 2734 2796 2859 697 84- 3233 3295 3357 342 3482 69S 84- 3855 3918 398 4042 4104 699 84- 4477 4539 460 1 4664 47 2 6 700 84- 5098 516 5222 5284 5346 701 84- 5718 578 5842 5904 5966 702 84- 6337 6399 6461 6523 6585 703 84- 6955 7017 7079 7141 7202 704 84- 7573 7634 7696 7758 7819 No. 5 6 7 8 9 6573 6639 6 705 6771 6838 7235 7301 7367 7433 7499 7896 7962 8028 8094 816 8556 8622 8688 8754 882 9215 9281 9346 9412 9478 9873 9939 — — — 0004 007 0136 053 °595 0661 0727 0792 1186 1251 1317 1382 1448 1841 1906 1972 2037 2103 2495 256 2626 2691 2756 3148 3213 3279 3344 3409 38 3865 393 3996 4061 4451 4516 4581 4646 4711 5101 5166 5231 5296 5361 575i 5815 588 5945 601 6399 6464 6528 6593 6658 7046 71 1 1 7175 724 7305 7692 7757 7821 7886 7951 8338 8402 8467 8531 8595 8982 9046 91 1 1 9175 9239 9625 969 9754 9818 9882 0268 0332 0396 046 0525 0909 0973 1037 1102 1166 155 1614 1678 1742 1806 2189 2253 2317 2381 2445 2828 2892 2956 302 3083 3466 353 3593 3^57 37 2 i 4103 4166 423 4294 4357 4739 4802 4866 4929 4993 5373 5437 55 55^4 5627 6007 6071 6134 6197 6261 6641 6704 6767 683 6894 7273 7336 7399 7462 7525 7904 7967 803 8093 8156 8534 8597 866 8723 8786 66 66 66 66 66 66 66 66 66 65 65 65 65 65 65 65 65 65 65 64 64 64 64 64 64 64 64 64 64 64 64 I 63 6 3 : 63 63 ; 63 9164 9227 9289 9352 9415 ; 63 9792 9855 9918 9981 — 63 — — — — 0043 63 042 0482 0545 0608 0671 ; 63 1046 II09 1172 1234 1297 63 1672 1735 1797 186 1922 | 63 2297 236 2422 2484 2547 62 2921 2983 *3046 3 108 3 1 7 62 3544 3 606 3669 373i 3793 62 4166 4229 4291 4353 44*5 62 4788 485 4912 4974 503 6 62 5408 547 553 2 5594 5656 62 6028 609 6151 6213 6275 62 6646 6708 677 6832 6894 62 7264 7326 7388 7449 75ii 62 7881 7943 8004 8066 8127 ] 62^ | c 6 7 8 9 I D LOGARITHMS OF NUMBERS. 325 No. 0 \ 2 3 4 5 6 7 8 9 D 705 84- 8189 8251 8312 8374 8435 8497 8559 862 8682 8743 62 706 84- 8805 8866 8928 8989 9051 9112 9174 9235 9297 9358 61 707 84- 9419 9481 9542 9604 9665 9726 9788 9849 9911 9972 61 708 85- 0033 0095 0156 0217 0279 034 0401 0462 0524 0585 61 709 85- 0646 0707 0769 083 0891 0952 1014 IO75 1136 1197 61 710 85- 1258 132 1381 1442 1503 1564 1625 1686 1747 1809 61 7 11 85- 187 i93i 1992 2053 2114 2175 2236 2297 2358 2419 61 712 85- 248 2541 2602 2663 2724 2785 2846 2907 2968 3029 61 7i3 85- 309 3i5 3211 3272 3333 3394 3455 3516 3577 3637 61 714 85- 3698 3759 382 3881 394i 4002 4063 4124 4185 4245 61 715 85- 4306 4367 4428 4488 4549 461 467 4731 4792 4852 61 716 85- 49i3 4974 5034 5095 5156 5216 5277 5337 5398 5459 61 7*7 85- 55i9 558 564 5701 5761 5822 5882 5943 6003 6064 61 718 85- 6124 6185 6245 6306 6366 6427 6487 6548 6608 6668 60 719 85- 6729 6789 685 691 697 7031 7091 7152 7212 to to 60 720 85- 7332 7393 7453 7513 7574 7634 7694 7755 7815 7875 60 721 85- 7935 7995 8056 8116 8176 8236 8297 8357 8417 8477 60 722 85- 8537 8597 8657 8718 8778 8838 8898 8958 9018 9078 60 7 2 3 85- 9138 9*98 9258 9318 9379 9439 9499 9559 9 6i 9 9679 60 724 85- 9739 9799 9859 9918 9978 — — — — — 60 724 86- — — — — — 0038 0098 0158 0218 0278 60 725 86- 0338 0398 0458 0518 0578 0637 0697 0757 0817 0877 60 726 86- 0937 0996 1056 1116 1176 1236 1295 1355 1415 1475 60 727 86- 1534 1594 1654 1714 1773 1833 1893 1952 2012 2072 60 728 86- 2131 2191 2251 231 237 243 2489 2549 2608 2668 60 729 86- 2728 2787 2847 2906 2966 3025 3085 3i44 3204 3263 60 730 86- 3323 3382 3442 3501 356 i 362 368 3739 3799 3858 59 73i 86- 39i7 3977 4036 4096 4155 4214 4274 4333 4392 4452 59 732 86- 45ii 457 463 4689 4748 4808 4867 4926 4985 5045 59 733 86- 5104 5163 5222 5282 534i 54 5459 5519 5578 5637 59 734 86- 5696 5755 5814 5874 5933 5992 6051 61 1 6169 6228 59 735 86- 6287 6346 6405 6465 6524 6583 6642 6701 676 6819 59 736 86- 6878 6937 6996 7055 7114 7173 7232 7291 735 7409 59 737 86- 7467 7526 7585 7644 7703 7762 7821 788 7939 7998 59 738 86- 8056 8115 8174 8233 8292 835 8409 8468 8527 8586 59 739 86- 8644 8703 8762 8821 8879 8938 8997 9056 9114 9173 59 740 86- 9232 929 9349 9408 9466 9525 9584 9642 9701 976 59 741 86- 9818 9877 9935 9994 — — — — — — 59 74i 87- — — — — 0053 OIII 017 0228 0287 0345 59 742 87- 0404 0462 0521 0579 0638 0696 0755 0813 0872 093 58 743 87- 0989 1047 1106 1164 1223 1281 1339 1398 1456 1515 58 744 87- 1573 1631 169 1748 1806 1865 1923 1981 204 2098 58 745 87- 2156 2215 2273 2 33i 2389 2448 2506 2564 2622 2681 58 746 87- 2739 2797 2855 2913 2972 303 3088 3146 3204 3262 58 747 87- 332i 3379 3437 3495 3553 3611 3669 3727 3785 3844 58 748 87- 39° 2 396 4018 4076 4i34 4192 425 4308 4366 4424 58 749 87- 4482 454 4598 4656 47i4 4772 483 4888 4945 5003 58 750 87- 5061 5ii9 5177 5235 5293 535i 5409 5466 5524 5582 58 75i 87- 564 5698 5756 5813 5871 5929 5987 6045 6102 616 58 752 j 87- 6218 6276 6333- 6391 6449 6507 6564 6622 668 6737 58 753 ; 87- 6795 6853 691 6968 7026 7083 7141 7199 7256 7314 58 754 j 87- 737i 7429 00 *■ 7544 7602 7659 7717 7774 7832 7889 58 No. | 0 1 2 3 4 5 6 7 8 9 D ' E E No. 755 756 757 758 758 759 760 761 762 763 764 765 766 767 768 765 77 C 771 77 " 77 : 77, 771 77 < 771 77 77 77 78 78 78 78 '78 logarithms of numbers. 7947 8004 8062 8119 8177 8522 8579 8637 8694 8752 9096 9153 92 11 9268 9325 9669 9726 9784 9841 9898 i- 1- v )■ 0242 0299 0356 0413 0471 0814 0871 0928 0985 1042 1385 1442 1499 1556 1613 1955 2012 2069 2120 2183 2525 2581 2638 2695 2752 3093 3*5 3 2 °7 3 26 4 33 21 3661 3718 3775 383 2 3888 4229 4285 4342 4399 4455 4795 4852 49°9 49^5 5 022 53 6x 5418 5474 553 1 55§7 5926 5983 6039 6096 6152 6491 6547 6604 666 6716 • 7054 71 1 1 7 i6 7 7223 728 ■7617 7674 773 7786 7842 - 8179 8236 8292 8348 8404 - 8741 8797 8853 8909 8965 - 9302 9358 9414 947 9526 9862 9918 9974 — — — — 003 0086 0421 0477 0533 0589 0645 O98 1035 1091 1147 ^03 1537 J 593 i6 49 i 7°5 i 7° 2095 215 2206 2262 2317 2651 2707 2762 2818 2873 3207 3262 3318 3373 3429 3762 3817 3873 3928 3984 4316 4371 4427 4482 4538 487 4925 498 5°3 6 5091 5423 5478 5533 5588 5 6 44 • 5975 603 6085 614 6195 - 6526 6581 6636 6692 6747 • 7°77 J 7 I 3 2 7 i 87 7 2 4 2 7 2 97 - 7627 7682 7737 7792 7847 - 8176 8231 8286 8341 8396 - 8725 878 8835 889 8944 - 9273 9328 9383 9437 9492 - 9821 9875 993 9985 — — 0039 - 0367 0422 0476 0531 0586 - 0913 0968 1022 1077 II 3 I - 1458 I5i3 1567 1622 1676 2003 2057 2112 2166 2221 >- 2547 2601 2655 271 2764 >- 309 3144 3 X 99 3253 33°7 )- 3633 3687 374i 3795 3849 >- 4174 4229 4283 4337 439 1 >- 4716 477 4824 4878 4932 >- 5256 531 5364 5 4 l8 5472 8234 8292 8349 8407 8464 8809 8866 8924 8981 9039 9383 944 9497 9555 9 612 9956 — — — — 0013 007 0127 0185 0528 0585 0642 0699 0756 1099 1156 1213 1271 1328 167 1727 1784 i 84 i ^98 224 2297 2354 2411 2468 2809 2866 2923 298 3037 3377 3434 349 1 354 8 36°5 3945 4002 4059 4115 4 X 7 2 4512 4569 4625 4682 4739 5078 513s 5192 5248 5305 5644 57 5757 5813 587 6209 6265 6321 6378 6434 6773 6829 6885 6942 6998 7336 7392 7449 75°5 75^1 7898 7955 8011 8067 8123 846 8516 8573 8629 8685 9021 9077 9134 9*9 9246 9582 9638 9694 975 9806 0141 0197 0253 0309 0365 07 0756 0812 0868 0924 1259 1314 137 T 4s6 1482 1816 1872 1928 1983 2039 2373 2429 2484 254 2595 2929 2985 304 309 6 3 1 5 ^ 3484 354 3595 3 ^ 5 1 37°6 4039 4094 415 4 2 °5 4261 4593 4 6 48 47°4 4759 4814 5146 5201 5257 5312 5367 5699 5754 5809 5864 592 6251 6306 6361 6416 6471 6802 6857 6912 6967 7022 7352 7407 7462 75 1 7 7572 7902 7957 8012 8067 8122 8451 8506 8561 8615 867 8999 9°54 9 io 9 9 i6 4 9218 9547 9 602 9656 97 11 9766 57 57 57 57 57 57 57 57 57 57 57 57 57 5 : 5 ' 5 * 5 < 5 < 5 ' 5 1 5 5 5 5 5 q c : c : l OO94 0149 0203 064 0695 0749 Il86 124 1295 I73I I785 184 2275 2329 2384 28l8 2873 2927 3361 3416 347 3904 3958 4 ° 12 4445 4499 4553 4986 504 5094 5526 558 5634 0258 0312 0804 0859 1349 1404 1894 1948 2438 2492 2981 3036 3524 3578 4066 412 4607 4661 5148 5202 5688 5742 5 6 7 LOGARITHMS OF NUMBERS. 327 No f 0 1 2 3 4|5 6 7 8 9 D 80s 806 807 808 809 810 81 1 812 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 851 852 < 853 ' 854 ' 90-5796 585 5904 5958 601: 90- 6335 6389 6443 6497 6551 90- 6874 6927 6981 7035 7085 90- 7411 7465 7519 7573 762 6 90- 7949 8002 8056 81 1 8163 90- 8485 8539 8592 8646 8699 90- 9021 9074 9128 9181 9235 90- 9556 9609 9663 9716 977 91- 0091 0144 0197 0251 0304 91- 0624 0678 0731 0784 0838 91- 1158 1211 1264 1317 1371 91- 169 1743 1797 185 1903 91- 2222 2275 2328 2381 2435 9 1 * 2 753 2806 2859 2913 2966 91- 3284 3337 339 3443 349 6 91- 3814 3867 392 3973 4026 9i- 4343 4396 4449 45°2 4555 9i- 4872 4925 4977 503 5083 9 r ~ 54 5453 55o 5 5558 5611 9i- 5927 598 6033 6085 6138 9 J - 6454 6507 6559 6612 6664 91- 698 7033 7085 7138 719 gi- 7506 7558 7611 7663 7716 91- 803 8083 8135 8188 824 9i- 8555 8607 8659 8712 8764 91- 9078 913 9183 9235 9287 91- 9601 9653 9706 9758 981 92- 0123 0176 0228 028 0332 92- 0645 0697 0749 °8oi 0853 92- 1166 1218 127 1322 1374 92- 1686 1738 179 1842 1894 92- 2206 2258 231 2362 2414 92- 2725 2777 2829 2881 2933 92- 3244 3296 3348 3399 345i 92- 3762 3814 3865 3917 3969 92- 4279 433i 4383 4434 4486 92- 4796 4848 4899 4951 5003 92- 5312 5364 5415 5467 5518 92- 5828 5879 5931 5982 6034 92- 6342 6394 6445 6497 6548 92- 6857 6908 6959 7011 7062 92- 737 7422 7473 7524 7576 92- 7883 7935 7986 8037 8088 92- 8396 8447 8498 8549 8601 92- 8908 8959 901 9061 9112 92- 9419 947 9521 9572 9623 92- 993 998 i — — — 93- — — 0032 0083 0134 93- 044 0491 0542 0592 0643 , 93“ 0949 1 1051 1102 1153 93" 1458 1509 156 161 1661 j 6066 6119 6173 6227 6281 6604 6658 6712 6766 682 > 7143 7196 725 7304 7358 ' 7o8 7734 7787 7841 7895 ; 8217 827 8324 8378 8431 1 8753 8807 886 8914 8967 9289 9342 9396 9449 9503 9823 9877 993 9984 — — — — — 0037 0358 0411 0464 0518 0571 0891 0944 0998 1051 1104 1424 1477 153 1584 1637 1956 2009 2063 2116 2169 2488 2541 2594 2647 27 3019 3072 3125 3178 3231 3549 3602 3655 3708 3761 4079 4132 4184 4237 429 4608 466 4713 4766 4819 5136 5189 5241 5294 5347 5664 5716 5769 5822 5875 6191 6243 6296 6349 6401 6717 677 6822 6875 6927 72 43 7295 7348 74 7453 7768 782 7873 7925 7978 8293 8345 8397 845 8502 8816 8869 8921 8973 9026 934 9392 9444 9496 9549 9862 9914 9967 — — — — 0019 0071 0384 0436 0489 0541 0593 0906 0958 101 1062 1 1 14 1426 1478 153 1582 1634 i 946 1998 205 2102 2154 2466 2518 257 2622 2674 2985 3037 3089 314 3192 3503 3555 3607 3658 371 4021 4072 4124 4176 4228 4538 4589 4641 4693 4744 5°54 5 i o6 5157 5209 5261 557 5621 5673 5725 5776 6085 6137 6188 624 6291 66 6651 6702 6754 6805 7114 7165 7216 7268 7319 7627 7678 773 7781 7832 814 8191 8242 8293 8345 8652 8703 8754 8805 8857 9163 9215 9266 9317 9368 9674 9725 9776 9827 9879 0185 0236 0287 0338 0389 0694 0745 0796 0847 0898 1203 1254 1305 1356 1407 1712 1763 1814 1865 1915 : 54 54 1 54 54 54 54 54 54 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 52 52 5 2 "52 52 52 52 52 52 52 52 52 52 52 52 52 52 5r 5i 5i 5i 5i 5 1 5i 5i 5i 5i 5i 5i 5i No. 0 ‘ 234 56789 D No. 855 856 857 858 859 860 861 862 863 864 865 866 867 868 865 87C 871 logarithms of numbers. V \ 1966 2017 2068 2474 2524 2575 2981 3031 3082 3487 3538 3589 3993 4044 4094 4498 4549 4599 5003 5054 5io4 5507 5558 5 6 °8 6011 6061 61 1 1 - 6514 6564 6614 - 7016 7066 7117 - 7518 7568 7618 - 8019 8069 8119 - 852 857 862 - 902 907 912 - 9519 9569 9^9 - 0018 0068 0118 - 0516 0566 0616 - 1014 1064 1 1 14 - 1511 i5 61 1611 - 2008 2058 2107 2504 2554 2603 r - 3 3049 3°99 ^ 3495 3544 3593 (.- 3989 4038 4080 [- 4483 4532 458 i [- 4976 5025 5074 t- 5469 55i8 5567 5961 601 6059 6452 6501 6551 4.- 6943 6992 7041 4“ 7434 7483 7532 4- 7924 7973 8022 4- 8413 8462 8511 4- 8902 8951 8999 4- 939 9439 94 88 ,4- 9878 9926 9975 ►5- — — ~ >5" 0365 0414 0402 >5- 0851 09 0949 )5- 1338 1386 1435 >5- 1823 1872 192 )5- 2308 2356 2405 )5 _ 2792 2841 2889 ) 5 ~ 3276 3325 3373 }5- 376 3808 3856 }5- 4243 4291 4339 15 - 4725 4773 4821 ? 5 ~ 5207 5255 5303 ?5 - 5688 5736 5784 95- 6168 6216 6265 8 2118 2169 2626 2677 3 T 33 3 i8 3 3639 369 4145 4195 465 47 5154 5205 5658 5709 6162 6212 6665 6715 7167 7217 7668 7718 8169 8219 867 872 917 922 9669 9719 0168 0218 0666 0716 1163 1213 166 171 2157 2207 2653 2702 3148 3198 3643 3692 4137 4186 4631 468 5124 5173 5616 5665 6108 6157 66 6649 709 714 7581 763 807 8119 856 8609 9048 9097 9536 9585 222 2271 2727 2778 3234 3285 374 3791 4246 4296 475 1 4801 5255 530 6 5759 5809 6262 6313 6765 6815 7267 7317 7769 7819 8269 8319 877 882 927 932 9769 9819 0267 0317 0765 0815 1263 1313 176 1809 2256 2306 2752 2801 3247 3297 3742 3791 4236 4285 4729 4779 5222 5272 5715 5764 6207 6256 6698 6747 7189 7238 7679 7728 8168 8217 8657 8706 9146 9195 9634 9683 2322 2372 2829 2879 3335 3386 3841 3892 4347 4397 4852 49° 2 5356 5406 586 59 1 6363 6413 6865 6916 7367 7418 7869 7919 837 842 887 892 9369 9419 9869 9918 0367 0417 0865 0915 1362 1412 1859 1909 2355 2405 2851 2901 3346 3396 3841 3 8 9 4335 4384 4828 4877 532i 537 5813 5862 6305 6354 6796 6845 7287 7336 7777 7826 8266 8315 8755 8804 9244 9292 9731 978 2423 5i 293 5i 3437 3943 4448 4953 5 C 5457 5< 596 0024 0073 0511 056 0997 1046 i4 8 3 J 53 2 1969 2017 2453 2502 2938 2986 342i 347 39°5 3953 4387 4435 4869 4918 535 1 5399 5832 588 6313 636 1 0121 017 0608 0657 1095 1143 158 1629 2066 2114 255 2599 3034 3°83 3518 3566 4001 4049 4484 4532 4966 5014 5447 5495 5928 5976 6409 6457 6463 6966 7468 7969 847 897 9469 9968 0467 0964 1462 1958 2455 295 3445 3939 4433 4927 5419 1 ?9 12 I 6403 I 6894 7385 7875 8364 8853 934i 9829 0219 0267 0316 0706 0754 0803 1192 124 1289 1677 1726 1775 2163 2211 226 2647 2696 2744 3131 3i8 3228 3615 3^3 37i 1 4098 4146 4194 458 4628 4677 5062 511 5 I 5 8 5543 5592 564 6024 6072 612 6505 6553 6601 5< 5< 5 ( 5 ( 5' 5 5 5 5 5 c e c { : 2 . LOGARITHMS OF NUMBERS. 329 No. | 0 1 2 3 4 5 6 7 8 9 D 905 95 - 6649 6697 6745 6793 684 6888 6936 6984 7032 708 48 qo6 95 - 7128 7176 7224 7272 732 7368 741.6 7464 7512 7559 48 907 95 - 7607 7655 7703 775 T 7799 7847 7894 7942 799 8038 48 qo8 95 ~ 8086 8134 8l8l 8229 8277 8325 8373 8421 8468 8516 48 909 95 - 8564 8612 8659 8707 8755 8803 885 8898 8946 8994 48 910 95 - 9 ° 4 I 9089 9*37 9 i8 5 9232 928 9328 9375 9423 947 i 48 9 11 95 - 95 i 8 9566 9614 9661 9709 9757 9804 9852 99 9947 48 912 912 95 - 96- 9995 0042 009 0138 0185 0233 028 0328 0376 0423 48 48 913 96- 0471 0518 0566 0613 0661 0709 0756 0804 0851 0899 48 914 96- 0946 0994 1041 1089 11 36 1184 1231 1279 1326 1374 47 915 96- 1421 1469 1516 1563 1611 1658 1706 1753 1801 1848 47 916 96- 1895 1943 199 2038 2085 2132 218 2227 2275 2322 47 9 X 7 96- 2369 2417 2464 2511 2559 2606 2653 2701 2748 2795 47 918 96- 2843 289 2937 2985 3032 3079 3126 3 J 74 3221 3268 47 9 T 9 96- 33 l6 33 6 3 34 i 3457 3504 3552 3599 3646 3693 374 i 47 920 96- 3788 3835 3882 3929 3977 4024 4071 4118 4165 4212 47 921 96- 426 4307 4354 4401 4448 4495 4542 459 4637 4684 47 922 96- 473 i 4778 4825 4872 4919 4966 5013 5061 5108 5155 47 9 2 3 96- 5202 5249 5296 5343 539 5437 5484 553 i 5578 5625 47 924 96- 5672 57 i 9 5766 5813 586 5907 5954 6001 6048 6095 47 925 96- 6142 6189 6236 6283 6329 6376 6423 647 6517 6564 47 926 96- 6611 6658 6705 6752 6799 6845 6892 6939 6986 7033 47 927 96- 708 7127 7 l 13 722 7267 7314 7361 7408 7454 75 oi 47 928 96- 7548 7595 7642 7688 7735 7782 7829 7875 7922 7969 47 929 96- 8016 8062 8109 8156 8203 8249 8296 8343 839 8436 47 930 96- 8483 853 8576 8623 867 8716 8763 881 8856 8903 47 93 1 96- 895 8996 9°43 909 9^6 9183 9229 9276 9323 9369 47 932 96- 9416 9463 9509 9556 9602 9649 9695 9742 9789 9835 47 933 96- 9882 9928 9975 0068 0161 47 933 91 - — — — 0021 0114 0207 0254 03 47 934 97 - 0347 0393 044 0486 0533 0579 0626 0672 0719 0765 46 935 97 - 0812 0858 0904 0951 0997 1044 109 ii 37 1183 1229 46 93 6 97 - 1276 1322 1369 1415 1461 1508 1554 1601 1647 1693 46 937 97 - i 74 1786 1832 1879 1925 I 97 i 2018 2064 211 2157 46 938 97 - 2203 2249 2295 2342 2388 2434 2481 2527 2573 2619 46 939 97 - 2666 2712 2758 2804 2851 2897 2943 2989 3035 3082 46 940 97 - 3128 3174 322 3266 3313 3359 3405 345 i 3497 3543 46 941 97 - 359 3636 3682 3728 3774 382 3866 39 i 3 3959 4005 46 942 97 - 4051 4097 4143 4189 4235 4281 4327 4374 442 4466 46 943 97 - 4512 4558 4604 465 4696 4742 4788 4834 488 4926 46 944 97 “ 4972 5018 5064 511 5156 5202 5248 5294 534 5386 46 945 97 - 5432 5478 5524 557 5616 5662 5707 5753 5799 5845 46 946 97 - 5891 5937 5983 6029 6075 6121 6167 6212 6258 6304 46 947 97 - 635 6396 6442 6488 6533 6579 6625 6671 6717 6763 46 948 91 - 6808 6854 69 6946 6992 7037 7083 7129 7175 722 46 949 91 - 7266 7312 7358 7403 7449 7495 7541 7586 7632 7678 46 950 97 " 7724 7769 7815 7861 7906 7952 7998 8043 8089 8 i 35 46 95 i 97 - 8181 8226 8272 8317 8363 8409 8454 85 8546 8591 46 952 97 - 8637 8683 8728 8774 8819 8865 8911 8956 9002 9047 46 953 97 - 9093 9138 9 i8 4 923 9275 9321 9366 9412 9457 9503 46 954 97 - 9548 9594 9639 9685 973 9776 9821 9867 9912 9958 46 .No. 0 1 2 3 4 5 6 7 8 9 D Ee* 330 LOGARITHMS OF NUMBERS. No. 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 977 978 979 980 981 982 9 8 3 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 01234 98- 0003 0049 0094 014 0185 98- 0458 0503 0549 0594 064 98- 0912 0957 1003 1048 1093 98- 1366 1411 1456 1501 I 547 98- 1819 1864 1909 1954 2 98- 2271 2316 2362 2407 2452 98- 2723 2769 2814 2859 2904 98- 3 J 75 3 22 3 2 65 33i 3356 98- 3626 3671 3716 3762 3807 98- 4077 4122 4167 4212 4257 98- 4527 4572 4617 4662 4707 98- 4977 50 22 5067 5 112 5*57 98- 5426 5471 5516 5561 5606 98- 5875 592 5965 601 6055 98- 6324 6369 6413 6458 6503 98- 6772 6817 6861 6906 6951 98- 7219 7264 7309 7353 7398 98- 7666 7711 7756 78 7845 98- 8113 8157 8202 8247 8291 98- 8559 8604 8648 8693 8737 98- 9005 9049 9094 9138 9183 98- 945 9494 9539 95 83 9 62 8 98- 9895 9939 9983 “ — 99- — — — 0028 0072 99- °33 9 0 383 0 428 0472 0516 99- 0783 0827 0871 0916 096 99- 1226 127 1315 1359 I 4°3 99- 1669 1713 1758 1802 1846 99- 21 11 2156 22 2244 2288 99- 2554 2598 2642 2686 273 99- 2995 3039 3083 3127 3172 99- 3436 348 3524 3568 3613 99" 3877 392i 3965 4009 4053 99- 43 1 7 43 61 4405 4449 4493 99- 4757 4801 4845 4889 4933 99- 5196 524 5284 5328 5372 99- 5635 5679 5723 5767 5811 99- 6074 6117 6161 6205 6249 99- 6512 6555 6599 6643 6687 99- 6949 6993 7037 708 7124 99- 7386 743 7474 75*7 75^i 99- 7823 7867 791 7954 7998 99- 8259 8303 8347 839 8434 99- 8695 8739 8782 8826 8869 99- 9131 9174 9218 9261 9305 99- 9565 9^ 9652 9696 9739 5 6 7 8 9 D 0231 0276 0322 0367 0412 0685 073 0776 0821 0867 1139 1184 1229 1275 132 1592 1637 1683 1728 1773 2045 209 2135 2181 2226 45 45 45 45 45 2497 2949 3401 3852 4302 2543 2994 3446 3897 4347 2588 304 349 1 3942 4392 2633 3085 3536 3987 4437 2678 3*3 3581 4032 4482 45 45 45 45 45 4752 4797 4842 4887 4932 5202 5247 5292 5337 5382 5651 5696 5741 5786 583 61 6144 6189 6234 6279 6548 6593 6637 6682 6727 45 45 45 45 45 6996 704 7085 713 7 j 75 7443 7488 7532 7577 7622 789 7934 7979 8024 8068 8336 8381 8425 847 8514 8782 8826 8871 8916 896 9227 9272 9316 9361 9405 9672 9717 9761 9806 985 0117 0161 0206 025 0294 0561 0605 065 0694 0738 1004 1049 1093 1137 1182 1448 1492 1536 158 1625 189 1935 *979 2023 2067 2333 2377 2421 2465 2509 2774 2819 2863 2907 2951 3216 326 3304 3348 3392 45 45 45 45 45 45 3657 370 1 3745 3789 3833 4097 4141 4 i8 5 4229 4273 4537 458 i 4625 4669 4713 4977 5021 5065 5108 5152 5416 546 5504 5547 559 1 5854 5898 5942 6293 6337 638 6731 6774 6818 7168 7212 7255 7605 7648 7692 8041 8085 8129 8477 8521 8564 8913 8956 9 9348 9392 9435 9783 9826 987 5986 603 6424 6468 6862 6906 j 7299 7343 ; 7736 7779 i 8172 8216 8608 8652 9043 9087 I 9479 9522 44 99*3 9957 | 43 8 6 7 9 | D *** ***** ***** ***** ***** HYPERBOLIC LOGARITHMS OF NUMBERS, 331 Hyperbolic XjOgaritliixLS of' nSTiam'bers. From 1 .01 to 30. In following table, the numbers range from 1.01 to 30, advancing by .01, up to the whole number 10 ; and thence by larger intervals up to 30. ” The hyperbolic logarithms of numbers, or Neperian logarithms, as they are some- times termed, are computed by multiplying the common logarithms of num- bers by the constant multiplier, 2.302585. The hyperbolic logarithms of numbers intermediate between those which are given in the table may be readily obtained by interpolating proportional differences. No. Log. No. Log. No. Log. No. Log. No. Log. I. OI .OO99 1. 41 •3436 I.8l •5933 2.21 •793 2.6l •9594 1.02 .OI98 I.42 •3507 1.82 .5988 2.22 •7975 2.62 .9632 I.03 .0296 i -43 •3577 I.83 .6043 2.23 .802 2.63 .967 I.04 .0392 1.44 .3646 I.84 .6098 2.24 .8065 2.64 .9708 1.05 .0488 i -45 .3716 1.85 .6152 2.25 .8109 2.65 .9746 1.06 .0583 1.46 •3784 1.86 .6206 2.26 .8154 2.66 •9783 I.07 .0677 1.47 •3853 1.87 .6259 2.27 .8198 2.67 .9821 1.08 .077 1.48 •392 1.88 •6313 2.28 .8242 2.68 .9858 I.09 .0862 i -49 .3988 1.89 .6366 2.29 .8286 2.69 •9895 1. 1 •0953 i -5 •4055 1.9 .6419 2-3 .8329 2.7 •9933 I. II .IO44 I * 5 I .4121 1.91 .6471 2.3I •8372 2.71 .9969 1. 12 •1133 1.52 .4187 1.92 •6523 2.32 .8416 2.72 1.0006 I * I 3 .1222 i -53 •4253 x -9 3 •6575 2-33 .8458 2.73 1.0043 1.14 •13 * i -54 .4318 1.94 .6627 2-34 .8502 2.74 1.008 I * I 5 .1398 i -55 •4383 i -95 .6678 2-35 •8544 2-75 1.0116 1. 16 .1484 1.56 •4447 1.96 .6729 2.36 .8587 2.76 1.0152 1. 17 •157 i -57 • 45 ii 1.97 .678 2-37 .8629 2.77 1. 0188 1. 18 •1655 1.58 •4574 1.98 .6831 2.38 .8671 2.78 1.0225 1. 19 .174 i -59 •4637 I *99 .6881 2.39 •8713 2.79 1.026 1.2 .1823 1.6 •47 2 .6931 2.4 •8755 2.8 1.0296 1. 21 .1906 1. 61 .4762 2.01 .6981 2.4I .8796 2.81 1.0332 1.22 .1988 1.62 .4824 2.02 .7031 2.42 .8838 2.82 1.0367 1.23 .207 1 63 .4886 2.03 .708 2-43 .8879 2.83 1.0403 1.24 .2151 1.64 •4947 2.04 .7129 2-44 .892 2.84 1.0438 1.25 .2231 1.65 .5008 2.05 .7178 2-45 .8961 2.85 1-0473 1.26 .2311 1.66 .5068 2.06 .7227 2.46 .9002 2.86 1.0508 1.27 .239 1.67 .5128 2.07 •7275 2.47 .9042 2.87 1-0543 1.28 .2469 1.68 .5188 2.08 •7324 2.48 .9083 2.88 1.0578 1.29 .2546 1.69 •5247 2.09 •7372 2.49 .9123 2.89 1.0613 i -3 .2624 i -7 •5306 2.1 .7419 2-5 • 9 i6 3 2.9 1.0647 I * 3 I .27 I * 7 I •5365 2. 11 .7467 2.51 .9203 2.91 1 .0682 1.32 .2776 1.72 •5423 2.12 •7514 2.52 •9243 2.92 1.0716 i -33 .2852 i -73 .5481 2.13 • 756 i 2-53 .9282 2-93 1-075 i -34 .2927 1.74 •5539 2.14 .7608 2-54 .9322 2.94 1.0784 i -35 .3001 i -75 •550 2.15 •7655 2-55 .9361 2-95 1.0818 1.36 •3075 1.76 •5653 2.16 .7701 2.56 •94 2.96 1.0852 I *37 .3148 i -77 • 57 i 2.17 •7747 2.57 •9439 2.97 1.0886 1.38 .3221 1.78 .5766 2.18 •7793 2.58 .9478 2.98 1. 0919 i -39 •3293 1.79 .5822 2 19 •7839 2-59 •9517 2-99 I -°953 1.4 •3365 1.8 .5878 2.2 .7885 , 2.6 •9555 3 1.0986 332 HYPERBOLIC LOGARITHMS OP NUMBERS. No. 1 Log. | j ’ No. Log. | No. Log. No. | Log. No. | : 3.OI 1. 1019 3 - 5 i 1.2556 i 4..OI I.3888 . 4.51 : I.5063 5.01 | 1. 3.02 1 .1053 3-52 I.2585 ' 4.02 I * 39 I 3 4-52 : 1.5085 5.02 1. 3-°3 1. 1086 3-53 I.2613 1 - 4.03 1.3938 4.53 1*5107 5-03 1 3-04 I.III 9 3-54 I.2641 4.04 1.3962 4-54 1.5129 5-04 i 1 3-°5 I.II 5 I 3*55 I.2669 4.05 1.3987 4-55 I. 5 I 5 I 5*05 | 1 3.06 I.I184 3-56 I.2698 1 4.06 1 .4012 4-56 I- 5 I 73 5.06 1 3.07 I. 1217 3-57 I.2726 1 4.07 1 .4036 4-57 I. 5 I 95 5-07 1 3.08 I. 1249 3.58 1.2754 4.08 1.4061 4-58 1*5217 5.08 1 3.09 1.1282 3-59 I.2782 4.09 1.4085 4-59 1.5239 5-09 1 3 - 1 I.I 3 H 3*6 I.2809 4.I 1. 41 1 4.6 I.5261 5 -i 1 3. 11 1.1346 3.61 I.2837 4.II I. 4 I 34 4.61 I.5282 5.11 1 3.12 1.1378 3.62 I.2865 4.12 I. 4 I 59 4.62 1.5304 5.12 1 3.13 1.141 3-63 I.2892 4.13 1.4183 4-63 1.5326 5-13 i 3.14 1. 1442 3-64 I.292 4.14 1.4207 4.64 1-5347 5-14 1 3-i S 1. 1474 3-65 I.2947 4.15 1. 4231 4-65 1.5369 5.15 3 3 * 16 1.1506 3.66 1.2975 4.16 1.4255 4.66 1-539 5. 16 i 3-17 i*i 537 3.67 I.3OO2 4.I7 1.4279 4.67 i* 54 i 2 5-17 ] 3.18 1.1569 3.68 I.3029 4.18 1-4303 4.68 1-5433 5. 18 ] 3.19 1. 16 3-69 1 . 3 0 56 4.19 1.4327 4.69 1.5454 5 .i 9 : 3-2 1.1632 3-7 1.3083 4.2 1. 435 1 4-7 I .5476 5-2 3.21 1.1663 3 - 7 1 I- 3 11 4.21 1-4375 4.71 1-5497 5.21 3.22 1.1694 3 - 7 2 I * 3 I 37 4.22 1.4398 4.72 i- 55 i 8 5-22 323 1. 1725 3-73 1.31:64 4.23 1.4422 4-73 1-5539 5.23 3.24 1.1756 3-74 i* 3 i 9 i 4.24 1.4446 4-74 1.556 5.24 3*25 1.1787 3-75 1.3218 4.25 1.4469 4-75 i. 558 i 5.25 3.26 1.1817 3-76 1.3244 4.26 1-4493 4.76 1.5602 5.26 3.27 1.1848 3-77 1.3271 4.27 1.4516 4.77 1.5623 5.27 3.28 1.1878 3-78 1.3297 4.28 1-454 4.78 1.5644 5.28 3- 2 9 1. 1909 3-79 1-3324 4.29 14563 4-79 1.5665 5.29 33 1 .*939 3-8 i -335 4*3 1.4586 4.8 1.5686 5-3 3.31 1.1969 3.81 I- 337 6 4.31 1.4609 4.81 1.5707 5 - 3 1 3.32 1. 1999 3 82 1-3403 4.32 1-4633 4.82 1.5728 5.32 3-33 1.203 3-83 1.3429 4*33 1.4656 4.83 1.5748 5-33 3-34 1.206 384 1-3455 4*34 1.4679 4.84 1-5769 5-34 3-35 1.209 3-85 1.3481 4-35 1.4702 4-85 1-579 5-35 3.36 1. 2119 3.86 1.3507 4.36 1.4725 4.86 1.581 5.36 3*37 1. 2149 3*87 1-3533 4-37 1.4748 4.87 1.5831 5-37 3-38 1. 2179 3.88 1.3558 4.38 1-477 4 88 1.5851 5.38 3.39 1.2208 3-89 13584 4-39 1-4793 4.89 1.5872 5-39 3-4 1.2238 3-9 1.361 4-4 1.4816 4.9 1.5892 5-4 3.41 1.2267 3 - 9 1 1.3635 4.41 1.4839 4.91 I - 59 I 3 541 3 - 4 2 1.2296 3 - 9 2 1.3661 4.42 1.4861 4.92 1-5933 5-42 3-43 1.2326 393 1.3686 4-43 1.4884 4.93 1-5953 5-43 3*44 1*2355 3-94 1.3712 4-44 1.4907 4.94 1-5974 5-44 3-45 1.2384 3-95 1-3737 4*45 1.4929 4-95 1-5994 5-45 3.46 ' 1.2413 396 1.3762 4.46 » 1.4951 4.96 1 1.6014 1 3.47 1.2442 3-97 1.3788 4-47 1.4974 4-97 1.6034 1 5-47 1.48 1 1.247 3 - 9 8 1.3813 4.48 ; 1.4996 4.98 : 1.6054 |i 5 48 3*49 1.2499 3-99 1 1.3838 4*49 ) i* 5 OI 9 4-99 1 1.6074 5-49 3-5 1 1.2528 4 1 1.3863 4-5 1.5041 5 1.0094 5.5 Log. I.6487 1.6506 I.6525 I.6514 I.6563 I.6582 1. 6601 1.662 I.6639 1.6658 I.6677 I.6696 1.6715 1.6734 I.6752 I.677I I.679 I.6808 I.6827 I.6845 1 .6864 1.6882 I.69OI I.6919 1.6938 I.6956 I.6974 1.6993 1. 7011 I.7029 1.7047 HYPERBOLIC LOGARITHMS OF NUMBERS. 333 No. Log. No. Log. | No. Log. No. Log. No. Log. 5*51 I.7066 6.01 1-7934 6.51 1.8733 7.OI 1-9473 7-51 2.0162 5 - 5 2 1.7084 6.02 I- 795 I ! 6.52 I.8749 7.02 1.9488 7.52 2.0176 5-53 1. 7102 6.03 1.7967 6-53 I.8764 7-03 1.9502 7-53 2.0189 5-54 1. 712 6.04 1.7984 6-54 1.8779 7-04 1.9516 7-54 2.0202 5-55 I.7138 6.05 1. 8001 6.55 1.8795 7-05 1-953 7-55 2.0215 5-56 1.7156 6.06 1.8017 6.56 I.881 7.06 1-9544 7.56 2.0229 5-57 i* 7 I 74 6.07 1.8034 6.57 1.8825 7-07 1-9559 7-57 2.0242 5-58 1. 7192 6.08 1.805 6.58 1.884 7.08 1-9573 7-58 2.0255 5-59 1. 721 6.09 I.8066 6-59 I.8856 7.09 1.9587 7-59 2.0268 5-6 1.7228 6.1 1.8083 6.6 I.8871 7 -i 1.9601 7-6 2.0281 5.61 1.7246 6. 1 1 1.8099 6.61 1.8886 7.11 1.9615 7.61 2.0295 5.62 1.7263 6.12 1. 8116 6.62 1.8901 7.12 1.9629 7.62 2.0308 5-63 1.7281 6.13 1.8132 6.63 1.8916 7.13 1.9643 7-63 2.0321 5*64 1.7299 6.14 1.8148 6.64 1.8931 7.14 i. 9 6 57 7.64 2.0334 5-65 I - 73 I 7 6.15 1.8165 6.65 1.8946 7.15 1.9671 7.65 2.O347 5.66 1-7334 6.16 1. 8181 6.66 1.8961 7.16 1.9685 7.66 2.036 5- 6 7 1-7352 6.17 1.8197 6.67 1.8976 7.17 1.9699 7.67 2.0373 5.68 i -737 6.18 1.8213 6.68 1.8991 7.18 I. 97 I 3 7.68 2.0386 5-69 I -7387 6.19 1.8229 6.69 1.9006 7.19 1.9727 7.69 2.O399 5-7 1-7405 6.2 1.8245 6.7 1. 902 1 7.2 1.9741 7-7 2.0412 5 - 7 i 1. .7422 6.21 1.8262 6.71 1.9036 7.21 1*9755 7 - 7 1 2.O425 5-72 1.744 6.22 1.8278 6.72 1. 905 1 7.22 1.9769 7.72 2.0438 5-73 1-7457 6.23 1.8294 6-73 1.9066 723 1.9782 7-73 2.O45I 5-74 1*7475 6.24 1.831 6.74 1.9081 7.24 1.9796 7-74 2.0464 5*75 1.7492 6.25 1.8326 6-75 1.9095 7-25 1.981 7-75 2.O477 5 - 7 ^ 1-7509 6.26 1.8342 6.76 1.911 7.26 1.9824 7.76 2.O49 5-77 1.7527 6.27 1.8358 6.77 !- 9 I2 5 7.27 1.9838 7-77 2.0503 5*78 1-7544 6.28 1.8374 6.78 1 * 91.4 7.28 1.9851 7.78 2.0516 5-79 1.7561 6.29 1.839 6.79 I- 9 I 55 7.29 1.9865 7-79 2.0528 5-8 1-7579 6-3 1.8405 6.8 1.9169 7-3 1.9879 7.8 2.O54I 5.81 1.7596 6.31 1.8421 6.81 1.9184 7 . 3 i 1.9892 7.81 2.0554 5.82 1.7613 6.32 1.8437 6.82 1. 9199 7-32 1.9906 7.82 2.0567 5.83 1-763 6-33 1.8453 6.83 1-9213 7.33 1.992 7.83 2.058 5-84 1.7647 6-34 1.8469 6.84 1.9228 7*34 1 ‘9933 7.84 2.0592 5.85 1.7664 6.35 1.8485 6.85 1.9242 7.35 1 -9947 7.85 2.0605 5.86 1.7681 6.36 1.85 6.86 1.9257 7-36 1.9961 7.86 2.o6l8 5-87 1.7699 6.37 1.8516 6.87 1.9272 7-37 1.9974 7.87 2.0631 • 5-88 1.7716 1 6.38 I -8532 6.88 1.9286 7.38 1.9988 7.88 2.0643 5-89 1-7733 6-39 1.8547 6.89 1. 9301 7-39 2.0001 7-89 2.0656 5-9 1-775 6.4 1.8563 6.9 i. 93 i 5 7-4 2.0015 7-9 2.0669 5 - 9 i 1.7766 6.41 1-8579 6.91 1-933 7.41 2.0028 7.91 2.o68l 5*92 1.7783 6.42 1.8594 6.92 1-9344 7.42 2.0042 7.92 2.0694 5-93 1.78 6-43 1. 861 6.93 1-9359 7-43 2.0055 7-93 2.0707 5-94 1.7817 6.44 1.8625 6.94 1-9373 7-44 2.0069 7-94 2.0719 5-95 I -7834 6.45 I.8641 6-95 1.9387 7-45 2.0082 7-95 2.0732 5-96 1.7851 6.46 1.8656 6.96 1.9402 7.46 2.0096 7.96 2.O744 5-97 1.7867 6.47 1.8672 6.97 1.9416 7-47 2.0109 7-97 2.0757 5-98 1.7884 6.48 1.8687 6.98 1-943 7-48 2.0122 7.98 2.0769 5-99 1.7901 6.49 1.8703 6.99 1-9445 7-49 2.0136 7-99 2.0782 6 1.7918 ! 6.5 1.8718 7 1-9459 7-5 2.0149 8 2.0794 334 HYPERBOLIC LOGARITHMS OF NUMBERS. No. Log. No. Log. No. Log. No. Log. No. 8.01 2.0807 8.41 2.1294 8.81 : 2.1759 9.21 : 2.2203 9.61 8.02 2.0819 8.42 2.1406 8.82 : 2.177 9.22 : 2.2214 9.62 8.03 2.0832 8-43 ' 2.1318 8.83 : 2.1782 9.23 i 2.2225 9- 6 3 8.04 2.0844 8.44 2.133 8.84 : 2.1793 9.24 : 2.2235 9.64 8.05 2.0857 8-45 2.1342 8.85 ! 2.1804 9.25 : 2.2246 9- 6 5 8.06 2.0869 8.46 2.1353 8.86 : 2.1815 9.26 : 2.2257 9.66 8.07 2.0882 8.47 2.1365 8.87 2.1827 9.27 2.2268 9.67 8.08 2.0894 8.48 2.1377 8.88 2.1838 | 9.28 2.2279 9.68 8.09 2.0906 8.49 2.1389 8.89 2.1849 9.29 2.2289 9.69 8.1 2.O919 8-5 2. 14OI 8.9 2.l86l 9-3 2.23 9.7 8.11 2.O93I 8.51 2. 1412 8.91 2.1872 9 - 3 1 2.23II 9.71 8.12 2.O943 8.52 2.I424 8.92 2.1883 9-32 2.2322 9.72 8.13 2.0956 8-53 2.1436 8-93 2.1894 9-33 2.2332 9-73 8.14 2.0968 8-54 2.1448 8.94 2.1905 9-34 2.2343 9-74 8.15 2.098 8-55 2.1459 8-95 2. 1917 9-35 2.2354 9-75 8.16 2.0992 8.56 2.I47I 8.96 2.1928 9-36 2.2364 9.76 8.17 2.1005 8-57 2.1483 8-97 2.I939 9-37 2.2375 9-77 8.18 2. 1017 8.58 2.1494 8.98 2.195 9-38 2.2386 9.78 8.19 2.1029 8-59 2.1506 8.99 2.1961 9-39 2.2396 9-79 8.2 2. IO4I 8.6 2.1518 9 2.I972 9.4 2.2407 9.8 8.21 2.1054 8.61 2.1529 9.01 2.1983 9.41 2.2418 9.81 8.22 2.1066 8.62 2.1541 9.02 2.I994 9.42 2.2428 9.82 8.23 2.IO78 8.63 2.1552 9-°3 2.2006 9-43 2.2439 9-83 8.24 2.109 8.64 2.1564 9.04 2.2017 9.44 2.245 9.84 8.25 2. 1102 8.65 2.1576 9-°5 2.2028 9-45 2.246 9-85 8.26 2 .III 4 8.66 2.1587 9.06 2.2039 9.46 2.2471 9.86 8.27 2.1126 8.67 2.1599 1 9-°7 2.205 1 9-47 2.2481 9.87 8.28 2.II38 8.68 2.l6l 9.08 2.2061 1 9.48 2.2492 9.88 8.29 2.II5 8.69 2.l622 9.09 2.2072 9.49 2.2502 9.89 8-3 2.H63 8.7 2.1633 9.1 2.2083 9-5 2.2513 9.9 8.31 2 .II 75 8.71 2.1645 9.11 2.2094 9 - 5 i 2.2523 9.91 8.32 2.H87 8.72 2.1656 9.12 2.2105 9-52 2.2534 9.92 8.33 2. II99 8.73 2.l668 9 -i 3 2.2Il6 9-53 2.2544 9-93 8-34 2 .I 2 II 8.74 2.1679 9- I 4 2.2127 9-54 2.2555 9.94 8-35 2.1223 8-75 2.1691 9 -i 5 2.2138 9-55 2.2565 9-95 8.36 2.1235 8.76 2.1702 9.16 2.2148 9-56 2.2576 9.96 8-37 2.1247 8.77 2 .I 7 I 3 9.17 2.2159 9-57 2.2586 9-97 8.38 2.1258 8.78 2.1725 9.18 2.217 9-58 2.2597 9 - 9 * 8.39 2.127 8.79 2.1736 9.19 2.2l8l 9-59 2.2607 9-99 8.4 2.1282 8.8 2.I748 9.2 2.2192 9.6 2.2618 10 10.25 2.3279 12.25 1 2.5052 14.25 ; 2.6567 17-5 2.8621 23 10.5 2.3513 12.5 2.5262 14-5 2.674 18 2.8904 24 10-75 ; 2.3749 12.75 : 2.5455 ; 14-75 i 2.6913 18.5 2.9173 25 11 2.3979 » 13 2.5649 1 15 2.7081 19 2.9444 . 26 11.25 ; 2.4201 13*25 > 2.584 i 5-5 2.7408 19-5 2.9703 ;l 27 11.5 2-443 13-5 2.6027 1 16 2.7726 1 2 2-9957 » 28 ii* 7 ! > 2.4636 > 13-75 ; 2.6211 : 16.5 2.8034 . 21 3 - 044 = ; 29 12 2.4845 > 1 14 2.6391 [ 17 2.8332 : 22 3 -° 9 I] : U Log- 2.2628 2.2638 2.2649 2.2659 2.267 2.268 2.269 2.2701 2. 27II 2.2721 2.2732 2.2742 2.2752 2.2762 2.2773 2.2783 2.2793 2.2803 2.2814 2.2824 2.2834 2.2844 2.2854 2.2865 2.2875 2.2885 2.2895 2.2905 2.2915 2.2925 2.2935 2.2946 2.2956 2.2966 2.2976 2.2986 2.2996 2.3006 2.3016 2.3026 3-1355 3.1781 32189 3- 2 58 i 3-2958 3-3322 33673 3 - 4° 12 MENSURATION OF AREAS, LINES, AND SURFACES. 335 MENSURATION OF AREAS, LINES, AND SURFACES. Parallelograms. Definition— Quadrilaterals, having their opposite sides parallel. To Compute Area of* a, Square, Rectangle, Plioxxi'bns, or !Rliom.looid..— Figs. 1,2, 3, and. 4r. Rule. — M ultiply length by breadth or height. Or, lxb = area , l representing length , and b breadth. Note i. — Opposite angles of a Rhombus and a Rhomboid are equal. 2. — In any parallelogram the four angles equal 360°. 3. —Side of a square multiplied by 1.52 is equal to side of an equilateral triangle of equal area. Definition. — Space included between, the lines forming two similar parallel- ograms, of which smaller is inscribed within larger, so that one angle in each is common to both, as shown by dotted lines, Fig. 1. To Compute Area of* a G-nomon.-Fig. 1. Rule. — Ascertain areas of the two parallelograms, and subtract less from Example. — Sides of a gnomon are 10 by 10 and 6 by 6 ins. ; what is its area ? 10X10 = 100, and 6 X 6 = 36. Then 100 — 36 = 64 square ins. Definition.— P lain superficies having three sides and angles. To Compute Area of a Triangle.— Pigs. 5, G, and V . Rule. — Multiply base by height, and divide product by 2. Note i. — Hypothenuse of a right angle is side opposite to right angle. 2. — Perpendicular height of a triangle = twice its area divided by its base. 3. — Perpendicular height of an equilateral triangle = a side X .866. 4. — Side of an equilateral triangle x .658 255 = side of a square of equal area, Or — 1.3468 = diameter of a circle of equal area. Fig. 5. Fig. 6. Fig. 7. Fig. 1. Fig. 3- a l >G Fig. 2. h Example. — Sides ab,bc, Fig. 1, are 5 feet 6 ins. ; what is area? 5. 5 X 5- 5 = 30- 25 square feet. GriioinorL, greater. Or, a — a' = area, a and a' representing areas. Triangles c G c Example. — Base a b, Fig. 5, is 4 feet, and height c b, 6 ; what is area ? 4 X 6 = 24, and 24-= 2 — 12 square feet. d bad a b a To Compute Area of a Triangle t)y Length of its Sides.- ITigs. 6 and *7, Kuut.-From half sum of the three sides subtract each side separately; then multiply half sum and the three remainders continually together, and take square root of product. 0r V («-«)X («-■ -&) s = area, a, 6, c representing sides, and S half sum of the three sides. *•»«**«« Example.— Sides of a triangle, Figs. 6 and 7, are 30, 40, and 50 feet; what is area . — 00 = ao 60—30 = 30) 60 — 40 = 20 J remainders. 30 + 40+50 ^ _ 6 0 or half sum of sides. — -r- . 2 2 60 — 50 = 107 Whence, 30 X 20 X 10 X 60 = 360000, and f 3 6 o 000 = 600 square feet. When all Sides are Equal. Bulk.— Square length of a side, and multi- ply product by .433. Or, S 2 X *433 — area i S representing length of a side. To Compute Length of One Side of a Right-Angled Triangle. When Length of the other Two Sides are given. To Ascertain Hypothenuse.-Fig. £5. Rule.— Add together squares of the two legs, and take square root of sum. Or Va 6 2 + b / I ( ab 2 = 5 c. Or, \J hyp- 2 — {/ t 2 = ^ 0r iy ac ~~\bc 2 = ab. 7 / ^/ ^ (l — */• y \ Example. -Base of a triangle, aft, Fig. 5, is 3 ° feet, and hypothenuse, a c, 30; what is height of it? so 2 __ 30 2 — j6oo, and V l6o ° = 4° f eet - To Compute Length. of a Side. When Hypothenuse of a Right-angled Triangle of Equal Sides alone is g { ven . — Fig. 8. Rule.— Divide hypothenuse by 1.414213. Or, -- — — = length of a side. Fig. 8. 1.414213 Example. — Hypothenuse acofa right-angled triangle, Fig. 8, is 300 feet; what is length of its sides ? ; 300-4- 1. 414 213 = 212. 1321 feet. Ur O To Compute Perpendicular or Height of a Triangle. When Base and Area alone are given.— Tig. 9. Rule.— Divide twice area by its base. Or, 20,-7- b = h. irea by its oase. or, 2 r — . Example. -Area of a triangle, Fig. 9, is 10 feet, and length of its base, a 6, 5, what is its perpendicular, cd? 10 X 2 = 20, and 20 -T- 5 = 4 f eet - MENSURATION OP AREAS, LINES, AND SURFACES. 337 To Compute Perpendicular or Height of a Triangle. When Base and Two Sides are given. Kule.-As base is to sum t)f the sides so is difference of sides to difference of divisions of base. Halt this difference bebu? added to or subtracted from half base will give the two di- v S thereof; Hence, as the sides and their opposite division of base con- stitute a right-angled triangle, the perpendicular thereof is readily ascertained by preceding rules. Or, b c + c a X b c a, c a b a — bd'X/da. Or, a c - -}- a b 2 b c __ a ^. -whence V a c 2 — ad 2 z=dc. 2db Fig. 9. Example.— Three sides of a triangle, a be, Fig. 9, are 9.928, 8, and 5 feet; what is length of perpeudicular on longest side? ’ As 9.928 : 8 + 5 : : 8 5 : 3.928 = difference of divisions of the base. Q.Q28 . Then 3.928 -f- 2 = 1.964, which, added to — — — 4 - 9 6 4 + !. 964 = 6 928 = length of longest division of base. ^TTpnre there is a right-angled triangle with its base 6.928, and its hypothenuse 8; consequenOyits remainingside or perpendicular is V(8 2 - 6. 928*) = 4 When anv Two of the Dimensions of a Triangle and One of the corresponding Dimensions of a similar Figure are given and it is required to ascertain the other corresponding Dimensions of the last r igure. Fig. 10. Fig. ix. , . Let a be, a ' b' c', be two similar triangles, Figs. 10 and 11. Then ab :bc :: a'b' : V c’ , or a'b ' : b' c' V. ab : be. Note — Same proportion holds with respect to the similar lineal parts of any other similar figures, whether a V a plane or solid. Example. Shadow of a vertical stake 4 feet in length was 5 feet; at same time, Shadow of a tree, both on level ground, was 83 feet; what was height of tree . 5 a' b’ : 4 b' c * :: 83 ab : 66.4 feet. To Compute Acreage. Divide area into convenient triangles, and multiply base of each triangle in links bv half perpendicular in links ; cut off 5 figures at the right, remain- ing figures will give acres ; multiply the 5 figures so cut off by 4, and aga cut off 5, and remainder will give roods ; multiply the 5 by 40, and a b am cut off 5 for perches. Tr ap e z i um . Definition.— A Quadrilateral having unequal sides of which no two are parallel. To Compute Area of a Trapezium.-Fig. 13 . Rule. — Multiply diagonal by sum of the two perpendiculars falling upon it from the opposite angles, and divide product by 2. db x a+c Or, • = area. Example. — Diagonal d b, Fig. 12, is 125 feet, and perpen- — diculars a and c 50 and 37; what is area? I2$ x 50 4- 37 = 10 875, and 10 875 - 4 - 2 = 5437- 5 square feet. F F 338 MENSURATION OF AREAS, LINES, AND SURFACES. When the Two opposite Angles are Supplements to each other, that is, when a Trapezium can be inscribed in a Circle, the Sum of its opposite Angles being equal to Two Right Angles , or i8o°. Rule. — From half sum of the four sides, subtract each side severally ; then multiply the four remainders continually together, and take square root of product. Example.— In a trapezium the sides are 15, 13, 14, and 12 feet; its opposite an- gles being supplements to each other, required its area. 15 -f 13 + 14 + 1 2 = 54> and — — 27. 27 27 27 27 2 15 13 14 12 12X14X13X 15 = 32760, and -^32760 = 180.997 square feet. Trapezoid. Definition. — A Quadrilateral with only one pair of opposite sides parallel. To Compute Area of a Trapezoid.— Fig. 13. Rule. — Multiply sum of the parallel sides by perpendicular distance be- tween them, and divide product. ^ ab~\-dcxah s-\-s' xh Or, ' . Or, — 1 = area , s and s! representing sides. 2 2 Fig. 13. a e p Example. — Parallel sides ab,cd , Fig. 13, are 100 and 132 feet, and distance between them 62. 5 feet ; what is area ? 100 -f- 132 x 62.5 = 14 500, and 14 500 -4- 2 = 7250 square 1 f^t. Polygons. Definition.— Plane figures having three or more sides, and are either regular or irregular, according as their sides or angles are equal or unequal, and they are named from the number of their sides and angles. li c h Regular Polygons. To Compute Area of* a, Itegixlar iPolygon.— Fig. 14. Rule.— Multiply length of a side by perpendicular distance to centre; multiply product by number of sides, and divide it by 2. Or, aJ> * ce = area , n representing number of sides. Example.— What is area of a pentagon, side ab , Fig. 14, being 5 feet, and distance ce 4. 25 feet ? 5 X 4.25 X 5 (») = 106.25 — product of length of a side , dis- tance to centre , and number of sides. Then, 106.25-4-2 = 53 -i 2 5 square feet. To Compute Radius of a Circle that contains a Given Polygon. When Length of a Perpendicular from Centre alone is given. Rule.— Multiply distance from centre to a side of the polygon, by unit in column A of following Table. Example — What is radius of a circle that contains a hexagon, distance to centre being 4.33 inches? < 4.33 X 1.156 = 5 mw- To Compute Length, of a Side of a Polygon that is con- \ tained in a Given Circle. When Radius of Circle is given. Rule.— M ultiply radius of circle, by unit in column B of following Table. Example.— What is length of side of a pentagon contained in a circle 8.5 feet in diameter? 8.5-4-2 = 4.25 radius, and 4.25 X 1.1756 = 5 feet. MENSURATION OF AREAS, LINES, AND SURFACES. 339 To Compute Radius of* a, Circumscribing Circle. When Length of a Side is given. Rule.— Multiply length of a side of the polygon, by unit in column C of following Table. Example. -What is radius of a circle that will contain a hexagon, a side being 5 inches? 5X1 — 5 ™ s - To Compute Radius of a Circle tliat can. be Inscribed in a Gfiven Polygon. When Length of a Side is given. Rule.— Multiply length of a side of polygon, by unit in column D of following Table. Example.— What is radius of the circle that is bounded by a hexagon, its sides being 5 inches? 5 X- 866 = 4.33 ms. To Compute Area of a Ttegnlar Polygon. When Length of a Side only is given. Rule. — Multiply square of side, by multiplier opposite to term of polygon in following Table : No. of Sides. Polygon. Area. A. Radius of Circumscribed Circle. B. Length of a Side. C. Radius of Circumscrib- ing Circle. D. Radius of Inscribed Circle. 3 Trigon •43301 2 1 732 -5773 .2887 4 Tetragon 1 I-4I4 1. 4142 .7071 • 5 5 Pentagon 1.720 48 1.238 1.1756 .8506 .6882 6 Hexagon 2. 598 08 I.I5 6 1 1 .866 7 Heptagon 3-^33 9 1 1. 11 .8677 I.I524 1.0383 8 Octagon 4.828 43 1.083 •7653 1.3066 1. 2071 9 Nonagon 6. 181 82 1.064 .684 1.4619 1.3737 10 Decagon 7.69421 1.051 .618 1. 618 1.5388 11 Un decagon 9.36564 1.042 •5634 1-7747 1,7028 12 Dodecagon 11.196 15 1037 •5176 I-93I9 1.866 Example. — What is area of a square (tetragon) when length of its sides is 7.071067 8 inches? 7.071 067 8 2 = 50, and 50X1 = 50 square ins. To Compute Length of a Side and If adii of a Regular [Polygon. When Area alone is given. Rule. — Multiply square root of area of poly- gon by multiplier in column E of the following table for length of side ; by multiplier in column G for radius of circumscribing circle, and by multiplier in column H for radius of inscribed circle or perpendicular. No. of Sides. Polygon. E. Length of Side. G. Radius of Circumscrib- ing Circle. H. Radius of Inscribed Circle. Angle. Angle of Polygon. Tangent. 3 Trigon I-5I97 .8774 •4387 120° 6o° •5774 4 Tetragon 1 .7071 -5 90 90 1 5 Pentagon .7624 .6485 •5247 72 108 1-3764 6 Hexagon .6204 -6204 •5373 60 120 1.7321 7 Heptagon .5246 .6045 •5446 51 25.71' 128 34.29' 2.0765 8 Octagon •4551 .5946 •5493 45 135 2.4142 9 Nonagon .4022 .588 •5525 40 140 2-7475 10 Decagon 3605 •5833 .5548 36 144 . „ 3.0777 11 Undecagon .3268 •5799 •5564 32 43-64' 147 16.36' 3-4057 12 Dodecagon .2989 •5774 •5577 30 150 3-7321 Example 1.— Area of a square (tetragon) is 16 inches; what is length of its side? f 16 = 4, and 4X1=4 ins. 2.— Area of an octagon is 70.698 yards; what is diameter of its circumscribing circle? V70.698 x .5946 = 5, and 5X2 = 10 yards. 340 MENSURATION OF AREAS, LINES, AND SURFACES. Additional Uses of foregoing Table.— 6th and 7 th columns of table facilitate con- struction of these 1 figures with aid of a sector. Thus, if it is required to describe an octagon, opposite to it in column 6th, is 45; then, with chord of 60 on sector as radius, describe a circle, taking length 45 on same line of sector; mark this dis- tance off on the circumference, which, being repeated around the circle, will give points of the sides. 7th column gives angle which any two adjoining sides of the respective figures make with each other; and 8th gives tangent of the angle in column 6th. To Compute Radius of Inscribed or Circumscribed Circles. When Radius of Circumscribing Circle is given. Rule.— Multiply radius given by unit in column E, in following Table, opposite to term of polygon for which radius is required. When Radius of Inscribed Circle is given. Rule.— Multiply radius given by unit in column F, in following Table, opposite to term of polygon for which radius is required. To Compute Area. When Radii of Inscribed or Circumscribing Circles are given. Rule. — Square radius given, and multiply it by unit in columns G or If, in following Table, and opposite to term of polygon for which area is required. When Length of a Side is given. Rule. — Square length of side and multiply it by unit in column I, in following Table, opposite to term of polygon for which area is required. To Compute Length. of a Side. When Radius of Inscribed Circle is given. Rule. — Multiply radius given by unit in column K, in following Table, and opposite to term of polygon for which length is required. No. of Sides. Polygon. E. Radius of Inscribed by Circum- sci-ibing Circle. F. Radius of Circumscrib- ing by Inscribing Circle. G. Area. By Radius of Inscribed Circle. H. Area. By Radius of Circum- scribing Circle. I. Area. By Length of Side. K. Length of Side. By Radius of Inscribed Circle. 3 Trigon •5 2 5- 19 6 1 5 1.299 04 •433 01 3.4641 4 Tetragon . 707 1 1 1. 414 21 4 2 1 2 5 Pentagon . 809 02 1.23607 3.63272 2.37764 1.72048 1.45308 6 Hexagon . 866 02 I - I 54 7 3.464 1 2.598 08 2. 598 08 I - I 54 7 7 Heptagon .90097 1. 109 92 3.371 02 2.73641 3-633 9 1 •96315 8 Octagon .923 88 1.082 39 3-3I37I 2.828 42 4.81843 .82843 9 Nonagon •939 6 9 1.064 1 8 3-275 73 2.89254 6.182 82 .72794 10 Decagon .95106 1. 051 46 3.2492 2-938 93 7.69421 . 649 84 11 Undecagon •959 49 1.042 22 3.229 89 2-973 53 9-365 64 • 587 25 12 Dodecagon •9 6 5 93 1.03528 3-215 39 3 11.196 15 •535 9 Regnlar Bodies. To Compute Surface or Linear Edge of Regular Body. Rule. — Multiply square of linear edge, or radius of circumscribed or in- scribed sphere, by units in following table, under head of dimension used : No. of Sides. Body. Surface by Linear Edge. Radius of Circumscribed Sphere. Radius of Inscribed Sphere. Linear Edge by Surface. 4 Tetrahedron 1.73205 1.632 99 4. 898 98 •759 8 4 6 Hexahedron 6 i- I 54 7 2 . 408 25 8 Octahedron 3.4641 1. 414 21 2.449 49 •537 29 12 Dodecahedron 20.64578 •71364 .89806 . 220 08 20 Icosahedron 8.66025 1. 051 46 1-323 i7 •339 81 Example. — W hat is surface of a hexahedron or cube, having sides of 5 inches? 5 2 x 6 = 25 X 6 = 150 square ins. MENSURATION OF AREAS, LINES, AND SURFACES. 241 To Compute Linear Edge. When Surface alone is given. Rule.— Multiply square root of surface, by multiplier opposite to term of body under head of Linear Edge by Sur- face in preceding Table. Example. -What is linear edge of a hexahedron, surface being 6 inches? V6 X .40825 = 1 inch. When Radius of an Inscribed or Circumscribed Sphere is given. Rule.— Multiply radius given, by multiplier opposite to term of body in preceding Table, under head of the Radius given. Example.— Radius of circumscribing sphere of a hexahedron is 10 inches; what is its linear edge ? 10 X 1. 1547 = h- 547 ins - To Compute Surface. When Linear Edge is given. Rule— Multiply square of edge, by multi- plier opposite to term of body in preceding Table, under head of Surface. Example.— Linear edge of a hexahedron is 1 inch; what is its surface? z 2 X 6 = 6 square ins. Irregular Polygons. Definition.— Figures with unequal sides. To Compute Area of an Irregular Lolygon.-Eigs. IS and. 16 . Fig. 15- Rule.— Draw diagonals and per- Fig. 16. pendiculars, as df dg , a. and c, Fig. / 15, and/ d, gd,gb,gu, and i, o r r, and j Fig. 16, to divide the figures into triangles and quadrilaterals: ascer- 0 tain areas of these separately, and take their sum. g Note —To ascertain area of mixed or compound figures, or 6> such as are composed of rectilineal and curvilineal figures to- gether computeareas of the several figures of which the whole is composed, then add them together, and the sum will give area of compound figure. In this manner any irregular surface or field of land may be measured by dividing it into trapeziums and triangles, and computing area of each separately. When any Part of a Figure is bounded by a Curve the Area may be ascer- tained as follows : Erect any number of perpendiculars upon base, at equal distances, and ascertain their lengths. . . ,, ... Add lengths of the perpendiculars thus ascertained together, mid their sum, divided by their number, will give mean breadth ; then multiply mean breadth by length of base. To Compute Area of a. Long, Irregular Eiguire.— Eig. V 7 . Fig. 17. Rule.— Take mean breadths at several places, at equal distances apart, as 1, 2, 3, b d, etc. ; add them together, divide their sum by number of breadths for total mean breadth, and multiply quotient by length of figure. + etc. Or, — - — A — l = area. F F* MENSURATION OF AREAS, LINES, AND SURFACES. 342 To Concipixte Fig. 18. a 1 2 3 4 5 a,x 1 Area "bonxided by a Curve.-Fig. 18. (Simpson's Rule.) Operation.— Divide line a b into any number of equal parts, by perpendiculars from base, as 1, 2, 3) etc., which will give an odd number of points of division. Measure lengths of _ 5 these perpendiculars or ordinates, and proceed as follows : To sum of lengths of first and last ordinates, add four times sum of lengths of all even numbered ordinates and twice sum of odd; multiply their sum by one third of distance between ordinates, and product will give area required. Illustration.— Water-line of a vessel has a length of 90 feet, and ordinates o, 1, 1.2, 1.5, 2, 1.9, 1.5, 1. 1, and o, each 10 feet apart; what is its area? Ordinates. Even. Odd. Sums. 1-5 1.9 1. 1 ITs x 4 = 22. i-5 4-7 X 2 = 9.4 first last o even 22 odd 9.4 31.4 X 10 = : 314, which -f- 3 = 104.66 square feet. Circle. Diameter is a right line drawn through its centre, bounded by its periphery. Radius is a right line drawn from its centre to its circumference. Circumference is assumed to be divided into 360 equal parts, termed degrees; each degree is divided into 60 parts, termed minutes ; each minute into 60 parts, termed seconds ; and each second into 60 parts, termed thirds , and so on. To Compute Circumference of a Circle. Rule— Multiply diameter by 3.1416. Or, as 7 is to 22, so is diameter to circumference. Or’ as 1 13 is to 355, so is diameter to circumference. Example.— Diameter of a circle is 1.25 inches; what is its circumference? 1.25 X 3.1416 = 3.927 ins. To Coxxxpxxte Diameter of* a Circle. Rule. — Divide circumference by 3.1416. Or, as 22 is to 7, so is circumference to diameter. Note. —Divide area by .7854, and square root of quotient will give diameter of circle. To Compute Area of* a Circle. Rule.— Multiply square of diameter by .7854. Or, multiply square of circumference by .079 58. Or, multiply half circumference by half diameter. Or, multiply square of radius by 3.1416. Or, p r 2 = area , r representing radius. Example.— The diameter of a circle is 8 inches; what is the area of it? 8 2 = 64, and 64 X -7854 = 50.2656 ins. Proportions of* a Circle, its ICcqxial, IixscriUecl, and Cii — cumscribed Scpxares. CIRCLE. 1. Diameter X .8862) _ g ide 0 f an pq Ua l Square. 2. Circumference X .2821) 3. Diameter X • 7 ° 7 I ) 4. Circumference x .2251 > = Side of Inscribed Square, e. Area X .9003 - 4 - diam. ) 6. Diameter X 1.3468 = Side of an Equilateral Triangle. 7. A Side 8 . SQUARE. X 1. 1442 = Diameter of its Circumscribing Circle. X 4.443 = Circumference of its Circumscribing Circle. 0. •• x 1.128 = Diameter ) jo. “ X 3-545 = Circumference > of an Equal Circle. 11. Square inches X 1-273 = Circle inches ) Note. Square described within a circle is one half area of one described without it. MENSURATION OF AREAS, LINES, AND SURFACES. 343 To Compute Side of Greatest Sqtiare that can "be In- scribed in a Circle. Rule. — Multiply diameter by .7071, or take twice square of radius. TJsefn .1 Factors. In wliicli p or n represents Circumference of* a Circle. Diameter — 1. p= 3.141592653589+ 2 p — 6.283 185 307 179+ 4 p — 12. 566 370 614 359+ pz=. 1.570796326794-f p— .785398163397+ |p = 4 .i88 79+ 14 p =. .523598-f % p=z .392699+ T2 P= .261 799+ ■3 60 -P— •' 008 726+ Diameter = 10. Vp- 1-772453 s/i= .797884 Log. p = .49714987 Y>y/P = .886 226+ 36 P — 11 3-097 335+ 1. Chord of arc of semicircle 2. Chord of half arc of semicircle 3. Versed sine of arc of semicircle. 4. Versed sine of half arc of semicircle 5. Chord of half arc, of half of arc of semicircle 6. Half chord, of chord of half arc 7. Length of arc of semicircle 8. Length of half arc of semicircle 9. Square of chord, of half arc of semicircle (2) 10. Square root of versed sine of half arc (4) 11. Square of versed sine of half arc (4) 12. Square of chord of half arc, of half arc of semicircle (5) 13. Square of half chord, of chord of half arc (6) =s 7.071067 = 5 = 1.464466 — 3.82683 = 3-535 533 = 15-797 963 = 7-853 9 Sl = 50 == 1.210151 =3 2.144664 == 14-644 67 = 12.5 Note.— In all computations p is taken at 3.1416, % p at .7854, % p at .5236; and whenever the decimal figure next to the one last taken exceeds 5, one is added. Thus, 3.141 59 for four places of decimals is taken as 3.1416. To Compute Length, of an Arc of a Circle.— Fig. 19. When Number of Degrees and Radius are given. Rule i. — Multiply number of degrees in the arc by 3.1416 times the radius, and divide by 180. 2.— Multiply radius of circle by .01745329, and product by degrees in the arc. If length is required for minutes, multiply radius by .000 290 889 ; if for seconds, by .000 004 848. Fig. 19. o 0 Example i.— Number of degrees in an arc, 0 ah, Fig. 19, are 90, and radius, 0 &, 5 inches; what is length of arc? 90 X (3.1416 X 5) = 1413.72, which -4- 180 = 7.854 ins. 2. — Radius of an arc is 10, and measure of its angle 44 0 30' 30"; what is length of arc? 10 X .017 453 29 = . 174 5329, which X 44 = 7.6794476, length for 44°. 10 X .000 290 889 = .002 908 89, which x 30 = .087 266 7 , length for 30'. 10 X .000004 848 = .000048 48, which x 30 = .001 454 4, length for 30". Then 7.679 447 6 1 .087 266 7 / == 7.768 168 7 ins. .0014544) Or, reduce minutes and seconds to decimal of a degree, and multiply by it. See Rule, page 93. 30' 30" = .5083, and .1745329 from above X 44.5083=: 7.768 163 ins. 344 MENSURATION OF AREAS, LINES, AND SURFACES. When Chord of Half A rc and Chord of A ro are given. Rule. — F rom eight times chord of half arc subtract chord of arc, and one third of remainder will give length nearly. Or, 8 c' - , c' representing chord of half arc, and c chord of arc. Example.— Chord of half arc, a c, Fig. 19, is 30 inches, and chord of arc, ah, 48; what is length of arc ? 30 x 8 = 240 = 8 times chord of half arc ; 240 — 48 = 192, and 192 -7- 3 = 64 ins. When Chord of Arc and Versed Sine of Arc are given. Rule. — Mul- tiply square root of sum of square of chord, and four times square of the versed sine (equal to twice chord of half arc), by ten times square of versed sine • divide this product by sum of fifteen times square of chord and thirty- three times square of versed sine ; then add this quotient to twice chord of half arc* and sum will give length of arc very nearly. Or, ^ c 2 A~ 4 v - s ^ n - X 10 v. sm. 2 v s ^ n representing versed sine. Hence : 7 . 1599, and 7.1599 + 100, or twice chord of half arc- 15 c 2 +33 sin - 2 Example.— Chord of an arc is 80, and its versed sine, c r, 30; what is length of arc? g 0 2 _ 6400 = square of chord ; 30 2 = 900 = square of versed sine. +(6400 + 900 X 4) = 100 = square root of square of chord and four times square of versed sine == twice chord of half arc. Then 100 X 30 2 X 10 = 900 000 == product of 10 times square of versed sine and root above obtained. And 80 2 X 1 5 = 96 000 = 1 5 tiries square of chord. 30 2 x 33 = 29 700 = 33 times square of versed sine. 125 700 100 X 900000 125 700 107. 1599 length. When Diameter and Versed Sine are given . Rule.— Multiply twice chord of half the arc by 10 times versed sine ; divide product by 27 times versed sine subtracted from 60 times diameter, add quotient to twice chord o± halt arc, and the sum will give length of arc very nearly. 0r ^ xiqt>.j^ + 2C> = c , ’ 60 d — 27 v. sm. Example.— Diameter of a circle is 100 feet, and versed sine, cr, of arc 25 ; what is length of arc? V25 X 100 = 50 = chord of half arc. See Rule, page 345. 50X2X25X10 = 25 000 = twice chord of half arc by 10 times versed sine. VooXbo — ~2sX2. 7 = 5325 = 27 times versed sine from 60 times diameter. Then 25 000 5325 4.6948, and 4.6948 + 50 X 2 = 104.6948 feet. To Compute CLord of an Arc. When Chord of Half the Arc and Versed Sine are given. Rule.— From square of chord of half arc subtract square of versed sine, and take twice square root of remainder. Or, + (c* 2 — v. sin. 2 ) X 2 = c. Example.— Chord of half arc, a c, is 60, and versed sine, c r, 36; what is length of chord of arc ? 60 2 — 36 s = 2304, and f 2304 X 2 = 96. * Square root of sum of square of ehord and four times square of the versed sine is equal to twice «hord of half arc. MENSURATION OF AREAS, LINES, AND SURFACES. 345 When Diameter and Versed Sine are given . Multiply versed sine by 2, and subtract product from diameter; subtract square of remainder from square of diameter, and take square root of that remainder. Or, a/ d' 2 — (d — v. sin. X 2 ) 2 = c. Example. — Diameter of a circle is 100, and versed sine of half arc is 36; what is length of chord of arc ? (36 X 2 — 100) 2 — 100 2 =z 9216, and y/9216 = 96. To Compute Chord of FXalf an Arc. When Chord of the Arc and Versed Sine are given. Rule i. — D ivide square root of sum of square of chord of the arc and four times square of versed sine by two. 2 —Take square root of sum of squares of half chord of arc and versed sine. Or, ■\/ c 2 -j- 4 v. sin . 2 Or, a/(t -J- v. sin. 2 = c r . When Diameter and Versed Sine are given. Rule.— Multiply diameter by versed sine, and take square root of their product. Or, a fd X v. sin. — c'. To Compute Diameter. Rule i. — Divide square of chord of half arc by versed sine. Or, c' 2 -rv. sin. = diameter. 2 —Add square of half chord of arc to the square of versed sine, and divide this sum by versed sine. Or, (C -r 2) 2 -f v. sin. 2 _ To Compute ~VersecL Sine. Rule.— Divide square of chord of half arc by diameter. When Chord of the Arc and Diameter are given. Rule.— F rom square of diameter subtract square of chord, and extract square root of remainder ; subtract this root from diameter, and divide remainder by 2. d — Vd 2 — c 2 Or, = v. sin. 2 When it is greater than a Semidiameter. Rule. — P roceed as before, but add square root of remainder (of squares of diameter and chord) to diam- eter, and halve the sum. ^ d 4 - Vd 2 — c 2 Or, — — = v. sm. 2 Example. — Diameter of a circle is 100, and chord of arc 97.9796; what is its versed sine? IOO + V IOO 2 97. 9796 s TOO 20 60. To Compute Ordinate of* a Circnlar Curve.—Fig. SO. 20 ‘ a/V 2 — x 2 — (r — v) = ordinate. Illustration. — Radius of circle 5 ins., versed sine \ 2, and distance x 2 ; what is length of ordinate 0 ? \ V'5 2 — 2 2 — (5 — 2) = 4. 58 — 3 — i. 58 ins. 346 MENSURATION OF AREAS, LINES, AND SURFACES. Sector oF a Circle. Definition. —A part of a circle bounded by an arc and two radii. To Compute Area of a Sector of a Circle. When Degrees in the Arc are given . — Fig. 21. Rule. — A s 360 is to num- ber of degrees in a sector, so is area of circle of which sector is a part to area of sector. Fig. 21. . Or, == area, d representing degrees in arc , and a area v 7 , 306 ' u of circle. Example. — Radius of a circle, o a, Fig. 21, is 5 ins., and number of degrees of sector, a b 0, is 22 0 30' ; what is area ? Area of a circle of 5 ins. radius = 78. 54 ins. Then, as 360° : 22 0 30' : 7 8 -54 : 4-9 o8 75 «*• When Length of the Arc , etc., are given. Rule.— M ultiply length of arc by half length of radius, and product is area. Or, 6 x r -f- 2 = area, b representing arc , and r radius. Segment of a Circle. Definition.— A part of a circle bounded by an arc and a chord. To Compute Area of a Segment of a Circle. When Chord and Versed Sine of Arc, and Radius or Diameter of Circle are given. When Segment is less than a Semicircle , as ah c, Fig. 21. Rule. Ascer- tain area of sector having same arc as segment ; then ascertain area of tri- angle formed by chord of segment and radii of sector, and take difference of these areas. Note.— S ubtract versed sine from radius; multiply remainder by one half of chord of arc, and product will give area of triangle. Or, a — a' — area , a and a' representing areas of sector and triangle. When Segment is greater than a Semicircle. Rule. Ascertain, by pre- ceding rule, area of iesser portion of circle 5 subtract it from area of whole circle, and remainder will give area. Or, a — a' — area , a and a' representing areas of circle and lesser portion. See Table of Areas of Segments, page 267. Fig. 22. , Example. — Chord, a c , Fig. 22, is 14.142; diameter, h e, is 20 ins. ; and versed sine, h r, is 2.929; "what is area of segment r 14. 142 -r- 2 = 7.071 = half chord of arc. V 7 . Q 7i 2 +' 2 W = 7.654 = square root of sum of squares of half chord of arc and versed sine , which is chord ah of half arc ah c. By Rule, page 346, . , 7.654 X 2 X 2.929 X 10 = 448.371 = twice chord of half arc hy xo times versed sine. 20 ><"6^2.929 x 27 = 1x20.917 = 60 times diameter subtracted from 27 times I versed sine. _ Then 448. 371 - 1120.917 = .4, and .4 added to 7.654 x 2 (twice chord of half arc) = 15.708 inches , length of arc. By Rule above, 15.708 X ” = 78.54 = ®* arc multiplied by half length of radius, — area of sector. . . . 10 — 2.929 — 7. 071 = versed sine subtracted from a radius , which is height oftn - angle aoc, and 7.071 X 50 = area of triangle. Consequently, 78. 54 — 50 = 28. 54. mensuration of areas, lines, and surfaces. 347 When the Chords of Arc, and of half of Arc, and Versed Sine are given. Rule.— To chord of whole arc add chord of half arc and one third of it more ; multiply this sum by versed sine, and this product, multiplied by .404 26, will give area nearly. Or, c _j_ c' — u. sin. X • 404 26 = area nearly. Example.— Chord of a segment, a c, Fig. 22, is 28 feet; chord of half arc, a b, is 15; and versed sine, b r, 6; what is area of segment? 2 0 _p l5 — = chord of arc added to chord of half arc and one third of it more. 4 g x 6 = 288=: product of above sum and versed sine. Hence 288 X . 4°4 26 = 1 l6 - 427 square feet. When the Chord of A rc and Versed Sine only are given. Rule.— A scer- tain chord of half arc, and proceed as before. r_p 0 Compute Cliorcl and. Ileiglit of a Segment of a Circle. When A rea is given. Rule.— D ivide area by square of diameter of circle, take tab. height for area from table of Areas of Segments of a Circle, p. 267, multiply it by diameter, and product will give required height. From diameter subtract height, multiply remainder by height, take square root of product and multiply it by 2 for required chord. Or — = (tab. area for height) X d = 4 , and Vd — h X h X 2 = c. ’ d 2 Circular Measure. (See Rule, page 1 13.) Sphere. Definition. — A figure, surface of which is at a uniform distance from centre. To Compute Convex Surface of a Spliere.— !Kig. S 3 . Rule.— M ultiply diameter by circumference, and prod- uct will give surface. Or, 4 p r 2 = surface. * Or, pd 2 = surface. • Example.— W hat is convex surface of a sphere, Fig. 23, hav- ing a diameter, a b, of 10 ins? 10 X 31-416 = 314.16 square ins. Segment of a Spliere. Definition.— A section of a sphere. To Compute Surface of a Segment of a Spliere.— If ig. 24 . Rule. — M ultiply height by the circumference of sphere, and add product to the area of base. Or, 2 prh — convex surface alone. Example. — Height, bo, of a segment, a b c , Fig. 24, is 36 ins., and diameter, b e , of sphere 100 ; what is convex surface, and what whole surface? 36 X 100 X 3- 1416 mu 309. 76 = height of segment multiplied by circumference of sphere. To ascertain area of base ; diameter and versed sine being given, diameter of base of segment, being equal to chord of arc, is, by Rule, page 347, 100 — 36 X 2 = 28 ; V100 2 — 28 s = 96- 96 s X • 7854 = 7238. 2464 = convex surface , and 7238. 2464 -f 1 1 309. 76 = 18 548. 0064 = convex surface added to area of base — square ins. Note. — When convex surface of a figure alone is required, area or areas of base or ends must be omitted. * j> or k represents in this, and in all cases where it is used, ratio of circumference of a circle to its diameter, or 3.1416. 348 MENSURATION OF AREAS, LINES, AND SURFACES. When the Diameter of Base of Segment and Height of it are alone given. Rule. — Add square of half diameter of base to the square of height ; divide this sum by height, and result will give diameter of sphere. 2 Or, d -T- 2 -f h 2 - 4 - h = diameter. Spherical Zone (or Frustum oF a Sphere). Definition.— The part of a sphere included between twa parallel chords. To Compute Surface of a Spherical Zone.— Fig. 25. Fig. 25. Rule. — Multiply height by the circumference of sphere, and add product to area of the two ends. Or, h c -f- a -\- a' = surface. Or, 2 p r h = convex surface alone. Example. — Diameter of a sphere, a b, Fig. 25, from which a zone, c g , is cut, is 25 inches, and height, eg, is 8 ; what is convex surface ? 25 X 3.1416 X 8 = 628.32 = height X circumference of sphere = square ins. When the Diameter of Sphere is not given. Rule. — M ultiply mean length of the two chords by half their difference ; divide this product by breadth of zone, and to quotient add breadth. To square of this sum add square of lesser chord, and square root of their sum will give diameter of sphere. Or, = — d. Spheroids or Ellipsoids. Definition. — Figures generated by the revolution of a semi-ellipse about one of its diameters. When revolution is about Transverse diameter they are Prolate, and when it is about Conjugate they arc Oblate. Fig. 26 To Compute Surface of a Spheroid..— Fig. 26. When Spheroid is Prolate. Rule. — Square diameters, and multiply square root of half their sum by 3.1416, and this product by conjugate diameter. Or, X 31416 Xd — surface, d and d ' represent- ing conjugate and transverse diameters. Example. — A prolate spheroid, Fig. 26, has diameters, cd and a b, of 10 and 14 inches; what is its surface? io 2 -j- 142 — 296 = sum of squares of diameters. 296-4-2 = 148. and Vi 48 = 12.1655 = square root of half sum of squares of diameters. 12.1655 X 3.1416 X 10 = 382.191 ins. = product of root above obtained X 3- I 4 I 6, : and by conjugate diameter. When Spheroid is Oblate. Rule.— S quare diameters, and multiply square j root of half their sum by 3.1416, and this product by transverse diameter. ) /d 2 A-d ' 2 1 Or, / — ~ — X 3. 1416 Xd' = surface. Example.— An oblate spheroid has diameters of 14 and 10 inches; what is its surface? , I2 2 _ j_ io 2 = 296 = su?n of squares of diameters. 296 -4- 2 = 148, and Vm 8 = 12. 1655 = square root of half sum of squares of di- ameter. 12.1655 X 3.1416 X 14 = 535-0679 ins.= product of root above obtained X 3 - j 4 i6 » and by transverse diameter. MENSURATION OF AREAS, LINES, AND SURFACES. 349 To Compute Convex Surface of a Segment of* a Sphe- roid.— Figs. ST and. S8. Rule— Square diameters, and take square root of half tlieir sum; then, as diameter from which the segment is cut is to this root, so is the height of segment to proportionate height required. Multiply product of other di- ameter and 3.1416 by proportionate height of segment, and this last product will give surface. Or, Vd 2 -M' 2 -^- 2 XhXd' or d X 3. 1416 — surface. d or d ' Example. — Height, a o, of a seg- ment, efoi a prolate spheroid; Fig. 27, is 4 inches, diameters being 10 and Fig. 28. what is convex surface of it ? . m\: — ~i& Square root of half sum of squares of diameters, 12.1655. Then- 14: 12.1655 1 : 4 : 3.4758 — height cl of segment, proportionate to mean of diameters , and 10 X 3- 14^ X 3-4758 = 109. 1957 ins. 2 Height co of a segment of an oblate spheroid, Fig. 28, is 4 inches, the diam- eters being 14 and 10; what is convex surface of it? 214.0272 square ms. To Compute Convex Surface of a, Frustum or Zone of a Spheroid. — Figs. 29 and 30 . Rule.— Proceed as by previous rule for surface of a segment, and obtain proportionate height of*' frustum ; then multiply product of diameter par- allel to base of frustum and 3.1416 by proportionate height of frustum, and it will give surface. Fig. 2Q. Example.— Middle frustum, 0 e, of Fig. 30. c a prolate spheroid, Fig. 29. is 6 inch- * es, diameters of spheroid being 10 and 14; what is its convex surface? Mean diameter, as per preceding example, is 12.1655. Diameter parallel to base of frus- tum is 10. ~d~ Then 14 : 12.1655 :: 6 : 5.2138, and 10 X 3.1416 X 5-2138 = 163.7967 square ins. 2.— Middle frustum of an oblate spheroid, as 0 e , Fig. 30, is 2 inches in height, diameters of spheroid, as in preceding examples, being 10 and 14; what is its con- vex surface ? io 7- 01 3 6 square ins. CircmlaT? Zone. Definition.— A part of a circle included between two parallel chords. To Compute Area of a Circular Zone. Rule. — F rom area of circle subtract areas of segments. Or, see Table of Areas of Zones, page 269. When Diameter of Circle is not given— Multiply* mean length of the two chords by half their difference ; divide this product by breadth of zone, and to quotient add the breadth. To square of this sum add square of lesser chord, and square root of their sum will give diameter of circle. Example.— Greater chord, h g, is 90 inches; lesser, a c, is 80; and breadth of zone. a 0, is 72.526; what is its diameter? 80 + 9 ° x — - = 85 X 5 = 425, and 425 ---f 72.526 = 78.385. 72.520 Then V 78.38s 2 -+- 80 2 —.f 12 544.2 == 112 = diameter. Gg 350 MENSURATION OF AREAS, LINES, AND SURFACES. Fig. 31. Cylinder. Definition.— A figure formed by revolution of a right-angled parallelogram around one of its sides. To Compute Surface of a Cylinder.— Fig. 31 . Rule. — Multiply length by circumference,, find acid product to area of the two ends. Or, l c -j- 2 a = s, a representing area of end. Note. — When internal or convex surface alone is wanted, areas of ends are omitted. Example. — Diameter of a cylinder, ho, Fig. 31, is 30 inches, and its length, a b , 50; what is its surface? 30 X 3.1416 = 94.248, and 94.248 X 50 = 4712.4. Then 30 2 X .7854 = 706.86 = area of one end; 706.86 X 2 = 1413.72 = area of both ends , and 4712.4-}- 1413.72 =6125.12 square ins. Prisms. Definition. — Figures, sides of which are parallelograms, and ends equal and parallel. Note.-— When ends are triangles, they are termed triangular prisms ; when they are square, square or right prisms ; and when they are a pentagon, pentagonal prisms , etc. To Compute Surface of* a iRiglut Frism.— Figs. 32 and 33. Fig. 32. Fig. 34- Rule. — Ascertain areas of ends and sides, and Fig. 33. add them together. ® Or, 2 a-j-na' =: s, a representing area of ends , a' area of sides, and n their number. Example. — Side, a b, Fig. 32, of a square prism is 12 inches, and length, b c, 30; what is surface? 12X12 = 144 = area of one end ; 144 X 2 = 288 = area of both ends ; 12X30 = 30 q = a rea of one side ; 360 X 4 = 1440 = area of four sides , and 288 -f- 1440 = 1728 sq. ins. To Compute Surface of an ODliqne or Irregular Prism.— Fig. 34. Rule. — Multiply circumference of one end, by perpendic- % a ular height, a 0. Or, multiply circumference, c, at a right angle to sides by actual length of figure, and add area of ends. Example. — Sides, q c, of an oblique hexagonal prism, Fig. 34, are 10 inches, and perpendicular height, a 0, is 5 feet; what is its sur- face ? To X 6 = 60 ins. — length of sides. 60 X 5 X 12 ±= 3600 square ins. — area of sides, and by table, page 342, 102 X 2.59808 X 2 = 519.616 square ins., which added to 3600 = 4x19.616 square ins. Wedge. is a prolate triangular prism, aiid its surface is computed by rule for that of a right prism. To Compute Surface of* a AV edge.— Fig. 35. Definition. — A we Fig. 35- Example. — Back of a wedge, abed, Fig. 35, is 20 by 2 inches, and its end, ef, 20 by 2; what is its surface ? 20 2 -f- 2 -= 1 = 401 = sum of squares of half base , af and height, ef, of triangle, efa. V401 = 20.025 = square root of above sum — length of e a. Then 20.025 X 20X2 = 801 = area of sides. And 20X2 = 40 = area of back ; and 20X2 = 2X2 = 40 = area of ends. Hence 801 -}- 40 -f- 40 = 881 square ins. MENSU iration of areas, lines, and subfaces. 351 Prismoids. Definition— Figures alike to a prism, having only one pair of sides parallel. To Compete Surface of a Prismoid.-Fig. 30. a 1 • -C Art n n n OllfiQ Q Fig. 36. Fig. 38. Fig- 37 - iT>u.te ^ ^ — ~ „ Rule - Ascertain area of sides and ends as by rules for squares, triangles, etc., and add them together. Example. -Ends of a prismoid, efg h and a ^^U^ts’s^rfa^^ 8 inches square, and its slant height, d h, 25, "“at is its suriace . jo X 10 = 100 = area of base ; 8 X 8 = 64 — area of top. i° + 8 x 25 = 225, and 225X4 = 9 ®- area °f sides - „ Then 100 + 64 4" 900 = 1064 = square ins. To compute Surface of aa Otolictue or Irregular Prismoid. Proceed undirected for an Oblique or Irregular Prism, page 350. TJrigi^las. tbe base> to compute Curved^ oPau ^ula.-Figs. 3 ., When Section is parallel to Axis of the Cylinder , Fig. 37 - *»« *• Mul ‘ tinlv len°th of arc of one end by height. 1 ‘ ° Example. Diameter of a cylinder, a c, from which an inmilV Eicr -27 is cut is io inches, its length, b d, 50, and ver S sed%fnf o 3 r ? depth of ungula is 5 inches; what Is curved surface ? - ^ rad{u$ 0 f C y Under . Hence radius and versed sine are, equal; the arc, there- fore of ungula is one half circumference of the cylinder, L whtcA is 31.416 15-708, and 15.708X50 = 785-4 " square ins. When Section passes obliquely through inder Fig. 38. Rule 2,-Multiply circumference of base of cylinder by half sum of greatest and 1 ^mdrica"^ 2 L Fig. 38, is 10 inches, and great- 35 and 15 inches; what is its --ed surface? 10 diameter =31. 416 circumference; 25 + 15 = 4 °) an< * 40—2 = 20. Hence 3M .6 X 20 = 628.32 square ijis. When See, in, pM *»«. If Sit WK^i&T4SWS5 s, 3 i«ii .w -i ». «i I--, i,/. cn cle, Fig. 39. ,.^ t q from this product subtract product of aic and“ r H e of’ fi;S?di« thus found by quotient of height, y c, the lpngest chord that can he drawn in basa what is curyed surface? ■ 7 . • . 5X10 = 5° = sine of half arc by diameter. , 5^08, ^d'^ver^tTsine ami radiu^are equalj cosine is^' SO x I^Fs = 5° X 2 = 100 square mS ' - * When the cosine is o, this product is o. 352 MENSURATION OF AREAS, LINES, AND SURFACES. ceeds Fig. 40. When Section passes through Base of Cylinder , and Versed Sine , a g, ex - ids Sine , '> ai - , ’ Or,lxc = mrfm. To Compute Length of Axis and Circumference. When Rina is Elongated. Rule.-To less diameter add the diameter of the body of 'the link, and multiply sum by 3.1416 ; subtract less diameter from greater, multiply remainder by 2, and sum of these products is length Fig. 45 - of axis. L liAlS. Example. -Link of a chain, Fig. 45. is 1 inch m diameter of body, a 6 , and its inner diameters, b c and ef are 12.5 UI UUUV , LO t/, wau. n Cl and 2. 5 inches ; what is its circumference ? 2.5-j-x x 3.1416 — 10. 9956 = length of axis of ends. J 2 . 3 __ 0. 5, X 2 = 20 = length of sides of body. Then 10.9956 + 20 = 30.9956 = length of axis of link , and — 30.9956 X 3- 1416, (cir. of 1 inch) = 97. 375 s square ins. When Ring is Elliptical, Fig. 46- Rule.— Square diameters of axes of ring, multiply square root of half their sum by 3.1416, and product is length of axis. _ Cones. Definition. - A figure described by revolution of a right-angled triangle about one of its legs. For Sections of a Cone, see Conic Sections, page 379. To Compute Snrface of a Cone.-Fig. . Rule —Multiply perimeter or circumference of base by slant height, or side of cone ; divide product by 2, and add the quotient to area of the base. 0r c X A +2 + d' == surface , c representing perimeter. FiS ' 47 ' A Example. -Diameter, a ft, Fig. 47, of base of a cone is 3 feet, ^ and slant height, a c, 15 ; what is surface of cone . Circum. of 3 feet = 9. 4248, and 9 - 4 2 4 8 * Jl = JQ . 68 6 - sur- ^ face of side; area of base 3=7.068, and 70.686+7.068-77-754 square feet. ft a* 354 MENSURATION OF AREAS, LINES, AND SURFACES. To Compute Surface of the Frustum of* a Cone.— Fig. 48. Rule. — M ultiply sum of perimeters of two ends by slant height of frus- tum ; divide product by 2, and add it to areas of two ends. - c -b c' X h Or, f -a-j-a = surface. 2 Example.— Frustum, abed , Fig. 48, has a slant height, cd, of 26 inches, and « circumferences of its ends are 15.71 and 22 inches respectively; g ' 4 ' A what is its surface? 15.71 + 22 X 26 - = 490. 23 = surface of sides ; X .7854 d + / 22 \3M 7) X .7854 = 58.119 = areas of ends. Then 496.23 -f- 4 I t>/ 58. 1 19 =548. 349 square ins. Fig. 49. Pyramids. Definition. — A figure, base of which has three or more sides, and sides of which are plane triangles. To Compute Surface of a Pyramid. — Figs. 4r0 and. £30. Rule. — M ultiply perimeter of base by slant height; divide product by 2, and add it to area of base. _ ch . . Fig. 50. Or, (- a = surface. 2 Example.— Side of a quadrangular pyramid, a b, Fig.. 49, is 12 inches, and its slant height, a c, 40; what is its surface? 12 X 4 = 48 = perimeter of base. 48 * — =. 960 = 2 !' area of sides, and 12 X 12 -}- 960 = 1104 square ins. To Compute Surface of Frustum of a Pyramid.- Fig. SI. Rule. — M ultiply sum of perimeters of two ends by slant height; divide product by 2, and add it to areas of ends. ~ c.tf c' xh , . , Or, \- a -\- a — surface. 2 !• . , ... . Example. — Sides a b,cd, Fig. 5*, of frustum of a quadrangular pyramid are 10 and 9 inches, and its slant height, a c, 20; what is its surface? 10X4 = 40, and 9 x 4 = 36 ; 40 -f 36 = 76 = sum of perimeters. 76 X 20 = 1520, and I ^ 2 - = 760 = area of sides ; 10 X 10 = 100, 2 and 9X9 = 81. Then 100 -f- 81 -f- 760 = 941 = square ins. When Pyramid is Irregular sided or Oblique. Rule. — The surfaces of each of the sides and ends must be computed and added together. Helix (Screw). Definition..— A line generated by progressive rotation of a point around an axis and equidistant from its centre. To Compute Length, of a Helix.— Fig. £5Q. Rule. — T o square of circumference described by generating point, add square of distance advanced in one revolution, extract square root of their sum, and multiply it by number of revolutions of generating point. mensuration of areas, lines, and surfaces. 355 Or, V(P 2 + Z 2 ) n = length, n representing number of revolutions. Fyampif What is lepgth of a helical line, Fig. 52, running 3.5 times around a cylinder of 22 inches in circumference, and advancing 16 inches in each revolution? -big. 52 Fig. 53 22 2 _i_ j(52 — 740 = sum of squares of circumference and of distance advawed. * Then V 74 ° x 3 - 5 = 95 - 21 ins. To Compute Length of a Revolution of Thread of a Screw. Rule.— P roceed as above for length and omit number of revolutions. Spirals. Definition.- Lines generated by the progressive rotation of a point around a fixed axis. , . . . . A Plane Spiral is when the point rotates around a central point. \ Conical Spiral is when the point rotates around an axis at a progressing dis- tance from its centre, as around a cone. To Compute , Ijengtli of* a Plane Spiral Line.-Fig. SO. r ule —Add together greater and less diameters ; divide their sum by 2 ; mtiltinlv Quotient by 3.1416. and again by number of revolutions. oSh ef circun/erlnces are given, take their mean length, and multiply it by number of revolutions. . . : Q r _j_ (p —A— 2 X 3.1416 n — length of line; PXw = radius, and p r 2 V 1— pitch. P representing the pitch. Fxample — Less and greater diameters of a plane spiral Spring, as a b e d, Fig. 53, are 2 and 20 inches, and number of revolutions d 10; what ’is length of it? f ^4^20-4- 2 = n == sum of diameters - 4 - 2 ; n X 3. 1416 = 34.5576 and 34.5576 X 3 - i 4 i 6 - Then 34.5576 X 10= 345-576 inches. v OTE -Above rule is applicable to winding engines, see page 662, where s it is ^re- qnhed to ascertain length of a rope, its thickness, number of revoluttons, dtameter of drum, etc. To Compute Length of a Conical Spiral Line.-Fig. 54. — Add together greater and less diameters ; divide their sum by 2 ’ To^square'o? produc” 1 onhis circumference and number of revolutions of spiral, add square of height of its axis, and take square root of the sum. Fig. 54. Or, V(d + d'±-2 X 3.1416 ^ + h 2 ) == length of line. Example.— Greater and less diameters of a conical spiral, Fig. 54, are 20 and 2 inches; its height, cd, 10; and number of revolutions 10; what is length of it? 20 + 2 -f- 2 == 11 X 3.14*6 = 34-5576 = sum of diameters - 4 - 2, and X 3.1416; 35.5576 X io = 345-576- Then V 345. 576 2 10 2 == 345- 7 2 inches. Spindles. Definition. -Figures generated by revolution of a plane area, when the curve is revolved about a chord perpendicular to its axis, or about its double ordinate, and they are designated by the name of the arc or curve from which they are generated, as Circular, Elliptic, Parabolic, etc. * When the »piral is other than a line, measure diameters of it from middle of body composing it. 356 MENSURATION OF AREAS, LINES, AND SURFACES. Circular Spindle. To Compute Convex Surface of a Circular Spindle, Zone, or Segment of it.— L'igs. 55 , 50 , and 5 * 7 . Rule. — Multiply length by radius of revolving arc ; multiply this arc by central distance, or distance between centre of spindle and centre of revolv- ing arc ; subtract this product from former, double remain- der, and multiply it by 3.1416. Or, l r — ( a of arc , and c the spindle le chord. ) 2 p = surface , a representing length 14.142 X 10 = = 15.708. Example. — What is surface- of a circular spindle, Fig. 55, length of it,/c, being 14.142 inches, radius of its arc, oc, 10, and central distance, 0 e, 7.071 ? 141.42 = length x radius. Length of arc, fa c, by Rules, page 344 15-708 X 7.071 = 111.071:3 — length ofarcx central distance; 141.42 — 111.0713 = 30. 3487 = difference of products. Then 30. 3487 X 2 X 3. 1416 = 190. 687 square ins. Fig> 56. Zone. Example.— What is convex surface of zone of a circular e spindle, Fig. 56, length of it, i c, being 7.653 inches, radius of its arc, o g, 10, central distance, o e, 7.071, and length of its side or arc, d b, 7.854 inches? \ | / 7.653X10=76.53 — length X radius ; 7.854X 7.071=55.5356 V /' = length of arc x central distance ; 76. 53 — 55. 5356 = 20. 9944 o — difference of products. Then 20.9944 X 2 X 3- 1416 = 131.912 square ins. dL— — ”"“**> 1 ^ Segment. Example. —What is convex surface of a segment of a cir- cular spindle, Fig. 57, length of it, ic, being 3.2495 inches, \. j / radius of its arc, 0 g, 10, central distance, 0 e, 7.071, and length of its side, id, 3.927 inches? 0 3.2495 X 10 = 32.495 — lengthx radius ; 3.927 X 7-071 = 27.7678 = length of arc X central distance ; 32. 495 — > 2 7- 7678 = 4. 7272 = difference of products. Then 4.7272 X 2 X 3.1416 = 29.702 square ins. General Formula. — S = 2 (lr — ac)p = surface , l representing length of spindle, segment , or zone, a length of its revolving arc, r radius of generating circle , and c ceiitral distance. Illustration. — Length of a circular spindle is 14.142 inches, length of its revolv- ing arc is 15.708, radius of its generating circle is 10, and distance of its centre from centre of the circle from which it is generated is 7.071 ; what is its surface? 2 X (14.142 X icr — 15.708 X 7-071) X 3.1416 = 190.687 square inches. Note. — Surface of a frustum of a spindle may be obtained by division of the surface of a zone. Cycloidal Spindle. To Compute Convex Surface of* a Cycloidal Spindle.— IX g. 58. Rule. — Multiply area of generating circle by 64, and divide it by 3. Or, - = surface. Example.— Area of generating circle, a be, of a cycloidal spindle, de, is 32 inches; what is surface of spindle? 32 X 64 = 2048 = area of circle x 64 , and 2048 ~ 3 = 682. 667 square ins. Note.— Area of groatest or centre section of a oycloidal spindle is twice area of the cycloid. MENSURATION OF AREAS, LINES, AND SURFACES. 357 Ellipsoid, IParaDoloid, or Hyperboloid of Bev- olntion. Definition. Figures alike to a cone, generated by revolution of a conic section around its axis. Note.— T hese figures are usually known as Conoids. When they are generated by revolution of an ellipse, they are termed Ellipsoids, and when by a parabola, Paraboloids, etc. Revolution of an arc of a conic section around the axis of the curve will gne a segment of a conoid. Ellipsoid. To Compute Convex Surface of an Ellipsoid.— Fig. GO. Pm, — Add together square of base and four times square of height; multiply square root of half their sum by 3.1416, *" d ^ P roduct radlus of the base. Fig- 59- Or, 'b 2 -\- 4/^ 3. 141 6r — surface. Example.— Base, a b, of an ellipsoid, Fig. 59, is 10 inches, and vertical height, cd, 7 ; what is its surface? to 2_l 7 2 x . — 2 n6 = sum of square of base and 4 times square of height; 296 %■ 2 2 *48, and = ^SS square root of half above sum. Then 12.1655 X 3- *4*6 X ™ square ms. To Compute Convex Snrfaoe of a Segment, Frustum, or Zone of an Ellipsoid.— Enr. 59. See Holes for Convex Surface of a Segment, Frustum, or Zone of a Spheroid or Ellipsoid, pages 348-9. d or d' X 3- 1416 Y.K — surface , and mean. diam. X h li— h ; then d X 3- 4^ X h = surface. d or d' 3?araL>oloid To Compute Convex Surface of* a Paraboloid.-Fi . 60. Rule.— From cube of square root of sum of four times square of height, and square of radius of base, subtract cube of radius of base ; multiply re- mainder by quotient of 3.1416 times radius of base divided by six times square of height. Fig. 60. Or, (4/4/1* + r*)3 - r3 x = surface. Example.— Axis, 6 d, of a paraboloid, Fig. 60, is 40 inches; ra- dius, a d , of its base is 18 inches; what is its convex surface . 4 o 2 X 4 = 6400 = 4 times square of height.; 6400+ i8 2 _ 6724 — 1 sum of above product and square of radius of base ; (V6724 ) 3 — 18 1 y 6 _ ren minder of cube of radius of base subtracted from cube w of square root of preceding sum ; 3. 1416 X (6 X 40 ) _ .005 890 5 — quotient of 3.1416 times radius of basest) times square of height. Then 545 536 X -005 890 5 = 3213.48 square ins. Fig. 61. Cylinder Sections. To Compute Surface of* a Cylinder Section. — Eig. 61. Rule. — From entire surface of cylinder a o subtract surface of the two lingulas; r o, 0 c, as per rule, page 351, and multiply result by 4. 358 MENSURATION OP AREAS, LINES, AND SURFACES. Any Riguxe of Revolution. To Ascertain Convex Surface of any Figure of Revolu- tion.— Figs. 62 , 63 , and. 64 . Rule.— M ultiply length of generating line by circumference described by its centre of gravity. Or, l 2 r p = surface, r representing radius of centre of gravity. Example i.— If generating line, a c, of cylinder, a cdf io inches in diameter, Fig. 62, is 10, then centre of gravity of it will be in b radius of which is b r = 5. ’ Hence 10 X 5 X 2 X 3.1416 = 314.16 ins. Again, if generating line is eacg , and it; is (ea=z 5, a c= 10 and c g — 5) — 20, then centre of gravity, 0, will be in middle of line joining centres of gravity of triangles € a c and ac q- from r. Fig. 62. r 3-75 Hence 20 X 3-75 X 2 X 3.1416 = 471.24 square ins.— entire surface. Verification, j ^®“ vex f sur [ ac ® as aboVD -- 31416 l Area of each end, io 2 X . 7854 X 2 = i 57 . 08 Fig. 63. 471.24 inches. 2.— if generating elements of a cone. Fig. 63, are a d = io dc — 10, and a c, generating line, = 14. 142, centre of gravity of which is in 0, and or ==5, Then 14.142 X 5 X 2 X 3.1416=444.285, con - ^ 4 - vex surface , and 10 X 2 x .7854 = 314-16, area of base. Hence 444.285 '-J- 314. 16 = 758.445, entire surface. 3-— generating elements of a sphere, Fig. 64, are a c = 10, a b c will be 15.708, centre of gravity of which is in 0, and by Rule, page 606,0 r = 3.183. 516 Hence 15.708 x 3-183X2 x 3-1416 = 314. 16 square ins. Capillary Tribe. To Compute Diameter of a Capillary Tube. Rule.— W eigh tube when empty, and again when tilled with mercury; subtract one weight from the other ; reduce difference to grains, and divide it by length of tube in inches. Extract square root of this quotient, multi- ply it by .0192245, and product will give diameter of tube in inches. I iv 0r i J jX .019 224 5 = diameter, w representing difference in weights in grains and l length of tube. Example.— D ifference in weights of a capillary tube when empty and when filled with mercury is 90 grains, and length of tube is 10 inches; what is diameter of it? 90 = 10 = 9 = weight of mercury - 4 - length of tube ; V9 and 3 X .019 224 5 = •057 673 5 —square root of above quotient X .019 224 5 inches = diameter of tube. Proof.— W eight of a cube inch of mercury is 3442.75 grains, and diameter of a circular inch of equal area to a square inch is 1.128 (page 342). • *£’ th ? n > 344 2 -75 grains occupy 1 cube inch, 90 grains will require .026 141 9 cube inch, which, -f- 10 for height of tube ^ .002 614 19 inch for area of section of tube. Then .002 614 19 = .051 129 = side qf square of a column of mercury of this area Hence .051 129 X 1.128 (which is ratio between side of a square and diameter of a circle of equal area) = .057 673 5 ins. , To ^Ascertain. Area of an. Irregular Figure. Rule.— T ake a uniform piece, of board or pasteboard, weigh it, cut out figure of which area is required, and w^eigh it; then, as weight of board or pasteboard is to entire surface, so is weight of figure as cut out to its surface. Or, see rule page 341, or Simpson’s rule, page 342. MENSURATION OF AREAS, LINES, SURFACES, ETC. 359 To Ascertain. Area of* any- Plane Figure. Rule. — Divide surfaces into squares, triangles, prisms, etc. ; ascertain their areas and add them together. Reduction of* an Ascending or Descendinj izontal Measurement. In Link and Foot. ; Line to Hor- Degrees. Link. Foot. Degrees. Link; 13 14 15 16 .000099 .00015 7 .004917 .00745 .000403 .00061 8 .006421 .00973 .000904 .00137 9 .008125’ .01231 .00161 .00244 10 .010025 .01519 .002515 .00381 11 .012124 .01837 .003617 .00548 12 .014421 .02285 Illustration i.— In an ascending grade of 14 0 , what is reduction in 500 feet? 14° = 500 X .0297 = 14.85 feet — 14 feet 10.2 ins . 2. — What is reduction in 500 links? 140 — 500 X -019 602 = 9.801 feetkkqfeet 9.6 ins . Foot. Degrees. W 18 Link. .016915 .019602 .022 486 .025 569 .028 925 .0323 Foot. .025 63 .029 7 .03407 .03874 •043 7 .048 94 Redaction ofGrrade ofan Ascending or Descending Dine to Degrees. Per 100 Links , Feet x etc. Grade. Degrees. Grade. Degrees. Grade. Degrees. j Grade. Degrees. •25 8 35-2 i-75’ 1 6 10.3 4-5 2 34 45-5' 10 5 44 20. 7 •5 *7 10.3 2 1 8 45-5 5 2 51 5 7- 6 11 6 18 55.8 •75 25 47-6 2-5 1 25 57- 6 6, 3 26 22.7 12 6 53 3i 1 34 22.7 3 1 43 8,3 7 4 0 49.6 13 7 28 10.3 1.25 42 57-9 3-5 2 0 20.7 8 4 35 18.6 14 8 251.7 i-5 5i 35-2 4 2 W 33- 1 9 5 9 49- 6 1 15 8 37 37-2 To Plot Angles witlioat a Protractor. On a given line prick off 100 with any convenient scale, and from the point so pricked off lay off at right angle with the same scale the natural tangent due to the angle, (see table of Natural Tangents and Sines); or strike out a portion of a circle with radius 100 and lay off a chord = 2 sin. of half the angle required. - " . v. ' ■ ■ To Compute Chord. of an Angle. Double sine of half angle. Illustration. — What is the chord of 21 0 30'? Sine of 21 30 = io° 45', and sine of io° 45' == . 186 52, which, X 2 = .373 04 chord . 2 To Ascertain Value of a Dower of a Qnantity. Rule. — Multiply logarithm of quantity by fractional exponent, and prod- uct is logarithm of required number. Example.— W hat is the value of 16^ ? % X log. ‘it = \ x 1. 204 12 = . 903 09. Number for which = 8. 360 MENSURATION OF VOLUMES. MENSURATION OF VOLUMES. Cubes and. 3?arallelopipedons. Cube. Definition. — A volume contained by six equal square sides. Fig. 1. To Compute “V olume of a Cube.— Fig. 1. Rule. — M ultiply a side of cube by itself, and that product again by a side. Or, s 3 — V, s representing length of a side , and V volume. Example.— Side, a 6 , Fig. r, is 12 inches; what 13 volume of it? 12X12X12 = 1728 cube ins. Farallelopipedon. Definition.— A volume contained by six quadrilateral sides, every opposite two of which are equal and parallel. To Compute Volume of a IParallelopipedon. —Fig. 2. Rule. — M ultiply length by breadth, and that product again by depth. Or, l bd = V. Prisms, Prismoids, and Wedges. Prisms. Definition. — Volumes, ends of which are equal, similar, and parallel planes, and sides of which are parallelograms. Note. — When ends of a prism or prismoid are triangles, it is termed a triangular prism or prismoid; when rhomboids, a rhomboidal prism , and when squares, a square prism , etc. Fig. 3 - To Compute Volume of a Prism.— IVigs. 3 and 4. Rule. — M ultiply area of*base by height. Or, a h = V. Example.— A triangular prism, a b c, Fig. 4, has sides of 2.5 feet, and a length, c b, of 10; what is its volume? Fig. 4. a i4v By Rule, page 339, 2.5 s X -433 = 2.70625 = end a b , and 2.706 25 X 10 = 27.0625 cube feet. - area of Fig. 6. When a Prism is Oblique or hn'egular. Rule. — Multiply area of an end by height, as ao\ or, multiply area taken at a right angle' to sides, as at c, by actual length. To Compute Volume of any Frustum of a Prism, whether Ltiglit or OUliqne. — IVigs. 0 and 7. Rule. — M ultiply area of base by perpendicular distances between it and centre of gravity of upper or other end. Or, area at right angle to side as at e by actual length. Example. — A rea of base, a o, of frustum of a rectan- gular or cylindrical prism, Fig. 6, is 15 inches, and height to centre of gravity, c, is 12; what is its volume? 10X 12 = 1 20 cube ins. MENSURATION OP VOLUMES. 36l Prismoids.* To Compute Volume of a, I^ismoid.— Fig. 8 . Rule.— T o sum of areas of the two ends add four times area of middle section, parallel to them, and multiply this sum bv one sixth of perpendicu- lar height. Note.— T his is the general rule, and known as the Prismoidal Formula and it applies equally to all figures of proportionate or dissimilar ends. F i g 8 Or, a + a' + 4 m X h = 6 = V, a and a' representing areas of ends, - and m area of middle section. 1 Example. — What is volume of a rectangular prismoid Fig 8 \d lengths and breadths, eg and g h, a b and b d, of two ends being 1 7X6 and 3X2 inches, and height 15 feet ? 7X6 + 3X2 = 42 + 6 = 48 = sum of areas of two ends ; 7 + 3 — 2 = 5 = length of middle section ; 6 + 2 = 2 = 4 = breadth of middle section ; 5X4X4=80 =four times area of middle section. Then 48 + 80 X ^ — == 128 X 30 = 3840 cube ins. Note 1. — Length and breadth of middle section are respectively eaual to half sum of lengths and breadths of the two ends. 2.— Prismoids, alike to prisms, derive their designation from figure of their ends as triangular, square, rectangular, pentagonal, etc. ’ When it is Irregular or Oblique and their ends are united by plane or curved surfaces , through which and every point of them , a right line may be drawn from one of the ends or parallel faces to the other.— Figs 9 10 and n Fig. 9. Fig. 11. = 120 = sum of areas of ends + 4 times middle section. And „ h^It:f~ AreaS ?- f cnds > ° c a 'l d 0 r s ' F 'S- IQ . « & c d, and i m n u, Fig. u and ab ce and vxi 0 z, l ig. 9, are each 10 and 30 inches, that of their middle section 20, and their perpendicular heights 185 what is their volume? 10 + 30 + 20 x 4 18 120 X = 360 cube ins. W edge. To Compute Volume of a Wedge.— Fig. is. Rule.-To length of edge add twice length of back; multiply this sum of p?oduct dlCUlar helgh ’ and then by breadth 0f back ’ a,Kl tak<5 one sixtl Or, (l + V X 2 X h b) = 6 = V. Example. —Length of edge of a wedge, eg, is 20 inches, back abed, is 20 b y 2, and its height, ef 20; what is its volume? 20 + 20 X 2 = 60 = length of edge added to twice length of oack ; 60 X 20 X 2 = 2400 = above sum multiplied by height , and that product by breadth of back. y ’ Then 2400 + 6 = 400 cube ins. Note. — When a wedge is a true prism, as represented by eaual to ATPa of an and l' * CU U J Fig. 12. Fig. volume of it is r.*. a prto"d“ ,i0D ° r embankme “ t «' * terminated by parallel cros. section., 1. a r sc tan- Hh 362 MENSURATION OF VOLUMES. To Compute F ms trim of a Wedge.-Fig. 13. Rule. — To sum of areas of both ends, add 4 times area of section parallel to and equally distant from both ends, and multiply sum by one sixth of length. Or, A 4-^ + 4 a ^ 5 " *"**^’ Eva-mpt v Lengths of edge and back of a frustum of a wedge a^iYcd\rl 2 T and 1 X a ms., and height or is 2 o ms. ; , d w hat is its volume ? 2 + 4 X (20 X 4 r) X ^ = 60+ >20 X “ = 600 cube ins. Note -When frustum is a true prism, as represented Fig. 13, volume of it is equal to mean area of ends multiplied by its length. Regular Bodies (Polyhedrons). Definition —A regular body is a solid contained under a certain number of simi- lar^and equal plane faces,* all of which are equal regular polygons. Note 1 -Whole number of regular bodies which can possibly be formed is five. * A sn here may always be inscribed within, and may always be circumscribe about a regular body or polyhedron, which will have a common centre. Fig. >4. Fig.; 5 . Fig. >6. Fjg. 17. > To Compute 41 V. To Compute Radius of a Sphere that will Circumscribe a given Regular 1 Body, or that may be Inscribed within it. ’mol.-L,.... or . h.„. .droo U I* '. » • - radii of circumscribing and inscribed spheres . inch — 2 X .86602 = 1.73204 inches — radius of circumscribing sphere, 2 X .5-1 >«- radius of inscribed sphere. element required. When Volume is given. opposite to body in columns E and F in following iaDie, miner uea ment required. ; — * Angle of adjacent faces of a polygon is termed diedral angle. MENSURATION OF VOLUMES. 363 When one of the Radii of Circumscribing or Inscribed Sphere alone is re- quired, the other being given. Rule. — Multiply given radius by multiplier opposite to body in columns G and H in Table, page 364, under head of other radius. To Compute Linear Edge. When Radius of Circumscribing or Inscribed Sphere is given. Rule. — Multiply radius given by multiplier opposite to body in columns I and Iv in Table, page 364. When Surface is given. Rule. — Multiply square root of it by multiplier opposite to body in column L in Table, page 364. When Volume is given. Rule. — Multiply cube root of it by multiplier opposite to body in column M in Table, page 364. To Compute Surface. When Radius of Circumscribing Sphere is given. Rule. — Multiply square of radius by multiplier opposite to body in column N in Table, page 364. When Radius of Inscribed Sphere is given. Rule. — Multiply square of radius by multiplier opposite to body in column O in Table, page 364. When Linear Edge is given. Rule. — Multiply square of edge by multi- plier opposite to body in column P in Table, page 364. When Volume is given. Rule. — Extract cube root of volume, and multi- ply square of root by multiplier opposite to body in column Q in Table, page 364. To Compute 'Volnine. When Linear Edge is given. Rule. — Cube linear edge, and multiply it by multiplier opposite to body in column R in Table, page 364. When Radius of Circumscribing Sphere is given. Rule. — Multiply cube of radius given by multiplier opposite to body in column S in Table, page 364. When Radius of Inscribed Sphere is given. Rule. — Multiply cube of radius given by multiplier opposite to body in column T in Table, page 364. When Surface is given. Rule. — Cube surface given, extract square root, and multiply the root by multiplier opposite to body in column U in Table, page 364. Fig. 18. Cylinder. To Compute Volume of a Solid. Cylinder.— Fi g. 18 . Rule. — M ultiply area of base by height. Example.— D iameter of a cylinder, be, is 3 feet, and its length, a b, 7 feet; what is its volume? Area of 3 feet = 7.068. Then 7.068 x 7 = 49. 176 cube feet. To Compute "Volnme of a Hollow Cylinder. Rule. — S ubtract volume of internal cylinder from that of cylinder. Fig. 19. Cone. To Compute Volume of a Cone.— Fig. 19 . Rule. — Multiply area of base by perpendicular height, and take one third of product. Example. — Diameter, a b , of base of a cone is 15 inches, and height, c e, 32.5 inches; what is its volume? Area of 15 inches = 176.7146. Then U 6 -7 1 5X32. _ 5 ; _ : 1914. 41 25 cube ins. TJnits for Elements of tlie Regular Bodies. 364 MENSURATION OF VOLUMES. •aaaqds paquos -ui jo snipnn ^3 •aSpa anauiq •aaaqds Suiquasuuna -jiO jo snipT?H ^3 •aSpa atauiq •aaaqds . paquasui ^3 W -aaaqds Suiquos -mnoaia jo snipx?a QnnO t>. OnOO co -t On N tOO fO 4 N N O' w -t-vO ONt^NNO^i- co 10 h m 10 IO IO H M o o -t t N (N CO CO co co 10 10 t^. c^. fj CJ co h M w M •aonjans ^3 aiuiqoA t^o HOO w CO co M IO S’? •aaaqds poquos ^ -ui jo snipua ^3 auirqoA •aaaqds . gaiquosumoaia -£3 O ’ -aaaqds paqiaosuijosnipna •ararqoA ^3 -aaaqds paquasui josmptn NO « 1000 C" (N NO 00 rj- IO IO IO IO NO •aunqoA -^3 rA -aaaqds Suiquos -umoaio jo snipua •aotjans ^3 rj -aaaqds paqiaosuijo snipta ‘§.$'8 S d 00 On r aaaqds . Suiquosumo m -JiO' jo snip^H ^3 amn[OA m On Ng rf N N O COCO nO IO iO CO 10 NO fO M « CO On co iO nO m co COCO CO iO vo CO NO ' H H ci N •a§pa anauiq £& ararqoA rf m r>. W co M C^nO 00 tTnO m 1 r* ci auirqoA ^3 •aonjans 10 ca i> cp m t>NO to »o to N lONH h M CO O 00 iO rf On iOnO 10 O H -t to H « N N N •aotjans ^3 q -aaaqds Suiquos -umoaio jo sniptn •aSpa a-eouiq ^3 pq -aaaqds paquosuijosnipta IO M On 00 CO to ON CO fl IO CO On 00 CO vO tO t' O N rf CO CO CO CO •eSp 3 a^auia ^3 •aonjaii's tJ-nO nO H NO CO 6 06 aaaqds paquos Q -ui jo snipx?a aoijans w NO 00 M t o 2 00 IONO C^nO m M xt 6 NO to m CO 00 O N o n to ci M M rt- to •aonjans ^3 to On CO N N O •339 8 MENSURATION OF VOLUMES. 365 To Compute Volume of Ifrustiixn of a, Cone.-Fig. SO. Rule.— Add together squares of the diameters or circumferences of greater and lesser ends and product of the two diameters or circumferences ; mul- tiply their sum respectively by .7854 and .07958, and this product by height ; then divide this last product by 3. Or, d 2 -j-d' 2 -j- d x d ' X -7854 h-r- 3 = V. Or, c 2 -l -c' 2 + c Xo' X .079 58 Ah- 3 = V. Example. — What is volume of frustum of a cone, diameters of greater and lesser ends, bd,ac , being 5 and 3 feet, and height e o, 9 ? f 5 2 -f“3 2 + 5 X 3 — 49; and 49 x .7854 = 38.4846 = above sum by -7854; an d X ^ =t 115.4538 cube feet. Fig. 20. IPyraixLid. Note.— V olume of a pyramid is equal to one third of that of a prism having equal bases and altitude. Fig. 21. To Compute Volume of a Pyramid.-Fig. SI. Rule.— M ultiply area of base by perpendicular height, and take one third of product. Example.— W hat is the volume of a hexagonal pyramid, Fig. 21 a side, a b , being 40 feet, and its height, e c, 60? ’ 40 2 X 2.5981 (tabular multiplier, page 341) = 4156.96 = area of base. 4156 96 X 60 3 = 83 139.2 cube feet. To Compute Volume of Ifriistyirn of a Pyramid . Wig. SS. Rule.— A dd together squares of sides of greater and lesser ends, and product of these two sides ; multiply sum by tabular multiplier for areas in Table, page 341, and this product by height ; then divide last product by 3. Or, s 2 + s' 2 + s x s' X tab. mult, x h -4- 3 = V. When A reas of Ends are known, or can be obtained without reference to a tabular multiplier , use following. Fi S- 22 - .7 Or, a + a' + fax a' X h +' 3 = V. Example. —What is the volume of the frustum of a hexagonal pyramid, Fig. 22, the lengths of the sides of the greater and lesser ends ab cd being respectively 3 . 75 and 2.5 feet, and its perpen- dicular height, e o, 7.5 ? * 3 - 75 2 ~i~ 2 - 5 2 = 20.312 5 = sum of squares of sides of greater and lesser ends; 20.3125 + 3.75 x 2.5 = 29.6875 = above sum added to product of the two sides; 29.6875 X 2.5081 X 7. s = ^S.aS V tab mult . , and again by the height , which, + 3 — 192. 83 cube feet. When Ends of a Pyramid are not those of a Regular Polygon , or when Areas of Ends are given Rule.— A dd together areas of the two ends and square root of their prod- uct ; multiply sum by height, and take one third of product. Or, a + a'+ fa a' X 7 t = 3 = V. nrS A ^f‘T What ^ iS ^ e volurae of an ^regular-sided frustum of a pyramid, the areas of the two ends being 22 and 88 inches, and the length 20? 22 + 88 = 110 -sum of areas of ends ; 22 x 88 = 1936, and fi 93 6 = 44 = square root of product of areas. Then 110 + 44 X 20 3 IT H* = 1026. 66 cube ins. 3 66 mensuration of volumes. Spherical Pyramid. a ^nherical Pyramid is that part of a sphere included within three or more ad- • tnff nhw! surfaces meeting at centre of sphere. The spherical polygon defined by Tlmse^pSpe^UKaces of pyramid is termed the base, and the lateral faces are ^NotL— ' T o compute the Elements of Spherical Pyramids, see Docharty and Hack- ley s Geometry. Cylindrical TTngnlas. Definition.— Cylindrical Ungulas are frusta of cylinders. Conical Ungulas are frusta of cones. To Compute -Volume of a Cylindrical XJiugula.-Fig. S3. i When Section is parallel to Axis of Cylinder. KuLE.-Multiply area Fi ‘ , of base by height of the cylinder. V 3 ' Or, a fc= V. EXAMFLE—Area of base, d ef. Fig. 23, of a cylindrical ungula is 15.5 inches, and its height, a e, 20; what is its volume . 15.5 x 20= 310 cube ins. 2 When Section passes Obliquely through opposite sides of Cylinder , Fig. 24. Kui.E.-Multiply area of base of cylinder by half sum of greatest and least lengths of ungula. Fig. 24. Or,aXl + i'-i-«=' r - Example -Area of base, c d, of a cylindrical ungula, Fig. 24, is 25 inches and the greater and less heights of it, a c, 6 d, are .5 and .7, what is its volume? \ Id f 25 X 15 + 17 . : 400 cube ins. 3. When Section passes through Base of Cylinder °A * t8 ff^a SKTStaS ri** 1 »' 7>~; j; v S r v,SS£ difference thus found by quotient arising from height div ided b> v ersea Fig. 25. 0r 2 _ — x . - — V, v. sin. representing versed sine. ( y F „ E 3 _Sine a d of half arc, d ef, of base of an ungula, Fig. 25, i ' "Ac is® inches, diameter ot cylinder is 10, and height, « g, of ungula 10, ! / J what is its volume? i H-A Two thirds of r 3 = 83. 333 = two tMrds of cube of sine. As versed e( : - 4 Ml sine and radius of base are equal, ° f X cosine = 0, and 83.333 X iod- 5 = 166.666 cube ins. 4. When Section passes through Base , of Cylinder and Srf&ASS haT/arc of base all protlTmt of area .of bas< 1 and cosine. Multiply sum thus found by quotient arising from height, div by versed sine. _ h Fig. 26. Or, --facX; — V. ^ • v. sin. Sine, a d, of half arc of an ungnliyFig. 26, is .'“'hes. Fvamptf —Sine a d, ot hall arc oi au uiiguw, a. * ver^ine, * £ to »«! height , pc, .0, and diameter of cylinder 25, what is its volume? __ f Two thirds of = 1152 = two thir ds of cube of sine of half arc of ^umof^w^thirds'of mbeof^^ofhaifdM^cfbasP v sin. — v. sin/~ h ’ U sin ' and v ’ sin/ re P resentin 9 versed sines of arcs of the tico ends, h height of cylinder, and h' height of part pro- duced. Example.— Versed sines, ae,do, and sines, e and o, of arcs of two ends of an ungula, Fig. 27, are assumed to be respectively 8.5 and 25 and n. 5 and o inches, length of ungula, ho , within cylinder, cut from one having 25 inches diameter, d o, is 20 inches; what is height of un- gula produced beyond cylinder, and what is volume of it? 25 oj 8.5 : 8.5 :: 20 : 10.303 — height of ungula produced beyond cyl- Greater ungula, sine o being o, versed sine == the diameter. Base of ungula being a circle of 25 inches diameter, area = 490. 875. Versed sine and diameter of base being equal (25), sine = 0. 490.875 X 25 a* ^ = 6135.9375 —product of area of base and cosine , or excess of versed sine over sine of base. 30. 303 -=-25 = 1. 2 12 12 = Quo- tient of height — versed sine. 1 Then 6135.9375 x 1.212 12 = 7437.4926 cube inches; and by Rules 3 and 4, volumes of less and greater ungulas = 515.444, and 6 922.0486 == 7437.4926 cube inches. Sphere. Definition. — A solid, surface of which is at a uniform distance from the centre. To Compute Volume of a Spliere.— Fig. 28. Kule. — Multiply cube of diameter by .5236. Or,

->4 cube in& Spkerical Sector. fcspsi* sector of a ; with the sector at the centre of the sphere. To Compute Volume of a- Spherical Seotor.-FigV 39 . Rule.— Multiply external surface of zone, which is base of sec or, \ third of the radius* of sphere. Or ar-r-3 = V, a representing area of base. Note. —Surface of a spherical sector = sum of surface of zone and sur aces o two cones. py ample —What is volume of a spherical sec * or ’ 3 9 E g X e A neraIed b> sector, 0 a k, heightofzoneu 6 e *, be- ing ao, .2 inches, and radius, g h, of sphere 15? 2 XQ , 24 8 = 1130.976 = height of zone X circumference Id of sphere = external surface of zone (see page 350). 1.30.976 X 1 7^1 = efface X one third of radius _ 5654.88 cube ins. Spindles. Definition. —Figures f n « rat ^' 1 ^ SetdS and they ^e^esigna^d^ ^h?name of arc from which they are generated, as Circular, Elliptic, Parabolic, etc. MENSURATION OF VOLUMES. 371 Circular Spindle. To Compute Volume of a Circular Spindle.— Fig. 40 . Rule. — Multiply central distance by half area of revolving segment; subtract product from one third of cube of half length, and multiply re- mainder by 12.5664. Or, — — ( c x ') x 12.5664 = V, a representing area of revolving segment. Fig. 40. a Example.— What is volume of a circular spindle, Fig. 40 when central distance, oe, is 7.071067 inches, length, /c, 14.142 U and radius, oc, 10? * ■ 5 Note.— A rea of revolving segment; fe, being = side of square that can be inscribed in a circle of 20, is 20 2 X .7854 — 14.142 n* 4 4 = 28. 54 area. 0 \ / 7.071 067 X 28. 54-^-2=100.9041 == central distance xhalf area of revolving segment ; - — 7 100. 9041 = 16. 947 = remainder of above product and one third of cube of half length. Then 16.497 x 12.5664 = 212.9628 cube ins. ITrnstnm or Zone of a Circnlar Spindle.* To Compute Volume of' a Frustum or Zone of a Circnlar Spindle.— Fig. 41. Rule.— F rom square of half length of whole spindle take one third of square of half length of frustum, and multiply remainder by said half length of frustum ; multiply central distance by revolving area which generates the frustum ; subtract this product from former, and multiply remainder by 6.2832. J Or, 1 4 2 — - ' Xr- — ( c x a) x 6.2832 = V, l and V representing lengths of spindle and of frustum, and a area of revolving section of frustum. Note. — Revolving area of frustum can be obtained by dividing its plane into a segment of a circle and a parallelogram. Example.— Length of middle frustum of a circular spindle ic, Fig. 41, is 6 inches; length of spindle,/#, is 8; central dis- tance, 0 e , is 3 ; and area of revolving or generating segment is 10; what is volume of frustum ? (8 4 2 ) 3 - (6 4 2)3 = * 3 > an d 13 X 3 =39 — product of - length of frustum, and remainder of one third square of half _ length of frustum subtracted from square of half length of spindle ; 39 — 3X10 = 9= product of central distance and area of segment subtracted from preceding product. Then 9X6. 2832 ±= 56. 5488 cube ins. Segment of a Circular Spindle. To Compute Volume of a Segment of a Circnlar Spindle. — Fig. 42. Rule.— S ubtract length of segment from half length of spindle • double remainder, and ascertain volume of a middle frustum of this length. Sub- tract result from volume of whole spindle, and halve remainder, f Or, C — c 4 2 = V, C and c representing volume of spindle and middle frustum. * Middle frustum of a Circular Spindle is one of the various forms of casks teingfirst obtained PliCable t0 8egment of any S P lnd,e or an J Conoid, volume of the figure and frustum 372 Fig. 42- MENSURATION OF VOLUMES. Example. — Length of a circular spindle, i a, Fig. 42, is 14.142 13 inches; central distance, o e, is 7.07107; radius of a arc, o a, is 10; and length of segment, i c, is 3.535 53; what is its volume? 14. 142 13 x 2 = 7.071 07 = double remainder of 2 Zenpi/i 0/ segment subtracted from half length of spindle = length of middle frustum. ft 0TE ._Area of revolving or generating segment of whole spindle is 28.54 inches, and that of middle frustum is 19.25. The volume of whole spindle is 212.9628 cube ins. “ “ middle frustum is 162.8982 “ “ Hence 50.0646-^2 = 25.0323 cube ins. Cycloidal Spindle.* To Compute Volume of a Cycloidal Spindle.— Fig. 43 . Rule.— Multiply product of square of twice diameter of generating circle and 3.927 by its circumference, and divide this product by 8. Fig. 43 - Or, 2 ^ X 3 - 9 2 7 X d X 3- 1 4 1 ^ _ y^ ^ representing diameter of circle , or half width of spindle. Example.— Diameter of generating circle, a b e, of a cy- cloid, Fig. 43, is 10 inches; what is volume of spindle, del jo xtx 3.927 = 1570.8 =: product of twice diameter squared and 3.927. Then 1570.8 X 10 X 3.1416-7-8 = 6168.5316 cube ins. Elliptic Spindle. To Compute Volume of an Elliptic Spindle.— Eig. 44. Rule. — To square of its diameter add square of twice diameter at one fourth of its length ; multiply sum by length, and product by .13094 Or, d 2 -f 7 d' l, 1309 = V, d and d' representing diameters as above. Example. — Length of an elliptic spindle, a b, Fig. 44, is 75 inches, its diameter, cd, 35, and diameter, ef at .25 of its length, 25; what is its volume? 35 2 -f- 25 X 2 = 3725 = sum of squares of diameter 0/ spindle and of twice its diameter at one fourth of its length ; 3725 x 75 = 279 375 = above sum X length of spindle. \ Then 279 375 X • 1309 = 36 570. 1875 cube ins. Note— F or all such solid bodies this rule is exact when body is fonned by a conic section, or a part of it, revolving about axis of section, and will always ne very near when figure revolves about another line. To Compute Volume of Middle Frustum or Zone of an Elliptic Spindle.— Fig. 45. Rule. — Add together squares of greatest and least diameters, and square of double diameter in middle between the two ; multiply the sum by length, and product by .13094 Or, d 2 -f d ' 2 -f Td'W. 1309 = V, d , d ', and d" representing different diameters. Fig- 44 - * Volume of a Cvcloidal Spindle ia equal to .625 of ita circumscribing cylinder, t See preceding Note. * See Note above. MENSURATION OF VOLUMES. 373 Fig. 45- Example. — Greatest and least diameters, ab and cd of tho frustum of an elliptic spindle, Fig. 45, are 68 and so inches, its middle diameter, gh, 60, and its length e f 7c • what is its volume? 6 ’ Ji75 ’ 2 68 2 + 50 2 -j- 60 x 2 = 21 524 = sum of squares of greatest and least diameters and of double middle diameter. Then 21 524 X 75 X 1309 = 211 311.87 cube ins. To Compute Volume of a Segment ofan Elliptic Spin- Ole.— Fig. 46 . Rule —Add together square of diameter of base of segment and square ot double diameter in middle between base and vertex ; multiply sum by length of segment, and product by .1309.* \ Or, d 2 -f- 2 d" l x • 1309 — V,d and d" representing diameters. Example. — Diameters, cd and g h, of the segment ofan elliptic spindle, Fig. 46, are 20 and 12 inches, and length oe, is 16; what is its volume? , 2 2 ° 2 .+ 12 x , 2 = 97 6 — sum of squares of diameter at base and m middle. Then 976 X 116 X -1309 = 2044. 134 cube ins. Parabolic Spindle. To Compute Volume of a Parabolic Spindle.— Fi~. 4.7-. 41888V _MUltiply S<1Uare ° f diameter by length ’ and the Product by Or, d 2 l x .41888 = V. Rule: 2.— To square of its diameter add square of twice diameter at one fourth of its length ; multiply sum by length, and product by .13094 Fig. 47- Or, d 2 -|~2 __Add together square of diameter of base of segment and square of double diameter in middle between base and vertex ; multiply sum by height of segment, and product by .1309. Or, d 2 -J- d " 2 l X -1309 = V. Example.— Segment of a parabolic spindle, Fig. 49, has diameters, ef and g h , of 15 and 8.75 inches, and height, c d, is 2.5; what is its volume? 2e.2-j-8.75 X 2 = 531.25 = sum of square of base and of double diameter in middle of segment. Then 531.25 X 2.5 X .1309 = 173- ?5 2 oube ins. Hyperbolic Spinclle. To Compute Volume of a Hyperbolic Spindle.— Fig. SO. Rule.— To square of diameter add square of double diameter at one fourth of its length ; multiply sum by length, and product by .1309. 2 Fig. 50. Or, d 2 -j-2 d' l X .1309 = V. Example.— Length, a b, Fig. 50, of a hyperbolic spindle is 106 inches, and its diameters, cd and ef are 150 and no,; what is its volume? 150 2 4-1 xo x 2 x 109 = 7 090000 = 'product of sum of squares of greatest diameter and of twice diameter at one fourth of length of spindle and length. Then 7090000 X . 1309 ~ r ~ 928 081 cube inches. To Compute Volume of Middle Frustum of a. Hyper- bolic Spindle.— Fig. ol. Rule —Add together squares of greatest and least diameters and square of double diameter in middle between the two ; multiply this sum by length, and product by .i 3 ° 9 -t Or, d 2 -f d' 2 -f (2 d ") 2 IX .i3o 9 = V. 5 1 - Example. — Diameters, a b and c d , of middle frustum of a hyperbolic spindle, Fig. 51, are 150 and no inches; diam- eter, g h, 140; and length, ef 50; what is its volume? x e; 0 2 _i_ j io 2 lfi 4 o x.2 = 1 13 000 — sum of squares of great- est and least diameters and of double middle diameter. Then 113000 X 50 X .1309 = 739585 cubeins. To Compute Volume of a Segment of a Hyperbolic Spin- die.— Fig. 52. Rule. — Add together square of diameter of base of segment and square of double diameter in middle between base and vertex ; multiply sum by length of segment, and product by .1309. Or, d 2 -\-d " 2 l X .1309 = V. Example. —Segment of a hyperbolic spindle, Fig. 52, has diameters, e f and gh, of no and 65 inches, and its length, a b, 25; what is ‘its volume? no 2 + = 29 000 = sum of squares of diameter of base and of double middle diameter. Then 29000 X 25 X .1309 = 94902.5 cube ins. * See Note, page 372. t Ibid. MENSURATION OF VOLUMES. 375 Ellipsoid, Paraboloid, and Hyperboloid of Revo- lution* (Conoids). &r— - * — — « Ellipsoid of Revolution (Spheroid). Definition. An ellipsoid of revolution is a semi-spheroid. (See page 368.) Paraboloid of Re volution. t To Compute Volume of a. Paraboloid, of devolution - Pig. 53. Rule.— M ultiply area of base by half height. Fig - 53 - c Or, a h ~ 2 = V. Note., - p This rule will hold for any segment of paraboloid whether base be perpendicular or oblique to axis of solid. ’ Example. -Diameter, a b , of base of a paraboloid of revolution Fl &- 53 > 20 inches, and its height, d c, 20; what is it& volume? ’ Area of 20 inches diameter of base - 3I4 . l6 . Then 314 l6 v 20 — 3141.6 cube ms. Prustum. of a Paraboloid of Revolution. To Compute Volume, of a. Frustum of a ; Paraboloid of Revolution.-Fig. 04= . Flg 54 Multl P l y Slln } of squares of diameters bv *’ ' ' height of frustum, and this product by .3927. Or, d 2 -\-d'2 h X .3927 = V. Example. -Diameters, a b and d c, of the base and vertex of frustum of a paraboloid of revolution, Fig. 54 are 20 and 11. 5 mches, and its height, ef 12.6; what is its volume ? 15 v So 25 ~? Um of diameters. Then 532.25 x 12.6 X -39 2 7 = 2633.5837 cube ms. Segment of a Paraboloid, of Revolution. To Compute Volume of Segment of a Paraboloid of Revo- lution.— - Wig. 55. Rule. Multiply area of base by half height. Or, a X h 4-2 = V X 7.4-^-2 = 384.315 ^Tins. ”- 5 iDCheS diameter of base = I o 3 .86 9 . Then ,03.869 Hyperboloid of Revolution. To Compute Volume of a Hyperboloid of Revolution. — Pig. 56. tm ° f middle diameter; mul ' F 'g- 55- / tn0Wn - C0n0id8 - For **■“« »f » Conoid, see KwellOu**. .5 of its circumference. turation, page 233. t Volume of a Paraboloid of Revolution is MENSURATION OF VOLUMES. Or, r 2 +d 2 & X .5236= V, d representing middle diameter Example. — Base, a 6, of a hyperboloid of revolution, Fig. 56, is 80 inches; middle diameter, c d, 66; and height, ef 60; what is its volume? g 0 _i_ 2 \ 66 2 = 5956 =sum of square of radius of base and middie diam. Then 5956 X 60 X • 5236 = 87 113.7 (tube ins. Segment of* a Hyperboloid of Revolution.. r ro Compute Volume of Segment of a Hyperboloid of He volution, as ITig. 56. r ule> To square of radius of base add square of middle diameter ; mul- tiply this sum by height, and product by .5236. Or r 2 -j- d " 2 h X • 5236 = V, r representing radius of base. Example. — Radius, a e, of base of a segment of a hyperboloid of revolution as Fig. 56, is 21 inches; its middle diameter, c d, is 30; and its height, ef 15; what is its volume? ot 2 4- ™ 2 X 1 s = 20 ns = product of sum. of squares of radius of base and middle diameter multiplied by height. Then 20x15 X .5236 = 10532.214 cube ins. Frustum of a Hyperboloid of Revolution. rp 0 Compute Volume of Frustum of a Hyperboloid of He volution. — Li S'- 57 . T? UIE __Add together squares of greatest and least semi-diameters and square of diameter in middle of the two; multiply this sum by height, and product by .5236. 0r? + cT 2 h X • 5236 — V, d. d\ and d" representing several diameters. . 2 . Example.— Frustum of a hyperboloid of revolution. Fig. tig- 57- J ig in d, 50 inches; diameters of greater and lesser ends, a b and c d, are 110 and 42 ; and that of middle ."^§1^ diameter, g /i, is 80; what is volume? g/.. -i— 110-7-2 = 55, and 42-^9 = 21. Hence 55^ + 2i 2 + 8o 2 = q 866 = $mwi of squares of semi- diameters of ends ana of ^ 0 ^ b middle diam. Then 9866 X 5° X • 5 2 3 6 = 258 291. 88 cube ins. .A-ray Figure of Revolution. To Compute Volume of any Figure of Revolution.— Fig. 58. Eui.b. — M ultiply area of generating surface by circumference described by its centre of gravity. Or a 2 rp = V,r representing radius of centre of gravity. Fig. 58. ’ Illustration i. - If generating surface, a. be d of cylinder /> b ed f Fig. =;8, is 5 inches in width and 10 in height, then will a b — 5 and b d = 10, and centre of gravity will be in 0, the ra- dius of which is r 0 = 5 -r- 2 = 2. 5. Hence xo X 5 = 50 - area of generating surface. \ Then 50X 2.5 X2X3- I 4i6 = 785-4 = c ”‘f a a i _ of generating surface X circumference of its _ / centre of gravity = volume of cylinder. Proof.— V olume of a cylinder 10 inches in diameter and 10 inches in height, io 2 X .7854 = 78.54, and 78.54 X 10 = 785.4. 2 _lf generating surface of a cone, Fig. 59, is a e = 10, d e = c then will «d = ii.i8. and area of triangle. = 10 X 5^2 = 25, centre of gravity of which is in 0, and 0 r, by Rule, page 607, = 1.666. Hence, 25 X 1.666 X 2 X 3- 1416 = 261.8 = area of generating surface X dream- ference of its centre of gravity = volume of cone. MENSURATION OF VOLUMES. 377 3.— If generating surface of a sphere, Fig. 60, is a b c, and a c -.10, abc will be ( 1P ~ * ~ 7854 ) = 39.27, cent \ which is in 0, and by Rule, page 607, or = 2. 122 Hen< ing sw sphere. — io } ab c will be ^ 7 54 j — 39.27, centre of gravity of y Rule, page 607, or = 2. 122. / Hence, 39. 27 x 2.122 x 2 x 3. 1416 = 523. 6 = area of general- / tn 9 surface X circumference of its centre of gravity = volume oj Irregular Bodies. To Compute Volume of* an Irregular Body. lhJ :U Ih;7^ S ' 1 i t - b ? t 'U: n out of fre?h water - and note difference in in body ’ 6 5 1S th ‘ S dlfference ’ s0 ls to number of cube inches difference in lbs. by 62.5, and quotient will give volume in tor^el I r 6 ir er ' S t0 be USed ’ aScertained wei ® ht of a cube foot of it, or 64, is b ° dy WCighS 15 IbS ' in water - and 3 o out; what 3 ° 15 = 15 = difference of weights in and out of water. 62.5 : 15 : : 1728 : 414.72 ^volume in cube ins. Or, 15 -r- 62.5 — .24, and .24 x 1728 = 414.72 == volume in cube ins. CASK GAUGING. "Varieties of Casks. To Compute Volume of a Cask. ls lJ ar ] ef y- Ordinary form of middle frustum of a Prolate Spheroid a SphericaI outliae of staves, as Rum ^ T f*“ To twic ® S( l uare °f bung diameter add square of head diameter • multiply this sum by length of the cask, and product by .2618, and it will fahons 111 C inCheS ’ WhiGhj bdllg divided 2 3i, will give resulUn 2d Variety. Middle frustum of a Parabolic Spindle. Th as Brandy ° f StaVeS q “ s at ^Ime; J? 257 Tq square of a head diameter add double square of bung diam- eter, and from sum subtract .4 of square of difference of diameters • multinlv remainder by length, and product by .2618, which, being divided will give volume 111 gallons. ’ ® 1 2 3L 3d Variety. Middle frustum of a Paraboloid Th hf.gtasimfcasts aU ° aSkS Wh ‘ Ch CU ™ ^ckens slightly at Rule.— T o square of bung diameter add square of head diameter • mul- wUi y ghTvXmTfn ganon S Pr0dUCt ^ ' 3927 ’ WWch ’ being dMded V 4 th Variety. Two equal frustums of Cones. “S CrnTpes 3 a " iD WUich Cu ™ “ f staves quickens sharply at Rule.— A dd square of difference of diameters to three times sauare of olLT ?; “f'P'y sum >-*>;. length, and product by .065 % audit wm give gallons. ° Ube ,nCheS> WhlCh ’ being divided V =3i, will give Sf L *• Weight of a cube foot of fresh water. I I* t Number of inches in a cube foot. 378 mensuration of volumes. Example.— Bung and head diameters of a cask are 24 and 16 inches, and length 36; what is its volume in gallons? t: rftf+ (24 + x6)°x T = 4864, which X 36 = >75 104, and ,75 104 X .065 66 = ii 497.329, which -r- 231 = 49-77 gallons. Generally. Bd+M 5 .001 692 L = V. S. gallons, and .001 416 2 = Imperial gallons. I), d, and M representing interior , head and bung diameters, and L length of cask in inches. To Ascertain. Mean Diameter of a Cask. Rine— Subtract head diameter from bung diameter in inches, and mul- tinlv difference by following units for the four varieties; add product to head diameter, and sum will give mean diameter of varieties required. SW;:::::::::::::: i I t Example.— Bung and head diameters of a cask of ist variety are 24 and 20 inc - es- what is its mean diameter? 24 — 20 = 4, and 4 X -7 = 2. 8, which, added to 20, = 22. 8 ins. ULLAGE CASKS. To Compute Volume of Tillage Casks. When a cask is only partly filled, it is termed an ullage cask, and is con- sidered in two positions, viz., as lying on its side, when it is termed a ment Lying, or as standing on its end, when it is termed a Segment Standing. To XJUage a Dying Cask. Rule.— Divide wet inches (depth of liquid) by bung diameter ; find quo- tient in column of versed sines in table of circular segments, page 267, and take i?s corresponding segment; multiply this segment by capacity of cask in gallons, and product by 1.25 for ullage reqtuied. Example -Capacity of a cask is 90 gallons, bung diameter being 32 inches ; what is its volume at 8 inches depth? 8 -4- 32 = .25, tab. seg. of which is .153 55, which X 90 = 13- 8195, and again X 125 _ 17.2744 gallons. _ To Ullage a Standing Cask. r ux F —Add together square of diameter at surface of liquor, square of head diameter, and square of double diameter taken m middle between the two; multiply sum by wet inches, and product by .1309, and divide by 3 for result in gallons. Cask by Four Dimensions. To Compute Volume of Rule.- A dd together squares of hung and head diameters and square of double diameter taken in middle between bung and head; multip y and product by .1309, and divide tins product by 231 for result in gallons. To Compute Volume of any Cask from Three Dime..- sions only. n ri F __\dd into one sum 39 times square of bung diameter, 25 times square of head diameter, and 26 times product of the two diameters ; mul- tiply sum by length, and product by .008726; and divide quotient by -31 for result iu gallons. . For Rules in Gauging in all its conditions and for descr.ptton and use of instruments, see HaswelCs Mensuration , pages 307-23. CONIC SECTIONS. 379 Fig. x. A CONIC SECTIONS. A Cone is a figure described by revolution of a right-angled triangle about one of its legs, or it is a solid having a circle for its base and terminated in a vertex. • Conic Sections are figures made by a plane cutting a cone. If a cone is cut by a plane through vertex and base, section will be a triangle, and if cut by a plane parallel to its base, section will be a circle revolving 8SS$ftSSi? triaDS ’ e reV ° 1VeS ' *** h Cirde Which is described An Ellipse is a figure generated by an oblique plane cut- tmg a cone above its base. Transverse axis or diameter is longest right line that can be drawn in it, as a b , Fig. i. Conjugate axis or diameter is a line drawn through centre of ellipse perpendicular to trans- verse axis, as c d. A Parabola is a figure generated by a plane cutting a cone parallel to its side, as a be, Fig. 2. Axis is a right line drawn from vertex to middle of base, as b 0. Note. — A parabola has not a conjugate diameter. A Hyperbola is a figure generated by a plane cutting a cone at any angle with base greater than that of side of cone, as a b c, Fig. 3. Transverse axis or diameter, o 6, is that part of axis e b which if continued, as at 0, would join an opposite cone, o fr. ’ ’ ’ Conjugate axis or diameter is a right line drawn through centre g > °f transverse axis, and perpendicular to it. Straight line through foci is indefinite transverse axis* that part pf it between vertices of curves, as o b , is definite transverse axis. ~ Its middle point, <7, is centre of curve. Eccentricity of a hyperbola is ratio obtained by dividing distance from centre to e 1 th er focus by semi -transverse axis. Parameter is cord of curve drawn through focus at right angles to axis. Asymptotes of a hyperbola are two right lines to which the curve continually arn proaches, touches at an infinite distance but does not pass; they are prolongations of diagonals of rectangle constructed on extremes of the axes. b Two hyperbolas are conjugate when transverse axis of one is conjugate of the other, and contrariwise. ^uujugaie 01 me General Definitions. An Ordinate is a right line from any point of a curve to either of diameters as and a 6 an abscissa*" 111 “ & a “ d df ' " e d ° Uble ° rdinates; c 6 . ri K- 5 , is an ordinate, An Abscissa is that part of diameter which is contained between vertex and an ordinate, as ce, go. Fig. 4, and a b Fig. 5. Ij Parameter of any diameter is equal to four times \ ( J ^’stance from focus to vertex of curve; parameter 1 of axis is least possible, and is termed parameter of curve. Parameter of curve of a conic section is equal to chord of curve drawn through focus perpendic- ular to axis. I ar a meter of transverse axis is least, and is termed parameter of curve ofcu™e m “ a “ d f0Ci are sufflcieBt elements for construction Fig. 4 cd Fig. 5. CONIC SECTIONS. 380 A Focus is a point on principal axis where double ordinate to axis, through point, 18 It^ may° thus: Divide sc l uare of ordinate by four times abscissa, and quotient will give focal distances, a 5 and 5, in preceding figures. Directrix of a conic section is a right line at right angles to major axis, and it is in such a position that /: g \\ u : o. Here a d, Fig. 6, is directrix, and o is offset to directrix. Latus Rectum , or principal parameter, passes through a focus; it is a double ordinate, which is a third proportion to the axis. Or, A : a:: a: L. A and a representing major and minor axes. (See Haswell's Mensuration , page 232. ) A Conoid is a warped surface generated by a right line being moved in such a manner that it will touch a straight line and curve, and continue parallel to a given plane. Straight line and curve are called di- rectrices, plane a plane directrix , and moving line the generatrix . Thus, let a b a\ Fig. 7, be a circle in a horizontal plane, and d d' projection of right lines perpendicular to a ver- tical plane, r' b e; if right lines, da,rs , r' b, r" s, and d' a, be moved so as to touch circle and right line d d and bo constantly parallel to plane r* b e, it will generate conoid w w dab a' d'. Iiadii restores arc lines drawn from the foci to any point in the curve; hence a radius vector is one of these lines. Traced angle is angle formed by the radii vectores and the transverse diameter. Ellipsoid , Paraboloid , and Hyperboloid of Revolution — 1 igures generated by the revolution of an ellipse, parabola, etc., around their axes. (See Men- suration of Surfaces and Solids , pages 357 " 75 *) Note 1.— All figures which can possibly be formed by cutting of a cone a™ men- tioned in these definitions, and are five following — viz., a £) tangle, a Circle , an El a Hyperbola; hut last three only are termed Como Sections. 2 in Parabola parameter of any diameter is a third trbportional Jo .abscissa and ordinate of any point of curve, abscissa and ordinate being referred to th-t diameter and tangent at its vertex. 3.— In Ellipse and Hyperbola parameter of any diameter is a third proportional to diameter and its conjugate. To Determine Parameter of an. Ellipse or Hyperbola Z7 Fig. 8. Rule. — Divide product of copjugate diameter, multiplied by itself, by trans- verse, and quotient is equal to para- meter. Fig. 9. ► In annexed Figs. 8 and 9, of an Ellipse 1 ^ and Hyperbola , transverse and conjugate diameters, ab,cd , are each 30 and 20. Then 30 : 20 :: 20 : 13 - 333 —parameter. Parameter of curve — elf a double ordinate passing through focus, s. Ellipse. To Describe Ellipses. (See Geomotry, page 226.) To Compute Terms of an Ellipse. When ami three of four Terms of an Ellipse are given , viz., Transverse and Conjugate Diameters, an Ordinate, and its Abscissa, to ascertain remain - ing Terms. CONIC SECTIONS. 381 To Compute Orclinate. Transverse and Conjugate Diameters and Abscissa being given. Rule. As trans- whTch^d^des^hem co ^ ugate ’ so is square ro °t of product of abscissae to ordinate Example. —Transverse diameter, a b , of an ellipse Fig. 10, is 25; conjugate, c d, 16; and abscissa, at, 7; what is length of ordinate, t'e? Fig. ia 25 — 7 = 18 less abscissa ; V7 X 18 = 11.225. Hence 25 : 16 ; : 11.225 • 7*184 ordinate. 0r > \J c * “ (fj~j ~ any ordinate , c and t representing Tgtr 7 nJU9CLte aUd tranSverse diamete rs, and x distance of ordinate from centre of figure. To Compute -Abscissae. Transverse and Conjugate Diameters and Ordinate being given. Rule —4s coniu S?i«“ eter - 18 t : an J sverse > 80 is square root of difference of squares of ordinate fn Hence, as 16 : : 25 :: 3.52 | abscissce. 5*5* —7*184 =3.519943. Then 25^2 = 12.5, and 12.5-f- 5.5 = 18 = 6 25 = 2 = 12.5, and 12.5 — 5.5 = 7 = at,. To Compute Transverse Diameter. Conjugate, Ordinate , and Abscissa being given. Rule.— To or from semi coniu fprpAp a p C nf rdm? M g / eat , or less abscissa is used, add or subtract square root of dif- f - square ? of 01 ! dinate and semi-conjugate. Then, as this sum or difference is to abscissa, so is conjugate to transverse. uinerence Example. — Conjugate^ dimeter, c d, of an ellipse. Fig. 10, is 16- ordinate ie 7. 184 , and abscissae, b 1, i a, 18 and 7 ; what is length of transverse diameter ? ’ ’ . . (l6 = 2) 2 — 7.1842=3.52. 1 • 2 "r 3 * 5 2 . 18 .. 16 : 25; 16 = 2 — *3.52 : 7 :: 16 : 25 transverse diameter. To Compute Conjugate Diameter. being given ■ Rvle — As square root of prod- uct 01 abscissae is to ordinate, so is transverse diameter to conjugate. P , E o XA ^lT TrknSX Y S . e diameter ,06, of an ellipse, Fig. IO is 2 <- ordinate ie 7.1 4, and^abscissae, b 1 and t a, 18 and 7 ; what is length of conjugate diameter? ’ 18 x 7 = 11225. Hence 11.225 : 7- 184 :: 25 : 16 conjugate diameter. Rrtp T J? , C ? mpUte Circumference of an Ellipse. 3 Rule. - Multiply square root of half sum of the squares of two diameters by « 6 *<*. of an ellipse, Fig. , 24 2 -f-2Q 2 2 = 488 ’ and ^ 488 = 22 °9- Hence 22.09 X 3. 1416 = 69.398 circumference. p T ° Corupilte Area of an Ellipse on Or, multiply jugate, cd%; ^hat'is^ts area?' ameter ° f a “ ellipse ’ a b ’ Fi & IO > ls I2 > aud its con- 12 X 9 X .7854=184.8232 area. ' ' ? ' ' ' / ~ U// CU/. dia m °It E er-„f A o^ ^ «■*>* o uiuor minor axis. area L of drcTe°of ° f £ ircle ° f 40 “ I2 5 6 6 4 5 area of ellipse 40X20- 628 32 • 382 CONIC SECTIONS. Segment of an Ellipse. To Compute Area of a Segment of an Ellipse. When its Base is parallel to either Axis , as e if. RimE -Divide heigM of seg- ment b i. by diameter or axis, a b, of which it is a part, and find in Table of Areas of Segments of a Circle, page 267, a segment having same versed sine as this quo- ® tient; then multiply area of segment thus found and the F'g 11 — e axes of ellipse together. Example. — Height, b t, Fig. u, is 5, and axes of ellipse are 30 and 20; what is area of segment? 5 -f- 30 = .1666 tabular versed sine , the area of which (page 267) is .085 54. Hence .085 54 X 3 ° X 20 = 51.324 area. To Ascertain Length of an Elliptic Curve which is less than lialf of entire Figure. Fig. 12. Let curve of which length is required be A b C, Fig. 12. Extend versed sine b d to meet centre of curve in e. n Draw line e C, and from e, with distance e b , describe b ft,; bisect ft, C in i, and from e, with radius e i , de- ■ * scribe 1 c i, and it is equal to half arc A b C. To Ascertain Length, when Curve is greater than half entire Figure. Ascertain by above problem curve of less portion of figure; subtract it from cir- cumference of ellipse, and remainder will be length of curve required. JParatoola. To Describe a Parabola. (See Geometry, page 229.) To Compute either Ordinate or Abscissa of a Parabola. When the other Ordinate and Abscissa, or other Abscissa and^Ordina^s are given. Rule. —As either abscissa is to square of its ordinate, so is othei abscissa to square of its ordinate. . Or, as square of any ordinate is to its abscissa, so is square of other ordinate to its abscissa. Example i.— Abscissa, a b , of parabola, Fig. 13, is 9; its ordi- nate, be, 6 ; what is ordinate, d e, abscissa of which, a d, is 16 . Hence 9 : 6 2 I*. 16 : 64, and V64 tt 8 length. 2.— Abscissae of a parabola are 9 and 16, and their correspond- ing ordinates 6 and 8; any three of these being taken, it is re- quired to compute the fourth. 62 x _ l6 = 8 ordinate. 2. = 6 ordinate. 3 , 16 ^ — — 9 less abscissa. 4- 9 = 16 abscissa. Laraholic Curve. To Compute Length. of Curve of a Para ’ lola cut off a. Lovable Ordinate.— F ig. 13 . Rule. -To square of ordinate add ± of square of abscissa, and square root of this sum, multiplied by two, will give length of curve nearly. Example. -Ordinate, d «, Fig. 13, is 8, and its abscissa, a d, 16; what is length of curve, fa e ? ga 4 * — — 405. 333, and V 4 ° 5 - 333 X 2 ^ 4 °- 2 ^7 lenffik* Fig. 13. 3 CONIC SECTIONS. 383 Fig. 14. To Compute Area of a Parabola. Rule -Multiply base by height, and take two thirds of product - vuuus ui pruauct alMogmm 7 - Parab °' a iS two thirds of its circumscribing par- Example. —What is area of parabola, a b c, Fhr ™ heiVht h, being 16, and base, or double ordinate, a c, 16 ? g ' 4 ’ n ghtj b e > 16 X 16 — 256, and of 25 6 = 170.667 area. To Compute Area of a Segment of a Parabola. heX * M at P d^ hefgh A “e P “is £?■ Wtaf toS* SSL" ‘ Parab °' a ’ ° c and ^ Fi & >4, are xo and 6, and io ^6^10X2 = 15 680, and 4- - __ 2 /v 6 2 x 3 = 81 .66j area. of whlchls Same height ’ the base the two ends and lesser end. g 5 creased by a third proportional to sum of Hyperbola. To Describe a Hyperbola. (See Geometry, page 230.) To Compute Ordinate of a Hyperbola, Fig. 15. 6 dia^efa <-? yperboIa - ? b Fig. 15, has a transverse 40+ 120 = 160 greater abscissa , and 120 : 72 : . -^/ (40 x 160) : 48 ordinate. i c ^ perbolas lesser abscissa, added to axis (the transverse diameter), gives greater. n o* * ^ & *• vutti. is : hyperbola to any point in curve To Compute ^Vbscissee, J ugat^dlame ter fs^o* u^veree*™o^s sma!ue°rMr Tr . ; 0 72 ’ and Qrdina te, a c, 48; wha t are’len’gths all'' Conjugate ’ ** To Compute Conjugate Diameter, tu CUUJU^cUt*. 4s ; ^s,ra^ V40 x 160 — 80 : 48 :: 120 : 72 conjugate. CONIC SECTIONS. To Compute Transverse Diameter, Conjugate , Ordinate , and an Abscissa being given. Rule.— Add square of ordinate . enn iro of se mi -conjugate, and extract square root of their sum. Take sum or difference of semi-conjugate and this root, accord mg as greater or lpocpr abscissa is used. Then, as square of ordinate is to product of abscis&a and coni ? a»te sum or difference above ascertained to transverse diameter required. Note. —When the greater abscissa is used, the difference is taken, and con- ^Example -Conjugate diameter, df. \ of a hyperbola, Fig. 15, is 72; ordinate, e c, 48; and lesser abscissa, a e, 40; what is length of transverse diameter, a t? -/To 2 1 — 60, and 60 + 72 -r- 2 = 96 lesser abscissa , and 40 X 7 2 — 2880. v 40 T- v7 • Hencej ^g2 . 2 88o :: 96 : 120 transverse diameter. To Compute Length, of any Arc of a Hyperbola, com- mencing at Vertex. RnTE —To iq times transverse diameter add 21 times parameter of Tn rf'times transverse diameter add 21 times parameter, and multiply each of these 9 sCs res“ely by quotient of lesser abscissa divided by transverse do a To te eacb of products thus ascertained add r 5 times parameter, and divide former bv^latter; then this quotient, multiplied by ordinate, will give length of arc, nearly. ' Xote. —To Compute Parameter, divide square of conjugate by transverse diam- fTv', 6 r. Example. -I n hyperbola, abc , Fig. 16, transverse diameter is jeo “■ ’ conjugate. 72, ordinate, e c, 48, and lesser abscissa, a e, 40, wba length of arc, a b ? Zl_ = 43.2 parameter. 120 X 19 + 43- 2 X 21 X — = 1062.4. 120 - ^^ + 43.2X 21 X ^ = 662.4. Then 1062.4 + 43.2 X 15^662.4 v] c +43.2X 15 = 1.305, which X 48 = 62.64 length. Note. -As transverse diameter is to conjugate, so is conjugate to parameter. (See Rule, page 380.) To Compute Area of a Hyperbola, transverse diameter and lesser abscissa 10 ‘''^vUieTtim^prodac^of conjugate^iameter and lesser abscissa by transverse diameter, + and m t^ hU^tqwtient, J m B ultip.ied by former, will give area, nearly. Example. -Transverse diameter of a hyperbola, Fig .6, is 60, conjugate 36, and lesser abscissa or height, a e, 20; what is area ot figure . 60 X no + !■ of 20= = 1485.7143, a«d VH 85 - 7 I 43 X 21 = 809.43, V^X 2° X 4 + 809. 43 = 901.02, which -4- 75 = 12-0136 and ” 6 X4 X I2 - OI 8 6 = 576 6 53 a > ea - Nora. — For ordinate, of » pnrnbol. in divUion. of eighths nnd tenth., see page 229- Delta Nletal. Delta Metal is an improved composition of Aluminimn a, ,d its alloys , it is non-corrosive, capable of being cast, forged, and hot rolled. Tensile Strength per Sq. Inch. Cast in green sand 48380 lb f . I KoUe, M^ahA ; ; ; ; ; ; ; ; ; ; Boiled, hard 75 260 | Wire, no. 22 tt vx. PLANE TRIGONOMETRY. 385 PLANE TRIGONOMETRY. By Plane Trigonometry is ascertained how to compute or determine four of the seven elements of a plane or rectilinear triangle from the other three, for when any three of them are given, one of which bein«- a side or the area, the remaining elements may be determined ; and this operation is termed Solving the Triangle. The determination of the mutual relation of the Sines, Tangents, Secants etc., of the sums, differences, multiples, etc., of arcs or angles is also classed under this head. For Diagram and Explanation of Terms , see Geometry, pp. 219-21. Right-angled. Triangles. For Solution by Lines and Areas , see Mensuration of Areas, Lines and Surfaces, pp. 335-39. To Compute a Side. When a Side and its Opposite Angle is given. Rule. — A s sine of an 4, and 5; what are the angles of the hypothenuse ? 20 __ (Log. 4 = .602 06 -f Log. 5 = ^698 97) = 18.698 97 ; Log. 3 + 4 + 5 • 2 4 ■50103; and Log. 3 + 4 + 5^~ 2 — 5 — °- . QO ,, 3 Then 18.69897 + . 30103 = 19, which^-2 = 9.5 = log. sin. of half angle _ x8 2 , which X 2 = 36° 52' angle. Hence 9 o° — 36° 52' = 53° 8' remaining angle. = B ? y ^Sin^TaiL, 1 Sec.', etc., A B^ etc.,. is expressed Sine, Tangent, Secant, etc., of angles, A, B, etc. To Compute Sides A C and BE. -Figs. 1 and 3. When Hyp., Side B A, and An Sin. B X B A Fig. i, Cosine. A Vers. - = AC. Sin. C BAX Cot. C = A C. Hyp. X Cos. C = AC. Hyp. X Sin. B = A C. BA Sin. C ’ AC Sin B - = BC. = B C. To Compute Side A C and. Angles. When Hyp . and Side B A are given.— Fig. i and 2. AC Hyp.' A C I Sin. B. -®A = S in.C. BA Jl S !j — = AC - Hyp. fein - C B C X Sin. B = AC. BA Fig. 3- To Compute Side B € and Hyp. or Angles When both Sides are given.— Fig. 2. = Tan. B. ~ = BC. VaC* + B A^BC. B A AC = Tan. C. ? 4 = sin ' c - B C ^ = Sin.B. To Compute Sides.— Figs. 3 and 4. When a Side and an Angle are J' given, Fig. 4- | \';h A C X Tan.C — Rad‘ Rad. BCX Cos. B = BA. B C X Sin. B = A C. A B X Sec. b=bc. lC BA AC X Sin.C A. Sin. B ;C -BC. A C X Bad. Sin. B Tangent. In B A C Fig. 5, a right-angled triangle, C A, is assumed to be radius ; B A tangent MC, and BC secant to that radius ; Or, dividing each of tttese by base, there is obtained the tangent and secant of C respectively to radius . Radius. PLANE TRIGONOMETRY. 387 O Radius, (f Radius C A = 1 Secant CB = i.4i42 Tangent AB=i Co secant CB = 1.4142 Co tangent e B = 1 a/ A C 2 + B A 2 — hyp. B C. AC-r Co s. C = hyp. B C. A Area Cos. C V tSTc =Rad - sn= = Cot.C, Sine d g= .7071 Cosine C g or od = .7071 Versed sine g A= .2929 Co- versed sine oe= .2929 Angle CAB = 9 o° BA-f- Sin. C = hyp. B C. 1 -f- Tan. C = Cot. C. B C2 x Sin. 2 C - —Area. BC X Cos. C = Rad. B A x Tan. B = Rad. BC 4 -B A = Sec. B. B C X Cos. B = B A. Sin. C ' BAxSec. B = RC. B A x Cot. C — Rad. B C x Sin. B = Rad. B C x Sin. C = B A. A C X Tan. 0 = B A 1 4 - Sin. C = Cosec. C. 1 - Sin. C ~ Co-ver. sin. Cos. C 4- Sin. C = Cot. C. C B x Sin. B = A C. Perp. 4 - hyp. = Sin. C. Base 4- hyp. = Cos. C. Base 4 - hyp. =Sin. B. Base 4 - perp. = Cotan. C. Trigonometrical Equivalents. V (1 — sin. 2 ) == Cos. Sin. -4- tan. = Cos. Sin. x cot. = Cos. Sin. 4 - cos. rz Tan. Cos. 4- cot. — Sin. Cos. ‘ 4 - sin. — Cot. Hyp. 4- base = Sec. C. Perp. 4- base = Tan C Base 4- perp. = Tan. B. Hyp. 4- perp. = Sec. B. Perp. 4- hyp. = Cos. B. Hyp. 4- perp. = Cosec. C. Hyp. — Base = Versin. Hyp. — Perp. — Co-ver. sin. CT. Tan. 4- sin. = Sec. Tan. 4- sec. — Sin. Tan. x cot. = Rad. •cos. 2 ) t= Sin. cot. == Tan. 4- sin. Cosec. 1 4 - cos. = Sec. 1 4- cosec. = Sin. 1 4 - sec. = Cos. x — cos. — Versin. 1 — sin. = Co-ver. sin. tan. = Cotan. T ' 1 ~ tan. zzr ootan. A B 0f a right - aD 8 ,ed tri “gJ« ^ *00, and angle C Fig. 6. B Ofolicpne-anglecL Triangles. To Compute Sides B A and B C. When Side A C and Angles are given.— Fig. 6. Sin. C x A C Sin. B = BA. Sin. A x A C Sin. 0 X B C Sin. A = BA. Sin B = BC. To Compute Angles and Side A C Wken 8ides A B, B C, and one of the Angles are given.- FtV 6 / Sin n o * B C X Sin. B AC = Sin. A. Sin. C x A C B A Sin. B x B C = Sin. B. A B x Sin. B AC : Sin. C. = AC. Fig. 7 . Sin. A To Compute Sides B A and B C. When Side A C and Angles are given.— Fio*. 7. Sin. C x B C _ . Sin. Ax AC Sin. A = BA. Sin. B -=BC. When Side B C and Angles are qiven. — Fig- 7 B C x Sin. C ~ Sin. (C + B) = B A. Sin. C x AC Sin. B :BA. pU™tI Sim! "* C ° Sine ° fan arc are each W ^ sine and cosine of their sup. Spherical Triangles , Right - angled and Oblique. Molesworth, Lond., 1878, pp. 435-6. * For full formulas see 388 PLANE TRIGONOMETRY. To Compute Angles and. Side AC. When Sides A B, B C, and Angle B are given.— Fig. 7. _ ^ . o,- T. RA v Sin TC B C X Sin. B AC BAX Sin. A BC = Sin. A. = Sin. C. BAX Sin. B AC! B C X Sin. C = Sin. C. = Sin. A. BC, ' AB To Compute all tlie Angles. When all the Sides are given , Figs. 6 arid 7. Rule.— Let fall a perpen- dicular, B d, opposite to required angle. Then, as A C : s^ °f A B, B C their difference : twice d g , the distance of perpendicular, B d, from middle ° f Hence S A* d, C a are known, and triangle, A B C, is divided into two right- angled triangles! BCd,B A < 2 ; then, by rules for right-angled triangles, ascertain angle A or 0 . . . „ „ r Operation. -A C, Fig. 6, . 5014 : A B + B C, 1. 1 174 + MH 2 = 2 - 53*6 . . A B a> B L, 1. 4142 — 1. 1174 = .2968 : 2 X d g — 1 4986. Hence A c! = ^ ^ • 4986, and C d = A d + A C = x. Consequently, triangle B d C, Fig. 6, is divided into two triangles, B A C and B d A. To Compute Side A B and Angles. When Two Sides and One Angle, or One Side and Two Angles, are given.- Fig. 6. AC X Sin. C B C X Sin. B AC A B X Sin. B = Sin. A. = Sin. C. A C X Sin. A AB— (ACxCos. A) AC X Sin. C = Tah. B. Tan. B. 2 Area B C, Sin. C BC— (ACxCos. C) To Compute Area of a Triangle. -Fig. S. BAxBCxSin. B AC X BC X Sin. C B A X A C X Sin. A 7 T, I ’ 2 Sin. 2 C, B C 2 A C-, Tan. C an(J B A a , Cot. C _ ^ ’ 2 ’ 2 Note.— For other rules, see Mensuration of Areas, Lines, and J A Surfaces, page 335. To Compute Sides. When Areas and Angles are given.— Figs. 6 and 7. 2 Area = AC. A C, Sin. A - = BA. J Si 2 Area, Sin. A _ g q S in. C, Sin. (A -pC) To Ascertain Distance of Inacces- sible Objects on a Level Diane.— Digs. 9 and IO Fig. 10. Fig. 9. Operation.— Lay off perpendic- ulars to line A B, Fig. 9, as B c, d e, on line A d , terminating on line e A. Then e d — c B : c B : : B d : BA. When there are Two Inacces- sible Objects , as Fig. 10. Operation. — Measure a base line, A B, Fig. 10, and angles cAB, £ dB A, Acd, Bdc,etc. Then pro- ceed by formulas, page 387, to deduce cd. Note. If course of cd is required, take difference of ang d c A and cd B from course A B. PLANE TRIGONOMETRY. B Fig. ”• When the Objects can be aligned . — Fig. ii. I Operation.— Align c B, Fig. n, at A, measure a base line at any angle there- to, as A o, and angles o A c, B A c, and Bo A. Then proceed as per formula page 386, to deduce c B. To Compute Distance from a Given Point to an In- accessible Object. — Die:. 13 . 389 Fig. 12. Operation.— Measure a level line, Ac, Fig. 12, and ascertain angles B Ac c A B SrmTne A B Dg ’ A *** tW ° angles ’ pr0Ceed as formula, page 38 ^to de- To Compute Height of an Elevated. Doint.— DP . 13 . , . , compute uisiance Ac, Fjc. t» 4 a asc r e p ^ ta,n an S le °f depression Aoc and of elevation JtS A o. I hen nrnnoorl o o n ^ . Operation. — Measure distance on a horizontal line, A c, Fig. 13; ascertain Angle B A c. Then pro- ceed as per formulas, pp. 386-8, to ascertain B c. When a Horizontal Base is not Attainable. —Fig. 14. Operation. — Measure or compute distance Ac, Fig. Fig. 15. When a Full Base Line is not A ttain- B able. Fig. 15. Operation. — Measure a base line, A c, Fig. 15, and ascertain angles A c B, c A B. Then proceed as per for- mula, page 386, to ascer- tain d B. Fig. 16. Without Use of an Instrument. —Fig. 16. t^^ft^a^UkTeWaUon^ro^^se^ni^rf d 'W’*A* e * up a staff at each ex- ojr ‘ Se “^ight ofeyjfrom Then D 7 x—y and D length of line d d. + A + i = height, s representing height of line of sight from base d d, K K* 39 ° O -£ 2 * « SV c- 29 natural sines and cosines. ISTatnral Sines 3 3 4 4 5 5 6 6 7 7 8 8 9 9 0 ° N. sine. 13 13 14 14 15 15 15 16 16 17 17 23 23 24 24 25 25 26 26 27 27 28 28 29 29 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 8 .00000 .00029 .00058 .00087 .00116 .00145 .00175 .00204 .00233 .00262 .00291 .0032 .00349 .00378 .00407 .00436 .00465 .00495 00524 .00553 .00582 .00611 .0064 .00669 .00698 .00727 .00756 .00785 .00814 .00844 .00873 .00902 .00931 .0096 .009'' 1 ° N. sine. .01018 .01047 .01076 ,01105 .OH34 .01164 .01193 .01222 .01251 .0128 .01309 .01338 .01367 .01396 .01425 01454 .01483 .01513 .01542 .01571 .016 .01629 .01658 .01687 .01716 .01745 .99999 .99999 .99999 •99999 .99999 .99999 .99999 .99999 .99998 .99998 .99998 .99998 .99998 .99998 •99997 •99997 •99997 .99997 .99996 .99996 .99996 .99996 •99995 •99995 •99995 •99995 •99994 .99994 •99994 •99993 •99993 •99993 .99992 .99992 .99991 .99991 .99991 •9999 •9999 .99989 .99989 .99989 .99988 .99988 .99987 .99987 .99986 . 99986 •99985 .99985 N. cos. N. sine. N. cos. .01745 •01774 01803 .01832 .01862 .01891 .0192 .01949 .01978 ,02007 ,02036 .02065 . 02094 ,02123 .02152 .02181 .02211 ,0224 .02269 .02298 ,02327 ,02356 .02385 .02414 .02443 .02472 .02501 .0253 .0256 .02589 ,02618 ,02647 .02676 .02705 .02734 .02763 .02792 .02821 .0285 02879 and. Cosines. 2 ° N. cos, N. sine, 89 ° .02908 .02938 .02967 02996 03025 .03054 .03083 .03112 .03141 .0317 .03199 .03228 .03257 .03286 .03316 •03345 •03374 .03403 •03432 .03461 •0349 .99985 .99984 .99984 .99983 .99983 .99982 .99982 .99981 •999? .9998 •99979 •99979 .99978 .99977 •99977 .99976 .99976 •99975 •99974 •99974 •99973 .99972 .99972 •9997 •9997 .99969 .99969 .99968 .99967 . 99966 . 99966 •999 6 5 .99964 ,99963 ,99963 , 99962 .99961 .9996 •99959 •99959 .99958 •99957 •9995 6 •99955 •99954 •99953 •99952 .9995 2 •9995 1 •9995 •99949 .99948 •99947 •99946 •99945 .99944 •99943 •99942 .99941 •9994 •99939 3 ° N. sine. •99939 •99938 •99937 •99936 •99935 •99934 •99933 .99932 •99931 •9993 I .99929 | .99927 .99926 .99925 .99924 .99923 .99922 .99921 .99919 •999 l8 .99917 .99916 •99915 •999*3 •999 12 •999 11 .9991 .99909 .99907 .99906 .99905 .99904 .99902 • 999 01 •999 .99898 •99897 . 99896 •99 8 94 .99893 .99892 .9989 .99886 .99885 •99883 .99882 .99881 .99879 .99878 .99876 .99 8 75 •99873 .99872 .9987 .99869 .99867 .99866 .99864 .99863 05234 .05263 .05292 .05321 •0535 05379 >5408 •05437 .05466 05495 .05524 •05553 .05582 .0561 ,0564 .05669 05698 ,05727 ■05756 .05785 .05814 05844 .05873 05902 •05931 0596 ,05989 06018 ,06047 .06076 .06105 .06134 .06163 .06192 .0622 ,0625 99863 99861 .9986 99858 •99857 .99855 .99854 .99852 ,99851 .06279 .06308 •06337 .06366 .06395 .06424 .06453 .06482 .0651 .0654 .06569 .06598 .06627 .06656 .06685 .06714 .06743 .06773 g* 5 P< t .99847 ,99846 .99844 .99842 .9984 .99838 99836 ■99834 .99833 99831 .99829 .99827 ,99826 99824 59 58 57 56 55 54 53 52 5i 50 49 48 47 46 45 44 43 42 4 1 40 39 38 36 35 34 33 32 3 1 30 29 28 27 26 25 24 99821 .99819 .99817 ,99815 .99813 .99812 ■99? 1 . 99808 . 99806 .99804 .99803 .99801 •99799 •99797 •99795 •99793 .99792 •9979 .99788 .99786 •99784 .99782 •9978 .9977? .99776 •99774 .99772 I -9977 .06802 .99708 .06831 .99766 .0686 I .99764 .06889 .99762 .06918 .9976 .06947 | -99758 .06976 ! .99756 , o N. cos. I N. sine. ! ' 860 I NATURAL SINES AND COSINES. 391 Prop. parts. 40 I 5° 6° 70 2 9 N. sine i. N. cos 1 N. sine . N. cos . N. sine 5. N. cos . N. sine }. N. cos. 0 .0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 2 12 2 13 2 13 2 14 2 14 2 15 3 15 3 15 3 16 3 16 3. 17 3^ 17 3< 18 3; 18 3* 19 3 c 19 4c 20 41 20 42 21 43 21 44 22 45 22 46 2 3 47 23 48 24 49 24 50 25 5i 25 52 26 53 26 54 27 55 27 56 28 57 28 58 29 59 29 60 0 .0697* 1 . 0700 2 0703^ 3 .0706; 4 .07095 5 .07121 6 .0715 7 .07175 0 .0720$ 9 -07237 10 .07266 11 .07295 12 .07324 r 3 .07353 14 .07382 15 .07411 16 .0744 17 -07469 r8 .07498 [9 .07527 20 .07556 21 -07585 '2 .07614 3 -07643 4 .07672 5 .07701 6 -0773 7 -07759 8 .07788 9 .07817 0 .07846 1 .07875 2 . 07904 3 .07933 \ .07962 5 .07991 3 . 0802 7 .08049 3 .08078* ) .08107 > .08136 .08165 ! .08194 1 .08223 .08252 .08281 •0831 -08339 .08368 08397 .08426 •08455 .08484 . •08513 . .08542 . -08571 . .086 .08629 * .08658 . .08687 • .08716 . .99756 •99754 •99752 •9975 •99748 .99746 •99744 •99742 •9974 •99738 .99736 •99734 •9973i •99729 .99727 •99725 .99723 • 99721 .99719 .93716 .99714 .99712 .9971 .99708 •99705 •99703 .99701 .99699 .99696 .99694 •99692 .99689 .99687 .99685 .99683 .9968 .99678 .99676 .99673 .99671 . 99668 .99666 .99664 .99661 .99659 •99657 .99654 .99652 .99649 .99647 .99644 .99642 .99639 99 6 37 99635 • 99632 9963 • 99627 . 99625 . 99622 99619 . .o87i .0874 .0877. .0880 .0883' .0886 .0888c .0891$ .08947 .08976 .09005 •09034 .09063 .09092 .09121 .0915 •°9 I 79 .09208 .09237 .09266 .09295 •09324 •09353 .09382 .09411 •0944 . 09469 .09498 .09527 •09556 •09585 .09614 .09642 .09671 .097 .09729 .09758 .09787 .09816 •09845 .09874 .09903 •09932 .09961 •°999 . 10019 . 10048 10077 . 10106 •10135 .10164 . 10192 .10221 • 1025 .10279 ■ . 10308 . ID 337 • 10366 . 10395 • 10424 . IQ 453 • 5 .9961c •9961; .99614 .99612 .99605 .99607 .99604 . 99602 •99599 •99596 •99594 •9959i .99588 .99586 •99583 •9958 •99578 •99575 •99572 •9957 •99567 •99564 •99562 •99559 •99556 •99553 •99551 .99548 •99545 .99542 •9954 •99537 •99534 •9953i •99528 •99526 •99523 •9952 •99517 •99514 •995II .995o8 .99506 •99503 •995 •99497 •99494 •9949 1 .99488 •99485 .99482 •99479 ■99476 •99473 9947 99467 99464 99461 99458 99455 • 99452 . ) - io 45; r . 1048: l -1051: ■ • 1054 ) • 1056c ' • 10597 . 1062^ ! -1065c i . 10684 . IO713 • IO742 .IO771 . 108 . 10829 . 10858 .10887 .IO916 • io 945 • io 973 . 1 1002 .11031 .1106 . 1 1089 .11118 .H147 .11176 .11205 .11234 .11263 .11291 .1132 • XI 349 .11378 .11407 .11436 .11465 .11494 • ri 5 2 3 •11552 .1158 .11609 .11638 .11667 .11696 .11725 • II 754 • H783 . 11812 .1184 .11869 .11898 .11927 .11956 .11985 . .12014 . • I2 043 . .12071 . . 121 .12129 • .12158 . .12187 • 3 -9945= 2 .9944c t .9944^ •99443 ) -9944 r -99437 > -99434 ; .99431 ■ .99428 • 99424 •99421 .99418 •99415 .99412 •99409 • 99406 .99402 •99399 •99396 •99393 •9939 .99386 •99383 •9938 •99377 •99374 •9937 •99367 •99364 •9936 •99357 •99354 •99351 •99347 •99344 •9934i •99337 •99334 •9933i •99327 •99324 •9932 .99317 •99314 •993i •99307 •99303 • •993 .99297 .99293 •9929 . 99286 .99283 .99279 99276 99272 . 99269 . 99265 . 99262 . 99258 . 99 2 55 • 2 .1218; ) . I22l( ) .1224c ! .12274 .12302 •12331 .1236 .12389 . 12418 .12447 .12476 .12504 •12533 . 12562 .12591 .1262 .12649 .12678 . 12706 • I2 735 .12764 • i2 793 .12822 .12851 . 1288 . 12908 • 12937 .12966 • I2 995 • 13024 •13053 .13081 .1311 • I 3 I 39 .13168 • I 3 I 97 •13226 •13254 .13283 •13312 • I 334 I •1337 •13399 • i 34 2 7 • i 3456 •13485 •I35I4 •13543 •13572 .136 .13629 .13658 , •13687 . •13716 . • 3 3744 • 13773 • 13802 . *3831 • 1386 . 13889 . I 39 I 7 • 1 -99255 60 > -99251 59 > .99248 58 !• -99244 57 ! .9924 56 : -99237 55 •99233 54 1 -9923 53 .99226 52 .99222 51 .99219 50 .99215 49 .99211 48 .99208 47 . 99204 46 •992 45 .99197 44 •99193 43 .99189 42 .99186 41 .99182 40 .99178 39 .99175 38 •99 I 7 I 37 •99 i6 7 36 •99163 35 .9916 34 .99156 33 •99 I 5 2 32 •99 j 48 31 •99 I 44 3° .99141 29 .99137 28 •99 J 33 27 .99129 26 •99 I2 5 25 .99122 24 : .99 j i8 23 : .99114 22 .9911 21 j .99106 20 ] .99102 19 ] .99098 18 1 •99094 17 1 .99091 16 1 .99087 l5 1 •99083 14 1 .99079 13 1 .99075 12 1 .99071 11 1 .99067 10 1 .99063 9 1 .99059 8 1 99°55 7 0 99051 6 0 99047 5 0 99043 4 0 99039 3 0 99°35 2 0 99031 1 0 99027 0 0 N. COB. I 85° sine. N. cos. I II 84° 4. sine. N. cos. I 83o 4. sine. ] N - . cos. P 82° 4. sine. ' Prop. parts. 39 2 natural sines and cosines. p*2 0 T. 8° e- 2 . 28 ' N . sine. I 0 " 0 .13917 ■ 0 1 .13946 . 1 2 •13975 • 1 3 .14004 . 2 4 •M033 * 2 5 .14061 . 3 6 .1409 . 3 7 .14119 • 4 8 .14148 . 4 9 .14177 • 5 10 .14205 . 5 11 .14234 . 6 12 .14263 . 6 13 .14292 7 14 .1432 7 i5 •14349 7 16 •14378 8 17 .14407 8 18 .14436 9 19 .14464 9 20 •14493 10 21 .14522 10 22 .14551 11 23 .1458 11 24 . 14608 12 25 •14637 12 26 .14666 13 27 • 14695 13 28 •14723 14 29 -X4752 1 \ 30 .14781 ,99019 •99 OI 5 .99011 .99006 . 99002 .98994 .9899 .98986 ,98982 ,98978 •98973 . 98969 .98965 .98961 •98957 .98953 .98948 .98944 .9894 .98936 .98931 .98927 .98923 .98919 .98914 •<*9? . 98906 . 98902 .98897 .98893 •15643 .15672 • 15701 •1573 •15758 •15787 .15816 •15845 •15873 .15902 • I 593 I •15959 .15988 .16017 . 16046 .16074 .16103 . 16132 . 1616 .16189 16218 [6246 16275 16304 i6333 6361 1639 16419 16447 6476 .98876 .98871 .98867 .98863 .98858 .98854 .98849 .98845 .98841 .98836 .98832 .98827 .98823 .98818 .98814 . 98809 .98805 .988 .98796 •9879 1 .98787 .98782 .98778 •98773 •98769 N. cos. N.sirte* 61 ° 6591 1662 .1682 D N. cos. j j 10° Ps T . sine. 1 1 S’, cos;* | N .98769 .17365 • 98481 J .1 .98764 •17393 ■ .98476 .1 .9876 .17422 . ,98471 .1 •98755 .17451 ■ ,98466 j .: •98751 •17479 .98461 j .: .98746 .17508 .98455 . .98741 •17537 .9845 •98737 •17565 .98445 • .98732 •17594 .9844 .98728 .17623 .98435 . •98723 .17651 .9843 .98718 .1768 .98425 . .98714 .17708 .9842 .98709 .17737 .98414 • .98704 .17766 .98409 •987 •17794 .98404 • .98695 .17823 .98399 • .9869 .17852 .98394 • .98686 . 1788 .98389 1 . .98681 .17909 .98383 .98676 •17937 •98378 .98671 . 17966 9 o 3 aq .98667 •17995 .98368 .98662 .18023 .98362 .98657 . 18052 .98357 .98652 .18081 .98352 .98648 .18109 •98347 .98643 .18138 .98341 .98638' .18166 •98336 ' -98633 .18195 •98331 ; .98629 .18224 .98325 1 .98624 .18252 .9832 - .98619 .18281 •98315 .98614 .18309 .9831 .98609 .18338 .98304 5 .98604 .18367 .98299 r .986 .18395 .98294 > -98595 .18424 .98288 \ -9859 .18452 .98283 3 .98585 .18481 .98277 2 .9858 .18509 .98272 •98575 .18538 .98267 ? -9857 .18567 .98261 8 .93565 .18595 .98256 6 .98561 .18624 •9825 5 -98556 . 18652 .98245 4 -9855 1 . r868i .9824 2 .98546 .1871 .98234 1 .9854 1 .18738 .98229 i .98536 | .18767 .98223 ’8 .98531 1 -18795 , .98218 >7 .98526 1 .18824 . .98212 >6 .98521 .18852 ! . 98207 34 .98516 > .18881 .982OI )3 -98511 .1891 .98196 22 .98506 > .1893? l .9819 5 .98501 t .18965 r .98185 79 -9 8 49 ( 3 .1899; 5 -98x79 08 .98491 i .19024 .90174 36 . 9848* S .19052 j .98168 65 .9848 1 j .19081 j .98163 11 ° N. sine. N. cos. 19281 19309 ^9338 19366 *9395 19423 19452 [9481 19509 '9538 19566 19595 19623 .19652 .1968 19709 19737 .19766 19794 .19823 19851 1988 19908 19937 • 19965 19994 .20022 .2005 . 20079 -20108 .20136 .20165 .20193 . 20222 .2025 . 20279 . 20307 .20336 . 20364 .20393 .20421 .2045 . 20478 .20507 .20535 .20563 . 20592 .2062 . 20649 .20677 .20706 .20734 .20763 .20791 N. cos. | N. sine, j 800 790 .98004 .97988 .97992 .97987 .97981 •97975 •97969 •97963 •97958 •97952 •97946 •9794 •97934 .97928 .97922 •979 l6 .9791 .97905 .97899 •97893 .97887 .97881 .97875 .97869 .97863 .97857 •97851 •97845 .97839 ! •97833 ! .97827 .97821 j .97815 N. cos. ! N. sine, j 780 ] Prop. Cn parts. NATURAL SINES AND COSINES. © t £ 2 7 I . ' N. sine 12° 13° I40 150 Prop. parts. o c .20791 •97815 •22495 •97437 .24192 •9703 to 00 •96593 60 9 o 1 .2082 .97809 •22523 •9743 .2422 .97023 .2591 •96585 59 9 I 2 . 20848 •97803 •22552 •97424 .24249 •97015 •25938 .96578 58 9 I 0 .20877 •97797 .2258 •97417 •24277 .97008 .25966 •9657 57 9 2 < .20005 .97791 .22608 .97411 •24305 .97001 •25994 .96562 56 8 2 5 .20933 .97784 .22637 •97404 •24333 •96994 . 26022 •96555 55 8 3 6 .20962 •97778 .22665 •97398 •24362 .96987 .2605 •96547 54 8 3 7 .2099 .97772 .22693 • 9739 1 •2439 .9698 .26079 •9654 53 8 4 8 .21019 .97766 .22722 •97384 .24418 •96973 .26107 .96532 52 8 4 9 .21047 .9776 •2275 •97378 .24446 .96966 •26135 •96524 5 i 8 5 IO .21076 •97754 .22778 • 9737 i •24474 •96959 .26163 •96517 50 8 5 11 .21104 .97748 .22807 •97365 •24503 •96952 . 26191 •96509 49 7 5 12 .21132 .97742 .22835 •97358 •24531 •96945 .26219 .96502 48 7 6 13 .21161 •97735 .22863 • 9735 i •24559 •96937 .26247 .96494 47 7 6 14 .21189 .97729 .22892 •97345 •24587 •9693 .26275 .96486 46 7 7 IS .21218 •97723 .2292 •97338 •24615 •96923 •26303 .96479 45 7 7 16 .21246 .97717 .22948 •97331 . 24644 .96916 .26331 .96471 44 7 8 1 7 .21275 .97711 .22977 •97325 .24672 .96909 •26359 •96463 43 6 8 18 .21303 •97705 •23005 • 973 i 8 •247 .96902 .26387 .96456 42 6 9 i 9 •21331 .97698 •23033 • 973 H .24728 .96894 •26415 .96448 4 i 6 9 20 .2136 .97692 .23062 •97304 .24756 .96887 • 26443 •9644 40 6 9 21 .21388 .97686 •2309 .97298 •24784 .9688 .26471 •96433 39 6 IO 22 .21417 .9768 .23118 .97291 •24813 •96873 •265 •96425 38 6 IO 23 .21445 •97673 •23146 .97284 .24841 .96866 .26528 .96417 37 6 II 24 .21474 .97667 •23175 •97278 . 24869 .96858 •26556 .9641 36 5 II 25 .21502 .97661 •23203 .97271 .24897 .96851 .26584 .96402 35 5 12 26 •2153 •97655 •23231 .97264 •24925 .96844 .26612 •96394 34 5 12 27 •21559 .97648 .2326 •97257 •24954 •96837 .2664 .96386 33 5 13 28 .21587 .97642 .23288 •97251 . 24982 .96829 .26668 •96379 32 5 13 2 9 .21616 •97636 •23316 .97244 .2501 96822 . 26696 •96371 3 1 5 14 30 .21644 •9763 •23345 .97237 •25038 .96815 .26724 •96363 3 ° 5 14 3 i .21672 .97623 •23373 •9723 . 25066 . 96807 ■26752 •96355 29 4 14 32 .21701 .97617 • 23401 •97223 .25094 .968 .2678 •96347 28 4 15 33 .21729 .97611 •23429 .97217 .25122 •96793 .26808 •9634 27 4 15 34 .21758 .97604 •23458 .9721 •25151 .96786 . 26836 •96332 26 4 l6 35 .21786 •97598 .23486 .97203 •25179 .96778 .26864 •96324 25 4 l6 36 .21814 •97592 •23514 .97196 .25207 .96771 .26892 .96316 2 4 4 *7 37 .21843. •97585 •23542 .97189 •25235 .96764 .2692 . 96308 23 3 17 38 .21871 •97579 •23571 .97182 •25263 •96756 . 26948 .96301 22 3 18 39 .21899 •97573 •23599 .97176 •25291 .96749 .26976 •96293 21 3 18 40 .21928 .97566 •23627 .97169 •2532 •96742 .27004 .96285 20 3 18 4 i .21956 •9756 •23656 .97162 •25348 •96734 .27032 .96277 J 9 3 *9 42 .21985 •97553 .23684 •97155 •25376 .96727 .2706 .96269 18 3 43 .22013 •97547 .23712 • 97!48 .25404 .96719 . 27088 .96261 17 3 20 44 .22041 •97541 •2374 .97141 •25432 .96712 .27116 •96253 16 2 20 45 .2207 •97534 •23769 •97134 •2546 .96705 .27144 -96246 15 2 21 46 .22098 .97528 •23797 .97127 .25488 .96697 .27172 .96238 !4 2 21 47 .22126 •97521 .23825 .9712 .25516 .9669 .272 •9623 13 2 22 48 •22155 •97515 •23853 • 97 H 3 •25545 .96682 .27228 .96222 12 2 22 49 .22183 .97508 .23882 .97106 •25573 •96675 .27256 .96214 11 2 23 50 .22212 .97502 .2391 .971 .25601 . 96667 .27284 . 96206 10 2 23 5 i .2224 .97496 •23938 .97093 .25629 .9666 .27312 .96198 9 1 23 52 .22268 •97489 .23966 .97086 •25657 •96653 •2734 .9619 8 1 24 53 .22297 •97483 •23995 •97079 .25685 •96645 .27368 .96182 7 1 24 54 .22325 .97476 . 24023 .97072 •25713 .96638 .27396 .96174 6 1 25 55 •22353 •9747 .24051 .97065 •25741 •9663 .27424 .96166 5 1 25 : , 56 .22382 •97463 .24079 .97058 .25769 .96623 •27452 .96158 4 1 26 57 .2241 •97457 .24108 •97051 •25798 •96615 .2748 .9615 3 0 26 1 58 .22438 •9745 .24136 .97044 .25826 .96608 .27508 .96142 2 0 27 ; 59 .22467 . .97444 .24164 •97037 .25854 , .966 •27536 .96134 1 0 27 | < 5 o .22495 . 97437 .24192 •9703 .25882 ■96593 .27564 .96126 0 X. cos. ] X. sine. N. cos. 1 j X. sine. X. cos. ] X. sine. N. cos. 1 1 S’, sine. ✓ » 770 750 75 ° 1 74 ° 1 Prop. 394 NATURAL SINES AND COSINES. 16° N. sine. 10 23 II 24 II 25 12 26 12 27 13 28 13 29 14 30 14 3i 32 15 33 15 34 l6 35 l6 36 17 37 17 38 18 39 18 40 18 4i i9 42 *9 43 20 44 20 45 21 46 21 47 23 •27564 .27592 .2762 .27648 .27676 .27704 •27731 •27759 .27787 .27815 .27843 .27871 .27899 .27927 •27955 .27983 .28011 . 28039 . 28067 .28095 .28123 .2815 .28178 .28206 .28234 .28262 .2829 .28318 .28346 .28374 .28402 .28429 .28457 .28485 .28513 .28541 .28569 .28597 .28625 .28652 .2868 .28708 .28736 .28764 .28792 .2882 .28847 .28875 . 28903 •28931 .28959 .28987 .29015 .29042 .2907 . 29098 .29126 •29154 .29182 .29209 .29237 .96126 .96118 .9611 .96102 . 96094 . 96086 .96078 .9607 . 96062 .96054 . 96046 .96037 . 96029 .96021 .96013 . 96005 •95997 .95989 .95981 •95972 .95964 •95956 •95948 •9594 •9593i •95923 •959 I 5 •959°7 .95898 •9589 .95882 •95874 .95865 •95857 .9584] •95832 .958: 958 i •9574 •9569 N. cos. 73° 17° N. sine, j N. cos. N, •29237 •9563 -3 .29265 .95622 .3 .29293 .95613 -3 .29321 •95605 -3 .29348 .95596 -3 •29376 •95588 .3 .29404 •95579 -3 .29432 •9557 1 -3 .2946 •95562 .3 .29487 •95554 -3 •295X5 •95545 -3 •29543 •95536 .3 •29571 •95528 .3 •29599 •95519 -3 .29626 •955ii .3 .29654 •95502 .3 .29682 •95493 -3 .2971 •95485 -3 •29737 •95476 •: .29765 •95467 •: •29793 •95459 •: .29821 •9545 •: .29849 •95441 •: .29876 •95433 •: .29904 • 95424 •: .29932 •95415 •: .2996 •95407 •: .29987 •95398 • .30015 •95389 • .30043 •9538 .30071 •95372 • .30098 •95363 • .30126 •95354 • •30154 •95345 • .30182 •95337 • .30209 •95328 . •30237 •95319 • .30265 •953i > . 30292 •953oi • r .3032 •95293 • ) .30348 .95284 • : . 30376 •95275 2 .30403 .95266 I- -30431 •95257 3 .30459 .95248 7 . 30486 •9524 9 -30514 .95231 •30542 .95222 2 .3057 .95213 4 -30597 •95204 5 • 30625 7 -30653 , .95186 8 . 3068 •95!77 . 30708 ! .95168 i -3073^ > -95I59 3 -30763 5 -9515 4 -3079 1 : -95142 ,6 .3081c ) -95133 .7 . 3 o84( ) .95124 19 • 3087^ t -95115 1 .309°' 2 .95106 ie. N. cos. | N. sine. 72° 18o N. cos. •31593 .3162 .31648 .31675 •3i7 0 3 •3173 95106 95097 95088 95079 9507 95061 95052 95043 95033 95024 950X5 95006 94997 94988 94979 9497 94961 94952 94943 94933 94924 949 1 5 94906 94897 94878 9486c 9486 94851 9474 9473 ,31868 31896 31923 3I95I 31979 .32006 .32034 .32061 .32089 .32116 •32144 .3217 .32199 .32227 •32254 .32282 •32309 ‘32337 •32364 • 32392 •32419 • 32447 •32474 .32502 •32529 •32557 I -94552 710 19° N. sine. | N. cos. •32557 •94552 6c •32584 •94542 59 .32612 •94533 5* .32639 •94523 5/ . 32667 •94514 5< .32694 •94504 5; .32722 •94495 5< •32749 .94485 5; •32777 •94476 5i .32804 .94466 5 : •32832 •94457 5< .32859 •94447 4< •32887 •94438 4' •32914 • 94428 4' •32942 .94418 4 { . 32969 •94409 4 .32997 •94399 4 •33024 •9439 4 •33051 •9438 4 •33079 •9437 4 . 33106 .94361 4 j -33134 94351 3 .33i6i •94342 3 •33189 •94332 3 .33216 •94322 3 •33244 •94313 3 1 -33271 •94303 3 •33298 •94293 3 •33326 .94284 3 : ! -33353 •94274 3 : -33381 .94264 3 i -33408 •94254 1 ^ -33436 •94245 2 5 -33463 •94235 2 > -3349. .94225 J 5 .335x8 .94215 : 7 -33545 . 94206 : 5 -33573 .94196 : 3 .336 .94186 9 *33627 .94176 .33655 •94i67 .33682 •94157 1 -3371 •94147 2 -33737 •94137 2 .33764 •94127 3 .33792 .94118 4 .33819 .94108 4 -33846 .94098 >5 -33874 . 94088 ,6 .33901 .94078 ^6 .33929 .94068 \7 -33956 .94058 57 -33983 •94049 c8 .34011 •94039 39 .34038 .94029 )9 -34065 .94019 9 -34093 .94009 3 .3412 7 1 -34H7 •93999 •93989 5i .34175 •93979 52 .34202 •93969 ne. N. cos. N. sine. 70° 9 9 9 9 8 8 8 8 8 8 8 7 ,7 7 7 7 7 6 6 6 6 6 6 6 5 5 5 5 5 5 5 4 4 4 4 4 4 3 3 3 3 3 3 3 NATURAL SINES AND COSINES, 395 c.® C Z 20 ° 210 220 27 ' N. sine. N. cos. N. sine. N. cos. N. sine. 1 j N. cos. o 0 .34202 •93969 •35837 •93358 • 3746 i .92718 o z .34229 •93959 •35864 •93348 •37488 . 92707 I 2 •34257 •93949 •35891 •93337 •37515 .92697 I 3 .34284 •93939 • 359 l8 •93327 •37542 .92686 2 4 • 343 ** .93929 •35945 • 933 x 6 •37569 •92675 2 5 •34339 • 939*9 •35973 • 933 o 6 •37595 .92664 3 6 .34366 •93909 •36 •93295 .37622 .92653 3 7 •34393 •93899 .36027 •93285 •37649 .92642 4 8 .34421 .93889 •36054 •93274 •37676 .92631 4 9 1 -34448 •93879 .36081 .93264 •37703 .9262 5 IO •34475 .93869 .36108 •93253 •3773 .92609 5 11 •34503 •93859 •36135 •93243 •37757 .92598 5 12 •3453 •93849 .36162 •93232 •37784 .92587 6 13 •34557 •93839 •3619 .93222 .37811 .92576 6 14 •34584 •93829 .36217 .93211 •37838 •92565 7 i 5 .34612 .93819 .36244 .93201 •37865 •92554 7 16 •34639 •93809 .36271 • 93 x 9 •37892 •92543 8 17 .34666 •93799 .36298 .9318 • 379*9 •92532 8 18 .34694 •93789 •36325 .93169 •37946 .92521 9 i 9 •34721 •93779 •36352 •93159 •37973 .9251 9 20 •34748 •93769 •36379 .93148 •37999 .92499 9 21 •34775 •93759 .36406 • 93 x 37 .38026 .92488 IO 22 •34803 •93748 •36434 •93127 •38053 .92477 IO 23 •3483 •93738 .36461 .93116 .3808 . 92466 ii 24 34857 •93728 .36488 .93106 •38107 •92455 ii 25 .34884 • 937*8 •36515 •93095 •38134 •92444 12 26 • 349 12 .93708 •36542 •93084 .38161 .92432 12 27 •34939 .93698 •36569 •93074 .38188 .92421 13 1 28 .34966 .93688 •36596 .93063 •38215 .9241 *3 29 •34993 •93677 .36623 •93052 .38241 •92399 14 30 .35021 •93667 •3665 ,93042 .38268 .92388 i 4 3 i •35048 •93657 •36677 •93031 •38295 •92377 14 32 •35075 •93647 •36704 •9302 •38322 .92366 15 33 •35102 •93637 •36731 .9301 •38349 •92355 15 34 •3513 .93626 •36758 .92999 •38376 •92343 16 3 | •35157 .93616 •36785 .92988 •38403 •92332 16 36 •35184 .93606 .36812 .92978 •3843 .92321 i 7 37 • 352 U •93596 •36839 •92967 •38456 .9231 i 7 3 8 •35239 •93585 .36867 .92956 •38483 . 92299 18 39 .35266 •93575 • 36894 .92945 •3851 .92287 18 40 •35293 •93565 .36921 •92935 •38537 .92276 1 8 41 •3532 •93555 .36948 .92924 •38564 .92265 ^9 42 •35347 •93544 •36975 .92913 .3859* .92254 19 43 •35375 •93534 .37002 . 92902 •38617 .92243 20 44 •35402 •93524 •37029 .92892 .38644 .92231 20 45 •35429 •93514 •37056 .92881 .38671 .9222 21 46 •35456 •93503 •37083 .9287 .38698 . 92209 21 47 •35484 •93493 • 37 11 .92859 •38725 .92198 22 48 • 355 ** •93483 • 37 x 37 .92849 •38752 .92186 22 49 •35538 •93472 •37164 .92838 •38778 • 9 2X 75 23 5 o •35565 .93462 • 37 I 9 I .92827 .38805 .92164 23 5 i •35592 •93452 .37218 .92816 •38832 .92152 23 52 •35619 •93441 •37245 92805 •38859 .92141 24 53 •35647 •93431 •37272 •92794 .38886 .9213 24 54 •35674 •9342 •37299 .92784 .38912 .92119 25 55 • 357 oi • 934 i •37326 •92773 •38939 .92107 25 56 •35728 •934 •37353 .92762 .38966 . 92096 26 57 •35755 •93389 •3738 •92751 •38993 .92085 26 58 •35782 •93379 •37407 •9274 .3902 • 92073 27 • 358 i •93368 •37434 .92729 •39046 . 92062 27 60 •35837 •93358 • 3746 i .92718 •39073 .9205 N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. 65 ° 68 ° 1 67 ° 23 ° N. sine. N. cos. 11 •39073 .9205 60 11 • 39 * •92039 59 11 • 39 I2 7 .92028 58 11 • 39*53 .92016 57 10 • 39*8 .92005 56 10 •39207 • 9*994 55 10 •39234 .91982 54 10 • 3926 9 * 97 * 53 10 .39287 • 9*959 52 10 • 393*4 .91948 5 * 9 • 3934 * • 9*936 5 o 9 •39367 • 9*925 49 9 •39394 • 9 * 9*4 48 9 .39421 .91902 47 9 •39448 .91891 46 8 •39474 • 9*879 45 8 • 395 oi .91868 44 8 •39528 .91856 43 8 •39555 • 9*845 42 8 • 3958 i • 9*833 4 i 8 .39608 .91822 4 o 7 •39635 .9181 39 7 .39661 • 9*799 38 7 .39688 .91787 37 7 • 397*5 • 9*775 36 7 • 3974 * .91764 35 6 •39768 • 9*752 34 6 •39795 .91741 33 6 .39822 .9*729 32 6 .39848 .91718 3 * 6 •39875 .91706 30 6 . 39902 .91694 29 5 •39928 .91683 28 5 •39955 .91671 27 5 .39982 .9166 26 5 . 40008 .91648 25 5 •40035 .91636 24 4 .40062 •9*625 23 4 . 40088 .9*613 22 4 •40**5 .91601 21 4 .40141 • 9*59 20 4 .40168 • 9*578 *9 3 .40195 .9*566 18 3 .40221 • 9*555 *7 3 .40248 • 9*543 16 3 •40275 • 9 * 53 * *5 3 .40301 • 9 * 5*9 *4 3 .40328 •9*508 *3 2 •40355 .91496 12 2 .40381 .91484 11 2 . 40408 .91472 10 2 •40434 .91461 9 2 .40461 • 9*449 8 1 .40488 • 9*437 7 1 .40514 • 9*425 6 1 •40541 • 9 * 4*4 5 1 •40567 .91402 4 1 •40594 • 9*39 3 1 .40621 • 9*378 2 0 .40647 .91366 1 0 .40674 • 9*355 0 0 N. cos. N. sine. ' 660 Prop. parts. NATURAL SINES AND COSINES, 39 6 Oh .2 2 a 24 ° 250 26 ° 270 Prop. parts. Ph p. 26 ' N. sine. N. cos. N. Bine. N, cos. N. sine. N. cos. N. sine. N. cos. 14 0 0 .40674 • 9 I 355 .42262 .90631 .43837 .89879 •45399 .89101 60 14 0 1 • 4°7 • 9 I 343 .42288 .90618 .43863 .89867 •45425 . 89087 59 o 4 1 2 .40727 • 9^331 •42315 . 90606 .43889 .89854 •45451 .89074 58 14 1 • 4°753 ■ 9 I 3*9 •42341 •90594 .43916 .89841 •45477 .89061 57 13 2 4 .4078 • 9 I 3°7 .42367 .90582 •43942 .89828 •45503 .89048 56 3 2 5 .40806 .91295 •42394 .90569 .43968 .89816 45529 •89035 55 13 3 6 • 4 o8 33 .01283 .4242 •90557 •43994 . 89803 •45554 .89021 54 13 3 7 .4086 .91272 .42446 •90545 .4402 .8979 4558 .89008 53 12 3 8 .40886 .9126 •42473 •90532 .44046 •89777 .45606 .88995 52 12 4 9 • 4 ° 9 I 3 .91248 •42499 •9052 .44072 .89764 •45632 .88981 5 i 12 4 10 .40939 .01236 •42525 .90507 .44098 •89752 •45658 .88968 50 12 5 11 . 40966 .91224 •42552 •90495 .44124 •89739 .45684 •88955 4 9 II 5 12 .40992 .91212 •42578 .90483 • 44 i 5 i .89726 • 457 i .88942 48 II 6 13 .41019 .912 .42604 •9047 •44177 •89713 •45736 .88928 47 1 6 14 • 4 io 45 .91188 .42631 .90458 •44203 •897 •45762 .88915 46 II 7 15 .41072 .91176 .42657 .90446 .44229 .89687 •45787 .88902 45 II 7 16 .41098 .91164 .42683 •90433 •44255 .89674 •45813 .88888 44 IO 7 17 • 4 II2 5 .91152 .42709 .90421 .44281 . 89662 •45839 •88875 43 IO 8 18 • 4 II 5 I • 9 II 4 .42736 . 90408 •44307 .89649 •45865 .88862 42 IO 8 19 .41178 .91128 .42762 .90396 •44333 .89636 •45891 . 88848 41 IO 9 20 .41204 .91116 .42788 •90383 •44359 .89623 •45917 .88835 4 ° 9 9 21 .41231 .91104 .42815 • 9037 1 •44385 .8961 •45942 .88822 39 9 10 22 •41257 . QIOQ 2 .42841 .90358 .44411 •89597 .45968 . 88808 38 9 10 23 .41284 .9108 .42867 .90346 •44437 .89584 •45994 .88795 37 g 10 24 .413 1 .91068 .42894 •90334 •44464 - 8957 X 4602 .88782 36 8 8 11 2 5 • 4 X 337 .91056 .4292 .90321 •4449 •89558 . 46046 .88768 35 11 26 .41363 .91044 .42946 •90309 .44516 •89545 .46072 •88755 34 8 12 27 • 4 I 39 .9IO32 •42972 .90296 •44542 •89532 46097 .88741 33 8 12 28 .41416 .9IO2 •42999 .90284 .44568 .89519 .46123 .88728 32 7 13 29 • 4 T 443 .91008 .43025 90271 •44594 . 89506 .46149 .88715 31 7 13 3 ° .41469 .90996 •43051 .90259 .4462 .89493 •46175 .88701 3 ° 7 13 31 .41496 . 90984 •43077 .90246 .44646 . 8948 46201 . 88688 29 ; 7 14 32 .41522 .QOQ 72 .43104 • 9 02 33 •44672 .89467 .46226 .88674 2S 7 14 33 • 4 i 549 .9096 •4313 .90221 .44698 .89454 .46252 .88661 27 6 15 34 • 4 i 575 .90948 •43156 . 90208 •44724 .89441 .46278 .88647 26 6 15 35 .41602 .90936 .43182 .90196 •4475 .89428 .46304 .88634 25 6 16 36 .41628 .90924 .43209 .90183 •44776 .89415 •4633 .8862 24 6 16 37 .41655 .909II •43235 .90171 . 44802 . 89402 •46355 .88607 23 5 16 38 .41681 .90899 .43261 90158 .44828 .89389 .46381 •88593 22 5 17 39 .41707 .90887 •43287 .90146 •44854 •89376 .46407 .8858 21 5 17 40 • 4 I 734 .90875 •43313 • 9 OI 33 . 4488 •89363 •46433 .88566 20 5 18 41 .4176 .90863 •4334 .9012 .44906 •8935 .46458 •88553 *9 4 18 42 .41787 .90851 .43366 .90108 •44932 •89337 .46484 .88539 18 4 19 43 .41813 .90839 • 4339 2 .90095 •44958 .89324 .4651 .88526 17 4 19 44 .4184 . 90826 .43418 90082 •44984 • 89 3 » •46536 .88512 10 i 4 20 45 .41866 .90814 •43445 .9007 .4501 .89298 .46561 .88499 15 i 4 20 46 .41892 . 90802 • 4347 i •90057 •45036 .89285 •46587 . 88485 14 1 3 20 47 • 4 I 9 I 9 .9079 •43497 .90045 .45062 .89272 .46613 .88472 13 1 l 3 21 48 •41945 .90778 •43523 .90032 .45088 .89259 .46639 .88458 12 3 21 49 .41972 .90766 •43549 .90019 • 45“4 .89245 .46664 •88445 II | 3 22 5 ° .41998 • 9°753 • 43575 . .90007 .4514 .89232 .4669 .88431 1° ] 2 22 51 .42024 • 9 ° 74 I . 43602 • 89994 .45166 .89219 .46716 .88417 9 8 2 23 * 52 .42051 .90729 .43628 .89981 .45192 . 89206 •46742 . 88404 2 2 3 53 .42077 .90717 •43654 . 89968 .45218 • 0 91 ? 3 •46767 .8839 7 2 2 3 54 .42104 .90704 4368 .89956 • 45 2 43 .8918 •46793 •88377 O 1 24 55 .4213 . 90692 .43706 •89943 .45269 .89167 .46819 .88363 1 24 56 .42156 .9068 •43733 •8993 •45295 •89153 .46844 .88349 \ 1 2 5 57 .42183 .90668 •43759 .89918 4532 i .8914 .4687 .88336 3 1 2 5 26 58 59 .42209 .42235 • 9 0 655 .90643 •43785 .43811 .89905 . 89892 •45347 •45373 .89127 .89114 .46896 .46921 .88322 88308 2 1 0 0 26 60 .42262 .90631 •43837 .89879 •45399 1 .89101 •46947 8829; 1 1 0 0 N. cos. ( N. sine. 55° N. cos. * N. siue. 64° N. cos. ^ N. sine. 1 63° N. cos. < N. sine 52° NATURAL SINES AND COSINES. 397 & 25 : 1 N. sine. 28° . N. cos. 0 0 .46947 .88295 0 1 •46973 .88281 I 2 •46999 .88267 I 3 .47024 .88254 2 4 •4705 .8824 2 3 .47076 .88226 3 6 .47101 .88213 3 7 .47127 .88199 3 8 •47153 .88185 4 9 .47178 ‘oo 172 4 10 .47204 .88158 1 5 11 .47229 .88144 5 12 •47255 .8813 5 *3 .47281 .88117 6 *4 47306 . 88103 6 i5 •47332 .88089 7 16 •47358 .88075 7 17 •47383 . 88062 8 18 .47409 .88048 8 *9 •47434 . 88034 8 20 .4746 .8802 9 21 .47486 .88006 9 22 •47511 .87993 10 23 •47537 .87979 10 24 .47562 .87965 10 25 .47588 •87951 11 26 .47614 •87937 11 27 •47639 .87923 12 28 .47665 •87909 12 29 .4769 .87896 *3 30 •477 l6 .87882 *3 3i •4774i I .87868 . *3 32 •47767 ! •87854 ■ 1 4 33 •47793 j .8784 J 4 34 •47818 ! .87826 . r 5 35 •47844 .87812 15 36 .47869 ■ .87798 . 15 37 •47895 .87784 . 16 38 .4792 | .8777 16 39 •47946 | .87756 . *7 40 •47971 1 •87743 • *7 4i •47997 ! .87729 . 18 42 .48022 j •87715 . 18 43 .48048 .87701 18 44 •48073 ! .87687 *9 45 .48099 .87673 . z 9 46 .48124 .87659 . 20 47 .4815 .87645 . 20 48 •48175 .87631 . 20 49 48201 .87617 21 5o .48226 . 87603 21 5i .48252 .87589 . 22 52 .48277 •87575 • 22 53 •48303 •87561 . 2 3 54 .48323 • 87546 • 23 55 •48354 .87532 . 2 3 56 •48379 •87518 . 24 57 •48405 .87504 ., 24 58 •4843 .8749 25 i 9 • 48456 .87476 .. 25 60 .48481 .87462 .; N. cos. N. sine. P 6lo .48481 .48506 .48532 •48557 .48583 . 48608 .48634 .48659 .48684 .4871 •48735 .48761 48786 48811 48837 .48913 ,48938 48964 48989 49014 4904 49065 4909 49116 49141 .49166 .49192 49217 .49242 49268 49293 .49318 •49344 493% 49394 49419 •49445 4947 49495 49521 .49546 4957i 4959 6 49622 49047 49748 290 N. sine. N. cos. .87462 .87448 •87434 .8742 ■ 87406 •87391 •87377 .87363 •87349 •87335 .87321 .87306 .87292 .87278 .87264 .8725 •87235 .87221 .87207 •87193 87178 .87164 •8715 .87136 .87121 .87107 87093 .87079 . 87064 .8705 . 87036 . 87021 . 87007 ,86993 .86978 .86964 .86949 •86935 .86921 . 86906 .86892 .86878 .85853 . 86849 •86834 .8682 . 86805 .86791 .86777 . 86762 .86748 •86733 •86719 . 86704 .8669 . 86675 .86661 .86646 .86632 .86617 .86603 300 •5 .50025 .5005 . 50076 .50101 .50126 •50151 .50176 . 50201 .50227 •50252 •50277 . 50302 • 50327 •50352 •50377 • 50403 . 50428 •50453 .50478 • 50503 •50528 •50553 •50578 •50603 . 50628 •50654 .50679 • 50704 • 50729 • 50754 •50779 . 50804 .50829 • 50854 ■ 50879 .50904 . 50929 •50954 •50979 .51004 .51029 •51054 51079 51104 •51129 •5ii54 .51179 .51204 •51229 •51254 .51279 •51304 ■51329 •51354 '51379 ,51404 51429 •51454 •5M79 •51504 .8653 .86471 .86427 .86413 .86398 .86384 . 86369 •86354 8634 86325 .8631 .86295 .86281 .86266 .86251 .86237 .86222 .86207 .86192 .86178 .86163 .86148 .86133 .86119 .86104 . 86089 . 86074 .86059 . 86045 .8603 .86015 .86 •85985 •8597 •85956 .85941 .85926 .85911 .85896 .85881 .85866 •85851 .85836 ( . 85821 .85806 .85792 •85777 .85762 •85747 •85732 • 8 57 r 7 310 I I 3 *51504 •85717 60 3 -51529 .85702 59 3 *51554 .85687 58 ? • 5*579 .85672 57 [ -51604 •85657 56 .51628 .85642 55 5 -51653 .85627 54 .51678 .85612 53 > -51703 •85597 52 .51728 .85582 5i p *51753 •85567 5o .51778 •85551 49 .51803 •85536 48 •5 i 828 •85521 47 .51852 .85506 46 •51877 .85491 45 .51902 .85476 44 •5i9 2 7 .85461 43 •51952 .85446 42 •51977 •85431 4i .52002 .85416 40 . 52026 • 85401 39 •52051 •85385 38 .52076 •8537 37 .52101 •85355 36 .52126 •8534 35 .52151 •85325 34 •52175 •8531 33 . 522 .85294 32 •52225 .85279 3i .5225 .85264 30 •52275 .85249 29 .52299 •85234 28 .52324 .85218 27 •52349 •85203 26 •52374 .85188 25 •52399 •85173 24 •52423 •85157 23 .52448 .85142 22 •52473 .85127 21 .52498 .85112 20 •52522 .85096 J 9 •52547 .85081 18 •52572 . 85066 17 •52597 85051 16 .52621 •85035 15 . 52646 • 8502 14 •52671 .85005 *3 . 52696 . 84989 12 •5272 .84974 11 •52745 .84959 10 •5277 .84943 9 •52794 .84928 8 •52819 .84913 7 .52844 .84897 6 .52869 .84882 5 •52893 .84866 4 .52918 .84851 3 • 52943 .84836 2 .52967 . 8482 1 .52992 .84805 0 ! n. cos. : N. sine, j ' I f 580 | 1 Prop. parts. NATURAL SINES AND COSINES. 39 8 #• « pH P- 23 o O 2 2 2 3 3 3 4 4 5 5 5 6 6 7 7 7 8 8 8 9 9 10 10 10 11 11 12 12 12 13 13 13 14 14 15 15 15 16 16 16 17 17 18 18 18 *9 *9 20 20 20 21 21 21 22 22 23 23 32 c 33 ° 34 ° 35 C < < p H • e* m P. II 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 5 16 17 18 *9 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4 1 42 43 44 45 46 47 48 49 50 5 1 52 53 54 55 56 57 58 £ 52992 • 53 oi 7 •53041 . 53066 •53091 • 53**5 • 53 i 4 •53164 •53189 •53214 •53238 •53263 .53288 •53312 •53337 • 5336 i •53386 • 534 ** •53435 •5346 •53484 •53509 •53534 •53558 •53583 •53607 •53632 •53656 •53681 •53705 •5373 •53754 •53779 •53804 .53828 •53853 •53877 .53902 •53926 • 5395 i •53975 •54 .54024 .54049 •54073 •54097 .54122 .54146 • 54 * 7 * •54195 .5422 .54244 .54269 •54293 •54317 •54342 •54366 • 5439 1 •54415 •5444 • 54464 .84805 .84789 •84774 •84759 •84743 .84728 .84712 .84697 .84681 .84666 .8465 •84635 .84619 . 84604 .84588 •84573 •84557 .84542 .84526 .84511 •84495 . 8448 . 84464 .84448 •84433 • 844 I 7 . 84402 .84386 •8437 .84355 •84339 .84324 .84308 .84292 .84277 .84261 .84245 .8423 .84214 .84198 .84182 .84167 .84151 •84135 .8412 .84104 . 84088 .84072 .84057 .84041 .84025 . 84009 .83994 •83978 83962 .83946 •8393 •83915 .83899 .83883 .83867 .54464 •54488 •54513 •54537 • 5456 i •54586 .5461 •54635 • 54659 • 54683 •54708 •54732 •54756 • 5478 i .54805 .54829 •54854 • 54878 • 549 ° 2 • 549 2 7 •54951 •54975 •54999 .55024 •55048 • 55072 •55097 •55121 •55145 •55169 •55194 •55218 •55242 .55266 •55291 • 553 i 5 •55339 •55363 •55388 •55412 •55436 •5546 •55484 •55509 •55533 •55557 • 5558 i •55605 •5563 •55654 •55678 •55702 •55726 •5575 •55775 •55799 •55823 •55847 •55871 •55895 • 559 I 9 83867 .83851 •83835 .83819 . 83804 .83788 .83772 •83756 •8374 .83724 .83708 .83692 •83676 .8366 .83645 .83629 .83613 •83597 .83581 .83565 •83549 •83533 •83517 .83501 .83485 .83469 •83453 •83437 .83421 .83405 •83389 •83373 .83356 •8334 •83324 .83308 .83292 •83276 .8326 •83244 .83228 .83212 •83195 •83*79 •83163 •83*47 •83131 •83115 .83098 .83082 . 83066 .8305 •83034 .83017 .83001 .82985 . 82969 •82953 .82936 .8292 .82904 * 559 I 9 • •55943 • •55968 • 5599 2 .56016 .5604 . 56064 . 56088 .56112 .56136 .5616 .56184 . 56208 .56232 .56256 .5628 •56305 •56329 • 56353 •56377 . 56401 •56425 •56449 •56473 •56497 .56521 •56545 • 56569 •56593 .56617 .56641 • 56665 .56689 • 567*3 1 .56736 •5676 .56784 .56808 .56832 .56856 .5688 •56904 .56928 .56952 .56976 •57 .57024 •57047 -5707* •57095 • 57**9 • 57*43 • 57*67 • 57 * 9 * •57215 •57238 .57262 .57286 • 573 * •57334 •57358 .82904 .82887 .82871 .82855 .82839 .82822 .82806 .8279 •82773 •82757 .82741 .82724 . 82708 .82692 .82675 .82659 .82643 .82626 .8261 •82593 •82577 .82561 •82544 .82528 .82511 .82495 .82478 .82462 .82446 .82429 .82413 .82396 .8238 - 82363 •82347 •8233 .82314 .82297 .822§I .82264 .82248 .82231 .82214 .82198 .82l8l .82165 .82148 .82132 .82115 . 82C98 . 820§2 . 82065 .82048 .82032 .82015 .81999 .81982 .81965 .81949 .81932 .81915 •57358 • 5738 i •57405 •57429 •57453 •57477 •57501 •57524 •57548 •57572 •57596 •57619 •57643 •57667 •5769* • 577*5 •57738 .57762 .57786 •5781 •57833 •57857 .57881 • 57904 •57928 •57952 •57976 •57999 .58023 • 58047 •5807 •58094 .58118 .58141 .58165 .58189 .58212 . 58236 .5826 .58283 • 58307 •5833 .58354 •58378 .58401 .58425 •58449 .58472 .58496 •585*9 •58543 •58567 •5859 • 58614 •58637 .58661 . 58684 . 58708 •5873* •58755 •58779 .81915 .81899 .81882 .81865 .81848 .81832 .81815 .81798 .81782 .81765 .81748 •81731 .81714 .81698 .81681 .81664 .81647 .81631 .81614 .81597 .8158 .81563 •81546 •8153 •81513 .81496 .81479 .81462 •8x445 .81428 .81412 •81395 .81378 .81361 .81344 .81327 .8131 .81293 .81276 .81259 .81242 .81225 .81208 .81191 .81174 .81157 .8114 .81123 .81106 .81089 .81072 .81055 .81038 .81021 .81004 .80987 .8097 •80953 .80936 .80919 .80902 60 59 58 57 56 55 54 53 52 5 * 50 49 48 47 46 45 44 43 42 4 * ; 40 39 3 8 37 j 36 35 34 33 32 3 * 3 ° 29 28 27 26 25 24 23 22 21 20 *9 18 17 1 6 15 *4 *3 12 11 10 l 7 6 5 4 3 2 1 0 l6 l6 *5 i 5 *5 *5 *4 *4 *4 *4 *3 *3 *3 *3 12 12 12 11 11 11 11 10 10 10 10 9 9 9 9 8 8 8 7 7 7 7 6 6 6 6 5 5 5 5 4 4 4 3 3 3 3 2 2 2 2 i N. cos. N. sine. 57 ° N. cos. 1 N. sine. 1 56 ° N. cos. N. sine. > 5 ° 1 N. cos. • N. sine, j ' | 64 ° I I NATURAL SINES AND COSINES. 399 Prop. parts. 36 ° 3 70 3 8° 39 ° Prop. parts. 23 ' N. sine. 18 O 0 •58779 . 80902 .60182 .79864 .61566 .78801 .62932 • 777*5 60 18 0 1 . 58802 .80885 . 60205 .79846 .61589 •78783 •62955 . 77696 59 18 I 2 .58826 . 80867 .60228 .79829 .61612 .78765 .62977 .77678 58 *7 I 3 .58849 .8085 .60251 .79811 •61635 •78747 •63 .7766 57 *7 2 4 •58873 .80833 . 60274 •79793 .61658 .78729 .63022 .77641 56 *7 2 S .58896 .80816 .60298 •79776 .61681 .78711 •63045 .77623 55 *7 2 6 .5892 .80799 .60321 •79758 .61704 .78694 . 63068 .77605 54 16 3 7 •58943 .80782 .60344 • 7974 * .61726 .78676' .6309 .77586 53 16 3 8 • 58967 . 80765 . 60367 •79723 .61749 .78658 •63113 •77568 52 16 3 9 •5899 . 80748 .6039 .79706 .61772 .7864 •63135 •7755 5 * *5 4 10 • 59 OI 4 •8073 .60414 .79688 • 6 i 795 .78622 •63158 • 77 53 * 50 *5 4 11 • 59°37 •80713 .60437 .79671 .61818 . 78604 .6318 • 775*3 49 *5 5 12 .59061 . 80696 .6046 •79653 .61841 78586 •63203 •77494 48 *4 5 13 • 59 o8 4 . 80679 .60483 •79635 .61864 .78568 .63225 .77476 47 *4 3 14 .59108 .80662 . 60506 .79618 .61887 •7855 .63248 •77458 46 *4 6 13 • 59 * 3 * . 80644 .6052Q •796 .61909 •78532 •63271 •77439 45 *4 6 16 • 59*54 .80627 •60553 79583 61932 •785*4 .63293 .77421 44 13 7 17 • 59*78 .8061 .60576 •79565 •61955 .78496 •63316 .77402 43 13 7 18 • 59201 •80593 .60599 •79547 .61978 •78478 •63338 •77384 42 *3 7 19 •59225 .80576 . 60622 •7953 .62001 .7846 63361 •77366 4 * 12 8 20 • 59 2 48 .80558 .60645 • 795*2 .62024 .78442 •63383 •77347 4 ° 12 8 21 • 59272 .80541 .60668 •79494 .62046 .78424 . 63406 •77329 39 12 8 22 •59295 .80524 .60691 •79477 . 62069 .78405 .63428 • 773 * 33 11 9 23 • 593*8 •80507 .60714 •79459 .62092 .78387 •63451 .77292 37 11 9 24 •59342 . 80489 .60738 .79441 .62115 .78369 •63473 •77273 36 11 10 25 •59365 .80472 .60761 •79424 .62138 •78351 ,63496 •77255 35 11 10 26 •59389 •80455 .60784 .79406 .6216 •78333 .63518 .77236 34 10 10 27 • 594*2 .80438 . 60807 •79388 .62183 •78315 •6354 .77218 33 10 11 28 •59436 .8042 .6083 • 7937 * . 62206 .78297 •63563 • 77*99 32 10 11 29 •59459 .80403 .60853 •79353 .62229 .78279 •63585 .77181 3 * 9 12 30 •59482 . 80386 .60876 •79335 .62251 .78261 . 63608 .77162 30 9 12 3 i •59506 . 80368 .60899 • 793*8 .62274 .78243 •6363 • 77*44 29 9 12 32 •59529 •80351 .60922 •793 .62297 .78225 •63653 • 77*25 28 8 13 33 •59552 •80334 .60945 .79282 .6232 .78206 •63675 • 77*07 27 8 13 34 •59576 .80316 . 60968 .79264 .62342 .78188 . 63698 .77088 26 8 13 35 •59599 . 80299 .60991 •79247 •62365 .7817 .6372 .7707 25 8 14 36 . 59622 .80282 .61015 .79229 .62388 78152 .63742 •77051 24 7 14 37 .59646 . 80264 .61038 .79211 .62411 •78134 •63765 •77033 23 7 15 38 . 59669 .80247 .61061 • 79*93 •62433 78116 •63787 •77014 22 7 15 39 •59693 .8023 .61084 • 79*76 .62456 . 78098 .6381 .76996 21 6 15 40 • 597*6 .80212 .61107 • 79*58 .62479 .78079 .63832 •76977 20 6 16 4 i •59739 •80195 .6113 • 79*4 .62502 .78061 •63854 •76959 *9 6 16 42 •59763 . 80178 .61153 .79122 .62524 .78043 •63877 •7694 18 5 16 43 •59786 .8016 .61176 .79105 .62547 .78025 .63899 .76921 *7 5 *7 1 44 .59809 .80143 .61199 .79087 .6257 . 78007 .63922 •76903 16 5 *7 45 .59832 . 80125 .61222 . 79069 62592 .77988 •63944 .76884 *5 5 18 : 46 .59856 . 80108 .61245 • 79 ° 5 * .62615 •7797 .63966 .76866 *4 4 *g| 47 •59879 .80091 .61268 • 79°33 .62638 •77952 .63989 •76847 *3 4 18 ; 48 . 59902 . 80073 .61291 .79016 .6266 •77934 .64011 .76828 12 4 *9 49 • 59926 . 80056 .61314 .78998 .62683 • 779*6 •64033 .7681 11 3 *9 50 •59949 .80038 •61337 .7898 .62706 •77897 .64056 7679* 10 3 20 5 i •59972 .80021 .6136 . 78962 .62728 •77879 . 64078 .76772 9 3 20 52 •59995 .80003 •61383 .78944 .62751 .77861 641 •76754 8 2 20 53 .60019 .79986 61406 .78926 .62774 •77843 .64123 •76735 7 2 21 54 .60042 .79968 .61429 . 78908 .62796 .77824 64145 .76717 6 2 21 55 .60065 • 7995 * .61451 .78891 .62819 . 77806 .64167 .76698 5 2 21 56 .60089 •79934 .61474 •78873 62842 .77788 .6419 .76679 4 1 22 57 .60112 .79916 • 6 i 497 •78855 .62864 •77769 .64212 .76661 3 1 22 58 .60135 •79899 • 6152 •78837 .62887 • 7775 * •64234 . 76642 2 1 23 59 .60158 .79881 •61543 78819 . 62909 •77733 .64256 .76623 1 j 0 23 60 .60182 .79864 .61566 .78801 .62932 • 777*5 .64279 . 76604 0 0 N. cos. N. sine. N. cos. 1 N. sine. N. cos. N. sine. N. cos. N. sine. ' 1 53° II 52° 1 51° 50° 1 400 NATURAL SINES AND COSINES. c r 40 ° 410 42 ° i 43 ° pi & 22 ' N. sine. N. eos. N. sine. N. C 08 . N. sine. | N. cos. j N. sine. N. COR. 0 0 .64279 . 76604 .65606 • 7547 z • 66913 • 743*4 .682 • 73*35 60 0 1 .64301 • 76586 .65628 •75452 •66935 •74295 ; .68221 • 73**6 59 1 2 •64323 .76567 •6565 •75433 .66956 •74276 .68242 .73096 58 1 3 .64346 •76548 .65672 • 754 z 4 .66978 74256 | .68264 .73076 57 1 4 .64368 •7653 .65694 '75395 .66999 •74237 j .68285 • 73056 56 2 5 •6439 .76511 .65716 •75375 .67021 •74217 | .68306 •73036 55 2 6 .64412 .76492 •65738 75356 .67043 74198 1 .68327 .730*6 54 3 7 •64435 •76473 •65759 •75337 . 67064 •74178 ; -68349 . 72996 53 3 8 •64457 76455 65781 • 753 z 8 . 67086 • 74*59 1 -6837 .72976 52 3 9 •64479 76436 •65803 •75299 .67107 • 74*39 •68391 •72957 5 i 4 10 .64501 .76417 •65825 .7528 .67129 • 74*2 1 .68412 •72937 50 4 11 .64524 •76398 •65847 •75261 .67151 .741 ! -68434 .72917 49 4 12 .64546 .7638 .65869 75241 .67172 .7408 ; 68455 .72897 48 5 13 .64568 .76361 •65891 .75222 .67194 .74061 i .68476 .72877 47 5 14 •6459 .76342 • 659*3 •75203 .67215 •74041 • 6 8 497 72857 46 6 15 .64612 •76323 •65935 •75184 •67237 . 74022 ■ 68518 ; .72837 45 6 16 •64635 •76304 65056 •75165 .67258 . 74002 i -68539 .72817 44 6 17 .64657 .76286 .65978 •75146 .6728 • 739 8 3 ! -68561 ! .72797 43 7 18 . 64670 .76267 .66 •75126 .67301 •73963 j .68582 •72777 42 7 *9 .64701 .76248 . 66022 •75107 .67323 •73944 1 i .68603 •72757 4 * 7 20 .64723 .76229 .66044 .75088 .67344 •73924 .68624 •72737 40 8 21 .64746 .7621 .66066 .75069 •67366 • 739°4 . 68645 .72717 39 8 22 . 64768 .76192 .66088 •7505 .67387 • 73885 .68666 . 72697 38 8 2 3 •6479 • 76 i 73 .66109 •7503 .67409 •73865 .68688 72677 37 9 24 .64812 •76154 .66131 .75011 •6743 73846 .68709 •72657 36 9 25 •64834 •76135 •66153 .74992 •67452 .73826 •6873 •7*637 35 10 26 .64856 .76116 •66175 •74973 •67473 .73806 .68751 .72617 34 10 27 .64878 .76097 .66197 •74953 •67495 •73787 .68772 •72597 33 10 28 .64901 .76078 .66218 •74934 .67516 •73767 .68793 •72577 32 11 29 .64923 .76059 .6624 • 749*5 •67538 •73747 .68814 •72557 3 * 11 30 .64945 . 76041 .66262 .74896 •67559 .73728 .68835 •72537 3 ° 11 3 i .64967 . 76022 .66284 .74876 •6758 .73708 .68857 •725*7 29 12 32 .64989 .76003 .66306 •74857 .67602 .73688 .68878 72497 28 12 33 .65011 •75984 .66327 .74838 .67623 •73669 .68899 .72477 27 12 34 •65033 •75965 • 66349 .74818 .67645 • 73649 .6892 •72457 26 13 35 •65055 •75946 .66371 •74799 .67666 • 73629 .68941 •72437 25 13 ; 36 .65077 .75927 •66393 .7478 .67688 • 736 i .68962 • 724*7 24 14 i 37 .651 • 759 ° 8 .66414 .7476 .67709 •7359 .68983 72397 23 14 ! 38 .65122 .75889 .66436 7474 * •6773 •7357 .69004 •72377 22 14 ; 39 .65144 •7587 .66458 •74722 •67752 • 7355 * .69025 •72357 21 15 i 40 .65166. •75851 .6648 • 747°3 •67773 • 7353 * . 69046 •72337 20 15 ; 41 .65188 75832 .66501 •74683 •67795 • 735 H .69067 • 723*7 *9 i 5 42 •6521 • 758 i 3 .66523 .74664 .67816 • 7349 * . 69088 .72297 18 16 i 43 •65232 •75794 •66545 74644 •67837 •73472 .69109 .72277 *7 16 i 44 •65254 1 -75775 .66566 .74625 .67859 •73452 •69*3 .72257 16 *7 45 .65276 ! 75756 66588 . 74606 .6788 •73432 .69151 .72236 *5 17 46 .65298 i -75738 .6661 .74586 .67901 • 734*3 .69172 .72216 *4 17 47 6532 I -75719 .66632 •74567 .67923 •73393 .69193 .72196 *3 18 48 •65342 | -757 .66653 •74548 •67944 •73373 .69214 . 72176 12 18 49 •65364 | 7568 .66675 •74528 .67965 •73353 •69235 72156 11 18 50 .65386 S -75661 . 66697 ^ •74509 .67987 •73333 .69256 .72136 10 *9 ! 51 .65408 75642 .66718 .74489 .68008 • 733*4 .69277 .72116 9 *9 ; 52 •6543 •75623 .6674 7447 . 68029 •73294 .69298 72095 8 1 9 ! 53 •65452 •75604 .66762 • 7445 * .68051 •73274 • 693*9 72075 7 20 54 •65474 •75585 .66783 • 7443 * .68072 •73254 .6934 •72055 | 6 20 55 .65496 •75566 . 66805 • 744*2 .68093 •73234 .69361 •72035 5 21 56 .65518 •75547 .66827 ■74392 .68115 •73215 .69382 .72015 4 21 57 .6554 •75528 .66848 i -74373 .68136 1 - 73*95 69403 1 - 7*995 3 21 58 • 65562 •75509 .6687 1 -74353 68157 • 73*75 .69424 I - 7*974 2 22 59 65584 •7549 .66891 •74334 68179 1 - 73*55 •69445 | - 7*954 1 22 60 .65606 •75471 •66913 • 743*4 .682 • 73*35 69466 • 7*934 0 I ' 1 49° 48° 470 11 46° l £9 *9 *9 18 18 18 *7 i7 17 16 16 16 16 i5 i5 i5 i 4 14 14 13 13 13 12 12 12 iz 10 10 10 10 9 9 1 8 8 7 7 7 6 6 6 5 5 5 4 4 4 3 3 3 3 2 2 2 o o parts. NATURAL SINES AND COSINES, 401 h Prop. M parts. , 4 ' N . sine. 1 ° N. cos. h Prop. p° parts. u Prop. w parts. , 44 N. sine. LO N. cos. H p« p, 9 , 0 0 . 69466 • 7 T 934 60 19 11 3 i .70112 •71305 29 9 O 1 .69487 .71914 59 19 12 32 .70132 .71284 28 9 I 2 .69508 • 7 i8 94 5 8 18 12 33 •70153 .71264 27 9 I 3 .69529 • 7 i 8 73 57 18 12 34 • 7 OI 74 •71243 26 8 I 4 .69549 • 7 l8 53 56 18 13 35 •70195 .71223 25 8 2 5 •6957 • 7 i8 33 55 17 13 36 .70215 .71203 24 8 2 6 .69591 • 7 i8i 3 54 17 14 37 .70236 .71182 23 7 3 7 .69612 .71792 53 17 14 38 .70257 .71162 22 7 3 8 .69633 .71772 52 16 14 39 .70277 .71141 21 7 3 9 .69654 • 7 W 52 5 i 16 15 40 .70298 .71121 20 6 4 10 .69675 • 7 W 32 5 o 16 15 4 i •70319 ■ 7 « x 9 6 4 11 . 69696 .71711 49 16 15 42 •70339 .7108 18 6 4 12 .69717 .71691 48 15 16 43 .7036 • 7 io 59 17 5 5 13 •69737 .71671 47 15 16 44 .70381 • 7 IQ 39 16 5 5 14 .69758 • 7 i6 5 46 15 17 45 . 70401 .71019 x 5 5 6 i 5 .69779 • 7 i 6 3 45 14 17 46 .70422 .70998 14 4 6 16 .698 . 7161 44 14 W 47 •70443 .70978 13 4 6 17 .69821 • 7*59 43 H 18 48 .70463 •70957 12 4 7 18 .69842 • 7 i 569 42 13 18 49 .70484 •70937 11 3 7 19 .69862 • 7 i 549 4 1 13 18 50 •70505 .70916 xo 3 7 20 .69883 •71529 40 13 x 9 5 i •70525 .70896 9 3 8 21 . 69904 .71508 39 12 1 9 52 .70546 .70875 8 3 8 22 .69925 .71488 S 8 12 x 9 53 .70567 •70855 7 2 8 23 .69946 .71468 37 12 20 54 •70587 .70834 6 2 9 24 . 69966 •71447 36 II 20 55 .70608 .70813 5 2 9 25 .69987 .71427 35 II 21 56 .70628 •70793 4 i 10 26 .70008 .71407 34 II 21 57 .70649 .70772 3 X 10 27 .70029 .71386 33 IO 21 S 8 .7067 .70752 2 1 10 28 .70049 .71366 32 IO 22 59 .7069 •70731 1 0 11 29 .7007 •71345 3 i IO 22 60 7071 1 .70711 0 0 11 30 .70091 •71325 30 IO N. cos. N. sine. ' N.cos. N. sine. ' 45 ° 450 Preceding Table contains Natural Sine and Cosine for every minute of the Quadrant to Radius 1. If Degrees are taken at head of columns, Minutes, Sine, and Cosine must be taken from head also ; and if they are taken at foot of column, Minutes, etc., must be taken from foot also. Illustration. — . 3173 is sine of 18 0 30', and cosine 0171° 30'. To Compute Sine or Cosine for Seconds. When Angle is less than 45 0 . Rule. — Ascertain sine or cosine of angle for degrees and minutes from Table; take difference between it and &ine- or cosine of angle next below it. Look for this difference or remainder,* if Sine is required, at head of column of Pi'oportional Parts , on left side ; and if Cosine is required, at head of column on right side ; and in these- respective columns, opposite to number of seconds of angle in column,, is- number or correction in seconds to be added to Sine, or subtracted from- | Cosine of angle. Illustration i. — What is sine of 8° 9' 10"? Sine of 8° 9', per Table = Sine of 8° 10', “ = •14177; .142 05; .00028 difference. In left side column of proportional parts, under 28, and opposite to 10', is 5, cor- rection for 10', which, being added to . 141 77 = .141 82 Sine. * The table in some instances will give a unit too much, but this, in general, is of little importance. . L L* 402 NATURAL SINES AND COSINES. 2. — What is cosine of 8° 9' 10"? Cosine bf 8° 9' per Table = - 9 8 9 9 ° 1 ) .000 04 difference. Cosine of 8° 10 , “ =.99986;) In right-side column of proportional x>arts, under 4, and opposite to 10 , is 1, the correction for 10', -which, being subtracted from .989 90 = .989 89 cosine. When Angle exceeds 45 0 . Rule— A scertain sine or cosine for angle in degrees and minutes from Table, taking degrees at the foot of it ; then take difference between it and sine or cosine of angle next above it. Look for re- mainder, if Sine is required, at head of column of Proportional Parts , on right side ; and if Cosine is required, at head of column on left side ; and in these respective columns, opposite to seconds of angle, is number or correction in seconds to be added to Sine, or subtracted from Cosine of angle. Illustration.— What is the Sine and Cosine of 8i° 50' 50"? Sine of 8i° 50' per Table = .989 86; ) OOQO difference. Sine of 8i° 51 , “ =- 9^995 ) In right-side column of proportional parts, and opposite to 50', is 3, which, added to . 989 86 = . 989 89 Sine. Cosine of 8i° 50', per Table = .142 05 ;) Q difference. Cosine of 8x° 51 , =-i 4 I 77 j) In left-side column of proportional parts , and opposite to 50', is 24, which, sub- tracted from .14205 ==.141 81 Cosine. T'o Ascertain or Compute jNTxixn'ber of Degrees, Minutes, and Seconds of a given Sine or Cosine. When Sine is given. Rule. — If given sine is in Table, the degrees of it will be at top or bottom of page, and minutes in marginal column, at left or right side, according as sine corresponds to an angle less or greater than 45 0 . If given sine is not in Table, take sine m I able which is next less than the one for which degrees, etc., are required, and note degrees, etc., for it. Sub- tract this sine from next greater tabular sine, and also from given sine. Then, as tabular difference is to difference between given sine and tabu- lar sine, so is 60 seconds to seconds for sine given. Example. — What are the degrees, minutes, and seconds for sine of .75? Next less sine is .74992, arc for which is 48° 35'. Next greater sine is .75011, difference between which and next less is .75011 — .749 92 ^=. 000 19. Difference be- tween kss tabular sine and one given is . 75 — .749 92 = 8. TLeai i-g : 8 il.fp ; 25+, which, added to 48° 35' = 48° 35' 25". When Cosine is given. Rule. — If given cosine is found in Table, degrees of it will be found as in manner specified when .sine is given. . If given cosine is not in Table, take cosine in T able which is next greater than one for which degrees, etc., are required, and note degrees, etc., for it. Subtract this cosine from next less tabular cosine, and also from given cosine. Then, as tabular difference is to difference between given cosine and tabu- lar cosine, so is 60 seconds to seconds for cosine given. Example. — What are the degrees, minutes, and seconds for cosine of .75? Next greater cosine is .750 n, arc for which is 41 0 24'. Next less cosine is ^749 92, difference between which and next greater is .75011 — .74992 = .000 19. Difference between greater tabular cosine and one given is .750 11 — .75000 — 11. Then 19 : n :: 60 : 35 — , which, added to 41 0 24' = 41° 24' 35". To Compnte Versed Sine of an Angle. Subtract cosine of angle from 1. Illustration.— What is the versed sine of 21 0 30' ? Cosine of 21 0 30' is .93042, which, — 1 = .06958 versed sine. To Compute Co-versed Sine of an Angle. Subtract sine of angle from 1. Illustration. — What is the co- versed sine of 21 0 30'? The sine of 21 0 30' is .3665, which, — 1 = .6335 co-versed sine. NATURAL SECANTS AND COSECANTS, 403 ]N"atnral Secants and. Co -secants. 0° 1° 2° I 30 ' Secant. Co-SKCANT. Secant. Co-sec’t. Secant. Co-sec’t. Secant. Co-sec’t. r 0 j Infinite. I. OOOI 57-299 1.0006 28.654 1. 0014 19.107 60 I I 3437*7 .OOOI 6-359 .0006 8.417 .0014 9.002 59 2 I 1718.9 .0002 5-45 .0066 8.184 .0014 8.897 58 3 I 145-9 .0002 4-57 .0006 7-955 .0014 8.794 57 4 I 859.44 .0002 3 - 7 i 8 .0006 7-73 .0014 8.692 56 5 I 687.55 1.0002 52.891 1.0007 27. 508 1. 0014 18.591 55 6 I 572.90 .0002 2.09 .0007 7.29 •0015 8.491 54 7 I 491. 1 1 .0002 i- 3 i 3 .0007 7-075 .0015 8-393 53 8 I 29.72 .0002 o -558 .0007 6.864 •0015 8.295 52 9 I 38 i -97 .0002 49.826 .0007 6.655 .0015 8. 198 5 * 10 I 343-77 1.0002 49. H4 1.0007 26.45 1. 0015 18. 103 50 11 I 12.52 .0002 8.422 .0007 6. 249 •0015 8.008 49 12 I 286. 48 .0002 7-75 .0007 6.05 .0016 7.914 48 13 I 64.44 .0002 7.096 .0007 5-854 .0016 7.821 47 14 I 45-55 0002 6.46 .0008 5.661 .0016 7-73 46 15 I 229.18 1.0002 45-84 1.0008 25.471 1.0016 17.639 45 16 I 14.86 .0002 5-237 0008 5-284 .0016 7-549 44 17 I 02.22 .0002 4-65 .0008 5 -i .0016 7.46 43 18 I 190.99 .0002 4.077 .0008 4.918 .0017 7-372 42 *9 I 80.73 .0003 3-52 .0008 4-739 .0017 7.285 4 * 20 I i 7 i - 8 9 I.OOO3 42.976 1.0008 24,562 1. 0017 17.198 40 21 X 63-7 .OOO3 2.445 .0008 4-358 .0017 7-**3 39 22 I 56.26 .0003 1.928 .0008 4.216 .0017 7.028 38 23 I - 49-47 .0003 1.423 .0009 4.047 .0017 6.944 37 24 I 43-24 .OOO3 4°-93 .0009 3-88 .0018 6.861 36 25 I I 37 - 5 I I.OOO3 40.448 1.0009 23.716 1. 0018 16.779 35 26 I 32.22 .0003 39.978 .0009 3-553 .0018 6.698 34 27 I . 27-32 .0003 9 - 5 i 8 .0009 3-393 .0018 6.617 33 28 I 22.78 .OOO3 9.069 .0009 3-235 .0018 6.538 3 2 29 I 18.54 .0003 8.631 .0009 3-079 .0018 6-459 3 * 30 I 1 14- 59 I.OOO3 38.201 1.0009 22.925 1. 0019 16.38 3 ° 3 i I 10.9 .0003 7.782 .001 2.774 .0019 6.303 29 32 I 07-43 .0003 7 - 37 * .001 2.624 .0019 6.226 28 33 I 04.17 .OOO4 6.969 .001 2.476 .0019 6.15 27 34 I 01. 11 .OOO4 6.576 .001 2-33 .0019 6.075 26 35 I 98.223 I.OOO4 36.191 1. 001 22. 186 1. 0019 16 25 36 I 5-495 .OOO4 5.814 .001 2.044 .002 5.926 24 37 I 2.914 .OOO4 5-445 .001 1.904 .002 5-853 23 38 I. OOOI 2.469 .OOO4 5-084 .001 1-765 .002 5-78 22 39 .OOOI 88. 149 .OOO4 4.729 .0011 1.629 .002 5.708 21 40 I. OOOI 85.946 I.OOO4 34-382 1. 001 1 21.494 1.002 *5-637 20 4 i .OOOI 3-849 .OOO4 4.042 .0011 1.36 .0021 5.566 *9 42 .OOOI 1-853 .OOO4 3.708 .0011 1.228 .0021 5-496 18 43 .OOOI 79-95 .OOO4 3 - 38 i .0011 1.098 .0021 5-427 *7 44 • OOOI 8-133 .OOO4 3.06 .0011 20.97 .0021 5-358 16 45 I. OOOI 76.396 I.OOO5 32-745 1. 001 1 20.843 1. 0021 15.29 i 5 46 • OOOI 4-736 .0005 2-437 .0012 0.717 .0022 5.222 14 47 .OOOI 3.146 .0005 2.134 .0012 o -593 .0022 5-*55 *3 48 .OOOI 1.622 .0005 1.836 .0012 0.471 .0022 5-089 12 49 .OOOI 1. 16 .0005 1-544 .0012 0-35 .0022 5-023 11 50 I. OOOI 68.757 I.OOO5 31-257 1. 001 2 20.23 1.0022 14.958 10 5 i .OOOI 7.409 .0005 30.976 .0012 0. 112 .0023 4-893 9 52 .OOOI 6. 1 13 .0005 0.699 .0012 19.995 .0023 4.829 8 53 .OOOI 4.866 .0005 0.428 .0013 9.88 .0023 4-765 7 54 .OOOI 3.664 .0005 0. 161 •0013 9.766 .0023 4.702 6 55 I. OOOI 62. 507 I.OOO5 29. 899 1. 0013 19-653 1.0023 14.64 5 56 .OOOI *• 39 * .0006 9.641 .0013 9 - 54 * .0024 4-578 4 57 .OOOI *- 3 i 4 .0006 9.388 •0013 9-431 .0024 4 - 5*7 3 58 .OOOI 59-274 .OO06 9 * 39 .0013 9.322 .0024 4-456 2 I 9 .OOOI 8.27 .0006 8.894 .0013 9.214 .0024 4-395 1 60 I. OOOI 57-299 I.OO06 28.654 1. 0014 19.107 1.0024 * 4-335 0 ' Co-SEC’T. Secant. Co-sec’t. Secant. Co-sec’t. Secant. Co-sec’t. Secant. 89 ° 88 ° | 87 ° 86 ° 404 NATURAL SECANTS AND CO-SECANTS, 40 , 50 6° 70 9 Secant. Co-sec’t. Secant. Co-sec’t. Secant. Co-sec’t. Secant. [ Co-sec’t. ' o 1.0024 14-335 1.0038 11.474 1.0055 9. 5668 1.0075 8.2055 60 i .0025 4.276 .0038 1.436 .0055 •5404 .0075 . 1861 59 2 .0025 4 - 2 i 7 .0039 1.398 .0056 • 5 Mi .0076 .1668 58 3 .0025 4 - *59 .0039 1.36 .0056 .488 .0076 .1476 57 4 .0025 4. IOI .0039 I -323 .0056 .462 .0076 • 1285 56 5 1.0025 14.043 1.0039 11.286 1.0057 9.4362 1.0077 8. 1094 55 6 .0026 3.986 .004 1.249 •0057 .4105 •°°77 • 0905 54 7 .0026 3-93 .004 1-213 .0057 .385 .0078 .0717 53 8 .0026 3-874 .004 1.176 .0057 •3596 .0078 .0529 52 9 .0026 3.818 .004 1.14 .0058 •3343 .0078 .0342 5 i IO 1.0026 i 3 - 7 6 3 1. 0041 11. 104 1.0058 9. 3092 1.0079 8.0156 50 ii .0027 3.708 .0041 1.069 .0058 .2842 .0079 7.9971 49 12 .0027 3-654 .0041 1-033 .0059 -2593 .0079 .9787 48 *3 .0027 3-6 .0041 0.988 .0059 .2346 .008 .9604 47 .0027 3-547 .0042 0.963 .0059 .21 .008 .9421 46 15 1.0027 13-494 1.0042 10.929 1.006 9-i855 1.008 7.924 45 16 .0028 3 - 44 i .0042 0. 894 .006 .1612 .0081 • 9°59 44 17 .0028 3-389 .0043 0.86 .006 •137 .0081 .8879 43 18 .0028 3-337 .0043 0.826 .0061 .1129 .0082 .87 42 19 .0028 3. 286 .0043 0.792 .0061 .089 .0082 .8522 4 1 20 1.0029 13-235 I.0043 10.758 1. 0061 9.0651 1.0082 7-8344 40 21 .0029 3.184 .0044 0.725 .0062 .0414 .0083 .8168 39 22 .0029 3-134 .0044 0. 692 .0062 .0179 .0083 •7992 38 23 .0029 3.084 .0044 0.659 .0062 8.9944 .0084 .7817 37 24 .0029 3-034 .0044 0.626 .0063 .9711 .0084 .7642 36 25 1.003 12.Q85 1.0045 10.593 1.0063 8.9479 1.0084 7.7469 35 26 .003 2-937 .0045 0. 561 .0063 .9248 .0085 .7296 34 27 .003 2.888 .0045 0. 529 .0064 .9018 .0085 .7124 33 28 .003 2.84 .0046 0.497 .0064 •879 .0085 •6953 32 29 .0031 2-793 .0046 0.465 .0064 .8563 .0086 .6783 3 i 30 1. 0031 12.745 1.0046 10.433 1.0065 8-8337 1.0086 7.6613 30 3 1 .0031 2.698 .0046 0. 402 .0065 .8112 .0087 •6444 29 3 2 .0031 2.652 .0047 0.371 .0065 .7888 .0087 .6276 23 33 .0032 2.606 .0047 o -34 .0066 .7665 .0087 .6108 27 34 .0032 2. 56 .0047 0. 309 .0066 •7444 .0088 •5942 26 35 1.0032 12.514 1.0048 10.278 1.0066 8.7223 1.0088 7-5776 25 36 .0032 2.469 .0048 0.248 .0067 .7004 .0089 .5611 24 37 .0032 2.424 .0048 0.217 .0067 .6786 .0089 •5440 23 38 • 0033 2-379 .0048 0.187 .0067 .6569 .0089 .5282 22 39 .0033 2-335 .0049 o.i 57 .0068 •6353 .009 •5119 21 40 1.0033 12.291 1.0049 10.127 1.0068 8.6138 1.009 7-4957 20 4 i .0033 2.248 .0049 0.098 .0068 •5924 .009 •4795 r 9 42 .0034 2.204 .005 0.068 .0069 .5711 .0091 •4634 18 43 • oc >34 2. 161 .005 0.039 .0069 •5499 .0091 •4474 17 44 .0034 2.118 .005 0.01 .0069 .5289 .0092 •4315 16 45 1.0034 12.076 1.005 9.9812 1.007 8.5079 1.0092 7 - 4 I 56 15 46 •0035 2.034 .0051 .9525 .007 .4871 .0092 • 399 s 14 47 •0035 1.992 .0051 •9239 .007 .4663 .0093 • 3 8 4 13 48 •0035 x -95 .0051 •8955 .0071 •4457 .0093 •3683 12 49 •0035 1.909 .0052 .8672 .0071 .4251 .0094 •3527 11 50 1.0036 11.868 1.0052 9.8391 1. 007 1 8. 4046 1.0094 7-3372 10 5 i .0036 1.828 .0052 .8112 .0072 •3843 .0094 •3217 9 52 .0036 1.787 •0053 .7834 .0072 .3640 .0095 .3063 8 53 .0036 1-747 •0053 •7558 .0073 •3439 .0095 .2909 7 54 .0037 1.707 .0053 • 7283 •0073 •3238 .0096 •2757 6 55 1.0037 11.668 1-0053 9.701 1.0073 8.3039 1.0096 7.2604 5 56 .0037 1.628 .0054 .6739 .0074 .2840 .0097 •2453 4 57 .0037 1.589 .0054 .6469 .0074 .2642 .0097 .2302 3 58 .0038 i -55 .0054 .62 .0074 .2446 .0097 .2152 2 59 .0038 i- 5 12 •0055 •5933 •0075 .225 .0098 .2002 1 60 1.0038 n -474 1.0055 9. 5668 1.0075 8. 2055 .0098 00 u > 0 r Co-sec’t. Secant. Co-sec’t. Secant. Co-sec’t. Secant. Co-sec’t. Secant. # 85° 84° 83° 82° NATURAL SECANTS AND CO-SECANTS. 405 80 II 9° | 10 ° | 11° 9 Secant. Co-sec’t. Secant. Co-sec’t. | Secant. Co-sec’t. ' Secant. Co-sec’t. ' 0 1.0098 7- i8 53 1. 0125 6.3924 i- 01 54 5-7588 1.0187 5. 2408 60 I .0099 .1704 .0125 .3807 •0155 •7493 .0188 •233 59 2 .0099 •1557 •0125 •369 •oi 55 • 739 8 .0188 .2252 58 3 .0099 .1409 .0126 •3574 .0156 •7304 .0189 .2174 57 4 .01 .1263 .0126 •3458 .0156 .721 .0189 .2097 56 5 1. 01 7 . m 7 1. 0127 6-3343 1.0157 5 - 7“7 1. 019 5.2019 55 6 .0101 .0972 .0127 .3228 .0157 .7023 .0191 .1942 54 7 .0101 .0827 .0128 • 3 ii 3 .0158 •693 .0191 .1865 53 8 .0102 .0683 .0128 .2999 .0158 .6838 .0192 .1788 52 9 .0102 •0539 .0129 .2885 .0159 •6745 .0192 .1712 5 i 10 1. 0102 7.0396 1. 0129 6.2772 1.0159 5-6653 1-0193 5.1636 50 11 .0103 .0254 .013 .2659 .016 .6561 .0193 • 156 49 12 • 0103 .0112 .013 .2546 .016 •647 .0194 .1484 48 13 .0104 6.9971 .0131 •2434 .0161 •6379 •0195 .1409 47 14 .0104 .983 .0131 .2322 .0162 .6288 .0195 •1333 46 15 1. 0104 6.969 1. 0132 6.2211 1.0162 5-6197 1.0196 5-1258 45 16 .0105 •955 .0132 .21 .0163 .6107 .0196 .1183 44 17 .0105 • 94 11 •0133 .199 .0163 .6017 •0197 .1109 43 18 .0106 •9273 •0133 .188 .0164 .5928 .0198 .1034 42 *9 .0106 • 9*35 .9134 .177 .0164 .5838 .0198 .096 4 i 20 1. 0107 6. 8998 1.0134 6.- 1661 1.0165 5-5749 I-OI 99 5.0886 40 21 .0107 .8861 •0135 •1552 .0165 .566 .0199 .0812 39 22 .0107 .8725 •0135 •1443 .0166 •5572 .02 •0739 38 23 .0108 .8589 .0136 •1335 .0166 • 54 8 4 .0201 .0666 37 24 .0108 •8454 .0136 .1227 .0167 •5396 .0201 •0593 36 25 1. 0109 6. 832 1.0136 6. 112 1.0167 5 - 53 o 8 1.0202 5-052 35 26 .0109 .8185 •QI 37 .1013 .0168 .5221 .0202 •0447 34 27 .on .8052 ’ -0137 . 0906 .0169 • 5 i 34 .0203 •0375 33 28 • on .7919 .0138 .08 .0169 •5047 .0204 .0302 32 29 .0111 .7787 .0138 • o6 94 .017 .496 .0204 .023 3 i 30 I.OIII 6-7655 1-0139 6.0588 1. 017 5-4874 1.0205 5.0158 30 3 i .0111 •7523 .0139 .0483 .0171 .4788 .0205 .0087 29 32 .0112 • 739 2 .014 •0379 .0171 .4702 .0206 •0015 28 33 .0112 .7262 .014 .0274 .0172 .4617 .0207 4-9944 27 34 •0113 .7132 .0141 .017 .0172 •4532 .0207 •9873 26 35 1. 0113 6.7003 1. 0141 6.0066 I - OI 73 5-4447 1.0208 4.9802 25 36 .0114 .6874 .0142 5-9963 .0174 .4362 .0208 •9732 24 37 .0114 •6745 .0142 ,986 • OI 74 .4278 .0209 .9661 23 38 .0115 .6617 .0143 •9758 •0175 .4194 .021 • 959 1 22 39 .0115 .649 .0143 •9655 • OI 75 .411 .021 .9521 21 40 1. 0115 6.6363 1. 0144 5-9554 1.0176 5.4026 1. 0211 4.9452 20 4 i .0116 •6237 .0144 •9452 .0176 •3943 .0211 .9382 19 42 .0116 .6m .0145 • 935 i • OI 77 .386 .0212 • 93 i 3 18 43 .0117 ' -5985 .0145 •925 .0177 •3777 •0213 •9243 17 44 .0117 .586 .0146 •915 .0178 •■3695 .0213 • 9 I 75 16 45 1. 0118 6.5736 1.0146 5.9049 1. 0179 5-3612 1. 0214 4.9106 15 46 .0118 .5612 .0147 •895 .0179 •353 .0215 • 9°37 14 47 .0119 • 5488 .0147 .885 .018 •3449 .0215 .8969 13 48 .0119 •5365 .0148 •8751 .018 •3367 .0216 .8901 12 49 .0119 •5243 .0148 .8652 .0181 . 3286 .0216 •8833 11 50 1. 012 6.5121 1. 0149 5-8554 1. 0181 5-3205 1. 0217 4.8765 10 5 i .012 •4999 • 015 .8456 .0182 .3124 .0218 .8697 9 52 .0121 .4878 .015 •8358 .0182 •3044 .0218 .863 8 53 .0121 •4757 .0151 .8261 .0183 .2963 .0219 .8563 7 54 .0122 •4637 .0151 •8163 .0184 .2883 .022 .8496 6 55 1 0122 6.4517 1. 0152 5.8067 1.0184 5.2803 1.022 4.8429 5 56 .0123 •4398 .0152 •797 .0185 •2724 .0221 .8362 4 57 .0123 4279 •0153 .7874 .0185 .2645 .0221 .8296 3 58 .0124 .416 •0153 ‘ 777 8 .0186 .2566 .0222 .8229 2 59 .0124 .4042 .0154 .7683 .0186 .2487 .0223 .8163 1 60 1-0125 6.3924 1.0154 5-7588 1.0187 5. 2408 1.0223 4.8097 0 ' 1 Co-sec’t. Secant. Co-sf.c’t. Secant. Co-sec’t. | Secant. O 0 4 * Secant. / 1 81 ° 80 ° | 790 780 406 natural secants and co-secants. 12° 13 ° 140 150 / Secant. Co-sec’t. Secant. Co-sec’t. Secant. Co-sec’t. Secant. 1 Co-sec’t. ' o 1.0223 4.8097 1.0263 4-4454 1.0306 4 -I 336 1-0353 3-8637 60 I .0224 .8032 .0264 •4398 •0307 . 1287 •0353 .8595 59 2 .0225 .7966 .0264 •4342 .0308 .1239 •0354 •8553 58 3 .0225 .7901 .0265 .4287 .0308 .1191 •0355 •8512 57 4 .0226 •7835 .0266 .4231 .0309 .1144 •0356 •847 56 5 1.0226 4-777 1*0266 4.4176 1.031 4.1096 1-0357 3. 8428 55 6 .0227 .7706 .0267 .4121 .0311 . 1048 •0358 -8387 54 7 .0228 .7641 .0268 .4065 .0311 . 1001 .0358 .8346 53 8 .0228 •7576 .0268 .4011 • 0312 •0953 •0359 •8304 52 9 .0229 •7512 .0269 •3956 •0313 .0906 .036 .8263 5 i IO 1.023 4. 7448 1.027 4 - 39 01 1.0314 4.0859 1.0361 3.8222 50 XI .023 •7384 .0271 •3847 .0314 .0812 .0362 j .8181 49 12 .0231 •732 .0271 • 379 2 •0315 .0765 .0362 j .814 48 *3 .0232 •7257 .0272 •3738 .0316 .0718 •0363 .81 47 i 4 .0232 •7193 .0273 .3684 •0317 .0672 •0364 i .8059 46 is 1.0233 4 - 7*3 1.0273 4-363 1-0317 4.0625 1.0365 3.8018 45 16 .0234 .7067 .0274 •3576 • 0318 •0579 .0366 •7978 44 i 7 .0234 .7004 .0275 •3522 .0319 •0532 .0367 •7937 43 18 •0235 .6942 .0276 •3469 •032 .0486 •0367 •7897 42 19 •0235 .6879 .0276 • 34 i 5 .032 •044 .0368 •7857 4 i 20 1.0236 4.6817 1.0277 4-3362 1. 0321 4-0394 1.0369 3.7816 40 21 .0237 •6754 .0278 •3309 .0322 .0348 .037 •7776 39 22 .0237 .6692 .0278 •3256 .0323 .0302 •0371 •7736 38 23 .0238 .6631 .0279 .3203 .0323 • 0256 •0371 •7697 37 24 •023Q .6569 .028 • 3 i 5 .0324 .0211 .0372 •7657 36 25 I.O239 4.6507 1.028 4.3098 1.0325 4.0165 1-0373 3 - 76 i 7 35 26 .024 .6446 .0281 •3045 .0326 .012 •0374 •7577 34 27 .O24I •6385 .0282 •2993 .0327 .0074 •0375 •7538 33 28 .0241 .6324 .0283 .2941 .0327 .0029 .0376 •7498 32 29 .0242 . 6263 .0283 .2888 .0328 3.9984 .0376 •7459 3 i 30 I.O243 4. 6202 1.0284 4.2836 1.0329 3-9939 1-0377 3 - 74 2 30 3 i .0243 6142 0285 •2785 •033 .9894 .0378 •738 29 32 .0244 608 X .0285 •2733 •033 •985 •0379 • 734 i 28 33 .0245 .6021 .0286 .2681 •0331 .9805 .038 .7302 27 34 .0245 .5961 .0287 .263 .0332 .976 .0381 •7263 26 35 I.O246 4.5901 1.0288 4-2579 1-0333 3.9716 1.0382 3.7224 25 36 .0247 .5841 .0288 .2527 •0334 .9672 .0382 .7186 24 37 .0247 .5782 .0289 .2476 •0334 .9627 •0383 • 7 i 47 23 38 .0248 .5722 .029 .2425 •0335 .9583 .0384 .7108 22 39 .0249 • 5663 .0291 •2375 .0336 •9539 -0385 • 7°7 21 40 I.O249 4. 5604 x.0291 4.2324 i-o 337 3-9495 1.0386 3 - 7 ° 3 i 20 4 i .025 •5545 .0292 .2273 •0338 •9451 •0387 •6993 *9 42 .0251 .5486 .0293 .2223 •0338 .9408 •0387 •6955 18 43 .0251 .5428 .0293 •2173 •0339 •9364 .0388 .6917 17 44 .0252 •5369 .0294 .2122 •034 •932 .0389 .6878 16 45 1-02.53 4 - 53 i 1 1-0295 4.2072 1. 0341 3 - 9 2 77 1.039 3.684 15 46 •0253 •5253 .0296 .2022 .0341 • 9 2 34 .0391 .6802 14 47 .0254 • 5 i 95 .0296 .1972 .0342 .9x9 .0392 •6765 13 48 •0255 • 5 i 37 .0297 .1923 •0343 .9x47 -0393 .6727 12 49 •0255 •5079 .0298 •1873 •0344 .9104 •0393 .6689 11 50 1.0256 4.5021 1.0299 4.1824 1-0345 3.9061 1-0394 3-6651 10 5 i .0257 .4964 .0299 •1774 •0345 .9018 •0395 .6614 9 52 .0257 • 49°7 •03 •1725 .0346 .8976 •0396 •6576 8 53 .0258 •485 • 0301 . 1676 •0347 •*933, •0397 ■6539 7 54 .0259 •4793 .0302 .1627 .0348 •fe 9 .0398 .6502 6 55 1.026 4-4736 1.0302 4 -I 578 1.0349 3.8848 1.0399 3.6464 5 56 .026 •4679 V0303 .1529 •0349 .8805 •0399 •6427 4 57 .0261 .4623 .0304 .1481 .035 .8763 .04 •639 3 58 .0262 .4566 •0305 •1432 •0351 .8721 .0401 •6353 2 59 .0262 • 45 i •0305 .1384 .0352 .8679 .0402 .6316 X 60 1.0263 4-4454 1.0306 4 -I 336 1-0353 3-8637 1.0403 3.6279 0 / Co-SEC’t. Secant. Co-sec’t. Secant. Co-sec’t. Secant. Co-sec’t. | Secant. 770 760 75 ° 740 NATUKAL SECANTS AND CO-SECANTS. 407 / 1 | Secant. L6° | Co-sec’t Secant. L7° Co-sec’t. 3 Secant. .8° Co-sec’t. 1 Secant. .90 Co-sec’t / 0 1.0403 3.6279 1-0457 3-4203 1-0515 3.2361 1.0576 3-0715 60 I .0404 .6243 .0458 .417 .0516 .2332 •0577 .069 59 2 .0405 .6206 •0459 .4138 •0517 .2303 .0578 .0664 58 3 .0406 .6169 .046 .4106 •0518 .2274 •0579 .0638 57 4 .0406 •6133 .0461 •4073 .0519 .2245 .058 .0612 56 5 1.0407 3.6096 1.0461 3.4041 1.052 3.2216 1.0581 3-0586 55 6 .0408 .606 .0462 .4009 .0521 .2188 .0582 •0561 54 7 .0409 .6024 .0463 •3977 .0522 .2159 .0584 •0535 53 8 .041 •5987 .0464 •3945 •0523 .2131 •0585 •0509 52 9 .0411 •5951 .0465 • 39 i 3 .0524 .2102 .0586 .0484 51 10 1. 0412 3 - 59 I 5 1.0466 3 - 388 i 1.0525 3.2074 1.0587 3-0458 50 11 .0413 •5879 .0467 •3849 .0526 .2045 .0588 •0433 49 12 .0413 ■5843 .0468 •3817 •0527 .2017 .0589 .0407 48 13 .0414 .5807 .0469 •3785 .0528 .1989 •059 •0382 47 14 .0415 •5772 •047 •3754 .0529 .196 .0591 •0357 46 15 1.0416 3-5736 1. 0471 3-3722 1-053 3 -I 932 1.0592 3-0331 45 16 .0417 •57 .0472 •369 •0531 .1904 •0593 .0306 44 17 .0418 .5665 •0473 •3659 •0532 .1876 •0594 .0281 43 18 .0419 .5629 •0474 .3627 •0533 .1848 •0595 .0256 42 *9 .042 •5594 •0475 •3596 •0534 .182 .0596 •0231 4 1 20 1.042 3-5559 1.0476 3-3565 3 -I 792 1.0598 3.0206 40 21 .0421 •5523 •0477 •3534 •0536 .1764 •0599 .0181 39 22 .0422 .5488 .0478 •3502 •0537 • I 73 6 .06 . •0156 38 23 .0423 •5453 .0478 • 347 i •0538 . 1708 .0601 .0131 37 24 .0424 .5418 .0479 •344 •0539 .1681 .0602 .0106 36 25 1.0425 3-5383 1.048 3-3409 1.054 3-1653 1.0603 3.0081 35 26 .0426 •5348 .0481 •3378 .0541 .1625 .0604 .0056 34 27 .0427 •5313 .0482 •3347 • 0542 .1598 .0605 0031 33 28 .0428 •5279 .0483 • 33 i 6 •0543 • I 57 .0606 .0007 32 29 .0428 •5244 .0484 .3286 •0544 • I 543 .0607 2.9982 31 30 1.0429 3-5209 1.0485 3-3255 I -°545 3 -I 5 I 5 1.0608 2-9957 30 3 i •043 •5175 .0486 .3224 .0546 .1488 .0609 •9933 29 32 .0431 •514 .0487 • 3 i 94 •0547 .1461 .0611 .9908 28 33 .0432 .5106 .0488 •3163 .0548 •1433 .0612 . 9884 27 34 •0433 .5072 .0489 •3133 •0549 .1406 .0613 •9859 26 35 36 I -°434 3-5037 1.049 3.3102 1-055 3-1379 1.0614 2-9835 25 •0435 .5003 .0491 .3072 •0551 •1352 .0615 .981 24 37 38 .0436 .4969 .0492 .3042 •0552 •1325 .0616 .9786 23 •°437 •4935 •0493 • 3011 •0553 .1298 .0617 .9762 22 39 .0438 .4901 •0494 .2981 •0554 .1271 .0618 •9738 21 40 1.0438 3.4867 1.0495 3-2951 i-o 555 3.1244 1.0619 2.9713 20 4 i •0439 •4833 .0496 .2921 •0556 . 1217 .062 .9689 19 42 .044 •4799 .0497 .2891 •0557 .119 .0622 .9665 18 43 .0441 .4766 .0498 .2861 •0558 .1163 .0623 .9641 17 44 .0442 •4732 •0499 .2831 •0559 • II 37 .0624 .9617 16 45 1.0443 3.4698 1.05 3.2801 1.056 3 - 111 1.0625 2-9593 15 46 .0444 .4665 .0501 .2772 • 0561 .1083 .0626 •9569 14 47 48 •0445 .4632 .0502 •2742 .0562 •1057 .0627 •9545 *3 .0446 •4598 •0503 .2712 •0363 .103 .0628 •9521 12 49 .0447 •4565 .0504 .2683 •0565 . 1004 .0629 •9497 11 50 1.0448 3-4532 1-0505 3-2653 1.0566 3-0977 1.063 2.9474 10 5 i .0448 •4498 .0506 .2624 •0567 .0951 .0632 •945 9 52 .0449 •4465 .0507 •2594 .0568 •0925 •0633 .9426 8 53 •°45 •4432 .0508 .2565 .0569 .0898 .0634 .9402 7 54 .0451 •4399 .0509 •2535 •057 .0872 •0635 •9379 6 55 56 1.0452 3*4366 1-051 3.2506 1. 0571 3.0846 1.0636 2-9355 5 •°453 •4334 .0511 •2477 •0572 .082 .0637 •9332 4 57 58 •°454 .4301 .0512 .2448 •0573 •0793 .0638 •9308 3 •°455 .0456 .4268 •0513 .2419 •° 5 74 - .0767 .0639 .9285 2 59 30 .4236 .0514 •239 •0575 .0741 .0641 .9261 j I -°457 3.4203 I -° 5 I 5 3.2361 1.0576 30715 1.0642 2.9238 0 Co-sec’t. i 73 Secant. 0 Co-sec’t. 72 Secant. 0 1 Co-sec’t. 71 < Secant, j 3 1 Co-sec’t. 70 < Secant. 3 ' NATURAL SECANTS AND CO-SECANTS. t 20< Secant. , II Co-sec’t 1 21 c Secant. Co-sec’t. 22' Secant. 3 Co-sec’t. 23° Secant. | Co-sec’t. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 1 4 1 5 16 17 18 20 21 22 23 24 25 26 27 28 29 3 ° 3 1 32 33 34 35 36 37 38 39 40 4 1 42 43 44 45 46 47 48 49 50 5 1 52 53 54 55 56 57 58 59 60 1.0642 .0643 .0644 .0645 .0646 1.0647 .0648 .065 .0651 .0652 1.0653 .0654 .0655 .0656 .0658 1.0659 .066 .0661 .0662 .0663 1.0664 .0666 .0667 .0668 .0669 1.067 .0671 .0673 .0674 •0675 1.0676 .0677 .0678 .0679 .0681 1.0682 .0683 .0684 .0685 .0686 1.0688 .0689 .069 .0691 .0692 1.0694 .0695 .0696 .0697 .0698 1.0699 .0701 .0702 .0703 .0704 1.0705 .0707 .0708 .0709 .071 1. 0711 2.9238 • 9 2I 5 .gigi .9168 • 9*45 2.9122 .9098 • 9°75 .9052 .9029 2. 9006 .8983 .896 •8937 .8915 2.8892 .8869 1 .8846 .8824 .8801 2.8778 .8756 •8733 .8711 .8688 2.8666 .8644 .8621 .8599 •8577 2.8554 .8532 .851 .8488 .8466 2.8444 .8422 .84 .8378 .8356 2.8334 .8312 .829 .8269 .8247 2.8225 .8204 .8182 .816 .8139 2.8117 .8096 .8074 .8053 . 8032 2.801 .7989 .7968 •7947 . 79 2 5 2.7904 1. 0711 .0713 .0714 .0715 .0716 1. 0717 .0719 .072 .0721 .0722 1.0723 .0725 .0726 .0727 .0728 1.0729 .0731 .0732 .0733 .0734 1.0736 .0737 .0738 •0739 .074 1.0742 .0743 •0744 •0745 .0747 1.0748 .0749 •075 • 075 1 •0753 1-0754 •0755 .0756 .0758 •0759 1.076 .0761 .0763 .0764 .'0765 1.0766 .0768 .0769 .077 .0771 1.0773 .0774 •0775 .0776 .0778 1.0779 .078 .0781 .0783 .0784 1.0785 2.7904 .7883 .7862 .7841 .782 2.7799 .7778 •7757 •7736 • 77 i 5 2.7694 17674 .7653 .7632 .7611 2 - 759 1 •757 •755 •7529 •7509 2.7488 .7468 •7447 •7427 .7406 2.7386 .7366 •7346 7325 •7305 2.7285 .7265 •7245 .7225 .7205 2.7185 • 7 i6 5 •7145 •7125 • 7 io 5 2.7085 .7065 • 7°45 .7026 .7006 2.6986 .6967 .6947 .6927 .6908 2.6888 .6869 .6849 .683 .681 2.6791 .6772 .6752 •6733 .6714 2.6695 1.0785 .0787 .0788 .0789 .079 1,0792 •0793 .0794 .0795 .0797 1.0798 .0799 .0801 .0802 .0803 1.0804 .0806 .0807 .0808 .081 1. 0811 .0812 .0813 .0815 .0816 1.0817 .0819 .082 .0821 .0823 1.0824 .0825 .0826 .0828 .0829 1.083 .0832 •0833 .0834 .0836 1.0837 0838 .084 .0841 .0842 1.0844 .0845 ,0846 .0847 .0849 1.085 .0851 .0853 .0854 .0855 1.0857 .0858 .0859 .0861 .0862 1.0864 2.6695 .6675 .6656 .6637 .6618 2.6599 .658 .6561 .6542 •6523 2.6504 6485 .6466 .6447 .6428 2.641 .6391 6372 •6353 •6335 2.6316 .6297 .6279 .626 .6242 2.6223 .6205 .6186 .6168 .615 2.6131 .6113 .6095 .6076 .6058 2.604 .6022 .6003 • 59 8 5 •5967 2- 5949 • 593 i • 59 I 3 • 5 B 95 •5877 25859 .5841 •5823 .5805 •5787 2-577 •5752 •5734 • 57 l6 •5699 2.5681 •5663 .5646 .5628 1 -561 2.5593 1.0864 .0865 .0866 *0868 *0869 1.087 .0872 .0873 .0874 .0876 1.0877 .0878 .08S .0881 .0882 1.0884 .0885 .0886 .0888 .0889 1.0891 .0892 .0893 .0895 .0896 1.0897 .0899 .09 .0902 .0903 1.0904 .0906 .0907 .0908 .091 1. 091 1 .0913 .0914 .0915 .0917 1.0918 .092 .0921 .0922 .0924 1.0925 .0927 .0928 .0929 .0931 1.0932 •0934 •0935 .0936 .0938 1.0939 .0941 .0942 •0943 •0945 .0946 2-5593 •5575 .5558 •554 •5523 2. 5506 .5488 • 547 i •5453 • 543 6 2.5419 .5402 • 53 8 4 •5367 •535 2-5333 .5316 •5299 .5281 .5264 2.5247 •523 •5213 .5196 •5179 2.5163 .5146 •5129 .5x12 •5095 2.5078 .5062 •5045 .5028 .5011 2.4995 .4978 .4961 •4945 .4928 2.4912 .4895 .4879 .4862 .4846 2.4829 .4813 •4797 .478 .4764 2.4748 •4731 • 47 i 5 .4699 .4683 2. 4666 .465 • 4 6 34 .4618 .4602 2.4586 60 59 58 57 56 55 54 53 53 5 i 50 49 48 47 46 45 44 43 42 4 i 40 39 38 37 36 35 34 33 32 3 i 30 - 29 28 27 26 25 , 24 23 22 21 20 x 9 18 17 16 15 14 13 12 ■ 11 \ i° i 9 i 8 . 7 i 6 ) 5 i 4 ' 3 2 1 0 ' Co-sec’t € . Secant. 390 Co-sec’t ( . Secant. 58° Co-sec’t. ! Secant. 67 ° Co-sec’t ( . Sbcant. 56° NATURAL SECANTS AND CO-SECANTS. 409 24 ° 250 I 260 I 270 f Secant. Co-sec’t Secant. Co-sec’t. . Secant. Co-sec’t. 1 Secant. | Co-skc’t . ' 0 1.0946 2.4586 1. 1034 2. 3662 1.1126 2.2812 1. 1223 2.2027 60 I .0948 •457 •1035 •3647 .1127 .2798 .1225 .2014 59 2 .0949 •4554 .1037 •3632 .1129 .2784 . 1226 .2002 58 3 .0951 .4538 .1038 .3618 .1131 .2771 . 1228 .1989 57 4 .0952 .4522 .104 .3603 .1132 •2757 .123 "977 56 5 1-0953 2.4506 1. 1041 2.3588 *•"34 2.2744 1. 1231 2. 1964 55 6 •0955 •449 .1043 •3574 •ii 35 •273 •"33 .1952 54 7 .0956 •4474 .1044 •3559 •"37 .2717 •1235 • 1939 53 8 .0958 •4458 . 1046 •3544 •ii 39 •2703 "237 .1927 52 9 •0959 .4442 .1047 •353 .114 .269 .1238 " 9 X 4 51 10 1.0961 2. 4426 1. 1049 2.3515 1. 1142 2. 2676 1. 124 2.1902 50 11 .0962 .4411 .105 • 35 oi .1143 .2663 . 1242 .1889 49 12 .0963 •4395 .1052 .3486 •"45 .265 .1243 .1877 48 13 .6965 •4379 • 1053 •3472 .1147 .2636 "245 .1865 47 14 .0966 •4363 •1055 •3457 .1148 .2623 .1247 .1852 46 15 1.0968 2-4347 1.1056 2-3443 i.n 5 2. 261 1.1248 2. 184 45 16 .0969 4332 .1058 •3428 .1x51 .2596 .125 .1828 44 17 .0971 .4316 .1059 • 34 i 4 •"53 •2583 .1252 .1815 43 18 .0972 •43 . 1061 •3399 •"55 •257 "253 .1803 42 *9 •0973 .4285 .1062 •3385 .1156 •2556 •1255 .1791 4 1 20 I,0 975 2.4269 1.1064 2-3371 1.1158 2.2543 1-1257 2.1778 40 21 .0976 •4254 .1065 ■3356 •"59 •253 .1258 .1766 39 22 .0978 •4238 .1067 •3342 .1161 •2517 .126 "754 38 23 .0979 .4222 . 1068 •3328 .1163 •2503 .1262 .1742 37 24 .0981 .4207 .107 • 33 i 3 ,1164 •249 . 1264 "73 36 25 1.0982 2. 4191 1. 1072 2.3299 1. 1166 2.2477 1.1265 2.1717 35 26 .0984 .4176 •1073 •3285 .1167 •2464 .1267 " 7°5 34 27 .0985 .416 •1075 •3271 .1169 .2451 . 1269 .1693 33 28 .0986 •4145 .1076 •3256 .1171 .2438 .127 .1681 32 29 .0988 • 4 i 3 .1078 .3242 .1172 •2425 .1272 . 1669 3 i 3 ° 1.0989 2. 41 14 1. 1079 2.3228 i - "74 2. 241 1 *• I2 74 2.1657 30 3 i 0991 .4099 .1081 •3214 .1176 .2398 "275 .1645 29 32 0992 •4083 . 1682 •32 .1177 •2385 • I2 77 "633 28 33 .0994 .4068 .1084 .3186 •"79 .2372 .1279 .162 27 34 •0995 •4053 .1085 •3172 .118 •2359 .1281 . 1608 26 35 1 0997 2.4037 1.1087 2.3158 1.1182 2. 2346 1.1282 2.1596 25 36 .0998 .4022 .1088 •3143 .1184 •2333 .1284 .1584 24 37 .1 .4007 .109 .3129 .1185 •232 .1286 " 57 2 23 38 .1001 •3992 .1092 •3115 .1187 .2307 .1287 .156 22 39 .1003 •3976 .1093 .3101 .1189 .2294 . 1289 .1548 21 +0 1. 1004 2.3961 1. 1095 2.3087 1.119 2.2282 1. 1291 2.1536 20 D • 1005 •3946 .1096 •3073 .1192 .2269 .1293 "525 1 9 \2 .1007 • 393 i .1098 •3059 •"93 .2256 .1294 "513 18 ♦3 .1008 • 39 j 6 .1099 •3046 •"95 •2243 .1296 • 1501 17 14 -IOI .3901 • IIOI •3032 •"97 •223 .1298 .1489 16 15 *6 I. IOII 2.3886 1. 1 102 2.3018 1. 1198 2.2217 1. 1299 2.1477 15 .1013 •3871 .1104 .3004 .12 .2204 .1301 .1465 14 \7 1-8 .1014 •3856 .1106 •299 .1202 • 2192 • 1303 "453 13 .1016 .3841 .1107 .2976 .1203 .2179 •1305 .1441 12 19 • 1017 .3826 • iio 9 .2962 .1205 .2166 .1306 <"43 11 >0 1* 1019 2.3811 1. hi 2.2949 1. 1207 2.2153 1.1308 2. 1418 10 .102 •3796 .1112 •2935 .1208 .2141 .1406 9 >2 . 1022 •3781 .1113 .2921 .121 .2128 .1312 "394 8 >3 .1023 .3766 .1115 .2907 .1212 .2115 " 3 X 3 .1382 7 >4 .1025 • 375 i .1116 .2894 .1213 •2103 " 3 X 5 " 37 1 6 >5 ;6 1.1026 .1028 2.3736 1.1118 2.288 1. 1215 2.209 i" 3 i 7 2"359 5 • 372 i .112 .2866 . 1217 .2077 "3x9 "347 4 >7 i8 . 1029 .3706 .1121 •2853 .1218 .2065 .132 "335 3 .1031 .3691 .1123 •2839 .122 .2052 .1322 .1324 2 9 0 .1032 !. IO34 •3677 2. 3662 .1124 1. 1126 .2825 2.2812 . 1222 1. 1223 •2039 2.2027 • .1324 1.1326 .1312 2. 13 1 0 Co-sec’t. 1 65 < Secant. 3 11 Co-skc’t. 64 c Secant. ) Co-sec’t. 63 c Secant. ) Co-sec’t. 62 c Secant. ) r M M 4io NATURAL SECANTS AND CO-SECANTS. 28 Secant. 0 Co-sec’t. 29 Secant. D Co-sec’t. 30 s Secant. < D Co-sec’t. 31 Secant. < 3 I Co-sec’t. f 0 1 2 3 4 5 6 7 8 9 10 11 12 *3 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3 1 32 33 34 35 36 37 38 39 40 4 * 42 43 44 45 4<3 47 48 49 5 0 5 1 52 5 3 54 5'5 56 57 58 60 1.1326 .1327 .1329 •I 33 i •1333 i - 1334 •I 33 6 .1338 •134 •1341 I-I 343 .1345 •1347 •1349 •135 I-I 352 •1354 ‘1356 •1357 •1359 1.1361 •1363 • i 3 6 5 .1366 .1368 i - 137 •1372 •1373 •1375 •1377 I - I 379 .1381 . 1382 .1384 .1386 1.1388 •139 .1391 •1393 •1395 i - 1397 •1399 .1401 .1402 .1404 1. 1406 .1408 .141 .141-1 •1413 1. 1415 .1417 .1419 .1421 .1422 1. 1424 . 1426 . 1428 •143 .1432 i- M 33 2.13 . 1289 .1277 . 1266 .1254 2. 1242 .1231 . 1219 .1208 .1196 2.1185 •ii 73 .1162 •ii 5 •1139 2.1127 .1116 .1104 .1093 .1082 2.107 .1059 .1048 .1036 1025 2. 1014 .1002 .0991 .098 .0969 2.0957 .0946 •0935 .0924 .0912 2. 0901 .089 .0879 .0868 .0857 2.0846 •0835 .0824 .0812 .0801 2.079 .0779 .0768 •0757 .0746 2 0735 .0725 .0714 .0703 .0692 2.0681 .067 .065-9 .0648 .0637 2.0627 i - 1433 •1435 •1437 •1439 .1441 i - 1443 •1445 .1446 .1448 -145 1.1452 •1454 .1456 .1458 •1459 1.1461 .1463 .1465 .1467 .1469 *• I 47 I •1473 .1474 • I 47 6 .1478 1. 148 .1482 .1484 .i486 .1488 1. 1489 • I 49 I •1493 •1495 .1497 1. 1499 .1501 •1503 •1505 •1507 1.1508 •151 .1512 •1514 .1516 1.1518 .152 .1522 .1524 .1526 1.1528 *i 53 •i 53 i •1533 •1535 i-i 537 •1539 •i 54 i •1543 •1545 1547 2.0627 .0616 .0605 •0594 •0583 2.0573 .0562 •0551 -054 -053 2.0519 .0508 .0498 .0487 .0476 2.0466 -0455 .0444 •0434 .0423 2.0413 .0402 .0392 .0381 •037 2.036 .0349 •0339 .0329 .0318 2.0308 0297 .0287 .0276 .0266 2.0256 .0245' .0235 .0224 .0214 2.0204 .0194 •0183 •oi 73 .0163 2.0152 .0142 .0132 .0122 .0111 2. 0101 .0091 .0081 .0071 .0061 2.005 .004 .003 .002 .001 2 i - 1547 •1549 •I 55 i -1553 •1555 I - 1557 •1559 .1561 .1562 .1564 1. 1566 .1568 •157 •I 57 2 •1574 i-i 57 6 •i 57 8 .i 5 8 .1582 .1584 1.1586 .1588 -159 .1592 -1594 i.i 59 6 .1598 .16 .1602 .1604 1. 1606 .1608 .161 .1612 .1614 1. 1616 .1618 .162 .1622 .1624 1.1626 .1628 .163 .1632 .1634 1.1636 .1638 . 164 .1642 .1644 1. 1646 . 1648 .165 .1652 .1654 1. 1656 • 1658 . 166 . 1662 .1664 1. 1666 2 1.999 .998 •997 .996 1-995 •994 •993 .992 .991 1.99 •989 .988 .987 .986 1.985 .984 • 9 8 3 .982 .9811 1.9801 .9791 .9781 .9771 .9761 1-9752 .9742 •9732 .9722 • 97 i 3 1 - 97°3 •9693 .9683 .9674 .9664 1.9654 •9645 • 9 6 35 .9625 .9616 1:9606 • 959 6 •9587 •9577 .9568 I .9558 •9549 •9539 •953 .952 i- 95 i .9501 • 949 1 .9482 •9473 i- 94 6 3 •9454 •9444 •9435 •9425 1.9416 1. 1666 .1668 .167 .1672 .1674 1 1676 .1678 .1681 .1683 .1685 1.1687 .1689 1691 .1693 .1695 1. 1697 .1699 .1701 •1703 •1705 1. 1707 .1709 .1712 .1714 .1716 1.1718 .172 .1722 .1724 . 1726 1. 1728 •173 •1732 •1734 •1737 i - 1739 .1741 •1743 •1745 •1747 i- 1749 •1751 •1753 •1756 •1758 1.176 .1762 .1764 .1766 .1768 1. 177 .1772 •1775 • 1777 .1779 1.1781 •1783 .1785 .1787 .179 1. 1792 1.9416 9407 •9397 • 9388 •9378 1 9369 •936 •935 • 934 i •9332 1.9322 • 93 i 3 •9304 • 9 2 95 .9285 1.9276 .9267 .9258 .9248 • 9 2 39 1.923 .9221 .9212 .9203 • 9*93 1.9184 • 9 T 75 .9166 • 9 I 57 .9148 I - 9 I 39 • 9 I 3 .9121 .9112 .9102 1.9093 .9084 •9075 .9066 •9057 1.9048 •9039 • 9°3 .9021 .9013 1.9004 .8995 .8986 .8977 .8968 1.8959 •895 .8941 .8932 .8924 1.8915 .8906 .8897 .8888 .8879 1.8871 60 59 58 57 56 55 54 53 52 5 i 50 49 48 46 45 44 43 42 4 i 40 !! 37 36 35 34 33 32 3 i 30 ’ 2 9 28 27 26 25 . 24 23 22 £ * :t ■ 1 7 16 15 14 . i 3 »i 12 i 11 { 10 j 9 l 8 : 7 i 6 'i 4 : 3 2. 1 0 0 Co-sec’t ( Secant. 51° Co-seO’t t Secant. 500 Go-sbc’t i . Secant. >9° Co-sec’t 1 .| Secant. 58° NATURAL SECANTS AND CO-SECANTS. 411 32 ° 33 ° I 34 ° 35 ° ' Secant. Co-sec’t. Secant. Co-sec’t. Secant. Co-sec’t. Secant. Co-sec’t. ' 0 1. 1792 1.8871 1. 1924 1.8361 1. 2062 1.7883 1.2208 1-7434 60 X .1794 .8862 .1926 •8352 .2064 •7875 .221 .7427 59 2 .1796 .8853 .1928 •8344 .2067 .7867 .2213 .742 58 3 .1798 .8844 •193 8336 .2069 .786 2215 •7413 57 4 .18 .8836 •1933 .8328 .2072 .7852 .2218 •7405 56 5 1. 1802 1.8827 i - 1935 1.832 1.2074 1.7844 1.222 1-7398 55 6 . 1805 .8818 •1937 • 83.11 2076 •7837 .2223 •7391 54 7 . 1807 .8809 .1939 •8393 .2079 .7829 .2225 •7384 53 8 .1809 .8801 .1942 .8295 .2081 .7821 .2228 •7377 52 9 . 1811 .8792 .1944 .8287 .2083 .7814 223 •7369 5 i 10 1. 1813 1.8783 1.1946 1.8279 1.2086 1.7806 1.2233 1.7362 50 11 .1815 .8785 .1948 .8271 .2088 .7798 •2235 •7355 49 12 .1818 .8766 •1951 .8263 .2091 • 779 1 .2238 •7348 48 13 . 182 •8757 •1953 •8255 .2093 •7783 .224 • 734 i 47 14 . 1822 •8749 •1955 .8246 .2095 .7776 .2243 •7334 46 i 5 1.1824 1.874 1.1958 1.8238 1.2098 1.7768 1.2245 I -7327 45 16 .1826 •8731 .196 .823 • 21 .776 .2248 •7319 44 17 .1828 .8723 .1962 .8222 .2103 •7753 .225 .7312 43 18 .1831 .8714 .1964 .8214 .2195 •7745 •2253 •7305 42 *9 •1833 .8706 .1967 • 8206 .2107 •7738 •2255 .7298 4 i 20 1-1835 1.8697 1.1969 1.8198 1. 21 1 i -773 1.2258 1. 7291 40 21 •1837 .8688 .1971 .819 .2112 •7723 .226 .7284 39 22 .1839 .868 .1974 .8182 .2115 •7715 .2263 •7277 38 23 . 1841 .8671 .1976 .8174 .2117 .7708 .2265 .727 37 24 .1844 .8663 .1978 .8166 .2119 •77 .2268 .7263 36 25 1. 1846 1.8654 1. 198 1.8158 I. 2122 1.7693 1.227 1.7256 35 26 .1848 .8646 .1983 • 815 .2124 .7685 .2273 .7249 34 27 .185 .8637 .1985 .8142 .2127 .7678 .2276 .7242 33 28 . 1052 .8629 .1987 .8134 .2129 .767 .2278 •7234 32 29 •1855 .862 • I 99 .8126 .2132 .7663 .2281 .7227 3 i 30 1.1857 1. 8611 1. 1992 1. 8118 1.2134 1-7655 1.2283 1.722 30 3 i .1859 .8603 .1994 .811 .2136 .7648 .2286 •7213 29 32 .1861 •8595 • *997 .8102 •2139 .764 .2288 .7206 28 33 .1863 .8586 .1999 .8094 .2141 •7633 .2291 .7199 27 34 .1866 .8578 .2001 .8086 .2144 .7625 .2293 .7192 26 35 1. 1868 1.8569 1.2004 1.8078 I. 2146 1.7618 1.2296 1.7185 25 36 .187 .8561 .2006 .807 .2149 .761 .2298 • 7 i 78 24 37 .1872 •8552 .2008 .8062 .2151 .7603 .2301 .7171 23 38 .1874 •8544 .201 .8054 •2153 • 759 6 .2304 .7164 22 39 .1877 •8535 .2013 .8047 .2156 .7588 .2306 • 7 I 57 21 40 1.1879 1.8527 1. 2015 1.8039 I.2158 1.7581 1.2309 i- 7 i 5 i 20 4 i .1881 .8519 .2017 .8031 .2l6l •7573 .2311 .7144 19 42 . 1883 .851 .202 .8023 .2163 •7566 •2314 • 7 J 37 18 43 .1886 .8502 .2022 .8015 .2166 •7559 .2316 - 7 I 3 17 44 .1888 •8493 .2024 .8007 .2l68 • 755 i .2319 .7123 16 45 1.8485 1.2027 X7999 I. 2171 1-7544 1.2322 1.7116 i 5 46 .1892 •8477 .2029 .7992 •2173 •7537 .2324 .7109 14 47 .1894 .8468 .2031 .7984 •2175 •7529 •2327 .7102 13 48 .1897 .846 .2034 .7976 .2178 .7522 .2329 •7095 12 49 .1899 .8452 .2036 .7968 .2l8 • 75 i 4 .2332 .7088 11 50 I. I9OI 1.8443 1. 2039 1.796 I.2183 1-7507 1-2335 1.7081 10 5 i .I9O3 •8435 .2041 •7953 .2185 •75 •2337 • 7°75 9 52 .I906 .8427 .2043 •7945 .2188 •7493 •234 .7068 8 53 . I908 .8418 .2046 •7937 .219 7485 .2342 .7061 7 54 .191 .841 .2048 •7929 .2193 •7478 •2345 •7054 6 55 I. 1912 1.8402 1.205 1. 7921 1.2195 1.7471 1.2348 1.7047 5 56 •* 9*5 •8394 •2053 .7914 .2198 •7463 •235 .704 4 57 .1917 •8385 •2055 .7906 .22 •7456 •2353 •7033 3 58 .1919 •8377 .2057 .7898 .2203 •7449 •2355 .7027 2 59 .1921 •8369 .296 .7891 .2205 •7442 •2358 .702 1 60 1. 1922 1.8361 1.2062 1.7883 1.2208 1-7434 1.2361 1. 7013 0 H Co-sec’t. Secant. Co-sec’t. Secant. Co-sec’t. Secant. Co-sec’t. Secant. / 57 ° 560 550 54 ° 412 NATURAL SECANTS AND CO-SECANTS. 36° 370 38° 390 ' 1 Secant. Co-sec’t. Secant. Co-sec’t. Secant. Co-sec’t. Secant. Co-sec’t. ' o 1.2361 i- 7 oi 3 1. 2521 1. 6616 1.269 1.6243 1.2867 1.589 60 I •2363 .7006 .2524 .661 .2693 .6237 .2871 • 5 88 4 59 2 .2366 .6999 .2527 .6603 .2696 .6231 .2874 • 5 8 79 58 3 •2368 •6993 •253 •6597 .2699 .6224 .2877 • 5 8 73 57 4 • 237 1 .6986 •2532 .6591 .2702 .6218 .288 •5867 56 5 1-2374 1.6979 x -2535 1.6584 1.2705 1.6212 1.2883 1.5862 55 6 .2376 .6972 .2538 .6578 .2707 .6206 .2886 •5856 54 7 •2379 .6965 .2541 .6572 .271 .62 .2889 •585 53 8 .2382 •6959 •2543 •6565 •2713 .6194 . 2892 j •5845 52 9 .2384 .6952 .2546 •6559 .2716 .6188 .2895 •5839 5 i IO 1.2387 1.6945 1-2549 1-6552 1. 2719 1.6182 1.2898 1-5833 50 ii •2389 .6938 *'•2552 .6546 .2722 .6176 .2901 ; . 5828 49 12 .2392 .6ch2 •2554 •654 •2725 .617 .29O4 .5822 48 i 3 •2395 .6925 •2557 •6533 .2728 .6164 . 2907 .5816 47 14 •2397 .6918 .256 .6527 . - 273 I •6159 .291 | .5811 46 i 5 1.24 1.6912 1-2563 1.6521 1-2734 1.6153 I- 29 I 3 1.5805 45 16 .2403 .6905 •2565 .6514 •2737 .6147 .2916 •5799 44 17 .2405 .6898 .2568 .6508 •2739 .6141 .2919 •5794 43 18 .2408 .6891 .2571 .6502 .2742 • 6 i 35 .2922 •5788 42 19 .2411 .6885 •2574 .6496 •2745 .6129 .2926 •5783 41 20 1. 2413 1.6878 1-2577 1.6489 1.2748 1.6123 I.2929 1-5777 40 21 .2416 .6871 •2579 .6483 •2751 .6117 .2932 •5771 39 22 .2419 .6865 .2582 .6477 •2754 .6111 •2935 •5766 38 23 .2421 .6858 •2585 .647 •2757 .6105 .2938 •576 37 24 .2424 .6851 .2588 .6464 .276 .6099 .2941 •5755 36 25 1.2427 1.6845 1.2591 1.6458 1.2763 1.6093 I.2944 1-5749 35 26 .2429 .6838 •2593 .6452 .2766 .6087 .2947 ■5743 34 27 .2432 .6831 .2596 •6445 .2769 .6081 •295 •5738 33 28 •2435 .6825 •2599 •6439 .2772 •6077 •2953 •5732 32 29 •2437 .6818 .2602 •6433 •2775 .607 •2956 •5727 3 i 30 1.244 1.6812 1.2605 1.6427 1.2778 1.6064 I.296 1. 5721 30 31 ■2443 .6805 .2607 .642 .2781 .6058 .2963 • 57 i 6 29 32 •2445 .6798 .261 .6414 .2784 .6052 .2966 • 57 i 28 33 .2448 .6792 • 2613 .6408 .2787 .6046 .2969 •5705 27 34 .2451 .6785 .2616 .6402 .279 .604 .2972 •5699 26 35 1-2453 1.6779 1.2619 1.6396 1-2793 1.6034 X -2975 1.5694 25 36 .2456 .6772 .2622 .6389 •2795 .6029 .2978 .5688 24 37 •2459 .6766 .2624 .6383 .2798 .6023 .2981 •5683 23 38 .2461 •6759 .2627 •6377 .2801 .6017 • 2 9 8 5 •5677 22 39 .2464 .6752 • 263 •6371 .2804 .6011 .2988 •5672 21 40 1.2467 1.6746 1.2633 1-6365 1.2807 1.6005 I. 299I 1.5666 20 4 i .247 •6739 .2636 •6359 .281 .6 .2994 • 5661 19 42 .2472 •6733 .2639 •6352 .2813 •5994 •2997 •5655 18 43 •2475 .6726 .2641 .6346 .2816 - 59 88 •3 •565 17 44 .2478 .672 .2644 •634 .2819 - 59 82 - 30°3 •5644 16 45 1.248 1.6713 1.2647 I -6334 1.2822 i -5976 1.3006 1-5639 i 5 46 •2483 .6707 .265 .6328 .2825 •5971 .301 •5633 14 47 .2486 .67 •2653 .6322 .2828 •5965 .3013 .5628 13 48 .2488 .6694 .2656 • 6316 .2831 •5959 .3016 .5622 12 49 .249 .6687 .2659 .6309 .2834 •5953 .3019 •5617 11 50 1.2494 1. 6681 1.2661 1.6303 1.2837 1-5947 1.3022 1.5611 10 51 .2497 .6674 .2664 .6297 .284 •5942 .3025 .5606 9 52 .2499 .6668 .2667 . 6291 .2843 •5936 • 3029 •56 8 53 .2502 .6661 .267 .6285 .2846 •593 .3032 •5595 7 54 •2505 ' -6655 .2673 .6279 .2849 .5924 •3035 •559 6 55 1.2508 1.6648 1.2676 1.6273 1.2852 I - 59 I 9 1-3038 1-5584 5 56 • 251 .6642 .2679 .6267 •2855 • 59 I 3 .3041 •5579 4 57 •2513 .6636 .2681 .6261 .2858 • 59°7 •3044 •5573 3 58 .2516 .6629 .2684 •6255 .2861 • 59 QI .3048 •5568 2 59 .2519 .6623 .2687 .6249 .2864 .5896 •3051 •5563 1 60 1. 2521 1. 6616 1.269 1.6243 1.2867 1.589 1-3054 1-5557 0 Co-sec’t. Secant. Co-skc’t. Secant. Co-sec’t. Secant. Co-sec’t. Secant. r 530 520 1 51° 1 500 413 NATURAL SECANTS AND CO-SECANTS. 40 ° I 41 ° 42 ° 439 , ' Secant. Co-sec't. ; Secant. Co-se»c’t. Secant. Co-sec’t. Secant. Co-sec’t. ' o i-3°54 1 - 55.57 i-325 1.5242 i-3456 1-4945 I -3673 1.4663 60 I ■ 3°57 •5552 •3253 •5237 •346 •494 •3677 .4658 59 2 .306 •5546 •3257 • 5232 .3463 •4935 .3681 •4654 58 3 • 3 o6 4 •5541 .326 •5227 •3467 •493 .3684 •4649 57 4 • 3 o6 7 • 5536 •3263 .5222 •347 •4925 .3688 •4644 56 5 i-3°7 1-553 1.3267 1-5217 1-3474 1. 4921 1.3692 1.464 55 6 •3073 •5525 •327 .5212 •3477 .4916 •3695 •4635 54 7 • 3 ° 7 6 •552 •3274 .5207 .3481 .49 11 •3699 .4631 53 8 .308 •5514 •3277 .5202 •3485 •4906 •3703 .4626 52 9 • 3 o8 3 •5509 .328 • 5197 .3488 .4901 •3707 .4622 5 i IO 1.3086 1-5503 1.3284 1.5192 1.3492 1.4897 I - 37 I 1.4617 50 ii .3089 •5498 .3287 .5187 •3495 .4892 • 37 i 4 .4613 49 12 .3092 •5493 •329 .5182 •3499 .4887 • 37 i 8 .4608 48 13 .3096 • 54 8 7 •3294 •5177 • 3502 .4882 .3722 .4604 47 14 •3099 .5482 •3297 • 5171 •3506 .4877 •3725 •4599 46 15 1. 3102 1-5477 i- 33 01 1.5166 1-3509 1-4873 1.3729 1-4595 45 l6 • 3 io 5 •5471 •3304 .5161 • 35 i 3 . 4868 •3733 •459 44 17 .3109 .5466 •3307 • 5156 • 35 i 7 .4863 •3737 .4586 43 l8 .3112 .5461 •3311 •5151 •352 .4858 •374 .4581 42 1 9 • 3 ii 5 •5456 • 33 I 4 .5146 •3524 •4854 •3744 •4577 4 i 20 1.3118 i -545 i- 33 i 8 1. 5141 *1-3527 1.4849 i-3748 1-4572 40 21 .3121 •5445 •3321 • 5136 • 353 i •4844 •3752 .4568 39 22 • 3 I2 5 •544 •3324 • 5131 •3534 •4839 •3756 •4563 38 23 • 3 I2 8 •5434 • 3328 .5126 •3538 •4835 •3759 •4559 37 24 •3131 •5429 •3331 .5121 •3542 •483 •3763 •4554 36 2 5 i- 3 i 34 I -5424 1-3335 1.5116 1-3545 1-4825 1-3767 i -455 35 26 •3138 •5419 •3338 .5111 •3549 .4821 • 377 i •4545 34 27 • 3 M 1 • 54 i 3 •3342 .5106 •3552 .4816 •3774 • 454 i 33 28 • 3 i 44 .5408 •3345 .5101 •3556 .4811 •3778 •4536 32 29 .3148 •5403 •3348 .5096 •356 .4806 •3782 4532 3 i 30 i- 3 i 5 i i - 5398 1-3352 1. 5092 1-3563 1.4802 1.3786 1 4527 30 3 i • 3 i 54 • 539 2 •3355 .5087 •3567 •4797 •379 •4523 29 32 • 3 i 57 •5387 •3359 . 5082 •3571 • 479 2 •3794 .4518 28 33 .3161 •5382 • 33 6 2 •5077 •3574 .4788 •3797 •4514 27 34 ; -3164 •5377 •3366 .5072 •3578 •4783 .3801 • 45 i 26 35 1.3167 i- 537 i 1-3369 1.5067 i-358i 1.4778 1.3805 i- 45 o 5 25 3 6 , - 3!7 •5366 •3372 .5062 •3585 •4774 .3809 .4501 24 37 ; - 3 I 74 536i •3376 • 5057 •3589 •4769 •3813 •4496 23 38 • 3 i 77 •5356 •3379 •5052 •3592 •4764 .3816 •4492 22 39 .318 •5351 •3383 •5047 •3596 •476 .382 •4487 21 40 1.3184 x -5345 1.3386 1.5042 1.36 1-4755 1.3824 1.4483 20 4 i •3187 •534 •339 •5037 •3603 •475 .3828 •4479 19 42 •319 •5335 •3393 • 5032 •3607 • 474*6 •3832 •4474 18 43 • 3 i 93 •533 •3397 .5027 .3611 .4741 •3836 •447 17 44 • 3 i 97 •5325 •34 .5022 .3614 •4736 •3839 •4465 16 45 1.32 I- 53 I 9 1.3404 1.5018 1.3618 1-4732 1-3843 1.4461 i 5 46 • 3203 • 53*4 •3407 • 5013 •3622 •4727 •3847 •4457 14 47 .3207 •5309 • 34 ii .5008 •3025 •4723 •3851 •4452 13 48 .321 •5304 • 34 H • 5003 •3629 .4718 •3855 •4448 12 49 •3213 •5299 .3418 .4998 •3633 • 47 i 3 •3859 •4443 11 50 1-3217 I -5294 1.3421 1-4993 1.3636 1.4709 1-3863 1-4439 10 5 i .322 .5289 •3425 .4988 •364 •4704 .3867 •4435 9 52 .3223 •5283 •3428 •4983 •3644 •4699 •387 •443 8 53 •3227 •5278 •3432 •4979 •3647 •4695 •3874 .4426 7 54 •323 •5273 •3435 •4974 •3651 •469 .3878 .4422 6 55 1-3233 1.5268 1-3439 1.4969 i-3655 1.4686 1.3882 I - 44 I 7 5 56 •3237 •5263 •3442 •4964 •3658 .4681 .3886 • 44 I 3 4 57 •324 •5258 •3446 •4959 .3662 .4676 •389 .4408 3 58 •3243 •5253 •3449 •4954 .3666 .4672 •3894 •4404 2 59 •3247 .5248 •3453 •4949 .3669 •3667 .3898 •44 1 60 1-325 1.5242 I -3456 1-4945 1-3673 1.4663 1.3902 1-4395 0 ' Co-sec’t. Secant. Co-sec’t. Secant. Co-sec’t. Secant. Co-sec’t. Secant. 49 ° 48 ° 47 ° 460 M M* 414 NATURAL SECANTS AND CO-SECANTS. 440 440 440 , t Secant. Co-sec’t. / t Secant. Co-sec’t. / / Secant. Co-sec’t. / 0 1.3902 1-4395 60 21 1.3984 1-4305 39 4 i 1.4065 1. 4221 19 I •3905 • 439 1 59 22 .3988 .4301 38 42 .4069 .4217 18 2 • 39°9 •4387 58 23 •3992 •4297 37 43 •4073 .4212 17 3 • 39*3 .4382 57 24 • 399 6 .4292 36 44 .4077 .4208 16 4 •39*7 •4378 56 25 1.4 1.4288 35 45 1.4081 1.4204 15 5 1.3921 1-4374 55 26 .4004 .4284 34 46 .4085 .42 14 6 • 39 2 5 •437 54 27 .4008 .428 33 47 .4089 .4196 13 7 3929 •4365 53 28 .4012 .4276 32 48 •4093 .4192 12 8 •3933 .4361 52 29 .4016 .4271 3 i 49 .4097 .4188 11 9 •3937 •4357 5 i 30 1.402 1.4267 30 5 o 1. 4101 1.4183 10 10 i- 394 i 1-4352 50 3 i .4024 .4263 29 5 i •4105 .4179 9 11 •3945 •4348 49 32 .4028 •4259 28 52 .4109 • 4 W 5 8 12 •3949 •4344 48 33 .4032 •4254 27 53 • 4 ii 3 .4171 7 13 •3953 •4339 47 34 .4036 <425 26 54 .4117 .4167 6 14 •3957 •4335 46 35 1.404 1.4246 25 55 1. 4122 1.4163 5 i 5 1.396 I - 433 I 45 36 .4044 .4242 24 56 .4126 •4159 4 16 • 3964 •4327 44 37 .4048 .4238 23 57 • 4 i 3 •4154 3 17 .3968 .4322 43 38 .4052 •4333 22 58 •4134 • 4*5 2 18 •3972 .4318 42 39 .4056 .4229 21 , 39 .4138 .4146 1 *9 •3976 • 43 i 4 4 1 40 1.406 # 1.4225 20 60 1.4142 1. 4142 0 20 1.398 I- 43 I 40 / Co-sec’t. Secant. t / Co-sec’t. Secant. / / Co-sec’t. Secant. t 45 ° 450 45 ° Preceding Table contains Natural Secants and Co-secants for every minute of the Quadrant to Radius i. If Degrees are taken at head of column, Minutes, Secant, and Co-secant must be taken from head also; and if they are taken at foot of column, Minutes, etc., must be taken from foot also. Illustration. — 1.05 is secant of 17 0 45' and co secant of 72 0 15'. To Compute Secant or Co-secant of any Angle. Rule. — Divide 1 by Cosine of angle for Secant, and by Sine for Co-secant. Example i. — What is secant of 25 0 25'? Cosine of angle = .903 21. Then 1 -f- .903 21 = 1.1072, Secant. 2. — What is co secant of 64° 35'? Sine of angle = .903 21. Then 1 -4- .903 21 = 1.1072, Co-secant. To Compute Degrees, IVTinntes, and. Seconds of a Secant or Co-secant. When Secant is given, Proceed as by Rule, page 402, for Sines, substituting Secants for Sines. Example. —What is secant for 1. 1607 ? The next less secant is 1. 1606, arc for which = 30 0 30'. Next greater secant is 1.1608, difference between which and next less is 1. 1608-— 1. 1 606 = .0002. Difference between less tab. secant and one given is 1. 1607 — 1. 1606 = .0001. Then .0002 : .0001 60 : 30, which, added to 30 0 30' = 30° 30' 30". When Co -secant is given, Proceed as by Rule, page 402, substituting Co-secants for Cosines. NATURAL TANGENTS AND CO-TANGENTS. INTatnral Tangents and. Co-tangents. 0° 1° 20 30 ' Tang. CO-TANG. Tang. CO-TANG. Tang* CO-TANG. Tang. CO-TANG. ' o .00000 Infinite. .017 46 5 l 29 C •°34 9 2 28.6363 .05241 I9.081I 60 I .000 29 3437-75 •017 75 6.3506 •03521 8.3994 •0527 8-9755 59 2 .000 58 1718.87 .018 04 5 - 44 I 5 •035 5 8. 1664 .05299 8.8711 58 3 .000 87 145.92 .01833 4 - 56 l 3 •035 79 7-9372 .053 28 8.7678 57 4 .001 16 859 - 43 6 .018 62 3.7086 .036 09 7.7117 •053 57 8.6656 56 5 .001 45 687.549 .018 91 52.8821 •036 38 27.4899 •053 87 18.5645 55 6 .001 75 572-957 .019 2 2.0807 .03667 7-2715 .054 16 8.4645 54 7 .00204 491. 106 .01949 1.3032 .036 96 7.0566 •054 45 8-3655 53 8 .002 33 29.718 .019 78 0.5485 03725 6.845 •054 74 8.2677 52 9 .002 62 381.971 .020 07 49.8157 •037 54 6.6367 •05503 8. 1708 51 IO .002 91 343-774 . 020 36 49 . IO39 •037 83 26.4316 •055 33 18.075 5 ° n .003 2 12.521 .020 66 8.4121 .038 12 6.2296 .05562 7.9802 49 12 .003 49 286.478 .020 95 7-7395 .038 42 6. 0307 •055 9 1 7.8863 48 13 .003 78 64.441 .021 24 7-0853 .03871 5-8348 .0562 7-7934 47 14 .00407 45-552 .021 53 6.4489 .039 5-6418 .056 49 7 - 7 OI 5 46 i 5 .004 36 229. 182 .021 82 45.8294 .039 29 25 - 45 I 7 .05678 17.6106 45 16 .004 65 14.858 .022 11 5.2261 •039 58 5-2644 .05708 7-5205 44 i 7 .004 95 02.219 .022 4 4.6386 .039 87 5.0798 •057 37 7 - 43 I 4 43 18 .005 24 190.984 .022 69 4.0661 .040 16 4.8978 .05766 7-3432 42 i 9 •005 53 80.932 .022 98 3 - 5 o 8 i . 040 46 4.7185 •057 95 7-2558 41 20 .005 82 i7!.885 .023 28 42.9641 .04075 24.5418 . 058 24 I 7 - i6 93 40 21 .006 11 63 -7 •02357 2-4335 .041 04 4-3675 •058 54 7.0837 39 22 .006 4 56.259 • 023 86 i - 9 i 58 •041 33 4-1957 -05883 6.999 38 23 006 69 49.465 •02415 1.4106 .041 62 4.0263 • 059 12 6.915 37 24 .006 98 43-237 .02444 0.9174 041 91 3-8593 -05941 6.8319 36 25 .007 27 137.507 .024 73 40.4358 042 2 23.6945 •059 7 16.749 6 35 26 .007 56 32.219 .025 02 399655 0425 3-5321 -059 99 6-6681 : 34 27 .007 85 27.321 •02531 9 - 5059 .042 79 3 - 37 i 8 .060 29 6- 5874 33 28 .008 14 22.774 .0256 9. 0568 .043 08 32137 .060 58 6-5075 32 29 .008 44 18.54 .025 89 8.6177 •043 37 3-0577 .060 87 6.4283 31 3 ° .008 73 114-589 .026 19 38.1885 .043 66 22.9038 .061 16 16.3499 3 ° 31 .009 02 10. 892 .02648 7.7686 •043 95 2-7519 .061 45 6.2722 29 32 .00931 07. 426 .026 77 7-3579 .044 24 2.602 •061 75 6. 1952 28 33 .009 6 04.171 .027 06 6.956 .044 54 2.4541 .062 04 6. 1 19 27 34 .009 89 01. 107 •02735 6.5627 .044 83 2. 3081 .06233 6-0435 26 35 .010 18 98.2179 .027 64 36.1776 .045 12 22. 164 .06262 15.9687 25 36 .01047 5-4895 •02793 5. 8006 .04541 2.0217 .06291 5-8945 24 37 .01076 2. 9085 .028 22 5-4313 0457 1.8813 . 063 2 1 5.8211 23 38 .011 05 0-46.33 .028 51 5.0695 •045 99 1.7426 •0635 5-7483 22 39 • OI1 35 88. 1436 .028 81 4 - 7 i 5 i . 046 28 1.6056 •063 79 5.6762 21 40 .011 64 85.9398 .029 1 34.3678 .046 58 21.4704 .06408 15.6048 20 4 i .01193 3-8435 •029 39 4.0273 .046 87 1-3369 •06437 5-534 19 42 .012 22 1.847 .029 68 3-6935 .047 16 1.2049 .06467 5.4638 18 43 .012 51 79-9434 .02997 3.3662 •047 45 1.0747 .064 96 5-3943 17 44 .012 8 8.1263 .030 26 3-0452 •047 74 0. 946 .065 25 5 - 3254 16 45 .01309 76-39 •03055 32.7303 .048 03 20.8188 •065 54 15-2571 15 46 •01338 4.7292 . 030 84 2.4213 .04832 0. 6932 .06584 5-1893 14 47 • 01367 3-139 •031 14 2. 1181 .048 62 0.5691 .066 13 5. 1222 I 3 48 •01396 1.6151 •03143 1.8205 .048 91 0.4465 .066 42 5-0557 12 49 .01425 0-1533 .031 72 1.5284 .0492 0.3253 . 066 7 1 4.9898 11 5 o 01455 68.7501 .632 01 31.2416 •049 49 20. 2056 .067 14.9244 10 5 i .014 84 7.4019 •0323 0-9599 .049 78 0.0872 .0673 4.8596 9 52 .01513 6.1055 •03259 0.6833 .05007 19.9702 •067 59 4-7954 8 53 .01542 4.858 .032 88 0.4116 •05037 9.8546 .067 88 4 - 73 I 7 7 54 .01571 3-6567 •03317 0. 1446 .05066 9 - 7403 .068 17 4.6685 6 55 .016 62.4992 •03346 29.8823 •05095 19.6273 .068 47 14.6059 5 56 .016 29 1.3829 •033 76 9.6245 .051 24 9 - 5 I 56 .068 76 4-5438 4 57 .016 58 0. 3058 •03405 9 - 37 11 •051 53 9.4051 .069 05 4. 4823 3 58 .016 87 59.2659 •034 34 9. 122 .051 82 9.2959 .069 34 4.4212 2 59 .017 16 8.2612 •°34 63 8-8771 .052 12 9.1879 .069 63 4. 3607 60 .01746 57-29 •034 92 28.6363 .05241 19.0811 . 069 93 14.3007 0 ' CO-TANG. | Tang. Co-TANG. Tang. CO-TANG. Tang. CO-TANG. Tang. ✓ 890 880 870 86 ° NATURAL TANGENTS AND CO-TANGENTS. , 40 Tang. | Co-tang. 6 Tang. D Co-TANG. 1 6 Tang. ? .' ■ 1 CO-TANG. 1 Tang. > CO-TANG. ' o .06993 14.3007 .087 49 i«- 43 ° 1 .1051 9 - 5 I 4 36 .122 78 8.144 35 60 X .070 22 4. 241 1 .087 78 i- 39 i 9 ■ 105 4 •487 8l . 12308 .12481 59 2 .070 51 4.1821 .088 07 1-354 • 105 69 .461 4I .123 38 .105 36 58 3 .070 8 4- 12 35 .088 17 1-316 3 • 105 99 •435 15 .123 67 .086 57 4 .071 1 4-0655 .08866 1.278 9 . 106 28 . 409 04 •123 97 . 066 74 56 5 •071 39 14.007Q .088 95 11. 241 7 . 106 57 9 - 383 07 . 124 26 8.O47 56 55 6 .071 68 3-9507 .089 25 1.204 8 . 106 87 •357 24 .124 56 .028 48 54 7 .07197 3-894 •089 54 1. 168 1 . 107 16 • 33 i 54 .12485 8.OO9 48 53 8 .072 27 3-8378 .089 83 1. 131 6 . 107 46 •305 99 •12515 7.99058 52 9 .072 56 3.7821 .09013 1.0954 •107 75 .28058 •125 44 .97176 5 i IO .072 85 13.7267 •ogo 42 n.0594 . 108 05 9-255 3 •125 74 7.95302 50 ii •073 J 4 3.6719 .09071 1.0237 .108 34 .230 16 . 126 03 •934 38 49 12 •07344 3 - 6 i 74 .091 01 0.988 2 .10863 .205 16 •126 33 .915 82 48 13 •073 73 3-5634 .091 3 0.9529 .10893 .18028 . 12662 •897 34 47 *4 .074 02 3-5098 •091 59 0.917 8 . 109 22 •155 54 . 12692 •87895 46 *5 •074 3 1 13.4566 .091 89 10.882 9 .10952 9-13093 .127 22 7.860 64 45 16 .07461 3- 4039 .092 18 0.8483 . 109 81 . 106 46 .12751 .842 42 44 17 .0749 3-3515 •09247 0.813 9 . no II .082 11 .127 81 .824 28 ! O ZT 1 43 18 •07519 3.2996 .092 77 0-779 7 .110 4 •057 89 .128 1 .806 22 1 42 J 9 •07548 3.248 .093 06 0-745 7 .110 7 •033 79 .1284 .78825 4 i 20 •075 78 13.1969 •09335 10. 71 1 9 .11099 9.009 83 . 128 69 7 - 77 ° 35 40 21 .076 07 3 - x 46 i .09365 0.6783 . in 28 8.98598 .12899 •752 54 39 22 .07636 3-0958 .09394 0.645 .111 58 .962 27 . 129 29 •734 8 38 23 .076 65 3- 0 458 .09423 0,611 8 .11187 .93867 .12958 • 7 i 7 i 5 37 24 •07695 2.9962 •094 53 0.5789 .11217 .9152 .12988 •699 57 36 25 .07724 12.9469 .09482 10.546 2 . 112 46 8.891 85 .13017 7.682 08 35 26 •07753 2.8981 .09511 CX5136 .112 76 . 868 62 •13947 . 664 66 34 27 .077 82 2. 8496 .09541 0.481 3 •11305 •84551 .13076 •647 32 33 28 .078 12 2.8014 •°95 7 0.4491 •H 335 .822 52 .13106 •63005 32 29 .078 41 2.7536 .096 0.417 2 .11364 •799 64 .13136 .612 87 3 i 30 •0787 12.7062 .096 29 10. 385 4 •II 394 8.776 89 •13165 7-595 75 30 31 .078 99 2,6591 .096 58 0.3538 .11423 •754 25 13195 .57872 29 32 .079 29 2.6124 .096 88 0.3224 .11452 • 73 i 72 .13224 • 561 76 1 28 33 •079 58 2.566 .09717 0.291 3 .11482 •70931 •132 54 •544 87 i 27 34 .07987 2.5199 .097 46 0. 260 2 .11511 .68701 .132 84 .52806 26 35 .080 17 12.4742 .097 76 10.2294 •11.541 8. 664 82 •13313 7 - 5 ii 32 1 25 36 .080 46 2.4288 .09805 0. 198 8 •ii 57 .64275 •133 43 .49465 24 37 .08075 2.3838 .098 34 0. 168 3 .116 .62078 •133 72 .478 06 1 23 38 .081 04 2-339 .098 64 0. 138 1 .11629 •59893 .13402 •461 54 ; 22 39 .081 34 2.2946 •09893 0. 108 .11659 • 577 i 8 •134 32 •445 09 21 40 .081 63 12.2505 .099 23 10.078 .11688 8-555 55 .13461 7.428 71 j 20 4 i .081 92 2.2067 .09952 0.048 3 '.117*18 •534 02 •i 349 i .4124 !:§ 42 .082 21 2. 1632 .099 81 0.018 7 •ii 747 •51259 •i 35 2 i .39616 43 .082 51 2. 1201 .10011 9.9893 .11777 .491 28 •1355 •37999 ! !7 44 .082 8 2.0772 .1004 . 960 07 . 1 1 8 06 .47007 •1358 • 363 89 16 45 .08309 12.0346 . 10069 9.931 01 .11836 8.448 96 .13609 7-347 86 15 46 •083 39 1.9923 . 100 99 . 902 1 1 . 118 65 •427 95 •136 39 •3319 14 47 083 68 1.9504 . 101 28 •87338 .11895 .40705 .136 69 •316 .30018 13 48 .083 97 1.9087 . 101 58 .844 82 .11924 • 38625 .136 98 12 49 .084 27 1 8673 . 101 87 .816 41 .119 54 •365 55 .137 28 .284 42 II 50 .084 56 11.8262 . 102 16 9.788 x 7 •11983 8-344 96 •137 58 7.268 73 1° 5 i .084 85 1-7853 . 102 46 . 760 69 .120 13 .32446 •13787 •253 1 9 52 .085 14 1.7448 .102 75 •732 17 .12042 .304 06 .13817 •237 54 8 53 .08544 1-7045 • 103 05 .70441 .120 72 .28376 .13846 .222 04 7 54 •08573 1. 6645 • 103 34 .6768 .121 01 •26355 .13876 .206 61 6 55 . 086 02 11.6248 • 103 63 9-649 35 - 121 31 8.243 45 .13906 7-i9i 25 5 56 .086 32 i i-5853 •10393 .62205 . 121 6 .223 44 •139 35 •175 94 4 57 . 086 61 1.5461 . 104 22 •594 9 . 121 9 .203 52 •13965 .16071 3 58 .086 9 1.5072 .IO452 •5679 1 .122 19 •1837 •13995 •145 53 2 59 .087 2 1.4685 . IO4 8l .54106 .12249 .16398 .140 24 • 130 42 1 60 .087 49 11. 4301 • 1°5 1 9-514 36 . 122 78 8.14435 .14054 7 -H 537 0 ' Co-tang. 1 Tang. 85 ° Co-tang. 1 Tang. 84° CO-TANG . E 1 Tang. 53° CO-TANG. i Tang. J2° NATURAL TANGENTS AND CO-TANGENTS. 417 8° 9 ° 10O 11* Tang. CO-TANG. Tang. CO-TANG. Tang. | CO-TANG. Tang. CO-TANG. ' 0 • 140 54 7 -H 537 .15838 6.313 75 •17633 5.671 28 .19438 5-144 55 60 I . 140 84 . 100 38 .15868 .30189 .17663 .66165 . 194 68 .136 58 59 2 .141 13 .085 46 .15898 .29QO7 .17693 ; 652 05 .19498 . 128 62 58 3 •141 43 •070 59 .15928 .278 29 •177 23 .642 48 •i 95 29 .120 69 57 4 •141 73 •055 79 •159 58 .266 55 •177 53 •632 95 •195 59 .II279 56 5 .142 02 7-04105 .15988 6.254 86 •177 83 5-623 44 •195 89 5. I0 4 9 55 6 .142 32 .026 37 .160 17 .24321 •17813 •613 97 . 196 19 .097 04 54 7 . 142 62 .011 74 . 160 47 .231 6 •17843 . 604 52 . 196 49 .089 21 53 8 .14291 6 .QQ 7 l8 • 160 77 . 220 03 •178 73 • 595 H . 196 8 .081 39 52 9 .14321 .982 68 .161 07 .208 51 •I 79°3 •585 73 •1971 .0736 51 10 143 5 i 6. 968 23 •161 37 6. 197 03 •179 33 5-57638 •197 4 5.06584 50 ii .14381 •953 85 .161 67 .185 59 •17963 . 567 06 •197 7 .05809 49 12 .1441 •939 52 .161 96 .17419 •179 93 •557 77 . 198 01 •050 37 48 13 .1444 • 9 2 5 25 .162 26 . 162 83 .18023 •54851 • 19831 .04267 47 14 144 7 .911 04 .162 56 •151 5i •18053 • 539 2 7 . 198 61 •034 99 46 *5 .144 99 6.896 88 .162 86 6.14023 .18083 5-53007 . 198 91 5.02734 45 16 •145 29 .882 78 .163 16 . 128 99 • 181 13 .5209 .19921 .019 71 44 17 •145 59 .868 74 .163 46 .11779 .18143 .511 76 •i 99 52 .012 I 43 18 .14s 88 •854 75 * .163 76 . 106 64 .18173 . 502 64 . 199 82 .OO451 42 19 . 146 18 . 840 82 .16405 •095 52 .18203 •493 56 .200 12 4.99695 4 i 26 .146 48 6.826 94 •16435 6.084 44 . 182 33 5-48451 . 200 42 4.9894 40 21 . 146 78 .813 12 . 164 65 •073 4 .18263 •475 48 . 200 73 .98188 39 22 • i47 07 •799 36 .16495 .062 4 .18293 .46648 .20103 •974 38 38 23 • r 47 37 .78564 • 165 25 •051 43 •18323 •45751 •201 33 .9669 37 24 .14767 .77199 •16555 .040 51 •18353 •448 57 . 201 64 •959 45 36 25 .147 96 6.758 38 •165 85 6.029 62 .183 83 5-439 66 .201 94 4-95201 35 26 . 148 26 •744 83 .!66i 5 .018 78 . 184 14 •430 77 . 202 24 .9446 34 27 •148 56 • 73 i 33 .16645 .007 97 .184 44 .421 92 • 202 54 •937 2 i 33 28 . 148 86 .71789 . 166 74 5-997 2 .18474 .41309 . 202 85 .929 84 32 29 •i 49 i 5 •704 5 .167 04 .986 46 .18504 .404 29 •20315 .92249 3 i 30 • r 49 45 6.691 16 •16734 5-975 76 •i85 34 5-395 52 •203 45 4 - 9 I 5 16 30 3 i •M 9 75 .677 87 .16764 •965 i .18564 •38677 . 203 76 •90785 29 32 .15005 .664 63 .16794 •954 48 •i85 94 •37805 .204 06 .900 56 28 33 •15034 .65144 .168 24 •943 9 .186 24 .369 36 . 204 36 •893 3 27 34 .15064 •63831 .16854 •933 35 .186 54 •3607 . 204 66 .886 05 26 35 .15094 6.625 23 . 168 84 5.92283 .186 84 5 - 352 o 6 .204 97 4.878 82 25 36 • 151 24 .612 19 .169 14 •9 12 35 .18714 •343 45 .205 27 .871 62 24 37 •I 5 I 53 .59921 •169 44 .901 91 •18745 •33487 •205 57 . 864 44 23 3 S .151 83 .586 27 .169 74 .891 51 •18775 .32631 . 205 88 .85727 22 39 .15213 •573 39 . 17004 .881 14 .18805 •31778 .206 18 •850 13 21 40 •15243 6.56055 •17033 5.8708 .18835 5.30928 . 206 48 4-843 20 4 i .15272 •547 77 .17063 . 860 51 .18865 .3008 . 206 79 • 8359 19 42 •15302 •535 03 •170 93 .850 24 . 188 95 • 292 35 . 207 09 .82882 18 43 •153 32 .52234 .17123 .84001 • 189 25 •283 93 •207 39 .821 75 17 44 15362 •509 7 •I 7 I 53 . 829 82 •18955 •275 53 .207 7 .81471 16 45 •I 539 1 6.497 1 .17183 5.819 66 .18986 5-267 15 .208 4.807 69 15 46 .15421 .484 56 .17213 •80953 . 190 16 .2588 . 208 3 .8co68 14 47 •i 54 5 i . 472 06 •17243 •799 44 . 19046 . 250 48 . 208 61 •793 7 13 48 • 15481 .45961 •17273 .789 38 .190 76 .242 18 .208 91 .786 73 12 49 •i 55 11 .4472 •17303 •779 36 . 191 06 • 2339 1 .209 21 .77978 11 50 •1554 6.434 84 •173 33 5-769 37 .191 36 5.225 66 . 209 52 4.772 86 10 5 i •1557 •42253 •17363 •759 4 i .191 66 •21744 .20982 •765 95 9 52 .156 .410 26 •173 93 •749 49 • I 9 I 97 .209 25 .210 13 •759 06 8 53 •1563 .398 04 •17423 •739 6 .192 27 .201 07 .21043 •752 19 7 54 .1566 •38587 •174 53 .72974 •19257 •19293 .21073 •745 34 6 55 .156 89 6-373 74 •174 83 5.71992 .19287 5.184 8 .211 04 4-738 5I 5 56 •157 19 .361 65 •i 75 i 3 .71013 •i 93 17 •17671 .21134 • 73 i 7 4 57 •157 49 .34961 •175 43 •70037 •193 47 . 16863 .211 64 •7249 3 58 •157 79 •3376i •175 73 . 690 64 •193 78 .16058 .21195 .71813 2 59 15809 •32566 .17603 .68094 . 194 08 .152 56 212 25 •7 ii 37 1 60 .158 38 6.313 75 •17633 5.671 28 •194 ?8 5 -M 4 55 .212 56 4.70463 0 ' Co-TANG. Tang. Co-tang. Tang. CO-TANG. Tang. | Co-TANG. Tang. 810 800 790 1 78° 41 8 NATURAL TANGENTS AND COTANGENTS. 120 130 UP 150 < Tang. CO-TANG. Tang. CO-TANO. Tang. Co-tang. Tang. Co-TANG. ' o .212 56 4.70463 .230 87 4 - 33 M 8 •249 33 4.01078 .26795 3.732 05 60 I .212 86 .69791 .231 17 •325 73 .24964 .005 82 .268 26 .72771 59 2 .213 16 .691 21 .23148 .320 01 •24995 .00086 .26857 •723 38 58 3 •213 47 .68452 •231 79 •3143 .250 26 3 - 9959 2 .268 88 .71907 57 4 •213 77 .677 86 .23209 .3086 . 250 56 •990 99 .269 2 • 7*4 76 56 5 .214 08 4.671 21 .2324 4.3029I .25087 3.98607 .26951 3.71046 55 6 .21438 .664 58 .232 71 .297 24 .251 18 .981 17 . 269 82 .706 16 54 7 .214 69 •657 97 .23301 .29159 •25149 .976 27 .270 13 .701 88 53 8 .21499 .65138 •23332 •28595 .251 8 •97139 .27044 .697 61 52 9 .215 29 .6448 • 233 63 .280 32 .252 11 .96651 .27076 •693 35 5 i IO .2156 4.638 25 •233 93 4.2747I .252 42 3 - 96 i 65 .27107 3.68909 50 ii .2159 .63171 .234 24 .269 II •25273 .956 8 • 271 38 .684 85 49 12 .216 21 .625 18 •234 55 .263 52 •25304 .95196 .271 69 .680 61 48 13 .21651 .61868 •23485 •257 95 •253 35 •947 J 3 .27201 .676 38 47 14 .216 82 .612 19 .235 i.6 .25239 .25366 .942 32 •27232 .672 17 46 15 .217 12 4.605 72 •235 47 4.246 85 •25397 3-937 5 i .27263 3.66796 45 16 .21743 •599 27 •23578 •241 32 .25428 .93271 .27294 .663 76 44 17 •21773 •592 83 .236 08 .2358 •254 59 • 9 2 7 93 .27326 •659 57 43 18 .21804 .58641 • 236 39 •2303 .2549, .923 16 •273 57 •65538 42 *9 .218 34 .58001 •2367 .22481 .25521 .91839 .273 88 .651 21 41 20 .218 64 4'573 63 •237 4-21933 •25552 3 - 9 i 3 6 4 .27419 3-647 05 40 21 .21895 • 567 26 •23731 •21387 •25583 .908 9 •27451 .642 89 39 22 .21925 .560 91 .23762 . 208 42 .25614 .90417 .27482 .638 74 38 23 .219 56 •554 58 •237 93 . 202 98 •256 45 •89945 •27513 .634 61 37 24 . 219 86 .548 26 .238 23 •197 56 .256 76 •89474 •275 45 .630 48 36 25 .220 17 4.54196 •23854 4.192 15 •257 °7 3. 89004 .27576 3.626 36 35 26 .22047 •535 68 •23885 .18675 •25738 .88536 .27607 .622 24 34 27 .22078 .52941 .239 16 .18137 •25769 .88068 .27638 .618 14 33 28 .22108 •523 16 .23946 .176 .258 .87601 .2767 •61405 32 29 .221 39. • 5^693 •239 77 . 17064 •25831 •871 36 .27701 .609 96 3 i 30 .221 69 4.51074 .24008 4-1653 .25862 3.86671 .27732 3.605 88 30 31 .222 •504 51 .24039 •15997 •25893 .86208 .27764 .601 81 29 32 .22231 .49832 . 240 69 •15465 .259 24 •857 45 •277 95 •597 75 28 33 .222 61 .492 15. .241 •14934 •259 55 .852 84 .278 26 •593 7 27 34 . 222 92 .486 .241 31 •14405 .259 86 .848 24 .27858 .589 66 26 35 .223 22 4.47986 .241 62 4 -I 38 77 . 260 1 7 3-84364 .278 89 3-58562 25 36 •22353 •473 74 • 24193 133 5 .26048 .839 06 2792 .581 6 24 37 .223 83 .467 64 .242 23 .128 25 . 260 79 •834 49 •27952 •577 58 23 38 .22414 •461 55 .24254 . 123 01 .261 1 .829 92 .27983 •573 57 2? 39 .224 44. •455 48 .24285 .11778 .261 41 .82537 .280 15 •569 57 21 40 .22475 4. 449 42 .24316 4.11256 .261 72 3.82083 . 280 46 3-565 57 20 41 .22505 •443 38 •243 47 . 107 36 .26203 .-816-3 .28077 •561 59 . *9 42 •225 36 •437 35 •243 77 . 102 16 •26235 .81177 .281 09 •557 6i 18 43 • 22567 • 43 i 34 . 244 08 .096 99 .26266 . 807 26 .281 4 •553 04 1 7 44 •22597 •425 34 •244 39 .091 82 .262 97 .802 76 .281 72 .54968 16 45 .226 28 4.41936 •244 7 4.086 66 .263 28 3.798 27 .282 03 3-545 73 i 5 46 .226 58 •4134 . 245 01 .081 52 •263 59 •793 7 8 • 282 34 • 54 i 79 14 47 . 226 89 •407 45 •24532 .07639 .2639 .78931 .282 66 •537 85 13 48 .227 19 .401 52 .24562 .071 27 .264 21 .78485 . 282 97 •533 93 12 49 .2275 •395 6 •245 93 .066 16 .264 52 .7804 . 283 29 • 53 ° OI 11 50 .227 81 4.38969 . 246 24 4.061 07 . 264 83 3-775 95 .2836 3.52609 10 5 i .228 11 •38381 .24655 •05599 .26515 •77152 .283 91 .522 19 9 52 .228 42 •377 93 .246 86 .050 92 .265 46 .76709 .28423 .51829 8 53 .228 72 • 372 07 •247 17 .045 86 •26577 .762 68 .284 54 • 5 i 4 4 i 7 54 .22903 . 366 23 *247 47 .04081 . 266 08 • 75828 . 284 86 •51053 6 55 •229 34 4.3604 .24778 4-03578 .266 39 3-753 88 .285 17 3. 506 66 5 56 .229 64 •354 59 .248 09 •030 75 . 266 7 •749 5 .28549 .502 79 4 57 •229 95 •348 79 .2484 .025 74 .267 01 •74512 .2858 .49894 3 58 .230 26 •343 .248 71 .020 74 •267 33 •740 75 .286 12 •495 09 2 59 .230 56 •337 23 .249 02 .015 76 .267 64 •7364 .28643 • 49 1 2 5 1 60 .23087 4 - 33 M 8 •249 33 4.010 78 •267 95 3-732 05 .28675 3.48741 0 ' CO-TANG. Tang. Co-tang. | Tang. Co-TANG. Tang. CO-TANG. Tang. ' 770 760 75° 74° NATURAL TANGENTS AND CO-TANGENTS, 419 > Tang. 16 ° CO-TANG Tang. I70 CO-TANG. Tang. 180 CO-TANG. Tang. l$o Co-TANG, c 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 9 20 21 22 23 24 25 26 27 28 2 9 30 3 * 32 33 34 35 36 37 3.8 39 40 4 1 42 43 44 45 46 47 48 49 50 5 1 52 53 54 55 56 57 58 59 60 .286 75 . 287 06 .28738 .287 69 .288 .288 32 .288 64 .28895 .289 27 .289 58 .2899 .290 21 .29053 . 290 84 .291 16 ■ 2 9 M 7 .291 79 .292 1 . 292 42 .292 74 • 2 93 05 • 2 93 37 .293 68 .294 .294 32 .294 63 • 2 94 95 .295 26 •29558 •2959 .296 21 •29653 .296 85 .297 16 .29748 .2978 .298 11 .298 43 •29875 .299 06 • 299 38 •2997 .30001 •30033 .300 65 •30097 . 301 28 .301 6 • 301 92 .30224 •30255 .302 87 •303 19 •30351 .30382 • 304 14 .304 46 .30478 •30509 ■30541 •30573 : 3.48741 •483 59 •479 77 •475 96 .472 16 3.46837 .46458 .4608 •457 03 •453 27 3-449 51 •445 76 . 442 02 .43829 •434 56 3-430 84 •427 13 •423 43 •41973 .416 04 3.41236 . 408 69 .405 02 .401 36 •397 71 3 - 394 06 •39042 .386 79 •383 17 •379 55 3-375 94 •372 34 •368 75 •36516 •361 58 3-358 •354 43 •35087 •347 32 •343 77 3.34023 •3367 •33317 •329 65 .326 14 3.32264 • 3 i 9 i 4 •31565 .312 16 .30868 3-30521 •301 74 . 298 29 •29483 .29139 3.287 95 •284 52 .281 09 .27767 . 274 26 3.27085 •305 73 .30605 •306 37 .30669 •307 •30732 • 307 64 .30796 .308 28 .3086 .30891 •30923 •309 55 •30987 .31019 .31051 .31083 3 ** 15 • 3 i i 47 .311 78 .312 1 •31242 •31274 .31306 •313 38 • 3 i 37 • 3 i 402 •314 34 .31466 .31498 • 3 i 53 •31562 •31594 • 316 26 •3165-8 .3169 • 3*7 22 • 3*7 54 .31786 .31818 •3*85 .31882 • 3 * 9*4 •3*946 • 3 * 9 . 7 8 .3201 • 320 42 •32074 .321 06 •32139 •32171 .322 03 •32635 .32267 •322 99 •32331 •32363 •32396 .324 28 •3246 .32492 ; 3.27085 •267 45 . 264 06 260 67 •25729 3-25392 •25055 .247 19 •24383 .240 49 3-237 *4 •23381 .23048 •227 15 .223 84, 3.22053 .217 22 .213 92 .210 63 .20734 3. 204 06 .200 79 •*97 52 .194 26 .191 3 - 187 75 .18451 .181 27 .17804 .17481 3 - * 7 * 59 .16838 .165 17 .161 97 •15877 3- *55 58 .1524 .14922 .14605 .14288 3 - 139 72 •13656 •*33 4 * .130 27 .127 13 3.124 . 120 87 •**775 .11464 •***53 3. 108 42 •105 32 . 102 23 .09914 .09606 3.09298 .089 91 . 086 85 .08379 .08073 3.07768 •324 9 2 •325 24 •32556 .32588 .326 21 •32653 •32685 •32717 •327 49 •327 82 .328 14 .32846 .32878 •329 11 •329 43 •329 75 •33007 •3304 •33072 •33*04 •33*36 •33*69 .33201 •33233 • 332 66 •33298 •333 3 •33363 •333 95 • 334 27 •334 6 •33492 •335 24 •335 57 •335 89 •33621 •336 54 .336 86 • 337 i 8 •337 51 •33783 •338 16 •33848 •33881 • 339*3 •33945 •339 78 .3401 • •340 43 •340 75 .341 08 • 34*4 • 34*73 •34205 •342 38 •342 7 •343 03 •343 35 •343 68 ■344 ■344 33 : : 3.07768 .07464 .071 6 .06857 •065 54 3.06252 •059 5 •05649 •053 49 •05049 3-047 49 •044 5 •041 52 •038 54 •035 56 3.0326 .029 63 .02667 .023 72 .020 77 3.01783 .014 89 .011 96 .00903 .O06 11 3.003 19 .00028 2.99738 •994 47 • 99 * 58 2.98868 .9858 .982 92 . 980 04 •977 17 2.974 3. • 97*44 .96858 • 9 6 5 73 .962 88 2.96004 •957 2 i •95437 • 95*55 .94872 2.9459 .94309 .940 28 •937 48 .93468 2.931 89 .9291 .926 32 •923 54 .92076 2.91799 • 9*5 23 .91246 •909 71 .906 96 2.904 21 •344 33 •34465 .34498 •345 3 •345 63 •345 96 •346 28 .34661 •34693 •347 26 •347 58 •347 9 * .34824 •348 56 .348 89 .34922 •349 54 .34987 • 35 o 19 •35052 •35085 • 35 * 17 • 35*5 •35* 83 •35216 •35248 •35281 •353 14 •35346 •35379 • 354*2 •354 45 •35477 •355 * •355 43 •355 76 •356 08 •35641 •35674 •357 07 •3574 •35772 •35805 •35838 •35871 •359 °4 •359 37 •359 69 .36002 • 360 35 .36068 .361 01 •36134 •361 67 •361 99 .362 32 •362 65 .36298 •3633* •36364 •363 97 2.90421 .9OI47 .89873 .896 .89327 2.89055 .88783 .88511 .882 4 •879 7 2.877 •874 3 .871 6l .868 92 . 866 24 2.863 56 . 860 89 • 85822 •855 55 •85289 2.85023 •84758 •844 94 .842 29 •83965 2.83702 •834 39 .831 76 .829 14 .826 53 2.823 9* .821 3 •8*87 .816 1 •8135 2.810 91 .808 33 .80574 .803 16 . 800 59 2.798 02 •795 45 .792 89 •790 33 •78778 2.785 23 .782 69 .780 14 •777 6 i •775 07 2-772 54 .77002 •7675 .764 98 .76247 2.75996 •757 46 •754 96 .75246 •749 97 2.747 48 60 59 58 57 56 55 54 53 52 5 * 50 49 48 47 46 45 44 43 42 4 * 40 39 38 37 36 35 34 33 32 3 * 30 29 28 27 26 25 24 2 3 22 21 20 *9 18 *7 16 *5 *4 *3 12 11 10 9 8 7 6 .5 4 3 2 1 0 1 Co-TANG. 1 1 73 ' Tang. ( 0 2 o-tang. 72 ( Tang. ( 3 /O-TANG. 71 < Tang. I ( 3 1 ? 0 -TANG. 70 Tang. 6 ' NATURAL TANGENTS AND CO-TANGENTS. 20° 21° 22° 230 I Tang. CO-TANG. Tang. I CO-TANG. Tang. | Co-tang. Tang. | CO-TANG. | 0 .36397 2.74748 .38386 2.605 09 • 404 °3 2.47509 •424 47 2.355 85 I •364 3 •744 99 .3842 . 602 83 .404 36 .47302 .424 82 •353 95 2 • 3 6 4 6 3 •74251 •384 53 .600 57 •404 7 •470 95 .425 16 •352 05 3 .364 96 .740 04 .38487 •598 31 .405 °4 .46888 •425 5 i •350 15 4 -365 29 •737 56 •3852 . 596 06 •405 38 . 466 82 •42585 •348 25 5 •36562 2-73509 •385 53 2.593 81 •405 72 2. 464 76 .426 19 2.346 36 6 •365 95 •73263 •38587 • 59 1 56 . 406 06 .4627 .426 54 • 344 47 7 .366 28 •73017 .3862 .58932 .4064 .46065 .426 88 •342 58 8 .366 61 .72771 •386 54 .587 08 .40674 .4586 .427 22 .34069 9 • 366 94 .725 26 .386 87 .58484 .40707 •456 55 •427 57 • 33881 10 .36727 2.722 81 .38721 2.58261 .40741 2.45451 .42791 2-33693 11 -3676 .720 36 •387 54 .580 38 •407 75 .45246 .42826 •335 05 12 •367 93 .71792 .38787 •57815 . 408 09 •450 43 .4286 •33317 13 .368 26 • 7 J 5 48 .38821 • 57 S 93 •408 43 .448 39 .42894 • 33 i 3 14 •36859 •71305 •38854 •573 7 i •40877 .446 36 .429 29 •329 43 15 .36892 2.71062 .38888 2 - 57 I 5 .40911 2-444 33 •42963 2.327 56 16 .36925 .708 19 .389 21 .56928 •40945 •442 3 .429 98 •325 7 17 •36958 •705 77 .389 55 •567 07 .409 79 .44027 43032 •323 83 18 .36991 •703 35 .38988 .56487 .41013 •43825 .43067 •32197 1 9 .37024 .70094 . 3 QO 22 .56266 .41047 •43623 ! • 43 * 01 .320 12 20 •370 57 2.698 53 .390 55 2.560 46 .41081 2.43422 .43136 2.318 26 21 •370 9 .696 12 .39089 •55827 .411 15 .4322 • 43*7 .31641 22 •37124 •693 7 1 • 39 1 22 .556 08 .41149 •43019 I • 432 05 •314 56 23 • 37 i 57 .691 31 • 39 1 56 •55389 • 4 11 83 .428 19 • 432 39 .31271 24 • 37 i 9 .688 92 • 39 1 9 • 55 i 7 .41217 .426 18 •432 74 .31086 2 5 .37223 2.686 53 .39223 2.54952 .41251 2.424 18 • 433 o 8 2.30902 26 •37256 .684 14 •392 57 •547 34 .41285 .422 18 •433 43 .30718 27 .37289 •681 75 . 39 2 9 • 545 i 6 • 4 * 3*9 .420 19 •433 78 •305 34 28 •373 22 •67937 •393 24 •542 99 • 4*3 53 .418 19 •43412 •30351 29 •373 55 •677 •393 57 . 540 82 •41387 .416 2 •43447 • 3 QI 67 3 ° •37388 2.674 62 .393 9 1 2.53865 .414 21 2.414 21 .43481 2.299 84 3 i .37422 •672 25 •394 25 •53648 •414 55 .41223 • 435 16 .29801 32 •374 55 . 669 89 •394 58 •534 32 .4149 • 4 IG 25 •435 5 .296 19 33 •374 88 •66752 .39492 •532 17 .41524 .408 27 •435 85 •294 37 34 •375 21 .66516 •395 26 .53001 •415 58 .406 29 .4362 .29254 35 •375 54 2.662 81 •395 59 2.527 86 .41592 2.404 32 •430 54 2.29073 36 •375 88 .660 46 •395 93 •52571 .416 26 •402 35 .43689 .288 91 37 .37621 .65811 . 396 26 •523 57 .4166 .400 38 •437 24 . .287 1 38 •376 54 •65576 •3966 .52142 .41694 .39841 •437 58 .285 28 39 •37687 •65342 •39694 .51929 .417 28 •39645 •437 93 .28348 40 •377 2 2.651 09 •397 27 2.517 i 5 •41763 2-394 49 .438 28 2.281 67 4 i •377 54 .64875 •39761 .51502 • 4*7 97 • 39 2 53 .43862 .27987 42 •377 87 .646 42 •397 95 .512 89 .41831 .390^8 •43897 .27806 43 .3782 .644 1 .398 29 .510 76 .418 65 .388 62 •43932 .276 26 44 •378 53 .641 7 7 .39862 . 508 64 .41899 .38668 .439 66 •274 47 45 •37887 2-63945 .398 96 2.506 52 •41933 2.38473 .44001 2.272 67 46 •379 2 •63714 •3993 •5044 .41968 .382 79 .44036 .270 88 47 •379 53 •63483 •39963 . 502 29 . 420 02 . 380 84 .44071 .26909 48 .37986 .63252 •39997 .500 18 . 420 36 .37891 .44105 .2673 49 .3802 .63021 .40031 .498 07 .4207 •37697 .4414 .265 52 50 • 380 53 2.62791 .400 65 2-495 97 .42105 2-375 04 • 44 i 75 2.263 74 5 i . 380 86 .62561 .400 98 •493 86 .421 39 •373 11 .4421 .261 96 52 .381 2 .623 32 .401 32 • 49 1 77 •421 73 .371 18 .44244 .260 18 53 •381 53 .621 03 .401 66 .48967 .42207 .36925 .44279 .2584 54 .38186 .618 74 .402 .487 58 .42242 •367 33 • 443 H .256 63 55 .3822 2.61646 .402 34 2.48549 .42276 2.36541 •443 49 2.25486 56 •382 53 .614 18 . 402 67 .4834 .4231 •363 49 •443 84 .25309 57 .382 86 .611 9 .40301 .481 32 .42345 .361 58 .44418 .251 32 58 •3832 .60963 •403 35 .47924 •423 79 •35967 •444 53 .24956 59 •383 53 .607 36 .403 69 .47716 .42413 •357 76 .44488 .2478 60 .38386 2. 605 09 .40403 | 2.47509 .42447 2.35585 •445 23 2.24604 CO-TANG. | Tang. Co-tang. ! Tang. C’O-TANG. | Tang. Co-TANG. Tang. 690 68° 67° 1 66° 60 59 58 57 56 55 54 53 52 5i 50 49 48 47 46 45 44 43 42 41 40 39 38 36 35 34 33 32 31 30 2 9 28 27 26 25 24 23 22 21 20 *9 18 17 16 i5 14 13 12 11 10 l 7 6 5 4 3 2 1 o NATURAL TANGENTS AND CO-TANGENTS. 421 24 ° 1 25 ° 26 ° 27 ° * Tang. CO-TANG. 1 Tang. CO-TANG. Tang. | CO-TANG. Tang. | CO-TANG. ' •445 23 2.24604 .466 31 2.144 51 •487 73 2.O5O3 •509 53 I.962 6l 60 I •445 58 .24428 : .466 66 . 142 88 . 488 09 .04879 •509 89 .961 2 59 2 •445 93 .24252 . 467 02 .141 25 .48845 .O47 28 .51026 •959 79 58 3 .44627 .24077 1 •467 37 .139 63 .48881 •045 77 •5^063 •95838 57 4 .446 62 .23002 j •46772 .13801 •489 17 . O44 26 .51099 .95698 56 5 .44697 2.237 27 1 .468 08 2.136 39 •489 53 2.042 76 •51136 1-955 57 55 6 •447 32 •235 53 i .46843 .134 77 .489 89 .04125 • 5 H 73 •95417 54 7 .447 67 •23378 1 .468 79 .J 33 16 .49026 •039 75 .51209 •952 77 53 8 .448 02 .23204 | .46914 •I 3 1 54 . 490 62 •O3825 .51246 •95137 52 9 •448 37 • 2303 .4695 .12993 . 490 98 •036 75 .51283 •94997 5 i 10 .44872 2.228 57 .469 85 2.12832 • 49 1 34 2.035 26 •513 19 1.948 58 50 11 •449 °7 .22683 .47021 .126 71 • 49 1 7 •033 76 •513 56 .94718 49 12 •449 42 .225 1 • 47 ° 56 .125 11 .492 06 .032 27 •51393 •945 79 48 *3 •449 77 • 22337 .47092 .1235 .49242 .O3O78 • 5 H 3 •944 4 47 14 .45012 .221 64 .47128 .121 9 .49278 .02929 •51467 .94301 46 15 •450 47 2.21992 .47163 2. 120 3 •49315 2.027 8 •51503 1. 941 62 45 16 .45082 .218 19 .47199 .11871 •493 5 i .026 31 •5154 .94023 44 17 • 45 1 *7 .21647 •472 34 .11711 •493 87 .02483 •51577 .93885 43 18 • 45 i 52 • 21475 •4727 .11552 •494 23 •02335 .51614 .93746 42 19 • 45 i 87 .21304 •47305 .11392 •494 59 .021 87 •51651 .93608 4 i 20 .452 22 2.211 32 •473 4 i 2.11233 •49495 2.020 39 .516 88 1-9347 40 21 •452 57 .20961 •473 77 .11075 •495 32 .018 91 .51724 •933 32 39 22 .452 92 .2079 .47412 .109 16 .495 68 • OI 7 43 .51761 •93195 38 23 •453 27 .206 19 .47448 .107 58 . 496 04 .01596 • 5 i 7 9 8 •930 57 37 24 •453 62 .20449 ! •474 83 .106 •4964 .01449 •51835 .929 2 36 25 •453 97 2.20278 - •47519 2. 104 42 •496 77 2.01302 • 51872 1.927 82 35 26 •454 32 .201 08 ! •475 55 . 102 84 .49713 55 .51909 .92645 34 27 •45467 .19938 •475 9 . 101 26 •497 49 .01008 .51946 .925 08 33 28 •455 02 .197 69 .47626 .099 69 .497 86 .008 62 •51983 • 9237 1 32 29 •455 37 • 19599 .47662 .09811 .498 22 •00715 .5202 •922 35 3 i 3 ° •455 73 2. 194 3 .47698 1 2.09654 .49858 2.005 69 • 520 57 1.92098 30 31 .456 08 .19261 •477 33 .094 98 .49894 .004 23 .52094 .91962 29 32 •456 43 .190 92 •47769 .093 41 •499 31 .002 77 •521 31 .918 26 28 33 .45678 .18923 •47805 .091 84 •499 67 .°oi 3I .521 68 •9169 27 34 •457 13 •18755 .4784 .09028 . 500 04 1-999 86 • 522 05 • 9 I 5 54 26 35 •457 48 2.185 87 .47876 2.088 72 .5004 1.99841 .52242 1. 914 18 25 36 •457 84 . 184 19 .47912 .087 16 .50076 •99695 .522 79 .912 82 24 37 •458 19 .182 51 •479 48 .0856 .501 13 •995 5 .523 16 • 9 11 47 23 38 •458 54 .18084 •479 84 .084 05 • 501 49 • 994 o 6 •523 53 .910 12 22 39 .45889 .179 16 .480 19 .082 5 .501 85 .99261 .5239 .908 76 21 4 ° •459 24 2.17749 •48055 2.080 94 . 502 22 1. 991 16 •52427 1.907 41 20 4 1 •459 6 .17582 .48091 •079 39 . 502 58 .98972 .52464 . 906 07 19 42 •45995 .17416 .48127 •07785 •502 95 • 98828 .52501 .90472 18 43 •4603 .172 49 .481 63 .0763 •50331 .986 84 •52538 9°3 37 17 44 .460 65 .17083 .481 98 .074 76 .503 68 ■98s 4 •525 75 .90203 16 45 .461 01 2.169 17 .482 34 2.073 21 .50404 1-98396 .526 13 1.90069 i 5 46 .461 36 .167 51 .4827 .071 67 .50441 .98253 •5265 •899 35 14 47 .46171 .16585 . 483 06 .070 14 •504 77 .981 j .526 87 .89801 13 48 . 462 06 .164 2 .48342 .068 6 •50514 •97966 .52724 .89667 12 49 .462 42 .16255 .48378 .067 06 .505 5 •97823 .527 61 •895 33 11 50 .46277 2. 160 9 .48414 2.065 53 •505 87 1.9768 •52798 1.894 10 51 •463 12 •i 59 2 5 •4845 .064 .50623 •975 38 .528 36 .892 66 9 52 .463 48 .1576 .484 86 .062 47 .5066 •97395 •52873 •891 33 8 53 •463 83 •15596 .48521 .06094 . 506 96 •972 53 • 529 1 .89 7 54 .464 18 •154 32 •485 57 .059 42 • 507 33 .971 11 •529 47 .888 67 6 55 • 464 54 2.152 68 •485 93 2057 9 .507 69 1.969 69 .529 84 1.88734 5 56 .464 80 .15104 .486 29 •05637 . 508 06 .968 27 .53022 .886 02 4 57 •465 23 .1494 .486 65 .054 85 • 508 43 .96685 •530 59 .884 69 3 58 .4656 •147 77 .487 01 •053 33 .508 79 .96544 •53096 •88337 2 59 •465 95 .146 14 •487 37 .051 82 .509 16 . 964 02 •53134 .88205 1 60 .466 31 2-14451 •487 73 2.0503 •509 53 1.962 61 • 53 i 7 1 1.880 73 0 Co-TANG Tang. C’O-TANG. . Tang. CO-TANG . 1 Tang. | Co-tang . 1 Tang. ' 65 ° 64 ° 630 ll 62 ° N N NATURAL TANGENTS AND CO-TANGENTS. 1 28 ° 29 ° 300 31 ° « Tang. I CO-TANG. Tang. CO-TANG. Tang. CO-TANG. Tang. CO-TANG. * o • 53 1 7 1 I.88073 •554 31 I.80405 •577 35 I -732 05 .600 86 I.664 28 60 I .53208 .87941 •554 69 . 802 8l •577 74 .73089 .601 26 .663 l8 59 2 .53246 .87809 •555 07 . 801 58 •57813 •729 73 .601 65 .66209 58 3 •532 83 .87677 •555 45 .800 34 •578 51 .72857 . 602 05 .660 99 57 4 •533 2 .87546 •555 83 .79911 •5789 .72741 .602 45 •659 9 56 5 • 533 58 I.874 15 .55621 1.797 88 • 579 2 9 I.726 25 . 602 84 I.658 8l 55 6 •533 95 .87283 •556 59 •79665 •57968 •725 09 .603 24 .65772 54 7 •534 32 .87152 •556 97 •795 42 .58007 •723 93 .603 64 .656 63 53 8 •534 7 .870 21 •557 36 .79419 .580 46 .722 78 . 604 03 •655 54 52 9 •535 07 .86891 •557 74 . 792 96 ,58085 .72163 .60443 •65445 5 i IO •535 45 I.8676 .55812 I, 79 I 74 .58124 I.72O 47 . 604 83 I -653 37 50 ii •535 82 .8663 •5585 • 79 ° 5 i .581 62 .719 32 .605 22 . 652 28 49 12 • 53 6 2 .864 99 .558 88 .78929 .58201 .718 17 .605 62 .6512 48 n • 53 6 57 .86369 • 559 26 .78807 .5824 .71702 .60602 .650 II. 47 14 •53694 .86239 •559 64 •786 85 •582 79 .715 88 .60642 .64903 46 15 •537 32 I. 86l 09 .560 03 1.78563 •58318 I- 7 I 4 73 .606 81 I.64795 45 16 •53769 •859 79 .56041 .78441 •583 57 •713 58 .607 21 .646 87 44 17 •538 07 .8585 •56079 •78319 •58396 •7 12 44 .607 61 .64579 43 18 •53844 .8572 •561 17 .781 98 •584 35 .711 29 .608 01 .64471 42 IQ •53882 •855 9 1 .561 56 .780 77 •584 74 .71015 .60841 .64363 4 i 20 .539 2 1.85462 .561 94 1-779 55 •58513 1.709 01 .608 81 I.642 56 40 21 • 539 57 •853 33 •562 32 •77834 •58552 •70787 .60921 .641 48 3 i 22 •53995 . 852 P4 .5627 •777 13 .5859 1 •706 73 .6096 .640 41 38 23 .54032 •85075 • 563 09 .775 9 2 •58631 •7056 .61 •639 34 37 24 • 54 ° 7 . 849 46 •563 47 • 7747 i .5867 .70446 .6104 .638 26 36 25 • 54 1 07 1.848 18 •563 85 i -773 5 i .58709 1.70332 .6108 1.637 19 35 26 • 54 1 45 . 846 89 .564 24. •772 3 .58748 . 702 19 .611 2 .636 12 34 27 1 .541 83 .845 61 .56462 771 1 .58787 .701 06 .611 6 •63505 33 28 •5422 •844 33 .565 .7699 .588 26 .699 92 .612 •633 9 8 32 29 •542 58 •843 °5 •565 39 . 768 69 .58865 .698 79 .612 4 .632 92 31 30 .542 9 6 1.84177 • 565 77 1.767 49 .58904 1.697 66 .612 8 1.631 85 30 3 1 •543 33 .840 49 . 566 16 .7663 •589 44 •696 53 •6132 .630 79 29 32 •543 7 i .83922 •566 54 •7651 •58983 .69541 •6136 . 629 72 28 33 • 544 09 •837 94 •566 93 •7639 . 590 22 .69428 .614 .628 66 27 34 •544 46 •83667 •56731 .762 71. . 590 61 .693 16 .6144 .627 6 26 35 •544 84 1-8354 .567 69 1.761 51 • 59 1 01 1.69203 .6148 1.626 54 25 36 •545 22 •834 J3 . 568 08. .760 32 • 59*4 .69091 .615 2 .625 48 24 37 •545 6 .832 86 .56846 •75913 • 59 1 79 .689 79 .61561 .62442 23 38 •545 97 •83159 • 56885 •757 94 • 592 18 .688 66 .61601 .62336 22 39 •546 35 •83033 • 569 23 •756 75 • 592 58 .687 54 .616 41 .622 3 21 40 •546 73 1.829 06 . 569 62 I -755 56 •592 97 1.686 43 .616 81 1.621 25 20 4 i • 547 11 .827 8 •57 •754 37 •593 36 •68531 .617 21 .620 19 42 •547 48 .826 54 •570 39 •753 19 •593 76 .684 19 .617 61 .619 14 18 43 •547 86 .825 28 .57078 •752 •59415 .68308 .61801 .618 08 l l 44 •548 24 .82402 .571 16 .750 82 •594 54 .681 96 .618 42 • 61703 ; 16 45 .548 62 1.822 76 • 57 i 55 1.749 64 •594 94 1.68085 .618 82 1.61598 i 5 46 •549 .821 5 •57193 .74846 •595 33 .67974 .619 22 .61493 H 47 •549 38 .82025 .57232 .74728 •595 73 .67863 .61962 .613 88 13 48 •54975 .81899 •57271 .7461 .596 12 •677 52 .62003 .612 83 12 49 • 55 oi 3 .81774 .57309 .74492 •59651 .676 41 .626 43 .611 79 11 50 •55051 1. 816 49 •573 48 1-743 75 .59691 I -675 3 .620 83 1.61074 10 51 • 55 ° 89 .81524 •573 86 •742 57 •597 3 •67419 .621 24 .6097 9 S 2 • 55 i 27 .81399 •574 25 .7414 •597 7 .67309 621 64 .608 65 8 53 • 551 65 .812 74 •57464 . 740 22 .59809 .67198 .62204 .607 61 7 54 •55203 .811 5 •575 03 •73905 •598 49 .670 88 .622 45 .60657 6 55 •55241 1. 810 25 •575 41 1.73788 . 598 88 1.66978 .622 85 1.605 53 5 56 •552 79 . 809 01 •5758 •73671 •599 28 .668 67 .62325 .604 49 4 57 • 553 1 7 • 807 77 •57619 •735 55 •59967 .66757 .623 66 •60345 3 58 • 553 55 .80653 •576 57 •734 38 .60007 .666 47 .62406 .602 41 2 59 •553 93 .805 29 .57696 •733 2 i .60046 .66538 .624 46 .601 37 1 60 • 5543 i 1.804 °5 •577 35 1.73205 .600 86 1.664 28 || .62487 1.600 33 0 " 7 " CO-TANG. Tang. Co-tang . Tang. Co-TANG. , 1 Tang. I CO-TANG . Tang. ' 610 60 ° 59 ° u 58 ° NATURAL TANGENTS AND CO-TANGENTS. 423 32° 330 340 350 -* Tang. Co-TANG. Tang. CO-TANG. Tang. CO-TANG. Tang. CO-TANG. f 0 .624 87 I.60033 .64941 1.539 86 •674 51 I.482 56 .700 21 I.42815 60 I .625 27 •599 3 .649 82 .53888 •674 93 .48163 .700 64 .427 26 59 2 .025 68 . 598 26 .65023 •537 9 1 •675 36 .4807 . 701 07 .426 38 58 3 . 626 08 •597 23 .65065 •53693 •67578 •47977 .70151 .425 5 57 4 .626 49 .5962 .651 06 •535 95 .676 2 .47885 • 7 °* 94 .424 62 56 S .626 89 I- 595 I 7 .651 48 1-534 97 .676 63 1.47792 .702 38 1.42374 55 6 .6273 ■594 14 .65189 •534 .67705 .47699 .702 81 .422 86 54 7 .6277 • 593 ** • 652 31 •533 02 •67748 .47607 70325 .421 98 53 8 .62811 . 592 08 .652 72 .53205 .6779 •475 *4 • 7°3 68 .421 1 52 9 .628 52 • 59 1 °5 •65314 .53107 • 67832 .47422 .704 12 .42022 5 * xo .628 92 1.59002 •653 55 1.530 r •67875 *•473 3 •704 55 *• 4*9 34 50 11 .629 33 •589 .653 97 •52913 .67917 .47238 •704 99 .41847 49 12 .62973 •587 97 •654 38 .528 16 .679 6 • 47*46 .705 42 • 4*7 59 48 13 .63014 •586 95 .6548 ,52719 .68002 •470 53 . 705 86 .416 72 47 14 •630 55 •585 93 .65521 .526 22 .68045 .469 62 . 706 29 •4*584 46 *5 •63095 1.5849 .65563 1-52525 .68088 1.4687 .70673 *• 4*4 97 45 16 .631 36 .58388 .656 04 .524 29 • 681 3 .46778 • 707*7 .41409 44 17 •631 77 .582 86 .656 46 •52332 .681 73 .466 86 .707 6 .413 22 43 18 .63217 .581 84 .656 88 .52235 .682 15 •46595 . 708 04 • 4*2 35 42 19 .632 58 .58083 .65729 •52139 .682 58 •46503 . 708 48 .411 48 4 * 20 •63299 1.57981 •65771 1.52043 .68301 1.464 11 .70891 1.41061 40 21 •633 4 •578 79 .65813 .519 46 •68343 .4632 •70935 .409 74 39 22 •6338 •577 78 •65854 •518 5 .683 86 . 462 29 •709 79 .40887 38 23 .63421 •576 76 .65896 .517 54 .684 29 •461 37 .71023 .408 37 24 .634 62 •575 75 •65938 .51658 .684 71 . 460 46 .71066 .407 14 36 2S •63503 *• 574 74 .6598 1.515 62 • 685 14 *•459 55 ,.711 * 1.406 27 35 26 •635 44 •573 72 .66021 .51466 •685 57 .45864 • 7**54 •405 4 34 27 .63584 •572 7 1 . 660 63 • 5 i 37 .686 •457 73 .71198 .404 54 33 28 •636 25 •5717 .661 05 •51275 .686 42 • 45682 .71242 • 4°3 67 32 29 .636 66 .57069 .661 47 • 5 ** 79 .68685 •455 92 .7*285 .402 81 3 * 30 •63707 1.56969 .661 89 1.51084 .687 28 i. 455 oi •7*329 1. 401 95 30 31 .63748 .56868 .662 3 .50988 .68771 • 454 * • 7*373 .401 09 29 32 .63789 •567 67 .66 2 72 •50893 .688 14 •453 2 • 7 * 4*7 . 400 22 28 33 .6383 . 566 67 •66314 •507 97 .688 57 .452 29 .71461 •399 36 27 34 • 638 71 •56566 •663 56 . 507 02 .689 • 45*39 • 7*505 .3985 26 35 .63912 1. 564 66 .663 98 1.506 07 .68942 1.45049 • 7*549 1.397 64 25 36 •639 53 .56366 .664 4 •505 12 .68985 •449 58 • 7*593 .396 79 24 37 •63994 .56265 .664 82 •504 *7 .69028 .448 68 •7*637 •395 93 23 38 •64035 .56165 .665 24 .50322 .69071 .44778 .716 81 •39507 22 39 . 640 76 .56065 .66566 . 502 28 .691 14 .44688 • 7*725 .39421 21 40 .641 i 7 1.55966 . 666 08 i- 5 oi 33 •69*57 1.44598 •7*769 *•393 36 20 4 i .641 58 .558 66 .6665 • 500 38 .692 .44508 .71813 •392 5 *9 42 .64199 •557 66 .666 92 • 499 44 • 69243 .44418 • 7*857 •39*65 18 43 .6424 . 556 66 .66734 •498 49 .692 86 •443 29 .71901 .39079 *7 44 .642 81 •555 67 .667 76 •497 55 .693 29 •442 39 • 7*9 46 .38994 16 45 .643 22 1-55467 .66818 1.49661 •693 72 1.44*49 • 7*99 1.389 09 *5 46 •643 63 .553 68 .668 6 .495 66 .694 16 .4406 .72034 .38824 *4 47 . 644 04 •55269 . 669 02 •494 72 694 59 •439 7 .72078 •387 38 *3 48 .644 46 • 55 i 7 . 669 44 •493 78 .69502 .43881 .721 22 .38653 12 49 .64487 .55071 . 669 86 .492 84 •695 45 •437 92 .721 66 .38568 11 5 o .645 28 1.54972 .67028 1 - 49*9 .695 88 *•437 03 .722 11 1.384 84 10 5 i .64569 •548 73 .67071 •490 97 •69631 .43614 •72255 •383 99 9 52 .646 1 •547 74 .671 13 .49003 •696 75 •435 25 .72299 •383*4 8 53 .646 52 •546 75 •67155 . 489 09 .697 18 •434 36 •723 44 .382 29 7 54 •64693 ■545 76 .67197 .488 16 .69761 •433 47 •72388 •38145 6 55 •647 34 1.544 78 .67239 1.487 22 . 698 04 1.432 58 •72432 1.380 6 5 56 •647 75 •543 79 .672 82 .486 29 .69847 • 43*69 •724 77 .37976 4 57 .648 17 .54281 •67324 •48536 .698 91 .4308 .725 21 •3789* 3 58 .64858 •54183 .673 66 .48442 •699 34 .42992 •725 65 .37807 2 59 .648 99 •54085 .67409 •483 49 •699 77 .42903 .726 1 •377 22 1 60 .64941 1.539 86 •67451 1.482 56 .700 21 1.42815 •726 54 1.37638 0 ' Co-TANG. Tang. CO-TANG. 1 Tang. Co-TANG. Tang. CO-TANG. Tang. 570 56° 55° 540 424 NATURAL TANGENTS AND CO-TANGENTS. 56 ° 37 ° I 38 ° 39 ° ' Tang. >| Co-TANG. Tang. Co-TANG. Tang. Co-TANG. Tang. CO-TANG. ' o .726 54 I- 37 6 38 •753 55 1.327 04 .781 29 I.279 94 .809 78 I.2349 60 I .726 99 •375 54 •75401 .326 24 •781 75 .27917 .81027 .234 l6 59 2 •727 43 •374 7 •75447 •325 44 .782 22 .27841 .81075 •233 43 58 3 .727 88 •373 86 •754 9 2 .324 64 .782 69 .277 64 .811 23 .232 7 57 4 .728 32 •373 02 •755 38 •323 84 .783 16 .276 88 .811 71 .231 96 56 5 .728 77 1.372 18 •755 84 I.32304 •78363 1.276 11 .812 2 I. 231 23 55 6 .729 21 • 37 1 34 .756 29 .322 24 .7841 •275 35 .812 68 •230 5 54 7 .72966 • 37 ° 5 •75675 .32144 •784 57 .27458 .813 16 .229 77 53 8 • 73 ° 1 .369 67 •7 5721 . 320 64 .78504 .273 82 .81364 .22904 52 9 • 73 ° 55 .36883 •75767 .319 84 • 7 8 5 5 i .27306 .81413 .228 31 1.227 58 5 i IO • 73 i 1.368 | .75812 I.31904 .785 98 1.272 3 .81461 50 ii • 731 44 .^67 16 .758 58 •3^825 .78645 •271 53 .8151 . 226 85 49 12 .731 89 •36633 •75904 •317 45 . 786 92 .27077 .81558 .226 12 48 13 •732 34 •365 49 •759 5 .316 66 •787 39 .27001 .81606 •225 39 47 14 .732 78 1.364 66 •759 96 .31586 .787 86 .26925 .81,655 .224 67 46 15 •733 23 •363 83 .76042 1-31507 •788 34 1.268 49 .81703 I.22394 45 l 6 • 733 68 •363 . 760 88 •31427 .788 81 •267 74 .81752 .223 21 44 17 •734 !3 .36217 •761 34 •31348 .789 28 . 266 98 .818 .222 49 43 18 •734 57 •361 33 .761 8 .31269 •789 75 . 266 22 .81849 .221 76 42 19 .73502 .36051 .762 26 .3119 .790 22 .265 46 .818 98 .221 04 41 20 •735 47 1.359 68 .762 72 I.- 3 11 1 •790 7 1.26471 .81946 1.220 31 40 21 •735 92 .35885 •763 18 • S^ 1 .791x7 .26395 .81995 .219 59 .218 86 39 22 .736 37 .35802 •763 64 •30952 .79164 .263 19 .820 44 38 23 .736 81 •357 *9 .7641 •30873 .792 12 .262 44 . 820 92 .218 14 37 24 • 737 26 .35637 •764 56 •307 95 • 79 2 59 .261 69 .821 41 .217 42 36 25 •737 7 1 i -355 54 , .76502 1.307 16 .79306 1.26093 .821 9 1.216 7 .21598 .215 26 35 26 .738 16 •35472' .76548 •30637 •793 54 .260 18 .822 38 34 27 .73861 .35389 •765 94 .305 58 .79401 •259 43 .822 87 33 28 .73906 •353 07 .7664 .3048 •794 49 .25867 • 823 36 .21454 .213 82 32 29 •739 5 1 •35224 .766 86 . 304 01 •794 96 .25792 .82385 3 i 30 •739 9 6 I- 35 M 2 •767 33 1-303 23 •795 44 1.257 17 .824 34 1213 1 30 3 1 74041 .3506 .767 79 .302 44 .795 91 .25642 . 824 83 .212 38 .211 66 29 32 .74086 •349 78 .768 25 .301 66 •796 39 .25567 • 825 31 28 33 • 74 1 31 .34896 .76871 . 300 87 . 796 86 .25492 • 8258 .210 94 27 34 .741 76 • 348 14 .769 18 . 300 09 •797 34 •25417 .826 29 .210 23 26 35 .742 21 1-347 32 .76964 1.299 31 .79781 1-253 43 .826 78 1.209 51 .208 79 .208 08 .207 36 . 206 65 25 36 .74267 •3465 .7701 .29853 .79829 .252 68 .827 27 24 37 •743 12 •345 68 •770 57 •297 75 •798 77 •25193 .827 76 23 38 •743 57 •34487 • 77 io 3 . 296 96 •79924 .251 18 .828 25 22 39 .74402 •344 05 • 77 M 9 .296 18 •799 72 .25044 .828 74 21 40 •744 47 1-343 23 .77196 1.295 41 .8002 1.249 69 .82923 1.205 93 20 41 .74492 •34242 •77242 •29463 .80067 .24895 .82972 .205 22 x 9 42 •745 38 .341 6 .77289 •29385 .801 15 .248 2 .83022 .20451 10 43 •74583 • 34 ° 79 •77335 .29307 .801 63 .247 46. .83071 .203 79 . 203 08 17 16 44 .746 28 •339 98 .77382 .292 29 . 802 1 1 .24672 .831 2 45 .74674 1.339 16 .77428 1. 291 52 . 802 58 1.24597 .83169 1.20237 . 201 66 15 46 •747 19 •338 35 •774 75 .29074 .803 06 •24523 .832 18 14 47 ■ 747 64 •337 54 •77521 .289 97 .803 54 .24449 .832 68 .20095 13 48 •748 1 •336 73 . •77568 .289 19 .804 02 •243 75 •83317 .2co 24 12 49 •748 55 •335 92 •77615 .288 42 .8045 .24301 .83366 .199 53 1.198 82 11 5 ° •749 i -335 11 •776 61 1.287 64 . 804 98 1.242 27 .83415 10 51 .74946 •334 3 .77708 .286 87 . 805 46 •241 53 .83465 .198 11 9 52 53 • 7499 1 .750 37 •333 49 .33268 •777 54 .77801 .286 1 .28533 .80594 .80642 .240 79 .24005 •83514 •83564 .1974 .196 69 0 7 54 .750 82 .331 87 •77848 .284 56 .8069 •23931 .83613 .195 99 6 55 .751 28 1. 331 07 •77895 1-383 79 • 807 38 1.238 58 .83662 1. 195 28 5 56 • 75 i 73 • 33° 26 •77941 .28302 . 807 86 •23784 .83712 •194 57 • 193 87 .193 16 .192 46 4 57 • 752 19 .32946 .779 88 .28225 .808 34 .237 1 .83761 3 58 •752 64 .32865 •78035 .28148 .808 82 .23637 .838 11 2 59 •753 1 •32785 .78Q82 .28071 .8093 .23563 l .8386 1 60 •753 55 1.32704 .781 29 | 1-27994 j .80978 1-234 9 1 -839 1 119 1 75 0 ' j Co-TANG. 1 i Tang. 53 ° Co-TANG : | Tang. 52 ° j j CO-TANG . | Tang. 510 I I CO-TANG. II i Tang. 50 ° NATURAL TANGENTS AND CO-TANGENTS. 425 40 ° 41 ° 420 43 ° ' Tang. CO-TANG. Tang. CO-TANG. Tang. CO-TANG. Tang. CO-TANG. ' 0 •839 1 *• I 9 I 75 .869 29 X -I 50 37 .9004 I.II061 •932 52 I.O7237 60 I •8396 .19105 .8698 .149 69 .90093 . IO9 96 •933 06 .071 74 59 2 .84009 •i 9 0 35 .87031 .149 02 .901 46 .IO93I .9336 .071 12 58 3 .840 59 . 189 64 .87082 .14834 .901 99 . 108 67 •93415 .070 49 57 4 .841 08 . 188 94 •87133 • I 47 67 .902 51 . 108 02 •934 69 .069 87 56 5 •841 58 1. 188 24 .871 84 1.146 99 • 9°3 04 I. IO7 37 •935 24 I.069 25 55 6 .84208 •18754 .87236 .14632 • 9°3 57 .10672 •935 78 .068 62 54 7 .842 58 . 186 84 .872 87 •14565 .9041 . 106 07 •93633 .068 53 8 .84307 . 186 14 .873 38 .14498 .90463 .105 43 .936 88 •067 38 52 9 •843 57 .18544 .87389 •144 3 .905 16 . IO4 78 •937 42 .066 76 5 i 10 .844 07 1.18474 .87441 *• x 43 6 3 • 9°5 69 I. IO4 14 •937 97 I.066 13 50 11 •844 57 .18404 .87492 . 142 96 .906 21 • 103 49 •938 52 .06551 49 12 .84507 •18334 •875 43 .142 29 .90674 .102 85 • 939 °6 . 064 89 48 13 •845 56 . 182 64 •87595 . 141 62 .907 27 . 102 2 .93961 .064 27 47 14 . 846 06 .18194 .876 46 • 140 95 .907 81 .IOI 56 .940 16 . 063 65 46 15 . 846 56 1. 181 25 .87698 1.14028 .908 34 I. IOO 91 94071 I.063 03 45 16 .847 06 .18055 •877 49 .13961 .908 87 . IOO 27 .941 25 .062 41 44 17 .847 56 . 179 86 .878 01 .13894 .9094 .09963 .941 8 .061 79 43 18 . 848 06 .179 16 •87852 .13828 .909 93 .098 99 •942 35 .061 17 42 *9 .848 56 . 178 46 .87904 .13761 .91046 .09834 •9429 .060 56 4 1 20 . 849 06 I -i 77 77 •879 55 1.136 94 .91099 I -°97 7 •943 45 I.05994 40 21 .849 56 .177 08 .88007 .136 27 • 9 11 53 .097 06 •944 .059 32 39 22 .850 06 .17638 .880 59 • x 356i .912 06 .096 42 •944 55 .0587 38 23 •85057 •17569 .881 1 • x 34 94 • 9 12 59 .09578 •945 1 .058 09 37 24 .85107 •175 .881 62 .134 28 • 9 1 3 *3 •095 14 •94565 •057 47 36 25 •851 57 i - 174 3 .882 14 1.13361 .91366 1 • °94 5 .9462 I.056 85 35 26 .85207 .17361 .882 65 •132 95 .91419 .093 86 .94676 .056 24 34 27 •85257 . 172 92 .883 17 .132 28 • 9*4 73 .093 22 •947 3 i .055 62 33 28 •85307 .172 23 .883 69 . 131 62 .915 26 .092 58 .947 86 .055 01 32 29 •85358 • I 7 I 54 .884 21 .130 96 .9158 .091 95 .94841 •054 39 3 1 30 .85408 1.17085 .88473 1. 130 29 •9 i6 33 1.09131 .948 96 *•05378 30 3 i .85458 . 170 16 .885 24 .129 63 .916 87 .09067 •949 52 •o 53 x 7 29 32 •85509 .16947 .885 76 . 128 97 .9174 . 090 03 •95007 .05255 28 33 •855 59 .16878 .88628 .12831 • 9 i 794 .089 4 •95062 .051 94 27 34 . 856 09 . 168 09 ,886 8 .12765 .91847 . 088 76 .951 18 .051 33 26 35 .856 6 1. 1 67 41 .88732 1. 126 99 .919 01 1.088 13 •95173 1.05072 25 36 •857 1 . 166 72 .887 84 .12633 • 9*9 55 .08749 .95229 .050 1 24 37 .85761 . 16603 .88836 .125 67 . 920 08 .086 86 .95284 .040 49 23 38 .85811 .16535 .888 88 . 125 01 .92062 .086 22 •953 4 .048 88 22 39 .858 62 . 164 66 .8894 .12435 .921 16 .085 59 •953 95 .048 27 21 40 .85912 1.16398 .88992 1.12369 • 9 21 7 1.084 96 •954 5 i 1.047 66 20 4 i •859 6 3 .163 29 .89045 .12303 .922 23 .08432 •955 06 •047 05 *9 42 .860 14 . 162 61 .89097 . 122 38 .92277 .08369 •955 62 .046 44 18 43 .860 64 . 161 92 .89149 .121 72 . 9 2 3 3 i .08306 •956 18 •04583 *7 44 .861 15 .161 24 .89201 . 121 06 • 9 2 3 85 .082 43 •95673 .045 22 16 45 .861 66 1. 160 56 .89253 1.12041 • 9 2 4 39 1. 081 79 •957 29 1.044 61 x 5 46 .862 16 • x 59 8 7 . 893 06 • IX 9 75 •924 93 .081 16 .95785 .044 01 x 4 47 . 862 67 • X 59 I 9 .893 58 . 1 19 09 •925 47 .08053 .95841 .043 4 *3 48 .863 18 •15851 .8941 .118 44 .92601 •0799 • 95897 .042 79 12 49 .863 68 •157 83 .89463 .11778 •92655 .07927 •959 52 .042 18 11 50 .864 19 i - 157 *5 •89515 1.117 *3 .927 09 1.07864 .960 08 1. 041 58 10 5 i .8647 .156 47 .89567 . 1 16 48 .927 6 3 .07801 .960 64 .04097 9 52 .865 21 •155 79 .8962 .115 82 .92817 .07738 .961 2 .04036 8 53 .865 72 . x 55 n . 896 72 • IX 5 17 .928 72 .076 76 .961 76 .03976 7 54 . 866 23 •154 43 .897 25 .11452 .929 26 • 076 13 .962 32 .039 x 5 6 55 .866 74 i - 153 75 .89777 1.11387 .9298 x -075 5 .962 88 1-03855 5 56 .867 25 .15308 .8983 .11321 •930 34 .074 87 .96344 •037 94 4 57 . 867 76 .1524 .898 83 .112 56 .93088 .07425 .964 •037 34 3 58 .868 27 .15172 •899 35 . hi 91 • 93 x 43 •07362 .964 57 .036 74 2 I 9 . 868 78 • 15104 . 899 88 . hi 26 • 93 x 97 .07299 . 9 6 5 x 3 .036 13 1 60 . 869 29 x - x 5 o 37 .9004 1. no 61 .932 52 1.07237 .96569 *•035 53 0 ' | Co-tang. Tang. CO-TANG. Tang. Co-tang. Tang. CO-TANG. Tang. / .. J 490 48 ° 470 460 N N* 426 NATURAL TANGENTS AND CO-TANGENTS. 440 | 440 r 44 ° ' Tang. CO-TANG. ' ' Tang. I Co-TANG. ' ' 1 Tang. I CO-TANG. t 0 .96569 1-035 53 60 21 •977 56 1.022 95 39 ! 41 .98901 1 i. on 12 19 1 .96625 1.03493 59 22 •978 13 1.022 36 38 42 .98958 | 1. 01053 18 2 .966 81 1-034 33 58 23 •978 7 1. 021 76 37 43 .99016 1.009 94 17 3 .967 38 1.033 72 57 24 .97927 1. 021 17 36 44 •990 73 1.00935 ID 4 .967 94 1.033 12 5 b 25 •979 84 1.020 57 35 45 • 99 1 3 i 1.008 76 *5 5 .968 5 1.042 52 ■ 55 26 .98041 I.OI998 34 46 .991 89 1.008 18 14 6 .q6q 07 I on Q2 54 27 . 980 98 I.OI939 33 47 .992 47 1.007 59 *3 7 . 969 63 1.03132 53 28 • 98 i 55 I.O1879 32 48 •993 04 1.007 01 12 8 .9702 1.03072 52 29 .982 13 I. Ol8 2 31 49 .99362 1.006 42 11 9 .97076 I.030 12 5 i 3 ° .982 7 I.OI761 30 50 •994 2 1.005 83 10 10 .971 33 1.029 52 50 3 1 .98327 1.017 02 2 9 51 •994 78 1.005 25 9 11 .971 89 1.028 92 49 32 .983 84 I.O1642 28 52 •995 36 1.00467 8 12 .97246 1.028 32 48 33 .98441 I.OI583 27 53 •995 94 1.004 08 l 13 .97302 1.027 72 47 34 .984 99 1. 015 24 26 54 .996 52 1.0035 0 14 •973 59 1.027 13 46 35 • 9 8 5 56 I.OI465 25 55 .997 1 1.002 91 5 15 •974 16 1.02653 45 36 .986 13 I.OI406 24 56 .99768 1.002 33 4 16 .97472 1.02593 44 37 .986 71 1.013 47 23 57 .998 26 1. 001 75 3 17 •975 29 1.02533 43 38 .987 28 1.012 88 22 58 . 998 84 1. 001 16 2 18 .97586 1.02474 42 39 .987 86 1. 012 29 21 59 .99942 1. 000 58 z 19 • 97 6 43 1.024 14 4 i 40 .98843 1. on 7 20 60 1 1 0 20 •977 1-02355 40 CO-TANG. Tang. t / CO-TANG. Tang. T 7 / CO-TANG. Tang. ' 45 ° 1 450 45 ° 1 Preceding Table contains Natural Tangents and Co-tangents for every minute of the quadrant, to the radius of 1. If Degrees are taken at head of columns, Minutes, Tangents, and Co-tan- gents must be taken from head also ; and if they are taken at foot of col- umns, Minutes, etc., must be taken from foot also. Illustration.— -.1974 is tangent for n° io', and co tangent for 78° 50'. To Compute Tangents and. Co-tangents for Seconds. Ascertain tangent or co-tangent of angle for degrees and minutes from Table ; take difference between it and tangent or co-tangent next below it. Then as 60 seconds is to difference, so are seconds given to result required, , which is to be added to tangent and subtracted from co-tangent. Illustration.— What is the tangent and co-tangent of 54 0 40' 40"? Tangent of 50° 40', per Table = 1.41061 ) ooo8 di ff erence . Tangent of 54 0 41', k =1.41148) Then 60 : .00087 :: 40 : 000 58, which, added to 1.41061 = 1.4x119 tangent. Co-tangent of 54 0 40', per Table = .708 91 ) ooo difference. Co-tangent of 54 0 41 , “ =.70848) 4:5 Then 6o° : .00043 *.*. 40 : 29, which, subtracted from .70891 = .70862 co-tangent. To Compute Tangent, or Co-tangent of* any ^Angle in Degrees, Minutes, and Seconds. Divide Sine by Cosine for Tangent, and Cosine by Sine for Co-tangent. Example.— What is tangent of 25 0 18'? Sine = .427 36 5 cosine == .904 08. Then q§ ~ ‘ 4 7 2 7 tangent. To Compute Number of Degrees, Minutes, and Seconds of a given Tangent or Co-tangent. When Tangent is given.— Proceed as by Rule, page 402, for Sines, substi- tuting Tangents for Sines. Example.— What is tangent for 1.411 19 ? Next less tangent is 1.41061, arc for which is 54 0 40'. Next greatest tangent is 1. 41 1 48, difference between which and next less is .000 87. Difference between less tabular tangent and one given is 1.41061 — 1.411 19 = .00058. Then .00087 : .00058 :: 60 : 40, which, added to 54° 40' = 54° 40 4 ° • When Co-tangent, is given.— Proceed as by Rule, page 402, for Cosines, substituting Co-tangents for Cosines. AEROSTATICS. 427 AEROSTATICS. Atmospheric Air consists, by volume, of Oxygen 21, and Nitrogen 79 parts; and in 10000 parts there are 4.9 parts of Carbonic acid gas. By weight, it consists of 77 parts of Oxygen, and 23 of Nitrogen. One cube foot of Atmospheric Air at surface of Earth, when barome- ter is at 30 ins., and at a temperature of 32 0 , weighs 565.0964 grains = .080728 lbs. avoirdupois, being 773.19 times lighter than water. Specific gravity compared with water , at 62.418 = .001 293 345. Mean weight of a column of air a foot square, and of an altitude equal to height of atmosphere (barometer 30 ins.), is 2124.6875 lbs. = 14.7548 lbs. per sq. inch = support of 34.0393 feet of water. Standard pound is computed with a mercurial barometer at 30 ins. ; hence, as a cube inch of mercury at 6o° weighs .4907769 lbs., pressure of atmos- phere at 6o° = 14.723 307 lbs. per square inch. 12.3873 cube feet of air weigh a pound, and its weight varies about 1 gr. for each degree of heat. Extreme height of barometer in latitude 30° to 35 0 N.= 30.21 ins. Rate of expansion of Air, and all other Elastic Fluids for all temperatures, is essentially uniform. From 32 0 to 212° they expand from 1000 to 1376 volumes = .002 088 or ^gth part of their bulk for every degree of heat. From 212 0 to 68o° they expand from 1376 to 2322 = .002021 for each de- gree of heat. Thus, if volume of air at 132 0 is required. 132 0 — 32 0 = 100, and 1000 -f- 100 X .002088= 1209 volumes. Height, at Equator is estimated at 300 feet greater than at Poles, its mean height at 45 0 latitude. In like latitudes, air loses about i° for every foot in height above level of sea. Below' surface of Earth, temperature increases. Elasticity of air is inversely as space it occupies, and directly as its density. When altitude of air is taken in arithmetical proportion, its Rarity will be in geometric proportion. Thus, at 7 miles above Surface of Earth, air is 4 times rarer or lighter than at Earth’s surface; at 14 miles, 16 times; at 21 miles, 64 times, and so on. Density of an aeriform fluid mass at 32 0 and at t° will be to each other as 1 .002 088 ( t° — 32 0 ) is to 1. For Volume, Pressure, and Density of Air, see Heat, page 521. Altitude of Atmosphere at ordinary density is = a column of mercury 30 ins. in height, divided by specific gravity of air compared with mercury. Hence 30 ins. = 2.5 feet, which, divided by .000094987, specific gravity of air compared with mercury, = 26 319 # /ee£ = 4.985 miles. Gay Lussac, Humboldt, and Boussingault estimated it at a minimum of 30 miles, Sir John Herschell 83, Bravais 66 to 100, Dalton 102, and Liais at 180 or 204 miles. The aqueous vapor ahvays existing in air, in a greater or less quantity, being lighter than air, diminishes its weight in mixing with it ; and as, other things equal, its quantity is greater the higher the temperature of the air, its effect is to be considered by increasing the multiplier of t by raising it to .002 22. Glaisher and Coxwell, in 1862, ascended in a balloon to a height of 37000 feet. AEROSTATICS. 428 At temperature of 32 0 , mean velocity of sound is 1089 feet per second. It is increased or diminished about one foot for each degree of temperature above or below 32 0 . Velocity of sound in water is estimated at 4750 feet per second. Velocity of Sound at Various Temperatures. 0 Per Second. 0 Per Second. O Feet. Feet. 68 5 1056 32 ON ■-O 0 14 1070 50 1102 77 86 23 1079 59 1 1 12 Per Second. Feet. 1122 1132 1142 95 104 i?3 Per Second. Feet. 1152 1161 1171 those of fluids. Sensation of hearing, or sound, cannot exist in an absolute vacuum. The human voice can be heard a distance of 3300 feet. Echo. At a less distance than 100 feet there is not a sufficient interval between the delivery of a sound and its reflection to render one perceptible. To Compute Velocity of Sound through .Air. io 8 9 x \ 3y / 1 4- [.002 088 (* — 32)1 = v in feet per second , t representing temperature of air. Illustration.— Flash of a cannon from a vessel was observed 13 seconds before report was heard; temperature of air 6o°; what was distance to vessel? io 8 9 xTffx + [.002 088 (6o° -^2)] = 1089 X 13 X 1.029 = 14 567 - 55 flet = 2. 76 miles. Theoretical velocity with which air will flow into a vacuum, if wholly un- obstructed, is VfJTi = 1347-4 feet per second. In operation, however, it is 1347.4 X .707 = 952.61 feet. To Compute Velocity of .Air Flowing into a Vacuum. \ / VgltX c = v in feet per second , c represent ing coefficient of efflux. Coefficients for openings are as follows : Circular aperture in a thin plate. ^ 65 to .7 Cylindrical adjutage 92 1 Conical adjutage 9 3 Velocity of Sound in Several Solids. Velocity in Air=- 1. 3 t « a 1 Pino is.? I Glass .... n. q I Steel Lead Gold 5-6 I Zinc . . . ... 9.8 I | Pine . 12.5 I Glass . . .. 11.9I | Oak | Copper . . . 11. 2 1 Pine... Iron . 4- 3 i5- 1 To Compute Elevations "by a Barometer. Approximately * 60000 (log. B-log. b)C=keight in feet ; heights of barometer at lower and upper stations, and C correction due to T -j- temperatures of lower and upper stations. Values of C or T-f-£. .996 •99 8 1 1.002 1.004 1.007 1.009 1. on 1-013 1. 016 1. 018 1.02 1.022 1.024 1.027 1.029 1. 031 1-033 1.036 1.038 114 116 118 1.04 1.042 1.044 1.047 1.049 1. 051 1-053 1.056 1.058 1.06 1.062 1.064 1.067 1.069 1. 07 1 1-073 1.076 1.078 1.08 1.082 1.084 1.087 1.089 1. 091 1.093 1.096 1.098 1. 1 1. 102 1. 104 1. 106 1. 108 1. in i-i*3 1-115 1.117 1. 12 1. 122 1. 124 1.126 * For more exact formulas, see Tables and Formulas, by Capt. T. S. Lee, U. S. Top. Eng., 1853. AEROSTATICS, 429 Their values vary approximately .0011 per degree. Upper Station. Lower Station. Illustration. — Thermometer 70.4 77.6 Barometer 23.66 30.05 0 = 77.6-1-70.4=: 1.093, log. B = i. 4778, log. 6=1.374. Then 60000 X (1.4778 — 1.374) X 1.093 = 6807.2 feet. To Compute Elevations Toy- a Thermometer. 520 B + B 2 X C= height in feet. B representing temperature of water boiling at elevated station deducted from 212 0 . Correction for temperatures of air at lower and upper stations, or T -f- t, to be taken from table, page 428, as before. Illustration.— Temperature of water boiling at upper station 192 0 ; temperature of air 50 0 and 32 0 . G = 1.02. Then 520 X 212 — 192 + 213 — 192 X 1.02 = 11010 feet. To Compute Capacity of a Balloon , etc., see page 218. Barometer. Elevations by Barometer Readings* (Astronomer Royal.) Mean Temperature of Air 50°. For correction for temperature, see note at foot. Height. Barom. Height. Barotn. Height. Barom. Height. Barom. Height. Barom. . Feet. las. Feet. Ins. Feet. Ins. Feet. Ins. Feet. Ins. 0 31 600 30*325 1500 29.34 4000 26.769 7 000 23.979 50 30-943 650 30. 269 1600 29.233 4250 26.524 7500 23.543 loo 30.886 700 30.214 1750 29. 072 4500 26.282 8000 23.115 150 30.83 750 30.159 1800 29.019 4750 26 042 8 500 22.695 200 30.773 800 30. 103 2000 28.807 5000 25.804 9000 22.282 250 30.717 850 30.048 2250 28.544 5250 25.569 9 5 oo 21.877 300 30.661 900 29.993 2500 28.283 5500 25.335 10060 21.479 350 30.604 1000 29.883 2750 28.025 5750 25.104 10 500 21.089 400 30.548 1100 29.774 3000 27.769 6000 24.875 11 000 20. 706 450 30.492 1200 29.665 3250 37.515 6250 24. 648 11 500 20.329 500 30.436 1300 29-556 3500 27.264 6500 24.423 12 000 19.959 550 30.381 1400 29.448 3750 27.015 6750 24. 2 12 500 19.952 Barometer. Correction for Capillary Attraction to be added in Inches. Diameter of tube I .6 I .55 I .5 I .45 i .4 ' I .55 I .3 I .25 I .2 I .1 Correction, unboiled I .004 005 .007 .01 .014 .02 .025 04 059 ,087 Correction, boiled | .002 | .003 | 004 | 065 | .007 | 01 | .014 | .02 | 029 | .044 To Compute Heiglit. Rule.— S ubtract reading at lower station from reading at upper station, difference is height in feet. Table assumes mean temperature of atmosphere to be 50 0 F. or io° C. For other temperatures following correction must be applied. Add together temperatures at upper and lower station. If this sum, in degrees in F., is greater than ioo°, increase height by part for every degree of excess above ioo°; if sum is less than ioo°, diminish height by part for every degree of defect from ioo°. Or if sum in C° is greater than 20 0 , increase height by ^-A^. part for every degree of excess above 20 0 ; if sum is less than 20 0 , diminish height by g part for every degree of defect from 20° Barometer Indications. Increasing storm.— If mercury falls during a high wind from S. W., S. S. W., W., or S. Violent but short— If fall be rapid. Less violent but of longer continuance.— If fall be slow. Snow. — If mercury falls when thermometer is low. Improved weather. — When a gradual continuous rise of mercury occurs with a falling thermometer. AEROSTATICS. 430 Heavy gales from N. — Soon after first rise of mercury from a very low point. Unsettled weather.— With a rapid rise of mercury. Settled weather.— With a slow rise of mercury. Very fine weather.— With a continued steadiness of mercury with dry air. Stormy weather with rain (or snow).— With a rapid and considerable fall of mer- ^Threatening, unsettled weather.— With an alternate rising and falling of mercury. Lightning only.— When mercury is low, storm being beyond horizon. Fine weather.— With a rosy sky at sunset. Wind and rain.— When sky has a sickly greenish hue. Rain.— When clouds are of a dark Indian red. Foul weather or much wind.— When sky is red in morning. "Weather Grlasses. Explanatory Card. Vice-Admiral Fitzroy , F. R. S. Barometer Rises for Northerly wind (including from N. W. by N. to E.), for dry, or less wet weather, for less wind, or for more than one of these changes— Except on a few occasions when rain, hail, or snow comes from N. with strong wind. Barometer Falls for Southerly wind (including from S. E. by S. to W.), for wet weather, for stronger wind, or for more than one of these changes — Except on a few occasions when moderate wind with rain (or snow) comes from N. For change of wind toward Northerly directions, a Thermometer falls. For change of wind toward Southerly directions, a Thermometer rises . Moisture or dampness in air (shown by a Hygrometer) increases before rain, fog, or dew. Add one tenth of an inch to observed height for each hundred feet Barometer is above half-tide level. Average height of Barometer, in England, at sea-level, is about 29.94 inches; and average temperature of air is nearly 50 degrees (latitude of London). Thermometer falls about one degree for each 300 feet of elevation from ground, but varies with wind. “ When the wind shifts against the sun, Trust it not, for back it will run.” First rise after very low I Long foretold— long last, Indicates a stronger blow. | Short notice — soon past. Rarefaction of Air . In consequence of rarefaction of air, gas loses of its illuminating power 1 cube inch for each 2.69 feet of elevation above the sea. (M- Bremond.) Clouds. Classification.— 1. Cirrus— Like to a feather, commonly termed Mare's tails. 2. Cirro-cumulus — Small round clouds, termed mackerel sky. 3. Cirro-stratus — Concave or undulated stratus. 4. Cumulus— Conical, round clusters, termed wool-packs and cotton balls. 5^ Cumulostratus - Two latter mixed. 6. Nimbus — A cumulus spreading out in arms, and precipitating rain beneath it. 7. Stratus — A level sheet. Note. — Cirrus is most elevated. Height. — Clouds have been seen at a greater height than 37000 feet. Velocity. — At an apparent moderate speed, they attain a velocity of 80 miles per hour. lAglvtiiing. Classification.— i. Striped or Zigzag— Developed with great rapidity. 2 . Sheet — Covering a large surface. 3. Globular — When the electric fluid appears condensed, and it is developed at a comparatively lower velocity. 4. Phosphoric — When the flash appears to rest upon the edges of the clouds. AEROSTATICS. — ATMOSPHERIC AIR, 431 WEATHER INDICATIONS. Weather. Clouds. Sky. Fine and Soft or delicate-looking and in- Gray in morning and light, Fair. definite outlines. delicate tints and low dawn. Wind. Wind only. Hard-edged, oily - looking, and tawny or copper-colored, and the more hard, “greasy,” and ragged, the more wind. Light scud alone. High dawn, and sunset of a bright yellow. Rain. Small and inky. Sunset of a pale yellow. Wind and Rain. Light scud driving across heavy masses. Orange or copper color. Rain and Wind. Change of Wind. Hard defined outlines. High upper, cross lower in a di- rection different to their course or that of wind. GeneraL Gaudy unusual hues. Fair.— When sea-birds fly early and far out, when dew is deposited, and when'a leech, confined in a bottle of water, will curl up at the bottom. Rain.— Clear atmosphere near to horizon and light atmospheric pressure, or a good “hearing day,” as it is termed. Storm.— When sea-birds remain near to shore or fly inland. Rain, Snow, or Wind. — When a leech, confined in a bottle of water, will rise ex- citedly to the surface. Thunder.— When a leech, confined as above, will be much excited and leave the water. Value of* Indications of* Hair Weather, in Days, Com- pared to one of Plain. From an extended series of observations. (Lowe.y Profuse Dew. White Stratus in a valley. . . Colored Clouds at sunset . . , Solar Halo Sun red and rayless Sun pale and sparkling White Frost Lunar Halo Lunar burr, or rough-edged, Moon dim Moon rising red 4. 5 Mock Sun or Moon. . , . . 7.2 Stars falling abundant. . 2.9 Stars bright 1.9 Stars dim 10.3 Stars scintillated 1 Aurora borealis 4.2 Toads in evening 1 Landrails noisy. 2.8 Ducks and Geese noisy. 2 Fish rising 7 Smoke rising vertically. 3-3 3-2 3-4 1- 5 6 1.8 2.4 13 2 - 3 i-5 5 For weather-foretelling plants, see page 185. ATMOSPHERIC AIR. Very pure air contains Oxygen 20.96, Nitrogen 79, and Carbonic Acid .04. Air respired by a human being in one hour is about 15 cube feet, produc- ing 500 grains of carbonic acid, corresponding to 137 grains carbon, and during this time about 200 grains of water will be exhaled by the lungs. During this period there would be consumed about 415 grains of oxygen. In one hour, then, there would be vitiated 73 cube feet pure air. A man, weighing 150 lbs., requires 930 cube feet of air per hour, in order that the air he breathes may not contain more than 1 per 1000 of carbonic acid (at which proportion its impurity becomes sensible to the nose); he ought, therefore, to have 800 cube feet of well ventilated space. ATMOSPHERIC AIR. — ANIMAL POWER. 432 An adult human being consumes in food from 145 to 165 grains of carbon per hour, and gives off from 12 to 16 cube feet of carbonic acid gas. An assemblage of 1000 persons will give off in two hours, in vapor, 8.5 gallons water, and nearly as much carbon as there is in 56 lbs. of bitumi- nous coal. proportion of Oxygen, and. Carbonic Acid at following Locations. Pure Air represented by Oxygen 20. 96. Street in Glasgow 20.895 Regent Street, London 20.865 Centre Hyde Park 21.005 Metropolitan Railway (underground) . . 20.6 Pit of a Theatre 20.74 Gallery of a Theatre. 20.63 Carbonic Acid .04 Per cent. Open field, Manchester 0383 Churchyard 0323 Market, Smithfield 0446 Factory mills 283 School- rooms 097 Pitt of theatre, n P. M 32 Boxes “ i,2 “ 218 Gallery “ 10 “ ..101 * Roscoe. Top of Monument, London 0398 Hyde Park ' 0334 Metropolitan Railway (underground).. .338 Lake of Geneva 046 Boys 1 school 3 1 * Girls’ “ 7 2 3t Horse stable 7 Convict prison °45 \ Pelteuhoffer. Consumption of Atmospheric Air. ( Coathupe .) One wax candle (three in a lb.) destroys, during its combustion, as much oxygen per hour as respiration of one adult. A lighted taper, when confined within a given volume of atmospheric air, will become extinguished as soon as it has converted 3 per cent, of grven volume of air into carbonic acid. Carbonic Acid Exhaled per Minute by a Man. [Or. Smith.) During sleep 4.99 per cent., lying down 5.91, walking at rate of 2 miles per hour 18. 1, at 3 miles 25.83, hard labor 44-97* animal power. Work. Work is measured by product of the resistance and distance through which its point of application is moved. In performance of work by means of mechanism, work done upon weight is equal to work done by ^ Unit of Work is the moment or effect of 1 pound through a distance of 1 foot, and it is termed a foot-pound. In France a kilogrammetre is the expression, or the pressure ot a kilogramme through a distance of 1 meter = 7.233 foot-pounds. Result of observation upon animal power furnishes the following as maximum daily effect: 1. When effect produced varied from .2 to .33 of that which could be produced without velocity during a brief interval. 2. When the velocity varied from .16 to .25 for a man, and from .08 _to .060 for a horse, of the velocity which they were capable for a brief intenal, and not involv- ing any effort. 3. When duration of the daily work varied from . 33 to . 5 for - a brief interval, during which the work could be constantly sustained without Prejudice s tc » b ea1 ^ of man or animal; the time not extending beyond 18 hours per day however lim- ited may be the daily task, so long as it involved a constant attendance. ANIMAL POWER. 433 Mien. Mean effect of power of men working to best practicable advantage, is raising of 70 lbs. 1 foot high in a second, for 10 hours per day 4200 foot- pounds per minute. Windlass. — Two men, working at a windlass at right angles to each Other, can raise 70 lbs. more easily than one man can 30 lbs. Labor.— A man of ordinary strength can exert a force of 30 lbs. for 10 hours in a day, with a velocity of 2.5 feet in a second = 4500 lbs. raised one foot in a minute = .2 of work of a horse. A man can travel, without a load, on level ground, during 8.5 hours a day, at rate of 3.7 miles an hour, or 31.25 miles a day. He can carry in lbs. 11 miles in a day. Daily allowance of water, 1 gallon for all purposes; and he requires from 220 to 240 cube feet of fresh air per hour. A porter going short distances, and returning unloaded, can carry 135 lbs. 7 miles a day, or he can transport, in a wheelbarrow, 150 lbs. 10 miles in a day. Cram. — The maximum power of a man at a crane, as determined by Mr. Field, for constant operation, is 15 lbs., exclusive of frictional resistance, which, at a velocity of 220 feet per minute = 3300 foot-pounds, and when exerted for a period of 2.5 minutes was 17.329 foot-pounds per minute. Pile-driving. — G. B. Bruce states that, in average work at a pile-driver, a laborer, for 10 hours, exerts a force of 16 lbs., plus resistance of gearing, and at a velocity of 270 feet per minute; making one blow every four minutes. Rowing. — A man rowing a boat 1 mile in 7 minutes, performs the labor of 6 fully-worked laborers at ordinary occupations of 10 hours per day. Drawing or Pushing. — A man drawing a boat in a canal can transport 110000 U)s ; for a distance of 7 miles, and produce 156 times the effect of a man weighing 154 lbs., and walking 31.25 miles in a day ; and he can push on a horizontal plane 20 lbs. with a velocity of 2 feet per second for 10 hours per day. Tread-mill. — A man either inside or outside of a tread-mill can raise 30 lbs. at a velocity of 1.3 feet per second for 10 hours, = 1 404 000 foot-pounds. Pulley— A man can raise by a single pulley 36 lbs., with a velocity of .8 of a foot per second, for 10 hours. Walking.— A man can pass over 12.5 times the space horizontally that he can vertically, and, according to J. Robison, by walking in alternate directions upon a platform supported on a fulcrum in its centre, he can, weighing 165 lbs., produce an effect of 3 984 000 foot-pounds, for 10 hours per day. Pump , Crank , Bell , and Rowing. — Mr. Buchanan ascertained that, in work- ing a pump, turning a crank, ringing a bell, and rowing a boat, the effective ' power of a man is as the numbers 100, 167, 227, and 248. Pumping.— A practised laborer can raise, during. 10 hours, 1000000 lbs. water 1 foot in height, with a properly designed and constructed pump. Crank. A man can exert on the handle of a screw-jack of 11 inches ra- dius for a short period a force of 25 lbs., and continuously 15 lbs., a net power of 20 lbs. Mr. J. Field’s tests gave 11.5 lbs. as easily attained, 17.3 as difficult, and 27.6 with great difficulty. Mowing. — A man can mow an acre of grass in 1 day. Reaping. — A man can reap an acre of wheat in 2 days. \ Ploughing. — A man and horse .8 of an acre per day. O o 434 ANIMAL POWER. Day’s "Worltk {D. K. Clark.) Laborer. — Carrying bricks or tiles, net load 106 lbs. = 6oo lbs. i mile. Carrying coal in a mine, liet load 95 to 115 lbs. = 342 lbs. 1 mile. Loading coke into a wagon, net load 100 lbs. = 270 lbs. 1 mile. • Loading a boat with coal, net load 190 lbs. = 1230 lbs. 1 mile, or 20 cube yards of ea piggino- stublde land .055 of an acre per day, or 2000 cube feet of superficial earth. Breaking 1.5 cube yards hard stone into 2 inch cubes. Quarrying.— A man can quarry from 5 to 8 tons of rock per day. A foot-soldier travels in 1 minute, in common time, 90 steps ==70 yards. He occupies in ranks a front of 20 inches, and a depth of 13, without a kuapsack; interval between the ranks is 13 inches. , . „ Average weight of men, 150 lbs. each, and five men can stand m a space of 1 square yard. Effective Power of NEeix for* a, Sliort IPeriod. Manner of Application. Force. Bench-vice or Chisel Drawing-knife or Auger. . Hand-plane Hand saw. Lbs. 72 100 50 36 Manner of Application. Force. Screw-driver, one-liand . Small screw driver Thumb and fingers, Windlass or Pincers Lbs. 84 14 H 60 The muscles of the human jaw exert a force of 534 lbs. Mr. Smeaton estimated power of an ordinary laborer at ordinary work was equiv- alent to 3762 foot-pounds per minute. But, according to a particular case made by him in the pumping of water 4 feet high, by good English laborers, their power was equivalent to 3904 foot-pounds per minute; and this he assigned as twice that of ordinary persons promiscuously operated with. Mr J Walker deduced from experiments that the power of an ordinary laborer, in turning a crank, was 13 lbs., at a velocity of 320 feet per minute for 8 hours per day. Amount of Labor produced by a IVIan. {Morin.) MANNER OF APPLICATION. - I Power, i •! Velocity per Second. Weight I raised. | Feet per j Minute, j I-P for Period given. Throwing earth with a shovel, a height of 5 feet. . Wheeling a loaded barrow up an inclined plane, Lbs. . . 6, 132 Feet. i-33 .625 Lbs. 480 495° No. 8.7 90 Raising and pitching earth in a shpvel 13 feet 6 2.25 810 14-7 Pushing and drawing alternately in a vertical 13 2-5 1950 35-5 Transporting weight upon a barrow, and return- 132 1 7 920 144 For 8 Hours per Day. 62 • j : ^ ~ 0 i;,At n l \ ad nnlondod i43 26 .5 4290 Ascending a siigni eiev (iiiuii, ...... . Walking, and pushing or drawing in a horizontal 2 3120 45-2 18 2.5 2790 39 140 • 5 4200 61. 1 26 5 7 800 1 13 For 7 Hours per Day. 88 2.5 13 200 160.5 Walking with a load upon his back For 6 Hours per Day. Transporting a weight upon his back, and return- 140 1-75 14700 160.5 Transporting a weight upon his back up a slignt 140 .2 1 680 19 elevation, and returning unloaded. ••••• 44 1 -5 1 320 14.4 individual case, at 140 lbs;, at a ve- * Morin gives amount of labor, of a man upon tread-mill, in an L— ----- feet JIS looity of .5 feet per second for 8 hours per day = 70 lbs. at 1 foot per second, hence 70 . 3 ANIMAL POWER, To Compute Number of IVLeix to Perform Work vipon a Tread-mill or P^e-driver, Rule.— T o product of weight to be raised and radius of crank, add fric- tion of wheel, and divide sum by product of power and radius of wheel. Fyaaiplf —How many men are required upon a tread-mill, 20 feet in diameter, to raise a weight of 9233.33 U*. crank 9 inches in length, weight of wheel and its load estimated at 5000 lbs., and friction at .015. Weight of a man assumed at 25 lbs. Radius of crank .7 5 feet. Effect of a man on a treadmill, page 433, 30 lbs. at a velocity of 1.3 feet per second, — 1. 3 X 60 = 78 feet per minute. 9233- 33 X .75 -j- 5000 X .015 = 7000 lbs. resistance of load and wheel, and 7000-4- 78 -X 10 X 3° 18.8 men. — — 7000 = load and weight -4- product of power increased by its 20 X 3- 1416 ^ J ... * 7 velocity over load, radius of wheel and power z - 7000-f- 1. 241 X 10 X 3° 1 Horse. Amount of Labor produced by a Horse under different Circumstances. {Morin.) For 10 hours per day. MANNER OF APPLICATION. Power. Velocity per Second. Weight drawn. Feet per Minute. l P for Period given. Drawing a 4- wheeled carriage at a walk Lbs. Feet. Lbs. I No. 154 ’ T 27 720 504 With load upon his back at a walk 264 3-75 59 4oo 1080 Transporting a loaded wagon, and return ng un- 184 800 336o 1540 2 Drawing a loaded wagon at a walk 1540 3-75 346 500 6300 For 8 Hours per Day. Upon a revolving platform at a walk 100 3 18000 260.8 For 4.5 Hours per Day. 218.7 Upon a revolving platform at a trot 66 6.75 26730 Drawing an unloaded 4-wheeled carriage at a trot. 97 7- 2 5 43 195 353-5 Drawing a loaded 4- wheeled carriage at a trot 770 7-25 334 950 2741 If traction power of a horse, when continuously at a walk, is equal to 120 lbs., and grade of road 1 in 30, resistance on a level being one thirtieth of load, he can draw a load of 120 x 30 -4- 2 = 1500 lbs. Street Rails or Tramways. Cars, 26 lbs. per ton, or 1 to 86 as a mean. Performance of Horses in France. [Henry Hughes .) [M. Charie-Marsaines.) SEASON. Road. Weight per Horse. Spaed per Hour. Work per Hour, drawn One Mile. Ratio of Pavement to Macadam. Winter ) Pavement ( Macadam | Pavement j Macadam Tons. 1.306 .851 1-395 Miles. 2.05 Ton-miles. 2.677) 1.62$ ) 3.027) 2.464) 1.644 to 1 Summer 1. 91 2.17 r.229 to 1 1.141 2. 10 Average daily work of a Flemish horse in North of France, where country is flat and loads heavy, is, on same authority, as follows: Winter, 21.82 ton-miles per day. ; Summer, 27.82 k ‘ “ i Mean for the year, 25. given in example = 53.8 lbs., from which a deduction is to be made for excess of amount of labor that can be performed in 8 hours over 10. Or, as 10 : 8 : : 53-8 : 43-04 lbs., which does not essentially differ from effect of 30 lbs. for that of an average performance. ANIMAL POWER. 436 Greatest mechanical effect of an ordinary horse is produced in operating a gin or drawing a load on a railroad, when travelling at rate of 2.5 miles per hour, where he can exert a tractive force of 150 lbs. for 8 hours per day. At a speed of 10 miles per hour, a horse will perform 13 miles per day for 3 years. In ordinary staging, a horse will perform 15 miles per day. To Compute Tractive Power of a Horse Team , see Traction , page 848. Assuming maximum load that a horse can draw on a gravel road as a standard, he can draw, On best-broken stone road 2 to 3 times. On a well-made stone pavement 3 to 5 “ On a stone trackway 7 to 8 “ On plank road 4 to 12 “ On a railway 18 to 20 “ Note.— Track of an iron railway compared with a plank-road is as 27 to 10. To Compute Power of Draught of a Horse at Different Hence, r represents force which horse must over- come to move his own weight. h 0 Then, by similar triangles, A C or l : B C or h : : 0 : r. Or, -j- = r. If t represents tractive power of horse, upon a level, of 100 lbs., t' tractive power upon a plane of inclination, and r that part of force exerted by horse which is expended upon his own body, then t ' = £ — or t ^ = t 'in lbs. Illustration. — If inclination is 1 in 50. Assume t = 100, weight of horse 900 lbs., and l = 50.01. Assuming load that a horse can draw on a level at 100, he can draw upon inclinations as follows : On his bade a horse can carry from 220 to 390 lbs., or about 27.5 per cent, of his weight. Labor . — The work of a horse as assigned by Boulton & Watt, Tredgold, Rennie, Beardmore, and others, ranges from 20 000 to 39 320 foot-pounds per minute for 8 hours, a mean of 27 750 lbs. A horse can travel, at a walk, 400 yards in 4.5 minutes ; at a trot, in 2 minutes ; and at a gallop, in 1 minute. He occupies in ranks, a front of 40 j ins., and a depth of 10 feet; in a stall, from 3.5 to 4.5 feet fron^; and at a picket, 3 feet by 9 ; and his average weight = 1000 lbs. Carrying a soldier and his equipments (225 lbs.) he can travel 25 miles in a day of 8 hours. A draught-horse can draw 1600 lbs. 23 miles a day, weight of carriage in- cluded. Horse upon Turnpike Road. Elevations. B Let ABC represent an inclined plane, 0 ^weight of a horse which, being resolved into two com- ponent forces, one of which, n , is perpendicular to r plane of inclination, and other, r, is parallel to it. ^ U vnrvvncnnto -pAVOD wllioh Vl ArCA mncf AVOl 1 1 in 100 91 1 in 75 88 1 in 50. 1 “ 90 90 1 “ 70 87 1 “ 45, 1 “ 80 89 1 “ 60 85 1 “ 40. 82 1 in 35 74 1 in 20 55 80 1 “ 30 70 1 “ 15 40 77 i “ 25 64 1 “ 10 10 AXIMAL POWER. 437 Ordinary work of a horse may be stated at 22 500 lbs., raised 1 foot in a minute, for 8 hours per day. In a mill, he moves at rate of 3 feet in a second. Diameter of track should not be less than 25 feet. Rennie ascertained that a horse weighing 1232 lbs. could draw a canal-boat at a speed of 2.5 miles per hour, with a power of 108 lbs., 20 miles per day. This is equivalent to a work of 23 760 foot-lbs. per minute, He estimated that the average work of horses, strong and weak, is at the rate of 22 000 foot-lbs. per minute. From results of trials upon strength and endurance of horses at Bedford, Eng., it was determined that average work of a horses 20000 foot-lbs. per minute. A good horse can draw 1 ton at rate of 2.5 miles per hour, from 10 to 12 hours per day. Expense of conveying goods at 3 miles per hour, per horse teams being 1, expense at 4.33 miles will be 1.33, and so on, expense being doubled when speed is 5.125 miles per hour. Strength of a horse is equivalent to that of 5 men, and his daily allowance of water should be 4 gallons. Amount of Labor a Horse of average Sfrengtli is capa- ble of performing, at different Velocities, 011 Canal, Railroad, and Turnpike. Traction estimated at 83.3 lbs. Veloci- Dura- Useful Effect, drawn i Mile. Veloci- Dura- Useful Effect, drawn 1 Mile. ty per tion of On a On a Rail- On a Turn- ty per tion of On a On a Rail- On a lurn- Hour. Work. Canal. road. pike. Hour. Work. Canal. road. pike. Miles. Hours. Tons. Tons. Tons. Miles. Hours. Tons. Tons. Tons. 2-5 n -5 520 115 14 6 2 30 48 6 3 8 243 92 12 7 i -5 19 0 4 i 5-1 4 4-5 102 72 9 8 1125 12.8 36 4-5 5 2-9 52 57 7.2 10 •75 6.6 28.8 3-6 Actual labor performed by horses is greater, but they are injured by it. Tractive Power of a horse decreases as his speed is increased, and within limits of low speed, or up to 4 miles per hour, it decreases nearly in an inverse ratio. For 10 Hours per Day. Miles. Traction. Miles. Traction. Miles. Traction. Miles. Traction. Per Hour. Lbs. Per Hour. Lbs. Per Hour. Lbs. Per Hour. Lbs. 75 33 ° i -5 165 2.25 IIO 3 82 250 1-75 140 2-5 100 3-5 70 1.25 200 2 125 2-75 90 4 62 For Ordinary or Short Periods. ( Molesworth .) Miles per hour 2 3 3-5 4 4-5 5 Power in lbs 166 125 104 83 62 41 Miles per hour 2 3 3-5 4 4-5 5 Power in lbs 166 125 104 83 62 41 Mule. (D.K. Clark.) Load on back , 170 to 220 lbs. day’s work = 6400 lbs. 1 mile ; 400 lbs. at 2.9 miles per hour =. 5300 lbs. 1 mile, and 330 lbs. at 2 miles per hour = 5000 lbs. 1 mile. Upon a revolving platform, at a velocity of 3 feet per second, = 11 880 lbs. raised one foot per minute, or 172.2 EP for 8 hours per day Ass. Load on back , 176 lbs. carried 19 miles day’s work = 3300 lbs. 1 mile. In Syria an ass carries 450 to 550 lbs. grain. Upon a revolving platform, at a velocity of 2.75 feet per second, = 5280 lbs. raised one foot per minute, or 76.5 EP for 8 hours per day. 0 0* 438 ANIMAL POWER. Ox. An Ox, walking at a velocity of 2 feet in a second (1.36 miles per hour), exerts a power of 154 lbs., = 18480 lbs. raised one foot per minute, or 268.8 IP for 8 hours per day. A pair of well-conditioned bullocks in India have performed work = 8000 foot-lbs. perminUte ' Camel. Load on back. 550 lbs. carried 30 miles per day for 4 days, 4 days’ work 16 500 lbs. 1 mile, for 5 days 13000 lbs. 1 mile = 44 IP for 10 hours per day. Load of a Dromedary , 770 lbs. Llama. Load on bach , no lbs., day’s work 2000 to 3000 lbs. 1 mile = .5 to .75 IP for 10 hours per day. Birds and Insects. Area of their wing surface is in an inverse ratio to their weight. Assuming weight of each of the following Birds to be one pound, and each Insect one ounce, the relative area of their wing surface proportionate to that of their act- ual weight would be as follows ( M . De Lucy): Sq. ft. Swallow 4.85 Sparrow 2.7 Turtle-dove.. 2.x 3 Sq. ft. Pigeon 1-27 Vulture 82 Crane, Australia, .41 Sq. ft. Gnat 3.05 Dragon-fly, sm’ll, 1.83 Ladybird 1.66 Sq. ft. Cockchafer ... 32 Bee 33 Meat-fly 35 Crocodile and Bog. The direct power of their jaws is estimated at 120 lbs. for the former and 44 for the latter, which, with the leverage, will give respectively 6000 and 1500 lbs. PERFORMANCES OF MEN, HORSES, ETC. Following are designed to furnish an authentic summary of the fastest or most successful recorded performances in each of the feats, etc., given. MAN. Walking. T874 Wm. Perkins , London, Eng., .5 mile, in 2 min. 56 sec.; 1, in 6 min. 23 sec.; 2 in 13 min. 30 see.; 1876, 8, in 59 min. 5 sec.; 1877, 20, in 2 hours 39 min. 57 sec. 1 i 83 o T. Smith , London, Eng., 12 miles, in 1 hour 31 min. 42.4 sec. 1881’ C. A. Harriman , Chicago, 111 ., 530 miles, in 5 days 20 hours 47 min. 1851’, J. Smith , London, Eng., 25 miles, in 3 hours 42 min. 16 sec 1878 W. Howes , London, Eng., 50 miles, in 7 hours 57 mm. 44 sec., 1880, 75 miles, in 13 hours 7 min. 27 sec., and 100, in 18 hours 8 min. 15 sec. x 88o John Dobler , Buftalo, N. Y., 150 miles 850 yards, in 24 hours. 1801’ Cant. R. Barclay , Eng., country road, 90 miles, in 20 hours 22 min. 4 sec., in- cluding rests; 1803, .2/ mile; in 56 sec., and Charing Cross to Newmarket, 64, in 10 hours ^including rests; 1806, 100, in 19 hours , including 1 hour 30 min. in rests; 1809, 1000 in 1000 consecutive hours , walking a mile only at commencement of each hour. 1877 D O' 1 Leary, London, Eng., 200 miles, in 45 hours 21 min. 33 sec. i8i8| Jos. Eaton] Stowmarket, Eng., 4032 quarter miles, in 4032 consecutne quar- t0 \siy™Wm. Gale, London, Eng., 1500 miles, in 1000 consecutive < >«»”■?• j-S nllles each hour; and 4000 quarter miles, in 4000 consecutive periods of 10 minutes. 1882 C/ias. Rowell, New York, N. Y., 89 miles 1640 yards, in 12 hours. 1882’, Geo. Hazael , New York, N. Y., 600 miles 220 yards, in 6 days. Ptviani ng. 1710 Levi Whitehead, Branham Moor, Eng., 4 miles, in 19 min. 1844’, Geo. Seward, of U. S., Manchester, Eng., 100 yards, in 9.25 sec. i860, Geo. Forbes , Providence, R. I.. 150 yards, in 15 sec. i8si Chas. Westhall , Manchester, Eng., 150 yards, in 15 sec., and 200, in 19.5 sec. 1864, Jas. Nuttall , Manchester, Eng., 600 yards, in 1 min. 13 sec. no, r r \fvrr. 3 3 000 H* „ r W_ A 1000 ’ v W ’ V ' t ' It P representing power or stress transmitted, W weight or stress on belt, t thickness of belt , S stress on belt per inch of width , A and a areas of coil and eye , and l length in feet. Note.— 70 square feet of good belting are capable of transmitting an indicated IP. India Rubber Belting. ( Vulcanized .) Results of Experiments upon Adhesion of India Rubber and Leather Belting.— (J. H. Cheever). Rubber. Leather. Leather belt slipped on iron pulley at 48 “ “• leather “ 64 “ “ rubber ‘‘ 128 Rubber belt slipped on iron pulley at 90 “ “ leather “ 128 “ “ rubber “ 183 Hence it appears that a Rubber Belt for equal resistances with a Leather Belt may be reduced respectively 46, 50, and 30 per cent. Iron Wire .— A wire rope .375 inch in diameter, over a pulley 4 feet in diameter, and running at a velocity of 1250 feet per minute, will transmit 4-5 . .. Diameter of pulley should not be less than 140 times diameter of rope, in order to avoid undue bending of wires. A sheet-iron belt 7 inches in width proved more effective than one of leather of like width. General Notes. Leather Belts — Are best when oak tanned, should be frequently oiled,* and when run with hair side over pulley will give greatest adhesion. Ordinary thickness .1875 inch, and weight 60 lbs. per cube foot Relative effect of different pulleys and belts: Pulleys.— Leather surface 1. I Turned iron. 64 Rough iron 41 I Turned wood 7 Tensile strength of calf and sheep skins is about one half that of beeve and horse. Morin assigns 50 lbs. as a proper stress per inch of width of good belting. Presence of small holes in a belt will prevent its slipping or squealing. Rubber Belts .— Best vulcanized rubber is stronger than leather, and its resistance is from 50 to 85 per cent, greater. To increase adhesion, coat driving surface with boiled oil or cold tallow, and then apply powdered chalk. ; When new, cut them .1875 inch short for each foot in length required, to admit of the stretch that occurs in their early operation. They should be kept free from contact with an animal oil. Three ply, .1875 inch thick, has a tensile resistance of 600 lbs. per inch of width, j Relative slipping of a vulcanized belt, over smooth or turned leather or rubber- faced iron pulleys is as .5,-7, and 1. Rubber, Gutta percha, and Canvas belts will stretch continuously. Memoranda. Belts should be set as near horizontal as practicable, in order that the sag may increase adhesion on pulley, and hence power should be communicated through under side. The “creeping” or lost speed by belts is about 2 per cent., hence, to maintain a uniform or required speed, driver must be increased in diameter pro rata with slip. * See Cements , etc., page 871, for compositions, etc. BELTS AND BELTING. — BLASTING. 443 A belt, ii ins. in width, over a driver 4 feet in diameter, running from 1200 to 2250 feet per minute, will transmit the power from two steam cylinders, 6 ins. in diam- eter and 11 ins. stroke, averaging 125 revolutions per minute, with a pressure of 60 lbs. per sq. inch. A double belt, 75 ins. in width and 153.5 feet in length, transmitted 650 IIP. Pulleys should have a slight convexity of surface. Authorities differ, from .5 inch per foot of breadth to .1 of breadth. Belts run at a high speed are less liable to slip than at low speed. The best speeds for economy are from 1200 to 1500 feet per minute, and the best for result not to exceed 1800. Belts.— Leather, hair-side 1 I Leather, flesh- side. . . .74 | Rubber 51 Gutta percha .44 | Canvas 35 Coefficient of Friction of a Belt in operation is assumed to be .423. Smooth surface belts are most endurable and soft most adherent. Round belts .25 and .5 inch in diameter are fully equal in operation to flat of 1 and 3 ins., and grooves in their pulleys should be angular or V shaped. The neutral point of a rope belt is at .33 of diameter from inside surface. Friction of driving and pulley bearings is about .025. A fan-blower No. 6*, driven by a belt 3.875 ins. in width and .18 in thickness, at a velocity of 2820 revolutions per minute, requires power of 9.7 horses. Area of belts per IP varies essentially, ranging from 25 to 100 square feet; the mean is 75. BLASTING. y In Blasting, rock requires from .25 to 1.5 lbs. gunpowder per cube yard, according to its degree of hardness and position. In small blasts 2 cube yards have been rent and loosened, and in very large blasts 2 to 4 cube yards have been rent and loosened, by 1 lb. of powder. Tunnels and shafts require 1.5 to 2 lbs. per cube yard of rock. Gunpowder has an explosive force varying from 40000 to 90000 lbs. per sq. inch. That used for blasting is much inferior to that used for projectiles, the proportion being fully one third less. NTitro-glycerin-e is an unctuous liquid, which explodes by concussion, an extreme pressure (2000 lbs. per sq. inch), or a temperature exceeding 6oo° if quickly applied to it ; it will inflame, however, and burn gradually. At a temperature below 40° it solidifies in crystals. Its explosion is so instantaneous that in rock-blasting tamping is not nec- essary ; its explosive power by weight is from 4 to 5 times that of gun- powder. Dynamite is nitro-glycerine 75 parts, absorbed in 25 parts of a sili- ceous earth termed kieselguhr; it also explodes so instantaneously as to render tamping in blasting quite unnecessary. It is insoluble in water, and may be used in wet holes ; it congeals at 40°, is rendered ineffective at 212 0 , and has an explosive force by weight of 3 times that of gunpowder, and by bulk 4.25 times. Gun-cotton is insoluble in water, and has an explosive force by weight of from 2.75 to 3 times that of gunpowder, and by bulk 2.5 times. It may be detonated in a wet state with a small quantity of dry material. Tonite is nitrated gun-cotton, and is known also as cotton powder. It is produced in a granulated form. T-jitLo-fracteur is a nitr o-glycerine compound in which a portion of the base or absorbent material is made explosive by the admixture therein of nitrate of baryta and charcoal. * For a table of Belts for Fan-blowers, etc., see J. H. Cooper, in “ Jour. Franklin Inst.,” vol. 66, p. 409. BLASTING. 444 Cellulose Dynamite is when gun-cotton is used as the absorbent for nitre-glycerine; it will explode frozen dynamite, and is more sensitive to percussion than it. To Compute Charge of* Gunpowder fbr Hoche Blasting. Rule. — Divide cube of line of least resistance by 25, as for limestone, to 32 for granite, and remainder will give charge of powder in lbs. Or, L 3 -4- 32 = lbs. Example.— When line of least resistance is 6 feet, what is charge required? 6 3 -4- 32 = 6. 75 lbs. Line of least resistance should not exceed .5 depth of hole. Tamping . — Dried clay is the most effective of all materials for tamping; Broken Brick the next, and Loose Sand the least. Relative Costs of a Tunnel and Shaft in England. (Sir John Burgoyne.) Iron and steel 8.98 Smiths and coal 6 Fuses 7- Powder 29.04 Labor Weight of Explosive Materials in Holes of Different Diameters . Diam. Powder or Gun- cotton. Dynamite. Diam. Powder or Gun- cotton. Dynamite. Diam. Powder or Gun- cotton. Dynamite. Ins. 1 1.25 x-5 Oz. .419 •654 .942 Oz. .67 1.046 I-507 Ins- i-75 2 2.25 Oz. 1.283 I-675 2.12 Oz. 2.053 2.68 3-39 2 Ins. 2-5 2-75 3 Oz. 2.618 3. 166 3-7 6 9 Oz. 4.189 5.066 6.03 Boring Holes in Granite. Depth of Hole. Diam. of Jumper. Depth of Hole. Men. Depth bored per Day. Ham- mer. Diam. of Jumper. Ins. Ins. No. Feet. Lbs. Ins. 1 1 to 2 1 8 6 2.25 *•75 2. 5 to 6 3 12 *4 2-5 2 4 t0 7 3 8 14 3 Ins. 5 to 10 9 tO 12 9 to 15 Men. Depth bored per Day. Feet. 6 5 4 Lbs. 16 16 18 Drill. - -Width of bit compared to stock .625. ORarges of* Powder. Usual practice of charging to one third depth of hole is erroneous, inasmuch as volume of charge increases as square of diameter of hole. Hence holes ofl *5* n ... .1 r .f n A„ni dontho uyuiIh rami ire chareres in proportion of 2. 25 and 4. Line of least re- sistance. Powder. Line of | least re- sistance. Powder. Line of least re- sistance. Powder. Line of least re- sistance. Powder. Feet. 1 2 Oz. •75 4 Feet. 3 4 Lbs. Oz. 13-5 2 Feet. 5 6 Lbs. Oz. 3 14-5 6 12 Feet. 7 8 Lbs. Oz. xo 11. 5 16 Effects. Gunpowder. — From its gradual combustion, rends and projects rather than Sh A Mta 5 . 5 ins. in diameter and „ feet 7 ins. in depth, filled to 8 feet ro ins. with 75 lbs. powder, has removed and rent 1200 cube yards, equal to 2400 to s. labor expended was that of 3 men for 14 days. Temperature of gases of explosion 4000 0 . Gun-cotton.— From the rapidity of its combustion, shatters. Dynamite. — From the greater rapidity of its combustion over gun-cotton, is more shattering in its explosion. BLASTING. BLOWING ENGINES. 445 C Drilling. Churn- drilling.— A. churn-driller will drill, in ordinary hard rock, from 8 to 12 feet, 2 inch holes of 2.5 feet depth, per day, and at a cost of from 12 to 18 cents per foot, on a basis of ordinary labor at $1 per day. Drillers receiving $2.50. One man can bore, with a bit 1 inch in diameter, from 50 to 100 inches per day of 10 hours in granite, or 300 to 400 inches per day in limestone. Tamping .— Two strikers and a holder can bore, with a bit 2 inches in diameter, 10 feet in a day in rock of medium hardness. Composition for waterproof charger or fuse consists by weight of Pitch, 8 parts; Beeswax and Tallow each 1 part. Alining. (Lefroy’s Handbook.) In demolition of walls line of least resistance L = half thickness, and C is a co- efficient depending on structure. Charge in lbs. = C X L3. In a wall without counterforts, where interval between the charge is 2 L, C — .15. In a wall with counterforts the charge to be placed in centre of each counterfort at junction with wall, C = .2. Where the charge is placed under a foundation, having equal support on both sides, C = .4. A leather bag, containing 50 to 60 lbs. powder, hung or supported against a gate or like barrier, will demolish it. For ordinary mines in average rock charge in ounces = L3-r- 160. BLOWING ENGINES. For Smelting. Volume of oxygen in air is different at different temperatures. Thus, dry air at 85° contains 10 per cent, less oxygen than when it is at tem- perature of 32 0 ; and when it is saturated with vapor, it contains 12 per cent. less. If an average supply of 1500 cube feet per minute is required in winter, 1650 feet will be required in summer. Smelting of Iron Ore. Colce or Anthracite Coal. — 18 to 20 tons of air are required for each ton of Pig Iron, and with Charcoal 17 to 18 tons are required. (1 ton of air at 34 0 = 29 751, and at 6o° = 31 366 cube feet.) Pressure . — Pressure ordinarily required for smelting purposes is equal to a column of mercury from 3 to 10 inches, or a pressure of 1.5 to 5 lbs. per square inch. Reservoir. — Capacity of it, if dry, should be 15 to 20 times that of cylin- der if single acting, and 10 times if double acting. Pipes. — Their area, leading to reservoir, should be .2 that of blast cylinder, and velocity of the air should not exceed 35 feet per second. A smith’s forge requires 150 cube feet of air per minute. Pressure of blast .25 to 2 lbs. per square inch. A ton of iron melted per hour in a cu- pola requires 3500 cube feet of air per minute. A finery forge requires 100000 cube feet of air for each ton of iron refined. A blast furnace re- quires 20 cube feet per minute for each cube yard capacity of furnace. A Ton of Pig Iron requires for its reduction from the ore 310000 cube feet of air, or 5.3 cube feet of air for each pound of carbon consumed Pressure, .7 lb. per square inch. Pp 446 blowing engines. To Compute Power Required, to Drive a Blowing Engine. .000 050 9 » = ./Z_ V .93 x . .93 X .7854 x V ' r ’(£+f L .b*” o “= B> ' v representing velocity of air in feet per sec - ond, d and d' diameters of pipe and of nozzle in feet, —\f- 35 93 X .7854 X 500 = - 309 - Illustration —What should he power of a steam-engine to drive 35 cube feet of air at a velocity of 500 feet per second, through a pipe 1 foot in diameter and 300 feet in length? . 77 . , , c = ratio -between power employed and effect produced by it = in a well-constructed engine . 5, and C = . 93. d = . 2974, assumed at . 3. .0000509 , ^5 ‘ d C = .93. a = .2974, afcbuuiuu at, .j. ^ 253 ^^ + 60-7-33000 = 22631.625 X 60 — 33000 = 41.15 EP. To Compute Required Dower of a Blowing Engine. F + fX a v __ jp p re p resen ting pressure of blast in lbs. per sq. inch; a arm of cylinder in sq. ins. ; v velocity of piston in feet per minute; f fac- tion of piston and from curvatures , etc., estimated at 1.25 per sq. inch oj piston. Note.— If cylinder is single acting, divide result by 2. Illustration.— Assume area of blast cylinder 5600 sq. ins., pressure of blast 2.25 lbs. per sq. inch, and velocity of piston 96 feet per second. 2.25 + 1.25 X 5600.X 96 _ 1 881 600 _ ^ Worses, the exact power developed in 33000 33000 this case. To Compute Dimensions of a Driving Engine. Rule i. — D ivide power in lbs. by product of mean effective pressure upon - piston of steam cylinder in lbs. per sq. inch, and velocity of piston in feet per minute, and quotient will give area of cylinder in sq. ins. 2. — Divide velocity of piston by twice number of revolutions, and quotient will give stroke of piston in feet. Volume of air at atmospheric density delivered into reservoir, in consequence of escape through valves, and partial vacuum necessary to produce a current, will be about . 2 less than capacity of cylinder. Example. — Assume elements of preceding case, with a pressure of 50 lbs. steam, cut off at .375, and with 12 revolutions of engine per minute, what should be area of cylinder of a non-condensing engine ? Mean effective pressure of steam with 5 per cent, clearance = 50 lbs., and 50 /*_j_ I4 . 7 — 50 — 2.5 + 3.33 + 14.7 = 29.47 lbs . , and velocity of piston *= 192 feet 5600 X 2.25 + 1.25 X 96 _ 1881600 ins and = 8 f eet strode. 29.47 X 192 5658 12 X 2 Area of cylinder in this case was 324 sq. ins. For Volume, Pressure, and Density of Air, see Heat, page 521. * See formula and note for power of non-condensing engine, page 733. BLOWING ENGINES. 447 To Compute Elements of a Blowing Engine. Single Stroke. V P+f „ r AsnP+/_ — Or — rj: , A ni F + / „ . V V + io L 230 33 000 D 2 s n TJp = a 3 D 2 s n Tr , -p, . , - — — = Y ; and 34 P -f 32 = t. 9 2 Y representing volume of air in cube feet per minute, P pressure of air and f f rictional resistance in lbs. per sq ; inch, A area of cylinder and a area of its valves in sq. ins., s stroke of piston in feet, n number of single strokes of piston per minute, L length of air-pipe from reservoir to discharge in feet, d diameter of air or blast pipe and I) diameter of cylinder in ins., v velocity of blast in feet per second, and t temperature of blast consequent upon cow - 2>ression in degrees. Illustrations. — Assume blowing cylinder 50 ins. in diam., stroke of piston 10 feet, number of single strokes 10 per minute, pressure by mercurial manometer 6.12 ins., frictional resistance .4 lb., length of pipe 25.25 feet, and area of valves 95 sq. ins. Y = 1363. 54 cube feet , P = 3 lbs. , A 1963. 5 sq. ms. H and h representing height of barometer and pressure of blast in ins. of mercury ; t temperature of blast ; and v velocity in feet per second. Illustration. — A furnace having 2 tuyeres of 5 ins. diameter, pressure and tem- perature of blast 3 ins. and 350 0 , and barometer 30 ins. ; what is volume of air trans- mitted per minute? C for a conical opening ==? .94. feet velocity per second. Then, area 5 ins. = 19. 635, which X 2 = 39. 27 ins. , and 39. 27 X 1 5- 14 X 60 - 4 - 144 = 247. 73 cube feet. To Compute Pressure of Blast from Water or jMCercarial Grange. Rule. — Divide Water and Mercurial Gauge in ins. by 27.67 and 2.04 re- spectively, and quotient will give pressure in lbs. per sq. inch. Proportions of Parts. Blades . — Their width and length should be at least equal to .4 or .5 radius of fan. Openings . — Inlet should be equal to radius of fan ; and outlet, or dis- charge, should be in depth not less than .125 diameter, its width being equal to width of fan. Eccentricity . — .1 of diameter of fan. Journals, 4 diameters of shaft. and 1963,5 X 10 X 10 X 3 + -4 _ = 20.23 IP* 33000 —95 sq. ins. To Compute Volume of Air transmitted by air Engine. When Pressure, Temperature , etc., are given. Then av x 60= Y in cube feet per minute. •94 = 34*5 Fan-blowers. BLOWING ENGINES. 448 By the experiments of Mr. Buckle, he deduced 1. That velocity of periphery of blades should be .9 that of their theoretical velocity ; that is, velocity a body would acquire in falling height of a homo- geneous column of air equivalent to required density. 2. That a diminution of inlet from proportions here given involved a greater expenditure of power to produce same density. 3. That greater the depth of blade, greater the density of air produced with same number of revolutions. To Compute Elements of a Fan-blower. v representing velocity of periphery of fan in feet per second , d inches of ■ mercury , V volume of air in cube feet , and a area of discharge in sq. ins. Illustration. — Assume velocity of periphery of fan 123 feet per second, density of blast .25 inch, volume of air 1845 cube feet, and area of discharge 40 sq. ins. .242 X 4° X 123 _ 2 g 7 independent of friction of blast in pipes and tuyeres. To Compute Power of a Centrifugal Fan. Y 2 -4- 97 300 = P. V representing velocity of tips of fan in feet per second . Operation of a blower requires about 2.5 per cent, of power of attached boiler. A11 increase in number of blades renders operation of fan smoother, but does not increase its capacity. Pressure or density of a blast is usually measured in ins. of mercury, a pressure of 1 lb. per sq. inch at 6o° = 2.0376 ins. When water is used, a pressure of 1 lb. = 27.671 ins. Cupola blast .8 lbs., and Smith's forge .25 to .3 lbs. per sq. inch. An ordinary Eccentric Fan, 4 feet in diameter, with 5 blades 10 ins. wide and 14 in length, set 1.5 ins. eccentric, with an inlet opening of 17.5 ins. in diameter, and an outlet of 12 ins. square, making 870 revolutions per min- ute, will supply air to 40 tuyeres, each of 1.625 ins. in diameter, and at a pressure per sq. inch of .5 inch of mercury. An ordinary eccentric fan-blower, 50 ins. in diameter, running at 1000 revolutions per minute, will give a pressure of 15 ins. of water, and require for its operation a power of 12 horses. Area of tuyere discharge 500 sq. ins. A non-condensing engine, diameter of cylinder 8 ins., stroke of piston 1 foot, press- ure of steam 18 lbs. (mercurial gauge), and making 100 revolutions per minute, will drive a fan, 4 feet by 2, opening 2 feet by 2, 500 revolutions per minute. Such a blower was applied as an exhausting draught to smoke-pipe of steamer Keystone State , cylinder 80 ins. by 8 feet, and evaporation was doubled over that of when wind was calm. In French blowing engines, volume of air discharged 75 per cent, that of volume of piston space in cylinder, stroke equal diameter of cylindci, and velocity of piston from 100 to 200 feet per minute. Area of admission valves from .066 to .083 of that of cylinder for speeds of 100 to 150 feet per minute, and from .1 to .111 for higher speeds. Area of exit valves from .066 to .05 of cylinder. (M. Claudel.) 40 x 123 X 60 = 1845 cub. ft. Memoranda, BLOWING ENGINES. — CENTRAL FORCES. 449 By some experiments lately concluded in England with boilers of two steamers, to determine relative effects of natural and forced draught furnaces, the results were as follows (R. J. Butler ) : Per Sq. Foot of Grate Surface,— Natural Draught , 10 to 10.87 IBP; Steam Blast , 12.5 to 13; Forced or Blast Draught , 15 to 16. Heating Surface per IIP. — Natural Draught , 2.44 to 2.61 ; Steam Blast , 1. 71 to 2.86; Forced or Blast Draught , 1.56 to 2.5. Tube Surface per IIP in Sq. Feet— Natural Draught , 2.03 to 2.18 ; Steam Blast , 2.02 to 2.08 5 Forced or Blast Draught , 1.3 to 2.8. IIP per Sq. Foot of Grate in these Trials. — Natural Draught , 10.15 to 10.87 ; Steam Blast, 12.76 to 13. 1 ; Forced or Blast Draught, 10.6 to 16.9. Root's Rotary Blower— Is constructed from .125 to 14 nominal IP, supplying from 150 to 10800 cube feet of air per minute. Delivery pipe 2.5 to 19 ms. in diameter. Efficiency 65 to 80 per cent, of power. For Ventilation of Mines — From 40 to 280 revolutions per minute, equal to discharge of 12 500 to 200 000 cube feet of air per minute. 15.5 to 189 IP. Steam cylinder from 14 X 18 ins. to 28 X 48 ins. For other details of Blowing Engines see page 898. CENTRAL FORCES. All bodies moving around a centre or fixed point have a tendency to fly off in a straight line: this is termed Centrifugal Force ; it is op- posed to & Centripetal Force, or that power which maintains a body in its curvilineal path. Centrifugal Force of a body, moving with different velocities in same circle, is proportional to square of velocity. Thus, centrifugal force of a body making 10 revolutions in a minute is 4 times as great as centrif- ugal force of same body making 5 revolutions in a minute. Hence, in equal circles, the forces are inversely as squares of times of revolution. If times are equal, velocities and force§ are as radii of circle of revolution. The squares of times are as cubes of distances of centrifugal force from axis of revolution. Centrifugal forces of two unequal bodies, having. same velocity, and at same dis- tance from central body, are to one another as the respective quantities of matter in the two bodies. Centrifugal forces of two bodies, which perform their revolutions in same time, the quantities of matter of which are inversely as their distances from centre, are equal to one another. Centrifugal forces of two equal bodies, moving with equal velocities at different distances from centre, are inversely as their distances from centre. Centrifugal forces of two unequal bodies, moving with equal velocities at different distances from centre, are to one another as their quantities of matter, multiplied by their respective distances from centre. Centrifugal forces of two unequal bodies, having unequal velocities, and at differ- ent distances from their axes are in compound ratio of their quantities of matter, squares of their velocities, and their distances from centre. Centrifugal force is to weight of body, as double height due to velocity is to radius of rotation. A Radius Vector is a line drawn from centre of force to moving body. P P* 450 CENTRAL FORCES. To Compute Centrifugal Force of any- Body. Rule i.— Divide its velocity in feet per second by 4.01, also square of quotient by diameter of circle ; this quotient is centrifugal force, assuming the weight of body as 1. Then this quotient, multiplied by weight of body, will give centrifugal force required. Example.— What is the centrifugal force of the rim of a fly-wheel having a diam- eter of 10 feet, and running with a velocity of 30 feet per second? 30-^4.01 = 7.48, and 7.482 = 10 = 5.59, or times weight of rim, O r W ft 2 VR 2 -j~^ 2 __ 0 r representing radius of inner diameter of ring. ’ 4100 Note.— Diameter of a fly-wheel should be measured from centres of gravity of rim. When great accuracy is required, ascertain centre of gyration of body, and take twice distance of it from axis for diameter. Rule 2. — Multiply square of number of revolutions in a minute by diam- eter of circle of centre of gyration in feet, and divide product by constant number 5217 ; quotient is centrifugal force when weight of body is 1. Then, as in previous Rule, this quotient, multiplied by weight of body, is centrif- ugal force required. Or n ^ — W. n representing number of revolutions per minute , d diameter of ’ 5217 circle of gyration in feet, and W weight of revolving body in lbs. Example.— What is centrifugal force of a grindstone weighing 1200 lbs., 42 inches in diameter, and turning with a velocity of 400 revolutions in a minute? Centre of gyration = rad. (42 = 2) X 7071 = 14.85 ins., which -7-12 and X2 = 2. 475 feet — diameter of circle of gyration. Then - ^ 47 X 1200 = 91 080 lbs. Formulas to Determine "Various Elements. C* = W v 2 W Rn 2 R = 2930 C _ 32.166 R ’ Wv : W R v' 1. 225 ; W = C 32.166 R Wn 2 2930 /2930 C _ _ / C R 32.166 32.166 c ’ * v w = 6.28 v f R. vv n 32. iuu v v ” ** * C representing centrifugal force, W mass or weight of revolving body , both in lbs., R radius of circle of revolving body in feet, n number of revolutions per minute, and v and v linear or circumferential and angular velocities of body in feet per second. . . /* 1 /» .. lho vnrrvhrir 1 ana v w ^ ^ ~ Illustration. -What is centrifugal force of a sphere weighing 30 lbs., revolving around a centre at a distance of 5 feet, at 30 revolutions per second . 5X2X3- 1416X30 _ feet T hen C 3 ° = 46.04 Ms. 6o — ** 1 J 32.166X5 Centrifugal forces of two bodies are as radii of circles of revolution directly, and as squares of times inversely. Ir lustration If a fly-wheel, 12 feet in diameter and 3 tons in weight, revolves in 8 seconds, 'and anoUnfr of like weight revolves in 6, what should he the dmmetcr of the second when their centrifugal forces are equal ? Then 3:3::“ : % ; or » = =6. 7s feet,x = unknown element. 5 •• g 2 Centrifugal forces of two bodies, when weights are unequal , are directly as squares of times. j umts. . Illustration.— What should be the ratio of the weights of the wheels m the pre ceding case, their forces being equal? Then 3 : 6 2 : 8 2 , or a; = = 5- 333 tons - Molesworth gives .000 34 W R n 2 = C. CENTRAL FORCES. — FLY-WHEEL. 451 FLY-WHEEL. A Fly-wheel by its inertia becomes a reservoir as well as a regulator of force, and to be effective should have high velocity, and its diameter should be from 3 to 4 times that of stroke of driving engine. Co-efficient of fluctuation of energy in a machine ranges from .015 to .035. Weight of a fly-wheel in engines that are subjected to irregular mo- tion, as in a cotton-press, rolling-mill, etc., must be greater than in others where so sudden a check is not experienced, and its diameter should range from 3.5 to 5 times length of the crank. A single acting engine requires a weight of wheel about 2.5 times greater than that for a double acting, and 5 times for double engines of double action. To Compute YV'eigh.t of 11 i m. of a, ITly- wheel. Rule. — Multiply mean effective pressure upon piston in lbs. by its stroke in feet, and divide product by product of square of number of revolutions, diameter of wheel, and .000 23. Note. — If a light wheel is required, multiply by .0003; and if a heavy one, by Example i. — A non-condensing engine (double acting), having a diameter of cyl- inder of 14 ins., and a stroke of piston of 4 feet, working full stroke, at a pressure of 65 lbs. mercurial gauge, and making 40 revolutions per minute, develops about 65 IP; what should be the weight of its fly-wheel, when adapted to ordinary work? Area of cylinder 154 sq. ins. Mean pressure assumed 50 lbs. per sq. inch. Diam- eter of wheel 4 feet stroke x 3 5 = 14 feet. 50 X 154 X 4 = 30 800, which -4- 40 2 X 14 X .00023 = 5978 lbs. 2. — If a fly-wheel, 16 feet in diameter and 4 tons in weight, is sufficient to regulate an engine (double acting) when it revolves in 4 seconds, what should be the weight of a wheel, 12 feet in diameter, revolving in 2 seconds, so that it may have like cen- trifugal force? Note.— The centrifugal forces of two bodies are as the radii of the circles of revo- lution directly, and as squares of times inversely. Then = nr _ 4 X .6 X 2* ^ 4 X 16 X 4 4 2 2 2 12X4 2 12X16 Assume elements of example i. = 1. 333 tons. 5978 X »i -r- 13.25 = 45.12 square ins. To Compute Dimensions of Rim. Rule. Multiply weight of wheel in lbs. by .1, and divide product by mean diameter of rim in feet ; quotient will give sectional area of rim in square inches of cast iron. PS vy — W, and — ^ = A. P representing pressure on piston and TV weight of xoheel in lbs., S stroke of piston and D mean diameter of wheel, both infect, and A area of section of rim in sq. ins. Or, 1 i6nPSC 60 D DO u C representing coefficient varying from 3 to. 4 ordinarily, and increasing to 6 when great regularity of speed is required , and n number of revo- lutions per minute. Note.— Maximum safe velocity for cast iron is assumed at 80 feet per second. For engines at high expansion of steam, or with irregular, loads, as with a rolling- mill, multiply W by 1.5, or put W 100 lbs. for each IIP. (Molesworth.) In corn or like mills, the velocity of periphery of fly-wheel should exceed that of the stones. 45 2 CENTRAL FORCES. — GOVERNORS.— PENDULUMS. GOVERNORS. A Governor or Conical Pendulum in its operation depends upon the principles of Central Forces. When in a Ball Governor the balls diverge, the ring on vertical shaft raises and in proportion to the increase of velocity of the balls squared, or the square roots of distances of ring from fixed point of arms, cor- responding to two velocities, will be as these velocities. Thus, if a governor makes 6 revolutions in a second when ring is 16 ins. from fixed point or top, the distance of ring will be 5.76 ins. when speed is increased to 10 revolutions in same time. For 10 : 6 : : V 16 : 2.4, which, squared = 5.76 ins., distance of ring from top . Or, 6 2 : 10 2 5-7^ • *6 ins. A governor performs in one minute half as many revolutions as a pendulum vibrates, the length of which is perpendicular distance be- tween plane in which the balls ipove and the fixed point or centre of suspension. To Compute Number of Revolutions of a Ball Governor per Minute to maintain Balls at any given Height. j88 .4- y/YL == revolutions. H representing vertical height between plane of balls and points of their suspension in ins. Illustration. -If the rise of the halls of a centrifugal governor is 22 ins., what are the number of revolutions per minute ? j 88 - 4 - V 22 ==. 4o-°9 revolutions. To Compute Vertical Height between Plane of Balls and their Points of Suspension. (l 8S + r)* = vertical height in ins. r representing number of revolutions per minute. Illustration.— If number of revolutions of a centrifugal governor is 100, what will be rise of balls ? 188 -r- 100= 1.88 2 — 3-53 ins • To Compute Angle of Arms or Blane of Balls with Centre SLaft. r -i-l = sin. / . r representing distance of balls from plane of centre shaft, and l distance betweU balls and point of suspension measured m plane of shaft. Illustration. -Distance of balls from plane of centre shaft is .0 inches, and their distance from point of suspension is 25; what is the angle . 10 -r- 25 = .4, and sin. .4=5 23° 35'. (54.l6-7-7l) 2 y When Number of Revolutions are given. — ; — = c05 - l Illustration. — Revolutions of a governor per minute are 50, and length of its arms 2 feet; what is their angle with plane of shalt . ( 54 . 16 -r- 5°) 2 _ ^173 = 5865 __ cos . 54 o 6 '. PENDULUMS. Pendulums are Simple or Compound , the former being a material point, or single weight suspended from a fixed point, about which it oscillates, or vibrates, by a connection void of weight ; and the latter, a like body or number of bodies suspended by a rod or connection. Any such body will have as many centres of oscillation as theie are given points of suspension to it., and when any one of these centres are determined the others are readily ascertained. CENTRAL FORCES. — PENDULUMS. 453 Thus, sox s g = a constant product , and sr=VsoXsg, s g o and r representing points of suspension, gravity, oscillation, and gyration. Or, any body, as a cone, a cylinder, or of any form, regular or irregular so suspended as to be capable of vibrating, is a compound pendulum, and distance of its centre of oscillation from any assumed point of suspension is considered as the length of an equivalent simple pendulum. The Amplitude of a simple pendulum is the distance through which it passes from its lowest position to its farthest on either side. Complete Period of a pendulum in motion is the time it occupies in making two vibrations. b All vibrations of same pendulum, whether great or small, are performed very nearly in same time. Number of Oscillations of two different pendulums in same time and at same place are in inverse ratio of square roots of their lengths. Length of a Pendulum vibrating seconds is in a constant ratio to force of gravity. Time of Vibration is half of a complete period, and it is proportional to square root of length of pendulum. Consequently, lengths of pendulums for different vibrations are— Latitude of Washington , 39.0958 ins. for one second. 9.774 ins. for half a second. 4.344 for third of a second. •.4435 for quarter of a second. lengths of Pendulums vibrating Seconds at Level of tlie Sea in several Places. Equator.. Ins. ins. . 39.0152 New York 39-1017 Paris on nntR T.nt . rO r . London. Ins. 39.1284 39- 1 393 xwnv 39.101 Washington 39-0958 I Lat, 45° 3 9 . 127 To Compute Length of a Simple Pendulum for a giver Latitude. 39. 127 — .099 82 cos. 2 L — l. L representing latitude. Illustration.— Required the length of a simple pendulum vibrating seconds ir toe latitude of 50 0 31 . L = 50° 3 i / cos. 2 L =3 2 x 50° 3 1 ' — cos. 180° — 50 0 31' x 2 = cos. 78° 5 8'r=.i 9 i 3 S 39- 12 7 + • 19 1 38 X .099 82 ( two — or negative == an affirmative or -f-) — 39. 1461 ins To Compute Length of a Simple Pendulum for a giver Nnmber of Vibrations. dulum ^inins^ re P resenl ™9 length for latitude , t time in seconds , and l length of pen Illustration.— Required vibrations of a pendulum in a minute at New York an 60; what should be its length? ’ 39. 1017 x i 2 =39. 1017, Or, -- = 1 . n representing number of vibrations per second To Compute Number of Vibrations of a Simple Pendu- lum in a given Time. y/L't _ t — n ) ~ representing time of one vibration in seconds. To Compute Centre of Gravity of a Compound Pendu- lnm of Two Weights connected in a Right Line. When Weights are both on one Side of Point of Suspension. zw+r w _ — 0 — distance oj centre of gravity from point of suspension. W + w 454 CENTRAL FORCES. — PENDULUMS. When Weights are on Opposite Sides of Point of Suspension. I w — l'w - q — distance of centre of gravity of greater weight from point of sus- W + w pension. ^ote.— T o obtain strictly isochronous vibrations, the circular arc must be sub- stituted for the cycloid curve, which possesses the property of having an inclina- tion, the sine of which is simply proportional to distance measured on the curve from its lowest point. For construction of a Cycloidal pendulum, see Deschaniel’s Physics, Fart I., pp. 71-2. To Compute Length of a Simple Pendulum, -Vibrations of which will be same in Number as Inches m its Length. -^(60 -v/L) 2 = i in inches. Illustration.— What will be length of a pendulum in New York, vibrations of which will be same number as the ins. in its length ? V W 39.1017 X 60) 2 = 7.21 1 2 = 52 ins. To Compute Time of Vibration of a Simple Pendulum, Length being given. y/l^-L = tin seconds. Illustration.— Length of a pendulum is 156.4 ins. ; what is the time of its vibra- tion in New York? I 5 ^’ 4 _ _ 2 seconds. h J 39.IOI7 Or /— x 3. 1416 = t. I representing length of a pendulum vibrating seconds in ^ \ Q • • ins., g measure of force of gravity, and t time of one oscillation. Illustration.— Length of a simple pendulum vibrating seconds, and measure of force of gravity at Washington, are 39.0958 ins., and 32.155 feet. 3.1416 39.0958 i 55 X 12 = 3.1416 x y/ 1.013 = 3- *4*6 X . 3 i8 3 = i second. To Compute Number of Vibrations of a Simple Pen- dulum in a given Time. Xt = n. n representing number of vibrations. ft Illustration.— The length of a pendulum in New York is 156.4 ins., and time of its vibration is 2 seconds; what are number of its vibrations? / 39- IOI 7 w 2 _ / 53 . X2 = . 5X2 = 1 vibration. Hence, 1 X — = 3 ° vl ~ v 156.4 V 12.506 2 brations per minute. To Compute Measure of Grravity, Length of Pendulum and. Number of its Vibrations being given. .82246 ln _ __ g g representing measure of gravity in feet. To Compute Number of Revolutions of a Conical Pen- dulum per NIinute. / 2 933:5 _ n representing distance between point of suspension and plane oj V h revolutions in ins. Note.— Number of revolutions per minute are constant for any given height, and the time of a revolution is directly as square root of height. CRANES. 455 CRANES. Usual form of a Crane is that of a right-angled triangle, the sides being post or jib, and stay or strut, which is hypothenuse of triangle. When jib and post are equal in length, and stay is diagonal of a square, this form is theoretically strongest, as the whole stress or weight is borne by stay, tending to compress it in direction of its length ; stress upon it, com- pared to weight supported, being as diagonal to side of square, or as 1.4142 to 1. Consequently, if weight borne by crane is 1000 lbs., thrust or com- pression upon stay will be 1414.2 lbs., or as a e to e W, Fig. 1. When Post is Supported at "both. Head and. Foot, as Fig. 1. Weight W is sustained by a rope or chain, and tension is equal upon both parts of it ; that is, on two sides of square, i a and e W. Conse- quently jib, i a, has no stress upon it, and serves merely to retain stay, a e. If foot of stay is set at w, thrust upon it, as compared with weight, will be as an to aw ; and if chain or rope from i to a is removed, and weight is suspended from a, tension on jib will be as i a to a W. # W foot of stay is raised to 0 , thrust, as compared with weight, will be as line a 0 is to a W, and tension on jib will be as line ar. By dividing line representing weight, as a W or a w ) into equal parts, to represent tons or pounds, and using it as a scale, stress upon any other part may be measured upon described parallelogram. Thus, as length of a W, compared to a e, is as 1 to 1.4142 : if a W is di- vided into 10 parts representing tons, a e would measure 14.142 parts or tons. 'W'h.erL Post is Supported: at Foot only. If post is wholly unsupported at head, and its foot is secured up to line. 0 \\ , then W, acting with leverage, e W, will tend to rupture post at e, with same effect as if twice that weight was laid upon middle of a beam equal to twice length of e W, e being at middle of beam, which is assumed to be sup-* ported at both ends, and of like dimensions to those of post. Or, force exerted to rupture post will be represented by stress, W, multi- plied by 4 times length of lever, e W, divided by depth of post in line of stress, squared, and multiplied by breadth of it and Value * of its material. Post of such a crane is in condition of half a beam supported at one end, weight suspended from other ; consequents, it must be estimated as a beam ot twice the length supported at both ends, “stress applied in middle. To Compute Stress on Jib, and on Stay or Stmt. -Fig. 2. On diagram of crane, Fig. 2, mark off on line of chain, a W, a distance, a b , representing weight on chain ; from point b draw a line, b c, parallel to jib, a e , and where this intersects stay or strut, draw a vertical line,, c 0, extending to jib, and distances from a to points & c and 0 c, measured upon a scale of equal parts, will represent proportional strain. Illustration. — In figure, weight being 10 tons, stress on stay or strut compressing, a c, will be 31 tons, and on jib or tension-rods, a 0, 26 tons. * For Value of Materials, see page 779. 456 CRANES. To Compute Dimensions of E*ost of a Crane. When Post is Supported at Feet only. Rule— Multiply weight or stress to be borne in lbs. by length of jib in feet measured upon a horizontal plane ; divide product by Value of material to be used, and product, divided by breadth in ins., will give square of depth, also in ins. Example. — Stress upon a crane is to be 22 400 lbs., and distance of it from centre of post 20 feet; what should be dimension of post if of American white oak? Value of American white oak 50. Assumed breadth 12 ins. 2? 409 -X 20 — 3q6 0j and — 746.67. Then ^746.67 = 27.32 ins. 50 12 When Post is Supported at both Ends. Rule.— M ultiply weight or stress to be borne in lbs. by twice length of jib in feet measured upon a horizontal plane • divide product by Value of material to be used, and product, divided bv four times breadth in ins., will give square of depth, also in ins. Example.— Take same elements as in preceding case. Assumed breadth 10 ins. i y ? 2 — — 448, and ^448 — 21. *66 ins. 4 X 10 In Fig. 3, angle a b e and ebc being equal, chain or rope is represented by a b c , and weight by W ; stress upon stay b d, as compared with weight, is as b d to a b or b c. In practice, however, it is not prudent to consider chain as supporting stay ; but it is proper to disregard chain or rope as forming part of system, and crane should be designed to support load independent of it. It is also proper that angles on each side of diagonal stay, in this case, should not be equal. If side a b is formed of tension-rods of wrought iron, point a should be depressed, so as to lengthen that side, and decrease angle a be; but if it is of timber, point a should be raised, and angle a b e increased. 22 400 x 20 x 2 Then — 1 = 17 9 2 °> 50 Fig- 3 * Fig. 4. Fig. 4 shows a form of crane very generally used; angles are same as in Fig. 3, and weight suspended from it, being attached to point d, is represented by line b d. The tension, which is equal to weight, is shown by length of line b c, and thrust by length of line b ^measured by a scale of equal parts, into which line b d, representing weight, is supposed to be divided. But if b e be direction of jib, then b g will show ten- sion and bf the thrust (df being taken parallel to b e), both of them being now greater than before; line b d representing weight, and being same in both cases. To Ascertain Stress on Jit), on Strnt of a Crane.— Fig. 5. Through a draw a s , parallel to jib or tension-rod 0 ?% and also s u parallel to strut a r ; then 1 s is a diagonal of parallelogram, sides of which are equal to r a and r u. If then r s represents a stress of 20 lbs., the two forces into which it is decom- posed are shown by r u and r a ; 0 r is equal to r «, as each of them is equal to a s, and r s is equal to 0 a. Hence, 20 represented by a 0, stress on jib will be represented by 0 r, and that on strut by r a. Assuming then or 3 feet, a r 3.5, and 0 a 1, stress on jib will be 60 lbs., and on strut 70. CRANES. 457 Thus, in all cases, stress on jib or tension-rod and on strut can be deter- mined by relative proportions of sides of triangle formed. To Compute Stress vipon Strxxt of a Crane. Rule —Multiply length of strut in feet by weight to be borne in lbs. ; di- vide product by height of jib from point of bearing of strut m feet, and quotient will give stress or thrust in lbs. Fxample.— Length of strut of a crane is 28.284 feet, height of post is 26.457 feet, and weight to be borne is 22 400 lbs. ; what is stress? 28.284X22400 = 6 33 56l 6 = 7 Z&5 . 26.457 26.457 Chains and. Tropes. Chains for Cranes should be made of short oval links, and should not ex- ceed 1 inch in diameter. ^hort-linked Crane Cliains and Ropes showing Di- melons and Weight of each, and Proof of Chain in Tons Diam. of Chain9. Weight per Fathom. Proof Strain. Circumf. of Rope. Weight of Rope per Fath. Diam. of Chains. Weight per Fathom. Proof Strain. Circumf. of Rope. Weight of Rope per Fath. Ins. .3l 2 5 •375 •4375 .5625 .625 Lbs. 6 8.5 II 14 18 24 Tons. •75 1-5 2.5 3- 5 4- 5 5- 25 Ins. 2- 5 3- 25 4 4- 75 5- 5 6.25 Lbs. 1- 5 2- 5 3- 75 5 7 8.7 Ins. .6875 •75 .8125 • 875 •9375 1 Lbs. 28 32 36 44 50 56 Tons. 6- 5 7- 75 9 - 25 io-75 12.5 14 Ins. 7 7-5 8.25 9 9-5 10 Lbs. 10.5 12 15 17-5 I 9-5 22 Ropes of circumferences given are cousiutueu iu uc ut " the chains, which, being short-linked, are made without studs. A crane chain will stretch, under a proof of 15 tons, half an inch per fathom. Machinery- of Cranes. To attain greater effect of application of power to a crane, the wheel- work must be properly designed and executed. If manual labor is employed, it should be exerted at a speed of 220 feet per minute. Proportions. — Capacity of Crane , 5 tons. Radius of winch or handle 15 to 18 ins. Height of axle from floor 36 to 39. 1st pinion, n teeth, 1.25 ins. pitch. I 2d pinion, 12 teeth, 1.5 ins. pitch. 1st wheel, 89 “ 1.25 “ “ | 2d wheel, 96 “ Barrel 8 ins. X n teeth X 12 teeth X n 200 lbs. — 30 800 _ ^ ^ ^ _ gtatical re _ Winch 17 ins. X 89 teeth x 96 teeth x 4 men == 1513 s'.stance to each of the 4 men at winches. An experiment upon capacitv of a crane, geared 1 to 105, developed that a strong man for a period of 2.5 minutes exerted a power of 27562 foot- pounds per minute, which, when friction of crane is considered, is fully equal to the power of a horse for one minute. In practice an ordinary man can develop a power of 15 lbs. upon a crane, handle moved at a velocity of 220 feet per minute, which is equivalent to 3300 foot-pounds. For Treatise on Cranes, see Woales’ Series, No. 33. Q Q 458 COMBUSTION. COMBUSTION. Combustion is one of the many sources of heat, and denotes combi- nation of a body with any of the substances termed Supporters of Com- bustion ; with reference to generation of steam, we are restricted to but one of these combinations, and that is Oxygen. All bodies, when intensely heated, become luminous. When this heat is produced by combination with oxygen, they are said to be ignited ; and when the body heated is in a gaseous state, it forms what is termed Flame. Carbon exists in nearly a pure state in charcoal and in soot. It com- bines with no more than 2.66 of its weight of oxygen. In its combus- tion, 1 lb. of it produces sufficient heat to increase temperature of 14 500 lbs. of water i°. Hydrogen exists in a gaseous state, and combines with 8 times its weight of oxygen, and 1 lb. of it, in burning, raises heat of 50000 lbs. of water 1 0 .* An increase in the rapidity of combustion is accompanied by a dimi- nution in the evaporative efficiency of the combustible. Mr. D. K. Clark furnishes the following: When coal is exposed to heat in a fur- nace, the carbon and hydrogen, associated in various chemical unions, as hydrocar- bons, are volatilized and pass off. At lowest temperature, naphthaline, resins, and fluids with high boiling-points are disengaged; at a higher temperature, volatile fluids are disengaged; and still higher, olefiant gas, followed by light carburetted hydrogen, which continues to be given off after the coal has reached a low red heat. As temperature rises, pure hydrogen is also given off, until finally, in the fifth or highest stage of temperature for distillation, hydrogen alone is discharged. What remains after distillatory process is over, is coke, which is the fixed or solid carbon of coal, with earthy matter or ash of the coal. The hydrocarbons, especially those which are given off at lowest temperatures, being richest in carbon, constitute the flame-making and smoke-making part of the coal. When subjected to heat much above the temperatures required to vaporize them, they become decomposed, and pass successively into more and more perma- nent forms by precipitating portions of their carbon. At temperature of low red- ness none of them are to be found, and the olefiant gas is the densest type that remains, mixed with carburetted and free hydrogen. It is during these trans- formations that the great volume of smoke is made, consisting of precipitated car- bon passing off uncombined. Even olefiant gas, at a bright red heat, deposits half its carbon, changing into carburetted hydrogen; and this gas, in its turn, may deposit the last remaining equivalent of carbon at highest furnace heats, and be converted into pure hydrogen. Throughout all this distillation and transformation, the element of hydrogen maintains a prior claim to the oxygen present above the fuel; and until it is satis- • fled, the precipitated carbon remains unburned. Summary of Products of Decomposition in tlie Furnace. Reverting to statement of average composition of coal, page 485, it ap- pears that the fixed carbon or coke remaining in a furnace after volatile portions of coal are driven off, averages 61 per cent, of gross weight of the coal. Taking it at 60 per cent., proportion of carbon volatilized in com- bination with hydrogen will be 20 per cent., making total of 80 per cent, of constituent carbon in average coal. Of the 5 per cent, of constituent hydrogen, 1 part is united to the 8 per cent, of oxygen, in the combining proportions to form water, and remaining 4 parts of hydrogen are found partly united to the volatilized carbon, and partly free. Mean effect. COMBUSTION. 4 These particulars are embodied in following summary of condition of elements of ioo lbs. of average coal, after having been decomposed, and prior to entering into combustion — ioo Lbs. of Average Coal in a Furnace. Composition Lbs. Carbon { Voiatiiized. . . . 20 Hydrogen • 5 Sulphur . 1.25 Oxygen . 8 Nitrogen Ash, etc • 4-55 ► forming Lbs. Decomposition. 60 fixed carbon. 24 hydrocarbons and free hydrogen. 1.25 sulphur. . 85.25 9 water or steam. 1.2 nitrogen. 4.55 ash, etc. IOO IOO showing a total useful combustible of 85.25 per cent., of which 25.25 per cent, is volatilized. While the decomposition proceeds, combustion proceeds, and the 25.25 per cent, of volatilized portions, and the 60 per cent, of fixed carbon, successively, are burned. It may be added that the sulphur and a portion of the nitrogen are dis- engaged in combination with hydrogen, as sulphuretted hydrogen and am- monia. But these compounds are small in quantity, and, for the sake of simplicity, they have not been indicated in the synopsis. Volume of Air chemically consumed in complete Combustion of Coal. Assume 100 lbs. of average coal. Then, by following 80 + 3 ^5 — ^ + -4 X 1.25 X 152 = 14 060 cube feet of air at 62° for 100 lbs. coal. For volatilized portion, Hydrogen (H), 4 lbs. x 457 = 1 828 cube feet. Carbon (C), 20 “ X 152= 3040 “ “ Sulphur (S), 1.25 “ X 57 = 71 “ “ 4939 u u For fixed portion, Carbon, 60 lbs. x 152= 9120 “ “ Total useful combustible, 85.25 “ 14059 “ “ for com- plete combustion of 100 lbs. coal of average composition at 62°. To Compute Volume of Air at 62°, under One At- mosphere, chemically consumed in Complete Com- bustion of 1 Lb. of a given ITnel. Rule. — Express constituent carbon, hydrogen, oxygen, and sulphur, as percentages of whole weight of fuel ; divide oxygen by 8, deduct quotient from hydrogen, and multiply remainder by 3 ; multiply sulphur by .4 ; add products to the carbon, and multiply sum by 1.52. Final product is volume of air in cube feet. To compute iveight of air chemically consumed. — Divide volume thus found by 13.14; quotient is weight of air in lbs. Or, 1. 52 (C -f- 3 (H — ~ ) -f . 4 S) = Air. 0 Oxygen. Note.— In ordinary or approximate computations, sulphur may be neglected. Example.— Assume 1 lb. Newcastle coal. 0 = 82.24, 11 = 5.42, 0 = 6.44, and S = i- 35- — — = 4 . 805, 5-42 — .805 = 4.615 X 3 = i3- 84S, 1-35 X .4 = -54» i3- 845 + -54 +82. 24 = 96.625, and 96.625 X 1.52 = 146.87 cube feet. Then 146.87-7-13.14 = 11.18 lbs. o COMBUSTION. To Compute Total Weight of Gaseous Products of Com- plete Combustion of 1 I/b. of a given Enel. Rule.— Express the elements as per-centages of fuel; multiply carbon by .126, hydrogen by .358, sulphur by .053, and nitrogen by .01, and add prod- ucts together. Sum is total weight of gases in lbs. Or, .126 C + .358 H + .053 S + .oi N = Weight. Example. —Assume as preceding case. N = 1. 61. 82.24 X 1.264-5.42 X .358 + i-35 X 053 + 1.61 X .01 = 12.39 lbs. To Compute Total -Volume, at 62°, of Gaseous Products of Complete Combustion of 1 Eh. of given Fuel. Rule.— Express elements as per-centages; multiply carbon by 1.52, hy- drogen by 5.52, sulphur by .567, and nitrogen by .135, and add products together. Sum is total volume, at 62° F., of gases, in cube feet. Or, 1.52 C + 5.52 H + .567 S -J- . 135 N = Volume. To Compute Volume of the several Gases separately from their Respective Quantities. Rule.— Multiply weight of each gaseous product by volume of 1 lb. in cube feet at 62°, as below. Volume of 1 Lb. of Gases at 62° under a Pressure of 14.7 Lbs. Cube feet. Cube feet. Cube feet. Aqueous Vapor or| Gaseous Steam . j 21.125 | Air. Nitrogen 13- 5 GI Carbonic Acid 8.594 I Oxygen 11.887 I I Hydrogen 190 | 13. 1 41 cube feet. For a lb. of oxygen in combustion, 4.35 lbs. air are consumed; or, by volume, for a cube foot of oxygen 4.76 cube feet of air are consumed. 1 lb. Hydrogen consumes 34- 8 lbs., or 457 cube feet, at 62°. 1 “ Carbon, completely burned, consumes 11.6 “ ‘ 152 * u x « “ partially “ “ .... 5-8 “ 76 , t Sulphur consumes. . 4-35 ' Composition and. Equivalents of Gases, Combustion of Euel. 57 combined in Oxygen . . . Hydrogen. Carbon Sulphur. . . Nitrogen . . COMPOUNDS. ^Atmospheric Air (mech. mixture) . . Aqueous Vapor or Water Equiv- alents. O. I H. 1 C. 1 S. I N. 1 0. 23 N. 77 O. 1 H. 1 By Weight. 6 16 14 26.8! GASES. Elements. By Weight. COMPOUNDS. Equiv- alents. Light Carburetted ) C. 2 I2 1 Hydrogen J 1 H. 4 A) l\ Carbonic Oxide j 1 0. 1 C. 1 Carbonic Acid ] 1 0. 2 C. I 1 ] Olefiant Gas (Bi-car- ] l C. 4 24 1 buretted Hyd. . . . ; 1 H. 4 4l Sulphurous Acid. . . J \ 0. 2 S. I 16) 16 } Weights of products in combustion of 1 lb. of given fuel, are— C == .0366. H = .09. S = .o2. N = . 0893 C + . 268 H + . 0335 S + . 01 N. Cube Feet. .02 X 5- 85 = .117 volume sulph. acid. .0893 + -268 + .0335 + .01 X i3-5°i =3 5. 409 volume nitrogen. Cube Feet. .0366 X 8.59= .315 volume carbonic acid. .09 X 190 =17.1 “ steam. Volume of Air or Gases at higher temperatures than here given (62°) is ascer- tained by V V 461 = V'. V representing volume of air or gas at temperature t, u t +461 and V' at temperature t'. By Volume 1 Oxygen, 3.762 Nitrogen. COMBUSTION. 46l Chemical Composition of some Compound Com- bustibles. Carbonic oxide Light carburetted hydrogen. . . Olefiant gas, Bicarburetted hyd. Sulphuric ether Alcohol.. . Turpentine . Wax Olive oil Tallow Combining equivalents. Car. Hyd. Oxy. Car. Hyd. Oxy. Per Cent. Per Cent. Per Cenl 1 — 1 42.9 — 57-i 2 4 — 75 25 — 4 4 — 85-7 J 4-3 — 4 5 1 64.8 13-5 21.7 4 6 2 52.2 13 34-8 20 16 ■ — 88.2 11. 8 — — — 81.6 i3-9 4-5 — — . — 77.2 13-4 9.4 — — — 79 11 -7 9-3 Heating powers of compound bodies are approximately equal to sum of heating powers of their elements. Thus, carburetted hydrogen, which consists of two equivalents of carbon and four of hydrogen, weighing respectively 2X6 = 12 and 1 x 4 = 4, in proportion of 3 to 1, or .75 lb. of carbon and .25 lb. of hydrogen in one lb. of gas. Elements of heat of combustion of one lb. are, then- units of heat. For carbon 14 544 X .75 = 10908 For hydrogen 62 032 X .25 = 15 508 Total heat of combustion, as computed 26416 Total heat, by direct trial 23513 Heating Powers of Coinbus 1 i L> 1 e s . (MM. Favre and Silbermann , D. K. Clark and others.) Hydrogen Carbon, making ) carbonic oxide, j Carbon, making 1 carbonic acid. . ) Carbonic oxide. . . . , Light carburetted 1 hydrogen j Olefiant gas Sulphuric ether Alcohol Turpentine Sulphur Tallow Petroleum Coal (average) Coke, desiccated. . . Wood, desiccated . . Wood - charcoal, 1 desiccated j Peat, desiccated Peat-charcoal, de - ) siccated j Lignite Asphalt Oxygen consumed per lb. of Com- bustible. Lbs. i-33 2.66 •57 4 3-43 j.6 2.78 3- 29 2-95 4.12 2-5 1.4 2.25 I *75 2.28 2.03 2-73 Weight and Volume of Air consumed per lb. of Combustible. Lbs. 34-8 5-8 11. 6 2.48 17.4 i 5 i 4-3 4-35 12.83 17-93 10.7 10.9 9.8 7.6 9.9 8.85 11.87 Cube Feet at 62°. 457 76 152 33 229 196 149 159 57 235 141 143 80 129 100 129 116 156 Total Heat of Combus- tion of 1 lb. of Combus- tible. Units. 62 032 4 452 14 500 4 325 23 513 21343 16 249 12929 19 534 4032 18 028 27 531 M 133 13 550 7792 13 309 9 951 12325 11 678 16655 Equivalent evaporative Power of 1 lb. of Com- bustible, under one At- mosphere. Lbs. of wa- ter at 62°. 55-6 13 3-88 21.07 19. 12 14.56 11.76 W-5 3.61 16. 1 q 12.67 12. 14 6.98 11.92 8.91 11.04 Lbs. of wa- ter at 212 0 . 64.2 4.61 15 4.48 24-34 22.09 16.82 I3-38 20.22 4.17 18.66 28.5 14.62 14.02 8.07 * 3 -i 3 10.3 12.76 12. 1 17.24 When carbon is not completely burned, and becomes carbonic oxide, it produc less than a third of heat yielded when it is completely burned. For heating pow of carbon an average of 14 500 units is adopted. Q Q* COMBUSTION”. 463 To Compute Heating Power of 1 Lt>. of a given Com- bustible. When proportions of Carbon, Hydrogen , Oxygen, and Sulphur are given . Rule. — Ascertain difference between hydrogen and .125 of oxygen; multi- ply remainder by 4.28 ; multiply sulphur by .28, add products to the carbon, multiply sum by 14 500, divide by 100, and product is total heating power in units of heat. Or, 145 (C + 4.28 H — Ox 125 + .28 S ) = heat. Illustration. — Assume as preceding case. 5.42 ^82.28 X -125 X 4.28 + 1.35 x .28 + 82.28 X i4 5°o-^-ioo— 15 005. To Compute Evaporative Power of 1 Lb. of a Griven. Combustible. When Proportions of Carbon , Hydrogen , Oxygen, and Sulphur are given . Rule. — Ascertain difference between hydrogen and .125 of oxygen, multiply remainder by 4.28 ; multiply sulphur by .28, add products to the carbon, and multiply sum by .13, when water is supplied at 62°, and .15 when at 212 0 ; product is evaporative power in lbs. of water at 212 0 . Or, When total heating power is known, divide it by 1116 when water is at 62°, or 996 when at 212 0 . Illustration. — By table, heating power of Tallow is 18028 units. Hence, 18028 -P 1 116 = 16.15 lbs. water evaporated at 62°. Temperature of’ CoiTADvistiorL. Temperature of combustion is determined by product of volumes and specific heats of products of combustion. Illustration.— 1 lb. carbon, when completely burned, yields 3.66 lbs. carbonic acid and 8.94 of nitrogen. Specific heats .2164 and .244. 3.66 X .2164 =•. .792 units of heat for i°. 8.94 X -244 == 2.x8i “ “ “ i°. 12.6 2.973 “ “ “ i°. Consequently, products of combustion of 1 lb. carbon absorbs 2.973 units of heat in producing i° temperature. Weiglit and Specific Heat of Products of Cornbnistion, and Temperature of Combustion. (2>. K. Clark.) 1 Lb. of Combustible. Gaseous Products for 1 Lb. of Combustible. Weight. Mean specific Heat. Heat to raise the Tempera- ture i°. Temperature of Combustion. Lbs. Water = 1. Units. 0 Ratio. bo .302 10.814 5744 100 11.97 . 256 3 - 063 5305 92 15-9 •257 4.089 5219 9 i 13.84 .256 3-54 5093 88.7 11.94 .246 2-935 4879 85 12.6 .236 2-973 4877 85 15.21 •257 3 - 9*4 4826 84 10.09 .27 2.68 4825 84 18.4 .268 4-933 4766 83 5-35 .211 1.128 3575 62 12. 18 •257 3.127 3470 60 22.64 .242 5-478 2614 45 Hydrogen Sulphuric ether Olefiant gas (Bi-carburetted hyd.) Tallow Coal (average) Carbon, or pure coke Wax Alcohol Light carburetted hydrogen Sulphur Turpentine Coal, with double supply of air. . Whence it appears, that mean specific heat of products of combustion, omitting hydrogen .302 and sulphur .211, is about .25. Hence, To Ascertain Temperature of Combustion . — Divide total heat of combustion in units by units of heat for x°, and quotient will give tem- perature. COMBUSTION. 463 Illustration.— What is temperature of combustion of coal of average composi- tion? Gaseous products as per preceding table 11.94, which X .246 specific heat = 2.935 units of heat at i°. Hence, 14 133 units of combustion (from table, page 461) - 4 - 2.935 = 48129 temper- ature of combustion of average coal. If surplus air is mixed with products of combustion equal to volume of air chem- ically combined, total weight of gases for one lb. of this coal is increased to 22.64. See following table, having a mean specific heat of .242. Then 22.64 X .242 = 5.478 units for i°. Hence, 14 133 total heat of combustion -4- 5.478 = 2614° temperature of combus- tion, or a little more than half that of undiluted products. Taking averages, it is seen that the evaporative efficiency of coal varies directly with volume of constituent carbon, and inversely with volume of constituent oxygen ; and that it varies, not so much because there is more or less carbon, as, chiefly, because there is less or more oxygen. The per-cent- ages of constituent hydrogen, nitrogen, sulphur, and ash, taking averages, are nearly constant, though there are individual exceptions, and their united effect, as a whole, appears to be nearly constant also. Ileat of ComlDvistion. Or, number of times in combustion of a substance , its equivalent weight of water would be raised i°, by heat evolved in combustion of substance. Alcohol 12930 I Ether 16 246 I Olefiant gas 21 340 Charcoal 14 545 I Olive oil 17750 | Hydrogen 62030 Combustion of IT.tiel, Constituents of coal are Carbon , Hydrogen , Azote , and Oxygen. Volatile products of combustion of coal are hydrogen and carbon, the unions of which (relating to combustion in a furnace) are Carburetted hydrogen and Bi-carburetted hydrogen or Olefiant gas , which, upon com- bining with atmospheric air, becomes Carbonic acid or Carbonic oxide , Steam, and uncombined Nitrogen. Carbonic oxide is result of imperfect combustion, and Carbonic acid that of perfect combustion. Perfect combustion of carbon evolves heat as 15 to 4.55 compared with imperfect combustion of it, as when carbonic oxide is produced. 1 lb. carbon combines with 2.66 lbs. of oxygen, and produces 3.66 lbs. of carbonic acid. Smoke is the combustible and incombustible products evolved in combustion of fuel, which pass off by flues of a furnace, and it is composed of such portions of hydrogen and carbon of the fuel gas as have not been supplied or combined with oxj 7 gen, and consequently have not been converted either into steam or carbonic acid; the hydrogen so passing away is invisible, but the carbon,. upon being sepa- rated from the hydrogen, loses its gaseous character, and returns to its elementary state of a black pulverulent body, and as such it becomes visible. Bituminous portion of coal is converted into gaseous state alone, carbonaceous portion only into solid state. It is partly combustible and partly incombustible. To effect combustion of 1 cube foot of coal gas, 2 cube feet of oxj 7 gen are required* and, as 10 cube feet of atmospheric air are necessary to supply this volume of oxy- gen, 1 cube foot of gas requires oxygen of 10 cube feet of air. In furnaces with a natural draught, volume of air required exceeds that when the draught is produced artificially. An insufficient supply of air causes imperfect combustion : an excessive supply, a waste of heat. COMBUSTION. 464 Volume of atmospheric air that is chemically required for combustion of 1 lb. of bituminous coal is 150.35 cube feet. Of this, 44.64* cube feet com- bine with the gases evolved from the coal, and remaining 105.71 cube teet combine with the carbon of the coal. Combination of gases evolved by combustion gives a resulting volume proportionate to volume of atmospheric air required to furnish the oxygen, as 11 to 10. Hence the 44.64 cube feet must be increased m this proportion, and it becomes 44.64 + 4.46 = 49.1. Gases resulting from combustion of the carbon of coal and oxygen of the atmosphere, are of same bulk as that of atmospheric air required to furnish the oxygen, viz., 105.71 cube feet. Total volume, then, of the atmospheric air and gases at bridge wall, flues, or tubes, becomes 105.71 + 49; 1 — *54-8i cube feet, assuming temperature to be that of the external air. Gonse- quentlv, augmentation of volume due to increase of temperature of a fur- nace is to be considered and added to this volume, hi the consideration of the capacity of flue or calorimeter of a furnace. There is required, then, to be admitted through the grates of a furnace for combustion of 1 lb. of bituminous coal as follows : Coal containing Soper cent, of cartoon, or .7047 per cent of coke. 1 lb. coal X 44.64 cube feet of gas = 44- 64 .7047 lb. carbon x 150 cube feet of air . . . = 105- 7 1 150.35 cube feet For anthracite, by observations of W. R. Johnston, an increase of 30 per cent, over that for bituminous coal is required = 195.45 cube feet. Coke does not require as much air as coal, usually not to exceed 108 cube feet, depending upon its purity. Heat of an ordinary furnace may be safely considered at iooo° ; hence air entering ash-pit and gases evolved in furnace under general law of expan- sion of permanently elastic fluids of ^ygths of its volume (or .002087) for each degree of heat imparted to it, the 154.81 is increased m volume from ioo° (assumed ordinary temperature of ai r at ash-pit) to 1 000 — 9°° » tnen 900X .002 087 = 1.8783 times, or 154.81 + 154-81 X 1.8783 = 445.59 cube feet. If the combustion of the gases evolved from coal and air was complete, there would be required to give passage to volume of but 445-59 cube teet over bridge wall or through flues of a furnace; but by experiments it ap- pears that about one half of the oxygen admitted beneath grates of a furnace passes off uncombined : the area of the bridge wall, or flues or tubes, must con- sequently be increased in this proportion, hence the 445.59 becomes 691.10. Velocity of the gases passing from furnace of a proper-proportioned boiler * rTI1 891.18 may be estimated at from 30 to 36 feet per second. Then ^ — .00687 sq. feet, or .99 sq. ins., of area at bridge wall for each lb. of coal con- sumed per hour. A limit, then, is here obtained for area at the bridge wall, or of flues or tubes immediately behind it, below which it must not be decreased, or com- bustion will be imperfect. In ordinary practice it will be found advan- tageous to make this area .014 sq. feet, or 2 sq. ins. for every lb. ot bitu- minous coal consumed per sq. foot of grate per hour, and so on 111 proportion for any other quantity. Volumes of heat evolved are very nearly same for same substance, what- ever temperature of combustible. COMBUSTION. 465 Relative Volumes Lbs. ! Warlich’s patent.. . . 13.1 I Charcoal n.16 Coke 11.28 I of Air required for Combustion of Fuels. Lbs. Lbs. Anthracite Coal 12.13 Bituminous “ 10.98 Bitum. Coal, average ic.7 Bitum. Coal, lowest. . 5.92 Peat, dry 7.08 Wood, dry 6 Perfect combustion of 1 lb. of carbon requires 11.18 lbs. air at 62 , and total weight = 12.39 lbs. Total heat of combustion of 1 lb. carbon or char- coal is 14500 thermal units; mean specific heat of products of combustion is .25, which, multiplied by 12.39 as above = 3 - 0975 , and 14 500* -- 3.0975 = 4681° temperature of a furnace, assuming every atom ot oxygen that was ignited in it entered into combination. ^ If however, as in ordinary furnaces, twice volume of air enters, then products of combustion of 1 lb. of coal will be 12.39 + I 1 :? 8 .— 2 ^ 57 ’ whlch ’ multiplied by its specific heat of .25 as before, and if divided into 14500, quotient will be 2641°, which is temperature of an ordinary furnace. Ratio of Combustion— Quantity of fuel burned per hour per sq. foot of grate varies very much in different classes of boilers. In Cornish boileis it is 3.5 lbs. per sq. foot ; in ordinary Land boilers, 10 to 20 lbs. ; (English) 13 to 14 lbs. ; in Marine boilers (natural draught), 10 to 24 lbs. ; (blast) 30 to 60 lbs. ; and in Locomotive boilers, 80 to 120 lbs. Volumes of air and smoke for each cube foot of water converted into steam, is for coal and coke 2000 cube feet, for wood 4000 cube feet ; and for each lb. of fuel as follows : Coal 207 | Canuel coal... 315 | Coke 216 | Wood 173 Calorific power of 1 lb. good coal = 14 000 X 77 2 = 10 808 000 lbs - Relative Evaporation of* Several Conabnstibles in. Lbs. of Water, Heated 1 ° by 1 Lb. of Material. Combustible. Composition. Water. Lbs. Alcohol 812 (Hyd. .12) i Carb. . 45 j 8 120 Bituminous coal. . . (Hyd. .04) I Carb. .75 J 9830 Carbon 14 220 9028 5 o 8 54 Cokfi Carb. .84 Hydrogen (mean). . Oak wood, dry (Hyd. .06) (Carb. .53} 6 018 “ “ green... (Hyd. .o8> ( Carb, .37/ 5662 Combustible. I Composition. Olive oil Peat, moist “ dry Pine wood, dry. . . . Sulphuric ether. .7 Tallow Hvd. .13 Carb. .77 ( Hyd. .04 (Carb. .43 f Hyd. .06 (Carb. .58 Hyd. .06 Carb. .7 Hyd. .13 Carb. .6 Water. Lbs. 14560 3481 3900 3618 8680 14 5 6 0 1 lb. Hydrogen will evaporate 62.6 lbs. water from 212 0 = 60.509 lbs. heated i°. 1 lb. Carbon “ 14.6 lbs. 212 0 , or raise 12 lbs. water at 6o° to steam at 120 lbs. pressure. 1 lb. of Oxygen will generate same quantity of heat whether in combustion with hydrogen, carbon, alcohol, or other combustible. Relative Volumes of Gases or Products of Combustion per Lb. of Fuel. Supply of Air per lb. of Fuel. Supply of Air per lb. of Fuel. Temp. 12 lbs. 18 lbs. 24 lbs. Temp. 12 lbs. 18 lbs. 24 lbs. Air. Volume Volume Volume Air. Volume Volume Volume per lb. per lb. per lb. per lb. per lb. per lb. O Cube Feet. Cube Feet. Cube Feet. O Cube Feet. Cube Feet. Cube Feet. 32 150 225 300 572 3H 471 628 68 161 241 322 752 369 553 738 104 172 258 344 1 1 12 479 718 957 212 205 3°7 409 1472 588 882 1176 39 2 259 389 5 i 9 2500 906 1359 1812 * Mean of all experiments 13964. 466 COMBUSTION. EXCAVATION AND EMBANKMENT. To Compute Consumption of Fuel to Heat ^Air. Rule.— Divide volume of air to be heated by volume of 1 lb. of it, at its temperature of supply ; multiply result by number of heat-units necessary to raise 1 lb. air through the range of temperature to which it is to be heated, and product, divided by number of heat-units of fuel used, will give result in lbs. per hour. Example.— What is required consumption per hour of coal of an average compo- sition to heat 776400 cube feet of air at 54 0 to 114 0 ? Coal of an average composition (Table, page 461) = 14 133 heat-units. Volume of 461 + 54. 1 lb. air at 54 0 (see formula, page 522) = - 39 - * - = 12.94 cube feet. 1 X 1 14 — 54 X *2377 (specific heat of air) = 14. 262 heat-units. 776 4 °° X 14- 262 -i- 14 133 — 60. 55 Ms. 12.94 Loss of heat by conduction of it to walls of apartment is to be added to this. excavation and embankment. LaPor aii cl Work upon Excavation and Embankment. Elements of Estimate of Worlc and Cost. Per Day of 10 Hours. Cart .— One horse. Distance or lead assumed at 100 feet, or 200 , feet for a trip , at a speed of 200 feet per minute . , V, V ^ Earths — Of gravelly, loam, and sandy, a laborer will load per day into a cart respectively io, 12, and 14 cube yards as measured in embankment, and if measured in excavation, .11 more is to be added, in consequence of the greater density of earth when placed in embankment than in excavation. Note. Earth when first loosened, increases in volume about .2, but when settled in embankment’it has less volume than when in bank or excavation Carting. — Descending, load .33 cube yard, Level, .28, and Ascending .25, measured in embankment ; and number of cart-loads in a cube yard of em- bankment are, Gravelly earth 3, Loam 3.5, and Sandy earth 4. Loosening. — Loam, a three-horsed plough will loosen from 250 to 800 cube yards per day. Trimming— Cost of trimming and superintendence 1 to 2 cents per cube yard, Scooping. — A scoop load measures about .1 cube yard in excavation; time lost in loading, unloading, and turning, 1. 125 mmutes per load ; in double scooping it is 1 minute. Time occupied for every 100 feet of dis tance from excavation to embankment, 1.43 minutes. Time —Time occupied in loading, unloading, awaiting, etc., 4 minutes per load. To Compute Number of Loads or Trips in Cube Yards > per Cart per Lay . / 60 \ h+-v = n. E representing average distance of carting from em - \E -T- 100 + 4/ _ - - hankmenufstalions of roofed each, y number of cart-loads to cube yard of excava- tion, and n number of cube yards in embankment, hauled by a cart per day to dis- tance E. EXCAVATION AND EMBANKMENT. 467 Illustration. — What is number of cube yards of loam that can be removed by one cart from an embankment on level ground for an average distance of 250 feet ? Substituting for 3, 3.5, and 4 number of cart-loads in a cube yard of embank* ment, 20, 17.14, and 15,= 60 minutes, divided respectively by these numbers. ascending, h representing number of hours actually at work. To Compute Cost of Excavating and. Embanking per Cube Yard. in different earths , as 10, 12, and 14, c of one cart and driver per day , l cost of loosen- ing material per cube yard , and s cost of trimming and superintendence , both per cube yard , and all in cents. Illustration.— Volume of excavation in loam 30000 cube yards. Level carting 650 feet = 6. s trips or courses. Loosening by plough 1.7 cents per cube yard, laborers 106 cents per day, carts 160, and trimming and superintendence 1.5 cents per cube yard. Then x*7 + i -5 = 8 - 8 33 + 9-797 + 1*7 + 1.5 = 21.83 cents per cube By Carts . — A laborer can load a cart with one third of a cube yard of sandy earth in 5 minutes, of loam in 6, and of heavy soil in 7. This will give a result, for a day of 10 hours, of 24, 20, and 17.2 cube yards of the respective earths, after de- ducting the necessary and indispensable losses of time, which is estimated at .4. It is not customary to alter the volume of a cart-load in consequence of any dif- ference in density of the earths, or to modify it in consequence of a slight inclina- tion in the grade of the lead. In a lead of ordinary length one driver can operate 4 carts. With labor at $1 per day, the expense of a horse and cart, including harness, repairs, etc., is $1.25 per day. A laborer will spread from 50 to 100 cube yards of earth per day. The removal of stones requires more time than earth. The cost of maintaining the lead in good order, the wear of tools, superintend- ence, trimming, etc., is fully 2.5 cents per cube yard. By Wheel-barrows . — A laborer in wheeling travels at the rate of 200 feet per min- ute, and the time occupied in loading, emptying, etc., is about 1.25 minutes, with- out including lead. The actual time of a man in wheeling in a day of 10 hours is .9 or 2.25 minutes per lead of 100 feet. Hence, To Compute Number of* Barrow-Loads removed by a By Carts.— Quarried rock will weigh upon an average 4250 lbs. per cube yard, and a load may be estimated at .2 cube yard, and weighing a very little more than a load of average earth. Hence, the comparative cost of carting earth and rock is to be computed on the basis of a cube yard of earth averaging 3.5 loads and one of rock 5 loads, with the addition of an increase in time of loading, and wear of cart. E = 250 -f- 100 = 2. 5, and y = 3. 5. X 10 -f- 3.5 = 26.37 cube yards. 6-5 + + 5 — L representing pay of laborers , v value or result of loading yard. E ar tbxvor Is , Laborer per Day. 10 X 60 X -Q — ■ ■ = n. n representing number of leads of 100 feet. 1.25 w A barrow- load is about .04 of a cube yard. Rock. 468 excavation and embankment. Labor. For labor of a man, see Animal Power, pp. 433-34- . mb. feet of By Wheel-barrow. - A barrow-load may be assumed at .75 «*• - a cube ^Blasting - When labor is * . per day, hard rock in ordinary position may be bl ld" hweve^in^onsequence'of condition, position, etc., may vary from so cents to $ 1. i^cubTyards^of hard rock may be carted per day over a lead of roo feet, at a cost Stone. Hauling Stone.- A cart drawn by horses over an ordinary road will travel ,.15 and ^unloading 1 by hand^vhen 'labcmte* $ 1 as per day and a horse 75 cents, ,s a 5 cents' per pe4=L 4 -75 cube whic b he moves is ,a 5 Work done by an ammans not impeded, and force then exerted .45 ff S^ffoTc^he animaS exert at a dead pull. Earthwork. ( Molesworth .) proportion vj culated at 50 yards run. Gett’s.| Fill’s, j Wheel’s. In loose earth, sand, etc. “ Compact “ Marl '■I Gett’s. Fill’s. Wheel’s. J In Hard clay I 1.25 1.25 “ Compact gravel I 1 1 1 “ Rock, from 1 3 1 z Sand . Average Weight of Earths, Rocks, etc. Per cube yard. Lbs. Marl Lbs. 1 Sandstone . . 1 minlp . . . . Lbs. . 4368 . 4480 Granite . Trap Clay J Chalk . . . . 1 Quartz Slate Lbs. 4700 4700 4710 ; liworii, sumed at 1. When in Embanlcment. Rock, large. Medium Metal I Sand and gravel. . J of T c ^ana auu ; \ A ,'L to 1.3 Clay and earth after subsidence. . 3 3 \ u u before 1.2 1 1.07 1.08 , 1.2 \ FRICTION. 469 FRICTION. Friction is the force that resists the bearing or movement of one sur- face over another, and it is termed Sliding when one surface moves over another, as on a slide or over a pin ; and Rolling when a body ro- tates upon the surface of some other, as a wheel upon a plane, so that new parts of both surfaces are continually being brought in contact with each other. The force necessary to abrade the fibres or particles of a body is termed Measure of friction ; this is determined by ascertaining what portion of the weight of a moving body must be exerted to overcome the resistance arising from this cause. Coefficient of Friction expresses ratio between pressure and resistance of one surface over or upon another, or of surfaces upon each other. Angle of Repose is the greatest angle of obliquity of pressure between two planes, consistent with stability, the tangent of which is the coefficient of friction. Experiments and Investigations have adduced the following observations and results : 1. Amount of friction in surfaces of like material is very nearly propor- tioned to pressure perpendicularly exerted on such surfaces. 2. With equal pressure and similar surfaces, friction increases as dimen- sions of surfaces are increased. 3. A regular velocity has no considerable influence on friction ; if velocity is increased friction may be greater, but this depends on secondary or inci- dental causes, as generation of heat and resistance of the air. M. Morin’s experiments afford the principal available data for use. Though con- stancy of friction holds good for velocities not exceeding 15 or 16 feet per second yet, for greater velocities, resistance of friction appears, from experiments of M.’ Poiree, in 1851, to be diminished in same proportion as velocity is increased. 4. Similar substances excite a greater degree of friction than dissimilar. If pressures are light, the hardest bodies excite least friction. 5. In the choice of unguents, those of a viscous nature are best adapted for • rough or porous surfaces, as tar and tallow are suitable for surfaces of woods, i and oils best adapted for surfaces of metals. 6. A rolling motion produces much less friction than a sliding one. 7. Hard metals and woods have less friction than soft. 8. Without unguents or .lubrication, and within the limits of 33 lbs. press- ure per sq. inch, the friction of hard metals upon each other may be esti- mated generally at about one sixth the pressure. 9. Within limits of abrasion friction of metals is nearly alike. 10. With greatly increased pressures friction increases in a very sensible ratio, being greatest with steel or cast iron, and least with brass or wrought iron. 11. With woods and metals, without lubrication, velocity has very little influence in augmenting friction, except under peculiar circumstances. 1 2. When no unguent is interposed, the amount of the friction is, in every case, independent of extent of surfaces of contact ; so that, the force with which two surfaces are pressed together being the same, their friction is the same, whatever may be the extent of their surfaces of contact. 13. Friction of a body sliding upon another will be the same, whether the body moves upon its face or upon its edge. Hr FRICTION. 470 14. When fibres of materials cross each other, friction is less than when they run in the same direction. 15. Friction is greater between surfaces of the same character than be- tween those of different characters. # ... 36 'With hard substances, and within limits of abrasion, friction is as pressure, without regard to surfaces, time, or velocity. 17 The influence of duration of contact (friction of rest) varies with the nature of substances; thus, with hard bodies resting upon each other, the £ a maximum very quickly ; with soft bodies, very slowly ; with wood upon wood, the limit is attained in a few minutes 5 and with metal on wood, the greatest effect is not attained for some days. Coefficient of Friction of Journals. Diameters from 2 to 4 ins. Speeds varied as 1 to 4. Pressure up to 2 tons. (From data of M. Morin.) Coefficient * Surfaces of Contact. Journals. Cast iron on cast iron. Bearings. Cast iron on gun metal Cast iron on lignum-vitse. . . . . Wrought iron on cast iron . . Wrought iron on gun metal. Wrought iron on lignum-vitse. Gun metal on gun metal Lignum-vitse on cast iron Lubrication. ( Olive oil, or tallow. . (Unctuous and wet. . ( Olive oil, or tallow. . (Unctuous and wet.. pressure = Ordinary Lubrication. (Slightly unctuous . {Oil, ( or lard. (Lard and plumbago. Olive oil, or tallow . . . ! Olive oil, or tallow . . Unctuous and wet.. Slightly unctuous. . . ( Oil. . . .* (Unctuous Oil Unctuous.. .07 to .08 .14 .07 to .08 .16 .18 .14 .07 to .08 .07 to .08 .19 •25 * Continuous lubrication reduces the coefficients fully one half. Surfaces of Contact. Oak on oak Wrought iron on oak . Cast iron on oak. . Leather on oak Leather belt on oak (flat). . . u “on oak pulley. Disposition of Fibres and Lubrication. Parallel and soaped “ wet it “ soaped 11 “ wet u “ soaped “ “ wet u “ dry Perpendicular u “ .47 of pressure, and over turned cast-iron pulleys Coefficient pressure = i. .16 .26 .22 .19 .29 .27 •47 Leather belts over wood drums .28 of pressure. Coefficients of Friction of Motion. Condition of Surfaces and Unguents. Hemp cords, etc. . Metal upon wood. Wood upon wood . P. id >>-g r % | 0 02 >» ■2 9 •g s S- -3 0 g Q O H Q On wood •45 •33 — — — — — On iron. — •15 — .19 Mean . . . .18 • 3 i .07 .09 .09 .2 •13 ; Raw •54 • 3 6 . 16 — .2 l Dry Mean . . . •34 .42 • 3 1 .24 .14 .06 -07 .14 .08 .2 .14 1 .36 •25 — 1 -°7 .07 •15 .12 FRICTION. 471 Relative Value of TJnguents to Reduce Friction. Unguents. Wood upon Wood. Wood upon Metals. Metals upon Metals. Unguents. Wood upon W ood. Wood upon Metals. Metals upon Metals. Dry soap •4 .82 •32 .85 .67 .27 •7 1 - 9 6 Olive oil Tallow 1 1 •93 .24 1 .8 Lard and plumbago. Water .22 .18 To Determine Coefficient of Friction of Bodies. Place them upon a horizontal plane, attach a cord to them, and lead it in a direction parallel to the plane over a pulley, and suspend from it a scale in which weights are to be placed until body moves. Then weight that moves the body is numerator, and weight of body moved is denominator of a fraction, which represents coefficient required. Illustration.— If, by a pressure of 320 lbs. friction amounts to 80 lbs., its coeffi- cient of friction in this case would be 80 - 4 - 320 = . 25. Hence, if coefficient of friction of a wagon over a gravel road was . 25, and the load 8400 lbs., the power required to draw it would be 8400 X 25 = 2100 lbs. Coefficients of Axle Friction. ( M . Morin.) Condition of Surfaces and Unguents. Substances. Dry and a iittle Greasy. Greasy and wet with Water. Oil, Tallov In usual way. > r , or Lard. Continu- ously. Very soft and puri- fied Car- riage Grease. Bell met ,1 upon bell metal .097 Cast iron upon bell metal .194 .161 •075 •054 .065 Cast iron upon cast iron .079 •075 •054 Cast iron upon lignum -vitse .185 . 1 .092 .109 Wrought iron upon bell metal .251 .189 •075 •054 .09 Wrought iron upon cast iron •075 •054 Wrought iron upon lignum-vitae 1 188 .125 Friction of a journal of an axle which presses on one side only, as in a worn bearing, is less than when it presses at all points, the difference being about .005. Friction of Axles. — With axles, friction of motion has alone been experi- mented upon. When weight upon axle and radius of its journal is given, mechanical effect of friction may be readily determined. The mechanical effect absorbed by, or of friction, increases with pressure or weight upon journal of axle and number of revolutions. Friction of an axle is greater the deeper it lies in its bearing. If journal of an axle lies in a prismatic bearing, as in a triangle, etc., friction is greater, as there is more pressure on, and consequently greater friction in contact: in a triangular bearing it is about double that of a cyl- indrical bearing. To Compute Mechanical Effect of Friction, on Journal of an Axle. — f. n representing number of revolutions , and r radius of journal . , 30 in feet. Illustration.— Weight of a wheel, with its axle or shaft resting on its journals, is 360 lbs. ; diameter of Journals 2 ins. ; and number of revolutions 30 ; what is me- chanical effect of the friction, the coefficient of it being .16? 3,1416 X 30 X X 360 X * _ 452.4 _ o8 lbc '30 30 FRICTION. 472 Bv amplication of friction-wheels (rollers) friction is much reduced and mechamcal effect then becomes, when weights of friction-wheels are disre- garded, p nf W r r ' __ p r f representing radii of axles of friction-wheels, a’ radii of frMioTwltls, and a angle of lines of direction between axis of roller and axis of friction-wheels. !J)r „ _ F = F , r When a single friction-wheel is used, ^ X/W — F, an r > and its shaft, making 5 revolutions per 2 . wSls ^chS^^^ ** use of the friction- wheel? x ; 32-^ 2 Xi2 _^ ^ circum. of wheel = z 8.4 times of axle. Coefficient of friction assumed at .075. Hence — 3 g* 7 - = 5 8 -59 lbs.= power 2 10 x 3-Uij „ 2 618 feet = distance passed at circum. to overcome friction at axle by friction. Consequently, 2 618 X _1 - .2181 feet = distance passed by friction in one second. Hence, .2181 X 2250 (30000 X .075) = 49 °J 2 5 - 3 - * faction refer red to circum. axle+by radius of friction-wheel, and 38. 4 X • 2 = 7 - 68 -F xctlon re J ei rea of wheel, and « 98.145 = mechanical effect by application of friction-wheel = a reduction of four fifths. Friction of Pivots. Friction on Pivots is independent of their velocity, inches m a degree than their pressures, and approximates very nea S and axle friction. Friction on Conical Bearings is greater than with like elements on plan ^Figure of point of a pivot, as to its acuteness, affects friction : with great pressure the most advantageous angle for the figure ranges from 30 to 45 , with less pressure it may be reduced to 10 ana 12 . Relative Value of Angles of Rivots. 6 o x | 1 5 0 66 | 45 ° 39 Relative Values of different Materials for use as Pivots. 83 I Granite i I Tempered steel 44 55 | Rock crystal 7 6 I Friction and. Rigidity of Cordage. Experiments by Amonton and Coulomb, with an apparatus of Amontons, furnish the following deductions : ... . i. That resistance caused by stiffness of cords about the same or like pul- leys varies directly as the suspended weight. . 2 That resistance caused by stiffness of cords increases not only in direct propor^n of suspended weights, but also in direct proportion of dtameter of the cords. Agate. Glass . FRICTION. 473 Consequently, that resistance to motion over the same or like pulleys, arising from stiffness of cords, is in direct compound proportion of suspend- ed weight and diameter of cords. 3. That resistance to bending varied inversely as diameter of sheave or drum. g I C t 4. That complete resistance is represented by expression — - — . S rep- resenting constant for each rope and sheave , expressing stiffness of rope ; T tension of rope which is being bent , expressed by C T ; C constant for each rope and sheave ; and d diameter of sheave , including diameter of rope. 5. That stiffness of tarred ropes is sensibly greater than that of white ropes. Extending results obtained by Coulomb, Morin furnishes following for- mulas : For White Ropes: 12 w-f-d (.002 15 -f-.ooi 77 n-\-. 0012 W) = R. For Tarred Ropes : 12 ra-r- d (.010 54 -j- .0025 n -J- .0014 W) — R. R representing rigidity in lbs., n number of yarns, d diameter of sheave in ins. and rope combined , and W weight in lbs. Illustration. — W liat is value of stiffness or resistance of a dry white rope hav- ing a diameter of 60 yarns, which runs over a sheave 6 ins. in diameter in the groove, with an attached weight of 1000 lbs. ? Assume diameter for 60 yarns to be 1.2 ins. Then 12 x (.002 15 -f-.ooi 77 x 7 * ^ 60 -f- .0012 X 1000) = 100 X 1*308 35 = 130.835 lbs. Value of natural stiffness of ropes increases as the square of number of threads nearly, and value of stiffness proportional to tension is directly as number of threads, being a constant number. Hence, having the rigidity for any number of threads, the rigidity for a greater or lesser number is readily ascertained. Wire Ropes. W eisbach deduced from his experiments on wire ropes that their rigidity for diameters capable of supporting equal strains with hemp ropes is con- siderably less. Wire ropes, newly tarred or greased, have about 40 per cent, less rigidity than untarred ropes. Rolling Friction. Rolling Friction increases with pressure, and is inversely as diameter of rolling body. For rolling upon compressed wood,./ = .019 to .031. When a Body is moved upon Rollers and Power applied at the Base of the Body , (/+/') — = F. / andf' representing coefficients of friction of two surfaces upon which rollers act. When Power is applied at Circumference of Roller, f W — r =. F. When Power is applied at Axis of Roller, / W -4- r -r- 2 := F. Bearings for Bropeller Shaft. (Mr. John Penn.) Bearings. Pressure per Sq. Inch. Time of Op- eration. Bearings. Pressure per Sq. Inch. Time of Op- eration. Babbit’s metal on iron*. . . Lbs. 1600 Min. 8 Brass on ironf Lbs. 675 4480 4000 4000 1250 Min. 60 Box on brass 4480 448 448 448 5 Brass on iron J Box on iron QO Lignum-vitse on brass . . Snake-wood on brass . . . Lignum-vitse on iron . . . Brass on brass QO 5 Brass on iron 3° 5 2160 * Rolled out. f Abraded. i Set fast. R R* FRICTION. 474 Result Of Experiments upon Friction of Several Instru- rn.en.ts. (iv- o. Hail. ) Instrument. Friction. Velocity ratio. Mechanical 1 efficiency. | Useful effect. F L 2.21 +.5453 2 1.8 Per Cent. V 64 3 6 2.36 +.238 3.87 -f“ • I5 1 .0 +.014 6 4 a 3 sheaves. .•••••••••••• 44 differential • . 16 6.1 193 70 3 6 Inclined plane, angle 17 0 2'. . . . .09 +.55 .66 -f.007 3-4 414 1.72 116 5i 28 WViof»l and Axlfi • • .204 + .043 •5 + -i 6 9 2.46 +. 21 .0 3 1 22 70 u u Barrel 5-95 5-55 93 u u Pinion 8 4.1 23 18 1 87 ^ r ? t . 185 -j-. 008 137 03 F representing friction, and L load. Illustration i.-If it is required to ascertain powCT necessary to raise aoojba 2 feet, by a single movable pulley, 200 X .5453 + *•"— ' Hence, for appli- t&btl&bs. m Hencet?o P r P applicaUon of I5 6. 9 6 lbs., roo or 63.7. per cent, are effectively employed. re ejjtciiutby , _ T he velocity ratio of a crane being 137, and its mechanical efficienc y 8 ?’ a man applying 26 lbs. to it can raise 87 X 26 = 2262 lbs. Application of preceding Results. illustration. - If a vessel, including beTe iron rollers, running on cast-iron rails? 1000 X 10 _ iqo fons =power required to draw vessel independent of faction. Ratio of friction to pressure of wrought iron on cast, in an axle and bearing, o„ Ratio of ditto of cast iron upon cast, say .005. ' Hence .075 + .005 = .08 of 1000 tons = 80 tons, which, added to 100 tons before de- ducted, gives 180 tons, or resistance to be overcome. Power or effect lost by friction in axles and their bearing may be ex- pressed by formula W/dr = p / represen ti„g coefficient of friction, d diameter of axle in ins. , and 23 ° . . r number of revolutions per minute. 8 ins. ; what is the effect ot friction ? 20 000 X .07 X 8X _20 ‘_ 973 qi i bS ' 2 3° Hence P u 4- 33000 = H>. v representing circumference of shaft in feet x by revo ■ , lutions per minute. The power or effect lost by friction in guides or slides may be expressed by following formula : W/s r ... _. p s representing stroke of cross-head, and l length of con- 6oXV(5 i 2 -* 2 )' necting rod in feet. FRICTION. 475 Frictional Resistances. Friction of* Steam-engines. Friction of Condensing Engines in Lbs. per Sq. Inch, of Piston. Diameter of Cylinder. Oscillating and Trunk. Beam and Geared. Direct- acting and Vertical. Diameter of Cylinder. Oscillating and Trunk. Beam and Geared. Direct- acting and Vertical. 10 5 6 7 50 2-5 2.7 3-3 15 4 5 6 60 2.4 2.6 3 20 3-5 4 5 70 2-3 2.5 2.7 25 3 3-6 4-5 80 2 2.3 2.6 30 •3 3-5 4 100 1.6 2.2 2-5 35 2.6 3 3-5 IIO i-5 2 2.1 Experiments upon different steam-engines have determined that friction, when pressure on piston is about 12 lbs. per sq. inch, does not exceed 1.5 lbs., or about one tenth of power exerted. Friction of double cylinder (50-inch diam.) direct-acting condensing pro- peller engine is 1.25 lbs. per sq. inch of piston = 10.3 per cent, of total power developed ; friction of load is .9 lbs. per sq. inch of piston = 7.5 per cent, of total pressure; and friction of propeller is 1.3 lbs. per sq. inch of piston = 10.8 per cent, of total power = 28.6 per cent. Friction of double cylinder (70-inch diam.) inclined condensing water- wheel engine with its load is 15 per cent, of total power developed. In general, when engines are in good order, their efficiency ranges from 80 per cent, for small engines to 93 per cent, for large. Power required to work air-pumps is 5 per cent., and to work feed-pumps 1 per cent. Results of Experiments upon Friction, of Machinery. (Davison . ) Steam-engine, vertical beam, one tenth its power ; 190 feet horizontal, and 180 feet vertical shafting, with 34 bearings, having an area of 3300 sq. ins., with 11 pair of spur and bevel wheels ; 7.65 IP. Set of three-throw Pumps, 6 ins. in diam., delivering 5000 gallons per hour at an elevation of 165 feet; 4.7 EP, or about 13 per cent. Two pair iron Rollers and an elevator, grinding and raising 320 bushels malt per hour ; 8.5 IP. Ale-mashing Machine, 800 bushels malt at a time ; 5.68 IP. Archimedes Screw (ninety-five feet), 15 ins. in diameter, and an elevator conveying 320 bushels malt per hour to a height of 65 feet ; 3.13 IP. Friction Clutch. — Driven by a leather belt 14 ins. in width ; face of clutch 5 ins. deep ; broke a cast-iron shaft 6.5 ins. in diameter. Flax Mill (M. Cornut, 1872). — Two condensing engines, cylinders, 12.9 ins. x 44.3 ins. stroke, and 22 ins. X 59.8 ins. stroke. Pressure of steam, 50 lbs. per sq. inch ; revolutions, 25 per minute. Friction of entire machin- ery, 20 per cent. With vegetable oil and hand oiling a steam pressure of 62 lbs. per sq. inch was required, and with mineral oil and continuous oiling a pressure of 50 lbs. only was required. By continuous oiling, a saving of 44 per cent, was effected over hand oiling. 476 FRICTION. ZETax Mill. Power required to Drive Engine, Shafting, and. entire Machinery. {M. Cornut.) Indicated Horse-power. One Machine Parts. Engines, shafting, and belts 4 cards •••••••, ; 14 drawing frames (29 heads or 156 slivers) 4 combing machines. 6 roving frames (330 spindles) 20 spinning frames. Dry (1480 spindles) Wet (2080 “ ) Total 1 50. 1 1 IP. Total. at work. empty. Machines. 30.41 — — — 8.42 2.105 1.423 32 7.19 •093 4 .0794 15 2.22 •555 .151 78 7.78 .026 27* 2-434 . 7-3 47-5 .032 1* 2.515 21.6 46-59 .022 4* 1.613 19 Effect of * Per 100 spindles. Estimate of Horse's Power— 2080 spindles, wet, 34.4 per IP, long fibre. 640 “ dry, 20.1 J40 “ “ 2±5 “ “ iow - 3560 “ average, 23.7 “ “ The IP per 100 spindles varies inversely as sq. root of their number. Winding Engine ( G . H. Daglish). Shafts 7^8 to 1740 feet in depth ; cylinder 65 X 84 ins. stroke ; pressure^ of steam, Ibsper^sq. inch; revolutions 12.5 per minute; mean diamete ) of di um , 26 feet. E> 313.4; effect 235 = 75 P er cent. Tools. {Dr. Hartig). Single shearing, . . + ^ = IP *> drive tool, n representing number of cuts per minute, t thickness of plate, and H> to shear, a representing v/m-oo - - 1980000 area of surface cut or punched per hour in sq. and F(ii66 + 1691 1 ) a factor ex- 2>ressing work required to cut or shear a surface of 1 inch squat e. Illustration. — A shearing machine cutting 4648 sq. ins. of surface per hour, in plates 4 inch thick, required .68 H> to run and 4.3 to operate it, equal to 5 horses. Iron Plate-bending 85 000 b t 2 l . 11 300 b t 2 l — P for cold plates , and _ p f or r ed-hot plates. b, t, and l representing breadth , thickness , and length of plate, r radius of curvature, all in ins., and P net power of bending. Power for large rolls when running only .5 to 6 IP. Ordinary Cutting Tools, in jVIeta. 1 . Materials of a brittle nature, as cast iron, are reduced most economically in power consumed by heavy cuts; while materials which yield tough curling sha 1 ® more economically reduced by thinner cuttings. Following formulas apply to g cutting work: Power required to plane cast iron is— t I> lallillg Cast iron, W l mss + £L^=km W representing weight of cast iron removed per hour, in lbs , and s average sectional area of shavings, in sq. ins. Steel, Wrought iron, and Gun-metal, with cuts of an average character- stoel 112 W = EP | Wrought iron, .o52W=IP | Gun-metal, .0127 IP N Planing and Molding. -Run without cutting. — = I?- N re P' resenting sum of revolutions of all the shafts per minute. FRICTION. 477 Molding. — Pine, .0566 -f — , and Red Beech, 088 95 -j- ■ 00 J 31 = IP. h rep- resenting depth of wood cut down to form molding. Turning. — Steel, .047 W = EP; Wrought iron, .0327 W = 2 *; Cast iron, .0314 W = BP. For turning off metals, power required is less than for planing, and it is ascer- tained that greater power is required for small diameters than large. Light Lathes , .05 -f- .0005 EP; 1 or 2 shafts, .05 -f- .0012 n = IP; 3 or 4 shafts, .05 -f- .05 n = EP. Heavy Lathes, ,025 -J- .0031 n; .025 -|- .053 w; .025-]-. 18 n. n representing number of revolutions of spindle per minute. Drilling.— Power required to remove a given weight of metal is greater than in planing. Volume being taken in place of weight. Holes from .4 to 2 ins. in diameter. Cast iron, dry. V ^.0168 -f : °°° 67 ^=IP. Wrought iron, oil. V ^.0168 + '-^^= IP. V representing volume removed in cube ins. per hour , and d diameter of hole. Without gearing, .0006 n-j-.ooos n'\ with gearing, .0006 w-f-.ooi n'; radial drills without gearing, .0006 n + .004 n'; radial drills with gearing, .04 -j- .0006 n -f- .004 n'. n representing number of revolutions per minute of gearing shaft , and n' of drill. Slotting. — Stroke 8 ins. .045 -j — IP. n representing number of strokes i per minute , and s stroke in ins. "Wood-sawing, Circular — A cube foot of soft wood and half a cube [ foot of hard, reduced to sawdust, requires 1 IP. Hard wood, = BP'. Soft wood, — — = BP'. A representing area in sq. feet and EP' horse-power per sq foot, both cut per hour , and c width of cut in ins. From .4 to 4 ins. in diameter.— Pine. V ^ 000 125 -f- - ° Q ^ = EP. Sc ' Dry pine timber. .004 28 -{-.0065 — = IP'. S representing stroke of saw in feet, and f feed per cut in ins. = H? for horse power to run only without cutting, d representing diameter of saw in ins. , and n number of revolutions per minute. Net power required to cut with a circular saw is proportional to volume of ma- terial removed. For a saw cutting hot iron, at a circumferential speed of 7875 feet per minute, and making a cut .14 inch wide, power is expressed by formulas .702 A = EP, for red-hot iron. 1.013 A:=EP, for red-hot steel. * A representing sectional area of surface cut through , in sq. feet. Vertical Saw. .004 28 + .0065 ~=IP in dry pine timber per sq. foot per hour. S representing stroke of saw in feet, c width of cut in ins., and f feed of cut in ins. Band Saw. 0034 + — EP'm Pine. .00483+ '-^-^ 4 = IP 'inOak. 10000/ T J 10000 / • 005 76 + ' ioooo f ~ IP ' in Beech, v representing velocity of saw, and f rate of feed, in feet per minute. Screw Catting. Screws, = IP- Taps, — = IP. d representing . . . , , 04 '29 0 diameter m ins. , and l length cut in feet per hour. 1 Machine of medium dimensions, .2 IP. 478 FRICTION. arindstones. pC - = IP. p representing pressure upon stone , v circum- ferential velocity of stone iTfeet per minute , and C coefficient of friction. Coefficients of Friction between Grindstones and Metals . Cast iron, .22 at high speed, .72 at low speed; Wrought iron, .44 at high speed, j at low; Steel, .29 at high speed, .94 at low. Power required to run them alone. Small. . Large 000 040 gdv = W or 000 128 d 2 n t= IP .16 + .000 089 5 d v — IP .16-]-. 000 28 d 2 n = IP Grrain Conveyers. , Mg tssa? «? 55?5FBJ& , Srjp.* A™. !fper Tent 0 ?Sof s 4 crew P At speeds above 60 turns per minute, the gram will not advance, but will revolve with screw. Sectional area of body of grain moved Dove 60 turns per minute, the g " Screw Steamer. {Vice-admiral C. R. Moorsom, E. N.) Moving friction of hull 07 Moving friction of load 003 Moving friction of rotation of) . . .09 blades of screw ) Slip of screw * 7 * Resistance of hull boo Side Lever Steam-engine. (J. V. Merrick.) In Pressure of Steam. Friction to work air-pump Friction of weight of parts Friction of cylinder packing £1^0^ .585 to .7 lb. •5 “ ’ 5 a • I5 , •3 .046 '° 9 l , .169 u .258 “ i -45 1.85 lbs . Hencc = ,.65 lbs. per sq. inch. If journals are kept constantly lubri- 1.45 + 1.85 . cated, as with automatic lubrications, friction of weight ^1 .^reducedto .*h«jd engln^without. 6 load^^Frfction "of ^ioadf according as journals are lubricated, ends ; keyed up, etc., will range from 2 to 5 per cent. Locomotives and Railway Trains. See Railways, page 682. Friction developed in Launching of Vessels. °“ne Railway.- To draw 3000 tons upon greased slides a power of 250 tons was no" toStovS it, but when started i 5 o tons would draw ,t. Woollen Machinery. (Dr. Hartig.) When runn.ng empty 8.15 IIP, and at work ^The efficiency of the various machines averaging 60.5 per cent. Friction of a Non-condensing Steam-engine. Friction of an Engine. Diameter of cylinder 20 ins. by 40 ins. stroke of p.ston Revolutions, 15 to 70 per minute. — T 86 to 8.60 IP. Fngine, unloaded, 2 lbs. per sq. inch — . •• _ ‘ 6 t0 IQ ( )1 “ Shafting, unloaded, 2.5 to 45 lbs. per sq. inch — ^ 9 - « Total 4. 5 to 6.5 lbs. per sq. inch - 4 22 3 FUEL. 479 FUEL. With equal weights, where each kind is exposed under like advan- tageous circumstances, that which contains most hydrogen ought, in its combustion, to produce greatest volume of flame. Thus, pine wood is preferable to hard, and bituminous to anthracite coal. When wood is used as a fuel, it should be as dry as practicable. To produce greatest quantity of heat, it should be dried by direct ap- plication of heat ; usually it has about 25 per cent, of water combined with it, heat necessary for evaporation of which is lost. Different fuels require different volumes of oxygen ; for different kinds of coal it varies from 1.87 to 3 lbs. for each lb. of coal. 60 cube feet of air is necessary to furnish 1 lb. of oxygen ; and, making a due allowance for loss, nearly go cube feet of air are required in furnace of a boiler for each lb. of oxygen applied to combustion. Lignite. Brown Coal or Bituminous Wood. — Presents a distinct woody structure ; is brittle, and burns readily, leaving a white ash, and contains and absorbs moisture in some cases fully 40 per cent. Caking. — Fractures uneven, and when heated breaks into small pieces, which afterwards agglomerate and form a compact body. When the pro- portion of bitumen is great, it fuses into a pasty mass. This coal is unsuit- ed where great heat is required, as the draught of a furnace is impeded by its caking. It is applicable for production of gas and coke. Splint or Hard. — Color black or brown-black, lustre resinous and glisten- ing. It kindles less readily than caking coal, but when ignited produces a clear and hot fire. Cherry or Soft. — Alike to splint coal in fracture, but its lustre is more splendent. Does not fuse when heated, is very brittle, ignites readily, and produces a bright fire with a yellow flame, but consumes rapidly. Cannel. — Color jet, or gray or brown-black, compact and even texture, a shining, resinous lustre. Fractures smooth or flat, conclioidal in. every di- rection, and polishes readily. Experiments upon practical burning of this description of coal in furnace of a steam-boiler give an evaporation of from 6 to 10 lbs. of fresh water, under a pressure of 30 lbs. per sq. inch per lb. of coal; Cumberland (Md., U. S.) coal being most ef- fective, and Scotch least. Limit of evaporation from 212 0 for 1 lb. of best coal, assuming all of heat evolved from it to be absorbed, would be 14.9 lbs. Coals that contain sulphur, and are in progress of decay, are liable to spontaneous combustion. There are very great variations in the chemical composition and proper- ties of coals. Cannel. Shaly. Asphalt. Hard. Semi or gaseous. 13 it it m i o s Coal. American. Carbon, from 75 to 80 per cent. Hydrogen, from 5 to 6. Oxygen, from 4 to 10. Nitrogen, from 1 to 2. Sulphur, from .4 to 3. Ash, from 3 to 10. Carbon, from 70 to 91 per cent. Hydrogen, from 3. 5 to nearly 7. Oxygen, from about .5 to 20. Nitrogen, from a mere trace to 2.2. Sulphur, from o to 5. Ash, from .2 to 15. Coke, from 49 to 93. British. Coke, from 48.5 to 79.5. For Volume of Air, etc., see Combustion, page 465. 480 FUEL. Coal. Anthracite. Anthracite or Glance Coal , or Culm — Is hard, compact, lustrous, and some- times iridescent, most perfect being entirely free from bitumen; it ignites with difficulty, and breaks into fragments when heated. Evaporative power, in furnace of a steam-boiler and under pressure, is from 7.5 to 9.5 lbs. of fresh water per lb. of coal. Coal from one pit will sometimes vary 6 per cent, in evaporative value. Elements of Various American Coals. Illinois, Warren Co Bureau “ Mercer “ Indiana, Clay “ Coopriders Pennsyl* \ Connellsville. . . vania j Youghiogheny . Fayette Co Kentucky, Sardric Mud River Ohio, Nelsonville Colorado, Carbon City Washington Territory Specific Gravity. 1.23 1.32 1.26 1.28 1.28 1.28 i-3 1.29 1.32 1.28 1.27 1.21 1.32 Fixed I Volatile j Carbon. I Matter. | Moist- ure. Earthy Matter. Per Cent. 5 i -7 57-6 54-8 56.5 50.5 65 58.4 58 5i 57 58.4 56.8 58.25 Per Cent. 43-1 28.8 31.2 3 2 - 5 42.5 24 35 34 4 2 -5 37 33 - 05 34 - 2 31-75 8-5 3 4-5 3- 5 6.65 4- 5 7 Per Cent. II. 2 8.4 Per Cent. 2.4 5-6 2-5 4 6.5 5-6 5 4-5 2-5 1.9 4-5 3 Per Cent. 5-2 Coke. Coke . — Coking in a close oven will give an increase of yield of 40 per cent, over coking in heaps, gain in bulk being 22 per cent. Coals when coked in heaps will lose in bulk. Cannel and Welsh (Cardiff) coals when coked in retorts will gain from 10 to 30 per cent, in bulk and lose 36.5 per cent, in weight. Relative costs of coal and coke for like results, as developed by an ex- periment in a locomotive boiler, are as 1 to 2.4. Evaporative power in furnace of a steam-boiler and under pressure, is from 7.5 to 8.5 lbs. of fresh water per lb. Bituminous coal will vield from 60 to 80 per cent, of coke. Averaging . 66 per cent. It is capable of absorbing 15 to 20 per cent, of moisture. Heat of combustion lost in coking of bituminous coal 40 per cent. Charcoal. Charcoal properly termed, is not made below a temperature of 536°. The best quality is made from Oak, Maple, Beech, and Chestnut. Wood will furnish, when properly burned, about 23 per cent, of coal. Charcoal absorbs, upon an average of the various kinds, from .8 P er c ^t* » of water for Beech, to 16.3 for Black Poplar, Oak absorbing about 4.28, and Pine 8.9. . „ Evaporative power, in furnace of a boiler and under pressure, is 5.5 los.^ of fresh water per lb. of coal. Volume of air chemically required for combustion of 1 lb. of charcoal is, when it consists of 79 carbon, 129 cube feet at 62°. 138 bushels charcoal and 432 lbs. limestone, with 2612 lbs. of ore, will pro- duce 1 ton of pig iron. FUEL. 481 Produce of Charcoal from Various Woods dried at 300° and Carbonized at 572 0 . {M. Violette.) Wood. Weight. Wood. Weight. Cork Per Cent. 62.8 46.09 44-25 41.48 40.9 Larch Per Cent. 40.31 36.06 34-69 34-59 34-17 Oak Chestnut .... Beech Apple .... Pine Elm Birch Poplar roots Poplar 31.12 per cent. In a Green or Ordinary State. {Weight per cent.) Wood. Weight. Maple Per Cent. *2 7. 7 S Willow DD* / D oo. n a Black elder. . . . . Ash DO* /4 33 - 61 33-28 31.88 Pear 23-8 1 I Birch 1 Oak . 22.85 I Red Pine . . , 2 6-7 Elm I “ young.. • 33-3 White Pine . 21. 1 | 1 Maple 1 Poplar . 20. 5 1 Willow Apple . . , Ash Beech. . . It appears from this that cork, the lightest of woods, yields largest percentage of charcoal, about 63 per cent. ; and that poplar yields lowest, about 31 per cent. There does not appear to be any definite relation between density of wood and volume of yield. J Produce by a slow process of charring is very nearly 50 per cent, greater than bv a quick process. 3 * & y Lignite. Lignite is an imperfect mineral coal. It is distinguished from coal by its large proportion of oxygen, being from 13 to 29 per cent. Its specific gravity ranges from 1.12 to 1.35. 1 Elements of Various American Lignites. {W. M. Barr.) Location. Kentucky Blandville . Washington Terr’y . . Vancouver’s Island. . Colorado, Carbon City Canon City Arkansas Texas, Robertson Co. Spec. Grav. 1. 17 1.27 1.28 1.23 Per Cent. 40 3i S8.25 62 4I-25 56.8 34-5 45 Per Cent. 23 48 3i-75 3i 46 34-2 28.5 39-5 Water. Per Cent. 30 H-5 7 4 3' 5 4-5 32 Per Cent. 7 9-5 3 3 9- 2 5 4'5 5 4-5 Per Cent. 53 59-5 38.75 35 49- 5 38.7 60.5 50- 5 Per C 47 40. i 61.5 65 50.= 61.3 39 - } 49-' Asplialt. Asphalt , alike to Lignite, contains a large proportion of oxygen. "Wood. Wood , as a combustible, is divided into two classes, the hard, as Oak, Ash Llm Beech, Maple, and Hickory, and soft, as Pine, Cotton, Birch, Sycamore and Chestnut. J Green wood subjected to a temperature ranging from 340° to 440 0 wil] lose 30 to 45 per cent, of its weight. 44 At a temperature of 300°, Oak, Asli, Elm, and Walnut, in a comparatively seasoned state, lost from 16 to 18 per cent. ^ Woods contain an average of 56 per cent, of combustible matter. an ? na M s ? f M * Violette it appears that composition of wood is about Qflmn throu » h . out tree ^ tha t of the bark also ; that wood and bark have about same proportion of carbon (49 per cent.), but that bark has more ash than wood 5 and7 per cem T °°^ haVG ^ carbon than W00(i (45 per cent.), and more ash| Leaves when dried at 212 0 lost 60 per cent, of water, and branches 45 per cent. S s FUEL. 482 Evaporative power of 1 cube foot of pine wood is equal to that of 1 cube foot of fresh water ; or, in the furnace of a steam-boiler and under pressure, it is 4.75 lbs. fresh water for 1 lb. of wood. Northern Wood.— One cord of hard wood and one cord of soft wood, such as is used upon Lakes Ontario and Erie, is equal in evaporative effects to 2000 lbs. of anthracite coal. Western Wood. — One cord of the description used by the river steamboats is equal in evaporative qualities to 12 bushels (960 lbs.) of Pittsburgh coal. 9 cords cotton, ash, and cypress wood are equal to 7 cords of yellow pme. Solid portion (lignin) of all woods, wherever and under whatever circum- stances of growth, are nearly similar, specific gravity being as 1.46 to 1.53. Densest woods give greatest heat, as charcoal produces greater heat than ^For every 14 parts of an ordinary pile of wood there are n parts of space; or a cord of wood in pile has 71.68 feet of solid wood and 56.32 feet of voids. Trees in the early part of April contain 20 per cent, more water than they do in the end of January. Ash. Woods. Wood. Leaves. Woods. Wood, j Leaves. Per Cent. Per Cent. Elm Per Cent. 1.88 Per Cent. II. 8 •35 •34 5-4 5 Oak .21 4 Birch Pitch Pine •25 3*15 Feat. Peat is the organic matter, or soil, of bogs, swamps, and marshes— decayed moss, sedge, coarse grass, etc.— in beds varying from i to 40 feet in depth. That near the surface, and less advanced in transformation, is light, spongy , and fibrous, of reddish-brown color; lower down, it is more compact, of a darker brown color ; and, in lowest strata, it is of a blackish brown, or almost black, of a pitchy or unctuous surface, the fibrous texture nearly or alto- gether transformed. . A 0 , . . „ Peat in its natural condition, contains from 75 to 80 per cent, of water. Occasionally its constituent water amounts to 85 or 90 per cent., in which case peat is of the consistency of mire. It shrinks very much in drying; and its specific gravity varies from .22 to 1.06, surface peat being lightest, and deepest peat densest. ..... . x « . • • A „ r When peat is milled, so that its fibre is broken up, its contraction m dry- ing is much increased, and in this condition it is termed condensed. # When ordinarily air dried, it will contain 20 to 30 per cent, of moisture, and when effectively dried at least 15 per cent. Products of Distillation of Peat. Water 31.4. Tar 2.8. Gas 36.6. Charcoal 29.2. The distillation of the tar will yield paraffine, oil, gas, water, and char- coal, and the water acetic acid, wood spirit, and chloride of ammonia. Evaporative power, in furnace of a steam-boiler and under pressure, is from 3.5 to 5 lbs. of fresh water per lb. of fuel. Taix. Tan, oak or hemlock bark, after having been used in the process of tan- ning, is combustible as a fuel. It consists of the fibre °f bark, an , according to M. Peclet, 5 parts of bark produce 4 parts of dry tan; ana heating power of it when perfectly dry, or containing but 15 per cent, of ash, is 6100 units ; while that of tan in an ordinary state of dryness, con- taining 30 per cent, of water, is 4284. Weight of water evaporated at 212 by 1 lb., equivalent to these units, is 6.31 lbs. for dry, and 4.44 for moist. FUEL, 483 Relative "Valxies of* different Fuels. Description. Lbs. of Steam from Water at 212 0 by 1 lb. of Fuel. Relative Evapora- tive Power for equal Weights. Relative Evapora- tive Power for equal Volumes. Relative Rapidi- ties of Ignition. Relative Freedom from Waste. Relative. Com- pleteness of Combustion. Relative Weights. Anthracites. Peach Mountain, Pa 10.7 X I •505 •633 •725 •945 Beaver Meadow 9.88 •923 .982 .207 .748 .6 1 Bituminous. Newcastle 8.66 .809 .776 •595 .887 •346 • 9°4 Pictou 8.48 .792 •738 .588 .418 1 .876 Liverpool 7.84 •733 .663 .581 1 •333 .852 Cannelton, Ind 7-34 .686 .616 1 .984 •578 . 848 Scotch 6-95 .649 .625 .521 •499 .649 •9°9 Pine wood, dry 4.69 •436 •175 — 16.417 — — ■Weights, Evaporative Rowers per Weight and Bulk, etc., of* different Fuels. [W. R. Johnson and others.) Fuel. Specific Gravity. Weight per Cube Foot. Steam from Water at 2X2° by 1 lb. of Fuel. Clinker from 100 lbs. Cube Feet in a Ton. Bituminous. Lbs. Lbs. Lbs. No. Cumberland, maximum I- 3 I 3 52.92 10.7 . 2.13 42.3 “ minimum 1-337 54-29 9.44 4-53 41.2 Duffryn 1.326 53-22 10.14 — 42.09 Cannel, Wigan 1.23 48.3 7.7 — 46.37 Blossburgh 1.324 53-05 9.72 3-4 42.2 Midlothian, screened 1.283 45-72 8-94 3-33 49 “ average 1.294 54-04 8-39 8.82 41.4 Newcastle, Hartley 1-257 50.82 8.76 3 -i 4 44 Pictou i- 3 l8 49-25 8.41 6.13 45 Q Pittsburgh 1.252 46.81 8.2 •94 47.8 Sydney i -338 47-44 7-99 2.25 47.2 Carr’s Hartley 1.262 47.88 7.84 1.86 46.7 Clover Hill, Va 1.285 45-49 7.67 3.86 49.2 Cannelton, Ind 1-273 47- 6 5 7 - 34 1.64 47 0 Scotch, Dalkeith i- 5 i 9 51.09 7.08 5-63 43-8 Chili — — 5-72 — — Japan 1-231 48.3 — — Anthracite. Peach Mountain 1.464 53-79 10. 1 1 3-03 41.6 Forest Improvement i -477 53-66 10.06 .81 41.7 Beaver Meadow i -554 56.19 9.88 .6 39-8 Lackawanna 1. 421 48.89 9-79 1.24 45-8 Beaver Meadow, No. 3 1. 61 54-93 9.21 1. 01 40.7 Lehigh i -59 55-32 8-93 1.08 40.5 Coke. Natural Virginia 1-323 46.64 8.47 5 - 3 i 48.3 Midlothian — 32-7 8.63 10.51 68.5 Cumberland — 31.6 8.99 3-55 70.9 Miscellaneous. Charcoal, Oak i -5 24 5-5 Ash. 3.06 104 Peat •53 30 5 — 75 Warlich’s fuel i- 15 6g.i xo.4 2.91 32-44 Wylam’s “ 65 8.9 . — - — Pine wood, dry — 21 4-7 • 3 i 106.6 484 FUEL. Weights and Comparative Values of different Woods. Woods. Shell-bark Hickory . . . Red-heart Hickory . . . White Oak Red Oak Virginia Pine Southern Pine Hard Maple Cord. Value. Woods. Cord. Value. Lbs. 4469 1 New Jersey Pine Lbs. 2137 •54 37°5 .81 Yellow Pine I9°4 •43 3821 .81 White Pine 1868 .42 3254 .69 Beech — • 7 2689 .61 Spruce — •52 3375 •73 Hemlock — •44 •33 2878 .6 Cottonwood Liquid. Fuels. Petroleum. Petroleum is a hydro-carbon liquid which is found in America and Europe. According to analysis of M. Sainte-Claire Deville, composition of 15 petro- leums from different sources was found to be practically constant. Average specific gravity was .87. Extreme and average elementary composition was as follows : Carbon 82 to 87. 1 per cent. Average, 84.7 per cent. Hydrogen 11.21014.8 “ “ 13.1 “ Oxygen 5 to 5.7 “ “ 2.2 “ 100 Its heat of combustion is 20240, and its evaporative power at 212 0 20.33. Petroleum Oils — Are obtained by distillation from petroleum, and are com- pounds of carbon and hydrogen, in average proportion of 72.6 and 27.4. Boiling-point ranges from 86° to 495 °. Schist Oil — Consists of carbon 80.3 parts, hydrogen 11.5, and oxygen 8.2. Pine Wood Oil — Consists of carbon 87.1 per cent., hydrogen 10.4, and oxygen 2.5. Coal-gas. Coal Gas — As furnished by Chartered Gas Co. of London is composed as follows ; Carbon. Hydrogen. Oxygen. Hydrogen. Nitrogen. Olefiant Gas, ) Bi-carb.hyd. ) * * Marsh gas, ) Carb. hyd. ( * • • • Carbonic oxide.... 3.096 •434 Hydrogen ..... Oxygen .08 51.8 - 26.445 8.815 Nitrogen .38 3-84 5-ii Total.. Heat of combustion at 212 0 52 961 units, and evaporative power 47.51 lbs. Coal-gas. (F Harcourt.) Carb. Hyd. Oxy. Nit. Carb. Hyd. Oxy. Nit. Per ct. Per ct. Per ct. Per ct. Per ct. Per ct. Per ct. Per ct. Olefiant gas 10.5 *•7 — — Hydrogen — ■ 8.1 — — Marsh gas 39-7 13.2 — — Nitrogen — — 5-8 Carbonic oxide. . 5-9 7-9 — Oxygen — — •3 Carbonic dioxide 1.9 — 5 — • Total 58 23 13.2 One lb. of this gas had a volume of 30 cube feet at 62° ; heat of combus- tion 22684 units; and of one cube foot 756 units, which is equivalent to evaporation of .68 lb. of water from 62°, or of .78 lb. from 212 0 per cube foot. FUEL. 485 Average C omp o s it i on. of Panels . Specific Grav- ity. Carbon. Hydro- gen. Nitro- gen. Oxygen. Sul- phur. Ash. Bituminous Coals. Per ct. Per ct. Per ct. Per ct. Per ct. Per ct. Australian X -3 X — — — — •5 8.38 Borneo 1.28 64.52 4-74 .8 20.75 i-45 7-74 British, lowest — 68.72 4.76 — 18.63 x-35 — Boghead, dry, average 1. 18 63-94 8.86 .96 4-7 •32 21.22 Chili, Conception Bay 1.29 70-55 5-76 •95 13.24 1.98 7-52 “ Chiriqui — 38.98 4.01 .58 13-38 6.14 36.91 Cannel, Wigan 1.23 79- 2 3 6.08 1. 18 7.24 x-43 4.84 Cumberland. Md 1.31 93.81 1.82 — 2-77 — 1.6 Coke, Garesfield — 97.6 — — — .85 i-55 “ Durham — 89-5 — — — 1.25 9-25 “ Average — 93-44 4.66 — — 1.22 5-34 Duffryn i-33 88.26 i-45 .6 1.77 3.26 Formosa Island 1.24 78.26 5-7 .64 10.95 •49 3-9 6 88.56 87-73 4.88 ( 4-38) ( 5-6 5 ) — 2. 19 “ caking 1.29 5.08 — i-54 “ long flame i-3 82.94 5-35 ( 8.63) — 3.08 “ average* i-3i 85 4-5 ( 7 ) — 3-5 Indian, average 47-3 — — I — — 22.9 “ Kotbec — 90 — — — — 4 Patagonia — 62.25 5-05 •63 1 17-54 x-i3 x 3-4 Russian, Miouchit — 9 I -45 4-5 ( 4-05) — ' — Sydney, S. W — 82.39 5-32 1.27 | I 8.32 .07 2.04 Splint, Wvlam il Glasgow — 74.82 6.18 ( 5.09) — I3-9 1 — 82.92 5-49 (10.46) — x* x 3 “ Cannel, Lancashire — 83-75 5-66 ( 8.04) — 2-55 ** “ Edinburgh — 67.6 5-4 ( 12.43) ■ — x 4-57 “ Cherry, Newcastle. — 84.85 5-o5 ( 8.43) — 1.67 “ Caking, Garesfield — 87-95 5-24 ( 5.42) — x-39 “ Ebbro Vale, Welsh “ Llangenneck u — 89.78 5-i5 2. 16 •39 1.02 x -5 — 84.97 4.26 i-45 3-5 .42 5-4 Vancouver’s Island — 66.93 5-32 if 02 8-7 2.2 x 5-83 Anthracites. Anthracite i-5 88.54 — — — •52 8.67 French i-5 86. 17 2.67 ( 2.85) — 8. 56 Russian 96.66 x-35 ( 1 99) — — Woods. Beech — 50.17 6.12 1.05 40.38 — 1.77 Birch — 48.12 6-37 1.15 43-95 . .48 Oak — 48.13 5-25 .82 44-5 . — x-3 White Pine — 49-95 6.41 — 43.65 — .31 Woods, average -r- 49-7 6.06 1.05 4 J -3 — 1.8 Charcoal. Oak — 87.68 2.83 — 6-43 — 3.06 Pine — 71-36 5-95 — 22.19$ — •4 Maple — 70.07 4.61 — 24.89$ — •43 Miscellaneous. Asphalt 1.06 79.18 9-3 ( 8.72) — 2.8 Lignite, perfect “ imperfect 1.29 69.02 5-05 ( 20 .12) • — 5.82 1.25 60. 18 5-29 (29.03) — ■ 5-57 “ bituminous 1. 18 74. 82 7-36 (13.38) — ■ 4-45 “ Colorado 1.28 56.8 — — — 4-5 “ Kentucky 1.2 40 — — — — 7 “ Arkansas — 34-5 — — -r- — 5 Peat, dense — 61.02 5-77 .81 32.4 — “ Irish, average .528 58.18 5-9 6 1.23 31.21 — 3-43 Patent, Warlich “ Wylam’s x-xS 90.02 5-56 — — 1.62 2.91S 1. 1 79-91 5-69 1.68 6.63 1.25 4.84 * Heat of Combustion of 1 Lb. 14 723. X Including Nitrogen. Ss* + Heat of Combustion of 1 Lb. 15 651. § Including Oxygen. 486 FUEL. Average Composition of Coals and. Fuels, Heat of Com- bustion, and Evaporative Power. Deduced from analysis and experiments of Messrs. De La Beche, Playfair , and Peclet. Coals an© Fuels. Specific Gravity. Carbon. Hydro- gen. Compos Nitro- gen. 5ITI0N. Sul- phur. Oxy- gen. Ash. Heat of Com- bustion of 1 lb. Evaporation from water at 212 0 . Per ct. Per ct. Per ct. Per ct. Per ct. Per ct. Units. Lbs. Derbyshire and) Yorkshire ) I.29 79.68 4.94 1.41 1. 01 0 M OO 2.65 13 860 14-34 Lancashire I.27 77-9 5-32 1-3 1.44 9-53 4.88 i39 l8 14.56 Newcastle 1.26 82.12 5-3i i-35 1.24 5-69 3-77 14 820 15-32 Scotch 1.26 78.53 5-6 i 1 1. 11 9.69 4-03 14 164 14-77 Welsh 1.32 83.78 4-79 .98 i-43 4-i5 4.91 14858 15-52 Average of British. 1.28 80.4 5-i9 1. 21 1.25 7.87 4-05 14320 14. 82 Patent fuels 1. 17 83-4 4 97 1.08 1.26 2.79 5.- 93. 15000 15.66 Van Diemen’s Land 65.8 3-5 i-3 1. 1 5-58 22.71 11 320 11.83 Chili — 63-56 5-43 .82 2-5 14.84 13-31 11 030 11.68 Lignite, Trinidad. . “ French Alps — 65.2 4-25 i-33 .69 21.69 6.84 10438 10.87 1.28 70. 02 5-2 — — — 3.01 11 790 12. 1 “ Bitum.,Cuba 1.2 75-85 7-25 — — . — 3-94 14562 14.96 “ Wash. Ter.*. — 67 4-55 — 1 — 3- 1 12538 12.91 Asphalt I.06 79.18 9-3 — — — 2.8 16655 17.24 Petroleum .87 84.7 I 3- 1 — — 2.2 — 20240 20.33 “ oils •75 — — — — 27 530 28.5 vOak bark Tan, dry. — — — — — i5 6 100 6.31 “ “ moist — — — — — i5 4284 4.24 "Charcoal at 302 0 . . . i-5 47-51 6.12 ( 0 and N 46.29 ) .8 8130 8.4 “572°... 1.4 73- 2 4 4-25 ( 0 and N 21.96) •57 11 861 12.27 “ “ 8io c . . . 1. 71 81.64 4.96 (0 and N 15.24 ) 1. 61 14916 15-43 Peat, dry, average. •53 58. 18 5-96 1.23 1 FT. 1 31-21 3-43 995i 10.3 “ moist, t “ 43- 1 4-3 ( 0 and N 21.4 ) 3-3 8917 9.22 Coal-gas 42 33-38 66.16 .38 1 - | .08 — 52961 47-51 * Water 7. Oxygen and Nitrogen 17.36. t Moisture 27.8. Sulphur .2. Elements of Enels not included in Preceding Tables. Heat of ' ~ Combustion Fuel. of 1 lb. Evaporative Power of 1 lb. at 212 0 . Weight ! of 1 I Cub. Foot. Volume of 1 Ton. Bituminous Coal. Welsh Newcastle Lancashire Scotch Boghead British, average. Irish, lowest Cumberland, Md American, average French, average Australian Anthracite. American French Miscellaneous. Units. 14858 14 820 13918 14 164 14478 14 133 14723 14038 Lbs. 9-05 8.01 7-94 7-7 7.87 8.13 9- 8 5 Per cent. 73 61 58 54 30-94 61 9° 83-7 82.5 64.2 68.27 94.82 88.83 Lbs. 82 78.3 79-4 78.6 79.8 99.6 8 4- 93 87-54 93- 78 Cube Feet. 42.7 45-3 45-2 42 44-52 35-7 42.4 43-49 40 42-35 Warlich’s fuel Coke Mickley Virginia, average Charcoal Lignite, perfect “ imperfect u Russian Asphalt Woods, dry, average 16495 15 600 13550 12325 11 678 9834 15837 16555 7792 14.02 12. 1 10. 18 17.24 8.07 47 37-5 9 73-5 45 34-5 80 69.8 12.76 114 FUEL. — GRAVITATION. 487 Miscellaneous. Experiments undertaken by Baltimore and Ohio R. R. Co. determined evaporating effect of 1 ton of Cumberland coal equal to 1.25 tons of anthra- cite, and 1 ton of anthracite to be equal to 1.75 cords of pine wood; also that 2000 lbs. of Lackawanna coal were equal to 4500 lbs. best pine wood. One lb. of anthracite coal in a cupola furnace will melt from 5 to 10 lbs. of cast iron ; 8 bushels bituminous coal in an air furnace will melt 1 ton of cast iron. Small coal produces about .75 effect of large coal of same description. Experiments by Messrs. Stevens, at Bordentown, N. J., gave following results: Under a pressure of 30 lbs., 1 lb. pine wood evaporated 3.5 to 4.75 lbs. of water. 1 lb. Lehigh coal, 7.25 to 8.75 lbs. Bituminous coal is 13 per cent, more effective than coke for equal weights; and in England effects are alike for equal costs. Radiation from Fuel.— Proportion which heat radiated from incandescent fuel bears to total heat of combustion is, From Wood 29 | From Charcoal and Peat 5 Least consumption of coal yet attained is 1.5 lbs. per IIP. It usually varies in different engines from 2 to 8 lbs. Volume of pine wood is about 5.5 times as great as its equivalent of bituminous coal. GRAVITATION. Gravity is an attraction common to all material substances, and they are affected by it directly, in exact proportion to their mass, and inversely, as square of their distance apart. This attraction is termed terrestrial gravity , and force with which a body is drawn toward centre of Earth is termed the weight of that body. Force of gravity differs a little at different latitudes : the law of variation, however, is not accurately ascertained ; but following theorems represent it very nearly : g (1 — .002 837 cos. 2 lat. ) 'j , representing force of gravity at lati- •tf |:±:£! $ ft ‘the equator j =* ,4 £ r, 32. 171 (lat. 45 0 ) (1 + .005 133 sin. L) ^1 — = £ Or, - g. L representing latitude , H height of elevation above level of sea, and R radius of Earth, both in feet. Note.— I f 2 L exceeds 90 0 , put cos. 180 — 2 L, and R at Equator = 20 926 062, at, Poles 20853429, and mean 20889746. Illustration. — What is force of gravity at latitude 45 0 , at an elevation of 209 feet, and radi us = 20 900 000 feet ? 32.171 (14-.005133 sin. 45 0 ) (x — 2o ~ ^ oq = 32.171 X 1.00363 X .99998 = 32.287. Gravity at Various Locations at Level of Sea. Equator 32.088 I New York 32.161 1 London 32.189 Washington 32.155 | Lat. 45 0 32-171 J Poles 32:253 In bodies descending freely by their own weight, their velocities are as times of their descent, and spaces passed through as square of the times. Times , then, being 1, 2, 3, 4, etc., Velocities will be 1, 2, 3, 4, etc. Spaces passed through will be as square of the velocities acquired at end of those times, as 1, 4, 9, 16, etc. ; and spaces for each time as 1, 3, 5, 7, 9, etc. GRAVITATION. 488 A body falling freely will descend through 16.0833 feet in first second of time, and will then have acquired a velocity which will carry it through 32.166 feet in next second. If a body descends in a curved line, it suffers no loss of velocity, and the curve of a cycloid is that of quickest descent. Motion of a falling body being uniformly accelerated by gravity, motion of a body projected vertically upwards is uniformly retarded in same manner. A body projected perpendicularly upwards with a velocity equal to that which it would have acquired by falling from any height, will ascend to the same height before it loses its velocity. Hence, a body projected up- wards is ascending for one half of time it is in motion, and descending the other half. Various Formulas here given are for Bodies Projected Upwards or Falling Freely , in Vacuo. When , however, weight of a body is great compared with its volume , and velocity of it is low , deductions given are sufficiently accurate for ordinary purposes. In considering action of gravitation on bodies not far distant from surface of the Earth, it is assumed, without sensible error, that the directions i a n which it acts are parallel, or perpendicular to the horizontal plane. A distance of one mile only produces a deviation from parallelism less than one minute, or the 60th part of a degree. Relation, of Time, Space, and. "Velocities. Time from Beginning of Descent. Velocity acquired at End of that Time. Squares of Time. Space fallen through in that Time. Spaces for this Time. Space fallen through in last Second of Fall. Seconds. Feet. Seconds. Feet. No. Feet. 1 32.166 1 16.083 1 16.08 2 64- 333 4 64-333 3 48.25 3 9 6 -5 9 144-75 5 80.41 4 128.665 16 257-33 7 112.58 5 160.832 25 402.08 9 144-75 6 193 36 579 11 176.91 7 225. 166 49 788.08 13 209.08 8 257-333 64 1029.33 i5 241.25 9 289.5 81 1302.75 17 273.42 10 321.666 100 1608.33 19 305-58 and in same manner this Table may be continued to any extent. ‘Velocity acquired due to given Height of Fall and Height due to given "Velocity. v 2 * 8.0 ^y/h — v', 32.2 t = v) - — =zh ; and 16.083 t 2 == h. 04.4 h representing height of fall in feet , v velocity acquired in feet per second , and t time of fall in seconds. To Compute ^Action of G-ravity. Time. When Space is given . Rule.— Divide space by 16.083, and square root of quotient will give time. Example. —How long will a body be in falling through 402.08 feet? V402.08 -5- 16.083 = 5 seconds. When Velocity is given. Rule. — Divide given velocity by 32.166, and quotient will give time. Example.— How long must a body be in falling to acquire a velocity of 800 feet per second ? 800 -r* 32. 166 = 24. 87 seconds. GRAVITATION. 489 Velocity. When Space is given. Rule. — Multiply space in feet by 64.333, and square root of product will give velocity. Example.— Required velocity a body acquires in descending through 579 feet. V 579 X 64.333 = 193 feet- Velocity acquired at any penod is equal to twice the mean velocity during that period. Illustration.— If a ball fall through 2316 feet in 12 seconds, with what velocity will it strike? 2316 -4- 12 = 193, mean velocity , which X 2 = 386 feet = velocity. When Time is given. Rule. — M ultiply time in seconds by 32.166, and product will give velocity. Example.— W hat is velocity acquired by a falling body in 6 seconds? 32. 166 X 6 = 192.996 feet. Space. When Velocity is given. Rule.— D ivide velocity by 8.04, and square of quotient will give distance fallen through to acquire that velocity. Or, Divide square of velocity by 64.33. Example. — If the velocity of a cannon-ball is 579 feet per second, from what height must a body fall to acquire the same velocity? 579 8.04 = 72.014, and 72.014 2 = 5186.02 feet. When Time is given. Rule. — Multiply square of time in seconds by 16.083, and it will give space in feet. Example.— R equired space fallen through in 5 seconds. 5 2 — 2 5, and 25 X 16.083 = 402.08 feet. Distance fallen through in feet is very nearly equal to square of time in fourths of a second. Illustration i. — A bullet dropped from the spire of a church was 4 seconds in reaching the ground; what was height of the spire? 4 X 4 — 16, and 16 2 = 25 6 feet. By Rule, 4 X 4 X 16.0833 = 257-33 f eet - 2.— A bullet dropped into a well was 2 seconds in reaching bottom; what is the depth of the well ? Then 2X4 = 8, and 82 — 64 feet. By Rule, 2 X 2 X 16.0833 = 64.33 f eet - By Inversion.— In what time will a bullet fall through 256 feet? V 256 = 16, and 16 -7- 4 = 4 seconds. Space fallen tlirongli in last Second of Fall. When Time is given. Rule.— S ubtract half of a second from time, and multiply remainder by 32.166. Example.— What is space fallen through in last second of time, of a body falling for 10 seconds ? 0 10 — . 5 x 32. 166 = 305. 58 feel. Promiscuous Examples. 1. If a ball is 1 minute in falling, how far will it fall in last second? Space fallen through = square of time, and 1 minute — 60 seconds. 60 2 X 16.083 = 57 898 feet for 60 seconds. 59 2 X 16.083 = 55984 “ “ 59 “ 19x4 1 descended 11 ^ °^£ enera ^ D £ a velocity of 193 feet per second, and whole space I 93~j“ 32-166 = 6 seconds; 6 2 X 16.083 = 579/^. GRAVITATION. 49 ° 3 . If a body was to fall 579 feet, what time would it be in falling, and how far would it fall in the last second? 579 X 2 _ , 6 _ 6 secon ds , and 6 — . 5 X 3 2 - 166 = 5 - 5 X 3 2 - 166 == *76-91 f eet - 32. 166 Formulas to determine the various Elements. S V 2 s . / S V 2 S / 2 » . _ T_ v ’ ~ s' ~ v ’ y 2 . s= (jLy. = 1 !; =1*; =£ 1 !; =T*. S i V25 g) 2 9 2 2 h±=T-. 5 g. v = Vsx.2fli; =Tr jV. 5S ,s ; = T representing time of falling in seconds, V udoctty acquired in feet per second S space or vertical height in feet, h space fallen through in last second, g 32.166 and .5 g and .25 g representing 16.083 and 8.04. Retarded. Nlotion. A body projected vertically upward is affected inversely to its motion when falling freely and directly downward, inasmuch as a like cause retards it in one case and accelerates it in the other. In air a ball will not return with same velocity with which it started. In vacuo it would. Effect of the air is to lessen its velocity both ascending and descending. Difference of velocities will depend, upon relative specific grav- ity of ball and density of medium through which it passes. Ihus, greater weight of ball, greater its velocity. To Compute ^Action of Grravity by a Body projected Upward or Downward with, a given. Velocity. Space. When projected Upward. Rule.— From the product of the given velocity and the time in seconds subtract the product of 32.166, and half the square of the time, and the remainder will give the space in feet. Or, Square velocity, divide result by 64.33, and quotient will give space in feet. Example.— If a body is projected upward with a velocity of 96.5 feet per second, through what space will it ascend before it stops? 96. 5^-32.166 = 3 seconds = time to acquire this velocity. Then, 96.5 X 3 — ^32.166 X = 289.5 — 144*75 = * 44-75 Time. Rule. — Divide velocity in feet by 32.166, and quotient will*give time in seconds. Example.— Velocity as in preceding example. 96. 5 -4- 32. 166 = 3 seconds. Velocity. Rule.— Multiply time in seconds by 32.166, and product will give velocity in feet per second. Example.— Time as in preceding example. 3 x 32. 166 = 96. 5 feet velocity. Space fallen throxigh in. last Second. Rule.— Subtract .5 from time, multiply remainder by 32.166, and product will give space in feet per second. Example. — T ime as in preceding example. 3 5 X 32. 166 = 2. 5 X 32. 166 = 80.41 6 feet. GRAVITATION. 491 When projected Downward. Space. Rule. — P roceed as for projection upwards and take sum of products. Example l— I f a body is projected downward with a velocity 0196.5 feet per sec- ond, through what space will it fall in 3 seconds? 9 6 - 5 X 3 + (32. i 66 X = 289. 5 -f 144. 75 = 434. 25 feet. Or, t 2 x 16.083 + v x t = s. 2. — If a body is projected downward with a velocity of 96.5 feet per second, through what space must it descend to acquire a velocity of 193 feet per second? 96.5 -r- 32. 166 = 3 seconds , time to acquire this velocity. 1 93 — i— 32.166 = 6 seconds , time to acquire this velocity. Hence 6 — 3 = 3 seconds , time of body falling. Then 96.5 X 3 = 289.5 = product of velocity of projection and time. 16.083 X 3 2 = 144-75 — product of 32. 166, and half square of time. Therefore 289.5-}- 144.75 = 434.25/^. Time. Rule.— S ubtract space for velocity of projection from space given, and remainder, divided by velocity of projection, will give time. Example. — In what time will a body fall through 434.25 feet of space, when pro- jected with a velocity of 96. 5 feet ? Space for velocity of 96. 5 = 144.75/ee^ Then, 434.25 — 144.75 = 96.5 = 289.5 — 96.5 = 3 seconds. V^elocity-. Rule.— D ivide twice space fallen through in feet by time in seconds. Example. — Elements as in preceding example. Space fallen through when projected at velocity of 96.5 feet— 144.75 feet, and 434.25 feet — space fallen through in 3 seconds. Then, 144.754434.25 = 579 feet space fallen through, and V579 = 16.083 = 6 seconds. Hence, 579 X 2 = 6 = 1158 = 6 = 193 feet. Space Fallen. tlirougTi in last Second. Rule.— S ubtract .5 from time, multiply remainder by 32.166, and product will give space in feet per second. Example.— E lements as in preceding example. 6 — .5 X 32.166 = 5.5 X 32.166 = 176.91 feet. Ascending bodies, as before stated, are retarded in same ratio that descending bodies are accelerated. Hence, a body projected upward is ascending for one half of the time it is in motion, and descending the other half. Illustration i. — If a body projected vertically upwards return to earth in 12 seconds, how high did it ascend ? The body is half time in ascending. 12 = 2 = 6. Hence, by Rule, p. 489, 6 2 X 16.083 = 579 feet = product of square of time and 16.083. 2.— If a body is projected upward with a velocity of 96.5 feet per second, it is required to ascertain point of body at end of 10 seconds. 96-5“f 3 2 I 66 = 3 seconds, time to acquire this velocity , and 3 2 x 16.083 = 144.75 feet, height body reached with its initial velocity. Then 10 — 3 = 7 seconds left for body to fall in. Hence, by Rule, as in preceding example, 7 2 X 16.083 = 788.07, and 788.07 — I *44- 75 — 643. 32 feet = distance below point of projection. \ 0r > 10 2 X 16.083 = 1608.3 feet, space fallen through under the effect of gravity, and 1 96. 5 X 10 = 965 feet , space if gravity did not act. Hence 1608. 3 — 965 = 643. 3 feet. 49 2 GRAVITATION. 3. — A body is projected vertically with a velocity of 135 feet; what velocity will it have at 60 feet ? I3 5 2 -4- 64.33 = 283.3 feet space projected at that velocity , 135 - 4 - 32. 16 = 4. 197 sec- onds = time of projection, and 283.3 — 60= 223.3 = space to he passed through after attainment of 60 feet. = 283.3/^. Hence, V 223. 3 X 64. 33 = 119.85 feet velocity , and 223.34-60 119.85. Hence, 1 * 9 ' 8 -- — 223.3 fat space , and 283.3 - 64-33 By Inversion . — Velocity : 223. 3 = 60 feet. Formulas to Determine Elements of Retarded IVtotioia. s gt 1. v=.V — gt. V = - t — t V — 9 + 9 4- S. s = Xt- 9 —. 3. V = v-f gt, 6. s = tv 2 9. h — T — t — t'—.sg. n , V /V 2 2S 8. t — — • — . f —z . 9 \ 9 2 9 v representing velocity at expiration of time , t any less time than T, t' less time than t, s space through which a body ascends in time t, V, T, S, and h as in previous formulas, page 490. Illustration.— A body projected upwards with a velocity of 193 feet per second, was arrested in 5 seconds. T = 6, t'=i. 1. What was its velocity when arrested ? (1.) 2. What was the time of its passing through 562.92 feet of space ? (7.) 3. What space had it passed through ? (5.) 4. What was the time of its projection, when it had a velocity of 96.5 feet? (4.) 5. What was the height it was projected in the last second of time? (8.) 1. 193 — 32. 166 X 5 = 32. 17 feet. 562.92 32. 166 X 5 5 i 93 X 5 = 193 velocity. 32.166 X 5 s = 562.92/ee^ 193 32. 166 3. 32. 17 + 32. 166X 5 — J 93 velocity. 193 — 9 6 - 5 _ 32. 166 6 193 — 32-17 . 32. 166 - = 3 seconds. - = 5 seconds. 193 2 2 x 562.9= 32. 166 2 32.166 !. 6 — 5 — 1 — . 5 x 32. 166 = 6 — V36 — 35 = 5 seconds. 48.25 feet. Grravity- and ZVTotion at an Inclination. If a body freely descend at an inclination, as upon an inclined plane, by force of gravity alone, the velocity acquired by it when it arrives at ter- mination of inclination is that which it would acquire by falling freely through vertical height thereof. Or, velocity is that due to height of in- clination of the plane. Time occupied in making descent is greater than that due to height, in ratio of length of its inclination, or distance passed, to its height. Consequently, times of descending different inclinations or planes of like heights are to one another as lengths of the inclinations or planes. Space which a body descends upon an inclination, when descending by gravity , is to space it would freely fall in same time as height of inclination is to its length ; and spaces being same, times will be inversely in this pro- portion. If a body descend in a curve, it suffers no loss of velocity. If two bodies begin to descend from rest, from same point, one upon an in- clined plane, and the other falling freely, their velocities at all equal heights below point of starting will be equal. GRAVITATION. 493 Illustration.— What distance will a body roll down an inclined plane 300 feet long and 25 feet high in one second, by force of gravity alone? As 300 : 25 :: 16.083 : 1.34025 feet. Hence if proportion of height to length of above plane is reduced from 25 to 300 to 25 to 600, the time required for body to fall 1.34025 feet would be determined as follows : As 25 • 600 ” 1.34025 : 32.166, and 32.166 = 16.083 X 2 = twice time or space in which it would fall freely required for one half proportion of height to length. Or as 1.34025 : 32.166, as above. ’ 25 25 Impelling or accelerating force by gravitation acting in a direction paral- lel to an inclination, is less than weight of body, in ratio of height of in- clination to its length. It is, therefore, inversely in proportion to length of inclination, when height is the same. Time of descent, under this condition, is inversely in proportion to accel- erating force. If, for instance, length of inclination is five times height, time of making freely descent at inclination by gravitation is five times that in which a body would freely fall vertically through height ; and impelling force down inclination is .2 of weight of body. Wlien bodies move down inclined planes, the accelerating force is ex- pressed by h -r- 1 , quotient of height length of plane $ or, what is equivalent thereto, sine of inclination of plane, i. e., sin. a. Illustration. — An inclined plane having a height of one half its length, the space fallen through in any time would be one half of that which it would fall freely. Velocity which a body rolling down such a plane would acquire in 5 seconds is 80.416 feet. Thus, 32.166 X 5 = 160.833 feet, and an inclined plane, having a height one half of its length, has an angle or sine of 30 0 . Hence, sin. 30 0 = .5, and 160.833 X .5 = 80.41 6 feet. Formulas to Determine various Elements of Gravita- tion 011 an Inclined Plane. *• S = . 5S rT*«n.a; =-5™ 4 - V^orTrtn. «. . 2 I 2. V = g T sin. a ; = V( 2 9 S sm. a) ; = — 6. H = l 2 : -50T2- 3- =Wh; 5. S = V T . 5 g T 2 sin. a. Or, 2 g sin. a v representing velocity of projection in feet per second , S space or vertical height of velocity and projection, a angle of inclination of plane, l length, and H height of plane. Illustration. — Assume elements of preceding illustration. V = 80.416, T = 5, and H = 201.04. 1. .5 X 32.166 X 5 2 X -5 — 201.04 /ee£. 2. 32.166 X 5 X .5 = 8o.4i6/ee£. 3. / = / V25 = 5 seconds. V \ 32. 166 x. 5/ V \16.083 / 283.422 - = 201.04 feet. — — — — - — — — - — ^1.^,/w. 7. 4 X 5 X V 20I -°4 — 283.42 /ee^. .5 X 16.083 X 5 If projected downward with an initial velocity of 16.083 feet per second. V-f-^. 4. 16.083-f-32.166x 5 X .5 = 9 6. 5 feet. 5. 80.416-}- 16.083 X 5 — -5 X 32.166 X 5 2 X .5 = 281.46/eefc t t GRAVITATION. 494 Illustration. —What time will it take for a ball to roll 38 feet down an inclined plane, the angle a= 12 0 20', and what velocity will it attain at 38 feet from its start- ing-point? X • 2136 = 22.88 feet per second. When a body is projected upward it is retarded in the same ratio that a descending body is accelerated. Illustration.— If a body is projected up an inclined plane having a length of twice its height, at a velocity of 96.5 feet per second, Then, T = 96. 5 -7-32.166 = 3 seconds. S = . 5 3 2 - j66 X 3 2 X .5 = 72.375 feet. v=z 32.166 X 3 X. 5 = 48-25/^. Problems on descent of bodies on inclined planes are soluble by formulas 1 to 9, page 495, for relations of accelerating forces. As a preliminary step, however, accelerating force is to be determined by multiplying weight of descending body by height of plane, and dividing product by length of plane. Illustration. —If a body of 15 lbs. weight gravitate freely down an inclined plane, length of which is five times height, accelerating force is 15^5 = 3 lbs. If length of plane is 100 feet and height 20, velocity acquired in falling freely from top to bottom of plane would be Whereas, for a free vertical fall through height of 20 feet, time would be, which is .2 of time of making descent on inclined plane. Velocities acquired by bodies in falling down planes of like height will all be equal when arriving at base of plane. When Length of an Inclined Plane and Time of Free Descent are given. Rule. — D ivide square of length by square of time in seconds and by 16 ; the quotient is height of inclined plane. Example. — Length of plane is 100 feet, and time of descent is 5.59 seconds; then vertical height of descent is If an Accelerating or Retarding force is greater than gravity, that us, weight of the body, the constant, g, or 32.166, is to be varied in proportion thereto, and to do this it is to be multiplied by the accelerating force, and product divided by weight of body. Thus, Let f represent accelerating force, and w weight of body. The same rules and formulas that have been given for action of gravity alone are applicable to the action of any other uniformly accelerating or retarding lorce, the numerical constants above given being adapted to the force. j 2 S _ V g sin. a~ 2 X 3 8 = 3. 33 seconds. V = g T sin. a = 32. 166 X 3- 33 32. 166 X -2136 Inclined Plane, Time occupied in making descent, making 32.166 5 - 59 2 X 16.08 == 20 feet. Accelerated and Retarded NIotion Then 64 ' 333 -^ or or 1 - 6 ° 8 ^ become the constants. ’ w ’ W 10 GRAVITATION. 495 Average Velocity- of a Moving Body uniformly Accel- erated or Retarded. Average velocity of a moving body uniformly accelerated or retarded, during a given time or in a given space, is equal to half sum of initial and final velocities ; and if body begin from a state of rest or arrive at a state of rest, its average speed is half the final or initial velocity, as the case may be. Thus, in example of a ball rolling, initial speed or velocity is, in either case, 60 feet per second, and terminal speed is nothing ; average speed is therefore - 6o ° namely, one half of that, or 30 feet per second. When a cannon-ball is projected at an angle to horizon, there are two forces act- ing on it at same time— viz., force of charge, which propels it uniformly in a right line, and force of gravity, which causes it to fall from a right line with an accel- erated motion; these two motions (uniform and accelerated) cause the ball to move in the curved line of a Parabola. w representing weight of hall and P of powder in lbs. ; l time of flight in seconds • h horizontal range , and h vertical height of range of projection of ball in feet. Illustration. — A cannon loaded to give a ball a velocity of 900 feet per second the angle a = 45 0 ; what is horizontal range, the time t and height of range h ? ’ Note.— As distance b will be greatest when angle a = 45°, product of sine and cosine is greatest for that angle. Sin. 45 0 x cos. 45 0 = . 5. 24 lb. ball with a velocity of 2000 feet per second at 45 0 range 7300 feet. G-eneral Bormnlas for Accelerating and Retarding Forces, Formulas for Flight of a Cannon-ball. V 2 sin. a, cos. a ^ n J V sin. a 9 ; 900 2 x sin. 45° X COS. 45O _ 900 2 X -5 -^^ = 12590 feet 32. l66 . 900 X. 7071 t = tv — = iQ- 70 seconds : h ~ 32.166 ; 900 2 x .7071 2 2 X 32.166 = 6295 feet. GRAVITATION. 496 Again same result may be arrived at, according to Note 1, by multiplying con- stant 64! 333, in Rule, page 494, for gravity, by ratio of force and weight, which in this case is and 64.333 X 3^ = 6-4333- Substituting 6.4333 for 64.333 in that rule, formula becomes ~ V 2 602 . . S = = = 559- 59 f eeL 6-4333 6.4333 The question may be answered more directly by aid of table for falling bodies, page 488. Height due to a velocity of 60 feet per second, is 55.9 feet; which is to be multiplied by inverse ratio of accelerating force and weight of body, or or 10; that is, 55 . 9 x 10 = 559 f eet If the question is put otherwise— What space will a weight move over before it comes to ci sttite of rest, with, an initial velocity of 60 feet per second, allowing fric- tion to be one tenth weight? The answer is that friction, which is retarding force, bein" one tenth of weight, or of gravity, space described will be 10 times as great as is necessary for gravity, supposing the weight to be projected vertically upwards to bring it to a state of rest. The height due to velocity being 55.9 feet; then 55.9X10 = 559 feet. Average velocity of a moving body, uniformly accelerated or retarded during a given period or space, is equal to half sum of initial and final velocities. To Compute "Velocity of a, [Falling Stream of Water per Second, at End of any- given Time. When Perpendicular Distance is given. Example.— What is the distance a stream of water will descend on an inclined plane 10 feet high, and 100 feet long at base, in 5 seconds? 5 2 X 16. 083 = 402.08 feet = space a body will f reedy fall in this time. Then, as 100 : 10 :: 402.08 : 40.21 feet = proportionate velocity on a plane of these dimensions to velocity when falling freely. [Miscellaneous Illnstrations. j.—What is the space descended vertically by a falling body in 7 seconds. S = 5 g X t 2 . Then 16.083 X 7 2 = 788.067 feet. 2 . —What is the time of a falling body descending 400 feet, and velocity acquired at end of that time? t = ~. Then l6 ° -- = 4.98 sec. v = V2 p X S. Then V64.333 X 4°° = 160.4 feet, g 32.166 3. — If a drop of rain fall through 176 feet in last second of its fall, how high was the cloud from which it fell? S = — . Then -^1^482.75/^. 2 g 64. 166 a. If two weights, one of 5 lbs. and one of 3, hanging freely over a sheave, are set free, how far will heavier one descend or lighter one rise in 4 seconds. x 16.083 x 4 2 = l X 257.328 = 64.33 feet. 5 + 3 8 5. — if length of an inclined plane is 100 feet, and time of descent of a body is 6 seconds, what is vertical height of plane or space fallen through? 6 2 X -5 l 579 - = 17.27/eek 6.— If a bullet is projected vertically with a velocity of 135 feet per second, what velocity will it have at 60 feet? Formula 7, page 492. i35 / i35 2 32.166 V 32.166 2 2 * _ .j seconds. 32 .166 GUNNERY. 497 GUNNERY. A heavy body impelled by a force of projection describes in its flight or track a parabola, 'parameter of which is four times height due to velocity of projection. Velocity of a shot projected from a gun varies as square root of charge directly, and as square root of weight of shot reciprocally. To Compute ‘Velocity of a Shot or Shell. Rule.— M ultiply square root of triple weight of powder in lbs. by 1600; divide product by square root of weight of shot ; and quotient will give ve- locity in feet per second. Example.— W hat is velocity of a shot of 196 lbs., projected with a charge of o lbs of powder? ^ V 9 X 3 X 1600 -r- y / 196 = 8320 = 14 == 594.3 lbs. To Compute Range for a Charge, or Charge for a Range. When Range for a Charge is given. — Ranges have same proportion as charges of powder ; that is, as one range is to its charge, so is any other range to its charge, elevation of gun being same in both cases. Consequently , To Compute Range. Rule. — M ultiply range determined by charge in lbs. for range required divide product by given charge, and quotient will give range required. * Example. If, with a charge of 9 lbs. of powder, a shot ranges 4000 feet, how far will a charge of 6.75 lbs. project same shot at same elevation? 4000 X 6. 75 -r- 9 = 3000 feet. To Compute Charge. . Rule.— M ultiply given range by charge in lbs. for range determined, divide product by range determined, and quotient will give charge required. Example.— I f required range of a shot is 3000 feet, and charge for a range of 4000 feet has been determined to be 9 lbs. of powder, what is charge required to project same shot at same elevation ? 3000 X 9 4°°° — 6-75 lbs. To Compute Range at one Elevation, when Range for another is given. Rule. — A s sine of double first elevation in degrees is to its range, so is sine of double another elevation to its range. Example.— I f a shot range 1000 yards when projected at an elevation of 45° how far will it range when elevation is 30 0 16', charge of powder being same? 1 Sine of 45 0 X 2 = 100 000 ; sine of 30° 16' X 2 = 87 064. Then, as 100000 : 1000 87064 : 870.64 feet. To Compute Elevation at one Range, when Elevation for another is given. Rule. — A s range for first elevation is to sine of double its elevation, so is range for elevation required to sine for double its elevation. Example.— I f range of a shell at 45 0 elevation is 3750 feet, at what elevation must a gun be set for a shell to range 2810 feet with a like charge of powder? Sine of 45 0 X 2 100000. Then, as 3750 : 100000 :: 2810 : 74933 = 51716 for double elevation — 2 16'. Approximate Rule for Time of Flight. Under 4000 yards, velocity of projectile 900 feet in one second ; under 6000 yards, velocity 800 feet ; and over 6000 yards, velocity 700 feet. Guns and Howitzers take their denomination from weights of their solid shot in round numbers, up to the 42-pounder; larger pieces, rifled guns, and mortars, from diameter of their bore. T T* GUNNERY. 498 Initial 'Velocity and. Ranges of Shot and Shells. The Range of a shot or shell is the distance of its first graze upon a horizontal ~i ~ *r>r»nnfArl nnnn its nroner carriage. Range. Arms and Ordnance. J Projectile. Description. | Weight. | Powder. Initial | Velocity.| Time of 1 Flight. 1 Eleva tion. Rifle Musket Elongated. Grains. 5io Grains. 60 Feet. 9 6 3 Seconds. Musket, 1841 Round. 412 no 1500 — — 6-Pounder .<< Lbs. 6.15 Lbs. 1.25 — 5 u 12.3 2-5 1826 i-75 1 24 “ 32 u 24.25 6 1870 — 2 u 32-3 8 1640 — 1 42 “ 42.5 10.5 — — 5 8-inch Columbiad. . . “ 65 10 — 14.19 i5 10 “ u • • • “ 127-5 15 — 14.32 i5 10 “ Mortar Shell. 98 10 — 36 45 13 “ “ “ 200 20 — — 45 k “ Columbiad... u 302 40 — ■ — 7 15 “ “ u 3 i 5 50 — 23.29 25 RIFLED. 10-pounder Parrott. . 20 u u “ 9-75 1 — 21 20 “ *9 2 — 17.25 i 5 30 “ u “ 29 3-25 — 27 25 * it u IOO Elongated. 100 10 — 2 9 25 100 “ “ Shell. IOI 10 1250 28 25 200 “ “ “ 150 16 — — 4 1 2- inch Rodman — 50 1154 5-5 Hall’s Rockets 3- inch. 16 - - 47 Yards. 1523 575 1147 7i3 1955 3224 3281 4250 4325 1948 4680 5000 4400 6700 6910 6820 2200 1720 Penetration of Shot and Shell. Mean Penetration. Mean Penetration . Ordnance. S, gS 1 A 2 e Ordnance. C/ tL 3 -2 • 2 ‘2 0 s ffig 5 O 5 £5 Lbs. Yds. Ins. Ins. Ins. Lbs. Yds. Ins. Ins. Ins. 32 Lbs. Shot. 8 880 15-25 3-5 8-inch Howitz.* 6 880 — 8-5 1 n IOO 60 8 “ Columb.if 12 200 — — — 32 42 ‘ 10.5 IOO 54-75 18 4 10 “ “ t iS 1 14 63-5 44 7-75 42 “ Shell. 7 100 40-75 — — 10 “ “ * 18 IOO 56.75 — — 1 24 ins. of Concrete. * Shell. t Shot Solid shot broke against granite, uut nut agamai* ^ ~ effect is less upon brick than upon granite. Shells broke into small fragments against each of the three materials. Penetration in earth of shell from a 10- inch Columbiad was 33 ins. Experiments — England. ( Holley . ) Ordnance. Charge. Projectile. Weight. Velocity. Range. Target and Effects. Lbs. Lbs. Feet. Yards. Iron plates, 14 ins. — loosened. n-inch U. S. Navy. 30 Shot. 169 1400 50 15-inch Rodman. . . 60 “ 400 1480 50 Iron plates, 6 ins. — destroyed. RIFLED. 7-inch Whitworth. . 25 Shot. 150 1241 200 Inglis'st— destr’d. 10.5-inch Armstrong 13-inch “ 45 90 * 41 3°7 344-5 1228 1760 200 200 Solid plates, n ins. thick— destr'd. * Steel. t 8-inch vertical an Blabs, 9X5 ins. ribs and 3-inch ribs. GUNNERY. 499 Elements of Report of Board of Engineers for Fortifications , U~. S. A. Professional Papa's RTo. 25. ( Brev . Maj.-Gen. Z. B. Tower.) Experimental firings for penetration during tlie past twenty years have determined that wrought iron and cast iron, unless chilled, are unsuitable for projectiles to be used against iron armor ; that the best material for that purpose is hammered steel or Whitworth’s compressed steel. 2. That cast-iron and cast-steel armor-plates will break up under the im- pact of the heaviest projectiles now in service, unless made so thick as to exclude their use in ship-protection. 3. That wrouglit-iron plates have been so perfected that they do not break up, but are penetrated by displacement or crowding aside of the material in the path of the shot, the rate of penetration bearing an approximately deter- mined ratio to the striking energy of the projectile, measured per inch of shot’s circumference, as expressed by the following formula : 2 '°V7 — — = penetration in ins. V representing velocity in V 2#X2 r it X 2240 X • 86 J feet per second , P weight of shot in lbs ., and r radius of shot in ins. That such plates can therefore be safely used in ship construction, their thickness being determined by the limit of flotation and the protection needed, 4. That, though experiments with wrouglit-iron plates, faced with steel, have not been sufficiently extended to determine the best combination of these two materials, we may nevertheless assume that they give a resistance of about one fourth greater than those of homogenous iron. 5. That hammered steel in the late Spezzia trials proved superior to anv other material hitherto tested for armor-plates. The 19-inch plate resisted penetration, and was only partially broken up by 4 shots, three of which had a striking energy of between 33 000 and 34000 foot-tons each. Not one shot penetrated the plate. Those of chilled iron were broken up, and the steel projectile, though of excellent quality, was set up to about two thirds of its length. "Velocity and. Ranges of Shot. [Krupff s Ballistic Tables.) IPenetration in. Wrought Iron. . / V 2 P V ~ ~ — Z 7 c * = penetration in ins. C = 2. 53. V 2 g X 2 r it X.2240 x C x Gun. Cali- ber. Powder. Shot. V at Muzzle per Sec. elocity Rat 3000 tge. 6000 at Muzzle Penet: 600 ration Range 3900 6000 Tons. Ins. Lbs. Lbs. Feet. Yds. Yds. Ins. Ins. Ins Ins. Armstrong, 100.. 17-75 550 2022 1715 1424 1191 34-76 33-2 27-55 22.04 “ “ . . 17-75 776 2000 1832 1518 1259 37-52 35 - 3 1 29.66 23-47 Woolwich, 81.. 16 445 1760 1657 1393 1181 32.6 31-23 26. 24 21-35 Krupp, 71.. 15-75 485 1715 1703 1434 1211 33-52 32. 12 27.04 21.89 18. . 9-45 165 474 1688 I 35 i 1113 20.42 I 9 - 3 1 15.46 12. 14 U. S.* 8- inch 8 35 180 1450 1036 840 10.23 9.22 6.72 5 -i 7 * Unchambered. Target .— For 100-ton gun, steel plate 22 ins. thick, backed with 28.8 ins. of wood 2 wrought-iron plates 1.5 ins. thick, and the frame of a vessel. Effect . — Total destruction of steel plate, and backing entered to a depth of 22 ins but not perforated. * * 500 GUNNERY. Summary of Record of 3Practi.ce in Europe witli Heavy Armstrong, Woolwich, and. Krupp Guns. Board of Engineers for Fortifications , U. S. A . , Professional Papers No. 25. Gun. Powder. Projectile. Charge of Powder. Weight of Projectile. Initial Velocity per Second. V. Initial P V2 En N per inch 01 <5 circumference 5^ of shot. P V 2 1 N O. Cl Lbs. Lbs. Feet. Ft. -tons. Foot-tons. Armstrong, 1 1. 5- inch cubes. . Shot.... 330 2000 1446 28990 544-05 100 Tons, caliber 1 Waltham Abbey u 375 2000 1543 33000 023 17 ins., bore 30.5 f Fossano u 400 2000 1502 31 282 5 ° 5-74 feet. J “ ....... 11 776 2000 1832 46 580 835-32 Woolwich, 81 "l .75-inch cubes. “ 170 1258 1393 16922 37 i -5 Tons, caliber 14.5 V 1.5 “ “ a 220 M 5 ° 1440 20 842 457-57 ins., bore 24 feet. J 2 “ “ “ . . . . 250 1260 1523 20 259 444.78 caliber 16 ins. . . . . 1.5 “ “ u 310 1466 1553 24 508 520.4 38 Tons, 'l 1.5 “ • “ Pall, shell 130 800 1451 ii 668 297.64 caliber 12.5 ins., 'r 1.5 “ “ “ 200 800 1421 11 210 2 o 5-4 bore 16.5 feet. J 1.5 “ “ . u. 180 800 1504 12545 3 I 9-4 Krupp, 71 Tons, 1 Prism A Plain . . . 298 1707 1184 16602 335-42 caliber 15.75 ins., ^ “ H Shrapnel 485 i 7 2 5 1703 34 503 697.91 bore 28. 58 feet. J “ 2 inch. . . Shell. . . . 441 1419 1761 30 484 616.14 18 Tons, 1 »! 1 hole... Plain . . . 132 300 1873 7 298 246.03 caliber 9.45 ins., y “ 2 inch. .. Shrapnel 145 474 1688 9 367 315.66 bore 17.5 feet. J Shell. . . . 165 300 1991 1 8 244 277.69 Penetration in Ball Cartridge Paper, No. 1. Musket, with 134 grains, at 13.3 yards 653 sheets. Common rifle, 92 grains, at 13.3 yards 5°° sheets. Penetration of* Lead 13 alls in Small Arm s . Experiments at Washington Arsenal in 1839, and at West Point in 1837. Distance. Musket Common Rifle Hall’s rifle Hall’s carbine, musket caliber Pistol Rifle musket Altered musket Rifle, Harper’s Ferry Pistol carbine. ...... Sharpe’s carbine Burnside’s “ .... Diameter of Bali. Inch. (.64 [.64 ■5775 •5775 .685 •5775 •5775 •55 •55 Charge Powder. Grains. 134 144 100 92 100 70 70 80 90* 100* 51 60 70 40 60 55 Yards. 9 5 5 9 5 9 5 5 5 5 5 200 200 200 200 30 30 Weight of Ball. Penetration. White Oah. White Pine. Grains. 397-5 397*5 219 219 219 219 219 500 730 500 450 463 350 * Charges too great for service. Ins. 1.6 3 2.05 1.8 2 .6 i-7 .8 1. 1 1.2 •7 2 5 11 10.5 9-33 5-75 7.1 7 6.15 Musket discharged at 9 yards distance, with a charge of 134 grains, 1 ball and buckshot, gave for ball a penetration of 1.15 ins., buckshot, .41 inch. GUNNERY. 501 Loss cf Force "by Windage. A comparison of results shows that 4 lbs. of po,wder give to a ball without wind- age nearly as great a velocity as is given by 6 lbs. having . 14 inch windage, which is true windage of a 24-lb. ball; or, in other words, this windage causes a loss of nearly one third of force of charge. Vents. — Experiments show that loss of force by escape of gas from vent of a gun is altogether inconsiderable when compared with whole force of charge. Diameter of Vent in U. S. Ordnance is in all cases .2 inch. Eject of dijerent Waddings with a Charge of 77 Grains of Powder. Wad. Velocity of Ball per Second. Ball wrapped in cartridge paper and crumpled Feet. 13 77 1 felt wad upon powder and 1 upon ball 2 felt wads upon powder and r upon ball *34^ T A Ro 1 elastic wad upon powder and 1 upon ball 1132 2 pasteboard wads upon powder. 2 elastic wads upon powder I IOO Felt wads cut from body of a hat, weight 3 grains. Pasteboard wads . 1 of an inch thick, weight 8 grains. Cartridge paper 3 X 4.5 ins., weight 12.82 grains. Elastic wads , “Baldwin’s indented,” a little more than .1 of an inch thick weight 5.127 grains. * Most advantageous wads are those made of thick pasteboard, or of or- dinary cartridge paper. In service of cannon , heavy wads over ball are in all respects injurious. For purpose of retaining the ball in its place, light grommets should be used. On the other hand, it is of great importance, and especially so in use of small arms, that there should be a good wad over powder for developing full force of charge, unless, as in the rifle, the ball has but very little windage. [Capt. Mordecai ) Weight and. Dimensions of Lead JBalls. Number of Balls in a Lb., from 1.3125 to .237 of an Inch Diameter. Diam. No. Diam. No. Diam. No.- Diam. No. Diam. No. Diam. No. Ins. Inch. Inch. Inch. Inch. Inch. 1.67 1 •75 11 •57 25 .388 80 .301 170 •259 270 1.326 2 •73 12 •537 30 •375 • 88 •295 180 .256 280 *•*57 1-051 3 4 •7 1 • 6 93 *3 x 4 •5 1 •505 35 36 •372 •359 90 IOO •29 .285 190 200 .252 .249 290 300 •977 5 .677 x 5 .488 40 •348 no .281 210 • 247 9 in •9 X 9 6 .662 16 .469 45 .338 120 .276 220 T / .244 320 •873 •835 7 8 .65 •637 17 18 •453 .426 50 60 •329 .321 130 I40 .272 .268 230 240 .242 •239 330 340 .802 9 .625 *9 •405 70 •3 X 4 150 .265 250 ■237 350 •775 10 .615 20 •395 75 •307 l6o .262 260 Heated shot do not return to their original dimensions upon coolin^ a permanent enlargement of about .02 per cent, in volume. but retain A A A B B, Number of Pellets in an Ounce 40 I B 75.1 No. 3.. 50 No. 1 82 4. . 58 I 2 1 12 I 5. . No. 14. . . of Lead Shot of the dijerent Sizes. 135 177 218 I No. 6 280 No. c • 34i . 600 - 3i5o 984 1726 2140 502 GUNNERY. Proportion of Powder to Sliot for following Numbers of Sliot. No. Shot. . Powder. No. | Shot. Powder. No. Shot. Powder. "t ' Oz. Drains. I Oz. Drams. Oz. Drams. 2 2 i -5 4 i -5 1-875 6 1.25 2-375 3 i -75 1.625 ;5 J i -375 2.125 7 1. 125 2.625 Note. 2 oz. of No. 2 shot, with 1.5 drams of powder, produced greatest effect. Increase of powder for greater number of pellets is in consequence of increased friction of their projection. Numbers of Percussion Caps corresponding with Birmingham Numbers. Eley’s 1 5 I 6 7 1 8 1 9 24 10 11 18 I 12 I 13 1 14 Birmingham. . | 43 44 46 i 48 j 49 [5o | 51 and 52 53 and 54 1 [ 55 and 56 ! 57 1 58 1 58 Where there are two numbers of Birmingham sizes corresponding with only one of Eley’s, it is in consequence of two numbers being of same size , varying only in length of caps. Comparison of Force of a Cliarge in various Arms. Arm. Lock. . Powder, A 5 - Windage. Weight of Ball. Velocity. Ordinary rifle Percussion. Grains. 100 Inch. .015 Grains. 219 Feet. 2018 “ 70 .015 219 1755 Hall’s rifle Flint. 70 .0 219 149° Hall’s carbine Percussion. 70 .0 219 1240 Jenks’s carbine “ 70 .0 219 1687 Cadet’s musket Flint. 70 •045 219 1690 Pistol Percussion. 35 .015 218.5 947 Ranges for Small Arms. Musket .— With a ball of 17 to pound, and a charge of no grains of powder, etc., an elevation of 36' is required for a range of 200 yards; and for a range of 500 yards, an elevation of 3 0 3c/ is necessary, and at this distance a ball w ill pass through a pine board 1 inch in thickness. Rifle . — With a charge of 70 grains, an effective range of from 300 to 350 yards is obtained; but as 75 grains can be used without stripping the ball, it is deemed better to use it, to allow for accidental loss, deterioration of powder, etc. Pistol.— With a charge of 30 grains, the ball is projected through a pine board 1 inch in thickness at a distance of 80 yards. Gunpowder. Gunpowder is distinguished as Musket, Mortar, Camion, Mammoth, and Sporting powder; it is all made in same . manner, of same proportions of materials, and differs only in size of its grain. Bursting or Explosive Energy — By the experunents of Captain Rodman U S. Ordnance Corps, a pressure of 45 000 lbs. per squaie inch was obtained with 10 of pow r der, and a ball of 43 lbs. Also, a pressure of 185000 lbs. per sq. inch was obtained when the powder ivas burned in its own volume , in a cast-iron shell having diameters of 3.85 and 12 in.. Proof of Powder. ( TJ . £. Ordnance Manual.) Powder in magazines that does not range over 180 yards is held to be unservice- able. Good powder averages from 280 to 300 yards; small grain, from 300 to 320 yards. Restoring Unserviceable Poiuder.- When powder has been damaged by being stored in damp places, it loses its strength, and requires to be worked o\er. If quantity of moisture absorbed does not exceed 7 per cent., it is sufficient to diy it to restore it for service. This is done by exposing it to the sun. When pow'der has absorbed more than 7 per cent, of w'ater it should be sent to a powder mill to be w’orked over. GUNNERY. 5O3 3 Properti.es and IResnlts of Gunpowder, determined "by Experiments. ( Captain A. Mordecai , U. S. A.) 24-PouNDER Gun. Weight of ball and wad 24.25 lbs. “ “ powder 6 “ Windage of ball 135 inch. Musket Pendulum. ball., powd Windage of ball. Weight of ball 397-5 grains. “ “ powder 120 “ 09 inch. Grain. C Salt- petre. ompositi Char- coal. on. Sul- phur. Manufacture. Where from. Number of Grains in io Troy Grains. Relative Quickness of Burning. W ater ab- sorbed by ex- posure to Air. Relative Force. Cannon, large... “ small Musket * Dupont’s Mills, 77 569 ii 34 6174 5 344 1 642 i 3 J 52 166 103 72 808 295 2378 275 314 214 142 282 Per c’t. 2.77 3-35 .677 .72 .808 Rifle ^76 TA 12 Rifle Del. 3-55 • 9°7 .728 •834 Musket Rifle Cannon, uneven. “ large... Sporting 75 12.5 T 7 12.5 IO ) t Dupont’s Mills, Del. * Dupont’s Mills, Del. Loomis, Hazard, & Co., Conn.* Waltham Abbey, 183 182 2.09 1. 91 •943 .788 •756 Blasting, uneven Rifle / / 70 1 D 15 . X?} ) 212 4.42 1 .82 qqq Sporting | 76 15 9 204 Rifle T £ ) 10 1 7 1 1 600 .OOO .865 / O A w> IO ) England. * | * Glazed. f Rough. Manufacture of Powder. —Powder of greatest force, whether for cannon or small arms, is produced by incorporation in the “cylinder mills.’ ’ Effect of Size of Grain.— Within limits of difference in size of grain, which occurs in ordinary cannon powder, the granulation appears to exercise but little influence upon force of it, unless grain be exceedingly dense and hard. Effect of Glazing. — Glazing is favorable to production of greatest force, and to quick combustion of grains, by affording a rapid transmission of flame through mass of the powder. 6 Effect of using Percussion Primers. — Increase of force by use of primers, which nearly closes vent , is constant and appreciable in amount, yet not of sufficient value to authorize a reduction of charge. Ratio of Relative Strength of different Powders for use under water differ but tittle from the reciprocal of the ratio between the sizes of the grains showing that the strength is nearly inversely proportional thereto.* Mammoth, .08; Oliver, .09; Cannon, .18; Mortar, 1; Musket, 1.57; Sporting 2.61, and Safety Compound 30.62. ’ 0 / ’ IDnalin is nitro-glycerine absorbed by Schultze’s powder. For other powders and explosive materials see Gunnery, page 443. Heat and. Explosive Power. ( Capt . Noble and F. A. Abel.) One gram of fired powder evolves a mean temperature of 730°. Temper- ature of explosion 3970°. Volume of permanent gas (which is in an in- verse ratio to units of heat evolved) at 32 0 = 250 °. The explosive power of powder, as tested in Ordnance, ranges, for volumes ot expansion of 1.5 to 50 times, from 36 to 17 o foot-tons per lb. burned. A charge of 70 lbs. gave to an 180 lbs. shot a velocity of 1694 feet per second, equal to a total energy of 3637 foot-tons, and a charge of 100 lbs. 0 a\e a velocity of 2182 feet, and an energy of 5940 foot-tons. aDd lQVe8tigati0n3 t0 devel °P a *y stera of ^marrae mines. Professional 504 HEAT. HEAT. Heat , alike to gravity, is a universal force, and is referred to both as cause and effect. Caloric is usually treated of as a material substance, though its claims to this distinction are not decided ; the strongest argument in favor of this position is that of its power of radiation. Upon touching a body having a higher temperature than our own, caloric passes from it, and excites the feeling of warmth ; and when we touch a body haying a lower temperature than our own, caloric passes from our body to it, and thus arises the sensation of cold. To avoid any ambiguity that may arise from use of the same expres- sion, it is usual and proper to employ the word Caloric to signify the principle or cause of sensation of heat. Heat U?iit — For purpose of expressing and comparing quantities of heat, it is convenient and customary to adopt a Unit of heat ov Thermal unit] being that quantity of heat which is raised or lost in a defined period of temperature in a defined weight of a particular substance. Thus a Thermal unit, Is quantity of heat which corresponds to an interval of i° in temperature of i lb. of pure liquid water, at and near its temperature of greatest density. Thermal unit in France, termed Caloric , Is quantity of heat which corresponds to an interval of i°,C. in temperature of i kilogramme of pure liquid water , at and near its temperature of greatest density. Thermal unit to Caloric, 3.96832; Caloric to Thermal unit, .251996. One Thermal unit or i° in 1 lb. of water, 772 foot-lbs. One Caloric or i° C. in 1 kilogramme of water, 423.55 kilogrammetres. i° C. in 1 lb. water, 1389.6 foot-lbs. Ratio of Fahrenheit to Centigrade, 1.8; of Centigrade to Fahrenheit, .555. Absolute Temperature , Is a temperature assigned by deduction, as an opportunity of observing it cannot occur, it being the temperature corre- sponding to entire absence of gaseous elasticity, or when pressure and vol- ume =0. By Fahrenheit it is— 461.2°, by Reaumur— 229.2 0 , and by Cen- tigrade — 274°. Heat is termed Sensible when it diffuses itself to all surrounding bodies ; hence it is free and uncombined, passing from one substance to another, affecting the senses in its passage, determining the height of the thermometer, etc. Temperature of a body, is the quantity of sensible heat in it, present at any moment. Heat is developed by water when it is violently agitated. Heat is developed by percussion of a metal, and it is greatest at the first blow. Quantities of heat evolved are nearly the same for same substance, with- out reference to temperature of its combustion. Mechanical power mav be expended in production of heat either by fric- tion or compression, and quantity of heat produced bears the same propor- tion to quantity of mechanical power expended, being 1 unit for power necessary to raise 1 lb. 772 feet in height. This number of 772 is termed the mechanical equivalent of heat (Joules). .HEAT. 505 Specific Heat. Specific Heat of a body signifies its capacity for heat, or quantity re- quired to raise temperature of a body i°, or it is that which is ab- sorbed by different bodies of equal weights or volumes when their temperature is equal, based upon the law, That similar quantities of different bodies require unequal quantities of heat at any given tempera- ture. It is also the quantity of heat requisite to change the tempera- ture of a body any stated number of degrees compared with that which would produce same effect upon water at 32 0 . Quantity of heat , therefore, is the quantity necessary to change the tem- perature of a body by any given amount (as i°), divided by quantity of heat necessary to change an equal weight or volume of water 32 0 by same amount. Note.— W ater has greater specific heat than any known body. Every substance has a specific heat peculiar to itself, whence a change of composition will be attended by a change of its capacity for heat. Specific heat of. a body varies with its form. A solid has a less capacity for heat than same substance when in state of a liquid; specific heat of water, for instance, being .5 in solid state (ice), .622 in gaseous (steam), and 1 in liquid. Specific heat of equal weights of same gas increases as density decreases ; exact rate of increase is not known, but ratio is less rapid than diminution in density. Change of capacity for heat always occasions a change of temperature. Increase in former is attended by diminution of latter, and contrariwise. Specific heat multiplied by atomic weight of a substance will give the constant 37.5 as an average, which shows that the atoms of all substances have equal capacity for heat. This is a result for which as yet no reason has been assigned. Thus: atomic weights of lead and copper are respectively 1294.5 and 205 7 and their specific heats are .031 and .095. Hence 1294.5 X .031 = 40.129, and 305 7 x •095 = 37- 59 1 - It is important to know the relative Specific Heat of bodies. The most conve- nient method of discovering it is by mixing different substances together at dif- ferent temperatures, and noting temperature of mixture; and by experiments it appears that the same quantity of heat imparts twice as high a temperature to mercury as to an equal quantity of water; thus, when water at ioo° and mercury at 40 0 are mixed together, the mixture will be at 8o°, the 20 0 lost by the water causing a rise of 40 0 in the mercury; and when weights are substituted for meas- ures, the fact is strikingly illustrated; for instance, on mixing a pound of mercury 1 mi 4 °° wit, k a P oun fi °f water at 160 0 , a thermometer placed in it will fall to ic S o Thus it appears that same quantity of heat imparts twice as high a temperature to mercury as to an equal volume of water, and that the heat which gives 5 0 to water will raise an equal weight of mercury 1150, being the ratio of 1 to 23. Hence if equal quantities of heat be added to equal weights of water and mercury their temperatures will be expressed in relation to each other by numbers 1 and 23- or in order to increase the temperature of equal weights of those substances to the same extent, the water will require 23 times as much heat as the mercury. Capacity for Heat is relative power of a body in receiving and re- taining heat in being raised to any given temperature ; while Specific applies to actual quantity of heat so received and retained. Specific Heat of ^Air and. other G-ases. Specific heat, or capacity for heat, of permanent gases is sensibly constant 1 for a11 temperatures, and for all densities. Capacity for heat of each gas is U u HEAT, 506 same for each degree of temperature. M. Regnault proved that capacity for heat for air was uniform for temperatures varying from — 22 0 to +437°; consequently, specific heat for equal weights of air, at constant pressure, averaged .2377. Metals from, 32 0 to 212 °. Antimony. . . .0508 Bismuth 0308 Brass °939 Copper 092 Cast iron 1298 Gold °3 2 4 Lead 0314 Mercury Nickel 1086 Platinum 0324 Specific Heat. Water at 32 0 = 1. Woods. Oak 57 1 Pear 5 Pine 65 Silver 056 Steel 1165 Tin 0562 Wrought iron .1138 Zinc 0955 Stones. Chalk .... Limestone Masonry. . Marble, gray. .2694 “ white. 2158 . 2149 . .2174 Mind Substances. Charcoal 2415 Coal 2411 Coke 203 Glass 1977 Gypsum 1966 Phosphorus.. 2503 Sulphur 2026 Liquids. Alcohol 6588 EtheV 4554 Linseed oil . . .31 Olive oil 3096 Steam ... .365 Turpentine . . -4 l6 Vinegar 72 Solid. Ice 504 Gases. Air. •2377 Hydrogen 2356 Carbonic Acid 3308 Hydrogen 2.4096 Carbonic Acid 1714 Oxygen 2412 For Equal Weights. Air 1688 Oxygen 1559 Metals have least, ranging from Bismuth .0308 to Cast Iron .1298. Stones and Mineral Substances have .2 that of water, and Woods about .5. Liquids, with ex- ception of Bromine, are less than water, Olive oil being lowest and Vinegar highest. Illustration.— If 1 lb. of coal will heat 1 lb. of water to ioo°, — — = — — of a lb. • 033 30.3 will heat 1 lb. of mercury to ioo°. To Compute Temperature of’ a Mixture of* lilie Sut>- stances. W T + w t ■.t w (t' — t) W: w ( t ' — t) + f = T. W representing weight W-fw ’ T — t' ’ W or volume of a substance of temperature T, w weight or volume of a like substance of temperature £, and t' temperature of mixture W -j- w. Illustration i. — When 5 cube feet of water (W) at a temperature of 150 0 (T) is mixed with 7.5 cube feet (10) at 50 0 ( t ), what is the resultant temperature of the mixture? 5 X 150° + 7.5 X 50 0 5 + 7-5 1125 ' 12.5 — 90^. 2. — How much water at (T) ioo° should be mixed with 30 gallons (w) at 6o c , for a required temperature of 8o°? 30(80° — 6o°) 600 • 5 — 0 - 0 - = — = 3° gallons. ioo° — 8o° 20 To Compute Temperature of* a Mixture of TJnlilre Substances. W S T + w .s t — t'; 10 S (t-r-t') — W; t' ( W S + w s) w s t = T. W and w W S-fu)i ’ S (T — t) WS representing weights , and S and s specific heat of substances. Illustration.— To what temperature should 20 lbs. cast iron (W) be heated to raise 150 lbs. ( w ) of water to a temperature ( t ) of 50° to 6o°? : 1, and S = . 1298. 6o° (20 X • 1298 + 150 X 1) 'v 150 X 1 X 50 0 1655.76 20 X .1298 2.596 = 638°. HEAT. 507 To Compute Specific Heat at Constant Volume. S p When Specific Heat at Constant Pressure is known. ~w = s - s represent- ing specific heat at constant pressure, p proportion of heat absorbed at constant vol- ume, H total heat absorbed at constant pressure, and s specific heat at constant volume. Or S (t' Q— -2.742 (V v) _ s t and t > re p re senting initial and final tempera- ture of the gas and that to which it is raised, and V and v initial and final volumes of the gas under 14.7 lbs. per sq. inch, and of it heated under constant pressure xn cube feet. Illustration. — Assume 1 lb. air at atmospheric pressure and at 32 0 , doubled in volume by heat. S = . 2377 *, t - 1' = 32° to evaporate H 1 lb. of wa- ter. Total lost per hour. Tempera- ture. Water evapor’d per sq. foot of surface p’r hour. lost by radia- tion from surface. HE; h ‘3 .5 to evaporate H 1 lb. of wa- ter. O Lbs. Units. Units. Units. Units. 0 Lbs. Units. Units. Units. 32 .027 — 1091 29 132 .706 182 202 1506 42 .04 270 424 1788 7 1 142 .916 158 162 1445 52 .058 375 581 2052 119 152 1.178 137 127 1392 62 .083 4°5 605 2110 i 74 162 I * 5°5 118 97 1346 72 .117 386 566 2055 239 172 1895 106 72 1312 82 . 162 358 504 1968 3*9 182 2-373 ¥ 50 1279 92 .223 3*9 434 l862 4 i 5 192 2.947 81 32 1253 102 • 3°3 280 366 1758 533 202 3-633 l l H 1228 1 12 .406 245 3°4 1664 671 212 4.471 63 — 1209 122 .528 211 250 1580 849 — — — ““ H e. Unit9. 1068 1326 1637 2039 2475 3°45 3685 4465 5397 To Compute Surface of a Refrigerator. Illustration of Table. — If it is required to cool 20 barrels, of 42 gallons each, of beer, from 202° to 82° in an hour. 4 Result to be attained is to dissipate 42 X 8.33 (lbs. U. S. gallons) X 20 X 202 — 82 = 840000 units of heat per hour. At 202°, 4465 units are lost, and at 82°, 319, hence, average loss for each temper- ature between oxtremes ;= 1850 units per sq. foot per hour. Then *4 ooc ?? ^ 4 ^ 4 sq.feet in a still air. 1850 The volume of air required per hour in this case would be about 100000 cube feet. HEAT. 513 To Compute Area of Grrate and. Consumption of* Eiiel for Evaporation. Illustration of Table .— If it is required to evaporate 6 Beer gallons (282 cube ins.) of liquid per hour, at a temperature not exceeding 152 0 . 6 gallons = 50 lbs. At 152 0 , water evaporated as per table = 1. 178 lbs. per hour. -- — = 42 sq.feet. Heat required to effect this = 1392 X 50 = 69 600 units. 1.178 Assuming 6000 units as average economic value of coals, then ^ 6 °° — 11.6 lbs. coal, on a grate of 1 sq.foot. When it is practicable to evaporate at a high temperature, as at or' above 212 0 , it is most economical. Thus, water requires only 1209 units per lb. if surface is exposed, but if enclosed, heat is reduced (1209 — 63) to 1146 units. Evaporative Powers of Different Tubes per Degree of Heat, per Sq. Foot of Surface. — In Units. Vertical tube, 230; Double-bottomed vessel, 330; Horizontal tube or Worm, 430. To Compute Volume of Water Evaporated in a given Time. Illustration.^- What is volume evaporated at 212 0 , in 15 minutes per sq. foot of surface, in a double-bottomed vessel having an area of heating surface of 17 feet, and subjected to steam at a pressure of 25 lbs. ? Temperature of steam at 25 -f 14.7 lbs. = 269°. 269° — 212 0 = 57°, and latent heat 927. Then 330X57X17X15 = g6 2 m wafer> 927 x 60 When Water is at a Lower Temperature than 212 0 . If 120 gallons or 1000 lbs. of water were to be evaporated from 42 0 in an hour, from same vessel and under like pressure as preceding : There would be required 1000 X (212 0 — 42 0 ) 170000 units of heat. Mean tempera- ture of water while being heated 1= 4 2 ° + 212° _ Q 2 ' Difference between temperature of steam and water = 267° 127 0 = 140 0 . 170000 Then, ^ w — • 216 hour =Z time to raise water to 212 0 ; hence 1 — . 216 = 33® s' 140 17 .784 hour left for evaporation, and quantity evaporated = 330 * 57 X 17 X -7 8 4 _, 270. 4 lbs. , or 32. 44 gallons. Dessiccation. Dessiccation , or the drying of a substance, is best effected in a drying chamber, and it is imperative that to attain greatest effect the hot air should be admitted at highest point of exposed substance and dis- charged at its lowest. W ood, submitted to an average temperature of 300° in an enclosed space for a period of 2.5 days, will lose its moisture at a consumption of 1 lb. of vood for 10.5 lbs. of wood dried, and evaporating 4 lbs. of water, equal to 2.66 lbs. of water per lb. of undried wood. Limit of. temperature for drying of wood is 340°. 5 14 HEAT. Evaporation of Water per Sq. Eoot of Surface per Hour. Temperature of Water. 1 Calm. Evaporation Light Air. (Dr. D Brisk Wind. 1 alton . ) Temperature of Water. 1 Calm. Evaporation Light Air. Brisk Wind. O 32 40 50 60 70 80 The rates ol Lbs. •0349 •0459 •0655 •°9 I 7 •1257 .1746 ? evaporat Lbs. .0448 .0589 .0841 •1175 .1616 .2241 :ion for tl Lbs. .055 .0723 .1032 .1441 .1983 •2751 iese cond 0 100 125 150 W 5 200 212 itions of the ai Lbs. .3248 .6619 1.296 2.378 4.128 5-239 r when pi Lbs. .4169 .8494 1.663 3- 053 5.298 6.724 erfectly d Lbs. .5116 1-043 2.043 3-746 6. 502 8.252 ry are as i, 1.28, and 1.57. To Compute Quantity of Water exposed to Air that would he evaporated as above . — Subtract tabulated weight of water corresponding to dew-point from weight of water corresponding to temperature of dry air, and remainder is weight of water that would be evaporated per sq. foot of surface per hour. Distillation. Distillation is depriving vapor of its latent heat, and, though it may be effected in a vacuum with very little heat, no advantage in regard to a saving of fuel is gained, as latent heat of vapor is increased propor- tionately to diminution of sensible heat. A temperature of 70° is sufficient for distillation of water in a vessel ex- hausted of air. Conduction or Convection of Heat. Air and gases are very imperfect conductors. Heat appears to be transmitted through them almost entirely by conveyance, the heated portions of air becoming lighter, and diffusing the heat through the mass in their ascent. Hence, in heating a room with air, the hot air should be introduced at lowest part. The advantage of double win- dows for retention of heat depends, in a great measure, upon sheet of air confined between them, through which heat is very slowly transmitted. Convection of heat refers to transfer and diffusion of heat in a fluid mass, by means of the motion of the particles of the mass. Relative Internal Conducting Rowers of Various Substances. Brass . .76 Cast Iron - -517 Copper . .89 Cement Chalk .. .6 Charcoal .. .07 Slate, Metals. Gold 1 I Porcelain .... .012 Lead 18 Silver •97 Platinum 98 1 Terra Cotta. . .011 Minerals. Coal, anth’cite 1.92 Fire brick. . . . .61 “ bitumin. 1.68 Fire clay . .76 Coke 1.98 Glass . .96 Wood ash. Tin Wrought Iron Zinc Gypsum, Lime. . . Marble . 08 Woods with Birch — .41 with Silver. •3 •44 •36 .2 .24 1.22 Apple Ash. . Cotton Eider down. . . .68 I •73 1 Birch 1 Chestnut 7 Ebony Elm :%\ I Oak Pine Hair and Fur with Air = 1. -55 I •44 1 I Flannel 2.44 1 Hemp Canvas. .28 | Hair | Hare’s fur — : 2 4sl I Silk 1 Wool ... .5 Liquids with Water. Alcohol 93 I Proof spirit 85 I Turpentine 3.1 Mercury 2.8 | Sulphuric acid 1.7 | Water z HEAT. 515 Practical Deductions from preceding Results. Asphalt is best composition for resisting moisture, and, being a slow con- ductor of heat, it is best adapted for economy of heat and dryness. Slate is a very dry material, but, from its quick conducting power, it is not adapted for retention of heat. Cements. — Plaster of Paris and Woods are well adapted for lining of rooms, having low conductive powers, while Hair and Lime , being a quick conductor, is one of the coldest compositions. Fire-brick absorbs much heat, and is well adapted for lining of fire-places, etc. ; while Iron , being a high conductor of heat, is one of the worst of sub- stances for this purpose. Common brick is not a very slow conductor of heat. CoixLinniaicat ion. Communication of Heat is passage of heat through different bodies with different degrees of velocity. This has led to the division of bodies into Conductors and Non-conductors of caloric; the former in- cludes such as metals, which allow caloric to pass freely through their substance, and the latter comprise those that do not give an easy pas- sage to it, such as stones, glass, wood, charcoal, etc. The velocity of cooling, other things being equal, increases with the extent of surface compared with volume of substance; and of two bodies of same material, temperature, and form, but differing in volume. Condensation. Tredgold ascertained by experiment that steam at pressure (absolute) of 17.5 lbs. per sq. inch, 22 1°, produced 1 cube foot of water per hour by condensation in 182 sq. feet of cast-iron pipe, at a uniform and qui- escent temperature of 6o°. Hence, condensation .352 lb. water per hour, or .0022 lbs. per degree of difference of temperature (221—60). From experiments of Mr. B. G. Nichol in England, 1875, it was deduced: That rates of transmission of heat, between temperature of steam and that of water of condensation at its exit, at the rate of 150 feet per minute, may be taken as 380 units for vertical tubes and 520 for horizontal. Condensation of Steam in Cast-iron IPipes. (M. Burnat.) Average Press, per Temperature. Condensation per sq . foot of external surface of pipe per hour. Sq. Inch. Steam. Air. Difference. Bare. Straw. Pipe. Waste. Plaster. Lbs. O 0 0 Lb. Lb. Lb. Lb. Lb. 22 233 36.5 i9 6 -5 ,.581 .2 .229 .286 •324 From these data, following constants are deduced for an absolute pressure of 22 lbs. per sq. inch of steam condensed, and heat passed off per sq. foot of external surface of pipe per hour of i° difference of temperature. Surface of Pipe, Steam condensed per Sq. Foot. Heat passed off. Surface of Pipe. I Steam condensed per Sq. Foot. Heat passed off. Bare pipe Lb. Units. 2.812 .968 1. 108 Cotton waste 1 inch. . Earth and hnir Lb. .001 46 .001 65 .001 56 Units. 1.384 1.S68 1.486 Straw coat .003 .001 02 Cased with clay pipe. . . .001 15 White paint HEAT. 516 Pipes were 4.72 ins. diameter, .25 inch thick, and had area of 58.5 sq. feet, Bart rough surface as cast. Straw coat — laid lengthwise .6 inch thick and bound. Pipe — laid in clay pipe with an air space between them, the whole covered with loam and straw. Waste cotton— 1 inch thick and bound with twine. Plaster— laid in clay and hair 2.36 ins. thick. A wrought-iron pipe 3.75 ins. in external diameter, .25 inch thick, and lagged with felt and spun yarn .5 inch thick, condensed steam at 245 0 at rate of .262 lb. per sq. foot per hour, in an external temperature of 6o°. Steam Condensed per Sep Foot and per Degree per Hour. Mean Results of several Experiments with hare Cast-iron Pipes , with Steam at Absolute Pressure of 20 lbs. per Sq. I?ich. .4 lb. per sq. foot, and .002 39 lb. per degree. Hence, to ascertain quantity of heat lost by condensation of .002 39 lb. = — of a lb. Difference of total and sensible heats of 1 lb. steam at 20 lbs. absolute pressure = II 5 I o_j_ 32 o — 228° = 955 units, and 955-4-420 = 2.274 units = heat condensed. The loss of heat from a naked boiler in air at 62°, under an absolute pressure of 50 lbs. per sq. inch, was 5.8 units. Congelation, and. Inqinefaction. Freezing water gives out 140° of heat. All solids absorb heat when becoming fluid. Particular quantity of heat which renders a substance fluid is termed its caloric of fluidity, or latent heat. Temperature of Solidification of Several Gases. [Faraday.) Cyanogen 31 0 I Ammonia 103° I Sulphuretted Hydrogen, 123° Carbonic Acid 72°! Sulphurous Acid. . . 105 0 | Protoxide of Nitrogen. . 148 Frigorific Mixtures. Mixtures. I I Fall of Parts. Temperature. Sea salt Nitrate of ammonia . . Snow, or pounded ice. Muriate of ammonia 1 Nitrate of potash ) Snow, or pounded ice. Phosphate of soda. . . . Nitrate of ammonia . . Dilute mixed acids . . . Snow Crystallized muriate 1 of lime j Snow Dilute sulphuric acid . . Phosphate of soda Nitrate of ammonia . . Dilute nitric acid Snow Dilute nitric acid —18 to —25 — 5 to — 18 -34 to -5o —40 to —73 — 68 to —91 o to —34 o to — 46 Nitrate of ammonia Water Snow Dilute sulphuric acid Sulphate of soda. . Diluted nitric acid Nitrate of ammonia Carbonate of soda. Water ... Sulphate of soda. . Muriate ofammoni Nitrate of potash. . Dilute nitric acid . Phosphate of soda. Dilute nitric acid . Snow Muriate of lime. . . Potash Snow Full of Temporature. -f5°t 0 -j-4 j — 10 to — 60 +50. to —3 +5° to — 7 -{-50 to — 10 -j-50 to — 12 -{-20 to — 48 +32 to —51 A Mixture of Solid Carbonic Acid and Sulphuric Ether, under receiver of an air- pump, under pressures of .6 lbs. to 14 lbs., exhibited a temperature ranging no I07 o iq — !66o which is the most intense cold as yet known. [Faraday.) HEAT. 517 Melting-points. Metals. Alloys. Aluminium at red heat. . Antimony Arsenic Bismuth Bronze Calcium at red heat Copper Gold, pure “ standard Iron, cast . 2d melting. Wrought malleable forge. Lead. . Lithium Mercury Platinum Nickel, highest forge heat. Potassium Silver Sodium Steel Tin Zinc Alloys. Lead 2, Tin 3, Bismuth 5. “ “ 3? “ 5- 810 365 476 1692 1996 ( 2282 (2590 2156 2000 2250 3479* 2200 2450 3700* 2700 2912 3509 * 608 356 —39 3080 •136 (1250 (1873 194 2500 446 680 212 210 Lead j Tin 4, Bismuth 5. “ 3 “ 2, Bismuth 5. (solder) (soft solder) . Tin 1, “ 1, Bism. 4, Cadm. 1 , Bismuth 1 Zinc 1, Tin 1. . Pnsi'ble Plugs. Lead 2, Tin 2 ■“ 6, “ 2 Various Substances. Ambergris Beeswax Carbonic acid Glass Ice Lard Nitro-Glycerine Phosphorus Pitch Saltpetre Spermaceti Stearine Sulphur Tallow Wax, white * Rankine. 240 334 199 552 475 360 368 155 286 336 39 2 399 372 383 388 410 145 151 — 108 2 377 32 95 45 112 9 1 606 112 114 239 9 2 142 Volume of Water, Antimony, and Cast iron, in the solid state, exceeds that of the liquid, as evidenced by the floating of ice on water, and of cold iron on iron in a liquid state. IB oiling-p oint s . Liquids. Alcohol, s. g. 813 Ammonia Benzine Chloroform Ether Linseed oil Mercury Milk Nitric acid, s. g. 1.42 “ “ “ 1.5 Oil of Turpentine Petroleum, rectified Phosphorus Sea water, average Sulphur Sulphuric acid, s. g. 1.848 “ ether 3 ( Under One Atmosphere.) Liquids. W3 140 W3 146 100 597 648 213 248 210 315 316 554 213.2 570 59° 240 100 Turpentine Water “ in vacuo Whale oil Saturated Solutions. Acetate of Soda “ “ Potash Brine Carbonate of Soda “ “ Potash Nitrate of Soda “ “ Potash Salt, common Various Substances. Coal Tar. Naphtha . 315 212 j* 630 255-8 336 226 220.3 275 250 240.6 227.2 ji8 heat. Boiling-points of Saturated -Vapors under "Various Pressures. ( Regnault .) Temper- ature. Water. Alcohol. Ether. Chloro- form. Temper- ature. Water. Alcohol. Ether. Chloro- form. 0 Lbs. Lbs. Lbs. Lbs. O Lbs. Lbs. Lbs. Lbs. 32 .089 .246 3-53 — 212 14.7 32.6 95-17 45-54 5° .178 .466 5-54 2.52 230 20.8 45-5 120.9 58.42 68 •337 • 8 5 i 8.6 3.68 240.8 25-37 -rr 137 Turp’tine. 86 .609 1-52 12. 32 5-34 248 29.88 62.05 — 4-97 104 1.06 2-59 17.67 7-°4 266 39-27 83.8 — 6.71 122 1.78 4.26 24-53 10.14 276.8 46.87 — — — 140 2.88 6.77 33-47 14.27 284 52.56 109. 1 — 8.94 158 4-5i 10.43 44.67 18.88 302 69.27 140.4 — 11.7 176 6.86 15.72 57- CI 26.46 3°5-6 73- °7 147-3 — 194 10.16 23.02 75-4i 35-°3 320 89.97 ” 13 - 1 Boiling-points of Water corresponding to Altitudes of Barometer between 62 and 31 Ins. Barom. Boiling-point. Barom. Boiling-point. Barom. Boiling-point. Barom. Boiling-point. 26 26.5 27 0 204.91 205.79 206.67 27-5 28 28.5 O 207. 55 208.43 209.31 29 29-5 3° O 210.19 211.07 212 3°-5 32 O 212.88 213.76 Boiling-point of Salt water, 213.2 0 . Water may be heated in a Digester to 400° without boiling. Fluids boil in a vacuum with less heat than under pressure of atmosphere. On Mont Blanc water boils at 187° ; and in a vacuum water bods at 98 to ioo°) according as it is more or less perfect. Water maybe reduced to 5 0 if confined in tubes of from .003 to .005 inch ^in diam- eter: this is in. consequence of adhesion of water to surface of tube, interfering^ a change in its state. It may also be reduced in its temperature below 32 if it is kept perfectly quiescent. Effect upon Various Bodies toy- Heat. Wedgewood’s zero is 1077° (Fahrenheit), and each degree = 130°. In designation of degrees of temperature, symbol -f- is omitted when temperature is above o; but when below it, symbol — must be prefixed. 78 Degrees Acetification ends Acetous fermen-) tation begins. . j Air Furnace 3300 Ammonia (liq.) freezes — 46 Blood (hum. ), heat of. 98 “ freezes. 25 Brandy freezes —7 Charcoal burns 800 Cold, greatest artific. —166 “ “ natural — 56 Common fire 790 Fire brick 4000 to 5000 Gutta-percha softens. . 145 Heat, cherry red 1500 11 “ (Daniell) 1141 “ bright red i860 “ red, visible by J IQ77 “ white 2900 117 293 752 Degrees Highest natural tern- 1 perature, Egypt . . j India-rubber and ) Gutta-percha vul- J canize ) Iron, bright red in| the dark ) Iron, red hot in twi- ) gg light ) 4 Iron, wrought, welds. .2700 Ignition of bodies 750 Combustion of do. . . 800 Mercury volatilizes. . . 680 Milk freezes 30 Nitric Acid (sp.grav. ) 1.424) freezes J ^ Nitrous Oxide freezes— 150 Olive-oil freezes 36 Petroleum boils 306 Degrees. Sea-water freezes 28 Snow and Salt, equal) Q parts ) Spirits Turpen. freezes 14 Steel, faint yellow 43° full “ 470 purple 53° blue 55° full blue 560 dark “ 600 polished, blue . . 580 “ straw color 460 Strong Wines freeze. . 20 Sulph. Acid (sp. grav. ) 1.641) freezes J ™ Sulph. Ether freezes. .—46 Vinegar freezes 28 Vinous ferment. ..60 to 77 Zinc boils 1872 Wood, dried 34° Proof Spirit freezes. . . —7 Volume of* Several Biqnids at tlieir Boiling-point. Water i i Alcohol. . . . *‘“28 | 1 Ether j . Turpentine HEAT. 519 TIeiglit corresponding to Boiling-point of* Pure Water. Boiling-point at Level of Sea — 212 0 . Degree. Feet. | Degree. Feet. Degree. | Feet. J] Degree. Feet. Degree. Feet. 2x1 52 i 207 2625 203 4761 199 6929 195 9129 210 1044 206 3156 202 5300 198 7476 194 9 684 209 1569 205 3689 201 5841 197 8025 193 10 241 208 2096 204 4224 200 6384 1 196 8576 192 10800 Correction for temperature of air same as given at page 428 for Elevation by a Barometer by multiplying by C. Illustration.— If water boils at a temperature of 200 0 and C = 136°, Then 6384 X 1.08 = 6894.72 feet. Underground Temperature. Mean increase of underground temperature per foot, from observations in 36 mines in various and extended localities, is .01565° = i° in 64 feet. Linear Expansion or Dilatation of* a Bar or Prism "by- Beat. For i° in a Length of 100 Feet. Metals. Minerals, etc. Inch. Antimony 00722 Bismuth 009 28 Brass 012 5 “ yellow 0126 Brick 001 44 Cast Iron 0074 Cement 009 56 Copper from o° to 212 0 on 5 u from 32 0 to 572 0 00418 Fire brick 003 3 Glass 005 74 “ flint 00541 “ tube 0612 Gold— Paris standard annealed.. .0101 “ “ “ unannealed .0103 Granite 005 25 Gun Metal— 16 copper -}- 1 tin... .0127 “ “ 8 copper -j- 1 tin. . . .0121 Ice 0333 Iron, forged 008 14 11 from o° to 212 0 007 88 Inch. Iron, from 32 0 to 572 0 00326 Iron wire 00823 Lead 019 Marble 005 66 Palladium 006 67 Platinum 005 71 “ from 32 0 to 572 0 00204 Sandstone 013 11 008 14 Silver 012 7 Speculum metal 013 Steel, rod 007 63 “ cast 007 2 “ tempered 00826 “ not tempered 007 19 Tin 0145 Water 000 222 9 White Solder— tin 1 + 2 lead. . .016 7 Zinc, forged 0207 “ sheet 0196 “ 8 -f- 1 tin 017 9 Superficial expansion is twice linear, and cubical, three times linear. To Compute Linear Expansion oP a Substance. Divide 1 by decimal given in above Table, and quotient will give pro- portion. Illustration l— A rod of copper 100 feet in length will expand between tem- peratures of 32 0 and 212 0 . 212 — 32 = 180 X -0115 = 2.07 ins. 2. — A cube of cast iron of 1 foot will expand in volume between temperatures of 62° and 212 0 . 212 — 62 = 150, and 150 X .0074 = i.n, which - 4 - 100 for 1 foot = .0111 inch, and i2-J-.om X 3 = 12.0333 ins. Some solids, as ice, cast iron, etc., have more volume when near to their melting- point than when melted. This is illustrated in floating of solid metal in the liquid. Expansion, of Water. Water expands from temperature of maximum density (see page 520), 39. 1 °, to 46°, at which degree it regains its initial volume of 32°, and from thence it expands under one atmosphere to 212° ; and its cubical expansion is .0466, that is, its volume is dilated from 1 at 32° to 1.0466 at 212 0 . Its expansion increases in a greater ratio than that of temperature. HEAT. To Compute Density of Water at a given Temperature, 62.5 X2 — approximate density , t representing temperature of water. £4-461 + 500 500 ' 1 4- 461 Illustration.— What is density . of pure water at 298° ? 2984-46 62.5 X 2 + - 500 500 ' 2984-461 Expansion of W a ter*. (Dalton.) = 57.42 lbs. or weight of 1 cube foot. Temp. Expansion. Temp. Expansion. Temp. Expansion. | Temp. Expansion. 0 22 1.0009 0 52 1. 000 21 0 112 1.008 8 0 172 1-02575 32 1 72 1. 001 8 132 1.01367 192 1.03265 *46 1 92 1.00477 152 1.01934 212 1.0466 * Greatest density 39.1 0 . Hence, at 72 °, water expands — ~ = 555- 55^ P art of its original bulk. Expansion of Eiqnids from 32° to 212°. Volume at 32 0 Liquids. Volume at 212 0 . Liquids. I Volume at 212 0 . 7 i i i 1. 11 1.08 1.0154 1.018433 1 1. 018 867 9 1. 11 Olive oil 1.08 1.06 1.07 1.07 1.046 6 1.05 Linseed oil Mercury “ 212 ° tO 392°. . . . . 11 39 2 ° to 57 2 0 Nitric acid Sulphuric acid “ ether.. Turpentine Water Water sat. with salt Expansion of Gases from 32° to 212°. Volume at 32°= 1. Gases. Volume at 212 0 . Gases. Volume at 212 0 . Air 1 Atmosphere. . 3-45 \\ Hydrogen 1 3-35 “ Carbonic acid, 1 3-32 “ 1.367 06 1.369 64 1.366 13 1.366 16 1.37099 i- 3 8 4 55 Nitrous oxide . . . 1 Atmosphere. . Sulphurous acid, 1 “ 1.16 “ Carbonic oxide .. 1 “ Cyanogen 1 “ I-3W9 i- 39°3 1. 398 1.3669 I-3S77 O’ O" - - - -t w Expansion of Gases is uniform for all temperatures. Volume of One Pound of Various Gases at 32 0 under one Atmosphere . ^ 1 _ r i fa/it Cube Cube feet. Air 12.387 Carbonic acid 8. 101 Ether, vapor 4.777 Expansion of .Adr Hydrogen. Nitrogen. . Olefiant. . . Cube feet. 78.83 12.753 12.58 Cube feet. Oxygen 12.205 Mercury 1.776 Steam 19. 913 (Dalton.) ’ Temp. Expan- sion. Temp. Expan- sion. Temp. Expan- Temp. Expan- sion. Temp. Expan- sion. Temp. | 0 0 0 0 0 0 3 2 1 4° 1. 021 60 1.066 80 1. no 100 1. 152 39 2 33 1.002 45 1.032 65 1.077 85 1.121 200 i-354 482 34 1.004 50 1.043 70 1.089 90 1. 132 212 i-37 6 680 35 1.007 55 1-055 75 1.099 95 1 142 302 i -558 772 i-739 1. 912 2.028 2.312 To Compute Volume of a Constant Weight of Adr or Permanent Gas for any Pressure. JT G at jlwa -A- A — When volume at a given pressure is Icnoicn , temperature remaining con- stant. Rule. — Multiply given volume by given pressure and divide by new pressure. lib W J II Loo U l C* Example. — P ressure at 212° = 18.92 lbs. per sq. inch, and volume 16.91 cube feet; what is volume at pressure of 13.86 lbs. 16.91 X 13.86-7-18.92 = 12.39 cube feet. HEAT, 521 Relative Densities of some Vapors. Water 1. Alcohol 2.59. Ether 4.16. Spirits of Turpentine 8.06. Sulphur 3.59. Volume, Pressure, and Density of Air at Various Tem- peratures. Volume and Atmospheric Pressure at 62° = 1. Volume of Density, or Temper- ature. 1 lb. of air at weight of one atmospheric pressure of weight of cube foot of air at 14.7 lbs. 14.7 lbs. O Cube feet. Lbs. per Sq. Inch. Lbs. 0 11583 12.96 .086331 32 12.387 13.86 .080 728 40 12.586 14. 08 •079 439 50 12.84 14.36 .077 884 62 13.141 14.7 .076097 70 13-342 14.92 •074 95 80 i 3 - 593 15.21 •073 565 90 13-845 15-49 .072 23 100 14.096 15-77 .070942 120 14.592 16.33 .068 5 140 i 5 - 1 16.89 .066 221 160 15-603 17-5 .064 088 180 16.106 18.02 . 062 09 200 16.606 18.58 .06021 210 16.86 18.86 •059 313 212 16.91 18.92 •059135 220 17.111 19.14 .058 442 240 17.612 19.7 •056774 260 18. 116 20.27 •0552 280 18.621 20.83 •053 71 300 19. 121 21.39 .052 297 320 19.624 21.95 .050959 340 20.126 22.51 .049 686 Temper- ature. Volume of 1 lb. of air at atmospheric pressure of 14.7 lbs. Pressure of a given weight of Density, or weight of one cube foot of air at 14 7 lbs. 0 Cube feet. Lbs. per Sq. Inch. Lbs. 360 20.63 23.08 .048 476 380 21. 131 23.64 •047 323 400 21.634 24.2 .046 223 425 22.262 24.9 .04492 450 22.89 25.61 .043 686 475 2 3 - 5 l8 26.31 •04252 500 24. 146 27.01 .041 414 525 24- 775 27.71 ■040364 550 25-403 28.42 •039365 575 26.031 29. 12 .038415 600 26.659 29.82 •03751 650 2 7 - 9 I 5 3 I -23 .035 822 700 29.171 32.635 •034 28 750 30.428 34-04 .032 865 800 31.684 35-445 •031 561 850 32.941 36.85 •030358 900 34-197 38.255 .029 242 950 35-454 39.66 .028 206 1000 36.811 41.065 .027241 1500 49-375 55 -H 5 .020 295 2000 61.94 69.165 .016 172 2500 74-565 83-215 .013441 3000 87-13 97.265 .011499 To Compute Volume of a Constant Wei "-lit of Air or other Permanent Gras for any other Pressure and Temperature. When volume is known at a given pressure and temperature. Rule. — M ul- tiply given volume by given pressure, and by new absolute temperature, and divide by new pressure, and by given absolute temperature. Example.— G iven volume 16.91 cube feet, pressure 13.86 lbs., and temperature 32 0 ; what is volume at this temperature? Temperature for volume 16.91 =212°. 16.91 x 13 86 X 32 -f- 461 -j- 13.86 X 212 -j- 461 == 12.39 f ee t. To Compute Pressure of a Constant Weight of Air or other Permanent Gras for any other Volume and Temperature. When pressure is known for a given volume and temperature. Rule. Multiply given pressure by new absolute temperature, and divide by given absolute temperature. Note.— A bsolute temperature is found by adding 461° to temperature. Example.— Given pressure r 3 .86 lbs., and temperature at this volume 32 °- what is pressure at temperature of 212 0 ? * 13.86 X 212-1-461 -4- 32 461 = 18.92 lbs. X x* HEAT. 522 To Compute Volume of a Constant Weight of Air or other Permanent Gras at any Temperature. When volume at a given temperature is known , pressure being constant . Rule.— Multiply given volume by new absolute temperature, and divide by given absolute temperature. Absolute zero-point by different thermometrical scales is: Fahrenheit — 461.2°; Reaumur —219. 2 0 ; Centigrade —274 0 . Example. — Volume of 1 lb. air at 32° = 12.387 cube feet; what is its volume at 212 0 ? 12.387 X 212 + 461-7-32 + 461 = 16.91 cube feet To Compute Increased. Volume of a Constant Weight of Air. When initial volume at 62° = 1 under 1 atmosphere . Rule.— T o given temperature add 461, and divide sum by 523 (32 + 4 ^ 1 )* Example. — Assume elements of preceding case. 2I2 o _j_ 4 6 1 -7- 523 = 1.287 comparative volume to 1. To Compute Pressure of a Constant Weight of Air or other G^s at 62 °, and at 14.7 lhs. Pressure per Sq. In., with Constant Volume, for a given Temperature. Rule. — Add 461 to given temperature, and divide sum by 35 . 58 . Example.— Temperature is 212 0 ; what is pressure? 212 + 461 -7- 35. 58 = 18. 92 lbs. To Compute Volume, Pressure, Temperature, and Density of Air. ^-f-46] p 2.71 P 2.71 ,__y. * 4 — — V • V 2.7074P — 461 — t\ and ’ 39- 8 v 2.71 __ — d. t representing temperature , p pressure in lbs. per sq. inch , V vol- ' ^ — l - 461 _ . 7 ume in cube feet, and D weight of 1 cube foot at 14.7 lbs. per sq. inch. Product of volume and pressure of a constant weight of air, or any other permanent gas, is equal to product of absolute temperature and a coefficient, determined for each gas by its density. Or, V » = C £ + 461. Coefficients, as determined by volumes and consequent densities.* Air 2.71 1 Hydrogen 1875 | Oxygen 2.99 Carbonic acid 4 -H Nitrogen 2.63 Mercury 18.88 Ether, vapor. 7.02 | Olefiant 2.67 | Steam i-°o * See D. K. Clark, London, 1877, page 349. From 1 Decrease of Temperature by Altitudes. to 1 000 feet, “ 10000 “ . “ 20000 “ In dear skt/. With cloudy sky. 1° in 139 feet i° in 222 feet. 1° 11 288 “ i° “ 33 i “ i° “ 365 “ i° “ 468 rp 0 Compute Temperature to which a Substance of a given Length or Dimension must he Submitted or Reduced, to give it a Greater or Less Length or vol- ume by Expansion or Contraction. Lineal. — When Length is to be increased. - ^ "H — T - L and l represent- ing lengths of increased and primitive substance in like denominations , T and t tem- peratures of L and l , and C expansion of substance for each degree of heat HEAT. 523 Illustration A copper rod at 32 0 is 100 feet in length ; to what temperature must it be subjected to increase its length 1.1633 ins - ? Expansion for a length of 100 feet of copper for i° = .0115. 100 X 12-4-1.1633 — 100 X 12 1.1633 , ^ + 3^ = — + 32 = 13 3-i6 0 . When Length is to be reduced. L — l C ■T = t. Illustration. — Take elements of preceding case. 1201.1633 — 1200 — 133. 16 0 = 101. 16 — 133. 16 = 32 0 . To Reduce Degrees of Fahrenheit to Reaumur and. Cen- tigrade, and Contrariwise. Fahrenheit to Reaumur. If above zero. — Multiply difference between number of degrees and 32 by 4, and divide product by 9. Thus, 212 0 — 32 0 = 180 0 , and 180 0 X 4-j- 9 = 8o°. If below zero . — Add 32 to number of degrees ; multiply sum by 4, and divide product by 9. Thus, — 40 0 -j- 32 0 = 72 0 , and 72 0 x 4 -r- 9 = — 32 0 . Reaumur to Fahrenheit. If above freezing-point. — Multiply number of degrees by 9, divide by 4, and add 32 to quotient. Thus, 8o° X 9 -r* 4 = 180 0 , and 18c 0 -j- 32 = 212 0 . If below freezing-point . — Multiply number of degrees by 9, divide by 4 and subtract 32 from product. J ’ Thus, — 32 0 X9-r4 = 72 0 , and 72 0 — 32 = — 40 0 . Fahrenheit to Centigrade. If above zero . — Multiply difference between number of degrees and 32 by 5, and divide product by 9. Thus, 212 0 — 32° X 5 -4- 9 = 180 0 X 5-^9 = ioo°. ..ff below zero . — Add 32 to number of degrees, multiply sum bv 5, and divide product by 9. Thus, — 4 o° -f- 32 0 x 5 -s- 9 = 72 0 X 5 -5- 9 = — 40° Centigrade to Fahrenheit. If above freezing-point . — Multiply number of degrees by 9, divide product by 5, and add 32 to quotient. Thus, ioo° X 9 -r- 5 = 180 0 , and 180 0 -f- 32 = 212 0 . If beloiv freezing-point . — Multiply number of degrees by 9, divide product by 5, and take difference between 32 and quotient. Thus, — io° X 9 -f- 5 = 18 0 , and 18 0 a, 32 = i 4 o. Reaumur to Centigrade. — Divide by 4, and add product. Thus, 8o° -r- 4 = 20 0 , and 20 0 + 8o° = ioo°. Centigrade to Feanmnr.— Divide by 5, and subtract product. Thus, ioo° -r- 5 = 20 0 , and ioo° — 20 = 8o°. Corresponding Degrees upon the Three Scales. Fahr. | Cent. I Reaum. jl Fahr. 1 I Cent. I Reaum. [1 Fahr. 1 Cent. j Reaum. 212 | 100 1 80 (| 32 | 1 0 1 0 11 —40 —40 1 —32 To Compute Expansion of Fluids in Volume, Rule.— Proceed by preceding formulas for computing length of a sub- stance. Substitute V and v for volume, instead of L and 7 , the lengths. 5 2 4 HEAT, VENTILATION, BUILDINGS, ETC. Illustration.— A closed vessel contains 6 cube feet of water at a temperature of 4 o° ; to what height will a column of it rise in a pipe 1.152 ins. in area, when it is exposed to a temperature of 130 0 ? 1. 152 ins. = .008 sq.foot. C for water = .000222 9. 6 (1 -I-.0002229 (130 — 40)) = 6.125 95, and ^ = 15-744 lineal feet. Temperature by Agitation. Results of Experiments with Water enclosed in a Vessel and violently Agitated. Temperature of Air, 60.5°; of Water, 59. 5 0 . Duration of Agitation. Increase of Temperature. Duration of Agitation. Increase of Temperature. Duration of Agitation. Increase of Temperature. Hour. O Hours. 0 Hours. O • 5 10 2 19*5 5 39-5 I 14- 5 3 29-5 6 42-5 VENTILATION. IB itil clings. Apartments, etc. In Ventilation of Apartments .— From 3.5 to 5 cube feet of air are required per minute in winter, and 5 to 10 feet in summer for each occupant. In Hospitals , this rate must be materially increased. Ventilation is attained by both natural draught and artificial means. In first case the ascensional force is measured by difference in weight of two columns of air of same height, the height being determined by total difference of level between entrance for warm air and its escape into the atmosphere. The difference of weight is ascertained from difference of temperatures of ascending warm air and the external atmosphere, as by table, page 521? or by formula, page 522. Volumes of Air discharged through a -Ventilator One IPoot Square of Opening, at Various Heights and Temperatures. Height of Ventilator from Base-line. Feet. 10 15 25 30 Excess of Temperature of Apartment above that of External Air. C.ft. 116 142 164 184 201 C. ft. 164 202 232 260 284 C. ft. 200 245 285 3 i 8 347 C.ft. 235 284 330 368 403 25° C.ft. 260 318 368 410 450 3° C.ft. 284 34 8 4°4 450 493 Height of Ventilator from Base-line. Feet. 35 40 45 50 55 Excess of Temperature of Apartment above that of External Air. C. ft. 218 235 248 260 270 C.ft. 306 329 348 367 385 15 ^ C. ft. 376 403 427 450 472 C.ft. 43 6 465 493 518 54i 25^ C. ft. 486 518 55i 579 605 3 <^ C. ft. 531 57° 605 635 663 Velocity of draft having been ascertained for any particular case, together with volume of air to be supplied per minute, sectional area of both air passages may be computed from these data. Heating "by Hot Water. One sq. foot of plate or pipe surface at 200° will heat from 40 to 100 cube feet of enclosed space to 70° where extreme depression of tempeiature is — io °. * The range from 40 to 100 is to meet conditions of exposed or corner buildings, of buildings less exposed, as intermediate ones ot a cluster or block, and of rooms intermediate between the front and rear. When the air is in constant course of change, as required for ventilation or occupation of space, these proportions are to be very materially increased as per following rules. HEAT, VENTILATION, AND HEATING. 525 In determining length of pipe for any given space it is proper to include in the computation the character and occupancy of the space. Thus, a church, during hours of service, or a dwelling-room, will require less service of plate or length of pipe than a hallway or a public building. Reduction of Heat by Surfaces of Glass or Metal.— In addition to the volume of air to be heated per minute for each occupant, 1.25 cube feet for each sq. foot of glass or metal the space is enclosed with must be added. The communicating power of the glass and metal being directly proportion- ate to difference of external and internal temperature of the air. Thus 80 feet of glass will reduce 100 feet of air per minute. When Pipes are laid in Trenches in the Earth. — The loss of heat is es- timated by Mr. Hood at from 5 to 7 per cent. Circulation of Water in Pipes.— In consequence of the complex forms of heating-pipes and the roughness of their internal surface, it is impracticable to apply a rule to determine the velocity of circulation, as consequent upon difference of weights of ascending and descending columns of the water. For a difference of temperature in the two columns of 30° (ioo° — 160°) and a height of 20 feet, the velocity due to the height would be 3.74 feet. In practice not .3, and in some cases but .1, would be attained. Volume of Air Required per Hour for each Occupant in an Enclosed Space. To Compute Length of Iron ripe required, to Heat Air [General Morin.) 1800 424 to 1060 Cube Feet. in an Enclosed. Space. By Hot Water. ? f a ™? m of a P rotec ted dwelling is 4000 cube feet; what ngin 01 a ms. nine, at onr»o . . 526 HEAT AND HEATING. Lengths of Four-Inch IPipe to Heat lOOO Cube Feet of Air per IMUntite. ( Chas . Hood.) Temperature of Pipe 200°. Temperature Temperature of Building. External Air. 45 ° 50° 55 ° | 6o° 65° 70 0 i 1 75 ° 80° 85° I 9 °° O Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. 10 126 150 174 200 229 259 292 328 367 409 16 105 127 I5i 176 204 223 265 3 °° 337 378 20 91 112 135 160 187 216 247 281 318 358 26 69 90 112 136 162 190 220 253 288 327 30 54 75 97 120 145 173 202 234 269 307 36 32 52 73 96 120 147 175 206 239 276 40 18 37 58 80 104 129 157 187 220 255 50 — — 19 40 62 86 112 140 171 204 Proper Temperatures of Enclosed. Spaces. Temper- ature required. Spaces. Temper- ature required. Work-rooms, manufactories, etc. Churches and like spaces Greenhouses Schools, lecture-rooms Halls, shops, waiting-rooms, etc. Dwelling-rooms 55 55 55 58 60 65 IB oiler. Dwelling-rooms Graperies Hothouses Drying-rooms, when filled “ “ for curing paper.. 70 70 80 80 70 120 Boiler for steam-heating should be capable of evaporating as much water as the pipes or surfaces will condense in equal times. Mr. Hood recom- mends that 6 sq. feet of direct heating-surface of boiler should be provided to evaporate a cube foot per hour. Adopt mean weight of steam of 5 lbs. above pressure of atmosphere, or 20 lbs. absolute pressure, condensed per sq. foot of pipe per degree of difference of temperature per hour, viz., .002 35 lb. (as given by D. K. Clark), the quantity of pipe or plate surface that would form a cube foot of condensed water per hour, weight of like volume of water 62.4 lbs., would be, per i° difference of temperature, 62.4-4.00235 = 26550 sq.feet , and for differences of 168 0 , 158°, 148°, and 108 0 , required surface would be respectively (26550-4-168 = 158) 158, 168, 179, and 246 sq. feet. ' Henoe, assuming, as previously stated, that 4 sq. feet of direct and effec- tive heating boiler-surface, or its equivalent flue or tube surface, will evap- orate 1 cube foot of water per hour, 158 sq. feet of steam-pipe or plate will require 4 sq. feet of direct surface, etc., for a temperature of 6o°, and cor- respondingly for other temperatures. Boiler-power .— One sq. foot of boiler-surface exposed to direct action of fire, or 3 sq. feet of flue-surface, will suffice, with good coal, for heating 50 sq. feet of 4-inch, 66 of 3-inch, and 100 of 2-inch pipe. Mr. Hood assigns the proportion at 40 feet of 4-inch pipe for all purposes. Usual rate of com- bustion of coal is 10 or n lbs. per sq. foot of grate-surface, and at this rate, 20 sq. ins. of grate suffice for heating 40 feet of 4-inch pipe. Four sq. feet of direct heating boiler-surface, or equivalent flue or tube surface, exposed to direct action of a good fire, are capable of evaporating 1 cube foot of water per hour. According to M. Grouvelle, 1 sq. meter of pipe-surface (10.76 sq. feet), heated to 6o° an ordinary room alike to a library or office, of from 90 to 100 cube meters (3178 to 3531 cube feet). HEAT, WARMING BUILDINGS, ETC. 527 If a workshop to be heated to a high temperature, 1 sq. meter (10.76 sq. feet) of surface is assigned to 70 cube meters (2472 cube feet) = 4.35 sq. feet or 5.1 1 lineal feet of 4-inch pipe per 1000 cube feet. For heating workshops, having a transverse section of 260 sq. feet, with a window- surface of one sixth total surface, it is customary in France to assign 1.33 sq. feet of iron pipe surface per lineal foot of shop = 5.2 sq. feet per 1000 cube feet. Illustrations of extensive Heating by Steam . ( R . Briggs , M. I. C. E.) 1. Total number of rooms, including halls and vaults 286 “ Area of floor surface i 37 37 o sq. feet. “ Volume of rooms 1 923 500 cube feet. “ Number of occupants 650 Maximum average of occupants at any time i 3 oo Volume per occupant, excluding vaults i 443 cube feet. Boilers.— 8 with 173 sq. feet of grate surface and 8000 sq. feet of heating surface. Furnishing steam in addition to the above, to operate the elevators and electric dynamos, elevating water, and supplying steam to heat a distant building, requiring one third of their capacity. IBy Steam. To Compute Length, of* Iron Eipe required, to Heat Air in an Enclosed Space, with Steam at 5 lhs. per Sq. Inch above Pressure of* Atmosphere. # Rule.-— M ultiply volume of air in cube feet to be heated per minute, by difference of temperature in space and external air, divide product by coeffi- cients in preceding table, and quotient will give length of 4-inch pipe in lineal feet, or area of plate-surface in sq. feet. Temperature of steam at 5 lbs.-f- pressure = 228°. Hence, if temperature of space required is 6o°, 70 0 , 8o°, or 120 0 , the differences will be 168 0 , 158°, 148°, and io8 u , which for a coefficient of .5, as given in rule for hot water, would be 336, 316, 296, and 216, for a pipe 4 ins. in diameter, and for 6o° 7 °° 8o° 120° 237 222 l62 .168 158 148 108 . 84 79 74 54 Illustration. — Volume of combined spaces of a factory is 50000 cube feet; what surface of wrought-iron plate at 200° is necessary to maintain a temperature of 50 0 ■when external air is at o° ? 50000 X 5° — o x 6666 square feet. 200 — 50 Coal Consumed per Hour to IT eat IOO Feet of IPipe. ( Chas . Hood.) Diam. of Difference of Temperature of Pipe and Air in Space, in Degrees. Pipe. 150 i 45 140 135 130 125 120 ii 5 IIO 105 IOO 95 90 8S 80 Ins. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. I 1. 1 1. 1 1. 1 1 1 •9 •9 •9 .8 .8 •7 •7 •7 .6 .6 2 2-3 2.2 2.2 2.1 2 i.g 1.8 1.8 1 -7 1.6 1.4 1.4 i -3 1.2 3 3-5 3-4 3-3 3 -i 3 2.9 2.8 2.7 2-5 2.4 2-3 2.2 2. 1 2 1.8 4 4-7 4-5 4.4 4.2 4.1 3-9 3-7 3.6 3-4 3-2 3 -i 2.9 2.8 2.6 2.5 To warm a fhctory, according to M. Claudel, 43 feet in width by 10.5 high, a single line of hot-water pipe 6.25 ins. in diameter per foot of length of room, appears to be sufficient, temperature in pipe being from 170 0 to 180 0 . Also, water being at 180 0 , and air at 6o°, making a difference of 120 0 , it is convenient to estimate from 1.5 to 1.75 sq. feet of water-heated surface as equivalent to one sq. foot of steam-heated surface, and to allow from 8 to 9 sq. feet of hot- water pipe-surface per 1000 cube feet of room. M. Grouyelle states that 4 sq. feet of cast-iron pipe-surface, whether heated by steam or by water at 176° to 194 0 , will warm 1000 cube feet of workshop, main- taining a temperature of 6o°. Steam is condensed at rate of .328 lb. per sq. foot per hour. HEAT, WARMING BUILDINGS, ETC. 528 2 (22 L. Greene.) Length of fronts of buildings Total volume of rooms Radiating surfaces, direct, 10804. indirect, 23 296, Boilers Grate-surface Heating surface ... 2 000 lineal feet. 2 574 084 cube feet. | 34 100 sq. feet. ... 180 “ ...5863 “ Volume of Air Heated by Radiators ; Consumption of Coal ; Areas of Grate and Heating-surface of toiler. ( RoVt Briggs.) Per 100 Sq. Feet of Warming-surface of Radiator . Pressure of steam per sq. inch + 1 atmosphere in lbs 1 Heat from radiators per minute \ in units Volume of air heated i° per min ute in cube feet - Efficiency of radiators in ratio . Coal consumed per hour in lbs . Area of grate consuming 8 lbs coal per hour in sq.feet do. 12 lbs Heating surface of boiler ; coal 1 consumed per hour X2.8 in sq.feet ) 8 lbs. X 2.8 12 lbs. X 2.8 — 3 10 30 456 486 537 642 25 IIO 26 772 29 570 35 352 I 1.066 1.178 1.408 3- c 4 3-24 3-58 4.28 •38 •405 .448 — — — .298 •357 8.512 9.072 10.02 11.98 22.4 22.4 22.4 — — — 33-6 33-6 60 74i 40 803 1.625 4.94 .412 13-83 33-6 By Hot-Air Furnaces or Stoves. A square foot of heating surface in a hot-air furnace or stove is held to be equivalent to 7 sq. feet of hot water pipe. M. Peclet deduced that when the flue-pipe of a stove radiated its heat directly to air of a space, the heat radiated per sq. foot per hour, for 1 difference of temperature, were, for: Cast iron, 3.65 units; Wrought iron, 1.45 units, and Terra cotta .4 inch thick, 1.42 units. In ordinary practice, 1 sq. foot of cast iron is assigned to 328 cube feet of space. Open ZETires. According to M. Claudel, the quantity of heat radiated into an apart- ment from an ordinary fireplace is .25 of total heat radiated by combustible. For wood the heat utilized is but from 6 to 7 per cent., and for coal 13 per cent. In combustion of wood, chimney of an ordinary open fireplace draws from 1000 to 1600 cube feet of air per pound of fuel, and a sectional area of from 50 to 60 sq. ins. is sufficient for an ordinary apartment. Proportions of fuel required to heat an apartment are: For ordinary fire- places, 100 ; metal stoves, 63 \ and open fires, 13 to 16. Furnaces. By D. K. Clark, from investigations of Mr. J. Lothian Bell Cupola. — M. Peclet estimates that in melting pig-iron by combustion of 30 per cent, of its weight of coke, 14 per cent, only of the heat of combus- tion is utilized. IVIetalHirgical. — According to Dr. Siemens, 1 ton of coal is consumed in heating 1.66 tons of wrought iron to welding-point of 2700°, and a ton of coal is capable of heating up 39 tons of iron; from which it appears that only 4.5 per cent, of whole heat is appropriated by the iron. . Similarly, lie estimates 1.5 per cent, of whole heat generated is utilized in melting pot HEAT AND HEATING. HYDRAULICS. 529 steel in ordinary furnaces, whilst, in his regenerative furnace, 1 ton of steel is melted by combustion of 1344 lbs. of small coal, showing that 6 per cent, of the heat is utilized. Blast-furnace.— Mr. Bell has formed detailed estimates of appro- priation of the heat of Durham coke in a blast-furnace; from which is de- duced following abstract : Durham coke consists of 92.5 per cent, of carbon, 2.5 of water, and 5 of a«li and sulphur. To produce 1 ton of pig-iron, there are required 1232 lbs. of limestone, and 5388 lbs. of calcined iron-stone ; the iron-stone consists of 2083 lbs. of iron, 1008 lbs. of oxygen, and 2509 lbs. of earths. There is formed 813 lbs. of slag, of which 123 lbs. is formed with ash of the coke, ami 690 lbs. with the limestone. There are 2397 lbs. of earths from the iron- stone, less 93 lbs. of bases taken up by the pig-irou and dissipated m fume, say 2314 lbs. Total of slag and earths, 3 12 7 16s. Mr. Bell assumes that 30.4 per cent, of the carbon of the fuel, which es- capes in a gaseous form, is carbonic acid; and that, therefore, only 5 X * 2 7 per cent, of heating power of fuel is developed, and remaining 48.73 per cent, leaves tunnel-head undeveloped. He adopts, as a unit of heat, the heat required to raise the temperature of 112 lbs. of water 33*8 . HYDRAULICS. Descending Fluids are actuated by same laws as Falling Bodies. A Fluid will fall through 1 foot in .25 of a second, 4 feet in .5 of a second, and through 9 feet in .75 of a second, and so on. Velocity of a fluid, flowing through an aperture in side of a vessel, reservoir, or bulkhead, is same that a heavy body would acquire by fall- ing freely from a height equal to that between surface of fluid and middle of aperture. Velocity of a fluid flowing out of an aperture is as square root of height of head of fluid. Theoretical velocity, therefore, in feet per sec- ond, is as square root of product of space fallen through in feet and 64.333 = V 2 gh\ consequently, for one foot it is V 64.333 = 8.02 feet. Mean velocity, however, of a number of experiments gives 5.4 feet, or .673. In short ajutages accurately rounded, and of form of contracted vein, ( vena contracta), coefficient of discharge = .974 of theoretical. Fluids subside to a natural level, or curve similar to Earth’s convexity; apparent level, or level taken by any instrument for that purpose, is only a tangent to Earth’s circumference; hence, in leveling for canals, etc., difference caused by Earth’s cur- vature must be deducted from apparent level, to obtain true level. Deductions from Experiments on Discharge of Fluids from Reservqirs. 1. That volumes of a fluid discharged in equal times by same apertures from same head are nearly as areas of apertures. 2. That volumes of a fluid discharged in equal times by same apertures, under different heads, are nearly as square roots of corresponding heights of fluid above surface of apertures. 3. That, on account of friction, small-lipped or thin orifices discharge pro- portionally more fluid than those which are larger and of similar figure, under same height of fluid. Y Y 53 ° HYDRAULICS. 4. That in consequence of a slight augmentation which contraction of the fluid vein undergoes, in proportion as the height of a fluid increases, the flow is a little diminished. 5. That if a cylindrical horizontal tube is of greater length than its di- ameter, discharge of a fluid is much increased, and may be increased with advantage, up to a length of tube of four times diameter of aperture. 6. That discharge of a fluid by a vertical pipe is augmented, on the prin- ciple of gravitation of falling bodies ; consequently, greater the length of a pipe, greater the discharge of the fluid. 7. That discharge of a fluid is inversely as square root of its density. 8. That velocity of a fluid line passing from a reservoir at any point is equal to ordinate of a parabola, of which twice the action of gravity (2 q ) is parameter, the distance of this point below surface of reservoir being the abscissa* Or, velocity of a jet being ascertained, its curve is a parabola parameter of which = 4 h, due to velocity of projection^ 9* Volume of water discharged through ah aperture from a prismatic vessel which empties itself, is only half of what it would have been during the time of emptying, if flow had taken place constantly under same head and corresponding velocity as at commencement of discharge ; consequently the time in which such a vessel empties itself is double the time in which all its fluid would have run out if the head had remained uniform. 10. Mean velocity of a fluid flowing from a rectangular slit in side of a reservoir is two thirds of that due to velocity at sill or lowest point, or it is that due to a point four ninths of whole height from surface of reservoir. 11. When a fluid issues through a short tube, the vein is less contracted than in preceding case, in proportion of 16 to 13 ; and if it issues through an aperture which is alike to frustum of a cone, base of which is the aper- ture, the height of frustum half diameter of aperture, and area of small end to area of large end as 10 to 16, there will be no contraction of the vein. Hence this form of aperture will give greatest attainable discharge of a fluid. 12. Velocity of efflux increases as square root of pressure on surface of a fluid. 13. In efflux under water, difference of levels between the surfaces must ; be taken as head of the flowing water. 14. To attain greatest mechanical effect, or vis viva, of water flowing through an opening, it should flow through a circular aperture in a thin plate, as it has less frictional surface. From Cond.viits or IPipes. ( Bossut .) 1. Less diameter of pipe, the less is proportional discharge of fluid. 2. Discharges made in equal times by horizontal pipes of different lengths, but of same diameter, and under same altitude of fluid, are to one another in inverse ratio of sq. roots of their lengths. 3. In order to have a perceptible and continuous discharge of fluid, the altitude of it in a reservoir, above plane of conduit pipe, must not be less than .082 ins. for every 100 feet of length of pipe. 4. I11 construction of hydraulic machines, it is not enough that elbow's and ) contractions be avoided, but also any intermediate enlargements, the in- jurious effects of which are proportionate, as in following Table, for like volumes of fluid, under like heads in pipes, having a different number of \ enlarged parts. No. of Parts. Velocity. II No. of Parts. Velocity. 1 No. 1 of Parts. Velocity. I No. of Parts. Velocity. 0 1 II 1 .741 1 3 1 •569 1 5 •454 * See D’Aubuisson, pnge 66. t Humber, page 57. HYDRAULICS. 531 Friction. Flowing of liquids through pipes or in natural channels is materially af- fected by friction. If equal volumes of water were to be discharged through pipes of equal diameters and lengths, but of following figures : Discharges from Compound or Divided. Reservoirs. Velocity in each may be considered as generated by difference of heights in contiguous reservoirs ; consequently, square root of difference will rep- resent velocities, which, if there are several apertures, must be inversely as their respective areas. Note.— When water flows into a vacuum, 32.166 feet must be added to height of it ; and when into a rarefied space only, height due to difference of external and internal pressure must be added. TELOCITY OF WATER OR OF FLUIDS. Coefficients of Discharge. Coefficient of Discharge or Efflux is product of coefficients of Contraction and Velocity. It is ascertained in practice that water issuing from a Circular Aperture in a thin plate contracts its section at a distance of .5 its diameter from aperture to very nearly .8 diameter of aperture, so as to reduce its area from 1 to about .61.* Velocity at this point is also ascertained to be about .974 times theoretical velocity due to a body falling from a height equal to head of water. Mean velocity in aperture is therefore .974, which, X .61 = .594, theoretical discharge ; and in this case .594 becomes coefficient of discharge , which, if expressed generally by C, will give for discharge itself C1V2 g hxC — V. a representing area of aperture, and V volume discharged per second. Or, 4.95 a y/h z=z V. Or, 3.91 d 2 y/h — V. d representing diameter in feet. Hence, for cube feet per second, 4.95 a y/h , or 3.91 d 2 y/h. Illustration. — Assume head of water 10 feet, diameter of opening 1.127 feet, area 1 sq. foot, and C == .62. Then 1V2 g 10X .62 = 15.72 cube feet. 4.97 X 1 X y/io = 15.72 cube feet , and 3.91 X 1.127 2 X y/ 10 =. 15.7 cube feet. For square aperture it is .615, and for rectangular .621. Volume of water or a fluid discharged in a given time from an aperture of a given area depends on head, form of aperture, and nature of approaches. V 2 64-333 h = v 2 , and ^ ^ == h. h representing height to centre of opening in feet. Note. — Head , or height , h, may be measured from surface of water to centre of aperture without practical error, for it has been proved by Mr. Neville that for cir- cular apertures, having their centre at the depth of their radius below the surface, and therefore circumference touching the surface, the error cannot exceed 4 per cent, in excess of the true theoretical discharge, and that for depths exceeding three Bayer, .61. Observed discharges of water coincide nearer to unit of Bayer than that of all others. 532 HYDRAULICS. times the diameter, the error is practically immaterial. For rectangular apertures it is also shown that, when their upper side is at surface of the water, as in notches the extreme error cannot exceed 4.17 per cent, in excess; and when the upper is three times depth of aperture below the surface, the excess is inappreciable. For notches , weirs , slits, etc., however, it is usual to take full depth for head, when .666 only of above equation must be taken to ascertain the discharge. Experiments show that coefficient for similar apertures in thin plates, for small apertures and low velocities, is greater than for large apertures and high velocities, and that for elongated and small apertures it is greater than for apertures which have a regular form, and which approximate to the circle. When Discharge of a Fluid is under the Surface of another body of a lilce Fluid . — The difference of levels between the two surfaces must be taken as the head of the fluid. Or, V2 g (h — hf) = v. When Outer Side of opening of a discharging Vessel is pressed by a Force . — The difference of height of head of fluid and quotient of pressures on two sides of vessel, divided by density of fluid, must be taken as heads of fluid. j ( 2) ‘ X 144A Or, 2 g l h J =v. S representing density of fluid. Illustration.— Assume head of water in open reservoir is 12 feet above water- line in boiler, and pressures of atmosphere and steam are 14.7 and 19.7 lbs. Then f g X ^.333 x („ = 5.56^ When Water flows into a rarefied Space , as into Condenser of a Steam- engine , and is either pressed upon or open to Atmosphere . — The height due to mean pressure of atmosphere within condenser, added to height of water above internal surface of it, must be taken as head of the water. Of, V2 g{h-\- h f ) = v. Illustration. — Assume head of water external to condenser of a steam-engine to be 3 feet, vacuum gauge to indicate a column of mercury of 26.467 ins. (= 13 lbs.), and a column of water of 13 lbs. = 29. 9 feet. Then V2 g (3 + 29.9) = V 64.333 X 3 2 -9 = V 21 16 - 57 = 46 feet. Relative "Velocity - of Discharge of Water through. differ- ent Apertures and. under lihe Heads. Velocity that would result from direct , unretarded action of the column of water tuhich produces it, being a constant . or 1 Through a cylindrical aperture in a thin plate 625 A tube from 2 to 3 diameters in length, projecting outward 8125 A tube of the same length, projecting inward 6812 A conical tube of form of contracted vein 974 Wide opening, bottom of which is on a level with that of reservoir; sluice with walls in a line with orifice; or bridge with pointed piers .96 Narrow opening, bottom of which is on a level with that of reservoir; abrupt projections and square piers of bridges 86 Sluice without side walls 63 Discharge or Efflux of* "Water for various Openings and Apertures. Rectangular 'W'eir. Weirs are designated Perfect when their sill is above surface of natural stream, and Imperfect , Submerged , or Drowned when it is below that surface. HYDRAULICS. 533 Height measured from Surface of Water to Sill. {Jas. B. Francis.) Mean Head. I Length of Opening. I Mean Discharge per Second. I Mean Coefficient. .62 to 1.55 feet. | 10 feet. 32.9 cube feet. 1 *623 Principal causes for variation in coefficients derived from most experi- ments giving discharge of water over weirs arises from, j Depth being taken from only one part of surface, for it has been proved that heads on, at, and above a weir should be taken in order to determine true discharge. 2. Nature of the approaches, including ratio of the water-way in channel above, to water-way on weir. When a weir extends from side to side of a channel, the contraction is less than when it forms a notch, or Poncelet weir, and coefficient sometimes rises as high as .667. When weir or notch extends only one fourth, or a less portion of width, coefficient has been found to vary from .584 to .6. When wing-boards are added at an angle of about 64°, coefficient is greater than even when head is less. Computation of* "Vol vim e of Discharge. Mean velocity of a fluid issuing through a rectangular opening in side of a vessel is two thirds of that due to velocity at sill or lower edo-e of opening, or it is that due to a point four ninths of whole height from surface of fluid. Height measured from Surface of Head of Water to Sill of Opening. Rule.— Multiply square root of product of 64.333 and height or whole depth of the fluid in feet, by area in feet, and by coefficient for opening, and two thirds of product will give volume in cube feet per second. Or,f 6 *^0 = V; = t representing time in seconds and V volume in cube feet. Example.— Sill of a weir is 1 foot below surface of water, and its breadth is 10 feet; what volume of water will it discharge in one second? C = .623, V64.33 X 1 X ioX^ - - 80.2, and % 80.2 X .623 = 33.32 cube feet. Note. Mean coefficient of discharge of weirs, breadth of which is no more than third part of breadth of stream, is two thirds of .6 = .4; and for weirs which extend whole width of stream it is two thirds of . 66 £ = - 444 - Or, 214 y/h? = V in cube feet per minute. When h is in ins . , put 5. 15 for 214. Or, G bhy/TgTi — V. C for a depth .1 of lengths .417, and for .33 of length = .4. Or, by formula of Jas. B. Francis: 3.33 (L-.mH) H^-V. L representing length of weir and H depth of water in canal, sufficiently fai fiorn weir to be unaffected by depression caused by the current , both in feet , and n number of end contractions. Note.— When contraction exists at each end of weir, n — 2 ; and when weir is of width of canal or conduit, end contraction does not exist, and n = o. This formula is applicable only to rectangular and horizontal weirs in side of a dam, vertical on water-side, with sharp edges to current; for if bevelled or rounded off in any perceptible degree, a material effect will be produced in the discharge; it is essential also that the stream, after passing the edges, should in nowise be restricted in its flow and descent. Y Y* 534 HYDRAULICS. In cases in which depth exceeds one third of length of weir, this formula is not applicable. In the observations from which it was deduced, the depth varied from 7 to nearly 19 ins. • With end contraction, a distance from side of canal to weir equal to depth on weir is least admissible, in order that formula may apply correctly. Depth of water in canal should not be less than three times that on weir for ac- curate computation of flow. Illustration.— If an overfull weir has a length of 7.94 feet and a depth of .986 (as determined by a hook gauge), what volume will it discharge in 24 hours? = -522444 3-33 (7-94 — -2 X .986) .986^ = 3.33 X 7.94 — .1972 X .97907 = 3.33 X 7-74 2 8 X .97907 = 25.243 875, which X 60 X 60 X 24 = 2 181 061 cube feet. By Logarithms.— Log. 3. 33 7.7428 3. _ .986 s = 1.993 877 3 2 ) 1.981 631 1.990815 = 1.990815 1.402 157 Log. 24 hours = 86 400 seconds. 4. 936 514 6.338 671 Log. 6.33867 = 2 181 073 cube feet. C in this case = .615. Or. 2i4\/H3 and 5.15V&3 — V, if stream above the sill is not in motion. H repres enting height of surface of water above sill in feet , h in inches ; and 214VH3-I-.035 v 2 H3 — V, if in motion, v representing velocity of approach of water in feet per second , and V volume in cube feet discharged over each lineal foot of sill per minute. In gauging, waste-board must have a thin edge. Height measured to level of sur- face not affected by the current of overfall. ( Molesworth .) To Compute Depth, of Flow over a, Sill that will Dis- charge a given Volume of Water. / 3 V \2 C b a/ 2 g - 7 c = cl. h = — representing height due to velocity (v) as it , 3 \ 2 - -f- lc2 \3 fows to the weir. Note.— When back-water is raised considerably, say 2 feet, velocity of water ap- proaching weir ( 7 c) may be neglected. Rectangular Notches, or ‘Vertical Apertures or Slits. A Notch is an opening', either vertical or oblique, in side of a vessel, reser- voir, etc., alike to a narrow and deep weir. Vertical Apertures or Slits are narrow notches or weirs, running to or near to bottom of vessel or reservoir. Coefficient for opening, 8 ins. by 5, mean .606 {Poncelet and Lesbros). Coefficient increases as deptli decreases, or as ratio of length of notch to its depth increases. When sides and under edge of a notch increase in thickness, so as to be converted into a short open channel, coefficients reduce considerably, and to an extent beyond what increased resistance from friction, particularly for small depths, indicates. Poncelet and Lesbros found, for apertures 8x8 ins., that addition of a horizontal shoot 21 ins. long reduced coefficient from .604 to .601, with a head of about 4 feet; but for a head of 4.5 ins. coefficient fell from .572 to .483. For Rule and Formulas, see preceding page. HYDRAULICS. 535 .Rectangular Openings or Sluices, or Horizontal Slits. Height measured from Surface of Head of Water to Upper Side and to Sill of Opening. c inch by i inch. Head, 7 to 23 feet. = .621. Coefficient for { Opening, 23 = .614. 3 feet 11 1 foot. “ 1 “ 2 11 = .641. Poncelet and Lesbros deduced that coefficient of discharge increases with small and very oblong apertures as they approach the surface, and decreases with large and square apertures under like circumstances. Coefficients ranged, in square apertures of 8 by 8 ins., under a head of 6 ins. to rectangular apertures, 8 by 4 ins. ; under a head of 10 feet, from .572 to .745. In a Thin Plate , C = .616 (Bossut) ; C = .'61 ( Michelotti ). To Compute Discharge. Rule. — M ultiply square root of 64.333 and breadth of opening in feet, by coefficient for opening, and by difference of products of heights of water and their square roots, and two thirds of whole product will give discharge in cube feet per second. Or, — b\ f 2g (fiT/h — h'y/h') C = V; —=z — 2 -~t\ and 3 f 6 V 2 g (h y/h — h' y/h') C V — v. h and h' representing depth to sill and opening in feet, and v velocity b ( k — k') in feet per second. Example.— S ill of a rectangular sluice, 6 feet in width by 5 feet in depth, is 9 feet below surface of water; what is discharge in cube feet per second? C rz .625, 9 — 5 = 4, and V2gX6x62$X(9V9— 4X V4) = 3 8o *95 cube feet. Or, V 2 g d a C == V. d representing depth to centre of opening in feet. d = 9 — 2.5 = 6.5, a = 6 X 5 = 30? and V64.33 X 6.5 X 3° X .625 = 383.44 cube ft. Sluice 'W'eirs or Sluices. Discharge of water by Sluices occurs under three forms— viz., Unimpeded , Impeded , or Partly Unimpeded. To Compute Discharge when Unimpeded. C d b V2 g h — V. d representing depth of opening and li taken from centre of opening to surface of water. If velocity, /j, with which water flows to sluice is considered, 1 / V \2 v V — (ftt) — =d; and — d. *9\CdbJ O6V20/1 „ , / d \ C V 2 9 \ h ~T) h' representing height to which water is raised by dam above sill. Illustration. — How high must the gate of a sluice weir be raised, to discharge 250 cube feet of water per second, its breadth being 24 feet and height, h f , 5 feet? C by experiment = .6. d approximately = 1. 2 5° 2 5° r . — - = 1.0204 jeez. .6X24 V 64-33(5— i) I4 ‘ 250 4 X 17.014 To Compute Discharge when Impeded. C d b yj 2 g h = V, and -~d. C b V 2 g h h representing difference of level between supply and back-water. ’ 536 HYDRAULICS. To Compute Discharge when partly Impeded. C 6 V 2 g yj h — ~ -f* <*V*) — y* d ' representing depth of back-water above upper edge of sill. Illustration.— Dimensions of a sluice are 18 feet in breadth by .5 in depth- height of opening above surface of water .7 feet, and difference between levels of supply and surface water is 2 feet; what is discharge per second? 6 X 18 X 8. 02 ^.7 yj 2 — — -f- . 5 y/2j = 86. 62 X • 896 -f- . 707 = 138. 85 cube feet. Coefficients of Circnlar Openings or Slnices. Height measured from Surface of Head of Water to Centre of Opening. Contraction of section from 1 to .633, and reduction of velocity to .074; hence • 633 X -974 = .617 (Neville). In a Thin Plate , C = .666 {Bossut)\ .631 (Venturi) ] .64 (Eytelwein). Cylindrical Ajutages , or Additional Tubes, give a greater discharge than apertures in a thin side, head and area of opening being the same ; but it is necessary that the flowing water should entirely fill mouth of ajutage. Mean coefficient, as deduced by Castel, Bossut, and Eytelwein, is .82. Short Tubes, Month. -pieces, and. Cylindrical Prolonga- tions or Ajutages. Fig* 4 * If an aperture be placed in side of a vessel of from 1.5 to 2.5 diameters in thickness, it is converted thereby into a short tube, and coefficient, instead of being reduced by increased friction, is increased from mean value up to about .815, when opening is cylindrical, as in Fig. 4 ; and when junction is rounded, as in Fig. 5, to form of contracted vein, coefficient increases to .958, .959, and .975 for heads of 1, 10, and 15 feet. Conically Convergent and Divergent Tubes. In conically divergent tube, Fig. 6, coeffi- cient of discharge is greater than for same tube placed convergent, fluid filling in both cases, and the smaller diameters, or those at same distance from centres, O 0, being used in the computations. A tube, angle of convergence, O, of which is 5 0 nearly, with a head of from 1 to 10 feet, axial length of which is 3.5 ins., small diameter 1 inch, and large diameter 1.3 ins., gives, when placed as at Fig. 6, .921 for co- efficient ; but when placed as at Fig. 7, co- efficient increases up to .948. Coefficient of velocity is, however, larger for Fig. 6 than for Fig. 7, and discharging jet has greater amplitude in falling. If a prismatic tube project beyond sides into a vessel, coefficient will be re- duced to .715 nearly. Form of tube which gives greatest discharge is that of a truncated cone, lesser base being fitted to reservoir, Fig. 7. Venturi concluded from his ex- HYDRAULICS. 537 periments that tube of greatest discharge has a length 9 times diameter of lesser opening base, and a diverging angle of 5 0 6' — discharge being 2.5 greater than that through a thin plate, 1.9 times greater than through a short cylindrical tube, and 1.46 greater than theoretic discharge. Compound IMovitli-pieces and. Ajutages. Coefficients for Mouth-pieces, Short Tubes, and Cyl- indrical Prolongations. Computed and reduced by Mr. Neville, from Venturi's Experiments. Description of Aperture, Mouthpiece, or Tube. C. for Diam. ab. C. for Diam. o 1. An aperture 1.5 ins. diameter, in a thin plate 2. Tube 1.5 ins. diameter, and 4.5 ins. long, Fig. 4 .823 •974 .823 3. Tube, Fig. 5, having junction rounded to form of contracted vein .611 •956 •934 4. Short conical convergent mouth-piece, Fig. 6 .607 5. Like tube divergent, with smaller diameter at junction with reservoir; length 3.5 ins., or = 1 in., and ab = 1.3 ins. . . . 6. Double conical tube, a 0, S T, r b, Fig. 9, when ab — S T = 1. 5 ins., or — 1.21 ins., a 0 — .92 in., and 0 S — 4.1 ins 7. Like tube w r hen, as in Fig. 8, a o rb = o S Tr, and aoS— ' 1.84 ins .561 .948 .928 1.428 .823 .823 1.266 8. Like tube when ST — 1.46 ins., and 0 S — 2.17 ins 1.266 9. Like tube when S T— 3 ins., and 0 S'— 9.5 ins .911 1.4 10. Like tube when 0 S — 6. 5 ins. , atid S T — 1. 92 ins 1.02 1.569 11. Like tube when ST — 2.25 ins., and 0 S — 12.125 ms 1-215 i -855 12. A tube, Fig. 10, when os = r t — 3 ins., -dr t=zSt ~ 1.21 ins., and tube oSTr, as in No. 6, ST = 1.5 ins., and sS=4. 1 ins. •895 i -377 Mean of various experiments with tubes of .5 to 3 ins. in diameter, and with a head of fluid of from 3 to 20 feet, gave a coefficient of .813 ; and as mean for circular apertures in a thin plate is .63, it follows that under similar circumstances, .813 -4- .63= 1.29 times as milch fluid flows through a tube as through a like aperture in a thin plate. Preceding Table gives coefficients of discharge for figures given, and it will be found of great value, as coefficients are calculated for large as well as small diameters, and the necessity for taking into consideration form of junction of a pipe with a reservoir will be understood from the results. Circular Slnices, etc. To Compute Discharge. Height measured from Surface of Head of Water to Centre of Opening. Rule. — Multiply square root of product of 64.333 and depth of centre of opening from surface of water, by area of opening in square feet, and this product by coefficient for the opening, and whole product will give discharge in cube feet per second. Or, V2 g d, a C = V. a representing area in sq. feet , and d depth of surface of Jluid f'om centre of opening in feet. HYDRAULICS. 538 Example.— Diameter of a circular sluice is 1 foot, and its centre is 1.5 feet below surface of the water; what is discharge in cube feet per second? Area of 1 foot = . 7854 ; C = . 64, and V64. 333X1.5 X . 7854 X . 64 = 4. 938 cube feet When Circumference reaches Surface of Water. 3/2 g r, .9604 aC = V. r representing radius of circle in feet. Illustration.— In what time will 800 cube feet of water be discharged through a circular opening of .025 sq. foot, centre of which is 8 feet below surface of water? C = .63. 800 800 3/2 gd X .025 X .63 2268 X .025 X .63 = 2239.58 = 37 min. 19.6 sec. Note.— F or circular orifices, the formula Vzgd aC = V is sufficiently exact for all depths exceeding 3 times diameter; the finish of openings being of more effect than extreme accuracy in coefficient. Semicircular Slnices. When Diameter is either Upward or Downward. V2 gd a C = V. d repre- senting depth of centre of gravity of figure from surface. When Diameter as above is at Depth d, below Surface. V2 g d 1. 188 a C = V. Circular, Semicircular, Triangular, Trapezoidal, 3 ?ris- matic Wedges, Sluices, Slits, etc. See Neville , London , i860 ,pp. 51-63, and Weisbach , vol. \.p. 456. For greater number of apertures at any depth below surface of water, product of area, and velocity of depth of centre, or centre of gravity, if practicable to obtain it, will give discharge with sufficient accuracy.* Discharge from "Vessels not Receiving any- Supply. For prismatic vessels the general law applies, that twice as much "would be discharged from like apertures if the vessels were kept full during the time which is required for emptying them. To Compute Time. 2 A fh 2 Ah Qay/zg v Illustration — A rectangular cistern has a transverse horizontal section of 14 feet, a depth of 4 feet, and a circular opening in its bottom of 2 ins. in diameter; in what time will it discharge its volume of water, when supply to it is cut off and cistern allowed to be emptied of its contents? h = \feet, a == 2 2 X .7854-4- 144 = . 0218, 0=3.613, and V2 gh x a X C = .2143 2 ^ 14 - X 4 cube foot per second. Then - = 522.6 seconds. .2143 To Compute Time and Fall. Depression or subsidence of surface of water in a vessel, corresponding to a given time of efflux, is h — h'. h' representing lesser depth. — 2 (fh — h f ) ■=. t. Inversely, (fh — — av ^ 2gf tY = h'. C a V2 <7 \ 2 A / Illustration. — In what time will the water in cistern, as given in preceding case, subside 1.6 feet, and how much will it subside in that time? A =3 14, C = .6, a = . 0218, f 2 g — 8.02, h = 4, h' = 4 — 1.63=2.4. .6 X. 0^8X8.02 x (V 4 — V*. • ♦) = ^ x (* - 1. 55 ) = 120.1 seconds, f . .6 X .0218 X 8.02 \ 2 . , t ^ v 4 2 x X120.1 j =2— .45 =32.4/^; hence, 4 - 2.4 =3 1.6 feet. When Supply is maintained. — Divide result obtained as preceding by 2. HYDRAULICS. 539 Discharge, wlien Form and. Dimensions of Vessel of Efflux are not known. Volume discharged may be estimated by observing heads of the water at equal intervals of time; and at end ot half time of discharge, head of water will be .25 of whole height from surface to delivery. ^ ^ ^ When t = such interval. For openings in bottom or side , Vaty/ig ^ 1 j = V, for I depth ; Cat fTy + + = V for 2 depths ; and GaW~y ^ + + + = v for 4 depths. Note. — A t end of half time of discharge, head of water will be . 25 of whole height from surface to delivery. Weirs or Notclies. - Cb t V2 ~g {y/hs -f- 4 y/h^ + y/h 3 2 ) = V. b representing breadth in feet. 9 Illustration. — A prismatic reservoir 9 feet in depth is discharged through a notch 2.222 feet wide, surface subsiding 6.75 feet in 935 seconds; what is volume discharged? C == .6, = 9 — 6.75 — 2. 25 feet, and ^ 6 X 2.222 X 935 X 8.02 (-/93 + 4 V2.253-}- Vo 3 ) = 2221.6 X 40.5 = 89974.8 cube feet. When there is an Influx and Efflux. If a reservoir during an efflux from it has an influx into it, determination of time in which surface of water rises or falls a certain height becomes so complicated that an approximate determination is here alone essayed. A state of permanency or constant height occurs whenever head of water is in- r decreased by — — fc. I representing influx in cube feet per second. 2 g \C a) Ai v Time ( t ) in which variable head » increases by volume (v) — 1 _c a > and time in which it sinks height, 7 c, by — fl _ =. — — Time of efflux, in which Ca V2 gx — I subsiding surface falls from A to A x , etc., and head of water from k to At, when k is represented by is C a V2 g h — h A / A 4 Ai ■ 2 A 2 4A3 A 4 \ t 12 C a VPg V ^3 V* V /( 7 Illustration.— In what time will surface of water in a pond, as in a previous example, fall 6 feet, if there is an influx into it of 3.0444 cube feet per second? — 3 -° 4 t r -,— .8. C . 537 and a = . 8836. V 537 X. 8836X8.02 537 4 X 495 o°° 1 2 X 41QOQQ . 4 X 325 00 0 8 ' 4.301— .8 4.123- 8 _t ~ 3-937 — - 8 creased or i -537 X .8836 X 8.02 20 — 14 / 600000 12 X -537 X .8836 X 8.02 265000 \ 6 X I \ 4 - 47 2 " — ^jvoo _\ _ — 6 — x j 480201 = 194486 seconds = 54 h., 1 min., 26 sec. 3.742 — .8/ 45-665 Prismatic Vessels. If vessel has a uniform transverse section, A. Then 2 A — [ y/h - V h z + V k X hyp. log. (^.! l 1 ~t- time in which ' a Vz g L \v tl i — V «7 J head of water flows from h to h t . 540 HYDRAULICS. Illustration.— A reservoir has a surface of 500000 sq. feet, a depth of 20 feet; it is fed by a stream affording a supply of 3.0444 cube feet per second, and outlet has an area of .8836 sq. foot; in what time will it subside 6 feet? y/lc, as before, = . 8, C = X2 ' 3 ° 3 ] = 238 . 2 X 500 000 537, and — n±- . C 7 < 50 X ly/zo — -/mP- 8 Xhyp.log. av 2 0 L To Compute Fall in a given Time. This is determining head h x at end of that time, and it should be sub- tracted from head h at commencement of discharge. Put into preceding equation several values of hi, until one is found to meet the condition. Illustration. — Take a prismatic pond having a surface of 38750 sq. feet, a depth to centre of opening of sluice of 10.5 feet, a supply of 33.6 cube feet, and a discharge of 40 cube feet per second. y/Jc — .84. Putting these numerical values into the equation, and assuming different values for hi' a value which nearly satisfies the equation is 4. Consequently, 10.5 — 4 — 6-5 feet, fall. A Ic 3 a hyp. lo „ ht+Vhik+Tc WK -y/k) + V» 2 arc (tang. = 2Vfc ^^j ] = t; 3 L V “7 \ V I V "1/ J ( - — p = &; arc (tang. = y, arc tangent of which = y, and I as preceding. f C by/ 2 g/ According as 1c is ^ h , and influx of water, I^fC ly/agh 3 , there is a rise or fall of fluid surface, the condition of permanency occurring when h z ==Jc, and time cor- responding becomes co. Illustration.— In what time will water in a rectangular tank, 12 feet in length by 6 feet in breadth, rise from sill of a weir or notch, 6 inches broad, to 2 feet above it, when 5 cube feet of water flow into the tank per second ? hi — 2, 7i — o, A = 12 X 6 = 72, 1 = 5, & = -5 j C = .6. 5 Jc = ( i \t= \f .6 X -5 X 8.02/ = 3. 117 2 = 2.1338. [.6 X -5 X 8.< ! f. , ... 2 -f- V2 X 2. 13384-2.1338 . , A [^hyp. logarithm arc (tang. = >me n 7?„X«-*33 , 8 | 3 X 5 — ~^ 3 n X 2 ) 1 = 10.2423 X hyp. log. 6,1 3-4641 X arc (tang. -) = 2 V 2 - I 33 8 + v 2 / J .002162 \ 4-3356/ 10.2423 x [7-961 — (3-461 X arc, tangent of which === .56497, or 29 0 28' = 29.466, length of which = .5143) = 1-781] = 10.2423 — 7-961 — 1.781 = 10.2423 X 6.18 = 63.297 seconds. Discharge of 'Water under Variable Pressures. To Compute Time, Tiise and. Fall, and Volume. - y/ 2 g x = v. x representing variable head , A and a areas of transverse horizon - tal section of vessel and discharge , and v theoretical velocity of efflux. To Compute Volume. A y — Y. y representing extent of fall , and V volume of water discharged , as h — h'. Illustration.— Assume elements of preceding case. A = 14. y — 4 feet. Then 56X4 = 224 cube feet. HYDRAULICS, 541 Discharge from Vessels of Communication. When Reservoir of Supply is maintained at a uniform Height. — Fig. 11. 2 A fh To Compute Tiipe. -= t. Ca f 2 g Illustration i. — In what time will level of water in a receiving vessel having a section of 14 sq. feet attain height of that in supply, through a pipe 2 ins. in diam- eter, placed 4 feet below level of supply ? ~ ^ 2 X 14 X a /4 56 , C = .613. -7 ■ n — — — — = 522.3 seconds. ** .613 X .0218 X 8.02 .1072 _A Fj g- «• 2 Assume C, vessel, Fig. 11, to be a cylinder 18 ins. in diameter, head of water in A = 4 feet, at A' 1 foot, and 2 feet below outlet o; in what time will water in vessel run out and over at o through a pipe, a, 1.5 ins. diameter? h — h' = 4 — 1 — 2=1 foot. C . 8. Hsr— ; v 288 t/i) — X 1. 73 — 1 = 32- 73 seconds. 0.424 When Vessel of Supply has no Influx, and is not indefinitely great compared with Receiving Vessel . 2 A A 'y/h ■ - ±=t. A representing section of receiving vessel , t time in which A h i ! A' iss a ■ - 1 2 A A' {y/h — fh') C a (A -f- A') f 2 g the two surfaces of water attain same level ; and C a ( A -f- A') sTTh which level falls from h to h'. Illustration. — Section of a cistern from which wa' feet, and section of receiving cistern is 4 sq. feet; initial and diameter of communicating pipe is 1 inch, in what in both vessels attain like levels? = t. ' • = .82. i"=- 7 8 54- 2 X 10 X 4 3/3 .82 X .7854 X - — X 8.02 144 — 276 seconds. Discharge from a Notch.* in 3 A Side of a Vessel. ■== ( —77 77" ) —t.b breadth of notch in feet. 2 q V A f h ) When it has no Influx. , , C b x V2 g Illustration.— If a reservoir of water, no feet in length by 40 in breadth, has a notch in end of 9 ins. in width; in what time will head of w r ater of 15 ins. fall to 6? C = .6. 9" = .75 foot. h'L 3 X no X 40 x /_ 1 1 \ _ 13200 > 8.02 \y / .5 fl.25/ ~ h = X.25. - X 1.414 — -894 = 1901 seconds. •6 X .75 X 8.02 \ * 5 3/1-25/ 3.61 Note. -For discharge of vessels in motion, see Weisbach, vol. 1, pp. 394-396. Reservoirs or Cisterns. To Compute Time of* billing and of Emptying a Reser- voir under Operation of Doth Supply and Discharge. V Y g £> — T > and j)\_g — fc V representing volume of vessel , S supply of water , and D discharge of water , both per minute , and in cube feet. T time of filling vessel and t time of discharging it, both in minutes. indefinite” €xtends to tlle bottom o{ tbe reservoir, etc., the time for the water to run out is Z z 542 HYDRAULICS. Irreguilar-Sliaped Vessels, as a Pond, Lake, etc. To Compute Time and. Volume Discharged. Operation .— Divide whole mass of u’ater into four or six strata of equal depths. , * 2 a 2 4 a3 a 4\ , . + VT2 + VA3+VAi)“’’’ h — hA 2 C ay/ : (a ,4 ar = X VVA+VA" 1 " Then, /or 4 Strata , etc., representing depths of strata at a, ai, etc., commencing at surface; a*, as, ft — etc., toeing areas of first, second , etc., transverse sections of pond . etc. ; and — — — Xa-f-4at-{-2a2-J-4a3-{-a4:=:V. Fig. 12. A Illustration. — In what time ?C will depth of water in a lake, A 6 C, Fig. 12, subside 6 feet, sur- faces of its strata having follow- ing areas, outline of sluice being a semicircle, 18 ins. wide, 9 deep, and 60 feet in length? a at 20 feet (h ) depth of water = area of 600000 sq. feet, ai “ 18.5 “ (7 m) “ “ = “ 495000 “ a 2 “ 17 “ (h?) “ “ = “ 410000 “ a3 “ 15.5 “ (^3) “ = “ 325000 “ a4 u 14 “ (/i4) “ = “ 265000 “ a = area of 18 -4- 2 = .8836 sq.feet; € = .537. Then -14 265 ooo\ 3-742 > 12 X-537 X .8836 X 8.02 /600000 . 4X495000 , 2 X 4 io o°o 4X325000 x {- r zr+ — -+ 4.472 4.3OI 4.123 3-937 : X 1 194 431 = 156 938 sec. = 43 h. , 35 min. 38 sec. ”45-665 And discharge ir'-— X (600 000 + 4 X 495 000 -f 2X 410 000 + 4 X 325 000 -f- 265 000) 12 = .5X4 965 000 = 2 482 500 cw&e feet. For 6 Strata, put 2a4, instead of a4, and 4 as and a6 additional, and divide by 18 instead of 12. Flow of Water in Beds. Flow of water in beds is either Uniform or Variable. It is uniform when mean velocit} r at all transverse sections is the same, and consequently when areas of sections are equal ; it is variable when mean velocities, and there- fore areas of sections, vary. To Compute Ball of Blow. C — x — = h. C representing coefficient of friction , l length of flow, p perimeter a . 2 g of sides and bottom of bed, and hfall in feet. Illustration. — A canal 2600 feet in length has breadths of 3 and 7 feet, a depth of 3 feet, with a flow of 40 cube feet per second; what is its fall ? C = as per table below .007 565 ; p — X 2 + 3 = 10.2; a = 15; and 2600X10.2 2.66 2 v — 40 -4- 15 = 2.66. Hence .007565 X X 7 = 1.4 7 f eet - * 3 15 64.33 / a q f I p 2 gli — v. Illustration. — A canal 5800 feet in length has breadths of 4 and 12 feet, a depth of 5, and a fall of 3 ; what is velocity and volume of flow? P = a/5^ + 4 2 X 2 + 4 = 16.8, and a = 40. Then A / ^ — — =r X 64. 33 X 3 = V.0542 X 193 = 3-23 f eet - Hence V .007 565 X 5800 X 10.8 volume = 40 x 3. 23 = 129. 2 cube feet. HYDRAULICS 543 Coefficients of Friction of Flow of Water in Beds, as in Rivers, Canals, Streams, etc. Forms of Transverse Sections of Canals, etc. Resistance or friction which bed of a stream, etc., opposes to flow of water, in consequence of its adhesion or viscosity, increases with surface of contact between bed and water, and therefore with the perimeter of water profile, or of that portion of transverse section which comprises the bed. Friction of flow of water in a bed is inversely as area of it. Of all regular figures, that which has greatest number of sides has for same area least perimeter ; hence, for enclosed conduits, nearer its trans- verse profile approaches to a regular figure, less the coefficient of its friction ; consequently, a circle has the profile which presents minimum of friction. When a canal is cut in earth or sand and not walled up, the slope of its sides should not exceed 45 °. Variable motion of water in beds of rivers or streams may be reduced to rules of uniform motion when resistance of friction for an observed length of river can be taken as constant. To Compute Volume of Water flowing in a River. Illustration. — A stream having a mean perimeter of water profile of 40 feet for a length of 300 feet has a fall of 9.6 ins. ; area of its upper section is 70 sq. feet, and of its lower 60; what is volume of its discharge? Friction in flow of water through pipes, etc., of a uniform diameter is in- dependent of pressure, and increases directly as length, very nearly as square of velocity of flow, and inversely as diameter of pipe. With wooden pipes friction is 1.75 times greater than in metallic. Time occupied in flowing of an equal quantity of water through Pipes or Sewers of equal lengths, and with equal heads, is proportionally as follows : In a Right Line as 90, in a True Curve as 100, and in a Right Angle as 140. In Feet per Second. Velocity. I C. C. Velocity. C. Velocity. C. Velocity. C. .3 .008 .4 .007 .5 .007 .6 .007 .00815 .7 .00773 i-5 .00797 .8 .00769 2 .00785 .9 .00766 2.5 .00778 1 .00763 3 2-5 3 2 •007 59 • 007 52 .007 51 .007 49 5 .00745 8 .00744 10 .00743 12 .00742 "V" ai*iaL>le VEotion. V2 gh - = V. A and A x representing areas of upper and lower transverse sections of flow. To obtain C for velocity due to this case, 92.35 coefficient for which, see Table above, = .007 44. 70 -f- 60 X — 12 V 64. 33 x (9.6^- 12) == 394. 6 cube feet ; and mean velocity = 394.6 X 2 70-j- 60 = 6.0 7 feet, C for which is .007 45. FRICTION IN PIPES AND SEWERS. HYDRAULICS. To Compute Head, necessary' to 'overcome Friction of Dipe. ( Weisbach . ) ( ols , I ' Q1 7 4 6 \ A x — = h'. h' representing head to overcome friction of V V v ) d 5-4 . flow in pipe, l length of pipe, and v velocity of water per second, all in feet , and d internal diameter of pipe in ins. Illustration. — Length of a conduit-pipe is 1000 feet, its diameter 3 ins., and the required velocity of its discharge 4 feet per second; what is required head of water to overcome friction of flow in pipe? (.0144 -f X X ~ = .023 13 X 333-333 X 2.963 = 22.845/eeZ. Head here deduced is height necessary to overcome friction of water in pipe alone. Whole or entire head or fall includes, in addition to above, height between surface of supply and centre of opening of pipe at its upper end. Conse- quently, it is whole height or vertical distance between supply and centre of outlet. To Compute wliole Head, or Heiglit from Surface of Supply to Centre of Discharge. (C> < f + , .5 ) X^ = ». 1.5 is taken as a mean, and is coefficient of friction for interior orifice, or that of upper portion of pipe. / .017 46\ ^ To obtain C or coefficient. ^0144 d — J = C. For facilitating computation, following Table of coefficients of resistance is introduced, being a reduction of preceding formula : Coefficients of Friction of YV ater. In IPipes at Different Velocities. V. C. V. C. V. C. V. C. V. C Ft. Ins. Ft. Ins. Ft. Ins. Ft. Ins. Ft. Ins. 4 •0443 2 8 .025 5 .0221 7 4 .0208 n 6 .0195 8 .0356 3 .0244 . 5 4 .0219 7 8 .0206 12 .0194 z .0317 3 4 .0239 5 8 .0217 8 .0205 12 6 .0193 1 4 .0294 3 8 .0234 6 .0215 8 6 .0204 13 .0191 1 8 .0278 4 .0231 6 4 .0213 9 .0202 14 .0189 2 .0266 4 4 .0227 6 8 .0211 10 .0199 15 .0188 2 4 .0257 4 8 .0224 7 .0209 11 .0196 16 .0187 Illustration 1.— Coefficient due to a velocity of 4 feet per second is .0231. 2.— Take elements of preceding case. 1000 X 12 , . (.0231 X b i-5) X ' — 93-9 X • - = 23.35 feet. Note. — In preceding formula Z was taken in feet, as the multiplier of 12 for ins. was cancelled by taking 5.4 for 2 g, but in above formula it is necessary to restore this multiplier. Ftadii of Curvatures. When Pipes branch off from Mains, or when they are deflected at right angles, radius of curvature should be proportionate to their diameter. Thus, Ins. Ins. Ins. Ins. Ins. Diameter 2 to 3 3 to 4 6 8 10 Radius 18 20 30 42 60 HYDRAULICS, 545 Curves and. Bends. Resistance or loss of head due to curves and bends, alike to that of friction, increases as square of velocity ; when, however, curves have a long radius and bends are obtuse, the loss is small. Curved Circular Pipe. ( Weisbach). x £.131 + 1-847 2 J X ^ = h- presenting angle of curve , d diameter of pipe , r radius of curve, and h height friction or resistance of curve, all in feet. facility of computations, following values of .131 -f 1.847 are intro- For duced. Coefficients of Resistance. In Curved Pipes with Section of a Circle. A 2 r ( - 1 1 • I31 II •25 I •145 II •4 | .206 | 1 - 6 ! •44 I. 1 -75 | .806 1] 1 *9 I 1 - 15 •3 - 1 5 8 •45 •244 • 6 5 •54 .8 •977 •95 l -2 1 .138 II •35 1 .178 || •5 ! 1 -294 1 1 -7 1 .661 | 1 -85 1 i-i 77 1 | 1 1.408 1.674 1.978 Illustration. — If in a pipe 18 ins. in diameter and 1 mile in length there is a right-angled curve of 5 feet radius, what additional head of flow should be given to attain velocity due to a head of 20 feet? a = 90°, v for such a pipe and head = 4 feet per second; 18 = 1.5 and • 1,5 — .15, and . 15 by table 1= . 133. — X -133 X = .5 X .133 X = -016 53 foot. a. „„ 64.33 2X5 Hence, 180 64.33 Note.— I f angle is greater than 90 0 , head should be proportionately increased. Bent or Angular Circular Pipes. Coefficient for angle of bend = .9457 sin. 2 x -}- 2.047 sin .4 x. Hence, X I 10° 20° I | 3 °° 1 4 °° 45 ° 50 ° | | 55 ° | 6o° 65 ° 0 0 C 1 • 046 I •139 •364 •74 1 1 -984 1 1-26 j i -556 | | 1. 861 1 2.158 | 2.431 and — x C = h. x representing half angle of bend. 2 9 Illustration. — Assume v = 4 feet, and angle =: 90 0 ; x 90'-' Then - 64-33 X -984 = .2447 foot additional head required. In Valve Grates or Slide Valves. In Rectangular Pipes. r 1 1 1 •9 .8 ■7 | .6 | •5 | .4 | .3 | .2 | .1 C 1 .0 I .09 •39 1 •95 1 2.08 I 4.02 | 8.12 | 17.8 | 44.5 1 193 r — ratio of cross section. In Cylindrical Pipes. li 0 .125 •25 •375 •5 .625 •75 •875 r 1 •948 .856 •74 .609 .466 • 3 i 5 •*59 C .0 .07 .26 .81 2.06 5-52 17 97.8 h — relative height of opening. In a Throttle Valve. In Cylindrical Pipes. A 5 ° IO° 15° 20° 1 25° 30 0 35 ° 40° 45 ° 0 0 IT) 6o° 70 0 r • 9 i 3 .826 .741 •658 I .577 •5 .426 •357 •293 •234 •i 34 .06 C .24 •52 •9 1.54 1 2-51 3 - 9 1 6.22 10.8 18.7 32.6 118 75 i A = angle of position. Z z* 546 HYDRAULICS. In a Clack or Trap "Valve. Angle of opening I5 ° 2 °° 1 25 ° 3 °° 35 ° 40° I 45 ° I 5 o° 55 ° 1 6 o° 65° 70 0 C 90 1 62 j 42 30 1 20 | 14 9-5 | 6.6 | 4.6 1 3-2 2-3 i *7 In a Cock. In Cylindrical Pipes. A 5° IO° I 5° 20° 25° I 30° 35° 40° 45° 50° 55° 6o° 65° r .926 .85 .772 .692 •6i3 -535 •458 .385 •3 X 5 •25 • x 9 • x 37 .091 C •05 .29 •75 1.56 3-i 15-47 9.68 *7-3 31.2 52.6 106 206 486 In a Conical "Valve. ( 1-645 ^ — 1 ^ = C. a and a ' = areas of pipe and opening. ( CL \ - — 1 j = C. c = a factor, rang- ing from .624 for = - 1 t° 1 f or ^7 = 1, b e i n 9 greater the greater the ratio. Illustration. — If a slide valve is set in a cylindrical pipe 3 ins. in diameter and 500 feet in length, is opened to .375 of diameter of pipe (hence, .625 diameter closed), what volume of water will it discharge under a head of 100 feet, coefficient of en- trance of pipe assumed at .5 ? C, by table, p. 54 5, pipe being .625 closed = 5.52. V 2 gfh V^-s+c+c^) C =from table, p. 5445/0?' an assumed velocity of 11 feet 6 ins. = .0195. Then V 64.33 x V 100 ; 8.03X10 803 an ,, 5M V( 7 -° 2 -f- 39 ) 6 -7 8 // . . 500 X I2\ ^i- 5 + 5- 52 + - OI 95 ) Hence, area of 3 ins. =7.07, and 7.07 X 12 X 11-85 = 1005.4 cube feet per second. Valves. ( Conical , Spherical, or Flap.) Conical or Splierical Valve IPnppet. v 2 Height due to resistance or loss of head of water— 11 — . v representing velocity of water in f ull dia meter of pipe or vessel. (—7—, 1^ = C. A and A' representing transverse areas of vessel and of valve opening , and (1.645 ~ — 1^ = C of contraction in general. Illustration. — If A' = .5 of vessel, C = (1.645 X ~ — = 2.292 = 5-24- Clack or Trap Valve. — C decreases with diameter of vessel. Illustration. If a single-acting force pump, 6 ins. in diameter, delivers at each stroke 5 cube feet of water in 4 seconds, diameter of valve seat 3.5 ins., and of valve 4.5; what resistance has water in its passage, and what is loss of mechanical effect? a — . 196. (=^ = . 34 ratio of transverse area of opening. 1 — (~^j — • 44 ratio of annular contraction to transverse area of vessel. Hence, .39 mean ratio, and coefficient of resistance corresponding 2 thereto = — iV = 3. 22 2 =10.37. 5 - — = 6.3 7 velocity per second. V -39 / 4 X -190 HYDRAULICS. 547 -- 37 2 - = .63 height due to velocity. Consequently, 10.37 X .63 = 6.53 height due to 64-33 resistance of valve, and - X 62. 5 X 6. 53 — 510.15 lbs. mechanical effect lost. Discharge of* YVater in. Pipes. For any Length and Head, and for Diameters from 1 Inch, to IO Feet. In Cube Feet per Minute. (Beardmore.) Diam. Tab. No. I Diam. Tab. No. Diam. Ins. Ft. Ins. Ft. Ins. I 4.71 9 1 147.6 I II 1.25 8.48 10 1493-5 2 i-5 13.02 11 1894.9 2 I i-75 19- 1 2 356 2 2 2 26.69 I 1 2 876.7 2 3 2-5 46.67 I 2 3 463-3 2 4 3 73-5 I 3 4 II 5-9 2 5 3-5 108. 14 I 4 4836.9 2 6 4 151.02 I 5 5628.5 2 7 4-5 194.84 I 6 6 493- 1 2 8 5 263.87 I 7 7 433 2 9 6 416.54 I 8 8449 2 10 7 612.32 I 9 9 544 2 11 8 854.99 1 10 10722 3 Tab. No. Diam. Tab. No. Diam. Tab. No. 11 9 8 3 Ft. Ins. 3 4 39 329 Ft. Ins. 4 9 115 854 13328 3 2 42 040 5 131 7°3 14 758 3 3 44 863 5 3 148791 16 278 3 4 47 794 5 6 167 139 17 889 3 5 5o 8 35 5 9 186786 19 592 3 6 53995 6 207 754 21 390 3 7 57265 6 6 253 781 23 282 3 8 60 648 7 305 437 25 270 3 9 64156 7 6 362 935 27 358 3 10 67 782 8 426 481 29 547 3 11 71526 8 6 496 275 31834 34 228 4 4 3 75 392 87730 S 6 572 508 655 369 36725 4 6 101 207 IO 745 038 This Table is applicable to Sewers and Drains by taking same proportion of tabular numbers that area of cross-section of water in sewer or drain bears to whole area of sewer or drain. Formula upon which the table is constructed is, 2356 y/i xd s = V in cube feet per minute , and 39.27 yj ~ X d^ — V in cube feet per second, h represent- ing height of fall of water and d diameter of pipe and l length , all in feet. To Compute Discharge. ( Eytelwein .) 4-7 T ~ ^ and . 538 = d. d=z diameter of pipe in ins., I length of pipe and h head of water, both in feet. ( Hawksley .) ~~ = d, and -y — ~ — G. G — number of Imperial gallons per hour, and l length of pipes in yards. {Neville.) 140 Vr s — n vr s = v in feet per second, r — hydraulic mean depth in feet, and s sine of the inclination or total fall divided by total length. v 47. 124 d 2 — V, and v 293.7286 d 2 = Imperial gallons per minute, d = diameter of pipe in feet. To Compute Volume discharged. When Length of Pipe, Height or Fall, and Diameter are given . Rule. — Divide tabular number, opposite to diameter of tube, by square root of rate of inclination, and quotient will give volume required in cube feet per minute. Example. — A pipe has a diameter of 9 ins., and a length of 4750 feet; what is its discharge per minute under a head of 17.5 feet? Tab. No. 9 ins. = 1x47.6, and ^Z-— - — — = 69.67 cube feet. / 475Q 16.47 V *7-5 548 HYDRAULICS. To Compute Diameter. When Length , Head, and Volume are given. Rule. — M ultiply discharge per minute by square root of ratio of inclination ; take nearest corresponding number in Table, and opposite to it is diameter required. Example. — Take elements of preceding case. 69.67 x < ' = 1147.61, and opposite to this is q ins. I 7-5 / v l ° r ’ V i 542 k ~ d in f eeL v representing velocity in feet per second and l length in feet. To Compute Head. When Length , Discharge , and Diameter are given . Rule. — Divide tabular number for diameter by discharge per minute, square quotient, and divide length of pipe by it ; quotient will give head necessary to force given volume of water through pipe in one minute. Example.— T ake elements of preceding cases. 1147.61 69. 67 16.47; I 6-47 2 -- 271.3; 4750-5-271.2 = 17.5 feet. To Compute whole Head necessary to furnish requisite Discharge. See Formula and Illustration, page 544. To Compute Velocity. When Volume and Diameter alone are given. Rule. — Divide volume when in feet per minute by area in feet, and quotient, divided by 60, will give velocity in feet per second. Example. — Take elements of preceding case. — 6 ^ 6? - 5- 60 = 2.63 feet. •75 2 X *7^54 When Volume is not given. Rule. — Multiply square root of product of height of pipe by diameter in feet, divided by length in feet, by 50, and product will give velocity in feet per second. (Beardmore.) To Compute Inclination of a Dipe. When Volume , Diameter , and Length are given. Illustration. — Take elements of preceding case. ( 6 q .67\ 2 i 17.5 7-] X — 7 — .000874 X 4.214 = .003 68, and = . 003 68, or 4750 x .00368 2 35° / -75° 475° = 17.49 feet head. To Compute Elements of Dong V 4 V / V \ 2 1 h I2356/ d 5 l' A 3. 1416 X d 2 = ..2732^=,; (i+« + oi).£=»; Dipes. V2 g h V i +«+ < 4 — v; and .4787 5Q5Xd-J-c l)~ = d in ins. This latter formula will only give an approximate dimension in consequence of 4 V unknown element d, and also of C, as v = ’ ’ 3.1416 xd 2 For Illustration, see Miscellaneous Illustration, page 556. HYDRAULICS. 549 To Compute Vertical Height of a Stream projected from IPipe of a Fire-engine or IPnmp. Rule.— A scertain velocity of stream by computing volume of water run- ning or forced through opening in a second; then by Rule in Gravi ation, pagf 488, ascertain height to which stream would be elevated if wholly un- obstructed, which multiply by a coefficient for particular case. In great heights and with small apertures, coefficients should be reduced. In consequence of the varying elements and conditions of operation of fne- engines, it is difficult to assign a coefficient for them. Difference between actual discharge and that as computed by capacity and stroke of cylinder, as ascertained by Mr. Larned, 1859, was 18 per cent. = a coefficient of .82. \ steam fire-engine of the Portland Company, discharging a stream 1.125 ms in diameter through 100 feet 2.5 inch hose, gave a theoretical head, computed from actual discharge, of 225 feet. Lid stream vertically projected was 200 feet; hence coefficient in this case was .88. . ^ , Example.— I f a fire engine discharges 14 cube feet of water vertically through a pipe .75 inch in diameter in one minute, how high will the water be projected. x < x 1728 -- .4417 area of pipe, -4- 12 ins. in a foot, -4- 60 seconds = 76 07 /^ ve- locity; and as coefficient of such a stream = at .85, then 114.1 X .85 — 96.98 feet Or H — '°° 22 H — h. H representing head at nozzle , and d height of jet , both in d feet, and d diameter of nozzle in ins. (R. F. Hartford.) Illustration. — A ssume head of no feet and diameter of nozzle .75 inch. IIO 1IO _ 35 . 5 = 74.5./M. •75 Note. — The loss of head is greater with ring than with smooth nozzles. E. B. Weston, Am. Soc. C. E., puts the difference at .000 171 v 2 . The loss of head increases with the absolute height of the jet, and is less with an increase of its diameter. This loss increases nearly in ratio of square of height ot jet, and varies nearly in inverse ratio to its diameter. Cylindrical Ajutage. Mean coefficient as determined by Mariotte and B°ssut = .oo3°66 square of effective head for cylindrical ajutages; hence, for conical, alike to that ot an engine pipe, coefficient ranges from .72 to .9, or a mean of .81. By formula of D’Aubuisson, .003047 h 2 = h'. Effective head, or h, in preceding example = 114.1. Then ” 4 -i — * 00 3 0 47 X ii 4 .i 2 = ii 4 -i — 39- 6 7 = 74-43 feet height of jet. Hence, fora conical or engine pipe, 74.43 X .81 = 60.29 feet, or a coefficient of .535. To Coin Fig- 13- p„te Distance a. Jet of Water will Toe projected A • _ Z „ B C Fig 13 is equal to twice square root of A 0 X 0 B. * is 4 times as deep below A as^a is, s will discharge If twice volume of water that will flow from a in same time, as 2 is yj of A s and 1 is yj of A a. Note —Water will spout farthest when 0 is equidistant from A and B ; and if vessel is raised above a plane, B must be taken upon plane. C B Volumes of water passing through equal apertures in same time are as square roots of their depths from surface. Rule. — M ultiply square root of product of distance of opening from sur- face of w r ater, and its height from plane upon which water flows, in feet by 2, and product will give distance in feet. Example.— A vessel 20 feet deep is raised 5 feet above a plane; how far will a jet reach that is 5 feet from bottom of vessel? 20 — 5 X 5 + 5 = I 5 °> and y/ l $° X 2 = 24.495 feet. 550 HYDRAULICS. Velocity of a jet of water flowing from a cylindrical tube is determined to be .974 to .98 of actual to theoretic velocity, or = .82 of that due to height of reservoir. Hence volume of discharge through a cylindrical opening Jets d’Eau. (Fig. 14.) That a jet may ascend to greatest practicable height, communication with supply should be perfectly free. Short tubes shaped alike to contracted fluid vein, and conically convergent pipes, are those w r hich give greatest velocities of efflux. Hence, to attain greatest effect, as in fire-engines, long and slightly conically convergent tubes or pipes should be applied. In order to diminish resistance of descending water, a jet must be directed with a slight inclination from vertical. Effect of combined causes which diminish height of a jet from that due to elevation of its supply can only be determined by experiments. Great jets rise higher than small ones. With cylindrical tubes, velocity being reduced in ratio of 1 to .82, and as heights of jets are as squares of these coefficients or ratios, or as 1 to .67, height of a jet through a cylindrical tube is two thirds that of head of water from which it flows. H C = h. H representing head of water , C coefficient , and h height of jet. ( Moles - worth. ) When d = H -r* 300, C = .96. When d = H -f- 1500, C = .8. “ “= 11 rr- 45°: “ = • 93 - “ “ = 11 -r- 1800, u _ • 7 - U U _ U _A_ (5 00? ‘• = .9. u “ = “ -i- 2800, u .6. “ 800, ‘‘=.87. “ “=“-^-3500, 11 = • 5 - U C< _ CC _L_ IOOO} “ = .85. “ “=“-f- 45 oo, u .25. FLOW OF WATER IN RIYERS, CANALS, AND STREAMS. Running Water . — Water flows either in a natural or artificial bed , or course. In first case it forms Streams, Brooks, and Bivers ; in second, Drains, Cuts, and Canals. Bed of a water-course is formed of a Bottom and two Banks or Shores. Transverse Section is a vertical plane at right angles to course of the ; flowing water ; Perimeter is length of this section in its bed. Longitudinal Section or Profile is a vertical plane in the course or thread of current of flowing water. Slope or Declivity is the mean angle of inclination of surface of the water to the horizon. Fall is vertical distance of the two extreme points of a defined length of the flowing course, measured upon a horizontal plane, and this fall assigns \ angle for defined length of the course. Line or Thread of Current is the point where flowing water attains its maximum velocity. Mid-channel is deepest point of the bed in thread of current. Velocity is ; greatest at surface and in middle of current ; and surface of flowing w r ater is highest in current, and lowest at banks or shore. A River, Canal, etc., is in a state of pemnanency w T hen an equal quantity of water flows through each of its transverse sections in an equal time, or when V, product of area of section , and mean velocity through whole extent of the stream , is a constant number . = .82 aV 2 g h. Fig. 14. hydraulics. To Compute Mean Depth of Flowing Water. bv number of divisions, and quotient is the mean depth. To Compute Mean Area of Flowing Water. Rule i.-Multiply breadth or breadths of the stream, etc., by the mean depth or depths, and product is the area. 2. Divide the volume flowing in cube feet per second by mean velocity in feet per second, and quotient is area m sq. iee . To Compute Volume of Flowing Water. Rule. — Multiply area of the stream, etc., in sq feet, by the mean velocity of its flow in feet, and product is volume in cube feet. To Compute Mean Velocity of Flowing Water. steamrefcfand 6 q^tenCSued^y coefficienT oTvelocity^lgive rolydtoSes'ftom ii^e of current toward banks, and, toobtainmean superficial velocity, ui + P2 + ^ 3 _ : hence, n To Compute Mean Velocity in whole Profile of a Navi- gamble River, etc., V+ 7 - 2 V V = tel ocity at bottom, and V+T 5 - V V = mean velocity. In rivers of low velocities multiply mean velocity by .8. Obstruction in Fivers. ( Molesworth .) 58.6 q_ 0 -x( — ) — 1 = R. v representing velocity in ins. per second previous to obstruction, Xand a areas of river unobstructed and at obstruction in sq.feet, and R rise in feet nTwtrnrted flow of a river is 6 feet per second, and - a “ d 5° «* wbat be rise in feet? . 2 JL + o S x V-i = -66 4 X.3 3 2 = .i54 f eet - 58.6^ 3 \ 9 °/ F lo W of Water in Lined Channels. (Bazin.) V C D _ v 1 n d representing mean hydraulic depth in feet, F p ’ / 1 \ fall or length of channel to fall of i,x and jcfy-hpJ J no jail, VI bc/tt/n/n wj VI y as per table , and C as per table p. 543 Plastered. Cut Stone . .0000045 .000013 y 10. 16 4-354 X I V .00006 1 1. 219 .000 35 I .214 p 2 D — u A = it — u » . - For Sections of Uniform Area, as Canals, Sewers, etc. f area of flow in sq. feet, P wet perimeter of section, and D fall of “iS^TKATiON.-Area of transverse section of a sewer is 50 sq. feet, its wet perim- eter 20 feet, and its fall 5 feet per mile. — — 7 (§2 x 2 x 5) = V25 = 5 f eet For Sections of Rivers. 12 JB p = v. Illcstratioh.— A ssume area 500 sq. feet, wet perimeter 200, and fall 5 feet per mile. 12 ./5X— = n Vi2-5 = 42-4 feet. V 200 552 HYDRAULICS. Hydraulic Fadius oi Me an Depth is obtained bv dividing qtcsl of trans- verse section by wet perimeter, both in feet. To Compute Fall per Mile for a required Mean Velocity. /VXl2\ 2 * _ y I2 - I -r* 2 r = D. r representing hydraulic radius in ins. Upper surface of flowing water is not exactly horizontal, as water at its surface flows with different velocities with respect to each other, and consequently exert on each other different pressures. If v and v i are velocities at line of current and bank of a stream, the difference ^.2 of the two levels rs — h. 2 9 2 Illustration.— If v = 5 feet, and u x 9 i>; then * 2 ~~' 9 X 5 _ ± 75 _ = 8 foof 2 9 64.33 A velocity of 7 to 8 ins. per second is necessary to prevent deposit of slime and growth of grass, and 15 ins. is necessary to prevent deposit of sand. Maximum velocity of water in a canal should depend on character of bed of the channel. Thus, Mean Velocity should not exceed per second over Fine clay 6 ins. I River sand. .. A slimy bed 8 “ Small gravel. Common clay 6 “ | Large shingle. 1 ft. 1 “ 3 “ Broken stones. Stones Loose rocks. . . 4 ft. 6 “ 10 “ To Compute Velocity of* Flow or Discharge of Water in Streams, Pipes, Canals, etc. 1. When Volume discharged per Minute is given in Cube Feet , and Area of Canal , etc., in Sq. Feet. Rule. — Divide volume by area, and quotient, di- vided by 60, will give velocity in feet per second. 2. When Volume is given in Cube Feet , and Area in Sq. Ins. Rule.— D i- vide volume by area ; multiply quotient by 144, and divide product by 60. 3- When Volume is given in Cube Ins., and Area in Sq. Ins. Rule. — Di- vide volume by area, and again by 12 and by 60. To Compute Flow or Volume of Discharge. 1. When Area is given in Sq. Feet. Rule.— M ultiply area of flow by its velocity in feet per second, and product, multiplied by 60, will give volume in cube feet per minute. 2. When Area is given in Sq. Ins. Rule. — Multiply area by its velocity, and again by 60, and divide product by 144. Note i. — Velocities find discharges here deduced are theoretical, actual results de- pending upon coefficient of efflux used. Mean velocity, however, as before given, page 529, may be taken at y/Yg .673 = 5.4 feet, instead of 8.02 feet. 2. — As a rule, with large bodies, as vessels, etc., their floating velocity is some- what greater than that of flow of water, not only because in floating they descend an inclined plane, formed by surface of the water, but because they are but slightly affected by the irregular intimate motion of water: the variation for small bodies is so slight that it may be neglected. To Coinpnte ITeight of Head, of Flowing Water. When Volume and Area of Flow are given in Feet. Rule. — Divide vol- ume in feet per second by product of area, and $ coefficient for opening, and square of quotient, divided by 64.33, will give height in feet. Example. — Assume volume 266.48 cube feet, area 40 sq. feet, and C = -623. Then ( 266 ^ . V h- 64.33 = 2^ = 4/^ \4° X £ -623/ 4 33 64.33 4 HYDRAULICS. 553 Su."bxxierged or Drowned Orifices and. Weirs. When wholly submerged (Fig. 1 5). —Available pressure at any point in depth of orifice is equal to difference of pressure on Fig. 15- « each side. Whence, C V2 g h = v, and C a V 2 g h = V. a representing area of sluice in sq.feet. Illustration. — Assume opening 3 feet by 5, Then, .5 X 3X5 V64.33 X 4 = 7-5 X 16.04 = ' — 120. 3 cube feet per second. When partly submerged (Fig. 16). h'-h = d = submerged depth, and h - P “ h" — d' — remaining portion of depth; whence Fig- l6 - m. d' -f- d = entire depth, a nd C l V2 ~g (d y/h -f f h y/h — h" y/h") = Y. Illustration. — Assume opening as above, h = 4 feet , fc' = 6, h" == 3, and C = .5. Then d = 6 — w \ ^ 2 f ee ^‘ o I — 1 I rri Then . ^ X «; X 8.02 (2 a /4 + X 4 V4 — 3 V3) ' — 1 — i k = 20.05 X 5-869 = 117.67 cube feet per second. Fig. 17. When drowned (Fig. 17). ClVz gh (d-\-%h) = V. Illustration. — Assume opening as above, h — 4 feet, d = 2, and C = -52. Then, .52 X 5 X V64.33 X 4 X (2 + f 4) == 2.6 IRD X 16.04 X 4-66 = 194.34 cube feet per second. CANAL LOCKS. Single Locks. When a fluid passes from one level or reservoir to another, through an aperture covered by the fluid in the latter, effective head on each point of aperture, and consequently head due to velocity of efflux at each instant, is the difference of levels of the two reservoirs at that instant. Hence C a -JTght = V per second. K representing difference of levels. To Compute Time of Filling and Discharging a Single Lock.-Fig. IS. When Sluice in Upper Gate is entirely under Water , and above Lower Level. A It' h — time of filling up to centre of sluice. C a V 2 gh . . ^r, h representing height of centre of sluice in upper gate from surface of canal or reservoir, and h height gj of centre of sluice in upper gate from lower sur- ^ face , or water in the lode or river, all in feet , ana 2 A h — time of filling the remaining space, C a V2 g h where a gradual diminution of head of water occurs. Consequently, - — ^ — — t time of filling a single loclc. C a V2 gh When Aperture or Sluice in Lower Gate is entirely under Water , and above Lower Level. 2 A __ ^ me 0 f emptying or discharging it. a ' representing C a' VTg area of lower sluice. 3 A 554 HYDRAULICS. Illustration. — Mean dimensions of a lock, Fig. 18, are 200 feet in length by 24 in breadth; height of centre of aperture of sluice from upper and lower surfaces is 5 feet; breadth of both upper and lower sluices is 2.5 feet; height of upper is 4 feet, and of lower— entirely under water— 5 feet; required the times of filling and dis* charging. h = 5, h' = 5 , A = 200 X 24 = 4800, C = .545, a = 4 X 2. 5 = 10, a' = 5 X 2.5 = 12.5. 4800 X 5 24 opo — ■ = = 245. 59 seconds = time of filling lock up to centre of .545X10 xVTJh 97-72 , . A 2 X4800X 5 48000 _ , sluice; and :==■ = 491.18 seconds — time of fillingremain- .545X10XV2 gh 97-72 ing space , or lock above centre of sluice , and 245. 59 -f- 491.18 = 736.77 seconds , whole time . Or (5 + 2 X 5) X 4800 _ 72 000 __ 736.77 sec. = time of filling. .545 X 12.5 X VTg .545X10XV2 gh 97 - 7 2 3° 358.08 = = 554.9 seconds — time of discharging. 54-7 When Aperture or Sluice in Upper Gate is entirely under Water and below Lower Level. r — — time of filling lock. Cay/zg When Sluice in the Lower Gate is in part above Surface of Lower Level i • 1 1 * 2A(/l-f/0 , . . , . and in part below it. 7 — = time of dis Cby/zg (ft^h + h' — ^ + d' y/h -j- h'^j charging, d and d' representing distances of part of aperture above and of below surface of lower water , b breadth of aperture, and h and h ' as before. Illustration. — Assume sluice in preceding example to be 1 foot above lower level of water, or that of lower canal; what is time of discharge of lock, distance of part of aperture 1 foot and of that below surface of water 4 feet? 2X4800(5 + 5) 96000 • 545 X 2.5 X 96000 171-95 02 [1 X V5 + 5 — (i-i-2) + 4X V5 + 5] 558.3 seconds. ‘ 10.93 X (3.082 + 12.65) " Double Lock. (J. D. Van Buren, Jr.) A double lock is not a duplication of a single lock in its operation, for in lower chamber supply of water w is from upper one, having no % 5 influx, instead of a uniform sup- » o Uppgr ply flowing directly from sur- M wi A face level of canal or feeder. Operation, therefore, of a double lock is complex, addition to formula for a single lock be- ing that of discharging of water in upper lock to All lower, the head of water gradually decreas- ing in the chamber, which is closed from upper reach during discharge into lower. To Compute Time required for Water to Fall from Upper to Uniform Water Level. 1. - r — — (v//-f v / 2 Tit—- y/z h — 2 d) — t, A representing horizontal area of lock, C a y/ g and a area of sluice opening, both in sq. feet, C coefficient of discharge = .545 for openings with square arrises , g acceleration of gravity, f depth of centre of sluice HYDRAULICS. 555 below uniform level , h depth of centre sluice opening belowuppei water level, and d height of centre of sluice above lower water level , alt in feet , and t time Jo) water to fall from upper to uniform water level , in seconds. a — 5 ; /= 6 ; = 14; and d = • 545 ; Illustration. — A == 2000 sq. feet ; C 2 feet. (Fig. 19.) Then, — — — = X 7~74 I= *9 = 367-6 seconds. .545X5X5-67 I 5-45 Tf 7 A V/ _ t . = _ 3 66. 34 seconds. 2 - Ifd = °> r.nTTn- 1 ' ^^XO^.67 15-45 CaV0 ri -545X5X5-67 Note -/is never greater than Z (lift in feet); it is equal to l when id = o; f> is equal to l when A ^o, never greater. In each case it is the unbalanced head above sluice, however far below the lowest water le\el the sluice is. To Fill Upper Lock or Empty Lower. To fill upper lock or empty lower, when the sluice is below the lowest water-line, in either case, takes the same time; for the head diminishes at the same rate, one from the upper surface, the other from the bottom. Aa/z /__£ Her e,f being below lowest water level of lock ~ Z feet, as d = o, Ca^/g ' ’ 2000 y/ 2 X 8 8000 7 and f— whole = ^ = — = 5 - 7-8 seconds. To Discharge a like Volume under a Constant Head. Ay If A //m.- _ C aV 7 g c *VsJ..-v Tl 545 X 5 V 64.33 -258.9 seconds, Or, one half the time given by preceding case. The times deduced by preceding formulas are in the following proportions in / a / 2 / 1 order, as i : V 2 : — , or i : v 2 : -~r • ’ 2 V 2 If sluice of upper lock, through which it is filled, is above lowest water level, then, by combining formulas 3 and 4, the time is thus deduced. To fill from Lowest Water Level of said Lock to Level of Centre of Sluice. 5 A /' representing height of centre of sluice above said lowest water Vay/^g level. To fill remaining Portion of Lock above Sluice. 6. 2 A ^ ^ - — t". f" representing depth below upper water level of centre of Cay/2 g A sluice or remaining portion of lift. Hence, t' -j- Z" == 77-— 7— iff' + 2 V f") =F. & V ft V 2 3 To fill Lower Lock under Constant Head from Upper Canal Level. C a^/—g\ +h V* ) ^ 8. Tf both lifts are the same, h — f=l. and — - ^ — 2 * / — ) = ^ If lower lock is filled from upper one under a constant head, when latter is drawn down to lowest level, formula 7 will apply by making h —fi and A — (2 V/+ -77); which is identical with 7, for/ =/ 2 and d —fi the cases Cay/2 g \ v/ / being the same. 556 HYDRAULICS. MISCELLANEOUS ILLUSTRATIONS. 1. If external height of fresh water, at 6o° above injection opening in condenser of a steam-engine, is 3 feet, and the indicated vacuum at 23 ins., velocity of water llowing into condenser is thus determined. [Formula page 532.) v = V 2 g [h + h'). li f representing height of a column of water equivalent to press- ure of atmosphere within condenser. Assuming mean pressure of atmosphere = 14.7 lbs. per sq. inch, height of a column of fresh water equivalent thereto = 33.95 feet. Then, if 1 inch = .4912 lbs., 23 ins. = 11.3 lbs.; and if 14.7 lbs. = 33.95 feet, 11.3 lbs. — 26. 1 feet. Hence v = V2 g (3 + 26. 1) = 43.27 feet, less retardation due to coefficient of both influx and efflux. 2. What breadth must be given to a rectangular weir, to admit of a flow of 6 cube feet of water, under a head of 8 ins. ? [Formula page 533.) 6 6 r / ' 1 • == 77-7 — = 2.21 Jeet. ^X. 625 V20 66 .417X6.55 3. It being required to ascertain volume of water flowing in a stream, a tem- porary dam is raised across it, with a notch in it 2 feet in breadth by 1 in depth, which so arrests flow that it raises to a head of 1.75 feet above sill of notch; what is volume of flow per second? [Formula page 533. ) C = .635. - X -635 X2X1.75 V2^X J-75 == 1-481 X 10.6 = 15.7 cube feet. 3 4. A rectangular sluice 6 feet in breadth by 5 in depth, has a depth of 9 feet of water over its sill, and discharges, as per example page 535, 380.95 cube feet per second ; what is velocity of flow ? [Formula page 535. ) 380.9s = 38095 ^ 6X (9 — 4) 30 2 ^/h 3 ■Jli' 3 If volume was not given : — C v 2 g X y —. — = v. C = . 625. 3 ti — ti Then — x .625 X 8.02 X ^ 729 — 3.341 X 3-8 = 12.7 feet. 3 9 — 4 5. If a river has an inclination of 1.5 feet per mile, is 40 feet in breadth with nearly vertical banks, and 3 feet depth; what is volume of its discharge ? [Formula p. 542.) Perimeter 40 - f - 2 X 3 = 46 feet; hydraulic mean depth — 2.61 feet; a = 120 feet; Then. 46 C per table , page 543, for assumed velocity of 2.5 feet — .0075. ■ X 64.33 X 1.5 = V.0659 X 96.5 = 2. 52 feet velocity. .0075 X 5280 X 46 ' Hence 120 X 2.52 = 302.4 cube feet. 6. What is head of water necessary to give a discharge of 25 cube feet of water per minute, through a pipe 5 ins. in diam. and 150 feet in length ? [Formula p. 548.) Tabular number for diameter 5 ins., page 547, = 263.87. < 2 Then 263.87 -r- 25 = 111.3, and 150-4- 111.3 = I -35 f ee t- If this pipe had 2 rectangular knees or bends, what then would be head of water required? [Formula page 545.) C, page 545, for — = .984, area of 5 ins. — . 136 feet, and -4- 60 = 3.06 feet velocity. Then — X -984 X 2 = .2863, which, added to 1.35 = 1.64 feet. 04-33 By formulas foot of page 548, C = .o24, and c .505 velocity — 3.06 feet ; head — 1.49 feet, and volume 26.38 cube feet. 7. Tf a stream of water has a mean velocity of 2.25 feet per second at a breadth of 560 feet, and a mean depth of 9 feet, what will be its mean velocity when it has a breadth of 320 feet, and a mean depth of 7.5 feet? [Rule page 548.) s6oX 9 Xa.2 5 = M34Q = ^ 320 X 7- 5 2400 HYDRAULICS. 557 8 What volume will a pipe 48 feet in length and 2 ins. in diameter, under a head of 5 feet, deliver per second ? ( Formula page 547.) Tabular number far diameter 2 ins., page 547, .= 26.69. , 2 ^ ,6 9 ; — 8.61, "which -f- 60 = . 143 cube feet. V 748 *= 3 1 * Then If this pipe had 5 curves of 9O 0 , with radii — = - = .5; wliat vvould be its dis ' charge per second ? ^ ^ V = .i 43 ; a = 2-4- 144 = .0139; Gper table = ~ = .294; v --^^ = 10.29 feet. Then .294 X^X = - M7.X 1-64 ff 241, w/itc/i X 5 /or 5 curves = 1.2 = * 180P 04.33 / lCl y t £ due to resistance of curves, h =. 5 — 1. 2 = 3. 8. Hence, if V 2 g 5 = .143; V 2 g 3.8 = .125 cube feet. a If a slide stop valve, set in a cylindrical conduit 500 feet in length and 3 ins. in diameter, is raised so as to close .625 of conduit; what volume will it discharge under a head of 4 feet ? ( Formula page 546. ) C for conduit = . 5, for friction . 025 r and for slide valve .375 open , table, page 545, 5.52, d = .25, and a = 7.07 sq. ins. Then - , •» * 9 -- — : ■ ■■■ ■ • : : l6; °^ - - = 2.13 feet velocity , and ,/ 500 \ V( 7 -° 2 + 5 °) f ('+.5+5-5- + ^S — •) 3.13 x 12X7-07 = 180.71 cube ins. 10. If a single lock chamber is 200 feet in length by 24 in breadth, with a depth of 10 feet, centre of upper gate, which is 4 feet in depth by 2.5 in breadth, is at middle of depth of chamber, lower gate, 5 feet in depth by 2.5 in breadth and wholly immersed; what is time required for filling and discharging it? (Formula p. 553.) C = .6 i 5, 7t = 5, h' = 5, A = 200 X 24 •= 4800, a = 4X2.5 — 10, and a = 5 X 2.5 = 12.5 (2X5 + 5) 48o^_ = 7^ = 652.8 seconds time of filling. • 615X10^64.33.* 5 ., 110-27 2 x 4800 X V 5 + 5. 3^83^ _ 491 . 4 seconds time of emptying. , ..-p=zr — 61.73 .615 x 12.5 V 2 g 1,. In a moderately direct and uniform course of a river, the depths and velocities are as follows; what is the volume of its flow and what, its mean velocity? (p. 551.) Feet. Feet. Feet. Feet. Feet. Area of profiles = 5 X 3 + 12 X 6 -f- 20 X n + 15 X 8 -f- 7 X 4 = 455 sq.feet. Distances 5 12 20 *5 7 Depths 3 6 11 8 4 Mean velocity 1.9 2.3 2.8 2.4 2.1 , - - . — . 1S x 1.9 ■ + 72 X 2.3 -f 220 X 2.8 + 120 x 2.4 + 28 X 2. 1 = 1156.9 cube feet volume, and 115 — 2 — 3.54 feet velocity. 455 IVtiner’s Inch. A “ Miner’s inch ” is a measure for flow of water, and is an opening one inch square through a plank two inches in thickness, under a head ox six inches of water to upper edge of opening. It will discharge 11.625 U. S. gallons water in one minute. Theoretical HP under different Heads. 18° I 7°' I 60 | 5° | 4° 30 1 20 1 15 |io | 5I 3| 1 4-°6| 1 4-641 1 5-4i 1 6.5 | 8.12 10.8I16.2I21.6I32.5I65I io8|; Water Inch ( Pouce d eau ).- —Circular opening of 1 inch in a thin plate is equal to a discharge of 19.1953 cube meters per 24 hours. 3 A* 558 HYDRODYNAMICS. HYDRODYNAMICS. Hydrodynamics treats of the force of action of Liquids or Inelastic Fluids, and it embraces Hydraulics and Hydrostatics: the former of which treats of liquids in motion, as flow of water in pipes, etc., and latter of pressure, weight, and equilibrium of liquids in a state of rest. Fluids are of two kinds, aeriform and liquid, or elastic and inelastic, and they press equally in all directions, and any pressure communicated to a fluid at rest is equally transmitted throughout the whole fluid. Pressure of a fluid at any depth is as depth or vertical height, and pressure upon bottom of a containing vessel is as base and perpendicu- lar height, whatever may be the figure of vessel. Pressure, therefore, of a fluid, upon any surface , whether Vertical , Oblique , or Horizontal , is equal to weight of a column of the fluid, base of which is equal to sur- face pressed, and height equal to distance of centre of gravity of sur- face pressed, below surface of the fluid. Side of any vessel sustains a pressure equal to its area, multiplied by half depth of fluid, and whole pressure upon bottom and against sides of a vessel is equal to three times weight of fluid. Pressure upon a number of surfaces is ascertained by multiplying sum of surfaces into depth of their common centre of gravity, below surface of fluid. When a body is partly or wholly immersed in a fluid, vertical press- ure of the fluid tends to raise the body with a force equal to weight of fluid displaced ; hence weight of any quantity of a fluid displaced by a buoyant body equals weight of that body. Centre of Pressure is that point of a surface against which any fluid presses, to which, if a force equal to whole pressure were applied, it would keep surface at rest. Hence distance of centre of pressure of any given surface from surface of fluid is same as Centre of Percussion. Centres of* Pressure. Parallelogram , Side, Base , Tangent , or Vertex of Figure at Surface of Fluid, is at ;i .66 of line (measuring downward) that joins centres of two horizontal sides. Triangle , Base uppermost, is at centre of a line raised vertically from lower apex, and joining it with centre of base; aud Vertex uppermost , it is at .75 of a line let fall perpendicularly from vertex, and joining it with centre of base. Right-angled Triangle , Base uppermost , is at intersection of a line extended from centre of base to extremity of triangle by a line running horizontally from centre of side of triangle. Vertex or Extremity uppermost , is at intersection of a line ex- tended from the centre of the base to the vertex, by a line running horizontally from . 375 of side of triangle, measured from base. Trapezoid , either of parallel Sides at Surface , 6 + 3 5 ' - X a = d. b and b' repre- 2 b -{- 4 b senting breadths of figure, d distance from surface of fluid, and a length of line join- ing opposite sides. Circle , at 1.25 of its radius, measured from upper edge. 3 p r Semicircle, Diameter at Surface of Fluid, — = d. r representing radius of circle 16 1 5 » r — 32 r and p = 3. 1416. Diam. downward, = d. J i2i> — 16 HYDRODYNAMICS. 559 Side, Base, or Tangent of Figure below Surface of Blviid. Rectangle or Parallelog'm. h'3 — hs X .... ^ = d ; 3 m o -f- wi 2 = d and — = d". 30 3 h ' * — /i 2 ~ 3 0 h and Ji' representing depths of upper and lower surfaces of figure and d depth , both from surface of fluid, m half depth of figure, o depth of centre of gravity of figure from surface of fluid, d' distance from upper side of figure, and d distance from centre of gravity. 1 -* 10 o * . Triangle. — Vertex Uppermost. --d\ - == d'. Base Uppermost. Z 2 + I 8 ° 2 _. 1 representing depth of figure, d distance from surface of fluid upon a line from vertex to centre of base, and d' distance from centre of gravity of figure. Circle 4 °~ + y2 _ d or :L_ = distance from centre of circle. • 4 ° 40 * 2 *6Z 2 - Semicircle. —Diam. Horizontal and Upward or Downward. -— — -^- Q + 0 = a 5 3 P l ~~ 4 1 . 4 l __ an( j i — c. d representing distance from op ’3J 3 ’ 4 o 9 p o surface of fluid, d' distance of centre of gravity from centre of arc, d ' distance of centre of gravity from diameter when it is uppermost, and c centre of pressure. Pressure. To Compute Pressure of a FlnicL upon Bottom of its Containing Vessel. Rule —Multiply area of base by height of fluid in feet, and product by weight of a cube foot of fluid. To Compnte Pressure of a Blnid upon a V ertical. In- clined, Curved, or any Surface. Rule.— Multiply area of surface by height of centre of gravity of fluid in feet, and product by weight of a cube foot of fluid. Example i. — W hat is pressure upon a sloping side of a pond of fresh water 10 feet square and 8 feet in depth ? Centre of gravity, 8-4-2 = 4 fed from surface. Then 10 2 X 4 X 62. 5 = 25 000 lbs. 2. —What is pressure upon staves of a cylindrical reservoir when filled with fresh water, depth beiDg 6 feet, and diameter of base 5 feet? 5 x 3-1416= 15-708 feet curved surface of reservoir, which is considered as a plane. 15.708 X 6 X 6-4-2 = 282.744, which X 62.5 = 17671.5 lbs. 3, a rectangular flood-gate in fresh water is 25 feet in length by 12 feet deep; what is pressure upon it? 25 X 12 X 12-4-2 = 1800, which X 62.5 = 112 500 lbs. When water presses against both sides of a plane surface, there arises^ from resultant forces, corresponding to the two sides, a new resultant, which is obtained by subtraction of former, as they are opposed to each other. Illustration -Depth of water in a canal is 7 feet; in its adjoining lock it is 4 feet, and breadtn of gates is 15 feet; what mean pressure have they to sustain, and what is depth of point of its application below surface ? 7 x 15= 105, and 4 X 15 =60 sq.feet. (105 X —60 X 2) X 62.5 = 1546.875 lbs., mean pressure. Then 1546.875-5-62.5 = 247.5 =cube feet pressing upon gates upon high side, and 247. 5 -4- 15 X 7 = 2. 35 feet = depth of centre of gravity of mean pressure. To Compute Pressure on a Slaice. Awd = P, and C P = P'. A representing area of sluice in sq. feet, w weight of water per cube foot, d mean depth of sluice below surface , in feet, P pressure on sluice , and P' power required to operate it, both in lbs. C = .68 when sluice is of wood, and .31 when of iron. HYDRODYNAMICS. 560 Example.— What is pressure on a sluice-gate 3 feet square, its centre of gravity being 30 feet below surface of a pond of fresh water? 3 X 3 X 30 = 270, which X 62.5 = 16 875 lbs. To Compute Pressure of a, Column of a pTliaid. per Sq. Inch. Rule. — M ultiply height of column in feet by weight of a cube foot of fluid, and divide product by 144 ; quotient will give weight or pressure per sq. inch in lbs. Note. —When height is given in ins., omit division by 144. PIPES. To Compute required Thickness oP a ripe. Rule.— Multiply pressure in lbs. per sq. inch by diameter of pipe in ins., and divide product by twice assumed tensile resistance or value of a sq! inch of material of which pipe is constructed. By experiment, it has been found that a cast-iron pipe 15 ins. in diameter, and .75 of an inch thick, will support a head of water of 600 feet; and that one of’ oak, of same diameter, and 2 ins. thick, will support a head of 180 feet ? Example 1.— Pressure upon a cast-iron pipe 15 ins. in diameter is 300 lbs. per sq. inch; what is required thickness of metal? 300 X 15 = 4500, which -r- 3000 X 2 . 7 5 inch. Note. — Here 3000 is taken as value of tensile strength of cast iron in ordinary small water-pipes. This is in consequence of liability of such castings to be im- perfect from honey-combs, springing of core, etc. 2. — Pressure upon a lead pipe 1 inch in diameter is 150 lbs. per sq. inch; what is required thickness of metal ? Here 500 is taken as value of tensile strength. 150 X 1 = 150, which rf- 500 X 2 = . 15 inch. Cast-iron IPipes. To Compute Thickness, etc., of* Flanged IPipes. For 75 lbs. Pressure. .025 D -f- .25 =c T .03 D-f .3 = t .05 D-f- 1. 15 —l •°3 R-f -35 —f 1.05 D -f- 4.25 d -f- 1.25 = o 1.05 D -f- 2 X d -}- 1 — o' A xp~ . 7 D -f- 2. 2 = n ; For 100 lbs. Pressure. •03 D+ -3 - •035 D + -45 = I>+ 1.15 6 - 4000 •05 .04 D + _ 11 D + 5 X d- \- 1.5 11 D + 2.5 X d-f- 1.4 and — T == t — I = f =z O — 0 \/-78S4 +■■■0 = d. D representing diam. of pipe , T thickness of metal, t thickness and l length of boss, f thickness of flange, o diam. of flange, o' diam. of centres at bolt holes, and d diam. of bolts, all in ins.; A area of pipe and a area of bolt at base of its thread, in sq. ins., p pressure in lbs. per sq. inch , and C a coefficient due to diam. of bolt. Thus, diam. . 125 -f- .032, .25 -h .064, .5 -{-• 107, i-f-.i6, 1.5 -f-. 214, and 2 -[-.285. Illustration.— W hat should be dimensions of a flanged pipe, 10 ins. in diameter, for a pressure of 100 lbs. per sq. inch? .7 X 10 -f- 2. 2 = 9. 2 = 10 number of bolts, and diam. 10 ins. = 78. 54 ins. area = A. 78.54 X .00^0 y and J.J96 35 + c = v . 25 = , 5 ; hence, .5 + . ro 7 = 4000 v .7054 .607 = .625 lbs. diameter of bolts ; .03 X 10 +.3 = .6 = thickness of metal; .035 X 10 — }- . 45 = . 8 — thickness of flange; .05 X 10 -f- 1. 15 = 1.65 = length of boss; .04X10 -f- . 6 = 1 s= thickness of flange ; i.iXio-f-5X.625-f-i.5 = i5.625 = diameter of flange; and 1.1 X 10 -f 2.5 X .625 -f 1.4 — 13.9625 diameter of bolt holes. For Tables of Cast-iron Pipes, see page 132. i \ HYDRODYN AMICS. 56l To Compute Elements of Water-pipes. 000 124 c; P d 4 - C = t: or, .000054 H d -J- C = t\ -4336H==P; and 142 d 2 v 2. A.11 = W. P representing pressure of water in lbs. per sq. inch, D andd external anil internal diameters of pipe, and t thickness of metal, all in ins., C coeffi- cient for diameter of pipe, and H head of water in feet. C = .37 for pipes less than 12 ins. in diameter, .5 from 12 to 30, and .6 from 30 to 50. To Compute Weight of Pipes. To Diameter add thickness of metal, multiply sum by 10 times thickness, and product will give weight in lbs. per foot of length. Weight of Faucet end is equal to 8 ins. of length of pipe. Hydrostatic Press. To Compute Elements of a Hydrostatic Press. PEA _ w . W E a — . W E a __ p P A E __ a p re p rescn ting power or press- l* a 1 P l * E A W l ure applied , W weight or resistance in lbs., I and V lengths of lever and fulcrum in ins. or feet, and A and a areas of ram and piston in sq. ins. Illustration. — A reas of a ram and piston are 86.6 and 1 sq. ins., lengths of lever and fulcrum 4 feet and 9 ins., and power applied 20 lbs. ; what is weight that may he sustained? *0X4 X12X86.6 = 83136 = 3 m 9X1 9 To Compute Thickness of HVTetal to Resist a given Pressure. Rule.— M ultiply pressure per sq. inch in lbs. by diameter of cylinder in ins., and divide product by twice estimated tensile resistance or value of metal in lbs. per sq. inch, and quotient will give thickness of metal required. Example.— P ressure required is 9000 lbs. per sq. inch, and diameter of cylinder is 5.3 ins. ; what is required thickness of metal of cast iron? 0000 X 5- 3 47 7°° Value of metal is taken at 6000. - z — — - — = 3-975 xns ' 6000 X 2 12 000 Values of Different Metals in Tons. ( Molesworth . ) Cast iron 41 | Gun metal 22 | Wrought iron.. .14 | Steel 06 Hydraulic Ram. Useful effect of an Hydraulic Ram, as determined by Eytelwein, varied from .9 to .18 of power expended. When height to which water is raised compared to fall is low, effect is greater than with any other machine ; but it diminishes as height increases. Length of supply pipe should not be less than .75 of height to which water is to be raised, or 5 times height of supply ; it may be much longer. To Compute Elements. .00113 VA = EP; ^^ = v ; i. 45VV = d ; -75 = ^ 5 and | x V^ = efficiency. V and v representing volumes expended and raised, in cube feet per minute, li and h' heights from which water is drawn and elevated in feet, D and d diameters of supply and discharging pipes in ins., and IP effective horsepower. Illustration. — H eights of a fall and of elevation are 10 and 26.3 feet, and vol- umes expended and raised per minute are 1.71 and .543 cube feet. .001 13 X 1. 71 X 10 = .0193 IP *, 881 ^ o ° 19j = 1 - 71 cubefeto; 1.45^/1.71 = 1.89 ins . ; . 75 Ji . 71 = . 975 ins . ; and §■ X ~ 543 X — . 696 efficiency . ' J v ' * ' 6 1. 71 X 10 HYDRODYNAMICS. 62 Results of Operations of Hydraulic Rams. Strokes per M. | Fall. Eleva- tion. Wai Expen'd. ter Raised. Useful Effect. j Strokes j per M. Fall. Eleva- tion. Wai Expen’d. ter Raised. Useful Effect. No. 66 50 36 3i Feet. 10.06 9-93 6.05 5.06 Feet. 26.3 38.6 38.6 38.6 C. Ft. 1.71 1-93 i-43 1.29 C. Ft. •543 .421 .169 •113 •9 .85 •75 .67 No. 15 10 Feet. 3.22 *•97 22.8 8-5 Feet. 38.6 38.6 196.8 52.7 C. Ft. 1.98 1.58 .38 2 C. Ft. .058 .014 .029 .186 .67 •57 V online 01 air vessel — volume of delivery pipe. One seventh of watei may be raised to about 4 times head of fall, or one fourteenth 8 times or one twenty eighth 16 times. ’ * WATER POWER. Water acts as a moving power, either by its weight or by its vis viva , and m latter case it acts either by Pressure or by Impact. Natural Effect or Power of a fall of water is equal to weight of its volume and vertical height of its fall. If water is made to impinge upon a machine, the velocity with which it impinges may be estimated in the effect of the machine. Result or effect however, is in nowise altered ; for in first case P = Vto h, and in latter = ~ V w. Y representing volume in cube feet, tv weight in lbs., and v velocity of flow in f eet per second. 62.5 V h — P, and 3.2* a fh — V. P representing pressure in. lbs., a area of open- ing in sq. feet, and h height of flow in feet per second. To Compute Rower of a Rail of Water. Rule. — Multiply volume of flowing water in cube feet per minute bv 62.5, and this product by vertical height of fall in feet. Note. — When Flow is over a Weir or Notch , height is measured from surface of tail-race to a point four ninths of height of weir, or to centre of velocity or pressure of opening of flow. When Flow is through a Sluice or Horizontal Slit, height is measured from sur- face of tail-race to centre of pressure of opening. Example.— What is power of a stream of water when flowing over a weir 5 feet in breadth by 1 in depth, and having a fall of 20 feet from centre of pressure of flow? By Rule, page 533, — 5 X 1 V2 ^ 1 X -625 = 16.68 cube feet per second. 16.68 X 60 X 62. 5 X 20 = r 251 000 lbs. , which - 4 - 33000 = 37.91 horses 1 power. Or, .1135 V h = theoretical IP. h representing height from race in feel. Illustration.— If flow of a stream is 17.9 cube feet per second, to what height and area of flow of 1 foot in depth should it be dammed to attain a power of 10 horses. 33 000 X 10 ssoo pc = 5500 lbs. per second, and = 88 cube feet per second. : 02.5 179 60 4. 92 feet height. Hence, — . 6 V2 g X 1 = 3. 2, and 17. 9 -4- 3. 2 = 5. 59 sq. feet. ' Water sometimes acts by its weight and vis viva simultaneously, by com- bining effect of an acquired velocity with fall through which it flows upon wheel or instrument. In this case V x 62. 5 = mechanical effect. * As determined by — C. HYDRODYNAMICS. 563 WATER-WHEELS. Water-wheels are divided into two classes, Vertical and Horizontal. Vertical comprises Overshot , Breast , and Undershot ; and Horizontal, Turbine , Impact , or Reaction wheels. Vertical wheels are limited by construction to falls of less than 60 feet. Turbines are applicable to falls of any height from 1 foot upward. Vertical wheels applied to a fall of from 20 to 40 feet give a greater effect than a Turbine, and for very low falls Turbines give a greater effect. Sluices. — Methods of admitting water to an Overshot or Breast Wheel are various, consisting of Overfall , Guide-bucket, and Penstock. An Overfall Sluice is a saddle-beam with a curved surface, so as to direct the current of water tangentially to buckets; a Guide-bucket is an apron by which water is guided in a course tangential to buckets; and a Penstock is sluice-board or ^ate placed as close to wheel as practicable, and of such thickness at its lower edge as to avoid a contraction of current. Bottom surface of penstock is formed with a parabolic lip. SL.roud.iiag of a wheel consists of plates' at*its periphery, which form the sides of the bucket. Height of fall of a water-wheel is measured between surfaces of water in penstock and in tail-race , and, ordinarily, two thirds of height between level of reservoir and point at which water strikes a wheel is lost for all effective operation. Velocity of a wheel at centre of percussion of fluid should be from .5 to .6 that of flow of the water. Total effect in a fall of water is expressed by product of its weight and height of its fall. Hatio of Effective Power of Water Motors. Overshot and high} from t 6 to _ Undershot, Poncelet’s, from .6 to. 4 to 1 breast j lrom t0 t0 1 Undershot “ .27 to .45 to 1 Turbine u .6 to .8 to 1 Impact and Reac - 1 u 0 to. 5 to 1 Breast “ .45 to .65 to 1 tion j Hydraulic Ram .6 to 1 Water-pressure engine .8 to 1 Overshot-wheel, Overshot- wheel. — The flow of water acts in some degree by impact, but chiefly by its weight. Lower the speed of wheel at its circumference, the greater will be mechan- ical effect of the water, in some cases rising to 80 per cent, ; with velocities of from 3 to 6.5 feet, efficiency ranges from 70 to 75 per cent. Proper ve- locity is about 5 feet per second. Humber of buckets should be as great, and should retain water as long, as practicable. Maximum effect is attained when the buckets are so numerous and close that water surface in the bucket commencing to be emptied should come in contact with the under side of the bucket next above it. Moles- worth gives 12 ins. apart. Curved buckets give greatest effect, and Radial give but .78 of effect of Elbow buckets. Wheel 40 feet in diameter should have 152 buckets. Small wheels give a less effect than large, in consequence of their greater centrifugal action, and discharging water from the buckets at an earlier period than with larger wheels, or when their velocity is lower. When head of water bears to fall or height of wheel a proportion as great as 1 to 4 or 5, ratio of effect to power is reduced. The general law there- fore is, that ratio of effect to power decreases as proportion of head to total head and f all increases. 564 HYDRODYNAMICS. Wheel with shallow Shrouding acts more efficiently than one where it is deep, and depth is usually made 10 or 12 ins., but in some cases it has been increased to 15. Breadth of a wheel depends upon capacity necessary to ffive the buckets to receive required volume of water. Form of Buckets. Radial buckets — that is, when the bottom is a right line in- volve so great a loss of mechanical effect as to render their use incompatible with economy; and when a bucket is formed of two pieces, lowrer or inner piece is termed bottom or floor, and outer piece arm or wrist. Former is usuallv placed in a line with radius of wheel. J 1 Line of a circle passing through elbow , made by junction of floor and arm is termed division circle, or bucket pitch, and it is usual to put this at one half depth of shrouding. F When arm of a bucket is included in division angle of buckets, that is, n representing number of buckets, the cells are not sufficiently covered, except for verv shadow shrouding; hence it is best to extend arm of a bucket over 1.2 of division angle, so as to cover or overlap elbow of bucket next in advance of it. Construction of Buckets (Fig. 1).— Capacity of bucket should be 3 times volume of water. Fig. Fairbairn gives area of opening of a bucket in a wheel of great diameter, compared to the volume of it as 5 to 24. Buckets having a bottom of two planes, that is, with two bottoms, and two division circles or bucket pitches and an arm, give a greater effect than with one bottom. When an opening is made in base of buckets so as D l to afford an escape of air contained within, without a ! : loss of water admitted, the buckets are termed ven- tilated, and effective power of wheel is much greater than with closed buckets. D — distance apart at periphery = d % d depth of shrouding, s length of radial start =. 33 'd, l length of bucket curves 1.25 a in large wheels, and 1 in wheels under 25 feet, a angle of radius of curve of bucket with radial line of wheel at points of bucket ~ 1^0’ (Molesworth.) To Compute Radius and Revolutions of* an Overshot- wheel, and Height of Fall of Water. h — h' - = When whole Fall and Velocity of Flow , etc., are given . hc v 2 .. o.i 4i 6 - = w. - 1. 1 — h'. and 1 -j- cos. a ~ h representing height of ivhole 3.1416 r 7 2 g 3 o {non h wM^L h tf W n en ^ C TK e °fO ravit y of discharge and half depth of bucket cntZZfni- wa j er fi° ws f velocity of flow in feet per second, a angle which point of nil 7 a , bu I cJce I t makes wit h summit of wheel, n number of revolutions per minute, c velocity of wheel at its circumference per second, and r its radius. iinSwWM 1 °f whole fall is distance between surface of water in flume and h ° Wer buckets are emptied of water, and as a proportion of velocity ot flow is lost, it is proper to assume height h' as above given. * fal1 ° t f w , ater is 30 feet, velocity of its flow' is 16 feet per second, JJSna 7? ,ai P act «Pf>n buckets is 12 0 , and required velocity of w heel is 8 feet per wheel? lat 1S required rad,u s, number of revolutions, and height of fall upon X 1.1=4.38 feet; cos. i 2 ° = . 97 8; = ^ = 12.95 fed radius; 24O 40.68 2 g 30 x s ^ 3.1416 X 12.95 = 5.9, revolutions. HYDRODYNAMICS. 565 When Number of Revolutions and Ratio between Velocities of Flow and at Circumference of Wheel are given. ^.000772 (»w) 8 AHr(r-H*>s, 0)2 — i-t-cos.ig) _ r x __ v ^ 3.1416 nr .000386 (xn) 2 ’ c’ 30 Illustration. — If number of revolutions are 5, x = 2, and fall, etc. , as in previous case; what is radius of wheel, velocity of flow, and height of fall? y.000772 (2 X 5) 2 X 3° + (*-97 8 ) 2 — 1 -97 8 _ ^ .000386 (2 X 5) 2 .0386 3.1416 X 5 X 13-4*. __ ~' 0 ?f ee t . Hence 7.03 X 2 = 14.06 velocity of flow, and - 1 - 4 - 6 - 30 04.33 X i.i = 3-37 /«*■ To Compute "Width, of an Overshot-wheel. C V — - — w. C representing a coefficient = 3, when buckets are filled to an excess, and 5 when they are deficiently filled , V volume of water in cube feet per second , s depth of shrouding, w luidth of buckets, both in feet, and c' velocity of wheel at centre of shrouding , in feet per second. Illustration. — A wheel is to be 31 feet in diameter, with a depth of shrouding of 1 foot, and is required to make 5 revolutions per minute under a discharge of 10 cube feet of water per second ; what should be width of buckets ? Assume C = 4, and c' = — — * X 3*4 l6 X 5 _ y Then 4X ^ Q ^f ee t. 60 1 x 7*054 To Compute Number of Buckets. 7(1 + -!^ -4- 12 = d, and — ^ = n. D representing diameter of wheel, d dis- \ • 83/ d tance between centres of buckets, in feet, and n number of buckets. Illustration. — Take elements of preceding case. 31 — 1 X 3- 1416 X Then ' ^1 -j- = 7X2.24-12 = 1.283, an( l 360® buckets ; hence — — = 5 0 , angle of subdivision of buckets. 7 2 1.283 1 = 73*4, ^7 72 To Compute Effect of an Overshot- wheel. TV "~(S T " +/ ) V h w - = P. w representing weight of cube foot of water in lbs., v' velocity of it discharged at tail of wheel, in feet per second, V volume of flow in cube feet, and f friction of wheel in lbs. Illustration. — A volume of 12 cube feet per second has a fall of 10 feet., wheel using but 8.5 feet of it, and velocity of water discharged is 9 feet per second; what is effect of fall ? Friction of wheel is assumed to be 750 lbs. 12 X 3.5 X 62.5— — X 12 X 62.5 + 750^ M . V64.33 / _ 6373 — (1.26 X 750 + 750 _ 4680 _ 12X10X62.5 — 7500 7500 .624 = ratio of effect to power ; and 4680 X 60 seconds -H 33 000 = 8.51 IP. To Compute Power of an Overshot- wheel. Pule. — Multiply weight of water in lbs. discharged upon wheel in one minute by height or distance in feet from centre of opening in gate to sur- face of tail-race; divide product by 33000, and multiply quotient by as- sumed or determined ratio of effect to power. Or, for general purposes, divide product by 50 000, and quotient is IP. Or, .0852 Vh = IP, and - - 7 — — V per second ; or, = V P er minute. 3 B HYDRODYNAMICS. 566 Mechanical Effect of water is product of its weight into height from which it falls. Example.— Volume of water discharged upon an overshot- wheel is 640 cube feet per minute, and effective height of fall is 22 feet; what is H?? 640 X 62. 5 X 22 __ 2 g >6 which, X -75 = assumed ratio of effect to power = 20 IP. 33000 TJ seful Effect of an OversRot- wheel. With a large wheel running in most advantageous manner, .84 of power may be taken for effect. Velocity of a wheel bears a constant ratio, for maximum effects, to that of the flowing water, and this ratio is at a mean .55. Ratio of effect to power with radial-buckets is .78 that of elbow-buckets. Ratio of effect decreases as proportion of head to total head and fall increases. Thus, a wheel 10 feet in diameter gave, with heads of water above gate, ranging from .25 to 3.75 feet, a ratio of effect decreasing from .82 to .67 of power. Higher an overshot-wheel is, in proportion to whole descent of water, greater will be its effect. Effect is as product of volume of water and its perpendicular height. Weight of arch of loaded buckets in lbs. is ascertained by multiplying .444 of their number by number of cube feet in each, and that product by 40. IT ndershot-wheel. Undershot-wheel is usually set in a curb, with as little clearance for escape of water as practicable ; hence a curb concentric to this wheel is more effective than one set straight or tangential to it. Computations for an undershot-wheel and rules for construction are near- ly identical with those for a breast-wheel. Buckets are usually set radially, but they may be inclined upward, so as to be more effectively relieved of water upon their return side, and they are usually filled from .5 to .6 of their volume. Depth of shrouding should be from 15 to 18 ins., in order to prevent overflow of water within the wheel, which would retard it. Velocity of periphery should equal theoretical velocity due to head of water X .57. Note.— When constructed without shrouding, as in a current- wheel, etc., buckets become blades. Sluice-gate should be set at an inclination to plane of curb, or tangential to wheel, in order that its aperture may be as close to wheel as practicable ; and in order to prevent partial contraction of flow of water, lower edge ot sluice should be rounded. Effect of an undershot-wheel is less than that of a breast-wheel, as the fall available as weight is less than with latter. 4 To Compute Power of an TJndersliot-wlieel. Proceed as per rule for an overshot-wheel, using 93 750 for 50 000, and .4 for .75. Or, V h .000 66 = IP ; or, — = V. V representing volume of water in cube feet per minute , and h head of water in feet. HYDRODYNAMICS. 567 TPoncelet’s Wheel. Ponce let’s Wheel. — B uckets are curved, so that flow of water is in course of their concave side, pressing upon them without impact ; and effect is greater than when water impinges at nearly right angles to a plane sur- face or blade. This wheel is advantageous for application to falls under 6 feet, as its effect is greater than that of other undershot wheels with a curb, and for falls from 3 to 6 feet its effect is equal to that of a Turbine. For falls of 4 feet and less, efficiency is 65 per cent., for 4.25 to 5 feet, 60 per cent., and from 6 to 6.5 feet, 55 to 50 per cent. In its arrangement, aperture of sluice should be brought close to face of wheel. First part of course should be inclined from 4 0 to 6° ; remainder of course, which should cover or embrace at least three buckets, should be car- ried concentric to wheel, and at end of it a quick fall of 6 ins. made, to guard against effect of back-water. Sluice should not be opened over 1 foot in any case, and 6 ins. is a suitable height for falls of 5 and 6 feet. Distance between two buckets should not exceed 8 or 10 ins., and radius of wheel should not be less than 40 ins., or more than 8 feet. Plane of stream or head of water should meet periphery of wheel at an angle of from 24 0 to 30°. Space between wheel and its curb should not ex- ceed .4 of an inch. Depth of shrouding should be at least .25 depth of head of water, or such as to prevent water from flowing through it and over the buckets, and width of wheel should be equal to that of stream of impinging water. Effect of this wheel increases with depth of water flow, and, therefore, other elements being equal, as filling of buckets, to obtain maximum effect, water should flow to buckets without impact, and velocity of wheel should be only a little less than half that of velocity of water flowing upon wheel. To Compute Ti’oportions of a, IPoncelet 'Wlieel. Note. — As it is impracticable to arrive at the results by a direct formula, they must be obtained by gradual approximation. Example. — H eight of fall is 4.5 feet; volume of water 40 cube feet per second, radius of wheel = 2 h, or 9 feet ; depth of the stream = . 75 feel ; and C assumed at .9. V representing volume of water in cube feet per second , h height of fall , d depth of shrouding = — . — — ]- d' ; d' opening of and e width of sluice, r radius of curva- 4 2 g ture of buckets == - ^ --- , and a of wheel , all in feet ; n number of revolutions = J cos. 2’ J , J > ^ pa per minute ; c velocity of circumference of wheel and v velocity of water, both in feet per second ; C coefficient of resistance of flow of water ; x angle between plane of flowing water and that of circumference of wheel at point of contact, sin. of - — 2 Vcos. z ; z angle made by circumference of wheel with end of buckets — 2 tang, y ; and y angle of direction of water from circumference of wheel — Then v — .9 yj 2 g (h — — j = .9 x 16.29 = 14-66 f tet velocity of wheel , being 568 HYDRODYNAMICS. 14.66 — .66 . . less than half velocity of water ; c = = 7 J eet / 4 .25, angle corresponding to which = 14 0 30'; n = to which = 14 0 30' ; n = 3 °^~ - = 7- 43 revolutions ; — 7- 43 revolutions ; sin. of 70 0 34' .'. x — 141 0 8'. Effect is a maximum when c = .5 v cos. y. back from perpendicular line wnicn passes mrougu wheel, the breast should then incline 1 in 10, or 1 in 15 towards sluice. After passing axis of wheel in tail-race, curb should make a '* sudden dip of 6 ins. Number of buckets 1.6 D -f 1.6, D = diameter of wheel in feet. Shrouding . 33 to ,5 depth of head of water, and D 2 7 t, and not less than 7 or more than 16 feet. Breast-wheel is designed for falls of water varying from 5 to 15 feet, and for flows of from 5 to 80 cube feet per second. It is constructed with either ordinary buckets or with blades confined by a Curb. Enclosure within which water flows to a breast-wheel, as it leaves the sluice is termed a Curb or Mantle. When blades are enclosed in a curb , they are not required to hold water ; hence thev may be set radial , and they should be numerous, as the. loss of water escaping between the wheel and the curb is less the greater their num- ber ; and that they may not lift or carry up water with them from tail-race, it is proper to give them such a plane that it may leave the water as nearly vertical as may be practicable. Distance between two buckets or blades should be from 1.3 to 1.5 times head over gate for low velocity of wheel and more for a high velocity, or equal to depth of shrouding, or at from 10 to 15 ins. It is essential that there should be air-holes in floor of buckets, to prevent air from impeding flow of water into them, as the -water admitted is nearly as deep as the interval between them ; and velocity of wheel should be such that buckets should be filled to .5 or .625 of their volume. When wheels are constructed of iron, and are accurately set in masonry, a clearance of .5 of an inch is sufficient. Fig. 2. Construction of Buckets (Fig. 2). ( Molesworth .) ' / cle, drawn at a’ distance’ of s X 1.17 from periphery of wheel, is 'jj centre from which bucket is struck with radius, b a. Radius of ^ wheel should not be less than 7, or more than 16 feet. From point of bucket, a, draw a line, a 6, at an angle of 26° , with radial line, point 6, where this line cuts an imaginary cir- c ur b should lit wheel accurately for 18 or 20 ins., measured '* sudden dip of 6 ins. To Compute 3?ower of a, Poncelet Wheel. V h .001 13 = IP, and 880 = V. V = velocity of theoretical -periphery — . 55.* 33 reast-wlieel. in feet per second. HYDRODYNAMICS. 569 High Breast-wheel is used when level of water in tail-race and penstock or forebay are subject to variation of heights, as wheel revolves in direction in which water flows from blades, and back-ivater is therefore less disad- vantageous, added to which, penstocks can be so constructed as to admit of an adjustable point of opening for the water to flow upon the wheel. Effect of this wheel is equal to that of the overshot, and in some instances, from the advantageous manner in which water is admitted to it, it is greater when both wheels have same general proportions. Under circumstances of a variable supply of water, Breast-wheel is better designed for effective duty than Overshot , as it can be made of a greater diameter; whereby it affords an increased facility for reception of water into its buckets, also for its discharge at bottom ; and further, its buckets more easily overcome retardation of back-water, enabling it to be worked for a longer period in back-water consequent upon a flood. In a well-constructed wheel an efficiency of 93 per cent, was observed by M. Morin, and Sir Wm. Fairbairn gives, at a velocity of circumference of wheel of 5 feet, an efficiency of 75 per cent. Velocity usually adopted by him was from 4 to 6 feet per second, both for high and low falls; a minimum of 3.5 feet for a fall of 40 and a maximum of 7 feet for a fall of 5 to 6 feet. When water flows at from io° to 12 0 above horizontal centre of wheel, Fairbairn gives area of opening of buckets, compared with their volume, as 8 to 24. The capacity between two buckets or blades should be very nearly double that of volume of water expended. To Compute [Proportions and. Effect of a Breast-wheel. Illustration. — Flow of water is 15 cube feet per second; height of fall, measured from centre of pressure of opening to tail-race, is -8. 5' feet; velocity of circumference of wheel 5 feet per second; and depth of buckets or blades 1 foot, filled to .5 of their volume. Width of wheel -= — ,d representing depth , and v velocity of buckets ; — - 5 ■■ — 3; and as buckets are but .5 filled, 3-=- .5 ‘=± 6 feet. Assume water is to flow with double velocity of circumference of wheel; v = $x 2 = 10 feet; and fall required to gen- erate this velocity = — X 1.1 = hf— Xn = 1.71 feet. 2 g 64.33 Deducting this height from total fall, there remains for height of curb or shroud- ing, or fall during which weight of water alone acts, h — h' = 8.5 — ■ 1.71 =6.79 feet Making radius of wheel 12 feet, and radius of bucket circle n feet, whole mechan- ical effect of flow of water = 15 x 62. 5 x 8.5 = 7968.75 lbs., from which is to be de- ducted from 10 to 15 per cent, for loss of water by escape. Theoretical effect, as determined by M. Morin, velocity of circumference about .5 of that of water, and within velocities of 1.66 to 6 feet. ( (y cos. a — v) v \ (- h J V 62.5. a representing angle of direction of velocity with which water flows to wheel at centre of thread of flow and direction of velocity of wheel at this Line , and h" h — h' in feet. a is here assumed at 20 0 . See Weisbacli, London, 1848, vol. ii. page 197, and for the necessarily small value of a, its cosine may be taken at 1. Cos. 20° = . 94. Then ^ + 6.79^ X 15 X 62.5 = 7.474 X 15 X 62.5 = 7006.9 Z6s., which is to be reduced by a coefficient of .77 for a penstock sluice, and .8 for an overfall sluice. Theoretical effect, as determined by Weisbach, 7273 lbs., from which are to be deducted losses, which he computes as follows : Loss by escape of water between wheel and curb = 916 Loss by escape at sides of wheel and curb — Friction and resistance of water = 2. 5 per cent — 160 1256 lbs. 3B* 570 HYDRODYNAMICS. Friction of wheel as per formula, page 571, =Wr nC .0086; '16500 . 5X6o and n = 5 ^ w -- = 4 revolutions. 12 X 2 X 3 -i 4 l6 Then 16 500 X 2. 18 X 4 X .08 X .0086 = 98.99 lbs. .o 4 s^= C = .o8. .048^^ = 4-36 ins.; r = 4. 36 -r- 2 — 2. whence 7 oo6 -9 I2 5 6 + 9; 9 __ efficiency, upon assumption of losses as com- ’ 79 68 -75 puted by Weisbach. To Compute Power of a Breast-wheel. Rule. — Proceed as per rule for an overshot-wheel, using 55000 and .65 with a high breast, and 62 500 and .6 for a low breast. Or, High breast, .0612 V h = H>, and = V; and Low breast .0546 V h = EP, andit|^ = V. .. I5X62.5X8.5X60 Illustration.— Assume elements of preceding case. Then — 14.49, which x -7 = 10. 14 horses. Or, 7006.9-1256+102-6X60 = ja27 horses 33000 Openinqs of Buckets or Blades.-High Breast, .33 sq. foot, and Lou, Breast, .2 sq foot for each cube foot of their volume, or generally 6 to 8 in opening in a high breast and 9 to 12 in a low breast. Forms of Buckets.-Two Part. d=D, s = . 5 d, l 1.25 d in large wheels, and =d in wheels ‘less than 25 feet in diameter. Three Part Buckets.— d divided into 3 equal parts ; Z = . 25 d, d = D, s = • 3 ? d,l — d in large wheels, and .75 d in wheels less than 25 feet in diameter. Ventilating Buckets (. Fairbairn's ). Spaces are about 1 inch in width. Notes —A Committee of the Franklin Institute ascertained that, with a high breast- wheel 20 feet in diameter, water admitted under a head of 9 ins., and at 17 feet above bottom of wheel, elbow-buckets gave a ratio of effect to power of .731 at a maximum, and radial-blades .653. With water admitted at a height 01 33 feet 8 ins., elbow-buckets gave .658, and radial blades .628. At 10.96 feet above bottom of wheel, with a head of 4.29 feet, elbow-buckets gave .544, and blades .329. At 7 feet above bottom of wheel, and a head of 2 feet, a low breast gave for elbow-buckets .62, and for blades .531. At 3 feet 8 ins. above bottom of wheel, and a head of 1 foot, elbow-buckets gave •555; antl b^des .533. Cmrrent-wlieel. Current-wheel. — D. K. Clark assigns the most suitable ratio of veloc- ity of blades to that of current as 40 per cent. Depth of blades should be from .25 to .2 of radius; it should not be less than 12 or 14 ins. Diameter is usually from 13 to 16.5 feet, with 12 blades ; but it is thought that there might be an advantage in applying 18 or even 24. The blades should be completely submerged at lower side, but not more than 2 ins. under water, and not less than 2 at one time. a s (u_s)2 — ip. a representing area of vertical section of immersed blades in sq. feet , s velocity of wheel at circumference, and v of stream, both in feet per second. Or .38 — V 62. 5 = useful effect. Hence, efficiency = . 38. ’ a g HYDRODYNAMICS. 571 Fritter- wheel. Flutter or Saw-mill Wheel — Is a small, low breast-wheel operating under a high head of water ; the design of its construction, water being plenty, is the attainment of a simple application to high-speed connections, as a gang or circular saw. In effect it is from .6 to .7 that of an overshot-wheel of like head of fall. Y $ — ( v — s) — IP. v and s as preceding. 150 Friction, of Journals or Gudgeons. A very considerable portion of mechanical effect of a wheel is lost in ef- fect absorbed by friction of its gudgeons. To Compute Friction, of Journals or Gudgeons of a W ater- wheel. WrnC .008 6 =f W representing weight of wheel in lbs., r radius of gudgeon in ins., and n number of revolutions of wheel per minute. For well-turned surfaces and good bearings, C = .c>75 with oil or tallow; when best of oil is well supplied = .054; and, as in ordinary circumstances, when a black- lead unguent is alone applied = .n. Illustration.— A wheel weighing 25000 lbs. has gudgeons 6 ins. in diameter, and makes 6 revolutions per minute; what is loss of effect? Assume C = . 08. Then 25000 X - X 6 X .08 X .0086 = 309.6 lbs. 'Weights. — Iron wheels of 18 to 20 feet in diameter will weigh from 800 to 1000 lbs. per IP Wood wheels of 30 feet in diameter, 2000 to 2500 lbs. per IP. To Compute Diameter and Journals of* a Shaft, Stress laid, uniformly along its Length. l Cast Iron, — = d. Wood, 6. ] 9- 6 . ^ lbs. , l length of shaft between journals in feet , and d diameter of shaft in its body in ins. '/F- d. W representing weight or load in Journals or Gudgeons. — Cast Iron , .048 d. When Shaft has to resist both Lateral and Torsional Stress . — Ascertain the diameter for each stress, and cube root of sum of their cubes will give diameter. To Compute Dimensions of Arms. Cast Iron, = w - d representing diameter of shaft, and w width of arm, both • w in ms . , n number of arms , — = t, and t thickness of arm. When Arm is of Oak, w should be 1.4 times that of iron, and thickness .7 that of width. Memoranda. A volume of water of 17.5 cube feet per second, with a fall of 25 ffeet, applied to an undershot- wheel, will drive a hammer of 1500 lbs. in weight from 100 to 120 blows per minute, with a lift of from 1 to 1.5 feet.* A volume of water of 21.5 cube feet per second, with a fall of 12.5 feet, applied to a wheel having a great height of water above its summit, being 7.75 feet in diame- ter, will drive a hammer of 500 lbs. in weight 100 blows per minute, with a lift of 2 feet 10 ins. Estimate of power 31.5 horses. * Volume of water required for a hammer increases in a much greater ratio than velocity to be given to it, it being nearly as cube of velocity. HYDRODYNAMICS. 572 A Stream and Overshot Wheel of following dimensions— viz., height of head to centre of opening, 24.875 ins. ; opening, 1.75 t>y 8° ins. ; wheel, 22 feet in diameter by 8 feet face; 52 buckets, each 1 foot in depth, making 3.5 revolutions per minute —drove 3 run of 4.5 feet stones 130 revolutions per minute, with all attendant ma- chinery, and ground and dressed 25 bushels of wheat per hour. 4.5 bushels Southern and 5 bushels Northern wheat are required to make 1 bar- rel of flour. A Breast-wheel and Stream of following dimensions— viz., head, 20 feet; height of water upon wheel, 16 feet; opening, 18 feet by 2 ins. ; diameter of wheel, 26 feet 4 ins. ; face of wheel, 20 feet 9 ins. ; depth of buckets, 15.75 ins. ; number of buck- ets, 70; revolutions, 4.5 per minute — drove 6144 self-acting mule spindles; 160 looms, weaving printing-cloths 27 ins. wide of No. 33 yarn (33 hanks to a lb.), and producing 24000 hanks in a day of n hours. Horizontal "Wlieels. In horizontal water-wheels, water produces its effect either by Impact , Pressure , or Reaction , but never directly by its weight. These wheels are therefore classed as Impact, Pressure, and Reaction, but are now designated by the generic term of Turbine. Turbines. Turbines, being operated at a higher number of revolutions than Ver- tical Wheels, are more generally applicable to mechanical purposes ; but in operations requiring low velocities, Vertical Wheel is preferred. For variable resistances, as rolling-mills, etc., Vertical Wheel is far preferable, as its mass serves to regulate motion better than a small wheel. In economy of construction there is no essential difference between a Vertical Wheel and a Turbine. When, however, fall of water and volume of it are great, the Turbine is least expensive. Variations in supply of water affect vertical wheels less than Turbines. Durability of a Turbine is less than that of a Vertical Wheel; and it is indispensable to its operation that the water should be free from sand, silt, branches, leaves, etc. With Overshot and Breast Wheels, when only a small quantity of 'water is available, or when it is required or becomes necessary to produce only a por- tion of the power of the fall, their efficiency is relatively increased, from the blades being but proportionately filled; but with Turbines the effect is con- trary, as when the sluice is lowered or supply decreased water enters the wheel under circumstances involving greater loss of effect. To produce maximum effect of a stream of water upon a wheel, it must flow without im- pact upon it, and leave it without velocity; and distance between. point at which the water flows upon a wheel and level of water in reservoir should be as short as practicable. Small wheels give less effect than large, in consequence of their making a greater number of revolutions and having a smaller water arc. In Ilwh-pressure Turbines reservoir (of wheel) is enclosed at top, and water is admitted through a pipe at its side. In Low-pressure, water flows into res- ervoir, which is open. In Turbines working under water, height is measured from surface of water in supply to surface of discharged water or race ; and when they work in air, height is measured from surface in supply to centre of wheel. In order to obtain maximum effect from water, velocity of it, whep. lead- ing a Turbine, should be the least practicable. HYDRODYNAMICS. 573 Efficiency is greater when sluice or supply is wide open, and it is less, af- fected by head than by variations in supply of water. It varies but little with velocity, as it was ascertained by experiment that when 35 revolutions gave an effect of .64, 55 gave but .66. When Turbines operate under water, the flow is always full through them ; hence they become Reaction-wheels , which are the most efficient. Experiments of Morin gave efficiency of Turbines as high as .75 of power. Angle of plane of water entering a Turbine, with inner periphery of it, should be greater than 90°, and angle which plane of water leaving reservoir makes with inner circumference of Turbine should be less than 90°. When Turbines are constructed without a guide curve *, angle of plane of flowing water and inner circumference of wheel = 90°. Great curvature involves greater resistance to efflux of water ; and hence it is advisable to make angle of plane of entering water rather obtuse than acute, say ioo° ; angle of plane of water leaving, then, should be 50°, if in- ternal pressure is to balance the external ; and if wheel operates free of water, it may be reduced to 25° and 30°. If blades are given increased length, and formed to such a hollow curve that the water leaves wheel in nearly a horizontal direction, water then both impinges on blades and exerts a pressure upon them ; therefore effect is greater than with an impact-wheel alone. Turbines are of three descriptions : Outward, Downward, and Inward flow. Outward-flow Turbines. Fourneyron Turbine, as recently constructed, may be considered as one of the most perfect of horizontal wheels; it operates both in and out of back-water, is applicable to high or low falls, and is either a high or low pressure turbine. In high-pressure, the reservoir is closed at top and the water is led to it through a pipe. In low-pressure, the water flows directly into an open res- ervoir. Pressure upon the step is confined to weight of wheel alone. Fourneyron makes angle of plane of water entering =go°, and angle of plane of water leaving = 30°. Efficiency is reduced in proportion as sluice is lowered, for action of water on wheel is less favorably exerted. M. Morin tested a Fourneyron turbine 6.56 feet in diameter, and he found that efficiency varied from a minimum of 24, to 79 per cent., when supply of water was reduced to .25 of full supply. In practice, radial length of blades of wheel is .25 of radius, for falls not ex- ceeding 6.5 feet, .3 for falls of from 6.5 to 19 feet, and .66 for higher falls. To Compute Elements and Results. High Pressure, 6.6 y/h — v; -=rA; V ' 1 ' 77 V — Df; 12.6 ^ = and v y/h ' h ’ ,079 V h = IP. h representing head of water , v velocity of turbine at periphery per minute , and D internal diameter of turbine, all in feet, V volume of water in cube feet per second , A sum of area of orifices in sq. feet, and IP effective horse-power. 1.2 D = external diameter of turbine in feet, when it is more than 6 feet, and 1.4 when it is less than 6 feet. Number of guides = number of blades t when less than 24, and number -i- 3 when greater than 24. Area of section of supply-pipe = .4 V. For construction of blades and guides, see Molesworth, London, 1882, page 540. * Guide curves are plates upon centre body of a Turbine, which give direction to flowing water, or to blades of wheel wnich surround them, t In extreme cases of very high falls diameter given by this formula may be increased. $ Fourneyron’s rule for the number of blades is constant number 36, irrespective of size of turbine. 574 HYDRODY N AM ICS. Operation of* Higli-Freasvire Turbines. I 120 h I 3 ° I 4° 1- 5° 60 1 70 | 1 80 V 4.2 3- 1 2 -5 1 2. 1 1.8 1.6 V [36 1 4 2 1 47 [51 1 55 159 1 x 4 ° 1 0 00 0 VO • 9 ! 00 1 78 1 84 | 89 | 94 — hp.a.a oj waier in jtti, v uvluiuo m/u.c,c ■> v and v velocity of periphery of turbine in feet per second. Boydkn Turbine. — Mr. Boyden, of Massachusetts, designed an outward-flow turbine of 75 BP, which realized an efficiency of 88 per cent. Peculiar features, as compared with a Fourneyron turbine, are, 1st, and most important, the conduction of the water to turbine through a vertical tiun- cated cone, concentric with the shaft. The water, as it descends, acquires a gradually increasing velocity, together with a spiral movement in direction of motion of wheel. The spiral movement is, in fact, a continuation of the motion of the water as it enters cone.— 2d. Guide-plates at base are inclined, so as to meet tangentially the approaching water— 3d. A “ diffuser ” or annu- lar chamber surrounding wheel, into which water Irom wheel is discharged. This chamber expands outwardly, and, thus escaping velocity of water, is eased off and reduced to a fourth when outside of diffuser is reached. Effect of diffuser is to accelerate velocity of water through machine ; and gain of efficiency is 3 per cent. Diffuser must be entirely submerged. (D. K. Clark.) Poncelet Turbine.— T his wheel is alike to one of his under shot- wheels set horizontally, and it is the most simple of all horizontal wheels. To Compute IClerneiits of CTeiieral IProportion and. Ttesnlts. (Lt. F. A. Mahan , TJ. S. A.) 5 D 2 V/i=t; .iD = H; 4.49^ = **, d .0425 D 2 h jh — BP; 4-85 sj ifjl — 13 ; D 4 D 3 (I) -p 10) = N ; ~ ™ ; — = W; D- V' and C coefficient for V' in terms of V = — . ^ = .5 N to .75 N = n ; - =W; f 71 D and d representing exterior and in- terior diameters of wheel, H and h heights of orifices of discharge at outer circum- ference and of fall acting on wheel , w and w' shortest distances between two adjacent blades and two adjacent guides , all in feet, V, V', and v velocities due to fall of watei passing through narrowest section of wheel, and of interior circumference of wheel, all in feet per second , N and n numbers of blades and guides, and BP actual horse- power. For falls of from 5 feet to 40, and diameters not less than 2 feet, n w should be equal to diameter of wheel. H equal to.iD ,nw' = d, and 4 w = width of crown. For falls exceeding this, H should be smaller, in proportion to diameter ot wheel. Downward-flow Turbines. In turbines with downward flow, wheel is placed below an annular series of guide-blades, by which water is conducted to wheel, llie water strikes curved blades, and falls vertically, or nearly so, into tail-race; consequently, centrifugal action is avoided, and downward flow is more compact. Fontaine Turbine vields an efficiency of 70 per cent., when fully charged. When supply of water is shut off to .75, by sluice, efficiency is cn per cent. Best velocity at mean circumference of wheel is equal to 55 per cent, of that due to height of fall. It may vary .25 of this either way, without materially affecting efficiency. In operation the water in race is in immediate contact with wheel, and its efficiencv is greatest when sluice is fully opened. Its efficiency, also, is less affected by variations of head of flow than in volume of water supplied; hence they are adapted for Tide-mills . HYDRODYNAMICS. 575 Jonval Turbine. — This wheel is essentially alike in its principal propor- tions to Fontaine’s, and in principle of operation it is the same. Water in race must be at a certain depth below wheel. For convenience, it is placed at some height above level of tail-race, within an air-tight cylinder, or “ draft-tube,” so that a partial vacuum or reduction of pressure is induced under wheel, and effect of wheel is by so much in- creased. Resulting efficiency is same as if wheel was placed at level of tail- race ; and thus, while it may be placed at any level, advantage is taken of whole height of fall, and its efficiency decreases as volume of water is di- minished or as sluice is contracted. To Coinpixte Elements ancL Ftesnlts. Low Pressure. — For falls of 30 feet and less. V 1/ 1.77 V IP 6 y/h = v ; = — = A; — --- — = D*; 12.7 — r:Y; and .079 V h = IP. h representing head of water, v velocity of turbine at periphery per minute , and D internal diameter of turbine, all in feet, V volume of water in cube feet per second , A sum of area of orifices in sq. feet, and IP effective horse-power. 1.2 D = external diameter of turbine in feet, when it is more than 6 feet, and 1.4 when it is less than 6 feet. Number of guides = number of bladesf when less than 24, and number -4- 3 when greater than 24. Area of section of supply-pipe = .4 V. For construction of blades and guides, see Mples worth, London, 1882. page 540. Low-Pressure Turbines. ( Molesworth .) •0 5 BP 10 IP 15 BP 20 BP 30 BP 40 IP 50 IP V V R V R V R Y R V R V R V R 2.5 9.48 25 34 50 24 75 20 100 *7 5 13-38 12.5 81 25 57 38 47 5 o 41 75 33 100 28 126 26 7-5 16.38 8.5 136 17 97 25 79 33 68 5 i 56 68 48 85 43 10 18.96 6-3 180 12.6 128 19 , 105 25 90 38 75 5 o 64 63 58 15 23.22 4.2 3 i 9 8.4 226 12.6 185 W 160 25 ! 3 i 33 113 42 100 20 26.82 — — 6-3 329 9-3 273 12.6 232 18.9 194 25 164 31 148 25 39 — — — — 7-5 358 10 310 i 5 253 20 220 25 196 30 32.88 8.4 380 12.6 310 17 268 21 240 v representing velocity of centre of blades in feet and V volume of water, in cube feet, both per second , R revolutions per minute , and IP effective horse-power. Vertical Shaft. 3 /— ' — diameter of shaft in ins. Inward-flow Tiarbiiie. Inward-flow Turbine. — Inward-flow or vortex wheel is made with radiating blades, and is surrounded by an annular case, closed externally, and open internally to wheel, having its inner circumference fitted with four curved guide-passages. The water is admitted by one or more pipes to the case, and it issues centripetally through the guide-passages upon circum- ference of wheel. The water acting against the curved blades, wheel is driven at a velocity dependent on height of fall, and water having expended its force, passes out at centre. This wheel has realized an efficiency as high as 77.5 per cent. It was originally designed by Prof. James Thomson. Swain Turbine. — Combines an inward and a downward discharge. Re- ceiving edges of buckets of wheel are vertical opposite guide-blades, and lower portions of the edges are bent into form of a quadrant. Each bucket thus forms, with the surface of adjoining bucket, an outlet which combines an inward and a downward discharge. One, 72 ins. in diameter, was tested * In extreme cases of very High falls diameter given by this formula may be increased, t Fourneyron’s rule for the number of blades is constant number 36, irrespective of size of turbine. HYDRODYNAMICS. 576 bv Mr J. B. Francis, for several heights of gate or sluice, from 2 to 13.08 ins., and circumferential velocities of wheel ranging from 60 to 80 per cent, of respective velocities due to heads acting on wheel. For a velocity of 60 per cent., and for heights of gate varying within limits al- ready stated, efficiency ranged from 47.5 to 76.5 per cent., and for a velocity of 80 r>er cent it ranged from 37.5 to 83 per cent. Maximum efficiency attained was 84 per cent., with a 12-inch gate and a velocity-ratio of 76 per cent. ; but from 9- inch to iq-inch gate, or from .66 gate to full gate, maximum efficiency varied within very narrow limits— from 83 to 84 per cent.,— velocity-ratios being 72 per cent, for o-inch gate, and 76.5 per cent, for full gate. At half-gate, maximum efficiency was 78 per cent., when velocity-ratio was 68 per cent. At quarter-gate, maximum effi- ciency was 61 per cent., and velocity-ratio 66 per cent. Tremor t Turbine, as observed by Mr. Francis, in his experiments at Lowell, Mass., gave a ratio of effect to^power as .793 to 1. Victor Turbine is alleged to have given an effect of .88 per cent, under a head of 18.34 feet, with a discharge of 977 cube feet of water per minute, and with 343.5 revolutions. Tangential Wlieel. Wheels to which water is applied at a portion only of the circumference are termed tangential. They are suited for very high falls, -where diameter and high tangential velocity mav be combined with moderate revolutions. The Girard turbine belongs to this class. It is employed at Goeschenen station for St. Gothard tunnel , it operates under a head of 279 feet. _ The wheels are 7 feet 10.5 ins. in diam., having 80 blades, and their speed is 160 revolutions per minute, with a maximum charge of water of 67 gallons per second. An efficiency of 87 per cent, is claimed for them at the I aris water-works ; ordinarily it is from 75 to 80 per cent. (D. A. Clark.) Impact and. Reaction "Wlieel. Impact-wheel. — Impact Turbine is most simple but least efficient form of impact-wheel. It consists of a series of rectangular buckets or blades, set upon a wheel at an angle of 50° to 70° to horizon; the water flows to blades through a pyramidal trough set at an angle of 20° to 40 , so that the water impinges nearly at right angles to blades. Effect is .5 entire me- chanical effect, which is increased by enclosing blades in a border or frame. If buckets are given increased length, and formed to such a hollow curve that the water leaves wheel in nearly a horizontal direction, the water then impinges on buckets and exerts a pressure upon them ; effect therefore is greater than with the force of impact alone. By deductions of Weisbach it appears that effect of impact is only half available effect under most favorable circumstances. Reaction-wheel.— Reaction of water issuing from an orifice of less capacity than section of vessel of supply, is equal to weight of a column of water , basis of which is area of orifice or of stream, and height of which, is twice height due to velocity of water discharged. Hence the expression is 2. — - a u> = R. w representing weight of a cube foot of 2 g water in tbs., and a area of opening in sq.feet. Wiiitelaw’s is a modification of Barker’s; the arms taper from centre towards circumference and are curved in such a manner as to enable the water to pass from central openings to orifices in a line nearly right and radial, when instrument is operating at a proper velocity ; in order that very little centrifugal force may be imparted to the water by the revolution ot the arms, and consequently a minimum of frictional resistance is opposed to course of the water. , HYDRODYNAMICS. 577 A Turbine 9.55 feet in diameter, with orifices 4.944 ins. in diameter, oper- ated by a fall of 25 feet, gave an efficiency of 75 per cent., including friction of gearing of an inclined plane. When a reaction wheel is loaded, so that height due to velocity, corresponding to velocity of rotation v, is equal to fall, or — 7 t, or v = f 2 gh, there is a loss of 17 v 2 per cent, of available effect; and when — =12/1, there is a loss of but 10 per cent. ; v 2 and when — = 4 h, there is a loss of but 6 per cent. Consequently, for moderate falls, and when a velocity of rotation exceeding velocity due to height of fall may be adopted, this wheel works very effectively. Efficiency of wheel is but one half that of an undershot-wheel. When sluice is lowered, so that only a portion of wheel is opened, efficiency of a Reaction-wheel is less than that of a Pressure Turbine. Ratio of Effect to Power of several Turbines is as follows : Poncelet 65 to .75 to 1 I Jonval 6 to 7 to 1 Fourneyron 6 to .75 to 1 | Fontaine .6 to .7 to 1 Barker’s Mill. — E ffect of this mill is considerably greater than that which same quantity of water would produce if applied to an undershot- wheel, but less than that which it would produce if properly applied to an overshot-wheel. For a description of it, see Grier’s Mechanics’ Calculator , page 234; and for its formulas, see London Artisan , 1845, page 229. IMPULSE AND RESISTANCE OF FLUIDS. Impulse and Resistance of "Water. — Water or any other fluid, when flowing against a body, imparts a force to it by which its condition of motion is altered. Resistance which a fluid opposes to motion of a body does not essentially differ from Impulse. Impulse of one and same mass of fluid under otherwise similar circum- stances is proportional to relative velocities c- p r of fluid. For an equal transverse section of a stream, the impulse against a surface at rest increases as square of velocity of water. Impulse against Plane Surfaces . — The impulse of a stream of water de- pends principally upon angle under which, after impulse, it leaves the water ; it is nothing if the angle is o, and a maximum if it is deflected back in a line parallel to that of its flow, or 180°, 2 ~ V ic — p* 9 When Surface of Resistance is a Plane , and = 90°, then S- V w = P, and for a surface at rest , 2 a h w = P. a representing area of opening in sq.feet . . ^~ 2 v j. if an< ^ v representing velocities of water and of surf ace upon which it impinges in feet per second , w weight of fluid per cube foot in lbs. , A transverse section of stream, in sq. ins., and cip v relative moiions of waiter and surface. .Normal impulse of water against a plane surface is equivalent to weight of a column which has for its base transverse section of stream, and for altitude twice height due to its velocity, 2 li == 2 — . 2 9 Resistance of a fluid to a body in motion is same as impulse of a fluid moving with same velocity against a body at rest. Weisbach, New York, 1870, vol. i. page 1008. 3C HYDRODYNAMICS. 578 Maximum Effect of Impulse. — Effect of impulse depends principally on velocity v of impinged surface. It is, for example, o, both when v = c and v — o ; hence there is a velocity for which effect of impulse is a maximum = ( c — v) v\ that &, v= ^ , and maximum effect of impulse of water is ob- tained when surface impinged moves from it with half velocity of water. Illustration.— A stream of water having a transverse section of 40 sq. ins., dis- charges 5 cube feet per second against a plane surface, and flows off with a velocity of 12 feet per second; effect of its impulse, then, is Vw,==p ; c— 5 *J 44 ==i8; g = 32-16; 10 = 62.5; 18- 32.16 x 5 X 62.5 = 58.28 lbs. Hence mechanical effect upon surface = P v = 58.28 X 12 = 699.36 lbs. Maximum effect would be v = - = X 5 X~— =gfeet, and - 40 X^-X 5X62.5 20 — - X 5.036 X 312.5 = 786.87 lbs.; and hydraulic pressure _ = 87. 44 lbs. 2 9 When Surface is a Plane and at an Angle, then (1 — cos. a) — V«? = P. Illustration. — A stream of water, having a transverse section of 64 sq. ins., dis- charges 17.778 cube feet per second against a fixed cone, having an angle of con- vergence from flow of stream of 50 0 , hydraulic pressure in direction of stream ; 40; cos. 50°= .642 79. (1 — .642 79) 32 i £ X 17- 77 s X 62.5 = th e„ c = 64—I44 .357 21 X 1382.2 = 494.26 lbs. When Surf ace of Resistance is a Plane at go°, and has Borders added to its Perimeter , effect will be greater, depending upon height of border and ratio of transverse section between stream and part confined. Oblique Impulse — In oblique impulse against a plane, the stream may flow in one, two, or in all directions over plane. When Stream is confined at Three Sides , (1 cos. a) ~ V w = V. When Stream is confined at Two Sides , C —~ sin. a 2 V w — P. Normal impulse of a stream increases as sine of angle of incidence ; par- allel impulse as square of sine of angle ; and lateral impulse as double the angle. When an Inclined Surface is not Bordered, then stream can spread over it in all directions, and impulse is greater, because of all the angles by which the water is deflected, a is least ; hence each particle that does not move in normal plane exerts a greater pressure than particle in that plane, , 2 sin. a 2 c — v and 1 -|- sin Ta 5 X ~g~ v«? = p. Impulse and Resistance against Surfaces. Coefficient of resistance, C, or number with which height due to velocity is to be multiplied, to obtain height of a column of water measuring this hydraulic press- ure, varies for bodies of different figures, and only for surfaces which are at right angles to direction of motion is it nearly a definite quantity. According to experiments of Du Buat and Thibault, 0 = 1.85 for impulse of air or water against a plane surface at rest, and for resistance of air or water against a surface in motion, C = 1.4. In each case about .66 of effect is expended upon front surface, and .34 upon rear. hydrodynamics. 579 Comparison between Xu-iToines and. otlier* "Water- wheels. Turbines are applicable to falls of water at any height, from i to 500 feet. Their efficiency for very high falls is less than for smaller, in consequence of the hydraulic resistances involved, and which increase as the square of the velocity of the water. They can only be operated in clear water. With Fourneyron’s, the stress and pressure on the step is that of the wheel in motion ; with Fontaine’s, the whole weight of the water is added to that of the wheel; they are well adapted, however, for tide-mills. Experiments on Jouval’s gave equal results with Fontaine’s. Vertical Water-wheels are limited in their application to falls under 60 feet in height. For falls of from 40 to 20 feet they give a greater effect than any turbine ; for falls of from 20 to 10 feet, they are equal to them ; and for very low falls, they have much less efficiency. Variations in the supply of water effect them less than turbines. Water-pressure Engine. By experiments of M. Jordan, he ascertained that a mean useful effect of .84 was attainable. Weisbach, London, 1848, vol. ii. page 349. PERCUSSION OF FLUIDS. When a stream strikes a plane perpendicular to its action, force with -which it strikes is estimated by product of area of plane, density of fluid, and square of its velocity. Or, A d v 2 = P. A representing area in sq. feet, d weight of fluid in lbs., and v velocity in feet per seeond. If plane is itself in motion, then force becomes Ad (v — v ') 2 = P. v' representing velocity of plane. If C represent a coefficient to be determined by experiment, and h height due to velocity v , then v 2 = 2 g k, and expression for force becomes A C 2 g h = P. CENTRIFUGAL PUMPS. (D. K. Clark.) Appold Pump, made with curved receding blades, is the form of centrifugal pump most widely known and accepted. M. Morin tested three kinds of centrifugal or revolving pumps : 1st, on model of Appold pump ; 2d, one having straight receding blades inclined at an angle of 45 0 with the radius, and 3d, one having radial blades. They were 12 ins. in diameter and 3.125 ins. in length, and had central open- ings of 6 ins. Their efficiencies were as follows : 1. Curved blades. . 48 to 68 per cent. | 2. Inclined blades. . 40 to 43 per cent. 3. Radial blades 24 per cent. Height to which water ascends in a pipe, by action of a centrifugal pump, would, if there were no other resistances, be that due to velocity of circum- v 2 ference of revolving wheel, or to — . Results of experiments made by the author on two pumps, in 1862, yielded following data, showing height to which water was raised, without any discharge : Gwyknk’s Pump (blades partly radial, curved at ends). Diameter of pump- wheel 4 feet. Revolutions per minute 177 Velocity of circumference per second. . . 37.05 feet. Head due to the velocity 21.45 “ Actual head 18.21 “ Do. do. in parts of head due to velocity, 85 per cent. Appold Pump (blades, curved). 7 ins. 580 HYDRODYNAMICS. IMPACT OR COLLISION. Mr. David Thomson made similar experiments with Appold pumps of from 1.25 to 1. 71 feet in diameter, the results of which showed that the actual head was about 90 per cent, of the head due to the velocity. M. Tresca, in 1861, tested two centrifugal pumps, 18 ins. in diameter, with a cen- tral opening of 9 ins. at each side. The blades were six in number, of which three sprung from centre, where they were .5 inch thick; the alternate three only sprung at a distance equal to radius of opening from centre. They were radial, except at ends, where they were curved backward, to a radius of about 2.25 ins. ; and they joined the circumference nearly at a tangent. Width of blades was taper, and they were 5.75 ins. wide at nave, and only 2.625 ins - at ends: so designed that section of outflowing water should be nearly constant. M. Tresca deduced from his experiments that, in making from 630 to 700 revolu- tions per minute, efficiency of the pump, or actual duty in raising water, through a height of 31.16 feet, amounted to from 34 to 54 per cent, of work applied to shaft; or that, in the conditions of the experiment, the pump could raise upward of 16200 cube feet of water per hour, through a height of 33 feet, with about 30 IP applied to shaft, and an efficiency of 45 per cent. According to Mr. Thomson, maximum duty of a centrifugal pump worked by a steam-engine varies from 55 per cent, for smaller pumps to 70 per cent, for larger pumps. They may be most effectively used for low or for moderately high lifts, of from 15 to 20 feet; and, in such conditions, they are as efficient as any pumps that can be made. For lifts of 4 or 5 feet they are even more efficient than others. At same time, larger the pump higher lift it may work against. Thus, an 18-inch pump works well at 20-feet lift, and a 3-feet pump at 30-feet lift. A 21-inch wheel at 40-feet lift has not given good results: high lifts demand very high velocities. Efficiency is influenced by form of casing of pump. Hon. R. C. Parsons made exper- iments with two 14-inch wheels on Appold’s and on Rankine’s forms. In Rankine’s wheel blades are curved backwards, like those of Appold’s, for half their length; and curved forwards, reversely, for outer half of their length. Deducing results of performance arrived at, following are the several amounts of work done per lb. of water evaporated from boiler : Work done per lb. of water evaporated. Foot-lbs. Ratio. Appold wheel, in concentric circular casing 11385 1.06 u “ in spiral casing 15996 1.5 Rankine wheel, in concentric circular casing 10748 1 “ “ in spiral casing 12954 1.2 These data prove: — 1st, that spiral casing was better than concentric casing; 2d, that Appold’s wheel was more efficient than Rankine’s wheel. IMPACT OR COLLISION. Impact is Direct or Oblique. Bodies are Elastic or Inelastic. The division of them into hard and elastic is wholly at variance with these properties ; as, for instance, glass and steel, which are among hardest of bodies, are most elastic of all. Product of mass and velocity of a body is the Momentum of the body. Principle upon which motions of bodies from percussion or collision are determined belongs both to elastic and inelastic bodies ; thus there exists in bodies the same momentum or quantity of motion, estimated in any one and same direction, both before collision and after it. Action and reaction are always equal and contrary. If a body impinge obliquely upon a plane, force of blow is as the sine of angle of incidence. ( When a body impinges upon a plane surface, it rebounds at an angle equal to that at which it impinged the plane, that is, angle of reflection is equal to that of incidence. Effect of a blow of an elastic body upon a plane is double that of an in- elastic one, velocity and mass being equal in each ; for the force of blow IMPACT OK COLLISION. 58i from inelastic body is as its mass and velocity, which is only destroyed by resistance of the plane ; but in an elastic body that force is not only destroyed, being sustained by plane, but another, also equal to it, is sustained by plane, in consequence of the restoring force, and by which the body is repelled with an equal velocity ; hence intensity of the blow is doubled. If two perfectly elastic bodies impinge on one another, their relative ve- locities will be same, both before and after impact; that is, they will recede from each other with same velocity with which they approached and met. If two bodies are imperfectly elastic, sum of their moments will be same, both before and after collision, but velocities after will be less than in case of perfect elasticity, in ratio of imperfection. Effect of collision of two bodies, as B and b , velocities of which are differ- ent, as v and v\ is given in following formulas, in which B is assumed to have greatest momentum before impact. If bodies move in same direction before and after impact, sum of their moments before impact will be equal to their sum after. If bodies move in same direction before, and in opposite direction after impact, sum of their moments before impact will be equal to difference of their sums after. If bodies move in opposite directions before, and in same direction after impact, difference of their moments before impact will be equal to their sum after . If bodies move in opposite directions before, and in opposite directions after impact, difference of their moments before impact will be equal to their difference after . To Compute Velocities of Inelastic Bodies after Impact. When Impelled in Same Direction. ^ v __ n b and b representing weights of the two bodies , V and v their velocities before impact , and r velocity of bodies after impact , all in feet. Consequently V — v ’B + 6 X b =. velocity lost by B, and V — v B + 6 X B = velocity gained by b. Note. — In these formulas it is assumed that V>v. If V<+ the result will be negative, but may be read as positive if lost and gained are reversed in places. Illustration.— An inelastic body, 6, weighing 30 lbs., having a velocity of 3 feet, is struck by another body, B, of 50 lbs., having a velocity of 7 feet; the velocity of b after impact will be , 5 ° X 7 + 3 ° X 3 __ 44 ° _ , , 50 + 30 80 S' 5 / • When Impelled in Opposite Directions. — ^ ~ ? V = r. B + 6 Illustration. — Assume elements of preceding case. 50 X 7 — 30 X 3 _ 260 50 + 3° _ 80 B V When One Body is at Rest. - = r. B + 6 Illustration.— Assume elements as preceding. 35o 80 ;= 3 . 2 s feet. 50 X 7 35 o - , = a7 = 4-75 M- 50 + 30 " When Bodies are inelastic, their velocities after impact will be alike AC* 5 82 IMPACT OR COLLISION. To Compute Velocities of Elastic Bodies* after Impact. B — bV - \-2bv 2BV — B — bv When Impelled in One Direction. and — r - B + 6 Illustration.— Assume elements as preceding. B -j- 6 50 — 30 X 7 + 2 X 3 °X 3 _ 32 °. ' 80 : 4 /eei, and 2 X 50 X 7 — 5Q— 3 °X 3 _ 6 4° 50 + 30 = ^ = 8 ** 50+30 Or, V — ~~~ -r V — v = velocity of A, and v + 2 jr V — v = velocity of r. B o B -J- 0 IFAen Impelled in Opposite Directions. B — b V (\) 2b v 2 BV — B — b v B++ - R ? an B + fc -• Illustration. — Assume elements as preceding. 50 — 30X7 ^2X30 X 3 _ 140 x8o ^ an ^ ? X 5° X 7 + 5(3 30 X 3 ' — 80 ’ 50 + 30 5o + 3o_ < 3° X 7 _ 50 -J- 30 80 7°°,+jg _ 9- 5 f ee t. Or, 2 & - = velocity lost by B. Ag 2 X 30 X 7 + 3 80 B + b = 7-5 feet. V B — 6 „ _ 2 B V One Body is at Rest. B & — R > and — r - Illustration. — Assume elements as preceding. 7*5°- 3° _n2 = I /«*, and 2X ? oX 7 = Z22 = 8.75 /<*(. 5O+3O 80 ’ 50+30 80 To Compute Velocities of Imperfect Elastic Bodies after Impact. Effect, of Collision is increased over that of perfectly inelastic bodies, but not doubled, as in case of perfectly elastic bodies ; it must be multiplied by id or m + ™, when - represents degree of elasticity relative to both per- fect inelasticity and elasticity. in -1- n B m -j- n Moving in same Direction. V X (V — v) = R ; and v -J X B (V _ V ) — r . m and n representing ratio of perfect to imperfect elasticity. t and n = 2 and 1 . 7 x — ^ — X V~~~3 = 7 — i '5 X^X 4 = 7 — 2 * 2 5 = 4*75 feet, and 3 + Illustration.— Assume elements as preceding. 7~ 2+1 ' 50 + 3° ' -X 50 50 + 30 X 7 — 3 = 3 + 3-75 = 6 -75/ ee *- When Moving in Opposite Directions , m + n ~ md m (b--&) V m / _ m + n b ( V + j _ R apd A- X (T + «)-»=i«-. m * B-f& m B + *> BV When One Body is at Rest. ■ — — — R > an d Illustration. — Assume elements of preceding case. 1 * ( 5 ° - ; * 3 °) ‘■(■+3 — r. 50 + 30 3|2 X A5 i= 6.s6 2 5/«t l XS °~^ = 3.6625/^, 80 and B + b 50X7X (1 + j) 50 + 30 LIGHT. 583 LIGHT. Light is similar to Heat in many of its qualities, being emitted in form of rays, and subject to same laws of reflection. It is of two kinds, Natural and Artificial ; one proceeding from Sun and Stars, the other from heated bodies. Solids shine in dark only at a temperature from 6oo° to 700°, and in daylight at iooo°. Intensity of Light is inversely as square of distance from luminous body. Velocity of Light of Sun is 185 000 miles per second. Standard of Intensity or of comparison of light between different methods of Illumination is a Sperm Candle “short 6,” burning 120 grains per hour. Candles. A Spermaceti candle .85 of a inch in diameter consumes an inch in length in 1 hour. Decomposition of* JLaglit. Colors. Maximum Ray. < Primary. Contrasts. Second’y. Tertiary. C Primary. 'ombinations. Secondary. Tertiary. Violet. . . . Chemical. __ _ Blue. . . 1 Indigo — — . Brown. Yellow. } ureen. . ; 1 Dark. Blue Electrical. Blue. — — Blue. . . ) Purple. ' ( Green. Green — — Green. Green. Red j Orange. ] 1 Yellow . . . Light. Yellow. — — Green. . j [ Gray. Orange . . . — — Orange. Broken. Yellow. 1 Purple. ] Red Heat. Red. Purple. Green. Red. ... j Orange. J f Brown. All colors of spectrum, when combined, are white. Consumption and Comparative Intensity- of* Liglit of Candles. Candle. No. in a Lb. Diameter. Length. Consumption per Hour. Light comp’d with Carcel. Wax Inch. Ins. Grains. 3 Q.h 15 i 135 .09 Spermaceti 3 •075 j u 3 .1 a. *5 i3-5 a c .09 a 4 6 Tallow .04 5 12.5 ) \ 3 1 | 204 •°7 U 3 4 •9 .8. *5 13-75 Compared with 1000 Cube Feet of Gas. Candle. Gas = i. Con- sump- tion. Light. Con- sumption for equal Light. Candle. Gas = i. Con- sump- tion. Light. Con- sumption for equal Light. Paraffine. Sperm . . . .098 •095 Lbs. 3-5 3-9 Lbs. 35-5 41. 1 103 120 Adamantine. Tallow .108 .074 Lbs. 5-i 5 -i Lbs. 47.2 53-8 137 155 In combustion of oil in an ordinary lamp, a straight or horizontally cut wick gives great economy over one irregularly cut. 584 LIGHT. Relative Intensity, Consumption, Illumination, and Cost of various Modes of Illumination. Oil at 11 cents, Tallow, at 14 cents, Wax at 52 cents, and Stearine at 32 cents per lb. 100 cube feet coal gas at 14 cents, and 100 cube feet of oil gas at 52 cents. Illuminator. Illumi- nation. CftTcel Lamp = 100. Actual Cost t/our. Cost for equal Inten- sity. Illuminator. Illumi- nation. Carcel Lamp = 100. Actual Cost per Hour. Cost for equal Inten- sity. Carcel Lamp 100 Cents. .87 Per H’r. .87 Stearine Candle 5 to lb. 66.6 Cents. •59 Per H’r. 4- J 3 Lamp with in - 1 verted reserv’r. J 57 - 8 .89 •99 Tallow “ 6 “ Sperm “ 6 “ 54 67-5 •25 .89 2-34 S ’ 7 * Astral Lamp 48.7 •56 1.78 Coal Gas — 1.22 .96 Wax Candle 6 to lb. 61.6 .92 6.31 Oil Gas — 1.25 .98 1000 cube feet of 13-candle coal gas is equal to 7.5 gallons sperm oil, 52.9 lbs. mold, and 44.6 lbs. sperm candles. Candles, I^amps, Rlmids, and Gas. Comparison of several Varieties of Candles, Lamps, and Fluids , with Coal * Gas , de- duced from Reports of Com. of Franklin Institute , and of A. Frye , M. D., etc. Candle. Intensity of ' Light. t Light at Equal ( Costs. Cost com- pared with Gas for Equal Light. Candle. Intensity of Light.f Light at Equal Costs. Cost com- pared with Gas for Equal Light. Diaphane •7 .8 •58 •5 •54 .85 15. 1 16.2 7-5 . Tallow, short 6’s, 1 double wick . . j Wax, short 6’s Palm oil 1 .8 1 7 1 .61 •77 7 - 1 14.4 10.5 Spermaceti, short 6’s. Tallow, short 6’s, ) single wick . . . } * City of Philadelphia \ Compared with a fish-tail jet of Edinburgh gas, containing 12 per cent, of condensable matter and consuming 1 cube toot per hour. Lamp and Fluid. Inten- sity of Light. Light at Equal Cost. Time of Burning 1 Pint of Oil. Lamp and Fluid. Inten- sity of Light. Light at Equal Cost. Time of Burning 1 Pint of Oil. Carcel. Sperm oil, max’m ‘ ‘ mean. 11 min'm Lard oil 1.8 I -35 1.2 •97 Hours. 6.32 9.87 14.6 XI *3 Has 1 I Hours. 2.15 1.22 .69 •77 Semi-solar, Sperm oil Solar, Sperm oil Camphene i -*5 1.76 i -75 •93 i -55 3.08 &75 8.42 9 - 3 1 Loss of Light by Use of Glass Globes . Clear Glass, 12 per cent. | Half ground, 35 per cent. | Full ground, 40 per cent. Refraction. Relative Index of Refraction- Is. Ratio of sine of angle of incidence to sine of angle of refraction, when a ray of light passes from one medium iuto another. Absolute Index or Index of Refraction— Is, When a ray passes from a vacuum into any medium, the ratio is greater than unity. Relative index of refraction from any medium, as A, into another, as B, is always oqual to absolute index of .B, divided by absolute index of A. Absolute index of air is so small, that it may be neglected when compared with liauids or solids; strictly, however, relative index for a ray passing from air into a given substance must be multiplied by absolute index for air, in order to obtain like index of refraction for the substance. Mean Indices of Refraction. Humors of eye 1.34 Salt, rock 1.55 Water, fresh 1.34 “ sea 1.34 — Air at 32 0 . Alcohol Canada balsam . , . . •• 1 -.54 Crystalline lens... .. 1.34 Glass, fluid X i-s8 “J 1-64 “ ernwa ...1 ‘-53 J x- 50 . LIGHT. 585 Gras. Retort. — A retort produces about 600 cube feet of gas in 5 hours with a charge of about 1.5 cwt. of coal, or 2800 cube feet in 24 hours. In estimating number of retorts required, one fourth should be added for being under repairs, etc. Pressure with which gas is forced through pipes should seldom exceed 2.5 ins. of water at the Works, or leakage will exceed advantages to be obtained from increased pressure. The average mean pressure in street mains is equal to that of 1 inch of water. When pipes are laid at an inclination either above or below horizon, a cor- rection will have, to be made in estimating supply, by adding or deducting .01 inch from initial pressure for every foot of rise or fall in the length of pipe. It is customary to locate a governor at each change of level of 30 feet. Illuminating power of coal-gas varies from 1.6 to 4.4 times that of a tallow candle 6 to a lb. ; consumption being from 1.5 to 2.3 cube feet per hour, and specific gravity from .42 to .58. Higher the flame from a burner greater the intensity of the light, the most effective height being 5 ins. Standard of gas burning is a 15-hole Argand lamp, internal diameter .44 inch, chimney 7 ins. in height, and consumption 5 cube feet per hour, giving a light from ordinary •coal-gas of from 10 to 12 candles, with Cannel coal from 20 to 24 candles, and with rich coals of Virginia and Pennsylvania of from 14 to 16 candles. In Philadelphia, with a fish-tail burner, consuming 4.26 cube feet per hour, illuminating power was equal to 17.9 candles, and with an Argand burner, consuming 5.28 cube feet per hour, illuminating power was 20.4 candles. Gas, which at level of sea would have a Value of 100, would have but 60 in city of Mexico. Internal lights require 4 cube feet, and external lights about 5 per hour. When large or Argand burners are used, from 6 to 10 are required. An ordinary single-jet house burner consumes 5 to 6 cube feet per hour. # Street-lamps in city of New York consume 3 cube feet per hour. In some cities 4 and 5 cube feet are consumed. Fish-tail burners for ordinary coal gas consume from 4 to 5 cube feet of gas per hour. A cube foot of good gas, from a jet .033 inch in diameter and height of flame of 4 ins., will bum for 65 minutes. Resin Gas . — Jet .033, flame 5 ins., 1.25 cube feet per hour. Purifiers .— Wet purifiers require 1 bushel of lime mixed with 48 bushels of water for 10 000 cube feet of gas. Dry purifiers require 1 bushel of lime to 10000 cube feet of gas, and 1 superficial foot for every 400 cube feet of gas. Intensity- of Light with Equal Volumes of Gras from different Burners. Equal to Spermaceti Candle burning 120 Grains per Hour. Burners. E l 1 enditu 'eet pe 2 ire in ( r Hou 3 Cube r. 4 Burners. I Single -jet, 1 foot Fish-tail No. 3 Bat’s wing 2.6 3-5 3 4 4.1 4.2 4-3 4-5 Argand, 16 holes Argand, 24 holes Argand, 28 holes •32 •33 •34 3.8 5-3 5.8 586 LIGHT, Material. Cube Feet. Material. Cube Feet, Material. Cube Feet. Boghead Cannel Wigan Cannel. . Cannel Cape Breton, ] u Cow Bay,” etc , ■{ I- 13 334 15426 8960 15 000 9500 Cumberland English, mean Newcastle j Oil and Grease Pictou and Sidney. . Pine wood 9 8 (DO 11 000 9500 10000 23000 8000 11 800 Pittsburgh Resin Scotch | Virginia “ West’n. Walls-end | 9520 15600 10300 15000 8960 9500 12000 i Chaldron Newcastle coal, 3136 lbs., will furnish 8600 cube feet of gas at a specific gravity of .4, 1454 lbs. coke, 14. 1 gallons tar, and 15 gallons am- moniacal liquor. Australian coal is superior to Welsh in producing of gas. Wigan Cannel, 1 ton, has produced coke, 1326 lbs. ; gas, 338 lbs. ; tar, 250 lbs. ; loss, 326 lbs. Peat , 1 lb. will produce gas for a light of one hour. Fuel, required for a retort 18 lbs. per 100 lbs. of coal. In distilling 56 lbs. of coal, volume of gas produced in cube feet when distillation was effected in 3 hours was 41.3, in 7, 37.5, in 20, 33.5, and 111 25, 31-7- Flow of Gras ill Pipes. Flow of Gas is determined by same rules as govern that of flow of water. Pressure applied is indicated and estimated in inches of water, usually from .5 to 1 inch. Volumes of gases of like specific gravities discharged in equal times by a horizontal pipe, under same pressure and for different lengths, are inversely as square roots of lengths. Velocity of gases of different specific gravities, under like pressure, are in- versely as square roots of their gravities. By experiment, 30000 cube feet of gas, specific gravity of charged in an hour through a main 6 ms. in diameter and 22.5 feet in lengtli. Loss of volume of discharge by friction, in a pipe 6 ins. in diameter and 1 mile in length, is estimated at 95 per cent. Diameter and Lengtli of Gas-pipes to transmit given Volumes of GJ-as to Branch-pipes. [Vi. Uie.) Volume per Hour. Diameter. Length. Volume per Hour. Diameter. Length. Volume per Hour. Diameter. Length. Cube Feet. 50 250 500 700 Ins. •4 1 1.97 2.65 Feet. 100 200 600 1000 Cube Feet. 1000 1500 2000 2000 Ins. 3.16 3-87 5-32 6.33 Feet. 1000 1000 2000 4000 Cube Feet. 2000 6000 6000 8000 Ins. 7 7 - 75 9.21 8 - 95 Feet. 6000 1000 2000 1000 Regulation of Diameter and Extreme Length °f Tub- ing, and Number of Burners permitted. Diameter of Tubing. Length. Capacity of Meters. Burners. Diameter of Tubing. Length. Capacity of Meters. Burners. Ins. •25 •375 •5 .625 Feet. 6 20 30 40 Light. 3 5 10 20 No. 9 i 5 30 60 Ins. •75 1 1.25 i -5 Feet. 50 70 100 150 Light. 30 45 60 100 No. 90 135 180 300 LIGHT. 587 Temperature of Gases. Combustion of a cube foot of common eras will heat 650 lbs. of water i°. & Services for Lamps. Lamps. Length from Main. Diameter of Pipe. Lamps. Length from Main. Diameter of Pipe. Lamps. Length from Main. Diameter of Pipe. No. 2 6 I Feet. 40 40 50 Ins. •375 •5 .625 No. 10 15 1 20 Feet. 100 130 150 Ins. •75 1 1.25 No. 25 30 Feet. 180 200 Ins. i -5 I -75 Volumes of Has Discharged per Hour under a Pressure of Half an Inch, of Water. Specific Gravity .42. Diam. of Opening. Volume. Diam. of Opening. Volume. 1 Diam. of Opening. Volume. I Diam. of Opening. Volume. Ins. •25 •5 Cube Feet. 80 321 Ins. •75 1 Cube Feet. | 723 1287 . 1 | Ins. 1-125 1 1-25 Cube Feet. 1625 2010 | Ins. *•5 1 5 Cube Feet. 2885 46 150 To Compute -Volume ofGas Discharged through a Dipe. 1000 \J G l ~ and 0(53 \f~~ti =d ' d representing diameter of pipe , and h height of water in ins., denoting pressure upon gas , l length of vine in yards G specif c gravity of gas, and V volume in cube feet per hour. P V 1 G may be assumed for ordinary computation at .42, and h .5 to 1 inch, of ^ pTi^ard^' ~ AssumQ diameter of Pipe 1 inch, pressure 1.68 ins., and length w A x 1.68 /1.68 1°°° X “ 2 x x - IQOO X = 2000 cube feet, »nd nd-r v c /4 000000 X -42 X i , /168000000 .063 X -g-. =!/—. L68 =I.°S ins. Note.— F or tables deduced by above formulas see Molesworth, 1878, page 226. Dimensions of Mains, with Weight of One Length. Diameter in ins. Length in feet Thickness in ins. . . . Weight in lbs 4 6 8 9 i° 1 14 18 9 9 9 9 9 9 9 ss' 375 •375 •5 •5 •5 -625 •75 288 224 400 454 489 1 868 13^ n ' ,0 1484 GAS ENGINES. * n 8 the i Lenoir Z n n h \ e ’ the best proportions of air and gas are, for common gas, 8 volumes of air to i of gas, and for cannel gas, n of air to i of gas. The time of explosion is about the 27th part of a second. An engine, havinga cylinder 4.625 ins. in diameter and 8.75 ins. stroke of piston, making 185 revolutions per minute, develops a half horse-power. Distribution of Heat Generated in the Cylinder. (. M ; Tresca.) Dissipated by the water and prcS^^'l Losses ucts of combustion 6 Q Converted into work ’ * * ^ | Hence efficiency as determined by the brake — 4 per cent. Atmospheric Gas Engine. opcd S1 ^6 e TP and ?he ins ' in diameter > making 81 strokes per minute, devel- flaming 2 rabe feet ( jf *** ^ CyUnder “ CUbe aDd for in ‘ Per eent. ••• 27 ICO 5 88 LIMES, CEMENTS, MORTARS, AND CONCRETES. LIMES, CEMENTS, MORTARS, AND CONCRETES. Essentially from a Treatise by Brig -Gen' l Q. A. Gillmore, U.S.A.* Lime. Calcination of marble or any pure limestone produces lime (quick- lime) Pure limestones burn white, and give richest limes. Finest calcareous minerals are rhombohedral pnsrns of calcareous spar, the transparent double-reflecting Iceland spar, and white or statu- ^ Property of hardening under water, or when excluded from air, con- ferred P upon a paste of lime, is effected by presence of foreign sub- stanees — as silicum, alumina, iron, etc.— when their aggregate presence amounts to .1 of whole. Times are classed: i. Common or Fat limes, which do not set in water. 2 Poor or Meagre, mixed with sand, which does not alter its condition. Hydraulic lime, containing 8 to 12 per cent, of silica, alumina, iron, pto set slowly in water. 4. Hydraulic, containing 12 to 20 per cent, of similar 1 ingreiUents'sets in wate/ in 6 or 8 days. 5. Eminently Hydraulic enntainimr 20 to to per cent, of similar ingredients, sets in water in 2 to 4 days. 6. g Hydraulic Cement, containing 30 to 50 per cent. °* yS 1 ’’ ^ few minutes, and attains the hardness of stone m a few ““"terras Artnes Pozzuolanas including pozzuolana properly so called, Tra&s 01 ienas, Arenes, Ochreous^rths, Basaltic sands, and a variety of similar substances. Indications of Limestones. They dissolve wholly or partly in weak acids with brisk effervescence, and are nearly insoluble in water. Kick Limes are fully dissolved in water frequently renewed and they the action of air. 3 They are rendered Hydraulic by admixture of pozzuolana or trass. , , - . Rich, fat, or common Limes usually contain less than 10 per cent, of im- purities. Hydraulic Limestones are those which contain iron and clay, so as to en- able them to produce cements which become solid when under watei. _ Poor Limes have all the defects of rich limes, and increase tut slightly in 1 . 11. t i.„ D oorer limes are invariably basis of the most rapidly - setting and most durable cements and mortars, and tliev ™ a, “ the f 2j“ which have the property, when in combination with silica, etc., ot lndurat in*, under water, and are therefore applicable for admixture of hydraulic cements f c Alike to rich limes tliev will not harden if in a state of paste underwater 0 ?£V? t 0 orTeSded from contact with the atmosphere o aH n ac d gL. They should be employed for mortar only when it is ini practicable 0 to procure common or hydraulic lime or cement, m which case it is recommended to reduce them to powder by grinding. Hydraulic Limes are those which readily Harden under water T, he most valuable or eminently hydraulic set from the ^dtothe^h day . ro^lrft^ ^e a cTp°^e They absorb less water than pure limes, and only increase m bulk from 1.75 to 2.5 times their original volume. _ . See .’.o hie Treat!... on Lime., Hydraulic Cement., and Mortar., in Paper, on Practical Engineer, ig, Engineer Department, U. S. A. LIMES, CEMENTS, MORTARS, AND CONCRETES. 589 Inferior grades, or moderately hydraulic , require a period of from 15 to 20 days’ immersion, and continue to harden for a period of 6 months. Resistance of hydraulic limes increase if sand is mixed in proportion of 50 to 180 per cent, of the part in volume ; from thence it decreases. M. Vicat declares that lime is rendered hydraulic by admixture with it of from 33 w> 4° per cent, of clay and silica, and that a lime is obtained which does not slake, and which quickly sets under water. Artificial Hydraulic Limes do not attain, even under favorable circum- stances, the same degree of hardness and power of resistance to compression as natural limes of same class. Close-grained and densest limestones furnish best limes. Hydraulic limes lose or depreciate in value by exposure to the air. Pastes of fat limes shrink, in hardening, to such a degree that they can- not be used as mortar without a large proportion of sand. Arenes is a species of ochreous sand. It is found in France. On account or the large proportion of clay it contains, sometimes as great as .7, it can be made into a paste with water without any addition of lime; hence it is some- times used m that state for walls constructed en pise, as well as for mortar. Mixed with rich lime it gives excellent mortar, which attains great hardness under water, and possesses great hydraulic energy. i ?° fvolcani J or[ S h }- , Xt comprises Trass or Terras, the Arenes, some of the ochreous earths, and the sand of certain graywackes, granites, schists, and basalts; their principal elements are silica’' and alumina, the former preponderating. None contain more than 10 per cent, of lime. nf ^ e r fiDe,y l 3ulverized > without previous calcination, and combined with paste °££ at m proportions suitable to supply its deficiency in that element itpos sesses hydraulic energy to a valuable degree. It is used in combination with X bKion ^.r^temperattreVr^o Ca ' CiU, “ g day aDd driviD « oflrthe wat »r ' ■»»- Bnclc or Tile Dust combined with rich lime possesses hydraulic energy. ? r V rrOS y ls - a blue " black and is also of volcanic origin, ‘it requires to be pulverized and combined with rich lime to render it fit for use, and to develop any of its hydraulic properties. assaaasar’-- " *~ 6 * * J Ul r iZ £ Slll< r? j Un \ e . d } vith rich lime P roduce s hydraulic lime of ex- cellent quality. Hydraulic limes are injured by air-slaking in a ratio vary- ing directly with their hydraulicity, and they deteriorate by age. cWeirSK 11 “ d “ mp S ° U ° r CXp ° SU,e - 1,ydraulic limea ■»«» be ex- Hydraulic Lime ofTeil is a silicious hydraulic lime; it is slow in setting requn ing a period of from 18 to 24 hours. Cerrients. *?*$&%* ■ Ce T nts c ?. ntain a . Iarger proportion of silica, alumina, magnesia, tion U nrf ° f P r ? cedln S varieties of lime ; they do not slake after oflcina- set under water P fr‘° r t0 a ‘® Y Cry best ° f h - vdra,l!ic limes - as s °me of them set under water at a moderate temperature (65°) in from 3 to 4 minutes- an Z S p pu qU ! re as man y hours. They do not shrink in hardening, and make an excellent mortar without any admixture of sand. i D 590 LIMES, CEMENTS, MORTARS, AND CONCRETES. When exposed to air, they absorb moisture and carbonic acid gas, and are ^tt’abont .33 strength of Portland, and is not adapted for use with sand. Kosendale Cement is from Rosendale, New York. , Portland Cement is made in hours from its ft, ^Property of setting s,ow may cement, as the Boulogne, when required f uctions having to be executed un- immediate causes of dest ™c tl0 *b J ot her hand a quick-setting cement is always der water and between tides. On the ‘ , active supervision. A slow- difficult of use ; it requires special w°Ame „ ossesses the advantage of being - Conclusions derived from Mr. Grant's Experiments. , Portland cement improves by age, if kept from moisture, j ^nd of a «S‘ sand is about .„ strength of neat cement; 5 . Strong cement- is ueav,,; u ^ e w . Less water used in mixing cement in • n we tted before use. 7. Bricks, stones, etc used with ‘ “mem should be v ^ , f kept dry 8 ' SrfSSnd wnent in a few months are equal to Blue bncks, Bramley-Fall stone, or Yorkshire landings. ^ picke d stock bri cUs . usIdfa^nTofVater Sill wash away the cement. Artificial Cement is made by a combination of slaked lime with unburned clay in suitable proportions. , - „ . t0 a sligh t calcination. Mortar. . Lime or Cement P aste . is l 1 1 that^Su^e^ portion should be determine y , ~ ^ ov S pa C es in sand or coarse iZSiiiZTSft SZ£ Ss M to -***• ] of the mass. . . . fm . mpd into a paste after having ! JSfStXS - >“ - “■ i 'ZtZSZSrTsx, ttfssa%iss» a : not very great, if the sea-water, mortars, however, is consid > 0 f finely-ground cement and clean Pointing Mortar is composed of » t ? ie Volume of cement paste is slig^ltlv'in^xce^'of the^olume h voids or spaces in the sand. The volume LIMES, CEMENTS, MORTARS, AND CONCRETES. [Jgi of sand varies from 2.5 to 2.75 that of the cement paste, or by weight, 1 of cement powder to 3 to 3.33 of sand. The mixture should be made under shelter, and in small quantities. All mortars are much improved by being worked or manipulated; and as rich imes gain somewhat by, exposure to the air, it is advisable to work mortar in then render it fit for use by a second manipulation White lime will take a larger proportion of sand than brown lime. Use of salt-water in the composition of mortar injures adhesion of it. . Wh , en a small quantity of water is mixed with slaked lime, a stiff paste is made, which, upon becoming dry or hard, has but very little tenacity, but by being mixed with sand or like substance, it acquires the properties of a cement or mortar. t 1 Proportion of sand that can be incorporated with mortar depends partly upon the decree of fineness of the sand itself, and partly upon character of the lime. I'or rich limes, the resistance is increased if the sand is in pro- portions varying from 50 to 240 per cent, of the paste in volume; beyond tins proportion the resistance decreases. 1, clean sharp sand, 2.5. An excess of water in slaking the lime swells the mortar, which remains light and porous, or shrinks in drying- an excess of sand destroys the cohesive properties of the mass. ’ It is indispensable that the sand should be sharp and clean. Stone Mortar .- 8 parts cement, 3 parts lime, and 31 parts of sand; or 1 cask cement 325 lbs., .5 cask 0: lime, 120 lbs., and 14.7 cube feet of sand== 10.5 cube feet of mortar. P arts cement, 3 parts lime, and 27 parts of sand; or 1 cask cement, 325 lbs., .5 cask of lime, 120 lbs., and 12 cube feet of sand=r 10 cube feet of mortar. Brown Mortar , —Lime 1 part, sand 2 parts, and a small quantity of hair. together. and ^ cement and sand > lessen about -33 in volume when mixed Calcareous Mortar , being composed of one or more of the varieties of lime whb" T 75 - r artlficiaJ ’ mixed with sand, will vary in its properties method of .Lnipuladom * “ " Sed ’ ** natU, ' e and ( l ua,it y of sand > Turkish. Plaster, or Hydraulic Cement. 1 00 lbs. fresh lime reduced to powder, 10 quarts linseed-oil, and i to 2 ounces cotton. Manipulate the lime, gradually mixing the oil and cotton, in a wooden vessel, until mixture becomes of the consistency of bread-dough ,na r m’ an ? wh(! u required for use, mix with linseed-oil to the ■consistency of nas’te reslsfthe ifffectVniumWity'to^ motal, joined or coated with it,’ Stucco. Stucco or Exterior Plaster is term given to a certain mortar designed for exterior plastering; it is sometimes manipulated to resemble variegated marble, and consists of i volume of cement powder to 2 volumes of dry sand. Li India, to water for mixing the plaster is added 1 lb. of sugar or molas- ses to 8 Imperial gallons of water, for the first coat ; and for second or finish- ing, 1 lb. sugar to 2 gallons of water. Powdered slaked lime and Smith’s forge scales, mixed with blood in suit- able proportions, make a moderate hydraulic mortar, which adheres well to masonry previously coated with boiled oil. 592 LIMES, CEMENTS, MORTARS, AND CONCRETES. that of the cement paste. Khorassar, or Turkish. Mortar, sistency, and lay between the courses of brick or stones. Mortars. Mortars used for inside plastering are termed Coarse, Fine, Gauge or bard finish, and Stucco. mcttick. 1 w\,S for 7 o upon “when full time for hardening cannot be allowed substitute from J5 to so per be *** coarse Staff. — Common lime mortar, as made for brick masonry, with a small quantity of hair ; or by volumes, lime paste (30 lbs. lime) part, sand 2 to 2.25 parts, hair .16 part. /lime nutty). — Lump lime slaked to a paste with a mod- eraTe volume of water! and afterward diluted to consistency of cream, and then to harden by evaporation to required consistency for w orking. In this state it is used for a slipped coat, and when mixed with sand or plaster of Paris, it is used for finishing coat. t., 1 TTivii<=;'h is composed of from 3 to 4 volumes fine each, fine stuff and plaster. Scratch Coat— First of three coats when laid upon laths, and is fiom .25 ^7? of an inch in thickness. One-coat IForifc.-Plastering in one coat without finish, either on masonry or laths— that is, rendered or laid. . Two-coat Work.— Plastering in two coats is done either in a laid coa and set, or in a screed coat and set. Screed coat is also termed a Floated coat. Laid ! first coat in two^oat work is resorted to in common work instead of screeding, when finished l sur Tee is not required to be exact to a straight-edge. It is laid m a coat of about .5 inch in thickness. Laid coat, except for very common work, should be hand-floated. Firmness and tenacity of plastering is very much increased by band- flea mg. Screeds are strips of mortar 6 to 8 inches in width, and of weired thick- ness of first coat, applied to the angles of a room, ”S° whe! thesehave become of a straight-edge, the inter- spaces between the screeds are filled out flush with them. ativelv even surface. This finish answers when the surface is to be nnislied in distemper, or paper. LIMES, CEMENTS, MORTARS, AND CONCRETES. 593 Concrete or Beton Is a mixture of mortar (generally hydraulic) with coarse materials, as gravel, pebbles, stones, shells, broken bricks, etc. Two or more of these materials, or all of them, may be used together. As lime or cement paste is the cementing substance in mortar, so is mortar the cementing substance in concrete or beton. The original distinction between cement and beton was that latter possessed hydraulic energy, while former did not. Hydraulic. — 1.5 parts unslaked hydraulic lime, 1.5 parts sand, 1 part gravel, and 2 parts of a hard broken limestone. 1 This mass contracts one fifth in volume. Fat lime may be mixed with concrete without serious prejudice to its hydraulic energy. concrete, "Various Compositions of Concrete. Hydraulic. 308 lbs. cement = 3.65 to 3.7 cube feet of stiff paste. 12 cube feet of loose sand = 9.75 cube feet of dense. For Superstructure.— n.75 cube feet of mortar as above, and 16 cube feet of stone fragments. Sea Wall. Boston Harbor. Hydraulic . — 308 lbs. cement, 8 cube feet of sand, and 30 cube feet of gravel. Whole producing 32.3 cube feet. lbs. cement, 80 lbs. lime, and 14.6 cube feet dense sands. Whole producing 12.825 cube feet. is , mad « ? f cla y or earth rammed in layers of from 3 to 4 ins. in depth In Tms Znnfr wljfa rcoiroTmtS. 1601 eXte ™ al SUrfa0e ° fa wa “ Oracled Asphalt Composition. 1. Mineral pitch i part, bitumen ii, powdered stone, or wood ashes 7 parts. ^ P arts ’ cla y 3 parts, and sand 1 part, mixed with a little oil makes a very fine and durable cement, suitable for external use. ’ Flooring.— % lbs. of composition will cover 1 sup. foot, .75 inch thick Asphaltum 55 lbs. and gravel 28.7 lbs. will cover an area of 10.75 sq. feet. ~£ u ! verized burnt clay 93 parts, litharge, ground very fine 7 narfs mixed with a sufficient quantity of pure linseed oil. 5 7 P j pa?ts S by C we'ight and pulverized calcareous stone , 4 , litharge 2, and linseed oil 4 appUecfmlmt'be^sa^urared with od. an ^ SUrfaCe Up0 “ which il is “> bo an 4 d parts ’ oil o^esin 6.25 parts, Artificial Mastic.— Composition of i square yard .9 inch thick: Mineral tar ao 5 cube ins. | Gravel. .'275 cube ins . «“ ii Slakedlime •• ^ • 1249 cube ins. nfSnMiioi morescence.— White alkaline efflorescence upon the surface X Mortom rad tKtei"/ natural h y draulic hme or cement is the basis, its formation * h * f 1 m the P ro P ortl <>n of .025 of its weight will prevent ■ CiystaHiratnm of these salts within ‘he pores of bricks, into which thev have been absorbed from the mortar, causes disintegration. ’ y a\e Distemper is term for all coloring mixed with water and size. Grmtmg.^lQYtar composed of lime and fine sand, in a semi-fluid state poured into the upper beds and internal joints of masonry * aJ)* 594 limes, cements, mortars, and concretes. slaking. Slaked Lime is a hydrate of lime, and it absorbs a mean of 2.5 times its volume and 2.25 times its weight of water. „ Lime (quicklime) must be slaked before it can be used as a matrix for ^SS£& , a3SaprS^ gSSSSSifSS consequent upon its reduction. . p^npcsive Quantity, is termed rawaffitsas "aSliXjUnfc. i> il'< "'.■“Ijf "*■?? tt'b.fcS'SSaS without essential deterioration. P Urn, mt W-A Cask * Lin« = »o «*,»■" — »- 7' 8 “ 8. 1 5 cube feet of stiff paste. - » A Cask of Cement = 300* lbs., will make from 3.7 to 3.75 c A Cask of Portland Cement = 4 bushels or 5 cube feet = 420 lbs. A Cask of Roman Cement = 3 bushels or 3.75 cube feet — 3 4 s - .5 inch. . 2.25 yards .75 inch. 1.5 yards * inch. 1. 14 yards. A Husliel of cement will cover. From experiments of General Totten, it appeared that r 1 volume oflime slaked with. 33 its volume of water gave 2.27 u u i “ “ “ -66 t . u u “ ' 0.: cube foot of dry cement, mixed with .33 cube foot of water, will make .63 to 635 cube foot of stiff paste. one volume or mass, for use as required. ; — ; — " ~ 3 oo lba. net is standard ; it usually overruns 8 lbs. LIMES, CEMENTS, MORTARS, AND CONCRETES. 595 Mortar, Cement, See. ( Molesworth .) Mortar.— -1 of lime to 2 to 3 of sharp river sand. Or, 1 of lime to 2 sand and 1 blacksmith’s ashes, or coarsely ground coke. Coarse Mortar. — 1 of lime to 4 of coarse gravelly sand. Concrete.— 1 of lime to 4 of gravel and 2 of sand. Hydraulic Mortar— 1 of blue lias lime to 2.5 of burnt clay, ground to- gether. J & Or, 1 of blue lias lime to 6 of sharp sand, 1 of pozzuolana and 1 of calcined ironstone. Beton. — 1 of hydraulic mortar to 1.5 of angular stones. Cement. — 1 of sand to 1 of cement.— If great tenacity is required, the ce- ment should be used without sand. Portland Cement Is composed of clayey mud and chalk ground together, and afterwards cal- cined at a high temperature— after calcining it is ground to a fine powder. Strength, of Mortars, Cements, and. Concretes. Deduced from Experiments of Vicat, Paisley , Treussart, and Voisin. Tensile Weight or Power required to Tear asunder One Sq. Inch. Cement Mortar. (42 days old.) Proportion of Sand to 1 of Cement. Roman . . , Portland. , 0 I 2 3 4 5 6 7 8 9 10 284 284 *99 166 142 128 116 106 99 92 95 lbs. 142 142 1L 3 92 79 67 57 42 35 25 — “ Bricls, Stone, and. Grranite Masonry. (320 days old.) Experiments of General Gillmore , U. S. A. Cement on Bricks. Pure, average . Sand 1 ) Cement 1 j Sand 1 j Cement 2 j Sand 1 ) Cement 3 j Lbs. 30.8 * 5-7 12.3 6.8 Pure. , Cement on Granite. Lbs. Sand 1 1 Cement 1 j Sand 1 j Cement 2 j Sand 1 1 Cement 3 j 27-5 20.8 12.6 9.2 Delafeld and Baxter. Lbs. Pure cement 68 Cement 4) Sand 1 4 } 68 Cement 8 ) Siftings 1 j 80 Cement 1 ) Siftings 1 j 82 Cement 1 ) Siftings 2 J 74 Lawrence Cement Co. Pure cement 87 Sand 1 1 Cement 4 j Water 1 ) Cement 2 J Water .42 ) Cement 1 j Water .33 ) Cement 1 j Lbs. 7-9 20.5 37-25 29.15 James River. Lbs. Pure cement 87 Cement 4 ) Sand 1 / 62 Newark Lime and Cement Co. Pure cement 04 Cement 1 ) Sand 2 j 4 ° Newark and Rosendale. Pure cement 75 Cement 1 ) Sand 1 j „ Lbs. Newark and Rosendale. Cement 1 ) Sand 3 j 7 Pure, without) mortar, mean) 45 Mortar. Lime paste 1, sand 2.5, 6 cement paste 5 u 596 LIMES, CEMENTS, MORTARS, AND CONCRETES. Pure Cement. Lbs. Portland, in sea- water, 45 days 266 “ English, 6 months 424 Roman “Septaria,” 1 year 191 “ masonry, 5 months 77 Rosendale, 9 months 7 00 Lawrence Cement Co 1210 Lbs. Boulogne 100, water 50 112 Portland, natural, 1 year 675 “ artificial, Eng., 1 year... 462 “ English, 320 days 1152 “ “ 1 month 393 Newark and Rosendale 339 Transverse. Reduced to a uniform Measure of One Inch Square and One Foot in Length . Supported at Both Ends. Experiments of General Grillmore. Formed in molds under a pressure of 32 lbs. per sq. inch, applied until mortar had set. Exposed to moisture for 24 hours, and then immersed in sea-water. Prisms 2 by 2 by 8 ins. between supports. Reduced by Formula 2 JJL. _ “ = C. C coefficient of rupture, and a weight of 3 4 v cl 2 portion of prism l. Cement . Mortar. James River. Thick cream Thin paste Stiff paste Rosendale “ Hoffman. ” Thin paste Stiff paste “ Delafield and Baxter.” Thin paste Stiff paste English. Portland, pure Stiff paste Cumberland, Md., pure . High Falls, U1-) ster Co. , N. Y. ) Complete calcination. Age. Pure. Days. Lbs. 59 3-9 320 5-8 59 6.9 320 9 320 8.9 , 320 8.5 , 320 12 ■ 320 16 . 320 13 . 320 13.2 • 95 8.4 . 95 4.2 1 Portland, Eng., stiff paste Roman, “ “ “ Cumberland, Md Akron, N. Y James River, Ya Pulverized and re- ) mixed after set — j Fresh Kingston and Rosendale. High Falls, Ul- ) sterCo.,N.Y. ) Fresh water to a stiff) paste ) Sea-water to a stiff paste Lawrence Cement Co. Fresh Days. 320 20 100 320 320 320 Ocfi Lbs. 7.8 8.4 8.8 Crusliixig. Cements, Stones, etc. ( Crystal Palace , London.) Reduced to a uniform Measure of One Sq. Inch. Material. Portl’d cem’t, area 1, height 1. “ cement ) “ sand. . . j “ stone ■ Destructive Pressure. Material. Lbs. 1680 Portland cement 1 ) “ sand 4 ) 1244 “ cement 1 ) 1144 “ sand 7 ) Roman cement, pure Lbs. [ 3 2- 5 6 12.8 8.8 8.6 3 - 6 9 7.6 6.6 3-2 4.4 2.6 I10.2 I — Destructive Pressure. Lbs. 1244 692 342 General deductions. , Particles of unwound cement exceeding .0125 of an inch in diameter may be allowed in cement paste without sand, to extent of 50 per cent, of whole, without detriment toTts properties, while a corresponding proportion of sand tnjures the strength of mortar about 40 per cent. LIMES, CEMENTS, MORTARS, ETC. — MASONRY. 597 2. When these unground particles exist in cement paste to extent of 66 per cent of whole, adhesive strength is diminished about 28 per cent. For a corresponding proportion of sand the diminution is 68 per cent. 3 - ^ d5 ^ iori siftings exercises a less injurious effect upon the cohesive than upon the adhesive property of cement. The converse is true when sand, instead of sift- ings, is used. ’ 4. In all mixtures with siftings, even when the latter amounted to 66 per cent of whole, cohesive strength of mortars exceeded their adhesion to bricks. Same're- sults appear to exist when siftings are replaced by sand, until volume of the latter exceeds 20 per cent, of whole, after which adhesion exceeds cohesion. 5. At age of 320 days (and perhaps considerably within that period) cohesive strength of pure cement mortar exceeds that of Croton front bricks. The converse is true when the mortar contains 50 per cent, or more of sand. 6. When cement is to be used without sand, as may be the case when qroulina is resorted to, or when old walls are to be repaired by injections of thin paste, there is no advantage in having it ground to an impalpable powder. 7. For economy it is customary to add lime to cement mortars, and this may be done to a considerable extent when in positions where hydraulic activity and strength are not required in an eminent degree. y 8. Hamming of concrete under water is held to be injurious. 9. Mortars of common lime, when suitably made, set in a very few days and with such rapidity that there is no need of awaiting its hardening in the prosecution of tn f a r ® Olay. The fusibility of clay arises from the presence of impurities, ^uch as lime, iron, and manganese. These may be removed by steeping the clay in bfiSfa to maMer WaBhlDS “ WitU Wat ° r - ° rucibles from common Votes by General GiUmore, U. S. A. -Recent experiments have developed that most American cements will sustain, without any great loss of strength a dose of lime paste equal to that of the cement paste, while l dose equal to 5 fo ’ 7 t the vol ume of cement paste may be safely added to any Rosendale cement without nro ducing any essential deterioration of the quality of the mortar NeYtois the hydraulic activity of the mortars so far impaired by this limited addlto of lime paste as to render them unsuited for concrete under water or other submarine economy ^ ^ US6 ° f iS secured tlle d °uble advantages of slow setting and Notes by General Totten , TJ. S. A.— 240 lbs. limez 8. 15 cube feet of stiff paste. cube^foot when^packe^'as at ^manufaemry 1 " 6 ^ WheQ l0 ° Se> WlU meaSUre * 8 *° ' 8 cherbourg ’ D ° ver - Aide ™^ t cask, will make from 7.8 to MASONRY. Brickwork. eof°ofh iS “ T™? ment ° f , bricks or stones > laid as i d e of and above dno» ^ er, . s0 . la ) he vertical joint between any two bricks or stones does not coincide with that between any other two. This is termed “breaking joints.” Header is a brick or stone laid with an end to face of wall. Stretcher is a brick or stone laid parallel to face of wall. Header Course or Bond is a course or courses of headers alone. Stretcher Course or Bond is a course or courses of stretchers alone. Closers are pieces of bricks inserted in alternate courses, in order to obtain 1 bond by preventing two headers from being exactly over a stretcher Ene/ksh Bond is laying of headers and stretchers in alternates courses. MASONRY. 598 Flemish Bond is laying of headers and stretchers alternately in each course. Gauged Work.— Bricks cut and rubbed to exact shape required. String Course is a horizontal and projecting course around a building. Corbelling is projection of some courses of a wall beyond its face, in order to support wall-plates or floor-beams, etc. Wood Bricks , Pallets , Plugs, or Slips are pieces of wood laid in a wall in order the better to secure any woodwork that it may be necessary to fasten to it. Reveals are portions of sides of an opening in a wall in front of the recesses for a door or window frame. Brick Ashlar. — Walls with ashlar-facing backed with brick. Grouting is pouring liquid mortar over last course for the purpose of filling all vacuities. Larrying is filling in of interior of thick walls or piers, after exteiior faces are laid, with a bed of soft mortar and floating bricks or spawls in it. Rendering (Eng.) is application of first coat on masonry, Laying if one or two coats on laths, and u Pricking up ” if three-coat work on laths. Bricks should be well wetted before use. Sea sand should not be used in the composition of mortar, as it contains salt and its grains are round, being worn by attrition, and consequently having less tenacity than sharp- edged giains. A common burned brick will absorb 1 pint or about one sixth of its weight ot water to saturate it. The volume of water a brick will absorb is inversely a test of ltS A^good^brick should not absorb to exceed .067 of its weight of water. The courses of brick walls should be of same height in front and rear, whether front is laid with stretchers and thin joints or not. In ashlar-facing the stones should have a width or depth of bed at least equal to Hard bricks set in cement and 3 months set will sustain a pressure of 40 tons P€ The compression to which a stone should be subjected should not exceed .1 of its The extreme stress upon any part of the masonry of St. Peter’s at Rome is com - puted at 15.5 tons per sq. foot ; of St. Paul’s, London, 14 tons ; and of piers of bew York and Brooklyn Bridge, 5.5 tons. ao . fioo ~ _ The absorption of water in 24 hours by granites, sandstones, and limestones ot a durable description is 1, 8, and 12 per cent, of volume of the stone. Color of Bricks depends upon composition of the clay, the molding sand, tem- perature of burning, and volume of air admitted to kiln. Pure clay free of iron will burn white , and mixing of chalk with the clay will Pl Presence of iron produces a tint ranging from red and orange to light yellow , “^large proportion of oxide of iron, mixed with a pure clay, will produce a bripld red, and when there is from 8 to ,o per cent., and the brick is exposed to an intense heat, the oxide fuses and produces a dark blue or purple, and w'lth .a small „ of manganese and an increased proportion of the oxide the color is darkened, e\en t0 Small 6 volume of lime and iron produces a cream color , an increase of iron pro- duces red , and an increase of lime brown. c\ay U (m n ta\n ing alkal i es anTbumedat a high temperature produces a bluish green. For other notes on materials of masonry, their manipulation, etc., see “Limes, Cements, Mortars, and Concretes,” pp. 588-597- Bointiiag.— Before pointing, the joints should be reamed, and in close ma- sonry they must be open to .2 of an inch, then thoroughly saturated with water, and maintained in a condition that they will neither absorb Water ^ or impart any to it. Masonry should not be allowed to dry rapidly after pointi g, but it should be well driven in by the aid of a calking iron and I11 pointing of rubble masonry the same general directions are to be obser\ ed. MASONRY. £gg Sand is Argillaceous , Siliceous , or Calcareous , according to its composition Its use is to prevent excessive shrinking, and to save cost of lime or cement. Or- d manly it is not acted upon by lime, its presence in mortar being mechanical and To thl 1 «nr? Ull ° 1 f mGS cen ? euts jt weakens the mortar. Rich lime adheres better modar SUrfaCe SaUd ^ an t0 ltS ° wn P art ^ c ^ es j hence the sand strengthens the It is imperative that sand should be perfectly clean, freed from all impurities and of a sharp or angular structure. Within moderate limits size of grain does not affect the strength of mortar; preference, however, should be given to coarse Calcareous sand is preferable to siliceous given 10 coarse. required for^mortTr^ from in Sharpn8SS saft'ean Whe “ rUbbed UP ° U th8m ’ aDd ,he pres8nce of aud Cinder ’ When pr0perIy crushed and used > make good V 18 I ? ixiD ? of concrete, slake lime first, mix with cement and then with the chips, etc., deposit in layers of 6 ins., and hammer down. ’ Bricks. Of Wenl;t°"of lh di - m i ns! °? s by 'T ioUS manufacturers, and different degrees of intensity of their burning, render a table of exact dimensions of different manufactures and classes of bricks altogether impracticable. averages are P given : h ° WeVer ’ ° f the ran ^ es of their dimensions, following Description. | I ns , Description. Ins. Baltimore front Philadelphia “ i Wilmington “ Croton “ Colabaugh Eng. ordinary. . . [ “ Lond. stock Butch Clinker. . . j 8.25 x 4-125 x 2.375 8-5 X4 X 2.25 8.25 X 3.625 X 2.375 9 X 4.5 X2.5 8.75X4.25 X2.5 6.25X3 X1.5 Maine Milwaukee North River Ordinary Stourbridge 1 fire-brick j Amer. do., N. Y. . 7 - 5 X 3-375X2.375 8.5 X 4.125X2.375 8 X3.5 X 2.25 { 7-75 X 3.625X2.25 (8 X 4.125X2.5 9- 12 5 X 4.625 X 2.375 8 - 8 75 X 4.5 X 2.625 tha l L 7 variations in aimensxons of bricks, and thickness of the layer of mortar or cement m which they may be laid, it is also impracti- R hpi° glVe any rU 6 + u f 8 l neral a PP H . cati °n for volume of laid brickwork, t becomes necessary, therefore, when it is required to ascertain the volume of bricks m masonry, to proceed as follows : To Compute Volume of Bricks, and Number in a Cube Foot of NX asonry. f ^ Ce dimensions of particular bricks used, add one half thick- ness of the mortar or cement in which thev are laid, and compute the area • divide width of wall by number of bricks of which it is composed ; multiply this area by quotient thus obtained, and product will give volume of the mass of a brick and its mortar in ins. cube foot. 1728 by thiS v0lume ’ and <™ in Philadelphia front brick, 8.25 X 2.375 ins. face. 825 + ^5X24-2 =8.25 + .25 = 8.5. _ length of trick and joint ; 2 „ 375 + ' 25 X 2 2 = 2 -375 + .25 = 2.625 = width of brick and joint. rvidt™/waU)=liji%' 3125 im=area °f Mo; 12.75 -1- 3 ( number of bricks in Hence 22.3125 x 4. 25 = 94.83 cube ins. ; and 1728=94,83 = 18.22 bricks. 6oo MASONRY. One rod of brick masonry (Eng.) = 11.33 cube yards and weighs 15 tons, or 272 superficial feet by 13.5 thick, averaging 4300 bricks, requiring 3 cube yards mortar and 120 gallons water. Bricklayers’ hod will contain 16 bricks or .7 cube feet mortar. I^ire— "briclis. Fire-clay contains Silica, Alumina, Oxide of Iron, and a small proportion 'of Lime, Magnesia, Potash, and Soda. Its fire-resisting properties depend- ing upon the relative proportions of these constituents and character of its grain. A good clay should be of a uniform structure, a coarse open grain, greasy to the hand, and free from any alkaline earths. The Stourbridge clay is black and is composed as follows : Silica 63.3 | Alumina 23.3 I Lime 73 I Protoxide of iron. ... 1.8 Water and organic matter 10.3 Newcastle clay is very similar. Stone Masonry. Masonry is classed as Ashlar or Rubble. Ashlar is composed of blocks of stone dressed square and laid with close joints. Coursed Ashlar consists of blocks of same height throughout each course. Fig. x. •i MM Fig. 2. Fig. 1.— Coursed, with chamfered and rusticated quoins and plinth. Fig. 2 . — Regular Coursed. Fig. 3 - pig. 3 . — Irregular Coursed. Fig. 5- Fig. 6. ri pig, 5 ._ Ranged Random , level, and broken courses. Fig. 6 .— Random, level, and broken. MASONRY. 601 Rn'b'ble -A^slilar Is ashlar faced stone with rubble backing. R.izb'ble Masonry Is composed of small stones irregular in form, and rough. [V‘% tt: .S, /'in.'' 1 ;. k:y, lg£s i\,vv f, V, l \ Vj KY: '"1 k* Fig. 7. Block Coursed.— Large blocks in courses (regular or irregular), Beds and Joints roughly dressed. Fig. 8. Fig. 8. — Coursed and Ranged Random. Fig. 9. Ranged Random. — Squared rubble laid in level and broken courses. Fig. 10. Coursed Random. — Stones laid in courses at intervals of from 12 to 18 ins. in height. Fig. 11. Fig. 11. Uncoursed or Random. — Beds and Joints undressed, projections knocked off, and laid at random. In- terstices filled with spalls and mortar. Fig. 12. Dry Rubble.— Without mortar or cement. Dry Rri'b'ble Is a wall laid without cement or mortar. Fig. 13 - Fig. 13. Laced Coursed. — Horizontal bands of stone or bricks, interposed to give stability. Fig. 14. Rustic or Rag. — Stones of irregular form, and dressed to make close joints. Note.— R ustic or Rag work is frequently laid in mortar. 3E 602 MASONRY. Terra Cotta. Terra Cotta in blocks should not exceed 4 cube feet in volume. When properly burned, it is unaffected by the atmosphere or by fumes of any acid. ^Arclies and AValls. Springing . — Point s i Fig. 1 5, on each side, Fig. 1 5. from which arch springs. Crown . — Highest point of arch. Haunches .— Sides of arch, from springing half-way up to crown. Spandrel . — Space between extrados, a hor- izontal line drawn through crown and a ver- tical line through upper end of skewback. Skewbaclc is upper surface of an abut- Pier Abutment ment or pier from which an arch springs, and its face is on a line radiating from centre of arch. Abutment is outer body that supports arch and from which it springs. Pier is the intermediate support for two or more arches. Jambs are sides of abutments or piers. Voussoirs are the blocks forming an arch. Key-stone is centre voussoir at crown. Span is horizontal distance from springing to springing of arch. pise . — Height from springing line to under side of arch at key-stone. Length is that of springing line or span. . Ring-course of a wall or arch is parallel to face of it, and in direction of its span. . String and Collar courses are projecting ashlar dressed broad stones at right angles to face of a wall or arch, and in direction of its length. c Camber is a slight rise of an arch as .125 to .25 of an inch per foot of span. Quoin is the external angle or course of a wall. Plinth is a projecting base to a wall. e . Footing is projecting course at bottom of a wall, in order to distribute it3 weight over an increased area. Its width should be double that of base of wall, diminishing in regular offsets .5 width of their height. Blocking Course.— A course placed on top of a cornice. Parapet is a low wall, over edge of a roof or terrace. Extrados— Back or upper and outer surface of an arch. Intrados or Soffit is underside of lower surface of arch or an opening. Groined is when arches intersect one another. Invert . — An inverted arch, an arch with its intrados below T axis or spring- ing line. Ashlar masonry requires .125 of its volume of mortar. Rubble , 1.2 cube yards stone and .25 cube yard mortar for each cube yard. Rubble masonry in cement, 160 feet in height, will stand and bear 20 000 { lbs. per sq. inch. Stones should be laid with their strata horizontal. When “ through” or “ thorough bonds ” are not introduced, headers should , overlap one another from opposite sides, known as dogs' tooth bond. Aggregate surface of ends of bond stones should be from .125 to .25 of area of each face of wall. . , Weak stones, as sandstone and granular limestone, should not -have a length over 3 times their depth. Strong or hard stones may ha\ e a length from 4 to 5 times their depth. MASONRY. 603 Gallets are small and sharp pieces of stone stuck into mortar joints, in which case the work is termed galleted. Snapped work is when stones are split and roughly squared. Quarry or Rock-/ heed.— Quarried stones with their faces undressed. Pitch-faced. — Stones on which the arris or angles of their face, with their sides and ends, is defined by a chisel, in order to show a right-lined edge. Drafted or Drafted Margin is a narrow border chiselled around edges of faces of a block of rough stone. Diamond-faced is when planes are either sunk or raised from each edge and meet in the centre. Squared Stones. — Stones roughly squared and dressed. Rubble. — Unsquared stones, as taken from a quarry or elsewhere, in their natural form, or their extreme projections removed. Cut Stones. — Stones squared and with dressed sides and ends. Dressed. Stones. The following are the modes of dressing the faces of ashlar in engineering : Rough Pointed. — Rough dressing with a pick or heavy point. Fine Pointed. — Rough dressing, followed by dressing with a fine point. Crandalled. — Fine pointing in right lines with a hammer, the face of which is close serried with sharp edges. Cross Crandalled. — When the operation of crandalling is right angled. Hammered. — The surface of stone may be finished or smooth dressed by being Axed or Bushed ; the former is a finish by a heavy hammer alike to a crandall, the latter is a final finish by a heavy hammer with a face serried 'with sharp points at right angles. Tliicliiiess of Briclc "Walls for Warehouses. ( Molesworth .) Length, Height. Thickness. Length. Height. Thickness. Length. Height. Thickness. Feet. Feet. Ins. Feet. Feet. Ins. Feet. Feet. Ins. Unlimited. 25 13 Unlimit’d. 100 34 45 30 13 do. 30 17-5 60 40 17-5 30 40 13 do. 40 21.5 70 50 21.5 40 50 17-5 do. 50 26 50 60 21.5 35 60 17.5 do. 60 26 45 70 21.5 30 70 17-5 do. 70 26 60 80 26 45 80 21.5 do. 80 30 70 90 30 60 90 26 do. 90 34 70 100 30 55 100 26 For drawings and a description of stone-dressing tools, see a paper by J. R. Cross, "W. E. Merrill, and E. B. Van Winkle, “A. S. Civil Engineer Transactions*” Nov. 1877. Walls not exceeding 30 feet in height, upper story walls may be 8.5 ins. thick. From 16 feet below top of wall to base of it, it should not be less than the space defined by two right lines drawn from each side of wall at its base to 16 feet from top. Thickness not to be less in any case than one fourteenth of height of story. Datlis. Laths are 1.25 to 1.5 ins. by 4 feet in length, are usually set .25 of an inch apart, and a bundle contains 100. 6 04 MAS 0 NKY. Plastering. Volumes required for Various Thickness. Material. Sq’ •5 uare Yar •75 ds. 1 Material. Square Yards. •5 I -75 I 1 Cube Feet. Cement 1 Ins. 2.25 4-5 6-75 Ins. i-5 3 4-5 Ins. I - I 5 2.25 3-33 Cube Feet. Lime 1, sand 2 , ) hair 3.75 j ** Ins. 1 Ins. | Ins. 75 yards, sup’l ren- dered and set on j brick or 70 on lath. Cement 1, sand 1. . . Cement 1, sand 2 . . . Estimate of Materials and. Labor for lOO Sq. Yards of Latli and Plaster. Materials and Labor. Three Coats Hard Finish. Two Coats Slipped. Materials and Labor. Three Coats Hard Finish. Two Coats Slipped. Lime 4 casks. 3. 5 casks. White sand 2.5 bushels. Lump lime .66 “ Kails 13 lbs. 4 days. 13 lbs. 3. 5 days. Plaster of Paris. . •5 “ Masons Laths 2000. 4 bushels. 2000. Laborer Hair 3 11 2 “ 3 ousneis. Sand 7 loads. 6 loads. Cartage 1 “ •75 “ Rough Cast is washed gravel mixed with hot hydraulic lime and water and applied in a semi-fluid condition. .A^rclies and vtIixl exit s . To Compute Depth of Keystone of Circnlar or Elliptic A.rcli. \/R + S-r- 2 _|_ .25 —d. R representing radius , s span , and d depth , all in feet This is for a rise of about .25 of span; when it is reduced, as to .125, add =5 instead of .25. Illustration. — Arch ofWasliington aqueduct at ‘‘Cabin John” has a span of 220 feet, a rise of 57.25, and a radius of 134.25 ; what should be depth of its keystone . Vi 34 . 25 + 220-^2 + 25 _ 15^3 _|_ 2 5 4 l6 f ee t. Depth is 4. 16 feet. 4 4 Viaducts of several arches increase results as determined above by add- ing .125 to .15 to depth. For arches of 2d class materials and work, and for spans exceeding 10 feet, add .125 to depth of keystone, and for good rubble or brick- work add .25. Note — It is customary to make the keystones of elliptic arches of greater depth than that obtained by. above formula. Trautwine, however, who is high authority in this case, declares it is unnecessary. To Compute Radius of an Arcla, Circular or Ellipse. 2 _|_ r 2 -f- 2 r = R. r representing rise. Railway Arches. For Spans between 25 and 70 feet. Rise .2 of span. Depth of arch .055 of span. Thickness of abutments .2 to .25 of span, and of pier .14 to .16 ol span. Abutments. When height does not exceed 1. 5 times base. R -^5 + - 1 r + 2 — thickness at spring of arch in feet. ( Trautwine. ) Batter.— From .5 to 1.5 ins. per foot of height of wall. MASONRY. — MECHANICAL CENTRES. GRAVITY. 605 To Compute Depth, of Arch. (Hurst.) c \/R = D. c = Stone (block) .3. Brick = .4. Rubblerir.45. When there are a series of arches, put .3 = .35, .4 = .45, and .45 ".5. NT i 11 i in vim Thickness of Abutments for Bridge and similar Arches of 120 °. (Hurst.) When depth of crown does not exceed 3 feet. Computed from formula. / 6 R+ (j-g) — = T. H representing height of abutment to springing in feet. Radius Height of Abutment to Springing. Radius Heig ht of Abutment to Springing. of Arch. 5 7-5 10 20 30 of Arch. 5 7-5 10 20 30 Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. 4 3-7 4.2 4-3 4.6 4-7 12 5-6 6.4 6.9 7.6 7-9 4-5 3-9 4.4 4.6 4.9 5 15 6 7 7-5 8.4 8.8 5 4.2 4.6 4.8 5 - 1 5-2 20 6-5 7-7 8.4 9.6 10 6 4-5 4-7 5-2 5-6 5-7 25 6.9 8.2 9.1 10.5 11. 1 7 4-7 5-2 5-5 6 6.1 30 7.2 8.7 9-7 11.4 12 8 4.9 5-5 5-8 6.4 6.5 35 7-4 9.1 10.2 11. 8 12.9 9 5 -i 5-8 6. 1 6.7 6.9 40 7.6 9.4 10.6 12.8 13.6 10 5-3 6 6.4 7 - 1 7-3 45 7.8 9-7 11 i 3-4 i 4-3 11 5-5 6.2 6.6 7-3 7-6 50 7-9 10 11.4 14 15 Note. — Abutments in Table are assumed to be without counterforts or wing- walls. A sufficient margin of safety must be allowed beyond dimensions here given. Culverts for a road having double tracks are not necessarily twice the length for a single track. For other and full notes, tables, etc., see Trautwine’s Pocket Book, pp. 341-356. MECHANICAL CENTRES. There are four Mechanical centres of force in bodies, namely, Centre of Gravity, Centre of Gyration, Centre of Oscillation, and Centre of Percussion. Centre of Grravity. Centre of Gravity of a body, or any system of bodies rigidly con- 1 nected together, is point about which, if suspended, all parts will be in equilibrium. A body or system of bodies, suspended at a point out of centre of gravity, will rest with its centre of gravity vertical under point of suspension. ’ A body or system of bodies, suspended at a point out of centre of gravity, and successively suspended at two or more such points, the vertical lines through these points of suspension will intersect each other at centre of gravity of body or bodies. Centre of gravity of a body is not always within the body itself. If centres of gravity of two bodies, as B C, be connected by a line, dis- tances of B and C from their common centre of gravity, c, is as the weiahts of the bodies. Thus, B : C :: C c : c B. y To Ascertain Centre of Gravity of any Plane Figure Mechanically . Suspend the figure by any point near its edge, and mark on it direction of a plumb-line hung from that point; then suspend it from some other point, and again mark direction of plumb-line. Then centre of gravity of surface will be at point of intersection of the two marks of plumb-line 3 E* 6o6 MECHANICAL CENTRES. GRAVITY. Centre of gravity of parallel-sided objects may readily be found in this way. For instance, to ascertain centre of gravity of an arch of a bridge, draw elevation upon paper to a scale, cut out figure, and proceed with it as above directed, in order to find position of centre of gravity in elevation of the model. In actual arch, centre of gravity will have same relative position as in paper model. . In regular figures or solids, centre of gravity is same as their geometrical centres. Line. Circular Arc. ™ = distance from centre , r representing radius , c chord , and l length of arc. Surfaces. Square , Rectangle , Rhombus , Rhomboid, Gnomon, Cube , Regular Polygon, Circle,. Sphere, Spheroid or Ellipsoid , Spheroidal Zone, Cylinder, Circular Ring, Cylindrical Ring, Link , Helix , Plain Spiral, Spindle , all Regular Fig- ures, and Middle Frusta of all Spheroids, Spindles , etc. The centre of gravity of the surfaces of these figures is in their geometri- cal centre. Triangle. — On a line drawn from any angle to the middle of opposite side , at two thirds of the distance from angle. Trapezium.-- Draw two diagonals, and ascertain centres of gravity of each of four triangles thus formed ; join each opposite pair of these centres, and it is at intersection of the lines. Trapezoid. X — = distance from B on a line joining middle of two parallel sides B b, m representing middle line. Circular Arc. ~ = distance from centre of circle. Sector of a Circle. . 4244 r = distance from centre of circle. Semicircle. . 4244 r = distance from centre. Semi- semicircle. .4244 r == distance from both base and height and at their inter- section. Segment of a Circle. = distance from centre , a representing area of segment. , Sector of a Circular Ring. ± X — ^4 X = distance from centre of 3 arc Z. arcs , r and r' representing the radii. Illustration.— Radii of surfaces of a dome are 5 and 3.5 feet, and angle «) at centre = 130 0 . 4 sin. 65° 125 — 42- 8 75 _4 y -9 o6 3 x 82.12$ __ feet. 7 X arc 130 0 X 25 — 12.25 3 X 2.2689 6.8067X12.75 Hemisphere , Spherical Segment , and Spherical Zone , At centre of their heights. Circular Zone.— Ascertain centres of gravity of trapezoid and segments comprising zone; draw a line (equally dividing zone) perpendicular to chords; connect centres of segments by a line cutting perpendicular to | chords. Then centre of gravity of figure will be on perpendicular , toward lesser chord , at such proportionate distance of difference between centres of gravity of trapezoid and line connecting centres of segments , as area of segments bears to area of trapezoid. MECHANICAL CENTRES. — GRAVITY. 607 Prism and Wedge . — When end is a Parallelogram, in their geometrical centres ; when the end is a Triangle, Trapezium, etc., it is in middle of its length, at same distance from base, as that of triangle or trapezoid of which it is a section. Parabola in its axis = .6 distance from vertex. Prismoid. — At same distance from its base as that of the trapezoid or trapezium, which is a section of it. Lune. — On a line connecting centres of gravity of arcs at a proportionate < point to respective areas of arcs. Solids. Cube , Parallelopipedon , Hexahedron, Octahedron, Dodecahedron, Icosahe- dron, Cylinder , Sphere, Right Spherical Zone, Spheroid or Ellipsoid, Cylin- drical Ring, Link, Spindle , all Regular Bodies , and Middle Frusta of all Spheroids and Spindles, etc. Centre of gravity of these figures is in their geometrical centre. Tetrahedron. — In common centre of centres of gravity of the triangles made by a section through centre of each side of the figures. Cone and Pyramid. .25 of line joining vertex and centre of gravity of base = dis- tance from base. (v “ 4 — ^ —l— 2 1 Frustum of a Cone or Pyramid. - — — - --- , X - h = distance from centre (i'-\-r’) z — r r' 4 J of lesser end , r and r’, in a cone representing radii, and in a pyramid sides, and h height. Cone, Frustum of a Cone , Pyramid , Frustum of a Pyramid, and Ungula . — At same distance from base as in that of triangle, parallelogram, or semicir- cle, which is a right section of them. Hemisphere. . 375 r := distance from centre. Spherical Segment. 3.1416 vs 2 (r — -T- v = distance from centre , vs repre- senting versed sine, and v volume of segment, vertex. Spherical Sector. .75 (r — .5 h) = distance ji'om centre. 2 r 3 ^ __ distance 8 from vertex. Spirals . — Plane, in its geometrical centre. Conical, at a distance from the base, .25 of line joining vertex and centre of gravity of base. Frustum of a Circular Spindle. v = distance from centre of spindle . 2 (h — u.z) ’ h representing distance Vetween two bases , D distance of centre of spindle from centre of circle, and z generating arc , expressed in units of radius. 7.2 Segment of a Circular Spindle. — — jfz) ~ ^ s ^ ance J rom centre of spindle. Semi-spheroids. — Prolate. .375 a. — Oblate. .375 a — distance from centre. Semi-spheroid or Ellipsoid and its Segment. — See HaswelVs Mensuration, pages 281 and 282. ’ ^ 6 ( 8 V 3 X h = distance from \i2 r — 4/7 J Frusta of Spheroids or Ellipsoids. Prolate. .75 gj 2 ** 2 ^ = distance from 3 CL^ ““ centre of spheroid, a representing semi-transverse diameter in a prolate frustum, and semi-conjugate in an oblate frustum. 608 MECHANICAL CENTRES. GRAVITY. Segments of Spheroids.— Prolate. .75 ^ ^ • — Oblate. .75 ^ — distance from centre of spheroid, d and d' representing distances of base of segments from centre of spheroid. Any Frustum. .75 + d ) X (2 a 2 d 4 -d ) __ distance f rom cm i re 0 j sphe- ' 3 a 2 — d z -\-d d-\-d 2 void, d and d' representing distances of base and end of segments from centre of the spheroid. Segment of an Elliptic Spindle at two thirds of height from vertex. Paraboloid of Revolution , at two thirds of height from vertex. Segment of a Hyperbolic Spindle, at 75 of height from vertex. Frustum of Paraboloid of Revolution. X — = distance from base, r and r' representing radii of base and vertex. Segment of Paraboloid of Revolution, at two thirds of height from vertex. Segments of a Circular and a Parabolic Spindle . — -See Haswell s Mensuration, pages 192 and 199. Parabola. .4 of height = distance from base. Hyperboloid of Revolution. ^ ^ ^ Xh = distance from vertex, b representing diameter of base. (d - 4 - d') (2 a 2 — d ' 2 -f- d 2 ) . Frustum of Hyperboloid of Revolution. .75 3 ” distance from centre of base, a representing semi-transverse axis, or distance from centre of curve to vertex of figure ; d and d’ distances from centre of curve to centre of lesser and greater diameter of frustum. Segment of Hyperboloid of Revolution. ^ ^ 4! X h = distance from vertex. Of Two Bodies. dV - — distance from V or volume or area of larger body, d rep- J V -|~ v resenting distance between centres of gravity of bodies, and v volume or area of less body. Cycloid. — . 833 of radius of generating circle = distance from centre of chord of curve. A ny Plane Figure.— Divide it into triangles, and ascertain centre of grav- ity of each ; connect two centres together, and ascertain their common cen- tre ; then connect this common centre and centre of a third, and ascertain the common centre, and so on, connecting the last-ascertained common centre to another centre till whole are included, and last common centre will give centre required. Of an Irregular Body of Rotatioil. Divide figure into four or six equidistant divisions ; ascertain volume of each, their moments with reference to first horizontal plane or base, and then connect them thus : (A + 4 Ar + e A 2 + 4 A 3 + A 4 ) ~ = v, a A„ etc., representing volume of dims- ions, and h height of body from base ; a (0A+1X4. Ax 4-2X2 A2 + 3X4 A3 + 4 A4) h _ d j s t ance 0 j cen tre of and A + 4 Ax-j-2 A 2 + 4 A3 + A4 4 gravity from base. MECHANICAL CENTRES. — GYRATION. 609 Centre of Gry-ration. Centre of Gyration is that point in any revolving body or system of bodies in which, if the whole quantity of matter were collected, the Angular velocity would be the same ; that is, the Momentum of the body or system of bodies is centred at this point, and the position of it is a mean proportional between the centres of Oscillation and Gravity. If a straight bar of uniform dimensions was struck at this point, the stroke would communicate the same angular velocity to the bar as if the whole bar was collected at that point. The A ngular velocity of a body or system of bodies is the motion of a line connecting any point and the centre or axis of motion : it is the same in all parts of the same revolving body. In different unconnected bodies, each oscillating about a common centre, their angular velocity is as the velocity directly, and as the distance from the centre inversely. Hence, if their velocities are as their radii, or distances from the axis of motion, their angular velocities will be equal. When a body revolves on an axis, and a force is impressed upon it suffi- cient to cause it to revolve on another, it will revolve on neither, but on a line in the plane of the axes, dividing the angle which they contain ; so that the sine of each part will be in the inverse ratio of the angular velocities with which the bodies would have revolved about these axes separately. Weight of revolving body, multiplied into height due to the velocity with which centre of gyration moves in its circle, is energy of body, or mechani- cal power, which must be communicated to it to give it that motion. Distance of centre of gyration from axis of motion is termed the Radius of gyration ; and the moment of inertia is equal to product of square of radius of gyration and mass or weight of body. The moment of inertia of a revolving body is ascertained exactly by as- certaining the moments of inertia of every particle separately, and adding them together ; or, approximately, by adding together the moments of the small parts arrived at by a subdivision of the body. To Compute Moment of Inertia of a Revolving Body. Rule. Divide body into small parts of regular figure. Multiply mass or weight of each part by square of distance of its centre of gravity from axis ot revolution. The sum of products is moment of inertia of body. NoTE -The value of moment of inertia obtained by this process will be more exact, the smaller and more numerous the parts into which body is divided. To Compute Radius of Gyration of a Revolvin al;> out its -A-xis of Revolution. Body Rule. Divide moment of inertia of body by its mass, or its weight, anc square root of quotient is length of radius of gyration. Note.— W hen the parts into which body is divided are equal, radius of gyratior ™ f ! determined by taking mean of all squares of distances of parts from axis ot revolution, and taking square root of their sum. Or, VR 2 -f- r 2 - 4 - 2 = G. R and r representing radii. example.— A straight rod of uniform diameter and 4 feet in length, weighs 4 lbs. vhat is its inertia, and where is its radius or centre of gyration ? f ° 0t of j en , gth wei S hs 1 lb -, and if divided into 4 parts, centre of gyration of each is respectively .5, 1.5, 2 . 5 , and 3.5 feet. Hence, *X .5^= .25I i 5 i ' 5 2 = l 21 == inertia, which -4-4 = 5. 25, and ^5. 25 = 2 . 291 1 X 2. 5 = 6. 25 f feet radius. 1 X 3.5 s = 12.25 J 6io MECHANICAL CENTRES. GYRATION. Following are distances of centres of gyration from centre of motion in various revolving bodies : Straight , uniform Rod or Cylinder or thin Rectangular Plate revolving about one end; length X -5773, and revolving about their centre; length x .2886. The general expression is, when revolving at any point of its length, ( l 3 + l' 3 \ \3 i*'+n/‘ 76 l and V representing length of the two points. Circular Plane , revolving on its centre; radius of circle X .7071 ; Circle Plane , as a Wheel or Disc of uniform Thickness , revolving about one of its diameters as an axis; radius X .5. Solid Cylinder, revolving abotit its axis; radius X •7°7 I - Solid Sphere, revolving about its diameter as an axis; radius X .6325. Thin, hollow Sphere, revolving about one of its diameters as an axis; radius X .8164. Surface of sphere .8615 r. Sphere and Solid Cylinder (vertical), at a distance from axis of revolution = y/t 2 -f . 4 r 2 for sphere , and Vl 2 -\~- 5 r 2 for cylinder , l representing length of connec- tion to centre of sphere and cylinder. Cone, revolving about its axis ; radius of base X -5447 5 revolving about its ver- tex = Vi2 /i 2 -f 3 r ' 2 -r- 20, li representing height , and r radius of base; revolving about its base = V2 h 2 -j- 3 r 2 -4- 20. Circular Ring , as Rim of a Fly-wheel or Hollow Cylinder, revolving about its diameter = Vlt 2 + r2 ' f2 , R representing radius of periphery, and r of inner circle of ring. 76 W (R 2 + r 2 ) 4- w (4 r» _ , .. ... , Flv-wheel = / — — — ! — : W and io representing weights of J V 12 (W-f w) nm and of arms and hub , and l length of arms from axis of wheel. Section of Rim. + r 2 -}- r d. d representing depth and c periphery of rim. j. ( 2 -i- b 2 Parallelopiped, revolving about one end, distance from end = W — — — , b rep- resenting breadth. Illustration. — In a solid sphere revolving about its diameter, diameter being 2 feet, distance of centre of gyration is 12 X -6325 = 7.59 ins. To Compute ]Elemen.ts of Gyration. GWv_. ?rtg_ GWv T>rtg _ G ¥j > =;( , rtg ’ ’ W« “ ’ P tg ’ Gv ’ ^ ? ^ ^ = u G representing distance of centre of gyration from axis of rotation , GW . W weight of body, t time power acts in seconds, v velocity in feet per second acquired by revolving body in that time, and r distance of point of application of power fi om axis of body, as length of crank, etc. Illustration i. — What is distance of centre of gyration in a fly-wheel, pover , 224 lbs., length of crank 7 feet, time of rotation 10 seconds, weight of wheel 5000 lbs., and velocity of it 8 feet per second? 224 X 7 X 10 X 32- 166 _ 504 373 a feet 5600 X 8 42 800 2.— What should be weight of a fly-wheel making 12 revolutions per minute, its diameter 8 feet, power applied at 2 feet from its axis 84 lbs., time of rotation 6 sec- onds, and distance of centre of gyration of wheel 3.5 feet? 84 X 2 X 6 X 32.166 8 X 3 -1416 X 12 60 = 5.0265 feet — velocity. Then - 3.5 X 5.0265 ; 1843.2 IbS. MECHANICAL CENTRES. — GYRATION. 6ll When the Body is a Compound one. Rule.— M ultiply weight of several particles or bodies by squares of their distances in feet from centre of mo- tion or rotation, and divide sum of their products by weight of entire mass • the square root of quotient will give distance of centre of gyration from centre of motion or rotation. J Example. -If two weights, of 3 and 4 lbs. respectively, be laid upon a lever (which is here assumed to be without weight) at the respective distances of 1 and 2 feet what is distance of centre of gyration from centre of motion (the fulcrum) ? ’ 3 + 16 3 Xi 2 = s; 4 X2 2 = i6; 3 + 4 _ ^9 _ " 7 2.71, and ^2.71 =+.64 feet. That is, a single weight of 7 lbs., placed at 1.64 feet from centre of motion and re respective places. time ’ WOUld have same mor * ent m as the two weights ’in their When Centre of Gravity is. given. Rule. -M ultiply distance of centre of oscillation from centre or point of suspension, by distance of centre of grav- ity from same point, and square root of product will give distance of centre 01 gyration. 0 Example.— Centre of oscillation of a body is 9 feet, and that of its gravitv d foot centre olr^ 011 ° F P ° int of ^pension; ^ It what distance fronfthis JotnUa 9 X 4 — 36, and +36 = 6 feet To Compute Centre of Gyration of a Water-wheel. Rule.— M ultiply severally twice weight of rim, as composed of buckets shrouding, etc., and twice that of arms and that of water in the buckets (when wheel is m operation) by square of radius of wheel in feet- divide by twice sum of these several weights, and square root of quotient will give distance m ieet. Example. — In a wheel 20 feet in diameter, weight of rim is 2 tons weight of ' in buckets 1 ton; what is aistanc ° ° f Buckets = 2 tons x 102 X 2= 4™ 3 + 2 + 1 X 2 = 12 sum of weights. Water =iton X 102 — IOO TT / IIOO noo Hence y. i f f^9* :6 T=9-57 feet. ^S^ 1 RA 7 L Formulas.— p ^Presenting power , H horses’ power, F force amlied to rotate m lbs., M mass of revolving body in lbs., r radius upon which F acts in feet, d distance from axis of motion to centre of gyration in feet, t time force is an plied in seconds , n number of revolutions in time t, x angular velocity, or number of revolutions per minute at end of time t, and G = 32-166 F r2 v - = t: i 4 pm G 153-5 t Fr '.lld 2 / : 2 pr 2 x 60 G 244 t P + 2 d 2 Mxd 2 * 53-5 l r = M; = F; lid 2 ' Hind 2 2+6 * 2 F = M d 2 _ 2.56 t 2 F r Hid 2 = 2 Hid 2 i = H. 244 t ’ 134 100 1 " reet^^fa°».r Ril ? ° f ? fly ' wheel weighing 7 ooo lbs. has radii of 6.5 and 5 7S from of mil- , tre of ? yration - and What force must be applied to it 2 feet ao Teconds? h™ “ t0 g,TC an an S u,ar Telocity of 130 revolutions per minute in power? * ° W many rcv °lutions will it make in 40 seconds? and what is its 130 2 X 7000 X 6. 14 2 134 100 X 40 4459 862680 5364000 : 829. 7 horses. 34306636 12 280 of gyration = -^Jr 1 Centre of gyration = + 5 ‘ 75 2 __ ^.f ee ^ Then F __ 130 X 7000 X 6. i 4 2 = 2793.7 lbs., and X 40 X 2793.7X2 153-5 X 40 X 2 7000X6.142 = 86 - e 7 revolutions. 612 MECHANICAL CENTRES. OSCILLATION, ETC. Centres of Oscillation and Percussion. Centre of Oscillation of a body, or a system of bodies, is that point in axis of vibration of a vibrating body in which, if, as an equivalent condition, the whole matter of vibrating body was concentrated it would continue to vibrate in same time. It is resultant point of whole vibrat- ing energy, or of action of gravity in producing oscillation. As narticles of a bodv further from centre of its suspension have greater velocity of vibration than those nearer to it, it is apparent^ that centre of oscillation is further from its centre than centre of gravity is from axisof susnension but it is situated in centre of a line drawn from axis of a body Sih centre of gravity. It further differs from centre of gyration in this that while motion of oscillation is produced by gravity of a body, that of gyration is caused by some other force acting at one place only. Radius of oscillation, or distance of centre of oscillation from axis of sus- pendoris a thlrd pro’portional, to distance of centre of gravity from axis of suspension and radius of gyration. Centre of Percussion of a body, or a system of bodies, revolving about a point or axis, is that point at which, if resisted by an imm - able obstacle all the motion of the body, or system of bodies, would be £ S without impulse on the point of suspension. It is also that point which would strike any obstacle with greatest effect, and from this property it has been termed percussion. Centres of Oscillation and Percussion are in same point.— If a blow is That is centre of percussion is identical with centre of osciUation ai d centre of percussion, its motion will be absolutely destroyed, so that the body will not incline either way. „ , . As in bodies at rest, the entire weight may be considered as conected m centre of gravity; so in bodies in vibration, the entire force may hecOTSid- ered as concentrated in centre of oscillation; and in bodies in motion, the whole force may be considered as concentrated m centre of percussion. If centre of oscillation is made point of suspension, point of suspension will become centre of oscillation. rill UtJbUlUC LCI1WO V/X. , , Angle of Oscillation or Percussion is determined by angle delineated by vertical plane of body in vibration, in plane of motion of body. Cl LlCctl piCtllVv V/X - ' • 1 1 equal in height to versed sine of the arc. To Compute Centre of Oscillation »r p er™^io,i of a Body of Uniform Density and F » Pin f —Multiply weight of body by distance of its centre of gravity JErom point |of' suspension ; W multiply a J weight of body by square of its length, 31 DkidfurS quotient by product of its centre of gravity, and quotient is distance of centres front po pension. MECHANICAL CENTRES. OSCILLATION, ETC. 613 W l~ Or, — -4- W X <7 = distance from axis. Or, square radius of gyration of body and divide by distance of centre of gravity from axis of suspension. Example.— Where is centre of oscillation in a rod 0 feet in length from its Doint of suspension, and weighing 9 lbs. ? F = 4°* 5 = product of weight and its centre of gravity ; 92 = 243 = quo- tient of product of weight of body and square of its length-- 3 ; --- — 6 feet 40-5 When Point of Suspension is not at End of Rod. Rule. — To cube of distance of point of suspension from top of rod or bar, add cube of its dis- tance from lower end, and multiply sum by 2. Divide product by three times difference of squares of these distances and quotient is distance of point of oscillation from point of suspension. ’ Example.— A homogeneous rod of uniform dimensions, 6 feet in length is sns E 0 e f suspensln°? m US UPP ® r ^ What iS distance of P oint of oscillation from . 2 (4-5 3 + i-5 3 ) i8q o 1. 5 — 4-5. — ■ ■ — - — — - — 3. 5 feet. ■ 3 (4-5 J I -5 ) 54 Centres of Oscillation and. Percussion in Bodies of Various Figures. When Axis of Motion is in Vertex of Figure, and when Oscillation or Motion is Facewise. Right Line , or any figure of uniform shape and density — 66 l Isosceles Triangle = .75 h. Circle =1 2 c r Parabola = .714 h. C one = . 8 h. When Axis of Motion is in Centre of Body. Wheel == . 75 radius. When Oscillation or Motion is Sidewise. Right Line, or any figure of uni- agonai aPe ^ densit 'J — 66 l - Rectangle, suspended at one angle — .66 of di- o’fTts + 33 parameter; IT suspended Sector of a Circle — c representing chord of arc, and r radius of base. Circle = .75 d. Sphere ~ - Cone = - axis-1 5 s axis 7^+7} + r + c ’ c representing length of cord by which it is suspended. To Ascertain Centres of Oscillation and IPercussion experimentally. Suspend body very freely from a fixed point, and make it vibrate in small arcs noting number of vibrations it makes in a minute, and let number made in a min’ ute be represented by n ; then will distance of centre of oSatlon from poin? of suspension be = - 4 - — ins. n 2 For length of a pendulum vibrating seconds, or 60 times in a minute beimr 39- i2 5 ins., and lengths of pendulums being reciprocally as the squares of number of vibrations made in same time, therefore n 2 : 60 2 : : 39. 125 : 6 ° 2 * 39- I2 5 _ 140850 *«"*“*• ordistance of Zntr’e 3 F 6 1 4 mechanical centres— mechanics. To Compute Centres of Oscillation or Percussion of a System of particles or Bodies. Rui e.— Multiply weight of each particle or body by square of its distance from point of suspension, and divide sum of their products by sum of weights, multiplied by distance of centre of gravity from point of suspension, and quotient will give centre required, measured from point of suspension. Or y ^ _ distance of centre. ’ W0 + W'0' Fkamplf i —Length of a suspended rod being 20 feet, and weight of a foot in length of it eqiial xoo oz has a ball attached at under end weighing 100 oz. ; at what point of rod from point of suspension is centre of percussion . IO o X 30 = 2000 = weight of rod ; 2000 X ?? = 20000 = momentum of rod, or prod- „ #1 2000 X 20 2 ,, ,,r __ uct of its weight, and distance of its centre of gravity ; - — 266 66 . force of rod ; 1000 X 20 2 = 400000 = force of hall. Then 266 666.66 400000 _ l6 ' 66 f eet 20 000-1-20 000 , , loncrth and weighing 2 lbs. for each foot of its length, th^^alV^of^bs'^ch-onl fixed 6 feet from the point of suspension, and the other aUhe end of the rod; what is the distance between the points of suspension and percussion? a I2 x 2 x = 144 = momentum of rod, = ^-=1152 =force of rod. 3 Xi2 =36= “ °o/^dhall. 3X 6 2 = 3 X 36 = i° 8 — li ofistball, ig 8 sum of moments. 3Xi2 2 =3Xi44 — of vd hall, n o a** 1602 sum of forces. Then 1692 198 = 8. 545 feet. MECHANICS. Mechanics is the science which treats of and investigates effects of forces, motion and resistance of material bodies, and of equilibrium, it is divided into two parts— S tatics and Dynamics. Statics treats of equilibrium of forces or bodies at rest. Dynamics of forces that prodmee motion, or bodies m motion. These bodies are further divided into Mechanics of Solid, Fluid , and Aert- form bodies ; hence the following combinations : 1 Statics of Solid Bodies, or Geostatics. .2! Dynamics of Solid Bodies , or Geodynamics. -I Statics of Fluids , or Hydrostatics. T Dynamics of Fluids , or Hydrodynamics. _ ce Statics of Aeriform BoSes, or Aerostatics, t. iLamics of Aeriform Mies, Pneumatics or Aerodynamics, mines me various, and are divided into moving forces or resistances; as ■ l > b is a pressure with which it cannot be compared. > -r ± - = 7.1215 feet. MECHANICS. DYNAMICS. 617 of 5 lbs. is impelled by a force of 40 lbs., accelerating force is 8 lbs. • but if a force of 40 lbs. act upon a body of 10 lbs., accelerating force is only 4 lbs., or half former, and will produce only half velocity. With equal masses, velocities are proportional to their forces. With equal forces, velocities are inversely as the masses. With equal velocities, forces are proportional to the masses. Work is product of force, velocity, and time. Motion .— The succession of positions which a body in its motion pro- gressively occupies forms a line which is termed the trajectory or path of the moving body. ’ A motion is Uniform when equal spaces are described by it in equal times, and Variable when this equality does not occur. When spaces described in equal times increase continuously with the time, a variable motion is termed accelerated , when spaces decrease, retarded , and when equal spaces are described within certain intervals only, the motion is termed periodic , and intervals periods. Uniform motion is illustrated m progressive motion of hands of a watch ; variable in progressive ve- ocity of falling and upwardly projected bodies ; and periodic by oscil- lation of a pendulum or strokes of a piston of a steam-engine. Formulas, fv, H 550, and — -P; * t Uniform Motion. P W =- , — =£_ and / and W 550 t H 55 ° / fs H 550 W . P t vt ' /’ V ■ and and H 550 t f t' /’ /’ " / = t] fs, H550 l, P t, and fv t = W; P 55o ! H 5 5 °_ , v ~ 7 ’ sf s p’ 7 ’ fv fs 550 ’ 550 1 t ’ W Jv' , and ^ „ = H - p representing power in effect , body , or momentum, f force in lbs. , v and s velocity and space in feet per second , t time in seconds, H horse-power , and W work \Yl jOOt-CuS, If two or more bodies , etc., are compared, two or more corresponding letters as l , p, p , \ , v, v 1 , etc., are employed. ^ Illustration i -Two bodies, one of 20, the other of 10 lbs., are impelled by same momentum, say 60. They move uniformly, first for 8 seconds, second for 6- what are the spaces described by both? 5 o, waai, 60 -f- 20 = 3 = Y, and 6o -f- io = 6 = v. Then T \ =3 X 8 = 24 = S, and tv — 6 x 6 = 36 = s, spaces respectively. 2. — If a power of 12 800 effects has a velocity of 10 feet per second, what is its lorce ' 12 800 - 4 - 10 = 1280 lbs. Uniform 'V'aria'ble Motion. Space described by a body having uniform variable motion is represented by sum or difference of velocity, and product of acceleration and time ac- cording as the motion is accelerated or retarded. ’ Illustration 1.— A sphere rolling down an inclined plane with an initial velocitv of 25 feet acquires in its course an additional velocity at each second of time of < feet; what will be its velocity after 3 seconds? 5 2 5 + 5 X 3 = 40 feet. ,ht,~ n A 2°i 0m0tl l h .V ine a " initla ' v elocity of 30 feet per second is so retarded that in each second it loses 4 feetj what is its velocity after 6 seconds? 30 — 4 x 6 = 6 feet. 6i8 MECHANICS. DYNAMICS. Uniform HVIotion. Accelerated. In this motion, velocity acquired at end of any time whatever is equal to prod- uct of accelerating force into time, and space described is equal to product of half accelerating force into square of time, or half product of velocity and time of ac- quiring the velocity. Spaces described in successive seconds of time are as the odd numbers, i, 3, 5, 7, 9, etc. Gravity is a constant force, and its effect upon a body falling freely in a vertical line is represented by 9, and the motion of such body is uniformly accelerated. The following theorems are applicable to all cases of motion uniformly acceler- ated by any constant force, F: v 2 2 s v I s .5 1 .,=,5 a F =. I7T . =.«• - = jg = yj .777- - *• 2 S V 2 S V 2 T= SF(=^ffF, = , - = — = — = F. When gravity acts alone, as when a body falls in a vertical line, F is omit- ted. Thus, V 2 . , v /2S •5 * t, = r, = t g = ^T =t t representing time in seconds , and s velocity in feet per second. v 2 s 1>2 If, instead of a heavy body falling freely, it be projected vertically upward or downward with a given velocity, v, then s = tv .5/7 1~ ; an expression in which — must be taken when the projection is upward, and + when it is downward. Illustration i. — Tf a body in 10 seconds has acquired a velocity by uniformly accelerated motion of 26 feet, what is accelerating force, and what space described, in that time? 2 6 26=10 = 2.6 = accelerating force ; -A- x io 2 = 1 30 feet = space described. 2. — A body moving with an acceleration of 15.625 feet describes in 1.5 seconds a 15.625 X (1- 5) 2 0 j. , space = = 17.578 feet, 3. — A body propelled with an initial velocity of 3 feet, and with an acceleration 7 2 of 5 feet, describes in 7 seconds a space — 3X7 + 5X — = 143- 5 feet. 2 4. — A body which in 180 seconds changes its velocity^ from 2.5 to 7.5 feet, trav- erses in that time a distance of 2-5 7 — x 180 = 900 feet. 5. — A body which rolls up an inclined plane with an initial velocity of 40 feet per second, by which it suffers a retardation of 8 feet, ascends only ^ = 5 seconds , and O 40 2 2 x 8 = 100 feet in height, then rolls back, and returns, after 10 seconds, with a velocity of 40 feet, to its initial point; and after 12 seconds arrives at a distance of 40 X 12 — 4 X i2 2 = 9 6 feet below point, assuming plane to be extended backward. Circular ^Motion. 2 p r n _ 2 p r n f _ ^ 5500 FP_ W f rn _ f 2pm __ jp. 60 — t ~~ ’ rn ~2 prn'~ J ' 5500 — 550 X 60 r f 2 p r n' — = W. r representing radius in feet, n number of revolutions 60 of circle per minute, n ' total revolutions, f force in lbs., i time in seconds, and H* horse-power. Fig. 6. nVXotion on an Inclined. Plane. To Ascertain Conditions of Motion by Gravity. 0 Assume A B, Fig. 6, an inclined plane, 13 C its base AC its height, and b a body descending the plane; from dot, centre ol gravity of body, draw b a perpendicular to BC, representing pressure of b by gravity; draw bo parallel and b r perpendicular to A 13, and complete j j K parallelogram; then force ba is equal to both b o. b r, . , i, ~ nf which b r is sustained by reaction of plane, and iorce b o is wholly effective in accelerating motion of body. Let tUis force be represented byf and b a, by g or force of gravity, then by similar A i \s s* J ^ '.will/ V v y y ut jut Go OJ f tr. angle, / : gy.bo : ba : AC: AB. Hence, A - L — 9 — f ’ A B ~ J ‘ Put A B _ AC — A and Z.ABC = fl, then force which produces motion on the plane on /becomes g - , and g sin .a. Therefore, accelerating force on an inclined plane is constant and equations of motion will be obtained by substituting its value of / for g in equations i 2 and 3, page 6 1 8. 5 ’ ght* lv 2 2 l ’ 2 g iC 2/ g h t j 2 g h s t ’ i ’ l v J2I s •5 tv, ■ 5 fft 2 sin.a, and 2 S lv /: V ’ g a’ V 9 /* <7 sin. a , and g t sin. a, V /? sin. a 2 g sin. a and V2 g s sin. a = v. = t. a representing /_ ABC. When a Body is projected doom or up an Inclined Plane, with a (riven Ve- limfwm be' 6 diStance wh “ h ( H wil * be from point of projection in a given t v ±- 2 t ■ , and — (2 Z v ± g li t) — s. Illustration 1 . —Length of an inclined plane is 100 feet, and its an^le of inclina- t ,on 60 ; what is time of a body rolling down it, and velocity acquired? s:n..6o°— 866. 2 X 100 3^6 x 7866 = V 7 - 18 = 268 seconds, and 32. 16 X 2.68 X 866 = 74.64 feet of 2 '7 f lL ab ^ ’ s Projected up an inclined plane, which rises i in 6, with a velocity 50 feet per second, what will be its place and velocity at end of 6 seconds ? 7 32.16X1X6^ / xX 2 ~ x 6 / — 2 ° 3 - 52 feet from, bottom , and 50 — ( 32 . 16 X 6 x - J = V: 6X50 50 — 32. 16 = 17. 84 feet. tnSESgK53” d r ’ ,ane in l0ast ,ime - » lcl ‘StlP to its height, Work Accumulated in Moving Bodies. Quantity of work stored in a body in motion is same as that which would AccunXte 6d "V by ffrav ! ,v Sfjt from the height due to the velocity. Accumulated work expressed in foot-lbs. is equal to product of hem-lit so found in feet, and weight of body in lbs. Height due to velocity is”equal dutSSdtwrt W ; ,0e,ty d i vid i! d l’ y 64 ' 4, a . nd "°'k and velocity may be de- duced oi.ectly from each other hv following rules: To Compute A dcumulated Work. arid dh'hiT^ r,d i iply w i igllt V 1 lb ?‘ bv S( l uare of velocity in feet per second, an.l divide by 64.4, and quotient is accumulated work in foot-lbs. r 2 x m ~b 4 . 4 • °L = w X h. W representing worlc, w weight in lbs., and Or. W = - h height due to velocity in feet per second. 6 20 MECHANICS. DYNAMICS. 60 Work ky Percussive Force. If a wedge is driven by strokes of a hammer or other heavy mass, effect of percussive force is measured by quantity of work accumulated in stricken body This work is computed by preceding rules, from weight of body and velocity with which a stroke is delivered, or directly from height of fall, if gravity be percussive power. Useful work done through a wedge is equal to work expended upon it, assuming that there is no elastic or vibrating reaction from the stroke, as if the work had been exerted by a constant pressure equal to weight of strik- ing body, exerted through a space equal to height of fall, or height due to its final velocity. If elastic action intervenes, a portion of work exerted is absorbed in an elastic stress to resisting body ; and the elastic action may be, in some cases, so great as to absorb the work expended. T he principle of action of a blow on a wedge is alike applicable to action of the stroke of a monkey of a pile-driver upon a pile. . If there be no elastic action, the work expended being product of weight of monkey by height of its fall, is equal to work performed in driving the pile: that is, to product of resistance to its descent by depth through which it is driven by each blow of monkey. Illustration. — If a horse draws 200 lbs. out of a mine, at a speed of 2 miles per hour, how many units of work does he perform in a minute, coefficient of friction .05 r 2 X 5280 __ x ^f ee t per minute. Hence, 176 X 200 .05 X 200 = 35 210 units. Decomposition of Force. By parallelogram of force it is il- lustrated how a vessel is enabled to be sailed with a free wind and against one. Assume wind to be free or in direction of arrows, Fig. 7, and perpendicular to line A B, the course of vessel. Let line m o represent direction and force of wind, and rs plane of sail; from 0 draw 0 u perpendicular to r s , and from m perpendicular, m v on r s, and mu on ou. By principle of parallelogram of forces, force m 0 may be decomposed into ov and ou. since they are the sides of parallelogram of which m o, representing force of wind, is diagonal. Force of wind, therefore, is measured by ou, both in magm- tude and direction, and represents actual pressure on sail. Q Draw u n and u x parallel to 0 A and o w, thus fornung parallclog am w n o x 1 Hence force 0 u is equal to the two, 0 n and o x. Force o n acts in a direction perpendicular to vessel’s course and that of 0 x is to drive vessel onward. It can thus be shown that when di- rection of sail bisects angle m o B, the effect of o x is greater than wffien sail is in any other position. Assume wind to be ahead as in direc- tion of arrow’s, Fig. 8. Let 0 m repre- sent direction and force of w'ind, and r s direction of sail; from 0 draw on, and proceed as before, and o u represents the effective force that acts upon the sail, 0 n that which drives her to leeward, and 0 x that which drives her on her course. For full treatises on this subject, see John C. Trautwine’s Engineer's Pocket-book 1872 ; Bull’s Ex- perimental Mechanics, London, 1871 ; and Dynamics, Construction of Machinery, etc., by G. Finden Warr, London, 1851. MECHANICS. MOMENTS OF STRESS ON GIRDERS, ETC. 621 MOMENTS OF STRESS. To Describe and Coixipate Moments of Stress in GJ-irders or Beams. Fig. i. Beam Supported at Both Ends. Loaded in Middle , Fig. i. — Assume A B beam. At middle- erect W c — Wl „ ~ . Connect A c and c B, and ver- 4 tical distances between them and A B “JB will give moment required. ib' w* I bus, — — - = M at any point. W rep- . , , „ resenting weight or load , l length of span, x horizontal distance from nearest support at which M is required , and M mo- ment of stress. Illustration. — A ssume l^z io feet, W = io lbs., and x — % feet. Then, W c = -°-- X 10 = 25 lbs. at centre of span ; ■— l5 at x. 2 Loaded at Any Point , Fig 2. Proceed as for previous figure. W a b — r — or vv c — maximum load. B l W xb „ — Y~ — M between A and W. a representing least distance of W to support , Wxa _ vr w „ and b greatest distance. 1 — ^ between W and B. Illustration. — T ake elements as before with a = 3 feet, and x = 1.5 and 3.5 feet. Then, We = ^3X7 = 2i ^ ^ ^ xoX ± s> Lz = ^ ^ ’ IO IO between A and W, and — * 3 5 * 3 — IO ^ a f x i e f ween an ^ g 10 Note. — x must be taken from the pier which is on the same side of W as * is. Loaded with Two Equal Weights at Equal Distances from Ends , alike to a Trans- verse Girder as for a Single Line of Railway.— Fig. 3. At point of stress of weights erect W c and W d , each ~ W a. Connect A c d and B, and vertical -t-j2 distances between A B, as defined ^ by cd, will give moments. W (l — 0) TTr = W a = w b = M at r, 1 2 any point between weights. Loaded with Four Equal Weights , symmetrically bearing from Centre , alike to a Transverse Girder as for a Double Line of Railway.— Fig. 4. Fig. 3 - t w IP 4 - d e At W and to" erect W c, and w" i = 2 W a, and at and re' erect w cZ, «/e, each == W (2 <2 + a'). Connect A c d ei and B, and or- dinates to A B will give mo- ments. W (2 a-\- a') =1 M at w and w'\ 2 W a — M at W and w". Illustration. — Assume W each 10 lbs. 2 feet apart, and 1 10 feet. Then, 10 (2 x 2 -f 2) = 60 at w or w\ and 2 x 10 x 2 = 40 at W or w". 3- T c c 1 i \ 622 MECHANICS. MOMENTS OE STRESS ON GIRDERS, ETC. Fj m Loaded at Different Points.— Fig. 5. JS-— Locate three weights, W, w , and /| ! \ w', as at a b , a z b x , a 2 b 2 . Draw A c B, A d B, and A e B, for three separate cases, as by formula, W ab „ — — , Fig. 2. Produce W c until Wo=Wr,W s, IB and Wc; W d until wu — wn , w v w and w d, and w' e to w' m in like ^ manner. Connect A oum and B, and an or- A/ m 0 s 1 s f y <2^ v \ \ r n. % Www'. f manner. 0--- f a 1 --—-y- b x -- :| i a 2 — x- b 2 — % dinate therefrom, to A B will give l moment or stress at the point taken. Illustration. — Take a = 2 feet, a z = 4, a 2 == 6, b = 8, Z>i = 6, = 4> x 2 > ty, and w' each 10 lbs., and l = 10 feet. Then y(Wax-)-wa I a:-l-w , &2x) = Mat as. __ 280 Q 7 , Take x = 2. Then — (10 X 2 X 2 + 10 X 4 X 2 + 10 X 8 X 2) = — — 28 s. 10 440 „ x = 4. Then — (10X2X2 + 10X4X2 + 10X8X4) = — = 44 Take a; = 5. Then ^ (10 X 2 X 5 + 10 X 4 X 5 + 10 X 4 X 5) = ~ = 5 ° lbs. Loaded with a Rolling Weight.— Fig. 6. Fig. 6. Define parabola A c B as deter- W l mined by — = the ordinate at c, and vertical distances between A B will S> ve moments. & W x(l — x) . . . == M at any point. Loaded Uniformly its Entire Length.— Define parabola as at Fig. 6, ordinate of which at c = — — • L representing stationary or dead load per unit of length. L x (l — x) — M at any and = M at centre. Loaded with Two Connected .Weights, moving in either Direction, alike to a Locomo- tive or Car on a Railway. — Fig. 7. pig 7 e Define parabola A c B as deter- ( W - 4 - w) l mined by = c, At A and B erect Ae, B i = rv d rJl JL \ i connect A i and B e, and vertical A , v—'—+r d • di % 1 „„ — — ij? w <|pr ~ Y distances between A 0 B and A c B will give moments. w d at any point. Position of W at greatest moment , when x=-± 2 ( W -f«q | [(W + w) {l — x) — wd] = M Or if W and w are l . d equal, when x = - ± — “ 4 Illustration.— Assume x = 3, d = 4, and W w each 10 lbs., and 1 10 feet. Then A (w+75 X io— ^ - 1^X4) = M any point, as at W r, to r. MECHANICS. MOMENTS OF STRESS ON GIRDERS, ETC. 623 Shearing,. Stress. To Determine Shearing Stress at any Dart of a Girder or and under any Distribution of Load. Fl £- 8 * c Required to determine stress of a A T | i -R beam at any point as c, Fig. 8. /y A J f|F Assume W = load between A and S c, and w that between B and c. Then Sx at c = P — W, or P' — iv. The greater of the two values to be taken. S x representing shearing stress at any point x, P and P' the reaction on «/„««.•/« any poM to ‘“* m bmm between su PP orts , W dnd » loads or stress concentrated at 'i’o Describe and Ascertain Shearing Stress in a Girder or Beam. Loaded Uniformly. Fig. 9. At A and B, erect A c, B e, each equal to — . Connect c and e at middle of span as at n, and vertical distances between A B and cne will give shearing stresses as determined by the ordinates tocne. l r be disregarded. L representing distributed load per unit of length. Illustration. — Assume W = 10 lbs. per foot, l — 10, and x = 2.5 feet. Then 10 ^ — 2.5^ = 25 lbs. S. Sign of result to Note.— T he moment of rupture at any point, produced by several loads aetino- “=teSieiy. eqUa ‘ t0 th0 — AD^EfLondonTs^ Diagrams see Straius in GirdCTS > by William Humber, Operation deduced by Graphic Delineation of Greatest Stress, with a Uniformly Distributed Load of 4000 Lbs.— Fig. 10. Determine moment of weights by formulas.^?, 211 and lill l l ’ l IB Assume W = 7 ooo lbs. , w = 4000, and w' = 3000, m — j feet, n = 13 r — T 3; s — 7 i °— 3 , v=i 7 , and 1 = 20. 7000 X 13 X 7 — = 31 850, Then W = w — 4000 X 13 X 7 , 3000X3X17 2o — 18200, and w __ — — 7650, and let fall perpendic ulars thereto, as 3 d, 2 c, and 1 b. upo^perpendimilars frl?’ “n,f Um ° f distances of intersections of these lines these points ’ 3 ’ ’ 1 res P ectlveI y. will give stress upon A B at To determine Greatest Stress at Greatest Load. Stress at 3d =31850 I Stress at 1 b = i 7 ; 7650 ; 3 - x „ 0 “20=13:18200:7=9800 I / 3 — 1 350 7XX3X4000X.5 „ 43000 52 100 lbs., concentrated load at W, and proportion of uniformly distributed load of 4000 lbs 624 MECHANICAL POWERS. LEVER. MECHANICAL POWERS. Mechanical Power is a compound of Weight , or Force and Velocity: it cannot be increased by mechanical means. The Powers are three in number — viz., Leyer, Inclined Plane, and Pulley. >^ OTEi a Wheel and Axle is a continuous or revolving lever , a Wedge a double in- clined plane, and a Screw a revolving inclined plane. LEVER. Levers are straight, bent, curved, single, or compound. To Compute Lengtli of a Lever. When Weight and Power are given. Rule.— D ivide weight by power, and quotient' is leverage, or distance from fulcrum at which power supports weight. Or, ^ —p. w representing weight , P power , and p distance of power from fulcrum. Example.— A weight of 1600 lbs. is to be raised by a power or force of 80; re- quired length of longest arm of lever, shortest being 1 foot. 1600 -T- 80 = 20 feet. To Compute Weight tliat can He raised, "by- a Lever. When its Length , Power , and Position of its Fulcrum . are given. Rule.-— Multiply power by its distance from fulcrum, and divide product by dis- tance of weight from fulcrum. Or — = W. w representing distance of weight from fulcrum. ’ w Example.— What weight can be raised by 375 lbs. suspended from end of a lever 8 feet from fulcrum, distance of weight from fulcrum being 2 feet? 375 X^2= 1500 lbs. To Compute ^Position. of Fulcrum. When Weight and Poioer and Length of Lever are given , and when Ful- crum is between Weight and Power. Rule— D ivide weight by power, add 1 to quotient, and divide length by sum thus obtained. j, -4- _j- — w. L representing entire length of lever. Example.— A weight of 2460 lbs. is to be raised with a lever 7 feet long and a power of 300; at what x>art of lever must fulcrum be placed ? 2460 - 4 - 300 = 8. 2, and 8. 2 + 1 = 9. 2. Then (7 X 12) 84 9. 2 = 9. 13 ins. When Weight is between Fulcrum and Power. Rule.— Divide length by quotient of weight, divided by power. To Compute Length. of Arm of Lever to which Weight is attached. When Weight , Power , and Length of Arm of Lever to which Power is ap- plied are given. Rule. — Multiply power by length of arm to which it is applied, and divide product by weight. MECHANICAL POWERS. LEVER. Example. — A weight of 1600 lbs., suspended from a lever, is supported by a power of 80, applied at other end of arm, 20 feet in length ; what is length of arm ? 80 x 20-i- 1600 = 1 foot. Note.— These rules apply equally When fulcrum {or support) of lever is between weight and power ;* when fulcrum is at one extremity of lever, and power , or weight , at the other ; t and when arms of lever are equally or unequally bent or curved. To Compute Power Required. to Raise a given Weight. When Length of Lever and Position of Fulcrum are given. Rule. — Mul- tiply weight to be raised by its distance from fulcrum, and divide product by distance of power from fulcrum. . W w Or, = P. P Example — Length of a lever is 10 feet, weight to be raised is 3000 lbs., and its distance from fulcrum is 2 feet; what is power required? 3000 x 2 6000 Z ~S~ Z : 750 lbs. To Compute Length of Arm of Lever to -which. Lower is applied. When Weight , Power, and Distance of Fulcrum are given. Rule. — Mul- tiply weight by its distance from fulcrum, and divide product by power. „ W w Or, -p- =p. Example.— A weight of 400 lbs., suspended 15 ins. from fulcrum, is supported by a power of 50, applied at other; what is length of the arm ? 400 X 15 -r- 50= 120 ins. Fig. 1. 'yf are computed directly as ah to b c. When Arms of a Lever are bent or curved , Distances taken from perpendiculars, drawn from lines of direction of weight and power, must be measured on a line running horizon- c tally through fulcrum, as a b c, Figs. 1 and 2. When A rms of a Lever are at Right A ngles , and Power and Weight are applied at a Right Angle to each other , Fig. 3, The moments Fig. 2. Thrust, or press- ure on fulcrum, is in this case less than sum of pow- er and weight ; and it may be i determined by drawing a paral- lelogram upon the two arms of O ^S lever, arms repre- | w P senting inverse- ly their respec- tive forces. That is, a b represents magnitude and direction of weight W, and b e of power P. Diagonal ob of parallelogram represents magnitude and direction of third force, or thrust upon fulcrum. * Pressure upon fulcrum is equal to sum of weight and power, t Pressure upon fulcrum is equal to difference of weight and power. 3 G 626 MECHANICAL POWERS. LEVER. WHEEL. Fig. 4. When same Lever is bovne into an Oblique Position , Power continuing to act Horizontally, Fig. 4, Draw vertical a v through end 0 of lever, and produce the power line p c to meet it at a. Complete parallelogram avbr ; then sides r b and b v are perpendiculars to direc- tions to power and weight, on which moments are computed. P Consequently, moment P x r b . = moment W X a v, and a diag( nal, b a, is resultant thrust at fulcrum. Fig- 5- When Power does not act Horizon- tally, Fig. 5, but in some other direc- tion, a p , produce the power - line p a and draw b c perpendicular to it ; draw b 0 , then moments are computed on perpendiculars b c,b 0, and Pxci = \V x b o. If several weights or powers act upon one or both ends of a lever, con- dition of equilibrium is V p -J- P ' p' -j- P " p", etc., = W w -f- W' w\ etc. In a system of levers, either of similar, compound, or mixed kinds, condition is P p p' p" ^ Illustration. — Let P = 1 lb., p and p ' each 10 feet, p" 1 foot; and if w and 10' be each 1 foot, and w" 1 inch, then == 1200; that is, 1 lb. will support 1200, with levers 1 X 120 X 120 X 12 172 800 12 X 12 X 1 144 of the lengths above given. Note. — Weights of levers in above formulas are not considered, centre of gravity being assumed to be over fulcrums. * • • bli'PJ ’ J i 1 f ‘ ' t General Rule, therefore, for ascertaining relation of Power to Weight in a lever, whether straight or curved, is. Power multiplied by its distance from fulcrum is equal to weight multiplied ly its distance from fulcrum. 0r> p . w . . w . or P p = W w ; and W w p. p p — w. W w =zp. P p W z WHEEL AND AXLE. A Wheel and Axle is a revolving lever. Power, multiplied bv radius of wheel, is equal to weight, multiplied by radius of axle. As radius of wheel is to radius of axle, so is effect to power. R R P Or, P R = W r. Or,PV = Wv. Or,R:r::W;P. Or,P---W; -^- = r — R. R and r representing radii , and V and v velocities of wheel and axle . MECHANICAL POWERS. WHEEL AND AXLE. When a series of wheels and axles act upon each other, either by belts or teeth, weight or velocity will be to power or unity as product of radii, or circumferences of wheels, to product of radii, or circumferences of axles. Illustration.— If radii of a series of wheels are 9, 6, 9, 10, and 12, and their pin- ions have each a radius of 6 ins., and power applied is 10 lbs., what weight will they raise? 10 X 9 X 6 X 9 X 10 X 12 __ 583 200 _ 6X6X6X6X6 7776 ~ 75 S ' Or, if 1st wheel make 10 revolutions, last will make 75 in same time. To Compute Tower of a Combination of Wheels and an Axle or Axles, as in Cranes, etc. # Rule.— D ivide product of driven teeth by product of drivers, and quo- tient is their relative velocity ; which, multiplied by length of lever or arm and power applied to it in pounds, and divided by radius of barrel, will give weight that can be raised. v IV w r ^ r , ~ = W ; Or, W r = vl P ; Or, ~j — P. I representing length of lever or arm , r radius of barrel, P power, v velocity, and W weight. Example i. — A power of 18 lbs. is applied to lever or winch of a crane, length of it being 8 ins., pinion having 6 teeth, driving-wheel 72, and barrel 6 ins. diameter. = 12, and 12X8X18 = 1728, which, -1- 3, radius of barrel, = 576 lbs. 2 . — A weight of 94 tons is to be raised 360 feet in 15 minutes, by a power, velocity of which is 220 feet per minute; what is power required ? 360 -r- 15 = 24 feet per minute. Hence — X 94 == 10. 2545 tons. Compound -Axle, ox* Oliixiese "Windlass. Axle or drum of windlass consists of two parts, diameter of one being less than that of the other. The operation is thus : At a revolution of axle or drum, a portion of sus- taining rope or chain equal to circumference of larger axle is wound up, and at same time a portion equal to circumference of lesser axle is unwound. Effect, therefore, is to wind up or shorten rope or chain, by which a weight or stress is borne, by a length equal to difference between circumferences^of the two axles. Consequently, half that portion of the rope or chain will be shortened by half difference between circumferences. To Compute Elements of a Wlieel and Compound Axle, or Cliinese Windlass.— Eig. 6. Rule.— M ultiply power by radius of wheel, arm, or f,v 6 bar to which it is applied, and divide product by half ° difference of radii of axle, and quotient is weight that a i - can be sustained. • P _ PR TTT ^ 0r > ~ r 7 ) ~ W ‘ R representing radius of wheel, etc., and r and r' radii of axle at its greatest and least diameters. Example.— What weight can be raised by a capstan, radius of its bar a 5 feet, power applied 50 lbs., and radii, r r', of axle or drum 6 and 5 ins. ? ’ 50 X 5 X 12 • 5 ( 6 - 5 ) 3 °°° * 7, — — = 0000 lbs. ■5 628 MECHANICAL POWERS. INCLINED PLANE. "Wheel and Pinion Combinations, or Complex Wheel-work. Power, multiplied by product of radii or circumferences, or number of teeth of wheels, is equal to weight, multiplied by product of radii or circum- ferences, or number of teeth or leaves of pinions. Or, PRR' R", etc., == W r r' r", etc. Note. — Cogs on face of wheel are termed teeth , and those on surface of axle are termed leaves ; the axle itself in this case is termed a pinion. Liacli and Pinion. Xo Compute Power of a Ftachc and iPinion. Rule. — Multiply weight to be sustained by quotient of radius of pinion, divided by radius of crank, and product is power required. Or, W^- = P. li When Pinion on Crank Axle communicates with a Wheel and Pinion. Rule. — Multiply weight to be sustained by quotient of product of radii of pinions, divided by radii of crank and wheel, and product is power required. r r' °r,W in? = P. Example. — If radii of pinions of a jack-screw are each one inch; of crank and wheel io and 5 ins. ; what power will sustain a weight of 750 lbs. ? 1 X 1 750 750 X — — - = = 15 Ms. 10X5 50 INCLINED PLANE. Xo Compute Length, of Base, Height, or Length. When any Two of them are given , and ivhen Line o f Direction of Power or Traction is Parallel to Face of Plane. — Proceed as in Mensuration or Trigonometry to determine side of a right-angled triangle, any two of three being given. Xo Compute Power necessary- to Support a "Weight on an Inclined. Plane. When Height and Length are given. Rule. — Multiply weight by height of plane, and divide product by length. W h Or, — — == P. h and l representing height and length of plane. Example. — What is power necessary to support 1000 lbs. on an inclined plane 4 feet in height and 6 feet in length? IOOO x 4 -r- 6 = 666.67 lbs. Xo Compute "Weight that may he Sustained by a given Lower on an Inclined Plane. When Height and Length of Plane are given. Rule. — Multiply power by length of plane, and divide product by height. Example.— What is weight that can be sustained on an inclined plane 5 feet in height and 7 feet in length by a power of 700 lbs. ? 700 X 7-^5 = 980 lbs. Note. — In estimating power required to overcome resistance of a body being drawn up or supported upon an inclined plane, and contrariwise, if body is de- scending; weight of body, in proportion of power of plane (i. e., as its length to its height), must be added to resistance , if being drawn up or supported, or to the mo- ment if descending. MECHANICS. INCLINED PLANE. To Compute Heiglit or Length of an Inclined. Plane. When Weight, and Power ancl one of required Elements are given , and when Height is required. Rule.— M ultiply power by length, and divide product by weight. When Length is required. Rule.— M ultiply weight by height, and divide product by power. ^ PI . . Wh , Or, — = h, and — p- — /. To Compute Pressure on an Inclined Plane. Rule. — M ultiply weight by length of base of plane, and divide product bv length of face. A W6 Or, — — —pressure. b representing length of base of plane. Example.— Weight on an inclined plane is ioo lbs., base of plane is 4 feet, and length of it 5; required pressure on plane. 100 X 4 -y* 5 = 80 lbs . When Two Bodies on Two Inclined Planes sustain each other , as by Connection of a Cord over a Pulley , their Weights are directly as Lengths of Planes. Illustration. — If a weight of 50 lbs. upon an inclined plane, of 10 feet rise in 100 of an inclination, is sustained by a weight on another plane of 10 feet rise in 90, what is the weight of the latter ? 100 : 90 : : 50 : 45 = weight that on shortest plane would sustain that on largest When a Body is Supported by Two Planes , as Fig. 7, pressure upon them 7 will be reciprocally as sines of inclinations of planes. D Thus, Aveight is as sin. A B D. A Pressure on A B as sin. 1 ) B i. Pressure on B D as sin. A B h. Assume angle A B D to be 90 0 , and D B i, 6o°; then angle A B h will be 30 0 ; and as sines of 90 0 , 6o°, and 30 0 are respec- tively .1, .866, and .5, if weight-- 100 lbs., then pressures on A B and B D will be 86.6 and 50 lbs., centre of gravity of weight assumed to be in its centre. When Line of Direction of Power is parallel to Base of Plane , power is to weight as height of plane to length of its base. Or, P : W :: h : b. Hence, P = W h P b ~h W = ; h = P h 7 W h 6 =-r- When Line of Direction of Power is neither parallel to Face of Plane nor to its Base , but in some other Direction , as P', Fig. 8, power is to weight as sine of angle of plane’s elevation to cosine of angle which line of power or traction describes with face of plane. Thus, P' : W : : sin. A : cos. P' e c. Sin. A : cos. P' e c : : P' : W. Cos. P ' e c : sin. A : : W : P'. Illustration. — A weight of 500 lbs. is required to be sustained on a plane, angle of elevation of which, c AB, is io°; line of direction of power or traction, P' e c, is 5 0 ; what is sustaining power required? Cos. P'ec (5 0 ) = . 996 19 : sin. A (io°) = . 173 65 : : 500 : 87. 16 lbs. Or, draw a line, B s, perpendicular to direction of power’s action from end of base line (at back of plane), and intersection of this line on length, A c. will determine length and height (n r) of the plane. 3 G* n B 630 MECHANICS. WEDGE. — SCREW. Illustration. — By Trigonometry (page 385), A B, assumed to be 1, A r and nr are = .985 and .171. Note.— When line of direction of power is parallel to plane, power is least. 1. When One Body is to be Forced or Sustained. Rule.— Multiply weight or resistance to be sustained by depth of back of wedge, and divide product I by length of its base. Example. — What power, applied to the back of a wedge 6 ins. deep, will raise a weight of 15000 lbs., the wedge being 100 ins. long on its base? 2. When Two Bodies or Two Parts of a Body are Forced or Sustained in a Direction Parallel to Bach of Wedge. Rule.— Multiply weight or resist- ance to be sustained by half depth of back of wedge, and divide product by length of wedge. Note. — The length of a single wedge is measured on its base, and of a double wedge, from centre of its head to its point. Example. — The depth of the back of a double-faced wedge is 6 ins., and the length of it through the middle 10; what power applied to it is necessary to sus- tain or overcome a resistance of 150 lbs. ? Note. — A s power of wedge in practice depends upon split or rift in wood to be cleft, or in rise of body to be raised, the above rules as regards length of wedge are only theoretical when a rift or rise exists. A Screw is a revolving inclined plane. To Compute Xj&ngtli and. Tleiglit of Plane of a Screw. As a screw is an inclined plane wound around a cylinder, length of plane is ascertained by adding square of circumference of screw to square of dis- tance between threads, and taking square root of sum. The Pitch or height of a screw is distance between its consecutive threads. To Compute Power. Rule. — M ultiply weight or resistance, to be sustained by pitch of threads, and divide product by circumference described by power. Example.— What is power requisite to raise a weight of 8000 lbs. by a screw of 12 Hence 500 X .171 • 9 8 5 = 86.8 lbs. = product of weight X height of plane -r- length of it. "W^edge. A Wedge is a double inclined plane. To Compute Power. 15 000X 6 QOOQO — = = 900 lbs. 100 100 Or, ^ d ' - - = P. d representing depth of back , and l length. ,50x6^450^.,^ 10 To Compute Elements of a Wedge. SCREW. ins. circumference and 1 inch pitch? 8000 X 1 12 = 666.66 lbs. To Compute 'Wei glut. Rule.— M ultiply power by circumference described by it, and divide product by pitch of threads. Or,— = W. P To Compute IPitcli. Rule.— M ultiply power by circumference described by it, and divide product by weight. To Compute Circumference. Rule.— M ultiply weight by pitch, and divide product by power. „ W p Wp ur j — c - Or , - p .= r. r representing radius. When Power is applied by a Lever or Wheel , substitute radius of power for circumference. Illustration.— I f a lever 30 ins. in length was added to circumference of screw in preceding example, Then, 12-7-3.416 = 3.819, and ^-^ + 30 = 31.9095 — radius of power. TT 8000 X 1 Hence Compound Screw. W : P : : R n : r n. Or, r n : R n : : P : W. n representing continued product of number of wheels or axles. Illustration.— If a power of 150 lbs. is applied to a crank of 20 ins. radius turn- ing an endless screw with a pitch of half an inch, geared to a wheel pinion of which is geared to another wheel, and pinion of second wheel is geared to a third wheel, to axle or barrel of which is suspended a weight; it is required to know what weight can be sustained in that position, diameter of wheels beinsr and pinions and axle 2 ins. 8 f 150X20X2X3-1416 ... "7 — = 37 699-2 lb s. = power applied to face of first wheel. Diameters of wheels and pinions being 18 and 2, their radii are 9 and 1 . Hence, 1 X 1 X 1 : 9 X 9 X 9 :*• 37 699.2 : 27 482 716.8 lbs. 632 MECHANICS. SCREW. PULLEY. Differential Screw. When a hollow screw revolves upon one of less diameter and pitch (as designed by Mr. Hunter), effect is same as that of a single screw, in which the distance between threads is equal to difference of distances between threads of the two screws. Therefore power, to effect or weight sustained, is as difference between distances of threads of the two screws to circumference described by power. Illustration. — If external screw has 20 threads, and internal one 21 threads in pitch of 1 inch, and power applied describes a circumference of 35 ins., the result or power is as — co — = , or .002 38. Hence — = 14 706. 1 21 20 420 .00230 PULLEY. Pulleys are designated as Fixed and Movable , according as cord is passed over a fixed or a movable pulley. A movable pulley is when cord passes through a second pulley or block in suspension ; a single movable pulley is termed a runner ; and a combination of pulleys is termed a system of pulleys. A Whip is a single cord over a fixed pulley. To Compute Dower Required to Raise a given Weiglit. When Number of Parts of Cord supporting Lower Bloch are given , and when only one Cord or Rope is used. Rule. — Divide weight to be raised by number of parts of cord supporting lower or movable block. Or, w -4- n — P. Or, n P = W. n representing number of parts of cord sustain- ing lower block. Example. — What power is required to raise 600 lbs. when lower block contains six sheaves? When Cord is attached to Upper or Fixed Bloch. — — go lbs. = weight -4- number of parts of rope sustaining lower block. 6X2 When Cord is attached to Lower or Movable Bloch. - 46.15 lbs. =a weight -7- number of parts of rope sustaining lower block. 600 6X2 + 1 To Compute Weiglit a given Dower will Raise. When Number of Parts of Cord supporting Lower Bloch are given. Rule. — Multiply power by number of parts of cord supporting lower block. Or, P n = W. To Compute 1ST umber of Cords necessary to Sustain Lower Block. When Weight and Power are given. Or, W -. Rule. — D ivide weight by power. -P = «. Fig. 10. When more than one Coifl is used. In a Spanish Burton , Fig. 10, where ends of one cord, a P, are fastened to support and power, and ends of the other, c 0, to lower and upper blocks, weight is to power as 4 to 1. In another, Fig. 11, where there are two cords, a and 0, two movable pulleys, and one fixed pulley, with ends of one rope fastened to sup- port and upper movable pulley, and ends of other fastened to lower block and power, weight is to power as 5 to 1. Fig. 12. Compound or Fast and Loose Fallens. When Cord is attached to Fixed Block, Fig. 12. Rule. — Multiply power by the power of 2, of which the index is number of movable pulleys. Or, P 2 * = W. Or, Multiply power successively by 2 for each pulley. Example i. — What weight will one pound support in a system of three movable pulleys, the cords being connected to a fixed block on Fig. 12. .. , 0 7 , 1 X 23 — 8 lbs. Example 2.— What would a like power support, fixed block be- ing made movable and cord attached thereto? 1 X 2* — 1 = 15 lbs. If fixed pulleys were substituted for hooks a b c, Fig. 12, power would be increased threefold; hence 1 x 33 = 27. In a System of Pulleys, Figs. 13 and 14, with any Number of Cords , 0 0, e e y Ends being fastened to Support. Fig. 13. W-f- 2 n = P; W “XP = W; ~=z 2 n . n rep- Fig. 14. resenting number of distinct cords. Illustration. — What weight will a power of 1 lb. sustain in a system of two movable pul- leys and two cords ? 1 X 2 X 2 = 4 lbs. When fixed Pulleys, e e, are used in Place p of Hooks, to Attach Ends of Rope to Sup- port. — Fig. 14. W-h 3 ” = P; 3 n X P = W; W=P = 3 « g Illustration. -What weight will a power of 5 lbs. sustain with two movable and three fixed pulle5 r s, and two cords? * v v 5 A 3 A 3 — 45 When Ends of Cord or Fixed Pulleys are fastened to Weight , as by an Inver- sion of the last Figures, putting Supports for Weights, and contrariwise. — Jb igs. 13 and 14. _. W w 7 T »_ - .» = p ; (2*-x)P = W; - = (2^-1). Fig. 14. (2* -I) w = P: ( 3 «-i)P = W; P W W=(3 w -i). ( 3 n ~ 1) Illustration —What weight will a power of 1 lb. sustain in a system of two mov- able pulleys and two cords, and one of two movable and two fixed pulleys and two 1X2X2- 1 = 0 lbs. 1 X 3 X 3 — 1 = 8 lbs. When Cords sustaining Pulleys are not in a Vertical Direction.— Fig. 15. Fig - x 5 - Fig. 15, is vertical line through which weight bears and from 0 draw or, os parallel to D e and A e. ; Forces acting at e are represented by lines e s, e r, and e o • and as tension of every part of cord is same, and equal to power P. sides o s and 0 r of parallelogram must be equal and ■ therefore diagonal e 0 divides the angle ros into two equal dk portions Hence the weight will always fall into the position V m which the two parts of cord A e and e D will be equally P inclined to vertical line, and it will bear to power same ratio aseotoei Therefore W : P : : 2 cos. .50:1. e representing angle A e D. P *„ C0S 7 5 p = W ' That J s ’ twice P° wer - multiplied by cosine of half angle of cord, at point of suspension of weight, is equal to weight. ° 634 METALS. — ALLOYS AND COMPOSITIONS. Illustration —What weight will be sustained by a power of 5 lbs., with an ob- lique movable pulley, Fig. 15, having an angle, A e D, of 30 0 ? 5 X 2 X -965 93 = 9-6593 lbs. = twice power X cos. 15 0 . When Direction of Cord is Irregular , Weight not resting in Centre of it. p sm. a W ““ sin. (a 4- b) ’ P sin. w . sin. a greater and lesser angles of cord at e. W sin. a sin. (a-f- b) — P. a and b representing METALS. ALLOYS AND COMPOSITIONS. A-lloy- is the proportion of a baser metal mixed with a finer or purer, as copper is mixed with gold, etc. Amalgam is a compound of Mercury and a metal— a soft alloy. Compositions of copper contract in admixture, and all Amalgams ex- pand. In manufacture of Alloys and Compositions, the less fusible metals should be melted first. In Compositions of Brass, as proportion of Zinc is increased, so is malleability decreased. Tenacity of Brass is impaired by addition of Lead or Tin. Steel alloyed with one five-hundredth part of Platinum, or Silver, is rendered harder, more malleable, and better adapted for cutting instru- ments. _ Specific gravity of alloys* does not follow the ratios of those of their components ; it is sometimes greater and sometimes less than the mean. Composition Tor "Welding Cast Steel. Borax qi parts; Sal-ammoniac, 9 parts. Grind or pound them roughly together; fufe them in a metal-pot over a clear fire, continuing heat until a 1 spume has dis- appeared from surface. When liquid is clear, pour composition out to cool and con- crete, and grind to a fine powder; then it is ready for use. To use this composition, the steel to be welded should be raised to a bright yellow heat then dip it in the welding powder, and again raise it to a like heat as before, it is then ready to be submitted to the hammer. ITxisible Compounds. Compounds. Rose’s, fusing at 200 0 Fusing at less than 200 0 Newton’s, fusing at less than 212 0 . Fusing at 150 0 to 160 0 33-3 Tin. 25 19 Lead. 23 33-3 3 1 25 Bismuth. 50 33-4 50 50 Cadmium. Solders. Solder is an alloy used to make joints between metals, and it must be more fusible than the metals it is designed to unite, and it is distinguished as hard and soft, according to the temperature of its fusing. The addition of a small portion o f Bismuth increases its fusibility. . For a table of Alloys, having densities different from a mean of their components, see D. K. Clark’s Manual, London, 1877, page 201. METALS. — ALLOYS AND COMPOSITIONS. 635 .Alloys and. Compositions. Copper. Zinc. Argentan 55 24 Aluminum, brown 95 — Babbitt’s metal * 3-7 — Brass, common 84-3 5-2 u 44 75 25 “ u hard 79-3 6.4 “ instruments 92.2 — “ locomot. bearings. 90 I 44 Pinchbeck 80 20 “ red Tombac 88.8 II. 2 “ rolled 74-3 22.3 “ Tutenag 50 31 11 very tenacious... 88.9 2.8 “ wheels, valves. .. . 9° — “ white 10 80 “ “ 3 90 il wire 7 — 67 33 11 yellow, fine 66 34 Britannia metal — — When fused add — — Bronze, red n rr 87 13 86 11. 1 “ yellow 67.2 31.2 “ Gun metal, large 9° “ 44 small 93 — “ 44 soft. 95 — “ Cvmbals 80 — “ Medals 93 — 44 Statuary 9 J, 4 5-5 Chinese silver 58.1 17.2 4i white copper. . . 40.4 25-4 Church bells 80 5-6 44 44 69 Clocks, Musical bells 87-5 — Clock bells 72 — German silver 33-3 33-4 “ 44 40.4 25-4 44 “ fine 49-5 24 Gongs 81.6 House bells 77 — Lathe bushes 80 ' Machinery bearings “ 44 hard. Metal that expands in) 87-5 — 77-4 7 cooling ) Muntz metal, 10 oz. lead. 60 40 Pewter, best — 44 Sheathing metal 56 45 Speculum “ 66 50 21 Telescopic mirrors 66.6 — Temper t 33-4 — Type metal and stereo- ) — type plates ] — — White metal 7-4 7-4 “ “ hard..'.,...,. 69.8 25-8 Oreide • 73, 12.3 Tin. Nickel. Lead. mony. muth. minum. — 21 _ — — — — — — — — 5 89 — 7-3 — 10.5 — — — — — — — — — — — 14- 3 — — ■ — — — 7.8 — — — — — 9 — — — ’ — — — — — — — — — — — — — 3-4 — — — — — — 19 — — — — 8-3 — — — — — 10 — — — — — 10 — — — — — — — — 7 — — — — 46 47 — — — — ' — — — — . — — — — — — 25 — — 25 — — — — — 25 25 — — — ■ — — •' — — 2.9 — — — ' — — 1.6 — — — : — c 10 — — — ' — 0 7 — — 0 5 — — — ' — ' £ 20 — — — 1 7 — — — > 0 1.4 • — i-7 — 02 — — 11. 6 — — 2 II. I 2.6 31.6 — — — — 10. 1 — 4-3 — — c 3i — — — — £ 12.5 — — — — — 26.5 — — •~T- — 1-5 — 33-3 — — — — — 31.6 — — , 2.6 — 24 — — — 2 -5: 18.4 — — — 23 — — _ • rO — . ■ 20 — — — 3 — 12.5 — — — a — 15-6 — — — S — — — 75 16.7 8.3 — — ,g 86 — — 14 — B 80 — 20 — — < — — — — — — 22 — — ' — 12 29 — — — — — 33-4 — — — — — 66.6 — — — — — , — 75 25 — — — — 87-5 12-5 — — 28.4 — 56.8 — — 4.4 — — — — Magnesia 4.4 Cream of tartar .6.5 Sal-ammoniac . 2.5 Quicklime .1.3 See page 636 for directions. t For adding small quantities of copper. METALS. ALLOYS AND COMPOSITIONS. Solders. Copper. Tin. Lead. Zinc. Silver. Bis- muth. Gold. Cad- mium. Anti- mony. Tin 25 75 — — 16 — — — “ — 58 16 — — — — 10 “ coarse, melts ) at 500 0 . . . j - 33 67 - - - - - — ordi’y, melts) at 360° . . . j - 67 33 - - — ■ — Spelter, soft “ hard So — — 50 — — — — — 65 — — 35 — ■ — — — — Lead 33 67 — 82 — — — — Steel 13 — 5 — — — — Brass or Copper . . . 50 — — 50 — — — — — Fine brass 47 — — 47 6 — — — — Pewterers’ or Soft . 33 45 — — ' 22 — — — ’ “ “ . — 50 25 — — 25 — — — Plumbers’ pot- 1 metal. . j - 33 67 - - - - — - “ coarse — 25 75 — — — — — — “ fine — 67 33. — — — — — — “ fusible... — • 50 50 — — — — — — ' “ very “ ... — 25 25 — — 50 89 — — Gold .4 — — — 7 — — — “ hard 66 — — 34 — — — — — “ soft — 66 34 — — — — — — Silver, hard 20 — — 80 — — — — “ soft 12 — — — 67 — — 21 — Pewter — 40 20 — 40 — — — Iron 66 — — j 33 — — — — 1 Copper 53 47 — — — — — — A Plastic Metallic Alloy.— See Journal of Franklin Institute, vol. xxxix., page 55, for its composition and manufacture. Soldering Fluid for use with Soft Solder. To 2 fluid oz. of Muriatic acid add small pieces of Zinc until bubbles cease to rise. Add .5 a teaspoonful of Sal-ammoniac and two fluid oz. of Water. By the application of this to Iron or Steel, they may be soldered without their sur- faces being previously tinned. Fluxes for Soldering or Welding. Iron Borax. Tinned iron Resin. Copper and Brass Sal-ammoniac. Zinc Chloride of zinc. Lead Tallow or resin. Lead and tin Resin and sweet oil. Baiybitt’s .Anti-attrition Metal. Melt 4 lbs. Copper; add by degrees 12 lbs. best Banca tin, 8 lbs. Regulus of anti- mony, and 12 lbs. more of Tin. After 4 or 5 lbs. Tin have been added, reduce heat to a dull red, then add remainder of metal as above. This composition is termed hardening ; for lining, take 1 lb. of this hardening , melt with it 2 lbs. Banca tin, which produces the lining metal for use. Hence, the proportions for lining metal are 4 lbs. of copper, 8 of regulus of antimony, and 96 of tin. Brass. Brass is an alloy of copper and zinc, in proportions varying with purpose of metal required, its color depending upon the proportions. It is rendered brittle by continued impacts, more malleable than copper when cold, but is impracticable of being forged, as its zinc melts at a low temperature. . _ . „ ... . Its fusibility is governed by its proportion of zinc ; a small quantity ol phosphorus gives it fluidity. METALS. — ALLOYS AND COMPOSITIONS. — IRON. 637 Bronze. Bronze is an alloy of copper and tin ; it is harder, more fusible, and stronger than copper. It is usually known as Gun-metal . Aluminum Bronze contains 90 to 95 per cent, of copper, and 5 to 10 per cent, aluminum. Phosphor Bronze contains copper and tin and a small proportion of phos- phorus. It wears better than bronze. IRON. Foreign substances which iron contains modify its essential proper- ties. Carbon adds to its hardness, but destroys some of its qualities, | and produces Cast Iron or Steel, according to proportion it contains. Thus, .25 per cent, renders it malleable, .5 steel, 1.75 is limit of weld- ing steel, and 2 is lowest limit of cast iron. Sulphur renders it fusible, difficult to weld, and brittle when heated, or “ hot short.” Phosphorus renders it “cold short” but may be present in proportion of .002 to .003, without affecting injuriously its tenacity. Antimony, Arsenic, and Copper have. same effect as sulphur, the last in a greater degree. Sili- con renders it hard and brittle. Manganese, in proportion of .02, ren- ders it “ cold short” and Vanadium adds to its ductility. Cast Iron. Process of making Cast Iron depends much upon description of fuel used ; whether charcoal, coke, bituminous, or anthracite coals. A larger yield from same furnace, and a great economy in fuel, are effected by use of a hot blast. The greater heat thus produced causes the iron to combine with a larger percentage of foreign substances. Cast Iron for. purposes requiring great strength should be smelted with a cold blast Pig-iron, according to proportion of carbon which it contains, is di\ ided into Foundry Iron and Forge Iron , latter adapted only to conver- sion into malleable iron ; while former, containing largest proportion of car- bon, can be used either for castings or bars. High temperature in melting injures gun-metal. There are many varieties, of . Cast Iron, differing by almost insensible shades ; the two principal divisions are gray and white, so termed from color of their fracture. Their properties are very different. Gray Iron is > softer and less brittle than white; it is in a slight degree malleable and flexible, and is insonorous ; it can easily be drilled or turned and does not resist the file. It has a brilliant fracture, of a gray, or some- times a bluish-gray, color; color is lighter as grain becomes closer, and its hardness increases. It melts at a lower heat than white, and preserves its flmdity longer. Color of the fluid metal is red, and deeper in proportion as the heat is lower; it does not adhere to the ladle; it fills molds well, con- tracts less, and contains fewer cavities than white; edges of its castings are sharp, and surfaces smooth and convex. It is used for machinery and ordnance where the pieces are to be bored or fitted. Its tenacity and specific gravity are diminished by annealing. ,,,2/''™ I s . very brittle and sonorons; it resists file and chisel, and is susceptible of high polish ; surface of its castings is concave ; fracture pre- s , ry appearance, generally fine grained and compact, sometimes radiating or lamellar. When melted it is white, throws off a great number of sparks, and its qualities are the reverse of those of grav iron ; it is there- in for machinery purposes. Its tenacity is' increased, and its specific gravity diminished, by annealing. 3 H METALS. — IRON. 638 Mottled Iron is a mixture of white and gray ; it has a spotted appear- ance; flows well, and with few sparks; its castings have a plane surface, with edges slightly rounded. It is suitable for shot, shells, etc. A fine mot- tled is only kind suitable for castings which require great strength. The kind of mottle will depend much upon volume of the casting. A medium- sized grain, bright gray color, fracture sharp to touch, and a close, compact texture, indicate a good quality of iron. A grain either very large or very small, a dull, earthy aspect, loose texture, dissimilar crystals mixed together, indicate an inferior quality. Besides these general divisions, the different. varieties of pig-iron are more particularly distinguished by numbers, according to their relative hardness. No. j —Fracture dark gray, crystals large and highly lustrous, alike to new surface of lead. It is the softest iron, possessing in highest degree the qualities belonging to gray iron ; it has not much strength, but on account of its fluidity when melted, and of its mixing advantageously with scrap iron and with the harder kinds of cast iron, it is of great use to a foundry. No. 2 is harder, closer grained, and stronger than No. 1 ; it has a gray color and considerable lustre. It is most suitable for shot and shells. No. 3 is harder than No. 2. Fracture white, crystals larger and brighter at centre than at the sides; color gray, but inclining. to white; has consid- erable strength, but is principally used for mixing with other kinds of iron and for large castings. No. 4 or Bright. — Fracture light gray, with small crystals and little lustre, and not being sufficiently fusible for castings it is used for conversion to wrought iron. No. 5. Mottled.' — Fracture dull white, with gray specks, and a line of white around edge or sides of fracture. No. 6. White. — Fracture white, with little lustre, granulated with radiat- ing crystalline surface. It is hardest and most brittle of all descriptions, and is*unfit for use unless mixed with other grades, cr for being converted to an inferior wrought iron. Qualities of these descriptions depend upon proportion of carbon, and upon state in which it exists in the metal ; in darker kinds of iron, where propor- tion is sometimes 7 per cent., it exists partly in state of graphite or plumbago, which makes the iron soft. In white iron the carbon is thoroughly com- bined with the metal, as in steel. Cast iron frequently retains a portion of foreign ingredients from the ore, such as earths or oxides of other metals, and sometimes sulphur and phos- phorus, which are all injurious to its quality. Foreign substances, and also a portion of the carbon, are separated by melting iron in contact with air, and soft iron is thus rendered harder and stronger. Effect of remelting varies with nature of the iron and character of ore from which it has been extracted ; that from hard ores, such as mag- netic oxides, undergoes less alteration than that from hematites, the latter being sometimes changed from No. 1 to white/ by a single remelting in an , air furnace. ' Color and texture of cast iron depend greatly upon volume of casting and . rapidity of its cooling ; a small casting, which cools quickly, is almost alwav s white , and surface of large castings partakes more of the qualities of white j metal than the interior. All cast iron expands at moment of becoming liquid, and contracts in cool- ing; gray iron expands more and contracts less than other iron. Remelting iron improves its tenacity; thus, a mean of 14 cases for two fusions gave, for 1st fusion, a tenacity of 29284 lbs.; for 2d fusion, 33 79° lbs. For two cases — for first fusion, 15 129 lbs. ; for 2d fusion, 35 7S6 lbs. METALS. — IEOX. 6 39 IVT a.llea'ble Castings. . Malleable cast iron is made by subjecting a casting to a process of anneal- in & by enclosing it m a box with hematite iron ore or black oxide of iron volume m a ° e< P ia ^ e * ieat ^ or a period depending upon form and Wrought Iron. Wrought iron is made from pig-iron in a Bloomeiy Fire or in a Puddllno Furnace — generally in latter. Process consists in melting and keeping it exposed to a great heat, constantly stirring the mass, bringing every part of it under action of the flame until it loses its remaining carbon! wh^it be- comes malleable iron- W hen, however, it is desired to obtain iron of best quality, pig-iron should be refined. ue&u Refining* —This operation deprives iron of a considerable portion of its carbon ; it is effected in a Blast Furnace, where iron is melted by means of charcoal or coke, and exposed for some time to action of a great heat • the metal is then run into a cast-iron mold, by which it is formed into a far>e Sftbir n " SUrfaCG ° f Pkte " Chmed ’ C ° Id Water is Po-edo n e A Bloomer l/ resembles a large forge fire, where charcoal and a strong blast are used ; and the refined metal or pig-iron, after being broken into pifces of 1S P J aced before the blast, directly in contact with charcoal • as the metal fuses, it falls into a cavity left for that purpose below the blast u here the bloomer works it into the shape of a ball , which he places ao ain before the blast, with fresh charcoal ; this operation is generally agaiif re- peated, when ball is ready for the “shingler.” b y ‘gam re Shingling is performed in a strong squeezer or under a trip-hammer. Its ho!f Ct *° Pr > SS , out / s P erfe ctly as practicable the liquid cinder which a ball contains ; it also forms a ball into shape for the puddle rolls A heavy hammer, we^hingfrem 6 to 7 tons, effects this object most Ihor^ughly^ t so cheaply as the squeezer. A ball receives from i s to 20 blows of a hammer, being turned from time to time as required: ft is now terned a Bloom, and is ready to be rolled or hammered ; or a ball is passed once ro[ig U ® 1 1G S( i ueezer ’ and is still hot enough to be passed through the puddle A Puddling Furnace is a reverberatory furnace, where flame of bituminous coal is brought to act directly upon the melted metal. The “ puddler ” then stirs it, exposing each portion in turn to action of flame, and continues this as ong as he is able to work it. When it has lost its fluidity, he forms i! into b t’- 18hlDg fr ° m 80 t0 IO ° lbs *’ which are then Passed ^ the “shingle/” Puddle Rolls.- By passing through different grooves in these rolls n bloom is reduced to a rough bar from% to 4 feet in length hi tlZ ™ ’ mg an idea of its condition, which is rough and imperfect.’ ° e " " n f ^W.-To prepare rough bars for this operation, they are cut by a pair of shears, into such lengths as are best adapted to the volume « , required ; the sheared bars are then piled' one over the Xr ■volume required, when pile is ready for balling. ° 1 to nnfe;r 7 h ‘ S ( P eration is performed in balling furnace, which is similar to tinm v F h rT’ v X “ Pt * h , at ltS • b ° ,t0m or 1,earth is made up, from tithe tearing. 11 f " UStd ‘° S ‘ Ve " W6lding h6at to pile " P-P-e Finishing Rolls — The balls are passed successively between rollers of va- ncus forms and dimensions, according to shape of finished bar required. u. in ', r0i . 1 depends upon description of pig-iron used skill of the ‘puddler,” and absence of deleterious substances in the furnace. 640 METALS.— IKOX. LEAD. STEEL. Strongest cast irons do not produce strongest malleable iron. For many purposes, such as sheets for tinning, best boiler-plates, and bars for converting into steel, charcoal iron is used exclusively ; and, generally, this kind of iron is to be relied upon, for strength and toughness, with greater confidence than any other, though iron of a superior quality is made from pigs made with other fuel, and with a hot blast. Iron for gun-barrels has been lately made from anthracite hot-blast pigs. Iron is improved in quality by judicious working, reheating hammering, or rolling : other things being equal, best iron is that which has been wrought the most. Best quality of iron has greatest elasticity. Tests. — It will not blacken if exposed to nitric acid. Long silky fibres in a fracture denote a soft and strong metal ; short black fibres denote a badly refined metal, and a fine grain denotes hardness and condition known as « co ltl short.” Coarse grain with bright and crystallized fracture, with dis- colored spots, also denotes “ cold short” and brittle metal, working easily and welding well. Cracks upon edges of a bar, etc., indicate hot short. Good iron heats readily, is worked easily, and throws off but few sparks.. A high breaking strain may not be conclusive as to quality, as it may be due to a hard, elastic metal, or a low one may be due to great softness. When iron is fractured suddenly, a crystalline surface, is produced, and when gradually, a fibrous one. Breaking strain of iron is increased by heat- ing it and suddenly cooling it in water. Iron exposed to a welding or white heat and not reduced by hammering or rolling is weakened. Specific gravity of iron is a good indication of its quality, as it indicates very correctly its relative degree of strength. LEAD. Sheet Lead is either Cast or Milled, , the former in sheets 16 to 18 feet in length and 6 feet in width ; the latter is rolled, is thinner than the former, is more uniform in its thickness, and is made into sheets 25 to 35 feet in length, and from 6 to 7.5 feet in width. Soft or Rain Water, when aerated, Silt of rivers, Vegetable matter, Acids Mortar and Vitiated Air will oxidize lead. The waters which act with greatest effect on it are the purest and most highly oxygenated, also nitrites, nitrates, and chlorides, and those which act with least effect are such as con- tain carbonate and phosphate of lime. Coating of Pipes , except with substances insoluble in water, as Bitumen and Sulphide of lead, is objectionable. Lead-encased Pipes.— An inner pipe of tin is encased in one of lead. STEEL. Steel is a compound of Iron and Carbon, in which proportion of latter is from 1 to 5 per cent, and even less in some descriptions It is dis- tinguished from iron by its fine grain, and by action of diluted nitric acid, which leaves a black spot upon it. There are many varieties of steel, principal of which are : Natural Steel , obtained by reducing rich and pure descriptions of iron ore with charcoal, and refining cast iron so as t0 de P n y® ll f ^ fiks and portion of carbon to bring it to a malleable state. It is used tor hies an other tools. , . . « Indian Steel , termed Wootz, is said to be a natural steel, containing a small portion of other metals. METALS. STEEL. 641 . Blistered Steel, ov Steel of Cementat ion, is prepared by direct combination of iron and carbon. For this purpose, iron in bars is put in layers, alternating with powdered charcoal, m a close furnace, and exposed for 7 or 8 days to a high temperature, and then put to cool for a like period. The bars, on being taken out, are covered with blisters, have acquired a brittle quality, and exhibit in fracture a uniform crystalline appearance. The degree of carbonization is varied according to purposes for which the steel is intended and the very best qualities of iron are used for the finest kinds of steel. Tilted Steel is made from blistered steel moderately heated, and subjected creased 11 ° f * ^ hatnmer ’ b - v which means its tenacity and density are in- Shear Steel is made from blistered or natural steel, refined by piling thin bars into fagots, which are brought to a welding heat in a reverberatorv furnace, and hammered or rolled again into bars; this operation is repeated several times to produce finest kinds of shear steel, which are distinguished by the terms of Half shear, Single shear, and Double shear, or steel of i 2 or 3 mans , etc., according to number of times it has been piled. hammered^' * S blister steel beated to al1 orange red color and rolled or Cr ™V e S ' eel is made h y breaking blistered steel into small pieces and melting it m close crucibles, from which it is poured into iron molds; Lmrt rf i'ri red l' c ® d , t0 a bar , b >' hammering or rolling. Cast steel is best kind of steel, and best adapted for most pu. posts; it is known by a very fine, er en, and close grain, and a silvery, homogeneous fracture; it is very of affi.v ff'v f tr |T bardness ’ bl,t is fMfflcult to weld without use 0t k i"l ds °l stee have a Slmilar appearance to cast steel, but gTam is coarser and less homogeneous : they arc softer and less brittle, and weld more read.lv. A fibrous or lamellar appearance in fracture indicates hnrdne^? 1 St T j ^ uatmal of K«‘ a t toughness and elasticity, as well as s ' ,adc b y. f ° r P "8 together steel and iron, forming the celebrated Damasked Steel, which is used for sword-blades, springs, etc. ; damask an- themed while th* produce ? b y adUuted acid , which gives a black tint to tne steel, while the iron remains white. with diem St Stee1 ’ kreakm £ 8tren gth is greater across fibres of rolling than P‘? C€SS j f, an improvement on this method, and consists in addins to molten metal a small quantity of carburet of manganese ® r e move > carlTOn <) and sulca!* 8 addi ” S Ditrate ° f soda t0 mo,te “ i«* order ,0 anf£coaV° CSM:_ “ alleabIe ir ° n iS meUed in crucib,es with oxide of manganese Puddled Steel is produced by arresting the puddling in the manufacture of the wrought iron before all the carbon has been removed, the small State, toL 1 remammg ’ - 3 t0 1 per cent -’ be “g sufficient to make an enUtiZ iTll'Z'sZ : 2 t0 ' 5 Per Cent ° f Carb ° n; Whe “ “ Fres- in ordctT S u e - iS made direbt f rom P'> iron - The car bon is first removed, in order to obtain pure wrought iron, and to this is added the exact quantity of carbon required for the steel. The pig should be free from sulphur and phosphorus It is melted in a blast or cSpola, and run into a Zl a pear-shaped iron vessel suspended on hollow trunnions and lined with fire- bnck or clay) where it is subjected to an air blast for a period of lo mtol gelds^ is adde°d SP ® “ ° n ’ after which fro; " 3 t0 P« cent, of spie- 3 H* METALS. — STEEL. The blast is then resumed for a short period, to incorporate the two metals, when the steel is run off into molds. The moment at which all the carbon has been removed is indicated by color of the flame at mouth of conv erter. The ingots, when thus produced, contain air holes, and it becomes necessary to heat them and render them solid under a hammer. Siemens Process.— Pig-iron is fused upon open hearth of a regenerative furnace, and when raised to a steel-melting temperature, rich and pure ore and limestone are added gradually, whereby a reaction is established between the oxvgen of the ferrous oxide and the carbon and silicon m the metal. Ihe silicon" is thus converted into silicic aild, which with the lime forms a fusible slag, and the carbon, combining with oxygen, escapes as carbonic acid, and induces a powerful ebullition. MnrNfi ration of this process. — The ore is treated in a separate rotatory furnace with carbonaceous material, and converted into balls of malleable iron, which are transferred from the rotatory to the bath of the steel-melting furnace. process is adapted to the production of steel of a very high quality, because th^SSKd ptoptoS of the ore are separated from the metal in the rotatory furnace. Siemens -Martin Process.- Scrap-iron or steel is highly heated condition to a bath of about .25 its weight, of highly heated pig and melted. Samples are occasionally taken from the bath, in order to ascertain the percentage of carbon remaining in * he } S a W d in small quantities, in order to reduce the carbon to about .1 per cent. At this stage of the process, siliceous iron, spiegeleisen, or ferro-manganese is added in sSch proportions as are necessary to produce steel of the required degree of hardness. The metal is then tapped into a ladle. Landore-Siemen’s Steel is a variety of steel made by the A^cation of Siemen’s Process. Its great value is due to its extreme ductility, and its having nearly like strength in both directions ot its plates. Whitworth'* Compressed Steel is molten steel subjected to a pressure of about 6 tons per square inch, by which all its cavities are dispelled, and it is compressed to about .875 of its original volume, its density and strength be- ing proportionately increased. Chrome and Tungsten Steel are made by adding a small percentage of Chromium or Tungsten to crucible steel, the result producing a steel of great hardness and tenacity, suitable for tools, such as drills, etc. Homogeneous Steel is a variety of cast steel containing .25 per cent, of carbon. Remarks on Manufacture of Steel , and Mode of Working it. (D. Chernoff ’ 1868 ). Steel when cast and allowed to cool quietly, assumes a . crj-stalline structure Higher temperature to which it is heated, softer it becomes, and gieater is libeity its particles possess to group themselves into crystals. . Steel however hard it may be, will not harden if heated to a temperature | lower than what may be distinguished as dark cherry-red, a, however quickly it is cooled, on contrary, it will become sensibly softer, and more easily woi ked w ith a tile. Steel heated to a temperature lower than red, but not sparkling, 6, does not chamre its structure whether cooled quickly or slowly. When temperature has reached b substance of steel quickly passes from granular or crystalline condition to amorphous, or wax-like structure, which it retains up to its melting-point, c. ^ Points a b and c have no permanent place in scale of temperature, but their posi tions vary’ with quality of steel; in pure steel, they depend directly on quantity of constituent^ carbon Harder the steel, lower the temperatures Tints above speci- fied have reference only to hard and medium qualities of steel; in very soft kinds of steel, nearly approaching to wrought iron, points a and b range very high, and m wrought iron point b rises to a white heat. METALS. — STEEL. 643 Assumption of the crystalline structure takes place entirely in cooling, between temperatures c and 6; when temperature sinks below b there is no change of struc- ture. For successful forging, therefore, heated ingot, after it is taken out of furnace must be forged as quickly as practicable, so as not to leave any spot untouched by hammer, where the steel might crystallize quietly, as formation of crystals should be hindered, and the steel should be kept in an amorphous condition until tem- perature sinks below point b. Below this temperature, if piece is cooled in quiet, mass will no longer be disposed to crystallize, but will possess great tenacity and homogeneousness of structure When steel is forged at temperatures lower than 6, its crystals or grains bein°- driven against each other, change their shapes, becoming elongated in one direction 0 and contracted in another; while density and tensile strength are considerably in- creased. But available hammer-power is only sufficient for treatment of small steel forgings; and object of preventing coarse crystalline structure in large forgings is more easily and more certainly effected, if, after having given forging desired shape, its structure be altered to an homogeneous amorphous condition by heating it to a temperature somewhat higher than 6, and the condition be fixed by rapid coo mg to a temperature lower than 6, the piece should then be allowed to finish cooling gradually, so as to prevent, as far as practicable, internal strains due to sudden and unequal contraction. Alloys of steel with Silver , Platinum , Rhodium , and Aluminum have been made with a view to imitating Damascus steel, Wootz, etc., and improving fabrication of some finer kinds of surgical and other instruments. Properties of Steel.— After being tempered it is not easily broken; it welds readily; does not crack or split; bears a very high heat/and preserves the capability of hardening after repeated working. Hardening and Tempering.— Upon these operations the quality of manu- factured steel in a great measure depends. Hardening is effected by heating steel to a cherry-red, or until scales of oxide are loosened on surface, and plunging it into a cooling liquid; decree of hardness depends upon heat and rapidity of cooling. Steel is thus ren- dered so hard as to resist files, and it becomes at same time extremely brittle. Degree of heat, and temperature and nature of cooling medium must be chosen with reference to quality of steel and purpose for which it is intended. Cold, water gives a greater hardness than oils or like sub- stances, sand, wet-iron scales, or cinders, but an inferior degree of hardness to that given by acids. Oil, tallow, etc., prevent cracks caused by too rapid cooling. Lower the heat at which steel becomes hard, the better. Tempering. Steel in its hardest state being too brittle for most purposes the requisite strength and elasticity are obtained by tempering— or “lettinq down the temper ” which is performed by heating hardened steel to a certain degree and cooling it quickly. Requisite heat is usually ascertained by color winch surface of the steel assumes from film of oxide thus formed. Degrees or heat to which these several colors correspond are as follows : At 43 oO very faint yellow.. (Suitable for hard instruments; as hammer -faces At 450 0 , pale straw color. . . . ( drills, lancets, razors, etc ’ At 47 oO, full yellow ; For instruments requiring hard edges without elastic!- At 5 4 ?o°; brown, with 'purple (A*’ “ ShearS ’ SCiSS ° rS ’ turai “gtools, penknives, etc. spots ) for tools for cutting wood and soft metals; such as At 538°, purple * l plane-irons, saws, knives, etc. it cSo’ minuni Ue f Fo , r t ? oIs re( * uirin S strong edges without extreme Tf I 6 o’ f 1 hardness; as cold-chisels, axes, cutlery etc A n 6 ^;f, ay l Sh b ue ’ verg ‘ { For s P r * n £-temper, which will bend before breaking- ing on black \ as saws, sword-blades, etc. ’ If steel is heated to a higher temperature than this, effect of the hardening process is destroyed. ° d t f hlgl ! br ^ akin g strain may not be conclusive as to quality, as it may be due to a hard, elastic metal, or a low one may be due to great softness. 644 METALS. TIN. ZINC. MODELS. Case-hardening. This operation consists in converting surface of wrought iron into steel, by cementation, for purpose of adapting it to receive a polish or to bear fric- tion, etc. ; it is effected by heating iron to a cherry-red, in a close vessel, in contact with carbonaceous materials, and then plunging it into cold water. Bones, leather, hoofs, and horns cf animals are generally used for this pur- pose, after having been burned or roasted so that they can be pulverized. Soot is also frequently used. The operation reduces strength of the iron. TIN. Tin is more readily fused than any other metal, and oxidizes very slowly. Its purity is tested by its extreme brittleness at high temperature. Tinplate is iron plate coated with tin. Block Tin is tin plate with an additional eoating of tin. ZINC. Zinc, if pure, is malleable at 220° ; at higher temperatures, such as 400°, it becomes brittle. It is readily acted upon by moist air, and when a film of oxide is formed, it protects the surface from further action. When, how- ever, the air is acid, as from the sea or large towns, it is readily oxidized to destruction. Iron, Copper, Lead, and Soot are very destructive of it, in consequence of the voltaic action generated, and it should not be in contact with calcareous water or acid woods. The best quality, as that known as u Vielle Montague,” is composed of zinc .995, iron .004, and lead .001. Its expansion and contraction by differences of temperature is in excess of that of any other metal. STRENGTH OF MODELS. The forces to which Models are subjected are, \ 1. To draw them asunder by tensile stress. 2. To break them by trans- verse stress. 3. To crush them by compression. The stress upon side of a model is to corresponding side of a structure as cube of its corresponding magnitude. 1 bus, if a structure is six times greater than its model, the stress upon it is as 6 3 to 1 = 216 to 1 : but resistance of rupture increases only as squares of the corresponding magnitudes, or as 62 i t0 j =3 6 to 1. A structure, therefore, will bear as much less resistance than its model as its side is greater. To Compute Dimensions of* a Beam, etc., which, a Structure can hear. Rule —Divide greatest weight which the beam, etc. (including its weight), in the model can bear,bv the greatest weight which, the structure is required to bear (including its weight), and quotient, multiplied by length ot beam, etc., in model, will give length of beam, etc., in structure. Example. — A beam in a model 7 inches in length is capable of bearing a weight of 26 lbs. but it is required to sustain only a weight or stress of 4 lbs. ; what is the greatest length that a corresponding beam can be made in the structuie / 26^4 = 6.5, and 6.5 X 7 = 45-5 ins • MODELS. — MOTION OF BODIES IN FLUIDS. 645 Resistance in a model to crushing increases directly as its dimensions ; but as stress increases as cubes of dimensions, a model is stronger than the structure, inversely as the squares of their comparative magnitudes. Hence, greatest magnitude of a structure is ascertained by taking square root of quotient, as obtained by preceding rule, instead of quotient itself. Example. — If greatest weight which a column in a model can sustain is 26 lbs., and it is required tt> bear only 4 lbs. ; height of column being 18 ins., what should be height of it in structure? .5 = 2.55, and 2.55 X 18 = 45.9 ins., height of column in structure. If, when length or height and breadth are retained, and it is required to give to the beam, etc., such a thickness or depth that it will not break in con- sequence of its increased dimensions, Then \/(T) = V6 ness required. •5 = 2- 55, which, x square of relative size of model = thick - To Compute Resistance of* a Bridge from a ]VLodel. n 2 W — (n — 1) ioJ = load bridge will bear in its centre. Example. — If length of the platform of a model between centres of its repose upon the piers is 12 feet, its weight 30 lbs., and the weight it will just sustain at its centre 350 lbs., the comparative magnitudes of model and bridge as 20, and actual length of bridge 240 feet ; what weight will bridge sustain ? [ 400 j — X (20 — 1) x 30 = 140 000 — 3800 X 30 — 26 000 lbs. MOTION OF BODIES IN FLUIDS. If a body move through a fluid at rest, or fluid move against body at rest, resistance of fluid against body is as square of velocity and density of fluid ; that is, R = d v 2 . For resistance is as quantity of matter or particles struck, and velocity with which they are struck. But quan- tity or number of particles struck in any time are as velocity and density of fluid ; therefore, resistance of a fluid is as density and square of velocity. v 2 a d v 2 _ — = h, and = R. h representing height due to velocity , d density of fluid, 2 g 2 9 and R resistance or motive force. Resistance to a plane is as plane is greater or less, and therefore resistance to a plane is as its area, density of medium, and square of velocity; that is, 'R=.adv‘ 2 . Motion is not perpendicular, but oblique, to plane or to face of body in any angle, sine of which is .9 to radius 1 ; then resistance to plane, or force of fluid against plane, in direction of motion, will be diminished in triplicate ratio of radius to sine of angle of inclination, or in ratio of 1 to s 3 . tt adv 2 s3 adv 2 s3 nence, = K, and = F. w representing weight of body , and F retarding force. 2 gw m Progression of a solid floating body, as a boat in a channel of still water, gives rise to a displacement of water surface, which advances with an un- dulation in direction of body, and this undulation is termed Wave of Dis- placement. MOTION OF BODIES IN FLUIDS. 646 Resistance of a fluid to progression of a floating body increases as velocity of body attains velocity of wave of displacement, and it is greatest when the two velocities are equal. In the motion of elastic fluids, it appears from experiments that oblique action produces nearly same effect as in motion of water, m the passage 01 curvatures, apertures, etc. Resistance to an Area of One Sq. Foot moving through Water, or Contrariwise. Angle of | . _ Surface Pressure per Sq. Foot for following V e- w jth j locities per Foot per Minute. Plane of 480 Angle of Surface with Plane of Current. Pressu lo 120 O Lbs. 6 .09 8 •133 9 .156 10 .179 15 •355 20 .608 25 •94 30 i -353 35 1.798 40 2.258 240 Lbs. •359 •53 .624 .718 1.42 2 - 434 3 - 7 6 5-4I3 7 - I 9 2 9.032 480 Lbs. 1-435 2.122 2.496 2.87 5.678 9-734 1 5 - 038 21.653 28.766 36- 1 3 900 Current. Lbs. 5.046 7 - 459 8 - 775 10.091 19.963 34.222 52.869 76.123 101.132 127.018 45 50 55 60 65 70 75 80 85 90 Lbs. 2.66 2 - 995 3 - 2 49 3-455 3.607 3.728 3.8! 3-857 3.892 3-9 240 Lbs. 10.639 11.981 12.995 13.822 14-43 1 4 - 9 I 4 15.241 15.428 15- 569 15.6 Lbs. 42-557 47-923 51-979 55.286 57-72 59- 6 54 60.965 61.714 62.275 62.4 Resistance to a plane, irom a niua acung m * ------ its face, is equal to weight of a column of fluid, base of which is plane and altitude equal to that which is due to velocity of the motion, 01 through which a heavy body must fall to acquire that velocity . Resistance to a plane running through a fluid is same as force of fluid m motion with same velocity on plane at rest. But force of fluid m motion is equal to weight or pressure which generates that motion, and this is equal to weight or pressure of a column of fluid, base of which is area of the plane, an (Fits altitude that which is due to velocity. Illustration.— If a plane i foot square be moved through water at rate of 32.166 feet per second, then ■ 32,i6 > - = 16.083, space a body would require to fall to acquire a velocity of 32.166 feet per second; therefore 1 X 62.5 (weight of a cube foot of wa ter) x 32 ' l66 -- = 1005 lbs. = resistance of plane. 64-333 Resistance of different Fig-ures at different Velocities in Air. Veloci- ty per Second. C01 Vertex. ae. Base. Sphere. Cylin- der. Hemi- sphere. Round. V eloci- ty per Second. Co: Vertex . ne. Base. Sphere. Cylin- der. Hemi- sphere. Round. Feet. 3 4 5 8 9 10 Oz. .028 .048 .071 . 168 .2x1 .26 Oz. .064 . 109 . 162 • 382 .478 •587 Oz. .027 .047 .068 .162 .205 •255 Oz. •05 .09 •143 •36 •456 •565 Oz. .02 •039 .063 .16 .199 .242 Feet. 12 14 15 x6 18 20 Oz. •376 .512 •589 •673 .858 1.069 Oz. .85 1. 166 1- 346 1.546 2.002 2 - 54 Oz. •37 .505 .581 .663 .848 i-o57 Oz. .826 1145 1.327 1.526 j I-986 ! 2.528 Oz. •347 .478 •552 ! .634 | .818 ' 1033 Diameter of all the figures was 6.375 ins., and altitude of the cone 6.625 ins. Angle of side of cone and its axis is, consequently, 25 0 42' nearly. From the above, several practical inferences may be drawn. 1. That resistance is nearly as surface, increasing but a very little above that proportion in greater surfaces. MOTION OF BODIES IN FLUIDS. 647 2. Resistance to same surface is nearly as square of velocity, but gradu- ally increasing more and more above that proportion as velocity increases. 3. TV hen after parts of bodies are of different forms, resistances are differ- ent, though fore parts be alike. 4. The resistance on base* of a cone is to that on vertex nearly as 2 a to I. And m same ratio is radius to sine of angle of inclination of side of cone to its path or axis. So that, 111 this instance, resistance is directly os sine of angle of incidence, transverse section being same, instead of square of sine. Resistance on base of a hemisphere is to that on convex side nearly as 2.4 to 1, instead or 2 to 1, as theory assigns the proportion. wed In w nter easiest when 648 MOTION OF BODIES IN FLUIDS. When a — — of a foot, as in all figures in table, * becomes ^ r when r _ re - 9 oietancc tw table to similar body. , Illustration.- Assume convex face of hemisphere resistance = .634 oz. at a ve- locity of 16 feet per second. Then r = .6 34 , and * = ^ r = *.ynsM = <***>* of column of air, pressure of which = resistance to a sphtrical surface at a velocity of 16 feet . -1 „ p rP c!sure of Air in. rear of a Projectile To Cot ]^ P j^| e ^ or to Pressure '!"<■ to its Velocity. Assume height of barometers. 5 feet, and weight of atmospheres 14.7 lbs. Weight of cube inch of mercury = ^ = -49 «*., “ d wei & ht of cube mch ° f “ r = .00004357 lbs.; hence, .49-000043 57 = » ' wbl «* X *-5 feet-zBns feet. Then 16.08 : V * «5 :: = *> a awstt x" n*.‘« H»«,V . VW* •••■ • — ** To Compute Velocity lost by a projectile. If a bod^ is projected with any velocity in a medium of same density with itsel , and it describes a space = 3 of lts diameters, ^ ^ Then * = 3 d, and b = g^ = gd* 9 nnd c-__ r _2 1 o8 = velocity lost nearly . 66 of projectile velocity. Hence, b x = -x- , ana bx — a o8 c bx ~ 1 _ c=base of Nap. system of log. ; hence c^ = number corresponding to Nap. log. I x. Hence, if b x X -4343i result — com. log. ol c . 6 z =5 = 1. 125, which x -4343 = -488 587 5, and number to this com. log.= 3 .°8o 3 . 3.0803 I 2.08 Hence, velocity lost g— - Illustration. -If an iron ba "“' c ^ 5’ anc >; f « r f vertical line drawn from centre intersects a line passing through centre of grawtj ot hull ot vessel perpendicular to plane of keel. Point of meta-centre may be the same, or it may differ slightly for different angles of heeling. Angle of direction adopted to ascertain position of n*eta- centre shouiji be greatest wjhich, iinder oroinary cir- cumstances, is of probable occurrence ; in different vessels tbit angle ranges from 20 to 60 .. ^ If meta-centre is above centre of gravity, equilibrium is Stable; if it coincides with it, equilibrium is Indifferent ; and if it is below it., equilibrium i$ Unstable. Comparative Stability of different hulls of vessels is proportionate to the distance of G M for same angles of heeling, or of distance G s. Oscillations of hull of a ves- sel may be resolved into a rolling about its longitudinal axis, pitching about its transverse axis, and vertical pitching, consisting in rising and sinking below and above. position of equilibrium. If transverse section of hull of a vessel is such that, when vessel heels, level of centre of gravity is not altered, then its rolling will be about a permanent longi- tudinal axis traversing its centre of gravity, and it will not be accompanied by any vertical oscillations or pitchings, and moment of its inertia will be constant while it rolls. But if, when hull heels, level of its centre of gravity is altered, then axis about which it rolls becomes an instantaneous one, and moment of its inertia will vary as it rolls; and rolling must then necessarily be accompanied by vertical os- cillations. Such oscillations tend to strain a vessel and her spars, and it is desirable, therefore, that transverse section of hull should be such that centre of its gravity should not alter as it rolls, a condition which is always secured if all water-lines, as ivl and ef are tangents to a common sphere described about G; or, in other words, if point of their intersections, o, with vertical plane of keel, is always equidistant from centre of gravity of hull. To Compute Statical Stability. stability ^ M = S ’ D re P resent ing displacement , M angle of inclination , and S Illustration r. Assume a ship weighing 6000 tons is heeled to an angle of q° distance c M = 3 feet, ® y ’ Sin. 9° = . 1564. Then 6000 X 3 X -1564 — 2815.2 foot-tons. Q«r>~T eigll ^ 0f a floatin „g bo(J y is 5515 l^s., distance between its Centre of gravity and meta-centre is 11.32 feet, and angle M — 20 0 . J Sin. M = .342 02. Hence 5515 x 11.32 x .34202 = 21 352.24 foot-lbs. Statical Surface Stability-. Moment of Statical surface stability at any angle is c z D. Assuming at anv^iSt' o^h^ COmcld , ed \ v ' ith o ; coefficient of a vessel’s stability ferMrnl r« 13 . e ^ resscd 'I he " the displacement is multiplied by gravity, or meta " cenU ' e for angle of heel above centre of Approximately . Rule.— D ivide moment of inertia of plane of flotation for upright position relatively to middle line bv volume of displacement- and quotient multiplied by sine of angle of heel will give result. Per F ° 0t °f Len V th °f Vessel < § (B 3 Sin. M). B representing half breadth. Dynamical Surface Stability. 0fvSr!ife miCa l Sl,r ^ stabilit y is expressed bv product of weight M^n:^s^^ epre8sio, “ of centre ° f b "°" anc ^ durhl ^ he To Compute Dynamical Stability of a Vessel. ab^^t^fi’ra^t ULE 'l MuI a Pl7 . d “^ cment b - v he 'g ht of meta-cHitre abov e centre of gravitv , and product by versed sine of angle of heel. Or multiply statical stability for given angle by tangent of .5 angle of heel. To Compute Elements of Stability of a Floating Body. A a — s > sin. M ~ r ’ sin. M ~ aD(i sin< Mr==c - A representing area of G c m; \zt ity and of line of disnlaremprtt nf’-t ° h ° 7 lzonia ^ ^stance, G s, between centre of grav- Of gravity and buoyancy, all “" s. Assumed elements of figure illustrated are A = 86, A' = 21. 5, b = 21. 5, and e = . 5. The deduced arc $ = 3.7, c = 3.87, g= 10.82, a = 14.9, and, r = 11.32. b repre- senting breadth at water-tine or beam in feet, and P weight or displacement in Lbs. or tons. TheQS = ? ^ Xl 4-9 = 3-7 /«*, = 10. 82 /eef, c = . 342 02 x 1 1. 3 2 — 3- 8 7 f eet Of Hull of a Vessel. 63 ( . b * d: p sin. M = S ; d cos. ,5 M = d \10.7 to 13* A ) ’ = p( fc “+Iinni)=Si and P (s dz e sin. M) = S. <2 representing depth of centre of gravity of displacement un- der water in equilibrium, and d' depth when out of equilibrium, both in feet. 10.7 to 13 (11.93) A = 9 , Illustration i. — Displacement of a vessel is 10000000 lbs. ; breadth of beam, 50 feet; area of immersed section, 800 sq. feet; vertical distance from centre of grav- ity of hull up to centre of buoyancy or displacement, 1.0 feet, and horizontal dis- tance a between centres of gravity of areas immersed and emerged, when careened to an angle of 9 0 10' = 33.4 feet, immersed area being 50 sq. feet. Sin. 9 0 10' = . 1593. Then s = ~ X 33-4 = 2.0875 feet, 800X2.0875 = 50X33-4) r = . 2 ' 39 . _ jh feet. g — — — — — = 13. 1 feet, S = ( - — \ -J- 1.9 X .1593 5 J 11.93X800 \ii.93X8oo/ 10 000 000 X • 1 593 — 2 3 9 ° 5 39 ^ lbs -> and e = (10 1 ! ocx! “ 2 ' 087 5 ) =1 ^ f eet 2.— Assume a ship having a displacement of 5000 tons, and a height of meta-centre of 3.25 feet, to be careened to 6° 12'. What is her statical stability? Sin. 6 6 12' = .1079. Then 5000 X 3-25 X . 1079 = 1753.37 foot-tons. 3. — Assume a weight, W, of 50 tons to be placed upon her spar deck, having a common centre of gravity of 15 feet above her load-line, Then 5000 X 3- 25 — 50 -f- 1 5 X • 1079 = 1745. 29 foot-tons. 4. — Assume 100 tons of water ballast to be admitted to her tanks at a common centre of gravity of 15 feet below her load-line, Then 5000 X 3-25 + 100 X 15 X .1079 = 1915. 22 foot- tons. 5. — Assume her masts, weighing 6 tons, to be cut down 20 feet, Then 10 X — = — foot = fall of centre of gravity, and 5000 X ( 3- 2 5 + ) X • io 79 5000 50 \ 5°/ = 1774.95 tons. To Compute Elements of Bower, etc., reqn ireci to Careen, a Body- or "Vessel. Sin. M (h - n sin. M) + » sec. M - * = l - - fj = m - W lrz=zFc, and W l = S. W representing weight or power exerted and l distance ; at which weight or power acts to careen body , taken from centre of gravity of displace- ment perpendicular to careening force, h vertical height from centre of gravity of dis- placement to centre of weight or power to careen body when it is in equilibrium , n horizontal distance from centre of vessel to centre of weight or power, L length of vessel, m meta-centre, and S as in preceding case , all in feet. * Unit for section of a parallelogram is 10.7 ; of a semicircle 12, and of a triangle 12.8. Illustration. — A weight is placed upon deck of a vessel at a mean height of o 87 feet from centre line of hull; height at which it is placed is n.32, and other ele- ments as in first case given. J Sec. 20 .342. Then h = n.32, n = 3.87, and l = .342 (1 1.3 —TfyX~w) + 3.87 x 1.0642 -3.7 = .342 x 10 + 4.12 — 3.7 = 3.84/^. Assume W = 5515. Then 5515 X 3-84 = 21 187.6 foot-lbs. Or P (iv cos. M + A sin. M) =. S. iv representing distance of weight from centre of vessel , and h height of w above water-line , both in feet. U J n re of H 0 !f ST ? A y°f' If a weight of 30 tons placed at 20 feet from centre of hull or deck, 10 feet above water-line, careens it to an angle of 2° 9', what is its stability? cos. 2° 9' = .9993 ; Sin. 2° 9' = .0375. 30 (20 X. 9993 + 10 X .0375) = 30 x 20.361 =610. 83 /oo^ons. Bottom, and. Immersed Surface of Hnll of ‘Vessels. To Compute Bottom a nd Side Surface of Bull Bottom and Side Rule. Multiply length of curve of amidship sec'tion, taken from top of tonnage or mam deck beams upon one side to same point upon other (omitting width of keel), by mean of lengths of keel and be- tween perpendiculars m feet, multiply product by .85 or .9 (according to the capacity of vessel), and product will give surface required in- sq. feet. Example.— L engths of a steamer are as follows: keel 201 feet, and between ner- pendiculars 210 feet, curved surface of amidship section 76 feet; what is surface? Coefficient .87. 210 + 201 2 = 205.5, and 76 X 205.5 X .87 = i 3 587 sq. feet. Note.— E xact surface as measured was 13650 sq. feet. Bottom Surface. Rule.— M ultiply length of hull at load-line by its breadth, and this product by depth of immersion (omitting the depth ol keel) m feet ; and. this product multiplied by from .07 to .08 (according tc capacity of vessel) will give surface required in sq. feet. Example.— L ength upon load-line of a vessel is 310 feet, beam 40 feet, depth of keel 1 foot, and draught of water 20 feet; what is bottom or wet surface? Coefficient assumed .073. 310 X 40 X 20 — 1 X .073 = 17 199 sq. feet. To Compute Resistance to Wet Surface of* Bull. C a v 2 — R C representing a coefficient of resistance, a area of wet surface in sa. feet, and v velocity of hull in feet per second. Values of C I- 00 /* clean copper. I .014, iron plate. ’ 1- oi j smooth paint. | .019, iron plate, moderately foul. 1 75 f ! “‘VWV.UVWJ Power required to propel one sq. foot of immersed amidship section at 53 is 071 that of smooth wet surface. ^ /j To Compute Elements of a Vessel. Displacement and its Centre of* Gravity. Displacement of a vessel is volume of her body below water-line. Centre of Gravity, or Centre of Buoyancy of Displacement , is centre of gravity of w^ater displaced by hull of vessel. For Displacement. Rule.— D ivide vessel, on half breadth plan, into a nU K5< r e Q u idistant sections, as one, tw r o, or more frames, commencing at & and running each side of it. Add together lengths of these lines in both fore and aft bodies, except first and last, by Simpson’s rule for areas (see page 344) ; multiply sum of products by one third distance between sections, and product will give area of water-line between fore and aft sections. Then compute areas contained in sections forward and aft of sections taken in- eluding stern and rudder-post, rudder and stem, and add sum to area of body-sec- tions already ascertained.* J * To Compute Area of a Water-line , see Mensuration of Surfaces, page 344. 3 I* NAVAL ARCHITECTURE. Compute area of remaining water-lines in like manner. Tabulate results, and multiply them by Simpson’s rule in like manner as for a water-line, and again by consecutive number of water-lines, and sum of products between water-line and product will give volume between load and lower water-line. Add area of lower water-line to area of upper surface of keel; multiply half sum by distance between them, and product will give volume; then compute areas con- tained in sections forward and aft of sections taken as before directed. If keel is not parallel to lower water-line, take average of distance between them. Compute volume of keel, rudder-post and rudder below water-line; add to volume already ascertained; multiply product by two, for full breadth, and product will give volume required in cube feet, all dimensions being taken in feet. Example. -Assume Fig. 2. a vessel ioo feet in length by 20 feet in extreme breadth, on load-line of 8 feet 9 inches immersion. Figs. 2 and 3. Distance between sections, for purpose of simplifying this example, is taken at 10 feet; usually frames are 18 to 30 ins. apart, and two or more included in a section. Water-lines 2 feet apart. 1 st Water-line. 2 d Water-line. 3 d Water-line. 4 5 = 5 4 2.7 = 2.7 4 i-5 = i-5 3 7-7 X 4 = 30.8 3 6.9 X 4 f=i 27.6 3 5 X 4 = 20 2 9-5 X 2 == *9 2 8.7 X 2 = 17.4 2 6.6 X 2 = 13.2 1 9.9 X 4 = 39-6 1 9-5 X 4 ~ 38 1 8.7 X 4 — 34-8 0 10 X 2 20 0 9.6 X 2 = 19.2 0 8.9 X 2 = 17.8 A 9.6 x 4 = 3 8 - 4 A 9 X 4 = 36 A 7.6 X 4 = 30-4 B 7.8 X 2 = 15.6 B 7 X 2 = 14 B 7 X 2 = 14 C 6.8 X 4 = 27.2 C 5 X 4 — 20 C 3 X 4 = 12 D 4 == 4 D 2 = 2 D 1.2 = 1.2 199.6 176.9 144.9 IO -r ' 3 = 3j 10 —7 "3 . = 10 -r "3 3^ Abaft section 4, rud- der and post Forward section D and stem 665.3 20.7 711 589-7 Abaft section 4, rud- der and post 13.2 Forward section D and stem 9.1 612 4 8 3 Abaft section 4, rud- der and post 7 Forward section D and stern 5.4 495-4 4 th Water-line. 4 5 •7 2 X 4 •7 8 2 4-3 X 2 . = 8.6 1 6-5 X 4 = 26 0 6.8 X 2 = 13.6 A 5 X 4 — 20 B 3.6 X 2 = 7.2 C •9 X 4 == 3-6 D •3 = •3 104-3 ^ 3 & 293.3 Abaft section 4, rud der and post 3.2 Forward section D and stem 8 297-3 Keel. Half breadth = .25 X length of 98 feet : Rudder-post and rudder 24 - 5 •3 24.8 Results. 1st water line 71 1 2d 3<* 4 th Keel 612 X 4 = 2448 X 1 = 2448 495.4 X 2 = 990.8 X 2 = 1981.6 297 3 X 4 = 1189.2 X 3 = 3567-6 24.8 _24- 8 X 4 — 99 -2 5363.8 8096.4 3) 10727.6 Displacement , 3575.9 X 2 = 7i5i.8cu&e;r. NAVAL ARCHITECTURE. To Compute Centre of Gravity of Displacement. Rule. — D ivide sum of products obtained as above, by consecutive water- lines, by sum of products obtained in column of products by Simpson’s mul- tipliers, and quotient, multiplied by distance between water-lines, will give, depth of centre below load water-line. Illustration ] Or, V an) 8096.4, from above, - 4 - 5363.8 = 1.5, which X 2 = 3 feet. — d. n representing draught of water exclusive of any drag of keel, a area of immersed surface of hull in sq. feet, and D displacement in cube feet. 2.— Assume draught of water 8 feet, displacement 7152 cube feet, and area of im- mersed surface of hull uoo sq. feet. Then 8 (»T — \ IIOI 2 X I.187 = 3 - 3 1 fat. 100 X 8/ To Compute Displacement -Approximately. Coefficient of Displacement of a vessel is ratio that volume of displacement bears to parallelopipedon circumscribing immersed body. V D — C. V representing volume of displacement in cube feet , L length at im- nursed water-line , B extreme breadth , and D draught in depth of immersion boih in feet. Coefficient of Area of A midship Section in Plane of a Water-line is ratio which their areas bear to that of circumscribing rectangle. L representing length of water-line, and D distance between water-lines , both in feet. Coefficients. [By S. M. Poole , Constructor t T . S. Navy. ) Rule.— M ultiply length of vessel at load-line bv breadth, and product by depth (from load-line to under side of garboard-strake) in feet, and this product by coefficient for vessel as follows : divide by 35 for salt water 36 for fresh water, and quotient will give displacement in tons. Amidship sections range from .7 to .9 of their circumscribing square and tneau of horizontal lines from . 55 to .75 of their respective parallelograms. Hence ranges for vessels of least capacity to greatest are .7y.ee — anH Merchant ship, very full 6 to .7 “ “ medium 58 to. 62 River steamer, stern-wheel. . . .6 to .65 Ship of the line 5 to 6 Naval steamer, first class 5 to .6 “ “ . • 52 to .58 Merchant steamer, sharp 54 to .58 Half clipper 52 to .56 Brigs, barksj etc 52 to .56 River steamer, tugboat, med’m .52 to. 56 Merchant steamer, medium. . . . c 2 to Clipper to .54 Schooner, medium 48 to .52 River steamer, tug boat, sharp .45 to .5 “ medium 45 to .5 “ “ sharp 42 to .45 Schooner, sharp 46 to .5 Yachts, sharp 4 to '45 “ very sharp .3 toil River steamers, very sharp. . . .36 to .42 In steam launch Miranda , when making 16.2 knots per hour, with a displace- ment of 58 tons, her coefficient was 3. 1 To Compute Change of Trim W d L IT X m~ d ' D re P resentin 9 displacement at line of draught in tons, L length at same line in feet, and m longitudinal meta-centre. at daughter 25.5 feet, lias 11=380 feet, >11 = 475 feet and D _ 8625 tons. If, then, a weight of 20 tons was shifted fore and aft 100 feet, ’ 20 X IOO 380 - x — = . 1856 feet = 2.22 ins. 475 8625 Illustration. — Vertical Plane at 53 and Horizontal at Load-line. ARCHITECTURE, NAVAL To Compute Centre of GJ-ravity or Buoyancy Approxi- mately. 2 Q j t° — °f mean draught of hull, using larger coefficient for full-bodied vessels. To Delineate Curve of Displacement. # This curve is for purpose of ascertaining volume of water or tons weight, displaced by immersed hull of a vessel at any given or required draught; or weight required to depress a hull to any given or required draught. From, results of computation for displacement of vessel, proceed as follows, Fig. 4 : Fi S- 4- On a vertical scale of feet and ins., as A B, set off depths of keel and water- lines, draw ordinates thereto represent- ing displacement of keel, and at each water-line, in tons. < Through points 1 , 2 , 3 , 4 , and 5 de- lineate curve A 5 , which will represent displacement at any given or required draught. Draw a horizontal scale correspond- ing to weight due to displacement at load-line, as A C. and subdivide it into tons and decimals thereof, and a ver- tical line let fall from any point, as x , at a given draught, will indicate weight of displacement at depth, on scale AC, and, contrariwise, a line raised from any point, as 2 , on A C will give draught at that weight. Illustration.— Displacement of hull (page 654) at load-line = 7151.8 cube feet, which - 4 - 35 for salt water = 204.3 tons, hence A C represents tons, and is to be sub- divided accordingly. Assume launching draught to have been 4 feet, then a vertical let fall from 4 will indicate weight of hull in tons on A C. Coefficients . {By C. MacJcrow , M. I. N. A.) Description of Vessel. Iron-Clads. Hail Steamers. , Merchant, small Gunboats Troop Ships Swift Naval Steamers. . Fast Steamers, R. N Length. 225 325 ^50 385 368.27 220 90 125 160 350 340-5 337-3 270 300 45 59 35 42 42.5 27 i5 23 3i-3 49.12 46.13 50.28 42 40.27 Mean Draught. 15 24-75 18.71 8 4 8 23-5 15-75 22.75 *9 14 Coefficient. Displace- Amidship Water- lines. •715 .64 .687 .516 .702 •637 •536 .466 •47 •4 •483 •497 .414 •932 .81 .85 .88 .812 .912 .914 .87 •745 .674 .68 •787 .792 .711 755 7i 84 8 635 742 704 616 603 7 582 614 62 » 711 Curve of Weight. To Compute Number of Tons required to Depress a vessel One Inch at any Draught of Water Parallel to a Water-line. ro ^V“' 7 Uivide area of plane by 12 , and again by 35 or 36 , as may be required for salt or fresh water. J ,.a^r E ^hi^Vwafer7 ter liDe ° fa ' V ff * 1423 ^ feet ; what is its 1422-7-12 = 118.5, which -4-35 = 3.38 tons. NAVAL ARCHITECTURE. To Compute Common. Centre of Grravity of* Hull, Ar- mament, Engine, Boilers, etc., of a "Vessel. Rule. — Compute moments of the several weights, relatively to assigned horizontal and vertical planes, by multiplying weight of each part by its horizontal and vertical distance from these planes. Add together these moments, according to their position forward or aft, or above or below these planes, and difference between these sums will give po- sition forward or aft, above or below, according to which are greatest. Divide results thus ascertained by total weight of vessel, and product will give horizontal and vertical distances of centre of gravity from these planes. It is customary to assume vertical plane at 0 , and horizontal plane at load-line. Note. — In following illustration, in order to simplify computation in table, com- mon centre of gravity of hull, machinery, etc., is taken, instead of centres of indi- vidual parts, as engine, boiler, propeller, etc. Illustration. — Assume half-girths as in following table, and distance between sections io feet. Sec- tion. Half- Girths. FOR^ Multi- pliers. fARD. Prod- uct. Multi- pliers. Mo- ments. Sec- tion. Half- Girths. ABJ Multi- pliers. lFT. Prod- uct. Multi- pliers. Mo- ments. No. &... Feet. 25 1 25 No. Feet. 23 4 92 1 92 A 23 4 92 1 92 2 . . . 20 2 40 2 80 B.... 21 2 42 2 84 3 ••• 18 4 72 3 216 C.... x 9 4 76 3 228 4 ... 16 2 32 4 128 D. ... E. ... 17 2 34 4 136 5 ••• H ,1 14 5 70 i5 1 15 5 75 615 534 586 Moments forward, 615 — moments abaft, 586 = 29 - 4 - sum of product 534 = .054, which X 10 feet == .54 feet forward of Centre of* Lateral Resistance. Centre of Lateral Resistance is centre of resistance of water, and as its po- sition is changed with velocity of vessel, it is variable. It is generally taken at centre of immersed vertical and longitudinal plane of vessel when upon an even keel. If vessel is constructed with a drag to her keel, the centre will be moved r proportionately abaft of longitudinal centre. Yacht America had a drag to her keel of 2 feet, and centre of lateral re- sistance of her hull was 8.08 feet abaft of centre of her length on load-line. Centre of* Effort. Centre of Effort is centre of pressure of wind upon sails of a vessel in a vertical and longitudinal plane. Its position varies with area and location i of sails that may be spread, and it is usually taken and determined by the ( ordinary standing sails, such as can be carried with propriety in a moderateh fresh breeze. Iii computing this position, the yards are assumed to be braced directly fore 5 and aft and the sails flat. Note.— Centre of effort of sails, to produce greatest propelling effect, must accord with capacity of vessel at her load-line, compared with fullness of her immersed body at its extremities. Thus, a vessel with a full load-line and sharp extremities below, will sustain a higher centre of effort than one of dissimilar capacity and con- struction. NAVAL ARCHITECTURE. 659 To Compute Location of Centre of Effort. Rule— M ultiply area of each sail in square feet by height of its centre of gra\ lty abo\ e centre of lateral resistance in feet, divide sum of these prod- ucts (moments) by total area of sails in square feet, and quotient will give height of centre in feet. b 2. Multiply area of each sail in square feet, centre of which is forward of a vertical plane passing through centre of lateral resistance, by direct dis- tance of its centre from that plane in feet, and add products together. 3. Proceed in like manner for sails that are abaft of this plane, add their products together, and centre of effort will be on that side which has greatest moment of sail. b Example.— Assume elements of yacht America as rigged when in U. S. Service. Distance of Centre of Gravity of Sails. • Foreward. I Abaft. Area. Flying Jib. Jib Foresail. .. Mainsail. . . Sq. Feet. 656 1087 *455 2185 I 5383 Vertical moments 172575 Area of sails 5383 sistance. Height of Cent, of Grav- ity of Sails. Feet. 28 26 34 35 Vertical Moments 18368 28 262 49470 76475 *72 575 52 32 Moments. Foreward. 34 ns 34 784 Abaft. 4 365 87 400 68 896 j 91 765 32.06 — height of centre above centre of lateral re- Xf j f 91 765 -v. 68 896 sistance 1 53*5 = 4 ' 25 “ distance °f c ™tre abaft centre of lateral re- Relative Positions of Centre of Effort and of Lateral Resistance. Square Riff. Ft?. ? f'+f' l ~= E. and 4 A 5 d Fore and Aft Rig. L E, I^F+O = E-. L representing length of load-line, d distance of centre of buoyancy fanTeof ^mhfofbnon^ ? f ? entre °f late l al resistance abaft centre of it, d" dis - Meta-Centre. Meta-centre of a vessel’s hull is determined by location of centre of erav 21^7 °f lm mersed bottom of hull, for it is that pltTn traSlfe section of hull, where a vertical line raised from its centre of gravitv o? F^page^o a ' paSSi " g through centre of g™vity of hullj as To Compute Height of Meta-Centre. By Moment of Inertia. ~ — M. I representing moment of inertia of area of water-line or plane of flotation, and D volume of displacement in cube feet. 'area 0 by square re r“ area '•* sa , m of Products of each element of that computed Ce fr0m ax,s > about vvhich moment of area is to be T ° Ascert£ ^ Moment of Inertia approximately. Rectangle = C L B 3 • c = when L = 4 B ; C = ~ when L = 5 B ; and C : 200 wlieu L 6B - Wit h ^ery fine lines and great proportionate length C = — . L and B measured at load-line. 25 66 o NAVAL ARCHITECTURE. Illustration.— Assume length of vessel 233 feet, breadth 43, draught 16, and displacement 2700 tons. Length = 5. 65 beams; hence C is taken at — . Volume of displacement = 2700 X 35 = 92 500 cube feet. Exact height of moment was 10.44 feet. Then 21 * *33 X 43^ ,., 400 X 92 500 By Ordinates. Rule.— D ivide a half longitudinal section of load water- line by ordinates perpendicular to its length, of such a number that area between any two may be taken as a parallelogram. Multiply sum of cubes of ordinates by respective distances between them, and divide two thirds of product by volume of immersion, in cube feet. Illustration.— Take dimensions from Figs. 2 and 3, page 654. Cube. 51 460 Length. Cube. Length. Cube. 5 . 125 A . . 7-7 • 456 B .. ....7.8.... •• 475 9-5 • 857 C .. 9-9 1 10 . 970 .1000 D . . ....4 .... .. 64 5146 X 10 3)102 920 7I5I-8) 3 4306.6 = 4.77 ft. If there are more ordinates, their coefficients must be taken in like manner, as 1 — 4 — 2 — 4 — 2 — 4 — 1. For operation of this method, see Simpson's rule for areas, page 342. Or, — I* X — M. y representing ordinates of half -breadth sections at load- line, d*x increment of length of load-line section or differential of x, and D displace- ment of immersed section in cube feet. — (a3 4 d>3 -J- 2 C3 -p 4 d 3 e3) — + F -f- A A 3 3 _ ft 5 c g By- Areas. — jy and e representing ordinates of 1 st or load water-line, F area of irregular section between 1st frame and stem , and A area of like section between ^ stern-post , both in sq. feet , D displacement , in cube feet, and l distance between frame* or sections of water-line, as may be taken, in feet. To Ascertain Areas of F and A. —Big. £5. — a&X&c3A-4 = F, and— deXc5'3-^ 4 = A. 3 3 Elements of Capacity and Speed of Several Types of Steamers of Ft. dST. (W. H. White.) T HP tO Length Displacement. Iron-clads. Recent types. do. twin sc. Unarmored. Swift cruisers Corvettes Ships Gun-vessels. . Gun-boats . . . Merchant. Mail, large. . . “ smaller. Cargo, large. . “ smaller. Feet. 300 to 330 280 to 320 270 to 340 200 to 220 160 125 to 170 80 to 90 400 to 500 300 to 400 250 to 350 200 to 300 Breadth. 5 - 25 * 05-75 4-5*0 5 6.5 to6-75 6 5 5. 5 to 6. 25 3 to 3-25 9 to 1 1 8 to 10 7.5 to 10 7 to 9 Tons. 7500 to 9000 6000 to 9 000 3000 to 5 500 1 800 tO 2 OOO 850 to 950 420 to 800 200 tO 250 7000 to IOOOO 5000 to 7000 3000 to 6 000 1 500 to 4 000 Speed. Knots. 14 to 15 to 15 15 to 16 12.75 to 13.25 to I to to 15 to 14 to 15 to II .9 to I . 7 to 1.3*01.5 [ to I. 2 I tO I. 2 .8 to 1.4 .8 to 1. 1 .5 to .6 . 4*0 -5 -3*o .5 • 2 to .4 Displace- [Displace- ment. ment ■%. 16 to 20 \ 151019 j 20 to 24 • 13 to 14 10 to 1 1 7 to 11 | 5 *° 7 10 to 11 7 to 10 5 to 9 3 to 6 NAVAL ARCHITECTURE. To Compute Power Required in a Steam Vessel, capac- lt V °T another Vessel "being given. In vessels of similar models. — = V ; — = y ' ; V __ q . an(1 _ R . VTOd T \°f volume i of given and required cylinders and reuo- lutions in cube feet, a and A areas o/ immersed section of given and required vessel in sq. feet at like revolutions and speed of given vessel , s and S speeds ofqiven and required vessel at revolutions of given vessel , , both in feet per minute rand r' ) e ?°lutwns of given and required vessel per minute , and C product of volume of com lined cylinder and revolutions for required vessel. J V0LUme °J com * i.l L ^f RA , TI T-- A / team Y essel having an area of amidship section of 67s sq feet has two cylinders of a combined capacity of 53? aa cube feet and a cnJod nf l knots per hour with , 5 revolutions of her e^in\\ Requhed volunfe of L\m 5 wVth 14‘^^luUons ° f IO feet ’ f ° r a SeCli °" ° f 700 feet and a s l’ eed of ’3 knots tt = 533-33 X 15 = 8000 mbefeel, —fl- = S 29 6. 3 cube feet, ; 3 3 X 8296.3 ^ i X ^*75 * 10.53 1574S -2 cube feet, — _* 5745 2 = 16388.! and = 5 6:.66 cu&e I 4-5 * 2-X 14.5 feet which +10 stroke of piston , 12 for ins., and X 1728 ins. in a cube foot — 561.66 X 1728 n n J - 8087.9 *«• area of each cylinder = diameter of 101.5 = v ’ V yy = V; and ^ = IIP - 0r > = v ; and L c-= IfP - 10 X 12 Approximate Rules to Compute Speed and IIP of Steam Vessels. V3 ®? = c- IIP - V3 Di V c .°^ ien l ofvasd. A area of immersed amidship section in sq feet \ velocity of vessel in knots per hour , and D displacement of vessel in tons. ^ J ’ Note.— W hen there exists rig, an unusual surface in free board, deck- houses etc or any element that effects coefficient for class of vessel given, a corresponding ad’ dition to, or decrease of, following units is to be made: 6 P d g ad * Range of Coefficients as deduced from observation is as follows : SIDE-WHEEL. Steamboat. Medium lines Fine lines. . . . Steamer. Medium full lines* Fine linest. Sq.F 43 150 136 675 3600 5233 * Full rigged. V 3 A v 3 d§ PPOPELLER. Steamboat. Medium lines. Fine lines . Steamer. Medium full. . . Torpedo bout.. A D y ( V 3 A v 3 dI IH? IH? 45 12 500 150 ~ 15 530 55o 2532 9 194 570 390 1475 IO 180 470 3600 13 210 — 27 20 170 500 t Bark rigged. Coefficients as Determined by Several Steamers of H. B. M. Service. (C. Mackrow , M. I. N. A.) Length. Length Beam. A rea of Section at Displace- ment. IH? Speed. V 3 A I FP ~ Feet. 185 212 360 270 380 400 362 400 6 - 53 5- 89 7 - 33 6*43 6.52 6 - 73 7 - 33 6-73 Sq. Feet. 236 377 814 632 1308 1198 778 1185 Tons. 775 I 554 5898 3057 9487 9*52 5600 9071 3K 782 1070 2084 2046 3205 5971 3945 6867 Knots. IO *34 10.89 ”•5 12.3 12.05 13.88 14.06 * 5-43 333 456 598 574 7 i 4 536 548 634 552 NAVAL ARCHITECTURE. Approximate Rule for Speed of Screw Propellers. i Moles worth.) PN S 3 v = v- and -*r-aP. 88 (Moles worth.) ; * 2 Lr = N; ^? = V; “i- = P; ^ = V and » representing velocities in knots and miles per hour, r pitch of propeller in feet and N number of revolutions per minute. This does not include slip, which ranges from io to 30 per cent. TPitcli of Screw IPropeller. Pitch ranges with area of circle described by diameter of screw to that of a midship section. Area of screw circle to amidsliip j I 6 section = 1 to ) 1 4-5 3-5 2 -5 Two Blades. | 1.02 | i.ii | 1.2 | 1.27 I I -3 I I i-4 I x *47 Pitch to diameter of screw = 1 to 1 1 * -7 ; ' j ~ j n 1 _ q8 Four Blades. I x.08 | 1.38 | 1 5 1 *.62 I x. 7 x I **77 1 *- 8 9 I J ‘9 8 Length = .166 diameter. Slip of Side-wlieels. Radial Blades. ?Aip^ = S. Feathering. length of arc of immersed circumference of blades, c length of chord of immersed arc. and S slip, alt in feet. Area c f Blades. 2(A-c)_ s Wenthemna. 1,5 (A ’ - = S. A representing 75 HT . A Sea Service. S = A. D representing diameter River Service. of wheel in fed, and A area of each blade in square feet. Length of Blades. .7 in Ri™r service and 6 in Sea service. Distances between Radial Blades. 2.25 in River serv.ee and 3 feet m service, between Feathering blades , 4 to 6 feet. Proportion of Bower Utilized iir a Steam Vesse . P — z __ c p representing gross IIP, 2 loss of diameter of wheels at centre of effect, ■O uV A noari r. mpMcient for vessel effect by slip ana ooixgue r revolutions per minute, and C coefficient for vessel Illustration. IIP of engines of of effect 'of wheels is aad what power applied to propel vessel ? c in this case is is 37 per cent., by wheels. „ . . Q _ _ 1120 — (1120X 28.74 — 100) _ 798^11 _ 6^.63 coefficient. Speed of to propel vessel at this speed = 65.63 X i7-°5 - 19 ‘V 0 - 1 * iq 076. 13 X i7-°5 X 60 2ft 33 000 Friction of engines 1.5 lbs. upon 3848 sq. ins. X x 3 -5 revolu- FrV^Uon^onoad 6percent^ upon pressuri of s^an^ess for friction of engine, as above ■V Oblique action 01 wheels Slip of wheels Absorbed by propulsion of vessel BP. Per cent, of Power. l *| 94-45] x8.8 3 5 ’} 60.45 j 1 18 X5-37 52.8 Screw IPropeller. Friction of engines. . “ of screw surface and resistance of edges of blades. Slip of propeller Absorbed by propulsion of vessel 663 IP. Per cent, of Power. 96. 06 ) 81.48} 18.83 53-44 6.83 205.55 26.27 375-92 48.04 782.45 100 Note. — From experiments of Mr. Froude, he deduced that, as a rule, only 37 to 40 per cent, of whole power exerted was usefully employed. With an auxiliary propeller, essential differences are in friction of surfaces and edges of blades of propeller and slip of propeller, being as 12 to 6 83 in excess in first case, and as 13.7 to 26.27 in second case, or 50 per cent less. Resistance of Bottoms of Hulls at a Speed of one Knot per Hour. Smooth wood or painted 01 lb. I Copper. 007 lb. Smooth plank 016 “ Moderately foul 019 k ‘ Iron bottom, painted 014 “ | Grass and small barnacles 06 “ Sailing. Ratio of* Effective Area of* Sails and of* 'Vessel’s Speed, under Sail to Velocity of Wind. Course. Ratio of Effective Area of Sails. Ratio of Speed of Vessel to Wind. Course. Ratio of Effective Area of Sails. Ratio of Speed of Vessel to Wind. 5 points of wind 2 abtftbeam.. •59 •33 Wind abeam .82 .6 • 9 1 •5 “ astern I .5 6 “ of wind .68 •5 “ on quarter •96 .66 Then = ^/i-5 2 = 1 139, hence area of sails a' = — - Rropnlsioix and Area of Sails. Plain sails of a vessel are standing sails, excluding royals and gaff topsails. Resistance of vessels of similar models but of different dimensions for equal speeds =-Dt Hence ^ . a and a' representing areas of sails of known and given ves- sels , and D and D' their displacements in tons. Illustration. — Assume D and D'==24 and 1600. 878 per centum. 1-139 In Vessels of Dissimilar Models . — Plain sail area should be a multiple of D*. Multiples for Different Classes of Vessels, R. N. Sailing. | Steamers. Ships of Line 100 to 120 j Ships, iron-clad 60 to 80 Frigates.. i I Frigates ) Sloops > 120 to 160 | Sloops ( 80 to 120 Brigs ) I Brigs ) English Yachts , designed for high speed, have multiples from 180 to 200, and when designed for ordinary speed from 130 to 180. When Area of Sail to Wet Surface of Hull is taken. — American yacht Sappho had a ratio of 2.7 to 1, and several English yachts nearly the same, while in some others it was but 2 to 1. 664 NAVAL ARCHITECTURE. Location of Masts, etc. Load-line = ioo. Vessel. Di Fore. [stance from Ster Main. n. Mizzen. Foot of Sail.* Height of Centre of Effect above Water-line = Breadth.* Ship Bark Brig Schooner Sloop 10 to 20 12 to 20 17 to 20 16 to 22 53 to- 58 54 to 60 64 to 65 55 to 61 36 to 42 80 to 90 81 to 91 125 to 160 130 to 160 160 to 165 160 t(M70 170 to 190 1.5 tO 2 1.5 to 1.95 1.5 to 1.75 1.5 to 1.75 1.25 to 1.75 * Measured from Tack of Jib to Clew of Spanker or Mainsail. Rake of Masts. Ships - Foremast o to .28 of length from heel, Main and Mizzen o to .25. Schooners.— Foremast .1 to .25, Mainmast .63 to .77. Sloops.— .08 to .11. Area of Sails. Sails. 3 Yards upon each Mast. 4 Yards upon 1 each Mast. Sails. J 3 Yards upon 1 each Mast. 4 Yards upon each Mast. J 4 b .08 .295 .417 .08 .295 •417 1 1 Mizzenmast 1 Spanker or ) ! j Driver. . . J .127 .081 .14 .068 Jfnyfi mast, Mainmast Proportional Area of Sails upon each Mast under above Divisions. Sail. | Fore. Main. Mizzen. Proportion to 1. Course Topsail Topgallant sail Royal Spanker or Driver Jib 115 105 • 075 .08 .097 .063 • 045 .08 . 162 .149 . 106 .138 .127 .089 .063 •075 .052 .081 .063 •045 .032 .068 •389 •358 •253 •33 •303 .215 .152 375 •375 .417 • 4*7 .208 .208 1 1 Balance of Sails .— lUiect 01 jiu is equal ^ ^ — mast, and sails upon mizzenmast balance those of foremast. Areas of sails upon masts of a ship should be in following proportion : Fore 1.4x4 I Main 2 | Mizzen. ....... . x When, therefore, main yard has a breadth of sail of ioo feet, fore yard should have 70.71 feet, and mizzen 50 feet -, topgallant and royal yards and sails being in same proportion. Angles of Heel for Different Vessels. Approximately. D representing displacement of vessel in lbs., M height of meta- centre above centre of gravity in feet, a angle of heel of Vf^tiUntor- culaf measure * and H height of centre of effect above centre of lateral resistance, in feet. Moment of sail should he equal to moment of stability at a defined angle of heel. Frigates, etc. 4° Corvettes • •• 5 ° . , Circular Angie- Measure. .07 .087 . . Circular Angle. Measure . Schooners, etc 6° .105 Yachts 6° to 9 0 .105 to .107 Illustration. -Assume displacement 170 . tons ' 6 ’ 75 feCt ’ H = 36 feet, and angle of heel 9 0 ; what should be area of sails . 170 X 2240 = 380 800 lbs. 9 0 = • i° 7 - 380800X6-75 x. 107 _ 7 g 39 . 8 sq.feet. 36 * See rule, page 113. NAVAL ARCHITECTURE. 665 Trimming of Sails. That a vessel’s sail may have greatest effect to propel her forward, it should be so set between plane of wind and that of her course, that tangent of angle it makes with wind may be twice tangent of angle it makes with her course. Or, tan. a — 2 tan. b. a representing angle of sail ivith wind, and b angle of sail and course of vessel. -Angles of Course and Sails with. Wind. Wind Ahead. Angle of Course. Tan- gent. Half Tan- gent. Angles with Wind. of Sail with Course. Wind Abaft. Angle Course. Tan- gent. Half Tan- gent. Angles with Wind. of Sail with Course. Points. 4 45° .562 .281 29 0 18' 15 0 42' Points. 2 112 0 30' 2.166 1.082 65° 13' 47° 17' 5 56° i 5 ' •732 •365 36° i2' 20° 3' 3 !23° 45' 2-737 1.368 69° 56' 53° 49' 6 67° 3°' •923 .461 42° 43' 24 0 45' 4 135° 3-562 1.781 74° 17' 6o° 43' Abeam 90 0 I-4I5 .707 54° 45' 35° 16' 6 157° 3o' 7-5 11 3-754 82° 25" 75° S' Kffective Impulse of Wind. -AX Fig. 6. Let P 0, Fig. 6, represent direction by com- x pass and force of wind on sail, AB; from P draw P C parallel to A B, from o draw 0 C per- pendicular to A B ; o C is effective pressure of wind on sail A B, and r C, perpendicular to plane of vessel, is component of o C, which pro- duces lateral motion, as heel and leeway, and r 0 is component of 0 C, which propels vessel. I sin. a=P; Pcos. * = L; and Psin. a = E. I representing direct impact and P effective pressure of wind on sail , L effective impact producing leeway , and E effective impact which propels vessel. Note. — The law as usually given is sin. 2 . This is manifestly incorrect, as it gives results less than normal pressure for angles of small incidence. At an angle of in- cidence of wind of 25 0 , the law of sin. is exact. Hence, although it may not be exact at all angles, it is sufficiently so for practical purposes. Illustration 1.— Assume wind 5 points ahead, and I = 100 lbs. By preceding table angle of course with wind 56° 15'; hence angle of sail a , with wind 36° 12', as tan. 36° 12' = 2 tan. 20 0 3', and angle x 56° 15' — 36° 12' = 20° 3'. Then, 100 X sin. 36° 12' = 100 X .5906 = 59.06; 59.06 X cos. 20 0 f — 59.06 X •9394 = 5S-48, and 59.06 X sin. 20 0 3' = 59.06 X .3426 = 20.23 lbs. 2.— Assume wind 4 points abaft, and I — - 100 lbs. Then, iooXsin. 2 74° 17'= 100 X .9626 s == 92.66 ; 92.66 X cos. 180 0 — 74O I7 '_j_ 45 o. = 6o° 43 r = 92.66 X -49 = 45.41, and 92.66xsin. 6o° 43' = 92.66 X .8722 = 80.82 lbs. To Compute Sailing Power of a "Vessel. F /sin. w, sin. 5 = P. To Compute Careening Power of a Sailing Vessel. F / sin. w, cos. s — P. F representing area of sails in sq. feet, f force of wind in lbs. per sq. foot, w angle of vjind to sails , and s angle of sails to course of vessel. To Compute Angle of Steady Heel. Within a Range of 8°. a PE . TT . J - j) = sin. H. a representing area of plain sail in sq.feet, P pressure of wind in lbs. per sq. foot , E height of centre of effect above mid-draught , in feet, D displace- ment of hull , in lbs . , and M height of meta-centre in feet. P assumed at 1 lb. per sq. foot , or that due to a brisk wind. Illustration.— Assume a = 15600, draught =±20, and M = 62 72, D = 6 800 000, and M = 3. Then x 5 600 X x X 72 1x23200 6 800 000 X 3 20 400 000 = -0555 = 3° 10'. 3K* hence 62 4- — = 10 666 NAVAL ARCHITECTURE. Course and. Apparent Course of Wind. Apparent course of a wind against sails of a vessel is resultant of normal course of wind and a course equal and directly opposite to that of vessel. Illustration. — I f P, Fig. 7 , repre- ss* 7- — sent direction by compass and force of wind, and a b direction and velocity of vessel, from P draw P c parallel and equal to a b , join c a and it will repre- sent direction and force of apparent wind. Or — — ratio of velocity of apparent ’ c P wind to that of vessel, ^ = ratio of velocity oj wind to that of vessel. Resistance of Air. {Mr. Fronde.) Resistance of wind to a vessel is estimated as equivalent to square of its In a calm resistance of air to a steamer = one thirty-fourth part of resist- ance of water, and when a steamer’s course is head-to, and combined veloc- ity of vessel and wind = 15 knots, resistance is one ninth of that of the water. Resistance of air to a sq. foot of surface at right angles to course of a ves- sel is about 33 lb., and when surface is inclined to direction of wind, press- ure varies as sine of angle of incidence. Mean of angles of surface of a steamer exposed to wind may be taken at 45 0 ; hence their resistance is about .25 lb. per sq. foot when wind has a locity of 10 knots per hour. If sectional area of a steamer’s hull above water .is 750 s^feet, fibf^nd to air at a speed of 10 knots in a calm would be 750 X .25 — 187.5 lbs., and resistance to smoke-pipe, spars, and rigging (brig rigged) would be 201 lbs. Leeway. Annie of Leeway in good sailing vessels, close hauled, varies from 8° to 12°, and in inferior vessels it is much greater. Ardency is tendency of vessel to fly to the wind, a consequence of the centre of effort being abaft centre of lateral resistance. Slackness is tendency of vessel to fall off irom the wmd, a consequence of the centre of effort being forward centre of lateral resistance. Results of Experiments upon Resistance of Screw ■propdlers, at High Velocities and Immersed at Varying Depths of \\ ate) . Immersion of Screw. Resistance. Immersion of Screw. Resistance. Surface. 1 2 feet. 7 1 foot. 5 3 “ 7-5 1 Immersion of 1 Resistance. Screw. 4 feet. 7.8 5 “ 1 8 Slip of Propeller, 15 per cem. ; 01 w ing axes of blades as the centre of pressure, 23 per cent. Freeboard. Measured from Spar-de.de stringer to surf ace of water D, epth of Hold from under • J side of spar deck to top of ceiling. Hold. II Hold. | Hold. | Feet. Ins. Feet. Ins. Feet. | 8 1-5 12 2.25 16 | *o 2 || 14 2-5 18 | Ins. 2.75 3 Ins. 3-125 3-25 Hold. Feet. 24 26 Ins. 3-375 3-5 Feet. 28 I 3° 3.625 3-75 NAVAL ARCHITECTURE. 667 _ „ VI a, ting Iron. Hulls. D L S oob d = T ‘ D re P resentin 9 displacement in tons , L length of hull, b breadth , and d depth. Or, .o$fy/d = T. f representing distance between centres of frames, and d depth of plate below load-tine, all in feet, and T thickness of plate in ins. NLasts and. Spars. Lower masts at spar deck. I Bowsprit “ stem. Topmasts “ lower cap. Topgallant masts “ topmast cap. | Fore and main masts, when of pieces, D „„ length. Mizzenmast .66 diameter of mainmast. Masts of one piece“i inch for each 3-5 t0 3-75 feet of whole length. Bowsprit, depth, equal diameter of mainmast; width, diameter equal to foremast. Main and fore topmasts 1 inch for each 3 to 3.25 ) Diameter for Dimensions. Jib-boom at bowsprit cap. Yards in middle. Gaft’s at inner end. Main and Spanker booms at taffrail. inch for each 3 to 3.25 feet of whole Mizzen topmast Topgallant masts 1 Royal masts 1 Topgallant poles 1 Jib-boom 1 Fore and main yards Topsail yards Cross -jack, Topgallant, and) Royal yards j Main and Spanker booms Gaffs Studding-sail yards and booms. 3-33 I 3.33 }-feet of whole length. 875 ‘ 3-25 “ 3-33 I 3-25 “ 3-33 Y 3.66 | 2.87 J 2 ft. of length beyond bowsprit cap. 4 Pd = T; .196 C D 3 4 5 3-5 3- 5 to 4 4 - 5 t0 4-75 Rudder Head. ( Mackrow .) T feet of whole length. = M; 3 / — T — — D; ’ V *196 C ’ , A v 2 and — P. P representing press- ure on rudder when hard over, in tons, d distance of geometrical centre of rudder from axis of motion, in ins., T stress on head, and M moment of resistance of head, both in inch-tons, A immersed area of rudder in sq. feet , v velocity of water passing rudder in knots per hour, and C coefficient = 3. 5 per sq. inch for Iron, and .125 for Oak. Illustration. — Assume area of wooden rudder 24 sq. feet, distance of its geomet- rical centre from centre of pintles 2 feet, and velocity of water 10 knots. 1X2X12 = 24 inch-tons. - == 9.93 ins. 2 4 °° ' V - *96 X .125" Memoranda. Weights. — A man requires in a vessel a displacement or 488 lbs. per month for baggage, stores, water, fuel, etc., in addition to his own weight, which is estimated at 175 lbs. A man and his baggage alone averages 225 lbs. A ship, 150 feet in length, 32 beam, and 22.83 in depth, or 664 tons, C. H. (0 M ) has stowed 2540 square and 484 round bales of cotton. Total weight of cargo I 254448 lbs., equal to 4.57 bales, weighing 1889 lbs., per ton of vessel. A full built ship of 1625 tons, N. M., can carry 1800 tons’ weight of cargo or stow 4500 bales of pressed cotton. Hull of iron steamboat John Stevens — length' 245 feet, beam 31 feet, and hold II feet; weight of iron 239440 lbs. And of one other— length 175 feet beam 24 feet, and 8 feet deep; weight of iron 159 190 lbs. Weight of hull of a vessel with an iron frame and oak planking (composite) com- pared with a hull entirely of wood, is as 8 to 15. An iron hull weighs about 45 per cent, less than a wooden hull. Iron ship, 254 feet in length, 42 beam, and 23.5 hold, 1800 tons register, has a stow- age of 3200 tons cargo at a draught of 22 feet. Weight of hull in service 1450 tons. Loss by Weight per Sq. Foot per Month of Metalling of a Vessel's Bottom in Service. Copper .0061 lb. ; Muntz metal .0045 lb. ; Zinc .007 lb. ; and Iron .0204 lb. Comparison between Iron and Steel plated Steamers.— In a vessel of 5000 tons displacement, hull of steel-plated will weigh 320 tons less = 6.^6 pep centum less. 668 OPTICS. OPTICS. Mirrors, in Optics, are either Plane or Spherical. A plane mirror is a plane reflecting surface, and a spherical mirror is one the reflecting surface of which is a portion of surface of a sphere. It is concave or convex, ac- cording as inside or outside of surface is reflected from. Centre of the sphere is termed Centre of curvature. Focus — Point in which a number of rays meet, or would meet if produced. Fig. i. Principal Focal Distance is half radius of curvature, and is generally termed the j focal distance. Line ac is termed the principal axis , and any other right line itz through c which meets the mirror is termed a Secondary axis. When the incident rays are parallel to the principal axis , the reflected rays converge to a point, F. Conjugate Foci are the foci of the rays proceeding from any given point in a spherical concave mirror, and which are reflected so as to meet in an- other point, on a line passing through centre of sphere. Hence, their relation being mu- tual, they are termed conjugate. Let P be a luminous point on principal axis. Fig. 2 , and P i a ray; draw the normal line c ?, which is a radius of the sphere; then c i P is an- gle of incidence, and ci 0 the angle of reflection, equal to it; hence c i bisects an angle of triangle i P c P P i 0, and therefore, ’ ’ i 0 c O When conjugate focus is behind a mirror, and reflected rays diverge, as if emanating from that point, such focus is termed Virtual, and a focus in which they actually meet is termed Real. As a luminous point, as P, Fig. 3 , is moved to the mirror, the conjugate focus moves up from an indefinite distance at back, and meets it at surface of mirror. If an incident ray converges to a point s, at back of mirror, it will be reflected to a point P in front. The conjugate foci P s having changed places, Pencil . — Rays which meet in a focus and are taken collectively. Objects .— As regards comparative dimensions or volumes, it follows, from similar triangles, that their linear dimensions are directly as their distances from centre of curvature. Xo Compute Dimension or "Volnme of ail Image. When Dimensions and Position of Object are Given , and for either Convex or Concave Mirrors . L 5. or _L — 0L L and l representing lengths of image and object , F focal l F ’ L F . . , length , and D and d respectively , distances of image and object from principal Jocus. , Refraction . Deviation. — Angle at which a ray is diverted from its original or normal course when subjected to refraction is thus termed. Indices of Refraction. — Ratio of sine of angle of incidence to sine of angle of refraction, when a ray is diverted from one medium into another, is termed relative index of refraction from former to latter. OPTICS. 669 When a ray is diverted from vacuum into any medium, the ratio is greater than unity, and is termed absolute index or index of refraction . Mean Indices of Refraction. Glass, lead, 3 flint 2.03 “ lead 2, sand 1 1.99 “ “ 1, flint 1 1.78 Ice 1. 31 Quartz........ 1.54 Eye, vitreous humor 1.339 crystalline lens, under 1.379 “ “ “ central 1.4 Diamond 2.6 Glass, flint 1.57 For indices of other substances, see page 584. Heat increases refractive power of fluids and glass. Critical Angle . — Its sine is reciprocal of index of refraction, the incident ray being in the less refractive medium. Visual Angle is measure of length of image of a straight line on the retina. Total Reflection is when rays are incident in the more refractive medium, at an angle greater than the critical angle. Mirage. — An appearance as of water, over a sandy soil when highly heated by the sun. Caustic Curves or Lines are the luminous intersections from curve lines, as shown on any reflective surface in a circular vessel. To Compute Index of Refraction. ^ = Index. I representing angle of incidence, and R that of refraction. To Compute Refraction. Concave-Convex and Meniscus. — Effect of a concave-convex in refracting light is same as that of a convex lens of same focal distance, and that of a meniscus is same as a concave lens of same focal distance. 2 R 7* Meniscus, with parallel rays — — - = F. Magnifying Power. — In Telescopes the comparison is the ratio in which it apparently increases length. In Microscopes the comparison is between the object as seen in the instrument and by the eye, at the least distance of vision, which is assumed at 10 ins., and the magnifying power of a micro- scope is equal to the distance at which an object can be most distinctly ex- amined, divided by the focal length of the lens or sphere. Linear po-wer is number of times it is magnified in length, and Super- ficial, number of times it is magnified in surface. Magnifying power of microscopes varies, according to object and eye- glass, from 40 to 350 times the linear dimensions of object, or from 1600 to 122500 times its superficial dimensions. Apparent Area. — As areas of like figures are as the squares of their linear dimensions, the apparent area of an object varies as square of visual angle subtended by its diameter. The number expressing Magnification of Apparent Area is therefore square of magnifying power as above described. Illustration. — If diameter of a sphere subtends i° as seep by the eye, and io° as seen through a telescope, the telescope is said to have a power of 10 diameters. OPTICS, 67O To Compute Elements or Mirrors and. Lenses. Or l r Mirrors. Spherical Concave.* • — — D ; =r = L. r — il r — zl Or Lr c2 2 Spherical Convex . t - y_^ - = D ; . - T - Parabolic Concave. Unequally. Convex, t 2 B r 2L-fr =r F. Plano- Convex. § 2 B - Sphere. 16 A .66 £ = F. I = F. ! 1 I i= F. K + r Hyperbolic Concave.W Elliptic Concave .If 0 representing object — 1, r radius of convexity, l and L length or distance of object from vertex of curve, and from external vertex, D dimension of object, d. diameter of base, ¥ focal distance, and h depth of mirror in lilce dimensions, I index of refraction, and t thickness of lens. Illustration i. — Before a concave mirror of 5 feet radius is set an object at 1.5 feet from vertex of curve; what is ratio of apparent dimension of image, and what is length of and distance of object from external vertex ? Object = 1. — 1 X5 — = 2.5 feet, and - X 5 = 3. 75 feet. 5 — 2X15 5 — 2X1.5 2.— If object is set at 4.5 feet from vertex of a like mirror, what is length of and distance of inverted object from internal vertex? *X 5 4 - 5 X 5 = 1.25 feet, and ; = 5.625 feet. 2X4-5 — 5 ' 2 X 4 f - 5 _ 3. — Before a convex mirror of 3.5 feet radius is set an object at 3 feet from ver- tex of curve; what is length of and distance of object from external curve? . . ** . 3 ‘ . 5 _^ — .368 foot, and 3X j 5 - ~ — * 2 X 3 + 3- 5 6 2 X 3 4" 3-5 : 1. 105 feet. 4. — A parabolic reflector has a depth of 1.25 feet and a diameter of 2 feet; what is its focal distance from vertex of internal curve? 16 X 1 25 Lenses. Double Convex. = .2 feet or 2.4 ins. R r 0 F = D; l ¥ ¥ — l 7 ¥ — l Double Concave. = L; Rr to — i X R + r S-fF OF F ~ ’ F — 0 " - , — F. When R — r = F; = V ; and Fo-D S F S + F = 0 . T A hF 7 = L; and - — r-= = l. -JXR+? u L + F Optical centres are in centres of lens. Plano - Convex and Plano - Concave. — F. Optical centres are respectively centres of convex and concave sur- Rr TO — I faces. Convex Concave ( Meniscus ) and Concavo-Convex. to — 1 X R — t z = ¥. Optical Centres. Convex Concave. Delineate lens in half section, draw R from its centre to circumference of lens (intersection of radii), draw r parallel thereto and extending to its circumference, connect R and r at these external points of contact with circumference and external curve, extend line to axis of lens, and point of contact is centre required. Concavo-Convex. Proceed in like manner, but in this case r extends to, or delineates, the inner surface of the lens, and point of con- tact with axis is centre required. * D or image disappears when l — .5 r. Lr t When 0 is beyond F, it will be inverted, as 2 I — r § When convex side is exposed to parallel rays and — - - = l. t When equally convex F = R. and when parallel rays fall upon plane side, F = 2 R. (1 Rays of light, heat, or sound, reflected from focus of a liyperbola, will diverge from its concave surface, ‘ft and when from the focus of an ellipse, will be refracted by surface of the other. OPTICS. — PILE-DRIVING. 67I When object is beyond focal distance (F), its image (D) will be inverted, as ^ — — = D, and j — = l. P representing magnifying power of lens, S limit of normal sight , 10 to 12 ins. for far-sighted eyes and 6 to 8 for near-sighted , ordinarily 10 ins . , V limit of distinct vision , 0 extreme distance of object from optical centre at distinct vision, and m index of refraction. Illustration i. — If a double convex lens of flint glass lias radii of 6 and 6.25 ins , 2. — If a double concave lens has a focal distance of 2 ins., and object is 6 ins. from vertex of curve, what is its dimension and what is its distance from vertex of inner curve ? 3.— If focal distance of a single microscope is 4 ins., what is its limit of distinct ing length of focal distance from object lens. Illustration. — Principal focal distance of ocular lens of a telescope is .9 in., of objective lens 90 ins. ; what is its magnifying poiver? Effect of blow* of a ram, or monkey, of a pile-driver, is as square of its velocity ; but the impact is not to be estimated directly by this rule, as the degree and extent of the yielding of the pile materially affects it. The rule, therefore, in application, is of value only as a means of com- parison. By my experiments in 1852, to determine the dynamical effect of a fall- ing body, it appeared that while the effect was directly as the velocity, it was far greater than that estimated bv the usual formula Vs 2 y, which, for a weight of 1 lb. falling .2 feet, would be 11.34 lbs., giving a momentum of 11.34 foot-lbs. ; vdiereas, by the effect shown by the record of actual obser- vations, it would be W v 4.426 = 50 lbs. Piles are distinguished according to their position and purpose: thus, Gauge Piles are driven to define limit of area to be enclosed, or as guides to the permanent piling. Sheet or Close Piles are driven between gauge piles to form a compact and continuous enclosure of the work. Weight which each pile is required to sustain should be computed as if the pile stood unsupported by any surrounding earth. A heavy ram and a low r fall is most effective condition of operation of a pile-driver, provided height is such that force of blow will not be expended in merely overcoming friction of leader and inertia of pile, and at same time not from such a height as to generate a velocity which will be essentially expended in crushing fibres of head of pile. what is its focal distance ? Index of refraction = 1.57, see page 584. 1.57 — 1 X 6 + 6.25 vision, and what its magnifying pow*er? 0 = 2. 857 ins. 2.857 X 4 • A IO + 4 ,. — — r= 10 ins., and — — = 3-5 times. 4 — 2.857 5 4 Telescopes, Opera-glasses, etc. 90-4- .9 = 100 times the object. PILE-DRIVING. *+ for telescopes and — for opera-glasse9, etc. PILE-DRIVING. 672 Refusal of a pile intended to support a weight of 13.5 tons can be : taken at 10 blows of a ram of 1350 lbs., falling 12 feet, and depressm 5 ^ pile .8 of an inch at each stroke.* Pneumatic Piles .- A hollow pile of cast iron, 2.5 feet in diameter, was depressed into the Goodwin Sands 33 feet 7 ins. in 5.5 hours. Nasmyth's Steam Pile-hammer has driven a pile 14 ins. square, and 18 feet m length 15 feet into a coarse ground, imbedded in a strong clay, in 17 seconds, with 20 blows of monkey, making 70 strokes per minute. Morin computed work of a ram in foot-lbs., in raising a monkey for 8 hours per day as follows: Tread-wlieel 3900, Winch 2600. French engineers estimate the safe load for a pile, when driven to refusal of .4 inch under 30 blows, to be 25 tons. Shaw's Gunpowder Pile-driver is operated by cartridges of powder on head of pile, which are ignited by fall of the ram. 30 to 40 blows per minute have been made under a fall of 5 aud 10 feet. 27 piles have been duven m rough gravel and clay 7.2 feet in one day. To Compute Safe Load tliat may toe Borne toy a Bile. (Maj. John Sanders , U. S. E.) Approximately. = W. E representing weight of ram in lbs . , h height of fall and d distance pile is depressed by blow , both in feet. Illustration.— A ram weighing 3500 lbs., falling 3.5 feet, depressed a pile 4.2 ins. Then 35oo X (42 4- a) _ 35og> _ 43?5 i bs ^ we ight which pile would bear with 8 8 safety. Molesworth gives this, but with a variation in symbols and their expression. To Compute Coefficient of Resistance of the Earth. — C. R representing resistance of the earth , and d as preceding. Weisbach gives following formula : Resistance of bed of earth being con- stant, mechanical effect expended in penetration of pile will be p P representing weight of pile in lbs. Illustration. — Assuming elements of preceding case, with addition of weight ot pile at ,500 lbs., ■ 3500^x 3-5. _ 43875 ooo_ 01 ^„,,,_ 1500 -f- 35°° X (4* 2 I2 ) *75° To Compute "Weight of* Ram. (Molesworth.) P (JlK. = R. P representing weight of pile in lbs ., h height of fall and L \ 5 A L / ; length of pile, both in feet, and A area of section of pile m sq. ins. Fall. 1000 Wei 1200 ght. 150a 2000 Fall. IOOO Wei 1200 gbt. 1500 Feet. 1 5 10 Lbs. 8000 17 920 25 360 Lbs. 9 600 21 504 3°432 Lbs. 12 000 26 880 3 3 °40 Lbs. 16000 35 840 50 720 Feet. 15 20 25 Lbs. 31 060 35860 40 100 Lbs. 37 272 43032 48 720 Lbs. 46 590 53 79 ° 60150 Lbs. 62 120 71720 80200 Sheet Riling. Bevelling 120 0 | Shoeing 25° Ringing Engine Requires 1 man to each 40 lbs. weight of ram, which varies from 450 to 900 lbs. PILE-DRIVING. PNEUMATICS. AEROMETRY. 673 Dile-sinliing. Mitchell's Screw Piles are constructed of a wrought-iron shaft of suitable diameter, usually from 3 to 8 ins., with 1.5 turns of a cast-iron thread of from 1.5 to 3 feet diameter. Hydraulic Process is effected by the direction of a stream of water under pressure, within a tube or around the base of a pile, by which the sand or earth is removed. Pneumatic and Plenum Process. — For illustration and details, see Traut- wine’s Engineer’s Pocket-book, page 326. Dr. Wkewell deduced the following results : 1. A slight increase in hardness of a pile or in weight of a ram will con- siderably increase distance a pile may be driven. 2. Resistance being great, the lighter a pile the faster it may bQ driven. 3. Distance driven varies as cube of the weight of ram. Relative Resistance of Formations to Driving a Pile. Cora l 100 I Hard clay 60 I Light clay and sand. . . 35 Clay and gravel 83 | Clay and sand 45 | River silt 25 PNEUMATICS.— AEROMETRY. Motion of gases by operation of gravity is same as that for liquids. Force or effect of wind increases as square of its velocity. If a volume of air represented by 1, and of 32 0 , is heated t degrees without assuming a different tension, the volume becomes (1 + .002088 t)=Y; and if it requires a temperature in excess of t' 32 0 , it will then assume volume (r -f .002088 t' — 32°). All aeriform fluids follow this law of dilatation as well as that of compression proportional to weight. When air passes into a medium of less density, its velocity is determined by difference of its densities. Under like conditions, a conduit will discharge 30.55 times more air than water. To Compute the Degree of Rarefaction that may "be ef- fected. in. a 'Vessel. Let quantity of air in vessel, tpbe, and pump be represented by 1, and proportion of capacity of pump to vessel and tube by .33 ; consequently, it contains .25 of the air in united apparatus. Upon the first stroke of piston this .25 will be expelled, and .75 of original quantity will remain ; .25 of this will be expelled upon second stroke, which is equal to .1875 of original quantity; and consequently there remains in apparatus .5625 of original quantity. Proceeding in this manner, following Table is deduced : No. of Strokes. Air Expelled at each Stroke. Air Remaining in Vessel. 1 to U\ II cln •75 — -75 3 _ 3 9 _3 X 3 2 16 4X4 16 4 X 4 9 _ 3X3 27 3X3X3 3 64 ~ 4 X 4 X 4 64 4X4X4 And so on, multiplying air expelled at preceding stroke by 3, and dividing it by 4; and air remaining after each stroke is ascertained by multiplying air remaining after preceding stroke by 3, and dividing it by 4. 2 L PNEUMATICS.— AEROMETRY. 674 Feet. Miles. 460 .087 15 840 3 16000 3.02 10 560 2 15 840 3 575 000 90 Distances at which. Different Sounds are -A/udible. A full human voice speaking in open air, calm In an observable breeze, a powerful human voice with the! wind can be heard • ... j Report of a musket. . . . . . Drum 10560 Music, strong brass band 15 840 Cannonading, very heavy In Arctic Ocean, conversation has been maintained over water a distance of 6696 feet. In a conduit in Paris, the human voice has been heard 3300 feet. For an echo to be distinctly produced, there must be a distance of 55 feet. Coefficients of Efflux of Discharge of Air. (D 1 Aubuisson.) Orifice in a thin plate. . 65 .751 Cylindrical ajutage .93 -958 Slight conical ajutage 94 1.09 To Compute Volume of Air Discharged through an Open- ing into a Vacuum, per Second. a C V2 g h = V in cube feet a representing area of opening in square feet, C co- efficient of efflux, and V 2 g h — 1347.4, as shown at page 428. Illustration. — Area of opening 1 foot square, and C = . 707. Then 1 X .707 X 1347.4 = 952.61 cube feet. Inversely, V - 4 - a = velocity in feet per second. ■Velocity and Pressure of Wind. Pressure varies as square of velocity, or P oc V 2 . V 2 X-oo5 = P; V200 P — V; d 2 x.oo 23 = P; and .0023 v 2 sin. x = P. V representing velocity in miles per hour , v in feet per second , P pressure in lbs. per sq.foot, and x angle of incidence of wind with plane of surface. Table deduced from above Formulas. Velocity Feet. 88 176 264 352 440 528 7°4 880 1320 1760 Pressure on a Sq. Foot. Lbs. .005 .02 ) • 045 ) .08 .125) .x8 .32 ; • 5 1.125 Character of the Wind. Barely observable. Just perceptible. Light breeze. Gentle, pleasant wind. Fresh breeze. Brisk blow. Stiff breeze. Velocity Miles. 25 30 35 40 45 50 60 80 9 ° Feet. 2200 2640 3080 3520 3960 4400 5280 7040 7920 8800 Pressure on a Sq. Foot. Lbs. 3 4o-5l 50 j * Character of the Wind. Very brisk. High wind. Very high wind. Gale. Storm. Great storm. Hurricane. Tornado. Illustration. — What is pressure per sq. foot, when wind has a velocity of 18 miles per hour? l8 2 x od5 _ l 6z ^ To Compute Force of Wind upon a Surface. - sin i — — p. v representing velocity of wind in feet per second, a area of 440 surface in sq.feet , and A angle of incidence of wind. At Mount Washington wind has been observed to have had a velocity of 150 miles per hour. Extreme pressure of wind at Greenwich Observatory for a period of 20 years was 41 lbs. per sq. foot. PNEUMATICS. — AEEOMETEY. 675 Force of wind upon a surface, perpendicular to its direction, has been ob- served as high as 57.75 lbs. per sq. foot; velocity = 159 feet per second. Dr. Hutton deduced that resistance of air varied as square of velocity nearly, and to an inclined surface as 1.84 power of sine X cosine. Figure of a plane makes no appreciable difference in resistance, but con- vex surface of a hemisphere, with a surface double the base, has only half the resistance. At high velocities, experiments upon railways show that the resistance becomes nearly a constant quantity. Direction in Northern Hemisphere. Course of AWixid.. Cyclones. Wind has its direction nearly at right angles to line between points of highest and lowest pressure of air, or barometer readings, and its course is with the point of lowest pressure at its left, and its velocity is directly as Direction in Southern Hemisphere. difference of the pressures. In Northern Temperate zone, winds course around an area of low pressure in reverse direction to course of hands of a watch, and they flow away from a location of high pressure, and cause an apparent course of the winds in di- rection of course of the hands. To Compute Resistance of* a Diane Surface to Air. .0022 av 2 = P in lbs. a representing area of plane in sq.feet , v velocity in direc- tion of wind in feet per second , -f- when it moves opposite , and — when with the wind. To Compute Resistance of a Plane Surface when moving at an Angle to Air. v a sin. x _ p ^ x re ^ resen ti n g angle of incidence. 45o To Compute Height of a Column of Mercury to induce an iCfflnx of Air through a given Nozzle. Barometer assumed at 2. 46 feet = 29. 52 ins. , and Temperature 52 0 . * , — — H, and 48.073 d 2 -^/H = P. d representing diameter of nozzle and H 48.073 2 d 4 height of column of mercury, both in feet , and P volume of air in lbs. per one second. Illustration. — Assume d — .19, and P = .7 lbs. - = .1511 foot. 48.073 X .iff .1511 = - 7 - 8.073 2 X • 19 To Compute Pressure or 'Weigh. t of Air under a given Height of Barometer and Temperature, Discharged in One Second. 30.787 d 2 B —pressure in lbs. Or, 48.073 d 2 fB — lbs. b representing height of barometer in external air , B manometer or pressure of air in reservoir in mercury , both in feet, and t temperature of air or gas in degrees. Illustration. — Assume b = 2. 5 feet ; d = . 25 foot ; B = . 1 foot ; and t = 1.055 f ee k Then 30.787 X .0625 f-i X 2.5-f.I = 1.924 x V- 2 4 6 5 = -9543 i-o55 676 PNEUMATICS. AEKOMETKY. To Compute Temperature for a. given. Latitude and Ele- vation. 82. 8 cos. I — .001 981 E — . 4 — t. E representing elevation in feet. Illustration.— Assume 1 = 45 0 ; cos. =.707; and E = 656 feet. Then 82.8 X -707 — .001 981 X 656 — .4 = 58-54 — 1-299 — -4 = 58-54 — -899 = 57.641. To Compute Volume of Air or Gras Discharged through, an Opening and under a Pressure above that of* Ex- ternal -A-ir. A ir. 1347.4 C ^ VB (6' + B) T = V in cube feet per second. T = 1 -f- .002 22 ( t — 32 0 ), and 6' = 2.5 — .00009 elevation. Or, 621.28 d 2 \/B = V. Illustration. — What would be volume of air that would flow through a nozzle .246 foot in diam. from a reservoir under a pressure of .098 foot of mercury, into air under a barometric pressure of 2.477 f ee b temperature of air 55. 4 0 , location 45 0 of latitude, and at an elevation of 650 feet above level of sea? C = . 75 ; b' = 2.5 — .00009 X 650 = 2.4415 (2.44); and T = i.c>502. Then 1347.4 X .75 V'098 (2.44 + .098) X 1.0502 = 24.689 X V -^7 - 12.63 2 -477 cube feet. When Densities of External Air and that in Reservoir are Equal. 1 347< 4 C ^ V B(6 + B)T = V. b' representing height of mercury in reservoir. Q as — — V. p representing specif c gravity of gas compared Vp V H4 2 X^ with air , and L length of pipe or conduit in feet. Illustration.— If a pipe .05 feet in diameter and 420 feet in length, communi- cates with a gasometer charged with carburetted hydrogen (illuminating gas), under a water pressure as indicated by a manometer of .1088 foot, what would be the dis- charge per second ? d = .05 foot ; L = 420 feet ; and B = = .008 foot. Specific gravity of gas .5625. 4231 / .008X .05 5 = 4231 /.poo 006 002 50^ 0 = .013 71 cube foot. V.5625 V 4 2o+i2XT^ -75 V 420H-2.X Resistance of Curves and Angles— Curves and angles increase resistance to discharge of air or gas very materially. By experiment of D’Aubuisson 7 angles of 45 0 reduced discharge of gas" one fourth. To Compute Diameter of* Discharge-pipe or Nozzle. When Length and Diameter of Pipe , Volume, and Pressure are given. . / 42 V 2 . 4 / a— . - — d in feet. \ 4230 2 Bd 5 — L V 2 Illustration. — If a pipe 1000 feet in length, and .4 foot in diameter, leads to a reservoir of air, under a mercurial manometric pressure of .18 foot, what diameter must be given to a nozzle to discharge 4 cube feet per second? Then « /■-• 42 X 4 2 X -4 s " 4/ 6-88»*8 = ^ = V 4*30* x 18 X -45— 1000 X 4 a V 32 980. 19- >6ooo .1418/00^ = 1.703 ins. Volumes of two gases flowing through equal orifices, and under equal pressures, are in inverse ratio of square roots of their respective densities. Specific gravity of mercury compared with water. RAILWAYS. 677 Fig. 1. RAILWAYS. To Define a Curve.— Dig. 1. ( Molesworth .) ^ &?*?-$ or Z tan. a? = R ; R (cotan. a?) == * ; 1719 c = a; R (cosec. x — i) = d-, R (cosin. *) = s ; R (coversin. a) = V ; — w, and (5400 — x) .000582 R — Z. c representing any chord , « length of tangent, d distance of centre of curve from, in- tersection of tangents, s half chord of curve, and l length of curve, all inlike dimensions a tangential angle ofc in minutes, n number of chords in curve, and x half angle of intersection, but in form'ulas for number of chords and length of curve to be expressed in minutes . Illustration. — Assume radius 900 and chord 400 feet; angle of intersection .= 12 0 44' = 764 minutes, and x = 56° 15' 5". Tangent of 56° 15' 5" = x. 496.33. Cotangent == .668 14. 1719 X 4 °° __ r — g QO f ee t ; I 7 I 9 X 4°^ _ ^4 minutes ; 900 X • 668 14 = t = 764 [ 9 00 601.33 feet; 900 X 1.20269 — 1 = d== 182,42 jM; 900 X • 555 55 — $ — $oofeet; 900 X .16833 = V = 161. 5 feet; — 2.645 times, and .000.582 X 900 X 764 5400 — 3379 = 1058 .6 feet Tangential Angles for Chords of One Chain . Radius of Curve. Tangential Angle. Radius of Curve. Tangential Angle. Radius of Curve. Tangential Angle. Radius of Curve. Tangential Angle. Chains. 5 8 9 10 12 5° 43-8', 3° 34-87 3° , 2° 51.9 / 2° 23.25 Chains. 15 20 25 30 35 i° 54-6' i° 25-95 1° 8.76' 57-3', 49 s11 Chains. 40 45 50 60 70 42.9/ 38.2 34.38' 28.65 24-55' 1 mile 1.25 mil’s 1.5 miles 1. 75 “ 2 21.48' I 4-33 12.28 10.74' INvJl £1. a ^ ^ . . 0 chain chords is .5 the angle for 1 chain chords. Curves of less than 20 chains radius should be set out in . 5 chain chords. Curves of more than 1 mile radius may be set out in 2 chain chords. Angles in above Table are in degrees, minutes, and decimals of minutes. Fig. 2. I Sidings. 2 y/d R — (.5 df = l. R representing radius of curve, l length of curve over points, and d distance between tracks, all in feet. Fig. 3- Turn-out of* Unequal Dadii. -y\ x—y = Z) o + b = Z; r — y = A; r x R ~+r~ y/ y {r -\- A) — a\ R 1 - z = B; y/z (R + B) _ 6. R and r representing radii of the curves re- spectively as to length , x distance between outer rails of tracks and other symbols as shown, all in feet. 3 L* 6y 8 RAILWAYS. Fig. 4. Points and. Grossings. . l G V (R-j-s) G z= l] — =sin. a; ’ R ver. sin. a — R. R repre- senting radius of curves, G gauge of road, a angle of crossing, and x — R — G, all in feet. In horizontal curves, width required for clearance of flange of wheel, and for width of rail at heel of switch, render it necessary to make an allowance in length of l, as ascertained by formula. For other diagrams and formulas, see Molesworth’s Pocket- book, pp. 208-18, 21st edition. To Compute Tangential Angle For Curves. — a. c representing chord in feet , and a angle in minutes. Illustration. — What is angle for a curve with a radius of 900 feet, and a chord of 400 feet? i7 T 9 X 4 °° — 764 minutes. 90Q Curving oF Hails. = v. I representing length of rail in feet , v versed sine at centre, when 1-56 l 2 R curved, in ins. Illustration. — What is curve for a rail 20 leet jn length, with a radius of 900 feet? 1.5 X 20^ 900 - =.666 ins. Curves 'by' Offsets in Equal Cliords. Fig. 5 - Chord 2 2R - — 0 offset. Chord 2 R = : 2, 0 offset. Illustration.— A ssume chords 150, and ra- f dius 900 feet. 6 " 22 500 . 22500 ,. 112.5 feet; — - — = -2$ feet. — o o (•scp 2 X 900 ~ ' 900 To Compute "Versed Sines and Ordinates oF Curves. Fig. Vw R — VR 2 — ( .5 C) 2 -f v = D ; and \/R 2 — ce 2 — (R — v) = o. D representing diameter of \ circle , and v versed sine of curve. I R 'x 1 Illustration. — Assume radius 900, and chord 400 feet. L — j D 900 — v 810 000 — 40 000 = 900 — 877. 5 = 12. 5 feet. Relation oF Base oF Driving or* Rigid ‘Wheels to Curve. R — — . B. R representing minimum radius of curve, G gauge of road, and B base, iri feet. To Compute Elevation oF Outer Rail. For any Radius or Combination o f Curve with Straight Line. •5 F y/G — c. V representing velocity of train in feet per second , G gauge of road, and c length of a chord , both in feet, the versed sine of which — elevation in ins. On Curves, — G = E. E representing elevation of outer rail in ins. alf’ RAILWAYS. 679 Radii of Carves set oat in Tangential Angles. Angle for Chord of 1 00 Feet. Radius of Curve. Angle for Chord of 100 Feet. Radius of Curve. Angle for Chord of 100 Feet. Radius of Curve. Angle for Chord of 100 Feet. Radius of Curve. 0 ' Feet. 0 ' Feet. 0 ' Feet. O r Feet. 30 5729.6 2 30 II 45-9 4 3 ° 636.6 6 30 440.7 1 2864.8 3 954-9 5 573 7 409'3 1 30 1909.9 3 30 818.5 5 3 ° 520.9 7 3 ° s 382 2 1432.4 4 716.2 6 447-5 8 358.1 Note.— I f chords of less length are used, radius will be proportional thereto. To Ascertain Radius of Curve in Inches for Scale , in Feet per Inch . Divide radius of curve in feet by scale of feet per inch. To Compate Repaired Weiglit of Rail. Rule.— Multiply extreme load upon one driving-wheel in lbs. by .005, and product will give weight of rail in lbs. per yard. To Compate Radias of Carve and. Wheel Base. q B G = R. = B. B representing maximum rigid wheel base of cars , and G 9 G gauge of way , both in feet. To Determine Elevation of Oater Rail. For any Radius or Construction of Curve with Straight. — Fig. 7. Fig. 7. Y .5 t/G = c. V representing speed of train in feet per sec- ond. G gauge of rails in feet, and c length of chord, versed sine v of which will give at its centre the elevation required. Thus, determine chord c, align it on inner c . side of rail, and distance of rail from it at -- cY~~ , centre of its length will give elevation re- ^ quired, whatever the radius of rail. For Cun [.782 V 2 (N D W)]— 4 P R = E ; V 2 Or, W = E. D representing 1.25 R N D R diameter of wheels, W width of gauge , P lateral play between flange and rail , and R radius of curve , all in feet , i-^N ratio of inclination of tire , V velocity of train in miles per hour , and E elevation of outer rail in ins. (Molesworth . ) WC(d+ 1 ) 2 R - = resistance due to curve, and W representing weight of body , both in lbs., C coefficient of friction of wheels upon rails = . 1 to .27, according to condition of weather , d distance of rails apart , l length of rigid wheel base, and R radius of curve, all in feet. (Morrison.) Illustration. — Assume weight of locomotive 30 tons, radius of curve 1000 feet, distance of rails apart 4 feet 8.75 ins., length of base 10 feet, and rails, dry, C == 1. 30 X 2240 X .1 X (4.73 + IO )_ — — — 494-93 wS. 2 X 1000 To Compute Resistance doe to Gravity' upon an In- clination. r — = lbs. per ton of train. gradient Rise per Male, and Resistance to Gravity-, in EDs. per Ton. Gradient of 1 inch. . 20 I 25 30 35 40 45 50 60 70 80 90 100 Rise in feet 264 21 1 176 151 132 117 106 88 75 66 59 53 Resistance 112 1 89.6 74-7 64 56 50 00 't- 37-3 32 28 24.8 22.4 68o RAILWAYS. To Coxnpuite Load which, a Locomotive will Draw up an Inclination. T-r- r-f- r' — W = L. T representing tractive power of locomotive in lbs., r re- sistance due to gravity , and r' resistance due to assumed velocity of train in lbs. per ton , W weight of locomotive and tender , and L load locomotive can draw, in tons , ex- clusive of its own weight and tender. Coefficients of Traction of Locomotives.— Railroads in good order, etc., 4 to 6 lbs. ; in ordinary condition, 8 lbs. In coupled engines adhesion is due to load upon wheels coupled to drivers. To Compute Traction, Retraction, and Adhesive Power of a Locomotive or Train. When upon a Level. asP-rD aT: a representing area of one cylinder in sq. ins., s stroke of piston and D diameter of driving-wheels, both in feet, P mean pressure of steam in lbs. per sq. inch , and T traction , in lbs. When upon an Inclination, a s P -4- D — r w h = T. r representing resistance per ton , w weight of locomotive upon driving-wheels , in tons , h height of rise in feet per 100 of road , and R z=.r w h — retraction,' in lbs. C w b -f- iqo — A. b representing base of inclination in feet per 100 of road. C io = A. C = coefficient in lbs. per ton, and A adhesion , in lbs. When Velocity of a Train is considered. When upon a Level, W (C + W) = R. When upon an Inclination, W(rh + C + VY) = R. V representing velocity of train in miles per hour. Illustration.— A train weighing 200 tons is to be driven up a grade of 52.8 feet per mile, with a velocity of 16 miles per hour; required the retractive power ? 52.8 per mile = 1 in 100 feet r'= 22.4 lbs. 0 = 5. 200 (22.4 X 1 + 5 V 1 ^) = 200 X 22.4 -f- 9 = 6280 lbs. Velocity of Trains. Miles per hour IC> 15 20 30 40 50 60 Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Resistance upon straight ) line per ton ) 8-5 9- 2 5 10.25 i3- 2 5 17-25 22.5 29 Do., with sharp curves) and strong wind* j 13 14 i5-5 20 26 34 43-5 7 ° Lbs. 36-5 . 55 * Equal to 50 per cent, added to resistance upon a straight line. Friction of locomotive engines is about 9 per cent., or 2 lbs. per ton of weight. Case-hardening of wheel-tires reduces their friction from .14 to .08 part of load. To Compute Maximum Load that can be drawn by an Engine, up the Maximum Grade that it can .Attain, Weight and Grade "being given. (Maj. McClellan , U.S.A.) .2 A — 8 L .2 A = L, and - = G. A representing adhesive weight of engine, .4242 G + 8 ’ .4242 L in lbs., G grade in feet per mile, and L load, in tons. Note 1.— When rails are out of order, and slippery, etc., for .2 A, put .143 A. 2. —With an engine of 4 drivers, put .6 as weight resting upon drivers; with 6 drivers the entire weight rests upon them. Illustration. — An engine weighing 30 tons has 6 drivers; what are the maximum loads it can draw upon a level, and upon a grade of 250 feet, and what is its maxi- mum grade for that load ? . 2 X 2240 X 10 i q 440 , ,7 • 2 X 2240 X 3 ° *3 44 ° ^ 4 J — ... P . 4 t _ — 1505.4 tons upon a level. — -7-7; = — 8.4242 ^ 4 • .4252X250 + 8 114.05 * X 2240 X 30 — 8 X 117 8 _ 12497 _ .4242 + 8 117. 8 tons Up a grade of 250 feet. “ 49 . 97 Adhesion of a 4-wheeled locomotive, compared with one of 6 wheels, is as 5 to 8. 250.1 feet. railways. 68 i OPERATION OF LOCOMOTIVES. (O. Chanute, Am. Soc. C. E.) zaires ion. Adhesion of a locomotive is friction of its driving-wheels upon the rails, varying with condition of the surface, and must exceed traction of the engine upon them, otherwise the wheels will slip. T ’ Wpfnfnrp made in the construction of locomotives and tracks^ave 1 gradually increased the proportion which the adhesion bears to the insistent weight upon the driving-wheels. The first accurate experiments were those of Mr Wood upon the early English coal radwaya He deduced the adhesion to be as follows : y . . ., ta of weight on drivers. Upon perfectly dry rails. . £ u u a “ “ damp or muddy rails u u u u “ very greasy rails 04 t o a n h tatrobe indicated .13 as a safe working adhesion, while modern In 1838, B. H. Latrooe inQlc »>~ u f iwht as maximum, and .11 as a minimum, European practice assumes q jo ° ns subject to mists. Thus, on the Seem- SS-SS Materially better results are obtained m United Snh^American =^£23H:ssf“ From these data the following tables have been computed. Coefficients of Adhesion upon Driving Wheels per Ton. European American European American Condition of Rails. European Practice. American Practice. Cdndition of Rails. European Practice. American Practice. Rails very dry Rails very wet Ordinary working. . C. •3 .27 .2 Lbs. 670 600 450 C. •33 •25 .222 Lbs. 667 500 444 In misty weather . In frost and snow. c. .015 .09 Lbs. 350 200 C. .2 . 16 Lbs. 400 333 Adhesion of Locomotives , in Lbs. (.222 in Summer and .2 in Winter). Type of Locomotive. No. of Drivers. W ei{ Locomotive. jbt. On Drivers. Adhe Summer. sion. Winter. A frmrinnn 4 wheels coupled 6 “ connected.. 6 “ 8 “ “ 6 “ “ 4 “ “ •• Lbs. 64000 78000 88 boo 100 000 68 000 48 000 Lbs. 42000 58 000 72 000 88 boo 68 000 48 000 Lbs. 9 350 13 000 16000 X 9 550 15 10 O 10650 Lbs. 8 400 11 600 14 000 17 600 13 600 9 600 Ten- wheeled \Togul Consolidation Tank switching Tractive Power. Traction of a locomotive is the horizontal resultant on the track of the pressure of the steam, as applied in the cylinders. r)2 p 1, — — T D representing diameter of cylinder, L length of stroke, and diameter of driving wheels, all in ins., P mean pressure in cylinder , m lbs. per sq. inch , and T tractive force on rails , in lbs. Iilustratiov —Assume a locomotive, cylinders 18 ins. in diarn., 22 ins. stroke, wheels 68 ins. in diam., and average steam pressure in cylinders 50 lbs. per sq. men. Then 18 X 18 X 50 X 22 - 4 - 68 = 5241 lbs. 682 RAILWAYS, Train Resistances. Usual formula for train resistances, on a level and straight line , is V2 V2 1- 8 = R per ton of train, and p 6 = R per ton of train alone. V repre- 171 . 240 senting velocity in miles per hour , and 8 constant axle friction. (D. K. Clark.) Note. — To meet the unfavorable conditions of quick curves, strong winds, and imperfection of road, Mr. Clark estimates results as obtained by above formula should be increased 50 per cent. Illustration.— At 20 miles per hour, the resistance would be: 20 2 -T- 171 8 = 10.3 lbs. per ion of train. This formula, however, is empirical. It gives results which are too large for freight trains at moderate speeds, and too small for passenger trains at high speeds. Engineers are not agreed as to exact measure and value of each of the elements of train resistances, but following approximations are sufficient for practical use: Analysis of Train Resistances. Resistance of trains to traction may be divided into four principal ele- ments: 1st. Grades; 2d. Curves; 3d. Wheel friction; 4th. Atmosphere. 1st. Grades. — Gradients generally oppose largest element of resistance to trains. Their influence is entirely independent of speed. The meas- ure of this resistance is equal to weight of train multiplied by rate of in- clination or per cent, of grade. Thus, a gradient of .5 per 100 feet (26.4 feet per mile) offers a resistance of 5 x 2000 — J12 ibs. p er ton, or 10 lbs. 1 y 10 X 100 per 2000 lbs., which is to be multiplied by weight in tons of entire train. Following table shows resistance, due to gravity alone, for the most usual grades, in lbs. per ton of train : 1st. Resistance due to Grades. Rate per 100 feet. . 1 .2 , •3 •4 •5 .6 •7 .8 Lbs. per ton of 2240 lbs. . . 2.24 4.48 6.72 8.96 1 1. 2 13-44 15.68 17.92 Rate per mile 5 11 16 21 26 32 37 42 Lbs. per ton of 2000 lbs. . . 2 4 6 8 10 12 16 Rate per 100 feet •9 1 1. 1 1.2 i -3 1.4 i -5 1.6 Lbs. per ton of 2240 lbs. . . 20.16 22.4 24.64 26.88 29.12 3 i -36 33-6 35-84 Rate per mile 47 53 58 63 68 74 79 85 Lbs. per ton of 2000 lbs. . . 18 20 22 24 26 28 30 32 2d. Curves . — Recent European formula is that given by Baron von Weber. . 6504 - 4 - R — 55 = W. R representing radius of curve in metres. This formula assumes that resistance due to curve increases faster than radius diminishes. It gives results varying from a resistance of .8 lb. per 2000 lbs. per degree for a curve of 1000 metres radius (3310 feet, or i° 44') to a resistance of 1.67 lbs. per 20C0 lbs. per degree for curves of 100 metres radius (331 feet, or 17 0 20')- Messrs. Vuillemin, Guebhard, and Dieudonne found curve-resistance to European rolling-stock to be from .8 to 1 lb. per 2000 lbs. per degree, on a gauge of 4 feet 8.5 ins., while Mr. B. H. Latrobe, in 1844, found that with American cars resistance on a curve of 400 feet radius did not exceed .56 lb. per 2000 lbs. per degree. Resistance of same curve varies with coning given tires of wheels, elevation of outer rail, and speed of train running over it, but both reasoning and experiment indicate that the general resistance of curves increases very nearly in direct pro- portion to degree of curvature, or inversely to the radius. Recent American experiments show that a safe allowance for curve resistance may be estimated at .125 of a lb. per 2000 lbs. for each foot in width of gauge. Thus, for 3 feet gauge resistance would be .375 lb. per degree of curve; for standard gauge of 4 feet 8.5 ins. .589, say .60, and for 6 feet gauge .75 lb. per degree. For standard gauge, when radius is given in feet, resistance due to this element is: .60 X 5730 -r- R = C in lbs. per ton of train. RAILWAYS. 683 This is somewhat reduced when curve coincides with that for which wheels are coned (generally about 3 0 ), and when train runs over it, at precise speed for which oX rSns elevated, an allowance of .5 lb. per ton per degree is found to give good results in practice. 2d. Resistance on Curves . It follows from above estimate of curve resistance that, in order to have the same resistance on a curve as on a straight line, the gradient should be diminished by cn per 100 feet of each degree of curve. Thus a 3 0 curve requires an easing of the grade by .09 per 100 feet, a io° curve an easing of .3 per 100, etc. This however need only be done upon the limiting gradients, and when sum of grade and curve’resistances exceeds resistance which has been assumed as limiting the trains. . , TTr , 7 ^ 3d. Resistance due to Wheel Friction. Experimenters are not agreed whether friction of wheels increases simply with weight which thev carrv, but also in some ratio with the speed. Originally as- sumed as a constant at‘8 lbs. per ton, improvements in condition of track (steel rails etc ) and in construction and lubrication of rolling-stock have reduced it to - * and 4 lbs. per ton for well-oiled trains. Under ordinary circumstances, m sum- mer it will be safe to estimate it at 5 lbs. per ton on first-class tracks, and 6dbs. per ton on fair tracks. It may run up to 7 or 8 lbs. per ton on bad tracks (iron rails) in summer, and all these amounts should be increased from 25 to 50 per cent, in cold climates in winter, to allow for inferior lubrication. 4th. Resistance due to Atmosphere. Atmospheric resistance to trains, complicated as it is by the wind which may be prevailing, has not been accurately ascertained by experiment. It consists of 1st. Head resistance of first car of train, which is presumably equal to its exposed area, in sq. feet, multiplied by air pressure due to speed. 2d Head resistance of each subsequent car. This varies with distance they are coupled apart, and so shield each other from end air pressure due to speed. 3d. Friction of air against sides of each car depending upon the speed. This is generally so small that it may be neglected altogether. 4 th. Effect due to prevailing wind, which modifies above three items of resistance. A head wind retards the train, a rear wind aids it, while a side wind increases re- sistance by pressing flanges of wheels against one rail, and, in consequence of curves, a train may assume all of these, positions to same wind. Recent experiments on Erie Railway seem to indicate that in a dead cairn re- sistance of first car of a freight train may be assumed at an exposed surface of 63 sq. feet* multiplied by air pressure due to speed, and that each subsequent car may be assumed to offer a resistance of 20 per cent, of that of first car, while in aj}as- senger train first car may be assumed at an area of 90 sq. feet,t multiplied by pressure due to speed, and that each subsequent car adds an increment equal to 40 per cent, that of first car, in consequence of greater distance they are coupled apart. This resistance is, of course, entirely independent of cars being loaded or ■ e nipt y. In practice it has been found that an allowance of 1.5 to 2 lbs. per ton of weight of a freight train covers atmospheric resistance, except in very high winds. In consequence of complexity of elements above enumerated, exact . formulas can- not probably be now given for train resistances, but following, if applied with judg- ment (and modified to fit circumstances), will be found to give fairly accurate results i n practice. They are for standard gauge, and in making them, resist^ been assumed at .5 lb. per degree, wheel friction at 5 lbs., exposed end area of first car at 90 sq. feet for passenger cars and 63 feet for freight cars, and increment for succeeding cars at .4 for passenger trains and .2 for freight trains. Passenger Train. W ( G + ^T+ s) + 9 ° P = R> Freight Train. W 5^ + ~ ^63P = R. * This is less than area of car, which generally measures about 71 sq.feet ; but part Is shielded by tender and narts beine convex, as wheels, bolts, etc., offer less resistance than a flat plane. t Not only is end area of passenger cars greater than that of freight cars, but in consequence of the projecting roof the end forms a hood in nature of a concave surface, and so opposes greater resistance than a flat plane. 684 RAILWAYS. W representing weight of train, without engine , in tons (2000 lbs.), G resistance of gradient per ton (2000 lbs.; see table, page 683), C° curve in degrees , n number of cars in train , P pressure per sq.foot due to speed, to which an allowance must be made for wind, if existing, R resistance of train, and 5, wheel friction, both in lbs. Illustration i. — Assume a passenger train of 5 cars, weighing 136 tons (2000 lbs.), ascending a grade .5 per 100 (26.4 feet per mile), with curves of 4 0 , at a speed of 60 miles per hour (for which the pressure is 18 lbs. per sq. foot), resistance will be: 136 (10-f 2 -j- 5) + (90 X iB) == 6524 lbs., of which 2312 lbs. are due to grade , curve, and wheels, and 4212 lbs. to atmospheric resistance. 2.— Assume a freight train of 31 cars, weighing 620 tons (2000 lbs.), turning a curve of 3 0 , up a grade of 52.8 feet per mile (1 foot per 100), at a speed of 21 miles per hour (pressure 2 lbs. per sq. foot), resistance Will be: 620 (20 -j- 1.5 + 5) + + y) (63 X 2) = 17 312 lbs., requiring a “Consolidation ” engine to haul it, allowance being made for possible winds, etc. Assume conversely, it is desired to know how many tons an American engine, with an adhesion of 10650 lbs., will draw up a grade of .9 per 100 (47 feet per mile), with curves of 4 0 , assuming atmospheric resistance between 1.5 to 2 lbs. per ton of train. Resistance from grade .9 x 2000 -4- 100 =18 lbs. ) “ curve 4-4-2 = 2 “ [27 lbs. “ “ wheel friction 5, atmosphere 2 = 7 “ ) Hence, 10650-4-27 = 395 tons, or about 20 cars, and in winter same engine will haul 9600-4-27 == 355 tons (2000 lbs.), or about 18 cars. Following table approximates to best modern practice. For freight trains it gives aggregate resistance, in lbs. per ton (2000 lbs ), for various grades and curves. In using it, it is sufficient to divide the adhesion in lbs. of locomotive used by number found in table, in order to obtain number of tons of train that it will haul at or- dinary speeds on gradient and curve selected. Of course, if grade has been equated for curves, only number found in first column (for straight lines) is to be used in computing tons of train on limiting gradient. Approximate Rreiglit-train. Resistances. Gauge 4 feet 8. 5 ins. In Lbs. per 2000 lbs. at Ordinary Speeds. Curve Resistance assumed at .5 lbs. per °, Wheel Friction at 5 lbs., Atmospheric Re- sistance at 2 lbs. per Ton. Gra Per Cent. DE. Per Mile. j Straight. i° 2° 3° 4° 5° 6° C 7° URV] 8° E. 9° 10° ii° 12° *3° 14 0 *5° lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. Level. Feet. 7 7-5 8 8.5 9 9-5 10 10.5 11 IXi 5 12 12.5 13 13-5 14 14-5 .1 5 9 9-5 10 10.5 11 II> 5 12 12.5 13 13*5 14 14-5 15 *5-5 16 16.5 .2 11 11 ii-5 12 12.5 13 I 3-5 14 i4- 5 15 i5- 5 16 16.5 17 17-5 18 18.5 •3 16 13 I 3*5 14 14-5 15 I 5-5 16 16.5 17 i7-5 18 18.5 1 9 i9-5 20 20.5 •4 21 15 i5-5 16 16.5 17 17-5 18 18.5 l 9 i9-5 20 20.5 21 21.5 22 22.5 •5 26 17 i7-5 18 18.5 *9 *9-5 20 20.5 21 21.5 22 22.5 23 23-5 24 24- 5 .6 3 2 J 9 i9-5 20 20.5 21 21.5 22 22.5 23 23-5 24 24-5 25 25-5 26 26.5 •7 37 21 21.5 22 22.5 23 23-5 24 2 4-5 25 25- 5 26 26.5 27 27-5 28 28.5 .8 42 23 23-5 24 24-5 25 25-5 26 26.5 27 27-5 28 28.5 29 29-5 30 30- 5 •9 47 25 25-5 26 26.5 27 275 28 28.5 29 29-5 30 3°-5 31 3i-5 32 32.5 1 53 27 27-5 28 28.5 29 29-5 30 30-5 3i 31-5 32 32-5 33 33-5 34 34-5 1. 1 58 29 29-5 30 30-5 3i 3i-5 32 32.5 33 33-5 34 34-5 35 35-5 36 3&-5 1.2 63 31 3i-5 32 32.5 33 33-5 34 34-5 35 35-5 36 3 6 -5 37 37-5 38 38-5 i-3 68 33 33-5 34 34-3 35 35-5 36 36.5 37 37-5 38 38-5 39 39-5 40 40- 5 i-4 74 35 35-5 36 36-5 37 37-5 38 38.5 39 39-5 40 40-5 41 4^-5 42 42.5 i-5 79 37 37-5 38 38-5 39 39-5 40 4° 5 4i 4i-5 42 42-5 43 43*5 44 44-5 1.6 85 39 39-5 40 40-5 41 4i-5 42 42-5 43 43-5 44 44-5 45 45-5 46 46.5 Illustration. — Assume a “Mogul” engine to have an adhesion of 16000 lbs. ; what weight will it haul up a grade of 74 feet per mile, combiued with a curve of 9 0 ? 16 000 — 4 — 39. 5 = 405 tons (2000 lbs . ). RAILWAYS. 685 Hence, To Compute Adhesion on a Given Grade and Curve , having Weight of Train. Rule— Multiply tabular number by weight of train in tons (2000 lbs.), and product will give adhesion, in lbs. Example. — Assume preceding elements. Then 39.5 X 4°5 = l( > 000 lbs. Note.— A “Consolidation” engine, by its superior adhesion (19550 lbs.) would haul up a like grade and curve 495 tons. Memoranda 011 English. Railways. Regulations [Board of Trade). Cast-iron girders to have a breaking weight = 3 times permanent load, added to 6 times moving load. , , WrouMit-iron bridges not to be strained to more than 5 tons per sq. men. Minimum distance of standing work from outer edge of rail at level of carnage steps, 3.5 feet in England and 4 feet in Ireland. Minimum distance between lines of railway, 6 feet. Stations —Minimum width of platform, 6 feet, and 12 at important stations. Minimum distance of columns from edge of platform, 6 feet, feteepest gradient tor stations, 1 in 260. Ends of platforms to be ramped (not stepped). Signals and dis- tant signals in both directions. Carriages. — Minimum space per passenger 20 cube feet. Minimum area of glass per passenger, 60 sq. ins. Minimum width of seats, 15 ins. Minimum breadth of seat per passenger, 18 ins. Minimum number of lamps per cairiage, 2. Requirements. -Joints of rails to be fished. Chairs to be secured by iron spikes. Fang bolts to be used at the joints of flat-bottomed rails. Construction. Width, single line 18 2 4 6 “ double line 3° 3 8 “ top of ballast, single line 13 6 15 ° “ “ “ double line 24 6 29 Slope of cuttings from centre, 1 in 30. Width of land beyond bottom of slope, 0 to 12 feet. Ditch with slopes, 1 foot at bottom, 1 to 1. Quick mound, 18 ins. in height. Post and rail-fence posts, 7 feet 6 ins. X 6 ins. X3.5 ins -> 9 feet apart, 3 : feet in ground. Intermediate posts, 5 feet 6 ins. X 4 ins. X i-5 i ns -? 3 f ee t apait. Rails 4 of 4 X 1.5 i»s. Parliamentary Regulations for Crossing Pioads. Turnpike Road. Public Occupation Road. Road. Feet. Ins. Feet. Ins. Feet. Ins. Clear width of under bridge, or approach Clear height of under bridge for a width of 12 ft. a u a “ “ o “ “ “ “ at springing Over bridge, height of parapets Approaches, inclination . . . 14 height of fencing 35 16 25 12 — 12 — 4 — 1 in 30 3 — 15 — 12 — 4 “ 1 in 20 3 ~ 14 — 4 — 1 in 16 3 — Limits of Deviation— In towns, 10 yards each side of centre line. In country, 100 yards, or 5 chains nearly. Level— In towns, 2 feet. In country, 5 feet. Gradient. — Gradients flatter than 1 in 100, deviation 10 feet per mile steeper. Do., steeper, 3 feet per mile. Curve. — Curves upwards of .5 a mile radius, may be sharpened to .5 mile radius. Curves of less than .5 mile radius mav not be sharpened. 3 M 686 ROADS, STREETS, AND PAVEMENTS. ROADS, STREETS, AND PAVEMENTS. Classification of Roads. i. Earth. 2. Corduroy. 3. Plank. 4. Gravel. 5. Broken stone (Mac- adam). 6. Stone sub-pavement with surface of broken stone (Telford). 7. Stone sub-pavement with surface of broken stone and gravel, or gravel alone. 8. Rubble stone bottom with surface of broken stone or gravel, or both. 9. Concrete bottom with surface of broken stone or gravel, or both. Oracle of Hoads. Limit of practicable grade varies with character of road and friction of ve- hicle. For best carriages on best roads, limit is 1 in 35, or 15 feet in a mile. Maximum grade of a turnpike road is 1 in 30 feet. An ascent is easier for draught if taken in alternate ascents and levels, than in one continuous rise, although the ascents may be steeper than in a uniform grade. Ordinary angle of repose is 1 in 40 if roads are bad, and 1 in 30, to 1 in 20. When roads have a greater grade than 1 in 35, time is lost in descending, in order to avoid unsafe speed. Grade of a road should be less than its angle of repose. Minimum grade of a road to secure effective drainage should be 1 in 80. In France it is 1 in 125. In construction of roads the advantage of a level road over that of an in- clined one, in reduction of labor, is superior to cost of an increased length of road in the avoiding of a hill. Alpine roads over the Simplon Pass average 1 in 1 7 on Swiss side, 1 in 22 on Italian side, and in one instance 1 in 13. In deciding upon a grade, the motive power available of ascent and avoid- able of waste of power in descending are to be first considered. When traffic is heavier in one direction than the other, the grade in as- cent of lighter traffic may be greatest. When axis of a road is upon side of a hill, and road is made in parts by excavation and by embankment, the side surface should be cut into steps, in order to afford a secure footing to embankment, and in extreme cases, sustaining walls should be erected. Constrvictioii. Estimate of Labor in Construction of Roads. (M. Ancelin.) A day’s work of 10 hours. of an average laborer is estimated as follows: In Cube Yards. WOKK. Ordinary Earth. Loose Earth. Mud. Clay and Earth. Gravel. Blasting Rock. Picking and digging 18 to 23 16 _ 9 7 to 11 2.4 Excavation and pitching \ 6 to 12 feet ) 8 to 12 8 7 to 16 4 2.2 Loading in barrows 22 — 8 — 19 — Wheeling in barrows per ) 100 feet j 20 tO 33 - - - 24 to 28 Loading in carts 16 to 48 — — 17 to 27 — Spreading and levelling. . . 44 to 88 — 2 5 — 30 to 80 — Time of pitching from a shovel is one third of that of digging. Ditches . — All ditches should lead to a natural water-course, and their min- imum inclination should be 1 in 125. Depressions and elevations in surface of a roadway involve a material loss of power. Thus, if elevation is 1 inch, under a wheel 4 feet in diameter, an inclined plane of 1 in 7 has to be surmounted, and, as a consequence, one seventh of weight has to be raised 1 inch. ROADS, STREETS, AND PAVEMENTS. 687 An unyielding foundation and surface are indispensable for a perfect roadway. Earth in embankment occupies an average of one tenth less space than in natural bank, and rock about one third more. Ruts. — Surface of a roadway should be maintained as intact as prac- ticable, as the rutting of it not only tends to a rapid destruction of it, but involves increased traction. The general practice of rutting a road displays a degree of ignorance of physical laws and mechanical effects that is as inexplicable as it is injurious and expensive. On compressible roadways, as earth, sand, etc., resistance of a wheel decreases as breadth of tire increases. Depressing of axles at their ends increases friction. • Long and pliant springs de- crease effect of shock in passing over obstacles in a very great degree. Transverse Section— Best profile of section of roadway is held to be one formed by two inclined planes meeting in centre of road and slightly rounded otf at point of junction. Roads having a rough surface or of broken stone should have a rise of 1 in 24, equal to a rise on crown of 6 ins., and on a smooth surface, as a block-stone or wood pavement, the rise may be reduced to 1 in 48. On roads, when longitudinal inclination is great, the rise of transverse section should be increased, in order that surface water may more readily run off to sides of roadway, instead of down its length, and consequently gullying it. Stone Breaking. A steam stone-breaking machine .will break a cube yard of stone into cubes of 1.5 ins. side, at rate of 1 to 1.5 IP per hour. NT acadamiz e d. Roads. In construction of a Macadamized road, the stones (road metal) used should be hard and rough, and cubical in form, the longest diameter of which exceed 2.5 ins., but when they are very hard this may be reduced to 1.25 and 1.5 ins. The best stones are such as are difficult of fracture, as basaltic and trap, and especially when they are combined with hornblende. Flint and sili- ceous stone are rendered unfit for use by being too brittle. Light granites are objectionable, in consequence of their being brittle and liable to disinte- gration ; dark granites, possessing hornblende, are less objectionable. Lime- stones, sandstones, and slate are too weak and friable. Dimensions of a hammer for breaking the stone should be, head 6 ins. in length, weighing 1 lb., handle 18 ins. in length ; and an average laborer can break from 1.5 to 2 cube yards per day. Stones broken up in this manner have a volume twice as great as in their original form. 100 cube feet of rock will make 190 of 1.5 ins. dimension, 182 of 2 ins., and 170 of 2.5 ins. A ton of hard metal has a volume of 1.185 cube yards. Construction of a Roadway . — Excavate and level to a depth of 1 foot, then lay a “bottom” 12 ins. deep of brick or stone spalls or chips, clinker or old concrete, etc., roll down to 9 ins, then add a layer of coarse gravel or small ballast 5 ins. deep, roll down to 3 ins., and then metal in 2 equal lay- ers of 3 ins., laid at an interval, enabling first layer to be fully consolidated before second is laid on and rolled to a depth of 4 ins. ; a surface or “ blind’* of .75 inch of sharp sand should be laid over last layer of metal and rolled in with a free supply of water. 688 ROADS, STREETS, AND PAVEMENTS. Proportion of Getters , Fillers , and Wheelers in different Soils. Wheelers computed at a Run of 50 Yards. (Molesivorth. ) Getters. Fillers. Wheelers. Getters. 1 Fillers. Wheelers. Loose earth, ) Hard clay 1 1.25 1.25 Sand, etc. ) I 1 1 Compact ) 2 j. Compact earth . . . I 2 2 gravel j Marl 1 2 2 Rock 3 X 1 T'elford. Roads. In construction of a Telford road, metalling is set upon a bottom course of stones, set by hand, in the manner of an ordinary block stone pavement, which course is composed of stones running progressively from 3 inches in depth at sides of road to 4, 5, and 7 inches to centre, and set upon their broadest edge, free from irregularities in their upper surface, and their in- terstices filled with stone spalls or chips, firmly wedged in. Centre portion of road to be metalled first to a depth of 4 ins., to which, after being used for a brief period, 2 ins. more are to be added, and entire surface to^be covered, “blinded,” with clean gravel 1.5 ins. in depth. Telford assigned a load not to exceed 1 ton upon each wheel of a vehicle, with a tire 4 ins. in breadth. Gf ravel or Ecirtli Roads. In construction of a gravel or earth road, selection should be made between clean round gravel that will not pack, and sharp gravel intermixed with earth or clay, that will bind or compact when submitted to the pressure of traffic or a roll. Surface of an ordinary gravel roadway should be excavated to a depth 01 from 8 to 12 ins. for full width of road, the surface of excavation conforming to that of road to be constructed. The gravel should then be spread in layers, and each layer compacted by the gradual pressure due to travel over it, or by a roller, the weight of it in- creasing with each layer. One of 6 tons will suffice for limit of weight.. If gravel is dry and will not readily pack, it should be wet, and mixed with a binding material, or covered with a thin layer of it, as clay or loam. In rolling, the sides of road should be first rolled, in order , to arrest the gravel, when the centre is being rolled, from spreading at the side. To re-form a mile of gravel or earth road, 30 feet in width between gutters, material cast up from sides, there will be required 1640 hours’ labor of men, and 20 of a double team. Corduroy Roads. A Corduroy road is one in which timber logs are laid transversely to its plane. Rlank Roads. A single plank road should not exceed 8 feet in width, as any greater width involves an expenditure of material, without any equivalent advantage. If a double track is required it should consist of two single and independ- ent tracks, as with one wide track the wear would be mostly in the centre, and consequently, wear would be restricted to one portion of its surface. Materials.— Sleepers should be as long as practicable of attainment, in depth 3 or 4 ins., according to requirements of the soil, and they should have a width of 3 ms. for each foot of width of road. Pine, oak, maple, or beech are best adapted for economy and wear. Planks should be from 3 to 3.5 ins. thick, and not less than 9 ins. in uidth, or more than 12 if of hard wood, or 15 if of soft. A plank road will wear from 7 to 12 years, according to service,. material, and location, and its traction, compared with an ordinary Macadamized road, is 2.5 to 3 times less, and Avith a common country road in bad order 7 tunes. For other elements, see Earth-work, page 466. ROADS, STREETS, AND PAVEMENTS. 689 .A^spkalt. Asphalt is a bituminous limestone, and is synonymous with bitumen; it consists of from 90 to 94 per cent, of carbonate of lime and 6 to 10 per cent, of bitumen. In forming a pavement the powder is heated to from 212 0 to 250 0 , and its par- ticles caused to adhere by pressure, or it is applied as a liquid asphalt or asphaltic mastic, which is thus manufactured. The powder is heated with from 5 to 8 per cent, of free bitumen for a flux, and the mixture when melted is run into molds. To be remelted, additional bitumen must be mixed with it, without which it would only become soft. For paving 60 per cent, of sand or gravel must be mixed with it. No chemical union takes place between the mastic and the sand or gravel, but cohesion is so complete that gravel will fracture with the mastic, and the admixture increases the resistance of the mass to heat of the sun. The roadway should have a convexity of .01 of its breadth. Artificial A sphait — Pleated limestone and gas tar, when mixed, possess some of the proportions of alphalt mastic, but it is very inferior for the purposes of a pavement. To repair surface of roadway, dissolve bitumen 1 part in 3 of pitch oil or resin oil, apply 10 oz. of mixture over each sq. yard of roadway, sprinkle on it 2 lbs. of asphalt powder, and then cover surface with sand. Wood Pavement. Close-grained and hard woods only are suitable, such as oak, elm, ash, beech, and yellow pine, and they should be laid on a foundation of concrete. Block Stone Pavement. Paving-blocks, as the Belgian, etc., where crest of street or area of pave- ment does not exceed 1 inch in 7.5 feet, should taper slightly toward the top, and the joints be well filled, “blinded,” with gravel. The common practice of tapering them downward is erroneous. The foundation or bottoming of a stone pavement for street travel should consist either of hydraulic concrete or rubble masonry in hydraulic mortar. The practice in this country of setting the stones in sand alone is at variance with endurance and ultimate economy, but when resorted to, there should be a bed of 12 ins. of gravel, rammed in three layers, covered with an inch of sand. Granite or Trap blocks should beg X 9 X 12 ins. Rubble Stone Pavement. Bowlders or Beach stone of irregular volumes and forms, set in a bed of sand, involves great resistance to vehicles and frequent repairs ; it is wholly at variance with requirements of heavy traffic or city use. Concrete Roads. Concrete roads are constructed of broken stones (road metal) 4 volumes, clean sharp sand 1.25 to .33 volumes, and hydraulic cement 1 volume. The mass is laid down in a layer of 3 or 4 ins. in depth, and left to harden during a period of 3 days, when a second and like layer is laid on and well rolled, and then left to harden for a period of from 10 to 20 days, according to temperature and moisture of the weather. Roads. ( Molesworth .) Ordinary turnpike roads . — 30 feet wide, centre 6 ins. higher than sides; 4 feet from centre, .5 inch below centre ; 9 feet from centre, 2 ins. below centre ; 15 feet from centre, 6 ins. below centre. Foot-paths— 6 feet wide, inclined 1 inch towards road, of fine gravel, or sifted quarry chippings, 3 ins. thick. Cross-roads — 20 feet wide. Foot-paths — 5 feet. Side drains — 3 feet below surface of road. Road material — bottom layer gravel, burned clay or chalk, 8 ins. deep. Top layer, broken granite not larger than 1.5 cube ins., 6 ins. deep, 3 M* 690 ROADS, STREETS, AND PAVEMENTS. Miscellaneous INTotes. Metalling should be from 6 ins. to 1 foot in depth, and in cubes of 1.5 to 1.75 ins. One layer of material of a road should be spread and submitted to traffic or roll- ing before next is laid down, and this process should be repeated in 2 or 3 layers of 3 ins. each. When new metal is laid on old, the surface of the old should be loosened with a pick. Patching is termed darning. Sand and Gravel, Blinding , should not be spread over a new surface, as they tend to arrest binding of metal. Mud should be scraped oft' of surface. Hoggin is application of a binding of surface of a metal road, composed of loam, fine gravel, and coarse sand. Metalled Roads should be swept wet. Rolling. — Steam rolls are most effective and economical. 1000 sq. yards of metal- ling will require 24 hours’ rolling at 1.5 miles per hour. A roller of 15 tons’ weight will roll 1000 sq. yards of Telford or Macadam pavement in from 30 to 40 hours, at a speed of 1.5 miles per hour, equal .675 and .9 ton mile per sq. yard. Sprinkling.— Go cube feet of water with one cart will cover 850 sq. yards. 100 cube feet per day will cover 1000 sq. yards; ordinarily two sprinklings are necessary. Granite Pavement.— The wear of granite pavement of London Bridge was .22 inch per year, and from an average of several streets in London, tbe w ear per ioo\ehicles per foot of width per day is equal to one sixteenth of an inch per year. Sweeping and Watering of granite pavement and Macadam road, for equal areas and under alike conditions in every respect, costs as 1 for former to 7 of latter. By men, \vith cart, horse, and driver, costs 3.25 times more than by a machine, one of which will sweep 16000 sq. yards of street per period of 6 houis. Asphalt Pavement. — Average cost per sq. yard in London: foundation, 50 cents; surface, $3.25; cost of maintenance per sq. yard per year, 40 cents. Wear varies from .2 to .42 near curb, and .17 to .34 inch on general surface per year. Washing. — Surface cleaning of stone or asphalt pavement by a jet can be effected at from 1 to 2 gallons per sq. yard. Wood Pavement. — W f ear of wood pavement in London, per 100 vehicles per day per foot of width, .083 inch per year. Macadamized Roads. — Annual cost of maintenance of several such roads in London was 62 cents per sq. yard. Block Slone Pavement— Stones should be set with their tapered or least ends up- wards, wdtli surface joints of 1 inch. Fascines , when used, should be in two layers, laid crosswise to each other and picketed down. Bituminous road may be made by breaking up asphalt, laying it 2 ins thick, covering with coal tar, and ramming it with a heavy beetle. To repair a bitumi- nous surface, dissolve one part of bitumen (mineral tar) in three of pitch oil or resin oil spread .625 of a lb. of solution over each sq. yard of road, sprinkle 2 lbs. pow- dered asphalt (bituminous limestone) and then sand, and sweep off the surplus. Slipping . — Granite safest when wet, and asphalt and wood when dry. Gravel , alike to that of Roa Hook, from its uniformity, Will bear an admixture of from .2 to .25 of ordinary gravel or coarse sand. Annual cost of a Telford pavement 4.2 cents per sq. yard, including sprinkling, repairs, and supervision. Voids in a Cube Yard of Stone. broken to a gauge of 2.5 ins 10 cube feet. Shingle. 9 cube feet. u u 2 u io . 66 “ “ Thames ballast 4.5 “ “ 1.5 “ ”-33 “ For further and full information, see Law and Clarke on Roads and Streets, New fork 1867- Weale’s Series. London. 1861 and 1877; Roads, Streets, and Pavements, >y Brev. Mai. -Gen. Q. A. Gilmore, U. S. A., New York. 1876; Engineering Notes, by <\ Robertson, London and New York, 1873; and Construction and Maintenance of Soads, by Ed. P. North, C. E., see Transactions Am. Soc: of C. E., vol. vm., May, 1879. SEWERS. 69I SEWERS. Sewers are the courses from a series of locations, and are classed as Drains, Sewers, and Culverts. Drains are small courses, from one or more points leading to a sewer. Culverts are courses that receive the discharge of sewers. Greatest fall of rain is 2 ins. per hour = 54 308.6 galls, per acre. Inclination of sewers should not be less than 1 foot in 240, and for house or short lateral service it should be 1 inch in 5 feet. * Fig. 1. Fig. 2. Circular. 55 V x 2 /= v, and v a = V. „ Egg. — = w, — w', and D = r. x representing 3 3 area of sewer -f- wetted perimeter , f inclination of sewer per mile , and v velocity of flow of contents in feet per minute ; a area of flow, in sq.feet , V volume of discharge, in cube feet per minute; D height of sewer, w and w' width at bottom and top, and r radius of sides , in feet. For diameter of sewer exceeding 6 feet. (T. Ilawlcsley.) D — -z=.iv\ D diameter of a circular sewer of area required. 9 Elliptic. — Top and bottom internal should be of equal diam- eters. Diameter .66 depth of culvert ; intersections of top and bottom circles form centres for striking courses connect- ing top and bottom circles. Pipes or Small Seicers. — Height of section = 1 ; diameter of arch = .66 ; of invert = .33, and radius of sides = 1. In culverts less than 6 feet internal depth, brickwork should be 9 ins. thick ; when they are above 6 feet and less than 9 feet, it should be 14 ins. thick. If diameter of top arch = 1, diameter of inverted arch = .5, and total depth =r sum of the two diameters, or 1.5 ; then radius of the arcs which are tangential to the top, and inverted, will be 1.5. From this any two of the elements can be deduced, one being known. Drainage of Lands by Dipes. Soils. Depth of Pipes. Distance apart. Soils. Depth of Pipes. Distance apart. Coarse gravel sand Light sand with gravel Light loam Ft. Ins. 4 6 A Feet. 60 eo Loam with gravel . . . Sandy loam Ft. Ins. 3 3 0 0 Feet. 27 r 3 6 Soft clay. 3 y 2 l 2 6 21 Loam with clay 3 2 1 21 Stiff clay 15 Minimum Velocity and GJ-rade of Sewers and Drains in Cities. (WicJcsteed.) Diam. Vel. per Minute. Grade, 1 in Grade per Mile. Diam. Vel. per Minute. Grade, 1 in Grade per Miie. Diam. Vel. per Minute. Grade, 1 in Grade per Mile. Ins. Feet. Feet. Ins. Feet. Feet. Ins. Feet. Feet. 4 240 36 146.7 15 180 244 21.6 42 180 686 7-7 6 220 65 81.2 18 180 294 18 48 180 784 6.8 8 220 87 60.7 24 180 39 2 13-5 54 180 882 6 IO 210 Ir 9 44.4 30 180 490 10.8 60 180 980 5-4 12 190 r 75 30-2 36 180 588 9 Area of Sewers or Pipes . — An area of 20 acres, miles, etc., will not re- quire 20 times capacity of pipes for one acre, mile, etc., as the discharge from the 19 acres, etc., will not flow into the main simultaneously with that from one acre, etc. Ordinarily in this country an area of sewer or pipe that will discharge a rainfall of 1 inch per hour (3630 cube feet per acre) is sufficient. 692 * SEWERS. Sewage . — The excreta per annum of ioo individuals of both sexes and all ages is estimated at 7250 lbs. solid matter and 94 700 fluid, equal to 1020 lbs. per capita , and in volume 16 cube feet, to which is to be added the volume of water used for domestic purposes. A velocity of flow of from 2.5 to 3 feet per second will discharge a sewer of its sewage matter and prevent deposits. The minimum velocity should not be less than 1.3 feet per second. Surface from which. Circular Sewers with proper Carves will discharge Water equal in. Volume to One Inch in Depth per Hour, including City Drainage. {John Roe.) Diameter of Sewers in Feet. Inclination in Feet. 2 None 1 in 480. . 1 in 240. . 1 in 160. , 1 in 120.. 1 in 80. . , 1 in 60. . . Acres. 38.75 48 50 63 78 90 125 Acres. 67.25 75 87 113 143 i 6 5 182 Acres. 120 135 155 203 257 295 Acres. 277 308 355 460 59 ° 570 730 Acres. 570 630 735 950 1200 1388 1500 Acres. 1020 1117 1318 1692 2180 2486 2675 Acres. 1725 I 9 2 5 2225 2875 37 °° 4225 4550 Acres. 2850 3025 3500 4500 5825 6625 7 I2 5 Surface of a Town from which small Circular Drains will discharge Water equal in Volume to Two Inches in Depth, per Hour. {John Roe.) n Ins. Inclination.! 7 I 8 Fall of 1 Inch. rCLINATION. Diameter of I .11 of 1 Inch. 3 4 5 Acres. Feet. Feet. Feet. ! .125 120 — — •25 20 120 — •4375 — 40 — ■ • 5 — 30 80 .6 — 20 60 1 — — 20 1.2 — — — i -5 — — — 1.8 — — — 2. 1 — — Feet. Feet. 'Feet. 60 120 80 — | 60 Acres. 2.1 2-5 2-75 4 - 5 5 - 3 5-8 7.8 9 10 17 Diameter of Drain in Ins. I 12 | 15 Feet. 120 80 60 Feet. 120 80 60 Feet. 240 120 80 60 Fe«t. 240 120 Dimensions, Areas, and Volume of Material per Lineal Loot of Egg-shaped Sewers of different Dimensions. Volume of Brick-work. Depth. Internal Dimensions. Diam. of Top Arch. Feet. 2.25 3 3 - 75 4 - 5 5 - 5 6 6 - 75 7 - 5 8.25 9 Feet. 1- 5 2 2 - 5 3 3 - 5 4 4 - 5 5 5 - 5 6 Diam. of Invert. Feet. •75 1 1.25 i -5 1- 75 2 2.25 2.5 2 - 75 3 Area. Sq. Feet. 2.53 4-5 7*°3 10. 12 13.78 18 22.78 28.12 34- °3 4°-5 4.5 Ins. thick. Cube Feet. 2. 81 3 - 56 4 - 3 1 5.06 5.81 6. 56 7 - 3 1 9 Ins. thick. 13 5 Ins. thick. ube Feet. Cube Feet. 9-56 — 10.87 — 12.75 — 14.25 — 15-75 24-75 17.06 27 18 28.41 19.69 3°-94 Area — product of mean diameter X height. Sewer Pipes should have a uniform thickness and be uniformly glazed, both internally and externally. Fire-clay pipes should be thicker than those of stone-clav. STABILITY. 693 STABILITY. Stability, Strength , and Stiffness are necessary to permanence of a structure, under all variations or distributions of load or stress to which it may be subjected. Stability of a Fixed Body— Is power of remaining in equilibrio without sensible deviation of position, notwithstanding load or stress to which it may be submitted may have certain directions. Stability of a Floating Body. — A body in a fluid floats, or is balanced, when it displaces a volume of the fluid, weight of which is equal to weight of body, and when centre of gravity of body and that of volume of fluid dis- placed "are in same vertical plane. When a body in equilibrio is free to move, and is caused to deviate in a small decree from its position of equilibrium, if it tends to return to its original position, its equilibrium is termed Stable ,* if it does not tend to de- viate further, or to recover its original position, its equilibrium is termed Indifferent ; and when it tends to deviate further from its original position, its equilibrium is Unstable. A body in equilibrio may be stable for one direction of stress, and unstable for another. Moment of Stability of a body or structure resting upon a plane is mo- ment or couple of forces, which must be applied in a plane vertically inclined to the body in addition to its weight, in order to remove centre of resistance of body upon plane, or of the joint, to its extreme position consistent with stability. The couple generally consists of the thrust of an adjoining struct- ure, or "an arch and pressure of water, or of a mass of earth against the structure, together with the equal and parallel, but not directly opposed, re- sistance of plane of foundation or joint of structure to that lateral thrust. It may differ according to position of axis of applied couple. Couple— Two forces of equal magnitude applied to same body or struct- ure in parallel and opposite directions, but not in same line of action, consti- tute a couple. Note.— For Statical and Dynamical Stability, see Naval Architecture, page 649. To Ascertain Stability of a Body on a Horizontal Blaxie. — Big. 1 . its moment would be 5 X 4 = 20 tons , although it is but half the weight. To Compute "Weigh, t of a Griven. Body "to Sastaiix a Griveix Thrust. and l distance of fulcrum from centre of gravity — as. T illustration. — Assume figure to be extended to a height of 20 feet, and required to be capable of resisting the extreme pressure of wind. b d 0 Illustration. — Stability of a body, A, Fig. 1, when a thrust is applied as at 0, to turn it on a , is ascertained by multiplying its weight by distance as, from fulcrum a to line of centre of gravity, cs. Hence, if cubical block weighed 10 tons and its base is ■e 6 feet, its moment would be 10 X — = 30 tons. 2 If upper part, abdc , was removed, remainder, a e d, F h = W. F representing thrust in lbs . , h height of centre of gravity of body = cs , STABILITY. KEVETMENT WALLS. 694 Pressure estimated at 50 lbs. F = 6 X 20 X 50 = 6000 lbs. at centre of gravity of surface of body. 6000 X 10 .. Then = 20 000 lbs. 3 Note i.— T his result is to be increased proportionately with the factor of safety due to character of its material and structure. 2.— If form of body has a cylindrical section, as a round tower, the thrust of wind would be but one half of that of a plane surface. When the Body is Tapered, as Frustum of Pyramid or Cone. — Ascertain centres of gravity of surface for pressure or thrust, and of body for its sta- bility, and proceed as before. Fig. 2. To Ascertain Stability of a Body 011 air Inclination.— Big. S. Illustration.— Stability of body, Fig. 2, when thrust is applied at c, is ascertained by multiplying its weight by distance a b from fulcrum, b, to line of centre of gravity, a g. If thrust was applied at 0, stability would be ascer- tained by distance s r from fulcrum r. Angles of Eq^nilibrinm at "wliicli various Substances ■will Repose, as determined Toy a Clinometer. Angle measured from a Horizontal Plane , and falling from a spout. Degrees. [ Degrees. Lime-dust 45 i Sand, less dry. . . , ... 39.6 Dry sand 40 1 Wheat Moist sand. . . 1 Degrees. Common mold. . . 37 Common gravel. . 35 to 36 Stones or Coal. . . 43 Weiglrt of a Cifbe Foot of Materials of Embankments, "Walls, and Barns. Concrete in cement. . . 137 Stone masonry 130 Brick “ 112 Gravel 125 Loam 126 I Sand 120 Clay. . Marl . 120 100 Revetment AW alls. When a wall sustains a pressure of earth, sand, or any loose material, it is termed a Revetment wall, and when erected to arrest the fall or subsidence of a natural bank of earth, it is termed a Face wall. When earth or banking is level with top of wall, it is termed a Scarp re- vetment, and when it is above it, or surcharged, a Counterscarp revetment. When face of wall is battered, it is termed Sloping, and when back is bat- tered, Countersloping. Thrust of earth, etc., upon a wall is caused by a certain portion, in shape of a wedge, tending to break away from the general mass. The pressure thus caused is similar to that of water, but weight of the material must be reduced by a particular ratio dependent upon angle of natural slope, which varies from 45 0 to 6o° (measured from vertical) in earth of mean density. Or, natural slope of earth or like material lessens the thrust, as the cosine of the slope. Angle which line of rupture makes with vertical is .5 of angle which line of natural slope, or angle of repose, makes with same vertical line, y V hen earth is level at top, its pressure may be ascertained by considering it as a fluid, weight of a cube foot of which is equal to weight of a cube foot of the earth, multiplied by square of tangent of .5 angle included between natural slope and vertical. STABILITY. — REVETMENT WALLS. 695 Therefore squares of the tangents of .5 of 45 0 and .5 of 6 o° = .iqi 6 and which are the multipliers to be used in ordinary cases to reduce a cube foot of material to a cube foot of equivalent fluid, which will have same effect as earth by its pressure upon a wall. Pressure of Earth, against Revetment "Walls. Let ABCD, Fig. 3, be vertical section of a revetment wall, behind which is a bank of earth, A D/e ; let Do represent angle of repose, line of rupture , or natural slope which earth would assume but for resistance of wall. In sandy or loose earth angle o D A is generally 30 0 ; in firmer earth it is 36°; and in some instances it is 45 0 . If upper surface of earth and wall which supports it are both in one horizontal plane, then the resultant, In, of pressure of the bank, behind a vertical wall, is at a dis- tance, D n, of one third A D. Line of Rupture behind a wall supporting a bank of vegetable earth is at a distance A 0 from interior face, AD = .618 height of it. When bank is of sand, A 0 = .677 h ; when of earth and small gravel = .646 h ; and when of earth and large gravel = .618 h. The prism, vertical section of which is A D 0, has a tendencv to descend along inclined plane, 0 D, by its gravity ; but it is retained in its plaoe by resistance of wall, and by its cohesion to and friction upon face 0 D. Each of these forces may be resolved into one which will be perpendicular to 0 D, and into another which will be parallel to o D. The lines c i , it represent components of the force of gravity, which is represented by vertical line c l , drawn from centre of gravity, c, of prism. Lines n r,lr represent compo- nents of forces of cohesion and friction, which is represented by horizontal line n l. Force that gives the prism a tendency to descend is i l, and that opposed to this is r l , together with effects of cohesion and friction. Thus, i l = r l + cohesion + friction. Consequently, exact solution of prob- lems of this nature must be in a great measure experimental. It has been found, however, and confirmed experimentally, that angle formed with vertical, by prism of earth that exerts greatest horizontal stress against a wall, is half the angle which angle of repose or natural slope of earth makes with vertical. Memoranda. Natural slope of dry sand = 39°, moist soil = 43°, very fine sand = 21° wet clay = 14 0 , and gravel = 35 0 . In setting or founding of retaining walls, if earth upon which wall is to rest is clayey or wet, coefficient of friction between wall and earth falls to . 3 ; hence it is necessary, in order to meet this, that the wall should be set to such a depth in the earth that the passive resistance of it on outer face of wall, combined with its fric- ticn on its bottom, may withstand the pressure or thrust on its inner face. Moment of a Retaining Wall is its weight multiplied by distance of its centre of gravity to vertical plane passing through outer edge of its base. Moment of Pressure of Earth against a retaining wall is pressure multiplied by distance of its centre of pressure to horizontal plane passing through base of wall. Equilibrium of Retaining Wall is when respective moments of wall and earth are equal. Stability of a Retaining Wall should be in excess of its equilibrium, according to character of thrust upon it, and the line of its resistance should be within wall and at a distance from vertical passing through centre of gravity of wall, at most .44 of distance of exterior axis of wall from this line. Coefficient of Stability varies with character of earth, location, exposure to vibra- tions, 'floods, etc. ; hence thickness of base of wall will vary from 1.4 to 2 6. Backs of retaining walls should be laid rough, in order to arrest lateral subsidence of the filling. Fig. 3- T A s £4 c / f'i C ID STABILITY. REVETMENT AVALLS. 696 When filling is composed of bowlders and gravel, the thickness of wall must be increased, and contrariwise; when of earth in layers and well rammed, it may be decreased. Courses of dry wall should be inclined inwards, in order to arrest the flow of water of subsidence in filling from running out upon face of wall. Less the natural slope, greater the pressure on wall. Sea walls should have an increased proportion of breadth, as the earth backing is not only subjected to being flooded, but the walls have at times to sustain the weight of heavy merchandise. Buttress.— An increased and projecting width of wall on its front, at intervals in its length. Counterfort.— An increased and projecting width of wall at its back and at in- tervals. Coefficient of Friction of masonry on masonry .67, of masonry on dry clay .51, and on wet clay .3. Face of wall should not be battered to exceed 1 to 1.25 ins. in a foot of height, in consequence of the facility afforded by a greater inclination to the permeation of rain between the joints of the courses. Footing of a wall, projecting beyond its faces, is not included in its width. Pressure.— Limit of pressure on masonry 12 500 to 16 500 lbs. per sq. foot wall. Thickness of Walls, in Mortar , Faces vertical. For Railways or Like Stress. Cut stone or Ranged rubble .35 | Brick or Dressed rubble 4 When laid dry, add one fourth. Friction in vegetable earths is .5; pressure in sand .4. When vegetable earths are well laid in courses, the thrust is reduced .5. When bank is liable to be saturated with water, thickness of wall should bo doubled. Centre of Pressure of earthwork, etc., coincides with centre of pressure of water, and hence, when surface is a rectangle, it is at .33 of height from base. The theory of required thickness of a retaining wall, as before stated, is, that the lateral thrust of a bank of earth with a horizontal surface is that due to the prism or wedge-shaped volume, included between the vertical inner face of the wall and a line bisecting the angle between the wall and the angle of repose of the material. To Compute Elements of Revetment Weills. — If ig. 4r. Fig. 4. B A a p? Let A D 0 represent angle of repose of material, resting against a wall, ABCD. ADn = .5 ADo. Tan. ADr h h 2 Tan. A D n h — , or — tan. A D n = V. = * 49 2 * wh 2 321 wh 2 tan. A D n = W h w h 2 tan. 2 ADn- C D W h x ' 2 W h X 2 wh 3 — — — tan. 2 A D n = E ; w A 3 6 W hx 2 tan. 2 ADn = P; W 2 tan. 2 ADn = S; / w h tan. AD n / --== V 3 W : x, and h tan. . _ /2 IV , • ADn Viw = x ' tan. 2 ADw = M; w h 3 J ~ 3 h representing height of wall in feet, V volume of section of prism of material AD n one foot in length in cube feet , W and w weights of a cube foot of wall and of material, P lateral pressure of prism of earth upon wall , M and m moments of pressure and weight on and of wall , E and S equilibrium and stability of wall, all in lbs. , and x and x, C D for weights of wall for equilibrium and stability. Illustration. — A revetment wall, Fig. 4, of 125 lbs. per cube foot and 40 feet in height , sustains a bank of earth having a natural slope of 52 0 24', and a weight of 89.25 lbs. per cube foot; what is pressure or thrust against it, etc. ? STABILITY. REVETMENT WALLS. 697 Tan. 2 A D n = .242. Then .492 X 40 X = 393-6 cube feet. 89.25 X 4 g_ x 35 12 g 3 lbs. 89.25 x ^ .4922 — 17 278.8 Z&s. 2 2 89-25 X 4° a ^ 492 2 x — = 230384 lbs. 125 X 40 X = 230400 lbs. 23 2 40 x. 492 a/— 1 ffg = 9-6/erf, and 40 X -49 For Rubble Walls in Mortar or Dry Rubble , add respectively to base as above obtained, .14 and .42 part. Note i. — When coefficient of friction is known, use it for tan. 2 A D n. h X C = base of wall for stability. ( Molesivorth .) 2. — When either relative weights of equal volumes of wall and bank of earth or their specific gravities are given, S and s may be taken for W and w. These equations involve simply the operation of a lever, the fulcrum being at the outer edge of wall C. The moment of pressure of bank is product of lateral pressure and perpendicular distance from fulcrum to line of direction of pressure. The moment of weight of wall is product of weight of wall and perpendicular distance from fulcrum to vertical line drawn through centre of gravity of wall. When Weights of Embankment and Wall are equal per Cube Foot. C for clay = .336, and for sand .267. When Weights are as 4/0 5. C for clay = .3, and for sand .239. When Wall has an Exterior Slope or Batter. — Fig. 5. 5- -g A — n_ p AA. D -j- E C — — M. M representing J / moment of weight of wall in lbs. / /' Illustration. — Assume weight of wall 120 lbs. per cube foot, and C D and E C respectively 10 and 2. 5 feet, /' and all other elements as in preceding case. E C Hence, x 2 w h 3 , / 2 2 - 5 2 \ ^10 -j- 2 - 5 — J ~ 37° 000 Z&S. W h ( — 2 n 2 h~\ wh 3 • z4 -nh ) = tan. 2 A D n = 2 V 3 / 3 - — tan. 2 ADn — nh = x. x representing A B or C D. n ratio of 3 3 W difference of widths of base and top to height. In absence of tan. 2 * A D n put C, co- efficient of material. C = .0424 for vegetable or clayey earth, mixed with large gravel; .0464 if mixed with small gravel; .1528 for sand, and .166 for semi-fluid earths. Illustration. — Assume elements of preceding case. n = one fortieth, and tan. A D n — .492. 40 a / — — - + 1- > ^ 8 9 :2 5 x .492 2 — 1 — 12.6 feet. V 3 X 40 2 3 X 125 ^ Hence, thickness of wall at base = 12.6 -f- 1 (one fortieth of height) = 13.6 feet. Note. — If n = one twentieth, Hence, wall at base = 11.63-}- 2 ( one twentieth of height) = 13. 63 / or L x tan. ADn — \~fr/ J 4-Ai D L X A n X sec. m D 0 JZ X sec. mDo. To Ascertain Point of Moment of Pressure of a Wall. —Fig:- 9. lg ' By its resisting lever l a, added to its weight. - Weight of wall as computed assumed as concentrated at its / centre of gravity • Draw a vertical line . 0 through its centre of gravity, and con- tinue line of pressure P * to l, take any distance r 0 by scale rep- resenting weight of wall, and r «, by same scale for amount of pressure or thrust against wall, complete parallelogram 1 o wm, then diagonal ru will give resultant of pressure in amount and direction to overturn wall. For stability this diagonal should fall inside of base at a point not less than one third of its breadth. E STABILITY. — REVETMENT WALLS, 699 Surcharged. Revetments. Fig. 10. f r 7 o as t / D C When the earth stands above a wall, as A B e, Fig. 10, with its natural slope, Ay, A B C is termed a Surcharged Revetment. If C r is line of rupture, A fr C is the part of earth that presses upon wall, which part must he taken into the computation, with exception of portion A B e, which rests upon wall; that is, the computation must be for part C efr, which must be reduced by multiply- ing weight of a cube foot of it by square of tangent of angle e C r = angle of line of rupture, or half angle e C 0, which natural slope makes with vertical, and then proceed as in previous cases for revetments. li' / = breadth or CD. W and w representing weights of wall and V 3 hW embankment in lbs. per cube foot, and h' height of embankment, as C e. Illustration. — Height of a surcharged revetment, BC, Fig. 10, is 12 feet, weight 130 lbs. per cube foot; what is its width or base to resist pressure of earth of a weight of 100 lbs. per cube foot, and a height, C e, of 15 feet, angle of repose 45°*? Tan. * 2 * * * (45° -4- 2) = .1716. Then 15 V-°55 = 3-52 feet. v 3 a 12 A I 7 * * 3° To Ascertain IPoint of* Moment of Pressure of a Sxir- cliarged Wall.— Wig. 11. Fig. 11. sj Draw a line, P *, parallel to slope, C r, through centre of gravity of sustained backing, BCr. When, as in this case, this section is that of a triangle, point * will be at .33 height of wall. When natural slope is 1.5 in length to 1 in height, as with gravel or sand, w x .64 = pressure P *. In a surcharged revetment, as/B 0, at its natural slope, the maximum pressure is attaiued when the backing reaches to r. When slope of maximum pressure, C nr, intersects face of natural slope, Bf so that if backing is raised to /, or above it, there is theoretically no addi- tional stress exerted at back of or against wall, but prac- tically there is, from effect of impact of vibration of a passing train, proximity to percussive actiQn, alike to that of a trip-hammer, etc. When backing rests on top of wall, as A B e. Fig. 10, small triangle of it is omitted in computations. Direction of pressure against wall is same as when wall is not surcharged. When Wall is set below Surface of Earth. — Fig. 12. Fig. 12 . 1.4 tan. 45 V -<■< \/ h * 45° ~ 2 ) 7 V — W ^2 /V - — d. a representing angle of repose of earth, w and W weights of earth and wall per cube foot, f friction of wall on base A B, and V weight of wall. Illustration. — I f a wall of masonry, Fig. 12, 8 feet in thickness and 13 in height, is to sustain earth ievel with its upper surface, earth weighing 100 lbs. per cube foot, weight of wall 150 lbs. per cube foot — 1 5 600 lbs., and angle of repose of earth 30 0 ; what should be the depth of wall below surface of earth? Tan. 45 — 30 2 = . 5774, and /= '9360 <-05634. 3 150 Then ,4 X 10 ° X - 577 f 5 ; 2 X - 3X 156 °° = -8°B4 xf = 4.027 feet. Note.— Coefficient of stability is assumed by French engineers for walls of forti- fications 1.4 h, and if ground is clayey or wet/=.3. STABILITY. EMBANKMENT WALLS AND DAMS. In Computing Stability of a Surcharged Waif Fig. 13, sub- stitute d for h , as in following illustration. ( Molesworth .) d, representing depth at distance l , = h. In slopes of 1 to 1, d — 1.71 h\ of 1.5 to 1,= 1.55; of 2 to 1,= 1.45 ; of 3 to ij== 1. 31, and 4 to 1,== 1.24. To Determine Form of a, Pier to Snstain eq.ti.al Pressure per TJiait of Surface at all its Horizontal Sections, or any Height. A nd = a, or A N = a. A and a representing areas of sections at summit of pier and at any depth , d, measured from summit , n a number the hyp. log. of which — 1 -4- height, H, of a column of the material of which pier is constructed , due to required •4343 ^ pressure , and N the number , com. log. of which = — -g — . Illustration. — Height of a pier is 20 feet, and area of section of its summit = 1 foot; what should be its areas at 10 feet and base? 1 -4- 20 — .05, and numbers 1. 0513; i X 1.0513 10 = 1.649 feet; and 1 X i.o5i3 20 = 2.719 feet. Counterforts are increased thicknesses of a wall at its back, at intervals of its length. Enilo ankin e nt "Walls and. Dams. Thrust of water upon inner face of an Embankment wall or Dam is horizontal. When Both Faces are Vertical, Fig. 14. Assume perpendicular embankment or wall, A BCD, Fig. 14, to sustain pressure of water, B C ef. Fig. 14- Let h i be a vertical line passing through 0, centre of gravity of wall, c centre of pressure of water, dis- tance C c being ==.33 B C. Draw c l perpendicular to B C ; then, since section A C of wall is rectangular, centre of gravity, 0 , is in its geometrical centre, and therefore D i — .5 DC. Now l D i is to be consid- ered as a bent lever, fulcrum of which is D, weight of wall acting in direction Of centre of gravity, o, on arm D i, and pressure of water on arm D /, or a force equal to that pressure thrusting in direction c l. Then P x D Z = P X — = W X — , or P = 3 D . P representing pressure 3 2 2 b o of water. Note. — When this equation holds, a wall or embankment will just be on the point of overturning; but in order that they may have complete stability, this equation should give a much larger value to P than its actual amount. The following formulas are for walls or embankments one foot in length ; for if they have stability for that length they will be stable for any other length. P = — w, also W = h b W, each value being for 1 foot in length, which, being sub- 2 stituted in the equations, there will result — w = 3 6 X W - , or h 2 w = 3 b 2 W; b /^- = h, and h /-^ = b. h rep - 2 2 h ’ J V w V 3 w resenting depth of water and ivall or embankment , which are here assumed to be equal , b breadth of wall or embankment , and W and w weights of wall and water per cube foot in lbs. Which gives breadth of a wall or embankment that will just sustain pressure of the water. STABILITY. — EMBANKMENT WALLS AND DAMS. J01 To Compute Equilibrium. h^J-^j—b. Illustration i.— Height of a wall, B C, equal to depth of water, is 12 feet, and re- spective weights of water and wall are 62.5 lbs. and 120 lbs. per cube foot; required breadth of wall, so that it may have complete stability to sustain the pressure of water. * 7 : 62.5 = 12 X .4166 = 5 feet, breadth that will just sustain pressure of the 3 X 120 water. Therefore an addition should be made to this to give the wall complete stability, say 2 feet; hence 54-2 = 7, required width of wall. 2. —Width of a wall is 3 feet, and weight of a cube foot of it is 150 lbs. ; required height of wall to resist pressure of fresh water to the top. /2 W To Compute Stability, h / = 0. Illustration.— Take elements of preceding case. 12. / 2 * 6 ^ = I2 X. 5 8 9 = 7.07 feet. V 3 x 120 Or, Divide 1, 2, or 3, etc., according as the nature of the ground, the mate- rial, and the character of the thrust of the water requires, by .05 weight of material of wall, per cube foot, extract the square root of quotient, and mul- tiply result by extreme height of water. Example. — What should be the thickness of a vertical faced wall of masonry, having a weight of 125 lbs. per cube foot, to sustain a head of water of 40 feet, and to have stability ? . v'(2-r-.o5 X 125) 40 = V- 3 2 X 4 ° == 22.63 fat. 0r , — 40 V-3472 = 23.56 feet. When Dam kas an Exterior Slope or Batter , as A D. — Fig. 15. Fig. 15 . a i E 0 n c Assume prismoidal wall, A B C D, to sustain press- jpjjjl ure of water, B C ef. jjggjg Draw A E perpendicular to D C ; h = B C, the top §§gg|§ breadth A B = E C = b, and bottom breadth, D E, "ggj of sloping part, AED = S. Ijlgf Then weights of portions A C and A E D respec- jflf tively for one foot in length are hb W and .5 W S 4 , f these weights acting at points n and i respectively. To Compute Moment. , h S W 2 S hb W X 4- — ^ = moment for A C, and — - — X = moment for A E D. Hence, — — l S 4- b r j = moment of dam, S representing batter or base E D. Illustration.— Height of a dam, B C, Fig. 15, is 9 feet, base C E 3, and E D 4 feet ; what is its moment ? A C = 9 X 3 X 120 X (44-7)=? 3 2 4° X 5- 5 = 17 820 Ms. ADE= 9 x 4 X 120 ^ 2 X 4 _ x 2-?- = 5760 lbs. 23 3 Hence, 17 820 4 - 5760 = 23 580 lbs. moment. Or, - 20 * 9 ^44-3 — = 540 X 43^ = 23 580 lbs. moment. 3N* STABILITY. EMBANKMENT WALLS AND DAMS. To Compute Elements of Walls or Dams witb. an Exterior Batter.— Big. 1£5. To Compute WicLtb. of Top. -S = &. . / 2 h 2 When Width of Batter is Given. -~ Illustration. — Assume height of wall 9 and batter 3 feet, and W and w 120 and 62. 5 lbs. per cube foot. _ "+§! 3 w T 3 4 ^2 X 9 2 X 62.5 3 f - x 3 = V28.125 + 3 — 3 = 2.s8/eei. 3 X 120 3 To Compute "Wid-tli of Base. When Width of Batter is Given. j'z h VT 3 w W = B. + — = 5 - 58 feet = S -f b. 7 2X9^X62.5 3 2 V 3 X 120 3 To Compute Widtli of Batter. 3 b . Ill 2 w 3 b 2 When Width of Top is Given. + — j = S. _ l&lS = V 4 2 .i8 + 4 -99 ~ 3-87 = 3 /^- lEAerc IFiefcA of Bottom is Given. 3 B 2 — - /t 2 10 W = S. To Determine Stability of a Detaining Wall or Dam "by Protraction.— Eig. 16 . Assume ABCD, section of a wall. On horizontal line 01 centre ui miuou u ~ ~ scale, lay off, from vertical line of centre of gravity • of wall, line or — thrust against wall, and on vertical line at centre of gravity of wall, at its intersection, 0, with centre of thrust, let fall os — weight of wall. Complete parallelogram, and if diagonal 0 u or its prolongation falls within C, the wall is stable, and W X distance from line os — moment of wall. W representing whole weight of wall in lbs. To Determine Centre of G-ravity of a Wall or Dam.— Eig. 16 . j / A B X C D\ .CD/2 A B + C D \ _ , By Ordinates. — (a B + C D- - B + c — *. and 3 Ub + Ou) 2 \ ' A B -J- C D/ 3 To Compute Base of Dam. When might , Rate of Balter , and Weight of Materials are given. Rule. —Multiply square of Widtli of batter by .0166 weight of material per cube foot, add i, 2 or 3 times square of depth of water, according as resistance due to equilibrium is required, divide result by .05 weight of materia pe cube foot, and extract square root of quotient. nr /xh 2 + b 2 X .0166 W _ x __ number of times 0 f resistance required. Ur ’V .05 W r T . MPTF Assume a dam 40 feet iu height, constructed of masonry weighing I2 olbs per *5 to batter 3 ins. per foot and to have twice the resistance due to its equilibrium ; what should be its breadth at its base, Dir 40X3. ^ — t/-i — hnitor and Ao* X 2 + io° X. 0166X120 _ ^ jrm - aJ . 8 feet. STABILITY. EMBANKMENT WALLS AND DAMS. 7°3 When Section of Dam is a Triangle , Fig. 17. — As- PjggS sume dam, A B C,* *to sustain a head of water, ef. Rule.— Proceed as by Rule for Fig. 14; multiply by |g|i .033 instead of .05. Example. — As before. f (2 -4- .033 X 125) 40 = f .485 X 40 = 27.84 feet. Hi 2 X w Or, Formula for S (C B), Fig. 1 5. / — = 28. 28 feet. To Determine Section of a Vertical Wall which shall have Equal Resist - ance of one having Section of a Triangle. (See J. C. Trautwine , Phila ., 1872.) To Compute Thickness of 13ase of a Wall or Dain.- Dig. 18. Fig. 18. Rule. — D ivide 1, 2, or 3 times square of depth of water . by .05 weight of material, add quotient to .5 batter on one jgsggg face, and square root of this sum, added to half batter on A|gg!jJ other side, will give thickness. Or, . / X - 4- 4- — = Base. b and b' representing it f ’ V .05 W \ 2 / 2 exterior and interior batters , and x, as before , number of times of resistance or square of depth. ^ Example. — Assume a dam 40 feet in height, to batter 5 feet on each side, constructed of masonry weighing 120 lbs. per cube foot, and to have twice the resistance due to its equilibrium; what should be breadth of base, DC? / j° - ^ X — -4- (A) 7539.584-2.5 = 25.73/^. V .05 X 120 \2 / 2 High. Masonry Dams. Rubble Masonry, well laid in strong cement, will bear with safety a load equivalent to weight of a column of it 160 feet in height. Assuming such Fig. I9 . masonry as twice weight of water, it is equivalent * to a pressure of 20 000 lbs. per sq. foot. Log. B 4- .434 294 X ~- — b. B representing width of wall at top , and d depth at any desired point below top, both in feet. Ordinarily, B may be taken at 18 feet, and in cases of extreme and exposed heights of dam at 20 and more, and when b is determined, .9 of it is to be on outer face of wall, as A B, and . 1 on inner face. Illustration. — Determine section of a dam, Fig. 19, 80 feet in height, at depths of 10, 20, 40, 60, and 80 feet. Log. B == 1.2553. Log. 1. 2553 4-. 4343 X^ = log. 1. 2553 4-. 0543 = 20. 4, which X .9 = 18.36. “ i- 2553 4- -4343 X ^ — log. 1. 2553 4-. 1086 = 23.11, which X .9 = 20.8. “ 1-2553 4- -4343 X log. 1. 2553 4-. 2172 = 29.68, which X .9 = 26.81. 00 x. 25534-. 4343 X ^ = log- 1-2553 4- -3257 = 3 8 - 11, which X -9 = 34-3- “ 1-2553 + -4343 X ^ = !°g- 1-2553 4" *4343 = 5°-°7, which X-9 = 45-o6. 704 STEAM. STEAM. Steam is generated by heating of water until it attains temperature of ebullition or vaporization, and elevation of its temperature is sensible to indications of a thermometer up to point of ebullition ; it is then converted into steam by additional temperature, which cannot be in- dicated by a thermometer, and is termed latent. (See Heat, page 508.) Pressure and density of steam, which is generated in free contact with water, rises with the temperature, and reciprocally its temperature rises with the press- ure and density, and higher the temperature more rapid the pressure. There is but one and a corresponding pressure and density for each temperature, and steam generated in free contact with water is both at its maximum density and pressure for its temperature, and in this condition it is termed saturated , from its being in- capable of vaporizing more water unless its temperature is raised. Saturated Steam is the normal condition of steam generated in free contact with water, and same density and same pressure always exist in conjunction with same temperature. It therefore is both at its condensing and generating points; that is, it is condensed if its temperature is reduced, and more water is evaporated if its temperature is raised. If, however, the whole of the water is evaporated, or a volume of saturated steam is isolated from water, in a confined space, and an additional quantity of heat is supplied to the steam, its condition of saturation is changed, the steam becomes superheated , and both temperature and pressure are increased, while its density is not increased. Steam, when thus surcharged, approaches to condition of a gas. With saturated steam, pressure does not rise directly with the temperature. Steam, at its boiling-point, is equal to pressure of atmosphere, which is 14.723 307 lbs, (page 427), at 6o° upon a sq. inch. In all computations concerning steam, it is necessary to have some or all of fol- lowing elements, viz. : Its Pressure , which is termed its tension or elastic force, and is expressed in lbs. per sq. inch. Its Temperature , which is number of its degrees of heat indicated by a thermometer. Its Density , which is weight of a unit of its volume compared with that of water. Its Relative volume , which is space occupied by a given weight or volume of it, compared with weight or volume of water that produced it. Under pressure of the atmosphere alone, temperature of water cannot be raised above its boiling-point. Expansive force of steam of all fluids is same at their boiling-point. A cube inch of water, evaporated under ordinary atmospheric pressure, is convert- ed into 1642* cube ins. of steam, or, in a unit of measure, very nearly 1 cube foot, and it exerts a mechanical force equal to raising of 14.723307 X 144 = 2120.156208 lbs. 1 foot high. A pressure of 1 lb. upon a sq. inch will support a column of mercury at a tem- perature of 6o°, 1 - 4 -. 4907769 (page 427) = 2.037 586 * ns - in height; hence it will raise a mercurial siphon gauge one half of this, or 1.018793 ins. Velocity of steam, when flowing into a vacuum, is about 1550 feet per second when at a pressure equal to the atmosphere ; when at 10 atmospheres velocity is increased to but 1780 feet; and when flowing into the air under a similar pressure it is about 650 feet per second, increasing to 1600 feet for a pressure of 20 atmospheres. Boiling-points of Water, corresponding to different heights of barometer, see Heat, page 517. Volume of a cube foot of water evaporated into steam at 212 0 is 1642 cube feet; hence 1 ~ 1642 = .000 609013, which represents density or specific gravity of steam at pressure of atmosphere. Elasticity of vapor of alcohol, at all temperatures, is about 2. 125 times that of steam. Specific Gravity , compared with air, is as weight of a cube foot of it compared with equal volume of air. Thus, weight of a cube foot of steam at 212 0 and at pressure of atmosphere is 266.124 grains; weight of a like volume of air at 32 0 is 565.096 grains, and at 62° 532.679 grains. Hence 266. 124 =532. 679 = .499 59, specific gravity of steam compared with air at 32 0 , and with water it is .000609013. * Pole’s Formuia makes it 1712. STEAM. 705 Total Heat of Saturated Steam. 1081.4 -f- .305 T = total heat T representing initial temperature of water. Illustration.— What is total heat of steam at 212 0 ? 1081. 4-}-. 305 x 212 = 1146.06. As specific heat of water is .9 greater at 212 0 than at 32 0 , hence the 212 0 would be 212.9, and 1146.33 the result Total Heat of Gaseous Steam 1074.6 + 475 T =±± total heat Absorption of Heat in Generation of 1 Lb. of Water from 32 0 to 212 0 . Sensible heat, or heat to raise temperature of water Units. Force, lbs. from 32 0 to 212 0 180.9 X 772 = i39 6 55 Latent heat to produce steam 892.9 “ tk to resist atmospheric pressure 14.7 lbs. per sq. inch 7^3 9 6 5-2 X 772 = 745 *34 Total or constituent heat 1146.1 884789 This number, 1146.1, is a Constant , and expresses units of heat in 1 lb. of steam from 32 0 up to temperature at which conversion takes place. Thus, 1 lb. water heated from 32 0 to 332 °, requires as much heat as would raise 300 lbs. i°. Hence 3°°° And 1 lb. water converted into steam at 332 0 (= 106 lbs. pressure), ab- sorbs as much heat for its conversion as would raise 846.1 lbs. water i°. Hence 846.1° 1146.1° Mechanical Equivalent of Heat contained in Steam. 1 lb. water heated from 32° to 212° requires as much heat as would raise 180 lbs. 1° Hence 180.9° 1 lb. water at 212 0 , converted into steam at 212° (= 14.7 lbs. pressure), absorbs as much heat for its conversion as would raise 966.6 lbs. water 1° Hence 9 6 5-2° 1146. 1° Mechanical Equivalent , or maximum theoretical duty of quantity of heat in 1 lb. of steam, is 772 lbs., which X 1146. 1 units of heat •'■==■ 884 789.2 lbs. raised 1 foot high. To Compute Pressure of* Steam. When Height of Column of Mercury it will Support is given. . Rule. — Di- vide height of column of mercury in ins. by 2.037 586, and quotient will give pressure per sq. inch in lbs. Example. — Height of a column of mercury is 203.7586 ins. ; what pressure per sq. inch will it contain ? 203. 7586 -r-- 2.037 586 = 100 lbs. To Compute Weiglit of a CiPfc>e Foot of Steam. Rule. — Multiply its density by 62.425. Example. — Density of a volume of steam is .000609013; what is its weight? .000609013 X 62.425 —-.038016 825 lbs. Note. — S ee table, page 708. 1 atmosphere or 14.723307 lbs. per sq. inch = 30 ins. of mercury. To Compute Temperature of Steam. Rule. — M ultiply 6tli root of its force in ins. of mercury by 177.2, sub- tract 100 from product, and remainder will give temperature in degrees. Example. — When elastic force of steam is equal to a pressure of 64 ins. of mer- cury, what is its temperature? Note. — To extract 6th root of a number, ascertain cube root of its square root. V64 = 8, and -^8 = 2. Hence, 2 X 177- 2 — 100 = 254. 4° t. Or, - 93^' 3 _ r 85 — t. p representing pressure in lbs. per sq. inch. 6.1993544 — log. p STEAM. 706 To Compute Y olume of' Water contained, in a given 'V'ol- nme of* Steam. When its Density is given. Rule. — Multiply volume of steam in cube feet by its density, and product will give volume of water in cube feet. Example. — D ensity of a volume of 16420 cube feet of steam is .000609; what is the weight of it in lbs. ? 16420 X .000609 = IO — volume of water, which X 62.425 = 624.25 lbs. To Compute Pressure of* Steam in Ins. of* Mercury, or X/bs. per Sq. Incli. When Temperature is given. Rule i. — Add 100 to temperature, divide sum proportionally by 177.2 for temperature of 212 0 , and by 160 for tem- peratures up to 445 0 ; or, 177.6 for sea-water, and 185.6 for sea-water sat- urated with salt, and 6th power of quotient will give pressure. Example. — T emperature of steam is 254 0 ; what is its pressure? 100 -f-254-r- 177.2 = 1.998, and 1. 998 s = 63.62 ins. When Ins. of Mercury are given. 2. — Divide ins. of mercury by 2.037 586, and quotient will give pressure. When Pressure in Lbs. is given. 3. — Multiply pressure by 2.037 586. To Compute Specific Gravity of* Steam compared with. .A.i r. Rule. — Divide constant number 829.05 (1642 X .5049) by volume of steam at temperature of pressure at which gravity is required. Example.— P ressure of steam is 60 lbs., and volume 437 ; what its specific gravity? 829.05 = 437 = 1.898. To Compute Volume of* a Cube Foot of Water in Steam. When Elastic Force and Temperature of Steam are given. Rule. — To 430.25 for temperature of 212 0 , and 332 for temperatures up to 445 0 , add temperature in degrees ; multiply sum by 76.5, and divide product by elastic force of steam in ins. of mercury. Note. — W hen force in ins. of mercury is not given, multiply pressure in lbs. per sq. inch by 2.037 586. Example.— T emperature of a cube foot of water evaporated into steam is 386°, and elastic force is 427.5 ins. ; what is its volume? Assume 369 for proportionate factor. 369 -f 386 X 76. 5 = 427. 5 =; 135. 1 cube feet. Or, for 1 lb. of steam, 2.519 — .941 log. j? = log. V in cube feet. Assume p = 14.7 lbs. 2.519 — .941 log. 14.7 = 2.519 — 1.098 = 1. 421 = log. 26.34 cube feet, which X 62.425 = 164 feet. Or, When Density is given. — Divide 1 by density, and quotient will give volume in cube feet. To Compute Density- or Specific (Gravity of* Steam. When Volume is given. Rule. — Divide 1 by volume in cube feet. Example. — V olume is 210; what is density? 1 -7- 210 = .004 761. Or, for 1 lb. of steam, .941 log. p — 2. 5 19 = log. D. When Pressure is given. — Take temperature due to pressure, and proceed as by rule to compute volume, which, when obtained, proceeds as above. To Compute Volume of Steam required to raise a (Given Volume of* Water to any (Given Temperature. Rule. — Multiply water to be heated by difference of temperatures between it and that to which it is to be raised, for a dividend ; then to temperature of steam add 965.2°, from that sum take required temperature of water for a divisor, and quotient will give volume of water. Example. - tO 212 ° ? STEAM. 707 -What volume of steam at 212° will raise 100 cube feet of water at 8o° i qq X 212 80^ _ cube f ee t water; or, (13.68 X 1642 — 212) = 22 463 of steam. 212+ 965.2 — 212 To Compute Volume of Water, at any- Given Temper- ature, that must he Mixed with Steam to Raise or ne- dace the Mixture to any Required Temperatare. Rule. — F rom required temperature subtract temperature of water ; then ascertain how often remainder is contained in required temperature sub- tracted from sum of sensible and latent heat of the steam, and quotient will give volume required. Sum of Sensible and Latent Heats for a range of temperatures will be found under Heat, pages 508 and 509. Example.— Temperature of condensing water of an engine is 8o°, and required temperature ioo°; what is proportion of condensing water to that evaporated at a pressure of 34 lbs. per sq. inch ? Sum of sensible and latent heats 1190.4°. 100 — 80 = 20. Then, 1190.4 — ioo-=-2o = 54.52 to 1. When .Temperature of Steam is given. 1 representing latent heat , T and t temperatures of steam and required temperature , w temperature of condensing water , and V volume of condensing water in cube feet. Illustration.— Temperature of steam in a cylinder is 257.6°, and other elements same as in preceding example; required volume of injection water? Latent heat of steam at 230° = 932.8°. _ *° 9 °- 4 _ ^ vo i umes . 932. 8 + 257. 6- 100 — 80 To Compute Temperature of Water in Condenser or Reservoir of a Steam-engine. 1 T - ~^ x w — t. Illustration.— Assume elements as preceding. V + i 932-8 + 257.6 + 54.52 X 80 __ 5,552 __ 54.52 + 1 55-52 To Compute Latent Heat of Saturated Steam. 1115.2 — .708 t = l. Illustration.— Assume temperature 257.6° as preceding. 1115.2 — .708 X 257.6 = 932.8°. To Compute Total Heat of Saturated Steam. 305 t-\- 1081.4 = H. Illustration. — Assume temperature as preceding. .305 X 257.6+ 1081.4 = 1160. Elastic Force and Temperature of Vapors of Alcohol, Ether, Sulphuret of Carbon, Petroleum, and Tur- pentine. Force in Ins. of Mercury. I Ins. 0 1 Ins. 0 1 1 Ins. 0 1 Ins. o | Ins. Alcohol. Alcohol. Ether. Sulphuret of 32 •4 140 13-9 34 6.2 Carbon. 50 .86 160 22.6 54 i 5-3 53-5 7-4 60 1.23 i 73 3 ° 74 16.2 7 2 -5 12.55 70 1.76 180 34-73 94 24.7 no 3 ° 80 2-45 200 53 96 1 3 ° 212 126 90 3-4 212 67-5 104) 279-5 300 100 4-5 220 78.5 120 39-47 347 606 120 8.1 240 111.24 15° 67.6 130 10.6 264 166. 1 212 178 Petroleum. 316 I 30 345 44-i 375 I 64 Oil of Turpentine. 315 I 3° 357 47-7^ 370 | 62.4 7 o8 STEAM, Saturated. Steam. Pressure, Temperature , Volume , and Density. Pressure per I in Sq. Mer- Inch. cury. Temperature. Total Heat from Water at 32 0 . Volume of 1 Lb. Lbs. j Ins. 0 0 Cub. ft. 1 | 2. 04 102. 1 1 1 12. 5 330.36 2 4.07 126.3 1119.7 172. 08 6. 11 141.6 1124.6 117.52 4 8. 14 * 53 * 1128. 1 89.62 5 10. 18 162.3 II 30-9 72.66 6 12.22 170.2 1 * 33-3 61.21 7 14.25 176.9 i* 35-3 52-94 8 16.29 182.9 1137.2 46.69 9 18.32 188.3 1138.8 4 *. 79 10 20.36 * 93-3 1140-3 37-84 11 22.39 197.8 ** 4*-7 34-63 12 24 - 43 202 **43 31.88 *3 26.46 205.9 1144-2 29-57 *4 28.51 209.6 1 * 45-3 27.61 14.7 29.92 212 1146. 1 26.36 i 5 30.54 213. 1 1146.4 25-85 16 32-57 216.3 1147.4 24.32 *7 34.61 219.6 1148.3 22.96 18 36.65 222.4 1149.2 21.78 *9 38.68 225.3 1150-1 20.7 20 40.72 228 1150.9 19.72 21 42.75 230.6 ** 5*-7 18.84 22 44-79 233 -* 1152.5 18.03 23 46.831235.5 1 * 53-2 17.26 24 48.86 | 237.8 ** 53-9 16.64 25 50.9 240. 1 ii 54-6 * 5-99 26 52.93 242.3 * 155-3 * 5 - 3 8 27 54-97 ; 244-4 1 * 55-8 14.86 28 57 -o* j 246.4 1156.4 * 4-37 2 9 59- °4 248.4 1 * 57-1 * 3-9 30 61.08 250.4 1157.8 13.46 3 * 63.11 252.2 1158.4 * 3-°5 32 65-15 254.1 1158.9 12.67 33 67.19 255-9 ** 59-5 12.31 34 69.22 257.6 1160 11.97 35 7*. 26 259-3 1160.5 11.65 36 73-29 260.9 1161 **•34 37 75-33 262.6 1161.5 11.04 38 77-37 264.2 1162 10.76 39 79-4 265.8 1162.5 10.51 40 81.43 267.3 1162.9 10.27 4 * 83-47 268.7 1163.4 10.03 42 85-5 270.2 1163.8 9.81 43 87-54 271.6 1164.2 9-59 44 89.58 273 1164.6 9-39 45 91.61 274.4 1165.1 9. 18 46 93-65 275.8 *165-5 9 0 47 95-69 277.1 11659 8.82 48 97.72 278.4 1166.3 8.65 49 99.76 279.7 1166.7 8.48 5 ° 101.8 281- 1167. 1 8.31 51 103.83 282.3 1167.5 8.17 52 105.87 283.5 H67.9 8.04 53 107.9 284.7 1168.3 7.88 54 *09.94 285.9 1168. 6 7-74 55 111.98 287. 1 1169 7.61 56 114.01 288.2 1169.3 7.48 57 116.05 289.3 1169.7 1 7-36 Lb. .003 .005 8 .008 5 .011 2 .013 8 .016 3 .018 9 .021 4 .0239 .026 4 .028 9 031 4 •° 33’8 . 036 2 .038 02 .0387 .041 1 •°4 35 •045 9 .0483 .0507 •053 1 .055 5 .058 .060 1 .062 5 .065 .0673 .0696 .07 1 9 •°74 3 .0766 .0789 .081 2 .0835 .085 8 .088 1 .0905 .092 9 .0952 .0974 .0996 .102 .104 2 .1065 .1089 .111 1 •1133 .1156 .1179 . 120 2 . 122 4 . 124 6 . 126 9 . 129 1 •* 3*4 •* 33 6 .1364 Inc! Lbs. 58 59 60 61 62 63 64 65 66 67 68 69 70 72 73 74 75 76 77 78 79 80 81 82 8.3 84 85 86 87 88 89 90 91 92 93 94 96 97 98 99 100 101 102 103 104 i °5 106 107 108 109 no in 112 **3 114 115 Mer- cury. Ins. 118.08 120. 12 122. 16 124.19 126.23 128.26 I30-3 I32-34 134-37 136.4 138.44 140.48 142.52 144-55 * 4 6 -59 148.62 150. 66 152.69 154-73 156.77 158.8 160.84 162.87 164.91 166.95 168.98 171.02 173-05 175-09 I 77 -I 3 179.16 181.2 183.23 185.27 187.31 189.34 191.38 i 93 - 4 i 195-45 197.49 199.52 201.56 203-59 205.63 207. 66 209.7 211.74 213.77 215.81 217.84 219. 88 221.92 223.95 225.99 228.02 230.06 232. 1 2 34 - *3 290.4 291.6 292.7 293.8 294.8 295-9 296.9 298 299 3°o 300.9 301.9 302.9 303-9 304.8 305-7 306.6 307-5 308.4 309-3 310.2 311.1 312 312.8 313- 6 314- 5 315 - 3 316.1 316.9 317-8 318.6 3I9-4 320.2 321 321.7 322.5 323-3 324.1 324.8 325-6 326.3 327-1 327-9 328.5 329.1 329-9 330.6 33i-3 331- 9 332 - 6 333- 3 334 334- 6 335- 3 336 336- 7 337- 4 338 53 o w 1170 1170.4 1 170. 7 1171.1 1171-4 1171.7 1172 1172.3 1 172. 6 1172.9 H 73-2 H 73-5 1173.8 H 74 - 1 ** 74-3 1174.6 H74.9 1175.2 H 75-4 ii 75 -7 1176 1176.3 1176.5 1176.8 1177.! 1 * 77-4 1177.6 1 * 77-9 1178. 1 1178.4 1178.6 1178.9 1179. 1 i* 79-3 i* 79-5 1179.8 1180 1180.3 1180.5 1 180. 8 1181 1181.2 1181.4 1181.6 1181.8 1182 1182.2 1182.4 1182.6 1182.8 1183 1183-3 1183.5 1183.7 1183.9 ! 1184.1 I 1184.3 | 1184.5 Cub. ft. 7.24 7. 12 7.01 6.9 6.81 6.7 6.6 6.49 6.41 6. 32 6.23 6.15 6.07 5-99 5-83 5-76 5.68 5.61 5-54 5-48 5 - 4 * 5-35 5-29 5-23 5- *7 5 -** 5-05 5 4.94 4.89 4.84 4-79 4-74 4.69 4.64 4.6 4-55 4 - 5 * 4.46 4.42 4-37 4-33 4.29 4- 2 5 4.21 4. 18 4*4 4. 11 4.07 4.04 4 3-97 3-93 3-9 3.86 3-83 3-8 1 ^ Lb. *38 1403 1425 *447 .1469 *493 .1516 1538 .156 .*583 .1605 .1627 . 1648 .167 . 1692 1714 .1736 •*759 .1782 .1804 .1826 . 1848 . 1869 1891 * 9*3 1935 *957 .198 .2002 2024 2044 .2067 .2089 .2111 •2133 •2155 .2176 .2198 .2219 .2241 .2263 .2285 .2307 .2329 . 235 * •2373 •2393 .2414 •2435 .2456 •2477 .2499 .2521 -2543 2564 .2586 . 2607 . 2628 STEAM. 709 Pr e Inch. SSSURE in Mer- cury. Temperature. Total Heat from Water at 32°. Volume of 1 Lb. Density, or Weight of one Cube Foot. Pressure per I in Sq. Mer- Inch. cury. Temperature. Total Heat from Water at 32 0 . Volume of 1 Lb. Density, or Weight of one Cube Foot. Lbs. Ins. 0 O Cub. ft. Lb. Lbs. Ins. 0 O Cub. ft. Lbs. 116 236.17 338.6 1184.7 3-77 .2649 149 303-35 357-8 1 1 90. 5 2.98 •3357 117 238.2 339-3 1184.9 3-74 .2652 150 . 305-39 358.3 1 190. 7 2.96 -3377 118 240.24 339-9 1185.1 3-7i .2674 155 3I5-57 361 1191.5 2. 87 •3484 119 00 OJ c3 cT 340-5 1185.3 3-68 .2696 160 325-75 363-4 1192. 2 2.79 •359 120 244.31 34 1 - 1 1185.4 3-65 •2738 165 335-93 366 1192.9 2.71 •3695 121 246.35 341-8 1185.6 3.62 •2759 170 346.11 368.2 XI 93-7 2.63 •3798 122 248.38 342-4 1185.8 3- 59 .278 175 356.29 370.8 1x94.4 2.56 .3899 123 250.42 343 1186 3-56 .2801 180 366.47 372-9 1195-1 2-49 .4009 124 252.45 343-6 1186.2 3-54 .2822 i8 5 376.65 375-3 1195.8 2-43 .4117 125 254.49 344-2 1186.4 3-5i .2845 190 386.83 377-5 1196.5 2-37 .4222 126 256.53 344-8 1186.6 3-49 .2867 i95 397-ox 379-7 1X97-2 2.31 •4327 127 258.56 345-4 1186.8 3-46 .2889 200 407. 19 .381.7 1197.8 2.26 •4431 128 260.6 346 1186.9 3-44 .2911 210 427-54 386 II99-I 2. 16 •4634 129 262.64 346.6 1187. 1 3-4i •2933 220 447-9 389-9 1200.3 2.06 .4842 130 264.67 347-2 1x87.3 3-38 •2955 230 468.26 393-8 1201. 5 1.98 •5052 131 266.71 347-8 1187.5 3-35 •2977 240 488.62 397-5 1202.6 1.9 .5248 132 268.74 348.3 1187.6 3-33 •2999 250 508.98 401. 1 1203.7 1.83 •5464 133 270.78 348-9 1187.8 3-3i .302 260 529-34 404-5 1204.8 1.76 .5669 134 272.81 349-5 1188 3-29 •304 270 549 -7 407.9 1205.8 : x-7 .5868 135 274.85 35o.i 1188.2 3-27 .306 280 570. 06 411.2 1206.8 1.64 .6081 136 276. 89 350.6 1188.3 3-25 .308 290 590. 42 4x4.4 1207.8 x-59 .6273 137 278.92 35i-2 1188.5 3.22 .3101 300 610.78 417-5 1208.7 I i-54 .6486 138 280. 96 35i-8 1188.7 3-2 • 3 121 350 7 12 - 57 43o.i 1212.6 i-33 .7498 139 282.99 352.4 1188.9 3.18 .3142 400 8x4-37 444.9 1217.1 1. 18 .8502 140 285.03 352-9 1189 3. 16 .3162 45o 916. 17 456.7 1220.7 1.05 •9499 141 287.07 353-5 1189.2 3-i4 .3184 500 1018 467-5 1224 -95 1.049 142 289. j 354 1189.4 3.12 .3206 550 1 1 19. 8 477-5 1227 •87 1. 148 143 291.14 354-5 1189.6 3 -i .3228 600 1221.6 487 1229.9 .8 1.245 144 293-17 355 1189.7 3.08 -325 650 1323-4 495-6 1232.5 •74 1.342 145 295.21 355-6 1189.9 3.06 •3273 700 1425-8 504.1 1235- 1 .69 x-4395 146 297.25 356 .i 1190 3- °4 •3294 800 1628. 7 5I9-5 1239.8 .61 1.6322 147 299.28 356-7 1 190. 2 3.02 •33i5 900 1832.3 533-6 1244.2 •55 1-8235 148 301.32 357-2 II 90-3 3- •3336 1000 J 2035-9 546-5 1248. 1 •5 2.014 Saturated. Steam from 32° to STS 0 . {Claudel.) Tem- pera- ture. PRE! Mercu- ry- 5 SURE. Per Sq. Inch. Weight of 100 Cub. Feet. Volume of 1 Lb. Tem- pera- ture. Pre Mercu- ry- SSURE. Per Sq. Inch. Weight of 100 Cub. Feet. Volume of 1 Lb. 0 Ins. Lbs. Lb. Cub. Feet. 0 Ins. Lbs. Lbs. Cub. Feet. 32 .181 .089 .031 3226 125 3-933 x -932 •554 180.5 35 .204 . 1 •034 2941 130 4-509 2.215 •63 158- 7 4 ° .248 .122 .041 2439 135 5-174 2.542 •7x4 140. X 45 .299 .147 •049 2041 140 5.86 2.879 .806 124. 1 50 .362 .178 •059 1695 145 6.662 3-273 .909 IIO 55 .426 .214 •07 X 4 2 9 150 7-548 3.708 1.022 97.8 60 •517 •254 .082 1220 155 8-535 4-193 i - 145 87-3 65 .619 •304 •097 1031 160 9-63 4-731 x -333 75 70 •733 •36 .114 877.2 165 10.843 5-327 x.432 69.8 75 .869 •427 ■134 746-3 170 12.183 5-985 1.602 62.4 80 1.024 •503 .156 641 175 13-654 6. 708 x -774 56.4 85 1.205 •592 .182 549-5 180 15.291 7 - 5 II x -97 50.8 90 1.41 •693 .212 47 X -7 185 17.041 8-375 2.181 45-9 95 1.647 .809 •245 408.2 190 19.001 9-335 2.411 4 X -5 100 I - 9 I 7 •942 .283 353-4 195 21.139 10.385 2.662 37-6 105 2.229 1.095 •325 307-7 200 23.461 11.526 2-933 34 - x IIO 2-579 1.267 •373 268.1 205 25.994 12.77 3.225 3 i xx 5 2.976 1.462 .426 234-7 210 28.753 14.127 3-543 28.2 120 3-43 1.685 .488 204.9 212 29.922 14.7 3.683 27.2 30 7io STEAM. GASEOUS STEAM. When saturated steam is surcharged with heat, or superheated, it is termed gaseous or steam-gas. The distinguishing feature of this condition of steam is its uniformity of rate of expansion above 230°, with the rise of its tem- perature, alike to the expansion of permanent gases. To Compute Total Heat of Gaseous Steam. 1074.6 -475 £ = H. t representing temperature , and H total heat in degrees. Hence, total heat at 212 0 , and at atmospheric pressure = 1175.3°. Specific gravity = .622. To Compute 'V'elocity of Steam. Into a Vacuum. Rule. — To temperature of steam add constant 459, and multiply square root of sum by 60.2 ; product will give velocity in feet per second. Into Atmosphere. 3.6 ffh = V. V representing velocity as above , and h height in feet of a column of steam of given pressure and uniform density , weight of which is equal to pressure in unit of base. Illustration. — Pressure of steam 100 lbs. per sq. inch, what is velocity of it# flow into the air? Cube foot of water = 62. 5 lbs., density of steam at 100 lbs. = 270 cube feet. Hence, 62. 5 : 100 : : 270 : 432 =1 volume at ico lbs. pressure , and 432 X 144 = 62 208 feet = height of a column of steam at a pressure 0/100 lbs. per sq. inch. Then 3. 6 y/62 208 = 898 feet. EXPANSION. To Compute Point of Cutting off to Attain Limit of Expansion. b _p/ L p — point of cutting off. b representing mean bach pressure for entire stroke , in lbs. per sq. inch , f friction of engine, P initial pressure of steam , all in lbs. per sq. inch , and L length of stroke, in feet. Illustration. — Assume stroke of piston 9 feet, pressure 30 lbs. . mean back press- ure 3 lbs., and friction 2 lbs. ’ ? 3 + 2 X 9-^ 30 = 1-5 fMt To Compute Actual Ratio of Expansion. I _1_ c -—A— — R. c representing clearance or volume of space between valve seat and 1 4~ c mean surf ace of piston, at one or each end in feet of stroke, l length of stroke at point of cutting off, excluding clearance in feet, and R actual ratio of expansion. Illustration. — Assume length of stroke 2 feet, clearance at each end 1.2 ins., and point of cutting off 1 foot. 1.2 ins. = . 1. Then ^ 4 — = !- 9 ratio. t I+-.X To Compute Pressure at any Point of Eeriod of Ex- pansion. When Initial Pressure is given. P l-±-s=p. p representing pressure at period t of given portion of stroke, both in lbs. per sq. inch , and s any greater portion of stroke than l. When Final Pressure is given. P'xL' 4 -s = j). P' representing final pressure, in lbs. per sq. inch, and L' length of stroke, including clearance, in feet. Illustration i. — Assume length of stroke 6 feet, clearance at each end 1.2 ins., pressure of steam 60 lbs., point of cutting off one third; what is pressure at 4 feet? 1.2 ins. .= . 1 foot. 60 X 2 -f- . 1 - 4 - 4 -f- . 1 = 30. 73 lbs. 2 .. — What is pressure in above cylinder at 2.8 feet, when final pressure is 21 lbs. ? 21 X 6 + .1 -r- 2.8 -f- .1 = 44-17 lbs. STEAM. /II To Compute M.ean or Total Average Pressure. P P* log ~ R — — — p' or mean or average pressure. I ' length of stroke at point of cutting off, including clearance. Illustration. — Assume elements of preceding cases: i -f hyp. log. R = 2.065. 60 (2. 1 X 2.065 — • 1) 254. 19 : 42.365 ll)S. To Compute Pinal Pressure. PXi'4-S = P'. Illustration. — Assume elements of preceding cases, steam cut off at 2 feet. 60 X 2 -f- . 1 -f- 6 -j- . 1 = 20. 65 lbs. To Compute Afean Effective Pressure. P (1 V 1 -fhyp. log. R — c — b, or (p' — b). Illustration. — Assume elements of preceding cases, b~ 2 lbs. per sq. inch. - — 2 = 40. 365 IbS . 60 (2.1 X 2.065 — . 1) 2 54- 1 9 To Compute Initial Pressure to Produce a Given .A.V— erage Effective or I^et Pressure. P' L l' ft +■ hyp. log. R) — c Illustration. — Assume elements of case 1. = P. 6 + -i 2 + .1 = 2. g ratio. 4 2 - 3 6 5 X 6 (2.1 X 2.065) — _ 254-19 ’4-2365" = 60 lbs. To Compute Point of Cutting off for a Given Patio of Expansion. 1 / - 4 - R — c. Or, L + c - 4 - R — c — l. Illustration. — Assume elements of preceding cases : R = — == 2.9, and ~ ^~‘ 1 2 -j- . 1 2.9 — .1 = 2 feet. To Compute Pressure in a Cylinder, at any- Point of Ex- pansion, or at End of Stroke. P r-r-H- 7 = P, or P 4 - R. Illustration. — Assume elements, of preceding cases: 60 X 2. 1 60 - = 60 lbs., and — = 20.69 lbs. 2.9 2-f .1 To Compute Initial Pressure for a Required 3 NTet Effec- tive Pressure for a Given Patio of Expansion. W+a&L ^ p' L Or, - = P. W representing net- a (1/ 1 — J— hyp. log. k — c) V 1 + hyp. log. R — c work in foot-lbs. = aLp' — b, and a area of piston, in sq. ins. Illustration.— Assume elements of preceding cases: area of piston = 100 sq. ins., back pressure 2 lbs., and net effective pressure = 42.365 lbs. 100 X 6 X 42. 365 — 2 = 24 219 foot-lbs. 24219 + 100X2X6 _ 25419 ^ 42.365X6 _ 254+9 = 60 lbs. 712 STEAM. Points of Expansion. i Relative points of expansion, including clearance 5 per cent., assuming stroke of piston to be divided as follows, and initial pressure = 1. Point 1 .75 -6875 .625 .5625 .5 .4375 -375 -333 - 2 5 - 2 .125 .1 Ratio 1 1. 31 1.43 i -55 I - 7 I *- 9 r 2 -i 5 2.43 2.74 3*5 4-4 6 - 7 - Hyp. Log. of above Ratios. .0 1.27 1.36 1.44 1.54 1-65 i -77 r -9 2 2 - 2 5 2 -43 2 -79 2 -95 Hyperbolic Logarithms. No. Log. No. Log. No. Log. No. Log. No. | Log. 1.05 i 00 00 2.65 • 974 6 . 4-25 1.447 5-8 x -758 7-4 2.001 •0953 2.66 •9783 4-3 i -459 5-85 1.766 7-45 2.008 1. 15 .1398 2.7 •9933 4-33 1.466 5-9 1-775 • 7:5 2.015 1.2 .1823 2-75 1. 0116 4-35 1.47 5-95 1-783 7-55 2.022 1.25 .2231 2. 8 1.0296 4.4 1.482 6 1.792 7.6 2.028 1.3 .2624 2.85 i-o 473 4-45 i -493 6.05 1.8 7-65 2.035 i -33 .2852 2.9 1.0647 4-5 1.504 6. 1 1.808 7.66 2.036 1.35* .3001 2-95 1. 0818 4-55 i- 5 i 5 6.15 1. 816 7-7 2.041 1.4 •3365 3 1.0986 4.6 1.526 6.2 1.824 7-75 2.048 i -45 • 3716 3°5 1.1151 4- 6 5 i -537 6.25 1-833 7.8 2.054 1.5 •4055 3 - 1 1. 1314 4.66 i -539 6-3 1.841 7-85 2.061 i -55 - 43 8 3 3^5 I * I 474 4-7 1.548 6-33 1.845 7-9 2. 067 1.6 •47 3-2 1.1632 4-75 i-558 6-35 1.848 7-95 2.073 1.65 .5008 3-25 1. 1787 4.8 1.569 6.4 1.856 8 2.079 1.66 .5068 3-3 *•*939 4-85 i -579 6-45 1.864 8.05 2.086 1.7 • 53 ° 6 3-33 1.203 4.9 1.589 6-5 1.872 8. 1 2.092 1.75 • 559 6 3-35 1.209 4-95 1-599 6-55 1.879 8.15 2.098 1.8 .5878 3-4 1.2238 5 1.609 6.6 1.887 8.2 2.104 1.85 .6152 3-45 1.2384 5-05 1.619 6.65 1.895 8.25 2.11 1.9 .6419 3-5 1.2528 5 -i 1.629 6.66 1.896 8-3 2.116 i -95 .6678 3-55 1.2669 5- I 5 1.639 6.7 1.902 8-33 2. 12 2 .6931 3 - 6 1.2809 5-2 j .6 4 9 6-75 1.909 8-35 2.122 2.05 .7178 3- 6 5 1.2947 5-25 1.658 6.8 1.917 8.4 2. 128 2. 1 .7419 3.66 1-2975 5-3 i.668 6.85 1.924 8-45 2.134 2.15 • 7 6 55 3-7 1-3083 5-33 1-673 6.9 I - 93 I 8-5 2.14 2.2 .7885 3-75 1.3218 5-35 1.677 6-95 1-939 8-55 2.146 2.25 .8109 3-8 i -335 5-4 1.686 7 1.946 8.6 2.152 2-3 .8329 3- 8 5 1.3481 5-45 1.696 7-05 i -953 8.65 2.158 2 -33 .8458 3-9 1.361 5-5 1-705 7 -i 1.96 8.66 2.159 2 -35 •B 544 3-95 1-3737 5-55 i- 7 x 4 7 -i 5 1.967 8.7 2. 163 2.4 .8755 4 1.3863 5-6 1-723 7.2 i -974 8-75 2. 169 2-45 .8961 4-°5 1.3987 5-65 1.732 7-25 1.981 8.8 2-175 2-5 •9 i6 3 4 - 1 1.411 5.66 i -733 7-3 1.988 8.85 2.18 2-55 • 93 6 4 - *5 1. 4231 5-7 1.74 1 7-33 1.992 8.9 2. IoO 2.6 •9555 4.2 I - 435 I 5-75 1.749 7-35 1-995 8-95 2.192 To Compute Me an Pressure of Steam upon a Piston ' "by- Hyperbolic Logarithms. t: Rule.— Divide length of stroke of a piston, added to clearance in cylinder at one end, by length" of stroke at which steam is cut off, added to clearance f at that end, and quotient will express ratio or relative expansion of steam or number. Find in table logarithm of number nearest to that of quotient, to which add 1. The sum is ratio of the gain. Multiply ratio thus obtained by pressure of steam (including the atmos- phere) as it enters the cylinder , divide product by relative expansion, and quotient will give mean pressure. Note.— Hyp. log. of any number not in tabic may be found by multiplying a common log. by 2.302*585, usually by 2.3. STEAM. 713 When Relative Expansion or Number falls between two Numbers in Table , proceed as follows : Take difference between logs, of the two numbers. Then, as difference between the numbers is to difference between these logs., so is excess of expansion over least number, which, added to least log., will give log. required. Illustration.— E xpansion is 4.84, logs, for 4.8 and 4.85 are 1.569 and 1.579, and their difference .01. Hence, as 4.85 4.8 = .05 : 1.579 00 1.569 = . 01 :: 4.84 — 4.8 = .04 : .008, and 1.569 -{-.008 = 1.577 = £0*7. required. Example. — A ssume steam to enter a cylinder at a pressure of 50 lbs. per sq. inch, and to be cut off at .25 length of stroke, stroke of piston being 10 feet; what will be mean pressure ? Clearance assumed at 2 per cent. — . 2 feet. 10 -f- • 2 = 10. 2 feet , stroke 10 -r- 4 -f- . 2 = 2. 38 feet. Then 10. 2 — 2.38 = 4. 29 rela- tive expansion. Hyp. log. 4.29 = 1.456, which -f- 1 = 2.456, and -- --- X 50 = 28.62 lbs. 4,29 Relative Effect of steam during expansion is obtained from preceding rule. Mechanical Effect of steam in a cylinder is product of mean pressure in lbs., and distance through which it has passed in feet. Effects of* Expansion. {Essentially from D. K. Clark.) Back Pressure is force of the uncondensed steam in a cylinder, consequent upon impracticability of obtaining a perfect vacuum, and is opposed to the course of a piston. It varies from 2 to 5 lbs. per sq. inch. It must be deducted from average pressure. Thus: assume pressure 60 lbs., stroke of piston as in preceding case, and back pressure 2 lbs. At termination of. ... . 1st, 2d, 3d, 4th, 5th, and 6th foot of stroke. Pressure 60 30 20 15 12 10 lbs. per inch. Back pressure 2222 2 2 “ “ “ Effective pressure 58 28 18 13 10 8 Total work done by expansion at termination of each foot or assumed division of stroke of piston is represented by hvp. log. of ratio of expansion, initial worker. ' 1 Thus, for a stroke of 10 feet and a pressure of 10 lbs. : At end of 1st, 2d, 3d, 4th, 5th, 6th, 7th, 8th, 9th, and ioth/oot. Steam is expanded ) into vols. , hyp. V— .69 1.1 1.39 1.61 1.79 1.95 2.o8« 2.2 2.3 log. of which ) Initial duty I 1 1 1 1 1 1 1 1 Total dutv 1.69 2.1 2- 39 2.61 2.79 2-95 3.08 3-2 3-3 Initial duty is rep- i resented by 10. . J 1 «o 16.9 21 23-9 26. 1 27.9 29-5 30.8 32 33 Resistance for each | 16 18 foot of stroke. . . ] ['=■ 2 4 6 8 10 12 14 20 Total effective ) duty j ;== 8 12.9 i5 15-9 16. 1 *5-9 15-5 14.8 14 13 Gain by expansion o 61.25 87.5 98.75 101.25 98.75 93.75 85 75 62.5 The same results would be produced if expansion was applied to a non-condens ing engine, exhausting into the atmosphere. Again, assume total initial pressure in a non-condensing cylinder 75 lbs. per sq. inch, expanded 5 times, or down to 15 lbs., and then exhausted against a back press- ure of atmosphere and friction of 15 lbs. At termination of. 1st, 2 d, 3 d, 4 th, and 5th foot of stroke Total duty 1 1.69 2.1 2.39 2 .6 i “ “ performed... 75 126.75 157.5 179.25 195.75 foot-lbs. “ back pressure 15 30 45 60 75 “ “ 11 effective duty.... 60 96.75 112.5 119.25 120.75 u Gain by expansion o 61.25 87.5 98.75 101.25 per cent. From which it appears that the total duty performed by expanding steam 5 times its initial volume is full 2.5 times, or as 75 to 195.75. 3 0 * STEAM. 7*4 Relative Effect of* Equal Volumes of Steam. Relative total effect or work of steam is directly as its mean or average pressure (A) and inversely as its final pressure (B), or volume of steam condensed. If former is divided by latter, quotient will give relative total effect or work (C) of a given volume of steam as admitted and cut off at different points of stroke of piston, with a clearance of 3.125 per cent. In following computations resistance of back pressure is omitted. If this press- ure is uniform with all the ratios of expansion, it is a^uniform pressure, to be de- ducted from the total mean pressure in column (A). (C) Relative Effect. Cut off at | Press (A) Mean. iure. (B) Final. (C) Relative Effect. Cut off at Press (A) Mean. sure. (B) Final. 1 1 I 1 •375 .761 •394 •75 .969 .787 1.28 •33 .702 •335 .6875 .946 .697 i-35 •25 .628 •273 .625 .924 .636 i-45 .2 •559 .224 •5 6 25 .889 •576 x-54 .125 •435 •15 •5 •857 .501 I -7 I - 1 .418 •13 1- 93 2.09 2- 3 2.05 2.9 3.21 To Compute Total Effective Work iix Oixe Strolre of Ris- ton, or as Griven Dy an Indicator Diagram. a P a + hyp. log. R — c) — w, and abh=z w'. w representing total work , and iv' back pressure. Note Pressure of atmosphere is to be included in computations of expansion; it is therefore to be deducted from result obtained in non-condensing engines. In condensing engines, the deduction due to imperfect vacuum must also be made, usually 2. 5 lbs. per sq. inch. Illustration.— A ssume cylinder of a condensing engine 26.1 ins. in diameter, a stroke of 2 feet, pressure of steam 95 lbs. (80.3 + 14-7) P e ** S( l- , inch i cut off at -5 stroke, with an average back pressure of 2 lbs. per sq. inch, and a clearance of 5 per cent. Area of piston* deducting half area of rod — 530 sq. ins. 2 X 5-r-ioo = .i clear- ance, and 2-j-.i-ri + .i='i.9 = ratio of expansion , and 1 -f hyp. log. 1. 9 — 1.642. Then 530 X 95 X i-i X 1.642 — . 1 — 53 ° X 2 X 2 = 50 35° X 1. 706 - 2120 - 83 777 lbs. Illustration.— A ssume cylinder of a non-condensing engine having an area of 2000 sq. ins., a stroke of 8 feet, steam at a pressure of 50 lbs. (35.3 + * 4 - 7 )» cut ott at .25 of stroke, and clearance .25 foot. Ratio of expansion 3.66, back pressure 17 lbs., and 1 -f hyp. l og. 3.66 — 2.297. 2000 X 50 (2.25 1 + hyp.' log. 3.66 — .25) = 100000 x 2.25 X 1 + 1-297 — c = 491 825 foot- lbs. 2000 x 17 X 8 = 272 oco foot-lbs. or negative effect , and 491 825 — 272 000 = 219 825 foot-lbs. Total Effect of One ED. of Expanded Steam. If 1 lb of water is converted into steam of atmospheric pressure = 14.7 lbs. per sq inch,' or 2116.8 lbs. per sq. foot, it occupies a volume equal to 26.36 pube feet ; and the effect of this volume under one atmosphere = 2116.8 lbs. X 26.3 6 Jeet — « 7Q0 foot-lbs. Equivalent quantity of heat expended is x unit per 772 foot- lbs , 5 1 55 7 ^ 9 7?2 — 72.3 units. This is effect of 1 lb. of steam of a pressure of one at- mosphere on a piston without expansion. Gross effect thus attained on a piston by 1 lb. of steam, generated at pressures varying from 15 to 100 lbs. per sq. inch, varies from 56000 to 62000 Joot-lbs. equiv- alent to from 72 to 80 units of heat. Effect of 1 lb. of steam, without expansion, as thus exemplified, is reduced by clearance according to proportion it bears to volume of cylinder. If _ clearance is 5 per cent, of stroke, then 105 parts of steam are consumed in the work of a stroke, which is represented by 100 parts, and effect of a given weight of steam vs 1 thou t ex- pansion, admitted for full stroke, is reduced in ratio of 105 to 100. Having deter- mined, by this ratio, effect of work by 1 lb. of steam without expansion, as reduced by clearance, effect for various ratios of expansion may be deduced from that, in terms of relative operation of equal weights of steam. STEAM. 715 Volume of 1 lb. of saturated steam of 100 lbs. per sq. inch is 4.33 cube feet, and pressure per sq. foot is 144X100= 14400 lbs.; then total initial work= 14400X4.33 — 62 352 foot-lbs. This amount is to be reduced for clearance assumed at 7 per cent. Then 62352 x 100 -r- 107 = 58273 foot-lbs., which, divided by 772 (Joule’s equiva- lent), = 75.5 units of heat. Total or constituent heat of steam of 100 lbs. pressure per sq. inch, computed from a temperature of 212 0 , is 1001. 4 units; and from 102 0 (temperature of condenser under a pressure of 1 lb.) the constituent heat is 1111.4 units. Equivalent, then, of net simple effect 75.5 units is 7.5 per cent, of total heat from 212 0 , or 6.7 per cent, from 102 0 . When steam is cut off at 1 -75 -5 -33 -25 -2 .125 and . 1 of stroke, comparative effects are as 1 1.26 1. 616 1.92 2.14 2.27 2.51 and 2.6. Total effects as given in table, page 718. Effect of 1 lb. of steam, without deduction for back pressure or other effects varies from about 60000 foot-lbs., without expansion, to about double that, or 120000 foot- lbs., when expanded 3 times, cutting off at about 27 per cent, of stroke- and to about 150000 foot-lbs. when expanded about 6 times, and cut off' at about 10 per cent, of stroke. Effect of Clearance. Clearance varies with length of stroke compared with diameter of cylinder, with form of valve, as poppet, slide, etc. With a diameter of cylinder of 48 ins., and a stroke of 10 feet, and poppet valves, clearance is but 3 per cent., and with a diameter of 34 ins. and a stroke of 4.5 feet and slide valves, it is 7 per cent. Illustration of Effect. — Assume steam admitted to a cylinder for .25 of its stroke, with a clearance of 7 per cent. Mean pressure for 1 lb. = .637, and loss by clearance — 7 -f- 100 = .07, which, added to .63 7, = .707, which is effect of a given volume of steam, if there was not any loss by clearance, or a gain of n per cent. When steam is cut off at 1 .71 Loss at 7 per cent, clearance. . — 7 7.2 •5 -33 * 2 5 .125 and . 1 stroke. 8.1 9.6 11 15.3 17 percent. To Compute Net Volume of Cylinder for Given Weight of Steam, Ratio of Expansion and One Stroke. Rule. Multiply volume of 1 lb. of steam, by given weight in lbs., by ratio of expansion and by 100, and divide product by 100, added to per cent, of clearance. Example. — Pressure of steam 95 lbs., cut off at .5, weight .54 lbs., volume of 1 lb. steam 4.55, and weight = .2198 lbs., stroke of piston 2 feet, and clearance 7 per cent.' Ratio of expansion 2 + . 14 -4- 1 -f- . 14 = 1. 88. 4*55 X -54 X i-88 X 100 461.92 , _ ; — - — = = 4.31 cube feet. 100+7 107 0 J To Compute Volume of Cylinder for Given Effect with a Given Initial Pressure and Ratio of Expansion. Rule. — Divide given effect or work by total effect of 1 lb. of steam of like pressure and ratio of expansion, and quotient will give weight of steam from which compute volume of cylinder by preceding rule. preceding E Assume S iven work at 50766 foot-lbs., and pressure and expansion as Total work by 1 lb., 100 lbs. steam, cut off at table of multipliers for 95 lbs. = .998, which x • 5» =by table 94 200 foot-lbs., and 94 200 = 94 012 foot-lbs. Then — -54 lbs. weight of steam. by 7 i6 STEAM. Consumption of Expanded Steam per IP of Effect per Hour. ~p~p . — nm ] which X 60^: 1980000 foot -lbs. per hour , which — 1 lb. steam, the quotient = weight of steam or water required per IP per hour. Illustration.— Effect of 1 lb., 100 lbs. steam, without expansion, with 7 per cent, of clearance = 58273 foot-lbs ., and 1 = 34 lbs - steam — weight of steam con- sumed for the effect per IP per hour. When steam is expanded, the weight of it per IP is less, as effect of 1 lb. of steam is greater and it may be ascertained by dividing 1 980 oco by the respective effect, or by dividing 34 lbs. by quotient of total mean pressure by final pressure, as given in table, page 718. When steam is cut off at 1 .75 .5 -375 -33 .25 and .2 of stroke. Volumes consumed per I . 2I l8 . 5 I? .6 16 14.9 lbs. IP per hour j Hence, assuming 10 lbs. steam are generated by combustion of 1 lb. coal per IP of total effect per hour, The coal consumed per) 6o 2 cut off at -5, mean pressure by rule (page 711) 86 lbs., and back pressure 3 lbs. V — x. v = 2. S = 4-33- * = 8.31. t and t' — 327.9° + 461. 2 0 and ioo° + 46i.2°. = 86. p' = 3- = 2 feet. Ti — i. a = 144 ins. L = IS7 74 8 - STEAM. 717 86 — 3 X 144 = 11 952 lbs. .1x54 cube feet. 2 -4- 1 _ . 2 ratio. 4. 33 -f- 8. 31 = . 526 effective cut-off. 33 000 I 2-2T Sr— — ■ — -2.76 cube feet. = .231 lbs. .. 3X144 4-33 2 2X4.33 772 X . 231 (789. i° — 561. 2 0 ) + 157 748 = 198 389 foot-lbs. ~ -f 9 = 99 i95 foot-lbs. 59?389 __ g 5 g g 27 f QoUbs 99i£5 _ 6g m • 2 3i 144 86 — 3 X 144 X 2 X 4. 33 = 103 504 foot-lbs. 198 389 -4-. 231 — 103 504 x 2.31 =1x74479 foot-lbs. 1744794-2 — 87239 foot-lbs. T ^5'5 X 100 X 144 X 4-33 = 966 456 foot-lbs. 9 66 456 — 103 504 = 862 952 foot-lbs. 966 456 2 X 4-33 = hi 600 lbs. 144X86 — 144 X3 tti6oo ‘ 10 7 1 080 000 Or io7 =18 504 673 foot-lbs. 2 3 9°4 . 980 000 X = 18 504 673 foot-lbs'. = . 725 IP, io 3 5°4 33 000 1 980 000 r . . = • 306 cw&e /ee£. 62. 5 X 103 504 J J 1 X 2 X 144 X 86 — 3 — 23 904 foot-lbs. 2X86 — 3X144 = 23904 foot-lbs. .1154X2.76X60 = 1 9 . n cube feet. 103 504 2 X 4-33 " c 952 foot-lbs. 33000 - = 2.761 cube feet. 86X144 — 3X144 0 f ft,^"““ f f? nn K eCti0n . 0f . e ^™ dit " re of available heat (A) and consumption resr ond f J? h i aTe , a total heat of colnl ' ustion of 10000000* foot-lbs., cor- f,® ap ™ d ‘“ g an equivalent evaporative power under i atmosphere at 212° of 13.4 lbs water and efficiency of furnace .5; then available heat of combustion of 1 lb? coai = 5 000 000 foot-lbs. tion e w?th th^r?i i0U ° f COal ,ri r IH> in an engiue of like dimensions and opera- tion with that here given would be 19223000=5000000 = 3.8444 lbs. Properties of Steam of Maximum Density. ( RanJcine .) Per Cube Foot. Temp o 3 2 4i 50 59 68 77 86 248 348 481 655 881 1171 1538 Temp. 95 104 113 122 131 140 149 X999 2 57i 3 2 77 4136 0 178 6430 79 21 Temp. 158 167 176 x8 5 194 203 212 9 687 11 760 14 200 17 010 20280 24 020 28 310 Temp. 221 230 2 39. 248 257 266 2 75 L 33 J 8o 38 700 44 930 51 920 59720 68 420 78050 Temp. L | Temp. L 0 0 284 88 740 347 197 700 293 100 500 356 219000 302 1 13 400 365 242 000 3ii 127 500 374 266 600 320 143 000 383 293 100 329 159 800 392 321 400 338 178000 401 351 600 SUPERHEATED STEAM. attuned by imparting to steam a temperature moderately in excess of that due to the volume or density of saturated steam are : Y 1. An increase of elasticity without a corresponding increase of water evaporated Both of these results, by increasing effect of the steam, economize fuel. Superheated steam should be treated as a gas re^hM 42140T . t pv 85. 44 T. T temperature of steam + 4 6x. 2 °, and t 32° + 4 6, 2 ° Iucstratiok— A ssum e temperature o f steam, 327. 9°, superheated to 341.1°. "T 32 + 461. 2° — 68 549 foot-lbs. - 100 lbs. per sq. inch, and at 341.1° 120. Then 42 140 X 461. 2 0 -f 34x . 1 Hence, as pressure of steam at 327. 9 0 120 100 = 1.2 to ; = a gain of one fifth. * Coal of average composition, 14 133 X 772 7 i8 STEAM. To Compute Energy and Efficiency of Superheated Steam. In following illustrations elements are same as those in preceding cases for satu- rated steam, with addition of the steam being superheated, so that I = n5 lbs., t = 338° -j- 461. 2 0 = 799. 2 0 , t' = 290 + 461.2° — 751-2°, S = 3.8, 5 = 7 - 4 - ?AtlS— Raf>'S = X; 7 , i 5-5 IaS = /i; a p — ap F" : E; h — X = 7i' ; ^ = P; - = H'"; - cw&e ; 1 980 000 US ’ RS 7 ap — ap' Efficiency of saturated steam (p. 716) .107, and, as above, . 109; hence — ^ = 1.02 to 1. If then, available heat of combustion of efficiency of furnace is assumed at 5 000000 foot-lbs ., as above, consumption of coal per IEP 18 183486 = 5000000 = 3.637 lbs. Note.— For further illustrations Rankine’s “ Steam-engine,” London, 1861, p. 436. Wire-drawing. Wire-drawing of steam is difference between pressure in boiler and pressure in cylinder, and is occasioned as follows: ... Resistance or friction in steam-pipe to passage of steam to steam-chest and piston. Resistance of throttle-valve to passage of steam, when it is partly closed or of in- sufficient area in proportion to steam-pipe. Resistance from insufficient area of valves or ports. Mr Clark, from his experimental investigation, declared, that resistance in a steam-pipe is inappreciable, when its sectional area is not less than .1 area of piston, and its velocity not exceeding 600 feet per minute. When velocity of a piston is from 200 to 240 feet per minute, area of steam may be .04th of piston. Effect of Expansion with Equal Volumes, and Effect of One Eh. of IOO Lbs. Pressure per Sq. Inch. Clearance at each End of Cylinder, including Volume of Steam Openings, 7 per cent, of Stroke, and 100 per cent, of Admission = 1. Ratio of Ex- pansion. Initial Volume Point of Cut-off. Stroke Totai Final. Initial Pressure r a Pressu Mean. Initial Pressure RES. Initial. Mean Pressure Weight of Steam of 100 Lbs. for one Stroke per Cube Foot. Actual By 1 Lb. of 100 Lbs. Steam. Effect. Per Sq. Inch per Foot of Stroke by 100 Lbs. Steam. Volume of Steam expended per IP of Work per Hour. Heat con- verted. Lbs. Foot-lbs. Foot-lbs. Lbs. Units. I j 1 X 1 • •247 58273 IOO 34 75-5 I. I .9 • 9°9 .996 1.004 .225 63850 99.6 3 1 82.7 1. 18 •83 .847 .986 1. 014 .209 67 836 98.6 29.2 87.9 1.23 .8 ..813 .98 1.02 .201 70 246 28.2 9 1 1.3 •75 .769 .969 1.032 ,I9 o 73513 96.9 26.9 95-2 1.39 .7 ■ 7 I 9 •953 1.049 .178 77 242 95-3 25.6 IOO. I 1.45 .66 .69 .942 1.062 • 17 79 555 94.2 24.9 102.9 i -54 .625 .649 • 9 2 5 1. 081 .161 83055 9 2 -5 23.8 107.6 1.6 .6 .625 • 9 I 3 1.095 • 155 85125 9 J -3 23-3 no. 3 1.88 .5 53-2 .86 1.163 • 131 94 200 86 21 122 2.28 •4 •439 •787 1. 271 . 108 104 466 787 X 9 1325 2.4 •375 .417 .766 I - 3°5 .103 107 050 76.6 l8.5 138.6 2.65 •33 •377 .726 i -377 •093 1 12 220 72.6 17.7 * 45-4 2 . Q .3 •345 .692 i -445 * .085 1x6855 69.2 16.9 i 5 i -4 3.35 .25 .298 •637 i- 57 - .074 124066 63-7 l6 160.7 4 .2 •25 •567 1.764 .062 132 770 56.7 I4.9 171.9 4.5 .16 .222 .526 1. 901 •055 138130 52.6 i 4 - 34 178.8 5 • 14 .2 .488 2.049 .049 142 180 48.8 13.92 184.2 5.5 .125 .182 •457 2.188 •045 146 325 45-7 i 3- 53 189.5 5.9 .11 . 169 •432 2 - 3!5 .042 148 940 43-2 13.29 192.9 6.3 . 1 .159 • 4*3 2.421 •039 15137° 4 i -3 13.08 196. 1 6.6 .09 .152 •398 2.513 •037 152955 39-8 12.98 197.7 7 .083 •143 , -381 2.625 •035 155200 38.1 12.75 201. 1 7.8 .066 .128 •348 2.874 .032 158414 34-8 12.5 205. 2 8 .0625 ; .125 •342 2.924 .031 159 433 34-2 11.83 206.5 STEAM. Alxiltipliers for Actual Weight 719 and Effect for other Pressure per Sq. Inch. Multi Weight. pliers. Actual Effect. Pressure per Sq. Inch. Multi Weight. pliers. Actual Effect. Pressure per Sq. Inch. Mult Weight. ipliers. Actual Effect. Lbs. 65 70 75 80 85 .666 .714 •763 .806 .855 •975 .981 .986 .988 .991 | Lbs. 90 95 IOO IIO 120 .901 •952 1 1.09 1.17 •995 •998 X.009 1. on Lbs. 130 140 150 160 170 1.28 I -37 1.46 !-55 1.64 1-015 1.022 1-025 1.031 *-°33 iu tuia liiu&tiauou, m connection witn preceding table, no deductions are made pressure 11011011 ° f temperature of steam while expanding, or for loss by back When steam is cut off at .0625, or one sixteenth, its expansion is 16 times but as / per cent, of stioke is to be added to it (.0625 - 1- .07) =: = 132 ? ner ppnt nr cedin'^ pa°e lG ° f ^ ° r ° Dly & liU1 ® ° Ver 7 times > as in 3 d column of table on pre- Column 7 is product of 58273 and ratio of total effect of equal weights of steam when expanded, or average total pressure divided by average final pressure. Thus, if steam is cut off at .5, with a clearance of 7 per ^rt ./ l' 86 X 100 ~ 86 1.6165, a od 58273 X 1.6165 = 94 200 foot-lbs. X ioo_ 53.2 Column 9 gives volume of steam consumed per IP per hour. Thus assume cvl " ba^area of 292 sq. ins., a stroke ot\ feet, ind pressure o/steumToo gj _. 2 ?k * 5% 400 foot-lbs., and 292 + 7 per cent, of stroke for clearance = ' 4 ’ X Q 2 - M M4 = 4- 34 cube feet, and weight of a cube foot of such steam Spe 3 r table 5 4 °° : 4 34 X ' 23 " 33000 : '5 6 4> which >X 60 minutes = 33.84, br 3 ” Tto pressures are computed on premise that steam is maintained at a uniform W terminluon S of s\rokT S ' 0n l ° Cylinder > and that expansion is operated correctly Column 10 is quotient of work in foot-lbs., divided by Joule's equivalent 772. Thus, 94 200 -r- 772 = 122. For percentage of constituent heat, converted from 102° and 21 2 0 assume 122 as in last case : ’ L X9 ^ IOO = 10,98 per cent for ID2 ° and 122 X 10-4-100 = 12.2 per cent. “Wire-drawing” will cause a reduction of pressure during admission and rlenr sho C rt"l'ide' ary fr °” 3 t0 8 PCr cent * acco ‘- di ”8 10 *■>*» ofvalve as po^et ion+r qtTOkr^Sni e ^ W i re ‘- d ? Win | 0f steara ’ and openin S of exhaust b'efore termination of stroke, involve deviations from a normal condition, for which deductions must ho made, added to which there is the back pressure, from insufficient condensation in condensing engines, and from pressure of air in non-condensing engines and com pression of exhaust steam at termination of stroke. s ’ ana com * r T 1,1 Feed at High Temperature. rLi ”iir W T H : T and 1 representing total heat in steam and temverature of Illustration. Assume steam at 248°, feed water 100° in one case and 150° in another, and density A, and total heat at 248° = II57 °; what is gain? 1 1 57 100 + 248 — 100 = IX 57 — 150 + 248 — 150= no w Then jL~ H ' _ I2Q 5 — 1x05 1 205° = total heat required from fuel. ieO= “ u = .083 = 8.3 per cent. H 1205 720 STEAM. COMPOUND EXPANSION. Compound Expansion is effected in two or more cylinders, and is prac- tised in three forms. ist. When steam in one cylinder is exhausted into a second, pistons of the two moving in unison from opposite ends — that is, steam from top or for- ward-end of first cylinder being exhausted into bottom or after-end of the other, and contrariwise— this is known as the Woolf* engine. 2 d. Steam from the ist cylinder is exhausted into an intermediate vessel, or “ receiver,” the pistons being connected at right angles to each other. 3d. Steam from receiver is exhausted into a 3d cylinder of like volume with 2d, pistons of all being connected at angles usually of 120 0 . The two latter types are those of the compound engine of the present time. Expansion from Receiver. The receiver is filled with steam exhausted from ist cylinder, which is then admitted to 2d, or 2d and 3d. in which it is cut off and expanded to termination of stroke. Initial pressure in 2d, or 2d and 3 d cylinders, is assumed to be equal to final press- ure in ist and consequently the volume cut off in the one or the other cylinders must be equal in volume to that of ist cylinder, for its full volume must be dis- charged therefrom. Inasmuch as 3d cylinder is but a division of the 2d, with addition of receiver, this engine, in following illustrations, will, for simplification, be treated as having but two cylinders. In illustration, assume ist and 2d cylinders to have volumes as 1 to 2, with like lengths of stroke, and that steam is cut off at .5 stroke, and equally expanded in both cylinders, the ratio of expansion in each cylinder being thus equal to their volumes. Volume received into 2 d cylinder would be equal to that exhausted from ist, as- suming there would not be any diminution of pressure from loss of heat by inter- mediate radiation, etc. This is based upon assumption that expansion occurs only upon a moving piston; but in operation, expansion occurs both in receiver and in intermediate passages, as nozzles and clearances; the 2d cylinder, therefore, receives steam at a reduced pressure, increased volume, and reduction of ratio of expansion. To meet this, and attain like effects, volume of 2d cylinder must be increased m proportion to increased volume of steam and its ratio of expansion. Consequently, there is no loss of effect aside from increased volume and weight of parts by inter- mediate expansion, provided primitive ratio of expansion is maintained by giving relative increased volume to 2d cylinder. Illustration.— Assume cylinders having volumes as 1 and 3, initial steam of ist cylinder to be 60 lbs. per sq. inch, stroke of piston 6 feet, cut oil' at one third, and clearance 7 per cent.* Final pressure, as per rule, page 711, = 22.62 lbs., and pressure as exhausted into receiver, reduced one fourth. = 16.97 lbs., assuming there is no intermediate fall of pressure. The steam, therefore, is expanded to 1.33 times volume of cylinder; a like volume, therefore, must be given to 2d cylinder, to admit of this at a like press- ure If therefore, the increased terminal volume of the steam in the ist cylinder was augmented, including a clearance of 7 per cent., the effect would be as follows: Volume admitted to 2d cylinder is equal to volume of ist added to its clearance, or to .33 volume of 2d cylinder added to its clearance; that is, to .33 of 107 per cent or 33.66 per cent., consisting of clearance, and 35.66 — 7 = 28.66 per cent stroke of 2d cylinder. The steam exhausted into 2d cylinder thus fills less than .33 of its stroke bv 4.67 (33.33 — 28.66). As steam is expanded from volume of ist cylinder, plus its clearance, to 2d cylinder, plus its clearance, ratio of expansion in 2d cylinder is equal to ratio of volume of both cylinders, which is 3, and 100 (represent ingfuU sfrofce) ± 7 = anU flna , ssure 22^2 = 7 . 54 lbs. per sq. inch. 28.66 + 7 * Iii 1825-28 James P. Allaire, of New York, adopted this design of engine in the steamboats Henry Eckford, Sun, Commerce, Swiftsure, Post Boy, and Pilot Boy. STEAM. 721 Assuming volume of receiver, or augmented terminal volume, for expansion in 2d cylinder, to have proportions of 1, 1.25, 1.33, and 1.5 times volume of 1st cylinder plus its clearance, the relations would be as follows: Augmented terminal volumes) in 2d cylinder . Final volumes in 2d cylinder) added to clearance J Ratio of expansion in 2d cyl’r. . Intermediate reductions of \ pressure Equal to Pressures in receiver and ini tial pressure in 2d cylinder. Final pressure in 2d cylinder I 1.25 i-33 i-5 ( times volume of { 1st cylinder. 1.07 1-337 1.427 1.605 ( do. do. J including clear- 3.21 3.21 3.21 3-21 ( ance. l times volume of ( 1st cylinder. 3 2.4 2.25 2 0 .2 •25 •33 (of terminal press- \ ure in 1st cyl’r. , 0 4-52 5-65 11.31 fbs. per sq. inch. 22.62 18. 1 16. 96 ix- 3* do. do. 7-54 7-54 7-54 7-54 do. do. To Compute Expansion in. a Compound Engine. RECEIVER ENGINE. Ratio of Expansion. In 1st cylinder , as per formula, page 710. In 2d cylinder. — — - r — ratio. Of Intermediate Expansion. — ratio, n representing ratio of intermediate reduction of pressure between 1st and 2d cylinder , to final pressure in 1st cylinder, and r ratio of area of 1st cylinder to that of 2d. Illustration. — Assume n = 4, and r = 3. Then - — - X 3 = 2.25 ratio , and — - — = 1.33 ratio. 4 4 — 1 Total or Combined Ratio of Expansion, r R' = product of ratio of 1st and 2d cyl- inders by ratio of expansion in 1 st cylinder. As when r — 3, and R' = 2.653, then 2-653 X 3 — - 7-959 tyfaZ ratio. Hence, Combined Ratio of Expansion in both cylinders. - — - r R'=R". R' rep- resenting ratio of expansion in 1 st cylinder , and R" combined ratio. Illustration.— Assume as preceding, and R' = 2.653. Then — — - X 3 X 2.653 1=1 5- 969 combined ratio. • 4 To Compute Effect for One Stroke and a Given Ratio of’ Expansion in Eirst Cylinder. Without Intermediate Expansion. Rule. — Multiply actual ratio of ex- pansion in 1st cylinder by ratio of both cylinders, and to hyp. log. of com- bined ratio add 1; multiply sum by period of admission to 1st cylinder plus clearance, and term product A. Divide ratio of both cylinders, less 1, by ratio of expansion in 1st cyl- inder ; to quotient add 1 ; multiply sum by clearance, and term product B. Subtract B from A, and term remainder C. Multiply area of 1st cylinder in sq. ins. by total initial pressure in lbs. per sq. inch, and by remainder C. Product is net effect in foot-lbs. for one stroke. With Intermediate Expansion. Add effect thereof to result obtained above, and by following formula : Or, V 1 -J- hyp. log. R" — c ^1 -j- aP = E. a representing area in sq. ins., P initial pressure in lbs. per sq. inch of 1 st cylinder , V length of admission or point of cutting off plus clearance , c clearance in feet , and E effect in foot-lbs. 3P 722 STEAM. Illustration.— Assume areas of cylinders i and 3 sq. ins., length of stroke 6 feet, pressure of steam 60 lbs. per sq. inch, cut off at 2 feet, clearance 7 per cent., and area of intermediate space, as receiver, one third volume of 1st cylinder. R"= ratio of expansion in 2d cylinder -X.3 X 2.653 = 5.969 hyp. log. 4 2.653 x 2.25 -fix 2.42— 3 — 1 = 2.653 4- 1 x .42 x i x 60 . == 1.7865 4- 1 x 2.42 — 2 = 2.6534-1 X .42 X 60 = 6.743 — .737 X 60 = 360.3 6 foot-lbs. 1st Cylinder . Effect on piston 60 lbs. X 1 inch x 2 feet = 120 foot-lbs. “ of clearance 60 lbs. x -42 foot = 25.2 “ Total initial effect = 60 x 2 X .42 =145.2 foot-lbs. Then 145.2 X i4-hyp. lo g- 2.653 or 1-976 . Less effect of clearance Net effect on piston above vacuum line. Less effect of back pressure 60 = 2.653 = 22.61, which, x 3 sq. ins. and 2 feet stroke Net effect on piston = 286.91 foot-lbs. = 25.2 “ = 261.71 foot-lbs. = 135.66 “ = 126.05 foot-lbs. 2 d Cylinder. 145.2 X 1 4~ hyp. log. 2.25 or 1. 81 =262.81 foot-lbs. Effect of clearance 22.61 X 3 X -42 = 28.49 “ =234.32 foot-lbs. 360. 37 foot-lbs. Intermediate reduction of pressure, as given at page 721, = .25 X 22.61 = 5.65 lbs. per sq. inch, which, x 3 sq. ins. and by 2 per foot of stroke, = 33.9 foot- lbs. Hence 360. 36 4- 33- 9 = 394- 26 foot-lbs. Or, by sum of the three results, viz. : 1st cylinder 126.05 foot-lbs. Intermediate expansion 33.9 2d cylinder 234.32 “ 394.27 foot-lbs. WOOLF ENGINE. D. K. Clark. Ratio of Expansion.— In 1st cylinder as per formula, page 710. In 2 d cylinder , r y 4-35 =i4- x = ratio, r representing ratio of area of 1st cylinder to that of 2d, l and V lengths of stroke and of stroke added to clearance , in ins. or feet, and x ratio value of intermediate volume. Illustration. — Assume 1 = 6 feet, V = 7 per cent. = .42, r = 3, and x = .333. 6 , 3X g — 4 -- 333 Then 4 = 2.353, ratio of expansion in 2d cylinder . 1 4" -333 Total Actual Ratio of Expansion. R' (r -~\-x S J= ratio. Illustration. — Assume preceding elements, R = 2.6s3. Then 2.653 (3 x 7— — h -333^ — 2-653 X 3. 137 = 8.322, total actual ratio. \ 6.42 / Combined Actual Ratio of Expansion. R' [r y 4- »^ = 1 4 - x = ratio. Illustration.— Assume preceding elements. 3 x 4 - -333 = 1 4 ~ -333 = ~~~ = 6 , 2 4 2 > com bincd actual ratio. STEAM. 723 To Attain. Combined Ratio of Expansion and. Einal JPressnre in Sd Cylinder. Assuming four cases as taken for Receiver Engine with a clearance of 7 per cent. The relations would be as follows : Intermediate spaces are Add to these 1.07, the volume of 1st ) cylinder plus its clearance, and. . . . J To same values of intermediate space"! add 3, the volume of 2d cylinder, I and the sums are the final volumes j by expansion in 2d cylinder J Ratios of expansion in 2d cyl’r are quo- ) tients of final by initial volumes.. ) Intermediate falls of pressure are, in ) parts of final pressure in 1st cylinder j The initial pressures for expansion in ' 2d cylinder are : •333 | part of volume of ist cylin- •5 1 ( der plus its clearance, or, , 0 •357 •535 1.07 f total initial volumes for ex- 1.07 1.427 1.605 2.14 < pansion in 2d cylinder or ( times volume of ist cyl’r. 3-357 3-535 4-°7 f times volume of ist cyl- 3 ( inder. 2.804 2-352 2.202 1.902 ratios of expansion. f of final pressure ; or, as- .25 •333 j suming initial pressure at •5 | 63 lbs., and final pressure ( at 23.75 lbs., they are 0 5-94 7.92 11.87 lbs. per sq. inch. 1 •75 .66 •5 ( of-final pressure 1 in ist eyl- ( inder, or 23-75 17.81 15-83 11.87 lbs. per sq. inch. . 8.47 7-57 7.19 6.24 tbs. per sq. inch. Hence, final pressures in 2 d cyl'r are . . Combined Ratios in these Four Cases. 1 St. ist ratio of expansion 1 to 2.653 Combined Ratio. 2d do. do 1 to 2. 804 = 2.653 X 2.804 = 7.44. 2d. ist do. do 1 to 2.653 2d do. do 1 to 2.352 = 2.653 X 2.352 = 6.24. 3 d- ist do. do 1 to 2.653 2d do. do 1 to 2.202 = 2.653 X 2.202 = 5.84. 4th. ist do. do 1 to 2.653 2.653 X 1.905 = 5.05. 2d do. do. . . . . 1 to 1.905 = 1st case. 2d case. 3d case. 4th case. Net Effect. 431.81 foot-lbs. Initial effect of steam at 63 lbs. pressure, admitted to 1st cylinder, for 2 feet, or one third of stroke of piston, and with a clearance of 7 per cent, or .42 feet, is as follows : Effect on piston 63 X 2 feet = 12 6 foot-lbs. j Total initial do. in clearance . . 63 X -42 foot = 26. 46 = 63 X 2. 42 = 1 52. 4 6 foot-lbs. ( effect. This sum is initial effect, on which effect by expansion is computed, while it is 26.46 foot-lbs. in excess of the initial effect on the piston. The total effect, then, is computed as follows: 152.46 X (i + hyp. log. 7.44) or 3.0069 = 458.27 Less effect of clearance 26. 46 152.46 X (i + hyp. log. 6.24) or 2.831 =431.47 Less effect of clearance ,. 26.46 405.01 “ 152.46 X (1 -f-hyp. log. 5.84) or 2.7647 = 421.35 Less effect of clearance 26. 46 394. 89 “ 152.46 X (i + hyp. log. 5.05) or 2.6294 = 399.29 Less effect of clearance 26. 46 372. 83 ‘ ‘ The reductions of net effect in 2d, 3d, and 4th cases are 6.2, 8.6, and 13.7 per cent, of effect in 1st case. To Compute Effect for One Stroke and. a G-iven Com- bined Actual Ratio of Expansion. Rule. — To hyp. log. of combined actual ratio of expansion (behind both pistons) add i ; multiply sum by period of admission of steam to ist cylin- der, added to clearance, and from product subtract clearance. Multiply area of ist cylinder in sq. ins. by initial pressure of steam in lbs. per sq. inch and by above remainder. Product is net effect in foot-lbs. for one stroke. STEAM. 724 Example.— Assume elements of 1st illustration page 723. Hyp. log. 6.24 + 1 = 2.831, which, X 2.42 = 6.85, and 6.85 — .42 and remainder X 60 = 385. 8 fool-lbs. Or, V (1 +hyp. log. R')-CxaP = E. Comparative Effect of Steam in Receiver and Woolf Engines. The effect of steam in a compound engine, without clearance and without any intermediate reduction of pressure, is the same whether operated in a receiver or Woolf engine. When, however, there is an intermediate space between the two cylinders, as a receiver, there is ’an intermediate reduction of the pressure of the steam, conse- quent upon the increase of its volume in the receiver; the reduction of pressure, therefore, being less rapid than with the Woolf engine, the effect is greater. In illustration, the following comparative elements of the effect of both engines is furnished. Receiver. (7 per cent, clearance.) Woolf. Ratio of Expansion. Net Effect. Ratio of Expansion. Net Effect. 1st case. . . . • - 7 - 9 6 ..422.3 foot-lbs. 1st case. . . . . .7.64 . .431.71 foot-lbs. 2d kt ... .•• 5-97 ..421.55 “ 2d “ .., . . .6.24. . . . . ..405.11 “ 3d “ ... . . *5- 3 1 ..417.96 “ 3d “ ... • ••5.84 ..394.99 “ 4th “ ... ••• 3 - 9 8 ..402.78 “ 4th “ ... ••• 5 ; 05 •• 372-93 From which it appears, that although the effect of a receiver engine is the great- est, its ratio of expansion is less than with the Woolf engine. Also that by the addition of clearance to the pistons of each engine, the actual ratios of expansion are sensibly reduced, as compared with the ratios without clearance. INDICATOR. To Compute Nlean Pressure "by" air Indicator. in „ m vo Rule.— Divide atmosphere line, o o in fig- ure, into any convenient number of parts, as feet of stroke of piston, and erect perpendic- ulars at each point. Measure by scale of parts (alike to that of diagram) the actual mean pressure, as defined between the two lines at top and bottom of diagram, add the results, divide sum by number of points, and 123456789 10 quotient will give mean pressure in lbs. per sq. inch upon piston. Example.— Pressures, as above given, are: 35 _j_ 35 4. 35 + 34 4. 32 + 25 + 16 -f 10 + 8 -f 6 = 236, which, -4- 10, = 23. 6 lbs. Note. — If it were practicable to run an engine without any load, and it some- times is. the mean pressure, as exhibited by an indicator, would be an exact meas- ure of the friction of the engine. Conclusions on Actual Efficiency of Steam. For development of highest efficiencies of steam, as used in an engine, means for protecting it from cooling and condensing in the cylinder must be employed. Super- heating of it prior to its introduction into a cylinder is probably most efficient means that may be employed for this purpose. Application to cylinder of gases hotter than it is next best means; and next is the steam-jacket. In cases of locomotive and portable engines, consumption of steam per IBP per hour is less than for that of single-cylinder condensing engines for like ratios of ex- piinsion, which is due to effect of temperature of non-condensing cylinders, always exceeding 212 0 . It is deduciblc from these results that the compound engine is more efficient than the single-cylinder, and that, of the two kinds of compound engines, the receiver- engine is more efficient than the Woolf. Average consumption of bituminous coal per IIP per hour, for compound engines in long voyages, as shown by Mr. Bramwell, ranged from 1.7 to 2.8 lbs. (Z>. K. Clark.) STEAM. 725 To Compute Volume of Water Evaporated, per Lb. of Coal. \ — v W __ VQ j ume 0 y wa i er ^ { n ib s% y an d v representing volume of steam and F d relative volume of water, in cube feet, W weight of cube foot of water, and F weight of fuel consumed , both in lbs., and d density of water, in degrees of saturation. Illustration.— Take case at foot of page. V = 449887 cube feet, ^ = 838 cube feet, W = 64.3, E = 1, and F = 4061 lbs. 449 887 . 838 X 64. 3 _ g 2 bs. per hour. 4061 X 1 0 Gain in Fuel , and Initial Pressure of Steam required , when Acting Expan- sively, compared with Non-Expansion or Full Stroke. Poiat of Catting off. Gain in Fuel. Cutting off. Point of Cutting off. Gain in Fuel. Cutting off. Point of Cutting off. Gain in Fuel; Cutting off. Stroke. Per Cent. Lbs. Stroke. Per Cent. Lbs. Stroke. Per Cent. Lbs. •75 22.4 1.03 •5 41 1. 18 •25 58.2 1.67 .625 32 1.09 •375 49.6 1.32 .125 67.6 2.6 Illustration.— What must be initial pressure of steam cut off at .5, to be equiv- alent to 100 lbs. per sq. inch at full stroke ? 100 at full stroke = 100, and 100 X 1.18 = 118 lbs. To Compute Grain in Enel. Rule. — Divide relative effect of steam by number of times the steam is expanded, and divide 1 by quotient ; result is the initial pressure of steam required to be expanded to produce a like effect to steam at full stroke. Divide this pressure by number of times the steam is expanded, and sub- tract quotient from 1, remainder will give gain per cent, in fuel. Example.— When steam is cut off at .5, what is gain in fuel, and what mechanical effect ? Relative effect, including clearance of 5 per cent.,= 1.64; number of times of ex- pansion = 2. 1. 64 -1-2 = .82, and 1 -4- . 82 = 1. 22 initial pressure. 1.22-4-2 = . 61’ and 1 —.61 = .39 per cent. Mechanical effects of steam at full and half strokes are 2 — 1.64 = .36 difference. Hence, 1.64 : .36 :: 50 (half volume of steam used) : 10.97 per cent, more fuel to produce same effect at half stroke, compared with steam at full stroke. To Compuite Consumption of Fuel in. a Furnace. When Dimensions of Cylinder , Pressure of Steam , Point of Cut-off, Revo- lutions, and Evaporation per Lb. of Fuel per Minute are given. Rule.— Compute volume of cylinder to point of cutting off steam, in- cluding clearance. Multiply result by number of cylinders, by twice number of strokes of piston, and by 60 (minutes) ; divide product by density of steam at its pressure in cylinder, and quotient will give number of cube feet of water expended in steam. Multiply number of cube feet by 64.3 for salt water (62.425 for fresh), divide product by evaporation per lb. of fuel consumed, and quotient will give consumption in lbs. per hour. Example.— Cylinder of a marine engine is 95 ins. in diameter by 10 feet stroke of piston; pressure of steam in steam-chest- is 15.3 lbs. per sq. inch, cut off at .5 stroke; number of revolutions 14.5, and evaporation estimated at 8.5 lbs. of salt water per lb. of coal; what is consumption of coal per hour, when density of water is maintained at 2-32? (See Saturation, page 726.) Volume of steam at above pressure, compared with w r ater (15.3 + I 4 - 7 )i = 8 3 8 - Area of 95 ins. 4-2.5 per cent, for clearance - 4 - 144 = 50.45 cube feet. Poin t of cut- ting off 5 feet -f 2. 5 per cent. = 5 feet 1.5 ins., and 50.45 X 5 feet *-5 ins - X 14 - 5 X 2 X 60 = 449 887 cube feet steam per hour. 3 P* yi6 STEAM. Hence, 449 887 -4- 838 = 536. 86 cube feet water , which, x 64. 3 = 34 520 lbs. , which, -4- 8. 5 = 4061 lbs. coal per hour. Note. — Elements given are those of one engine of steamer Arctic, and consump- tion of clean fuel (selected) for a run of 12 days (one engine) was 3820 lbs. per hour. Utilization of Coal in a Marine Boiler. Experiment gives from .55 to .8 per cent, of the heat developed in the combustion of coal, as utilized in the generation of steam. Ordinarily it may be safely taken at .65. SALINE SATURATION IN BOILERS. Average sea-water contains per 100 parts : Chloride of sodium (com. salt) . .2.5; Chloride of magnesium .33. . . . = 2.83 Sulphuret of magnesium 53; Sulphuret of lime = .54 Carbonate of lime and of magnesia 02 Saline matter 3. 39 Water 96.61 Hence, sea-water contains .0339th part of its weight of solid matter in solution, and is saturated when it contains 36.37 parts. Mean quantity of salts, or solid matter, in solution, is 3.39 per cent., three fourths of which is common salt. Removal of Incrustation of Scale or Sediment. Potatoes, in proportion of .033 weight of water. Molasses, in proportion of 1.6 lbs. per IP of boiler. Oalc, suspended in the water, and Mahogany or Oak sawdust , and Tanner's and Slippery Elm bark , renewed frequently, according to volume of it, and the evaporation of the water. Muriate of Ammonia and Carbonate of Soda, in quantity to be determined by observation. BLOWING OFF. To Compute Loss of Heat by Blowing Off of Saturated. Water from a Steam-boiler. - — * = proportion of heat lost, S — T X heat required from fuel for water evaporated in degrees , and - — ^ - ■ ■ = loss of heat per cent. S representing S — T E 1 sum of sensible and latent heats of water evaporated , T temperature of feed water, t difference in temperature of water blown off and that supplied to boiler, E volume of water evaporated, proportionate to that blown off, the latter being a constant quan- tity, and represented by 1. Values of E at following degrees of saturation, and volumes to be blown off: 32. Value E. Volume to Blow off. 32 . Value E. Volume to Blow off. 32. Value E. Volume to Blow off. 32. Value E. Volume to Blow off. 1.25 •25 4 1.65 •65 i-54 2 1 1 2-75 i-75 •57 i-35 •35 3 1-75 •75 I -33 2.25 1.25 .8 3 2 •5 i-5 •5 2 1.85 .85 1. 18 2-5 i-5 .66 2.25 2.25 •45 Thus, when water in a boiler is maintained at a density of — , 1 volume of it is 3 2 evaporated, and an equal volume, or 1, is to be blown off. Hence 1 + 1 — 1 == 1 — ratio of volume evaporated to volume blown off. Illustration i. — Point of blowing off is 2 (32), pressure of steam is 15.3 lbs., mer- curial gauge, and density of feed water 1 (32) ; what is proportion of heat lost? S = 1157.8°. Then *= *5-3+ *4- 7 = 250.4° = 150.4“ E = 1. 1157.8 — 100 X 1 -f- 150.4 150.4 = 8.03 proportion of heat lost. STEAM. — STEAM-ENGINE. J2J 2. — Assume point of blowing off 1.75 (32); preceding case? E = * 75 * i5o-4 1157.8 — 100 X .75+150.4 what would be loss of heat per cent, in = 15.9 per cent, lost by blowing off. 3 . — Assume elements of preceding case. What is total heat required from fuel for water evaporated ? 1157.8 — 100 X -75 = 793 - 35 ° To Compute Volume of Water Blown Off to that Evaporated. When Degree of Saturation is Given. Rule. — Divide 1 by proportionate volume of water evaporated to that blown off, or value of E as above, for degree of saturation given, and quotient will give number of volumes blown off to that evaporated. Illustration. — Degree of saturation in a marine boiler is — ^ ; what is volume 32 of water blown off? E == 1.25. Then 1 - 4 - 1.25 — .8 blown off. Proportional Volumes of Saline Matter in Sea-water . Baltic 1 in 152 | British Channel. . . .1 in 28 I Atlantic, South. .1 in 24 Black Sea 1 “ 46 Mediterranean 1 “ 25 “ North. .1 “ 22 Red Sea 1 “ 131 | Atlantic, Equator. . 1 “ 25 | Dead Sea... 1 “ 2.59 When saline matter at temperature of its boiling-point is in proportion of 10 per cent., lime will be deposited, and at 29.5 per cent. salt. Temperature of water adds much to extent of saline deposits. STEAM-ENGINE. The range of proportions here given is to meet the requirements of variations in speed, pressure, length of stroke, draught of water, etc., in the varied purposes of Marine, River, and Land practice. CONDENSING. Eor a Range of Pressures of from 30 to SO Tbs. (IVIeren- rial Grange) per Sq. Inch., Cnt Off at Half Stroke. Piston-rod. Cylinder or Air-pump ( Wrought /row), .1 to .14 of its diam. ; (Steel), .8 diam. ; and ( Copper or Brass), .11 and .125. Condenser (Jet). Volume, .35 to .6 of cylinder. (Surface.) Brass tubes, 16 to 19 B W G, .625 to .75 in diameter by from 5 to 10 feet in length, and .75 to 1.25 in pitch, condensing surfaces, .55 to .65 area of evaporating sur- face of boiler with a natural draught; .7 to .8 with a blower, jet, or like draught. Or, for a temperature of water of 6o°, 1.5 to 3 sq. feet per IIP. With a very effective and sufficient circulating pump, areas may be reduced to .5 and .6. Effect of vertical tube surface, compared to horizontal, is as 10 to 7. Air-pump (Single acting and direct connection). Volume from .15 to .2 steam cylinder. Or, 2.75. For Double acting put 4 for 2.75. V and v representing volumes of condensing and condensed water per cube foot, and n strokes of piston per minute. Foot and Delivery Valves. Area, .25 to .5 area of air-pump. Delivei'y Valve (Out-board). With a Reservoir. Area from .5 to .8 Foot valve. Note.— Velocity of water through these valves should not exceed 12 feet per second. STEAM-ENGINE. 728 Steam and Exhaust Valves. — (Poppet'). - sn — area s or steam. - s n for v 1 c /7 24000 J ’ 20000 J exhaust; (Slide), for steam, and —— for exhaust, a representing jrea of steam cylinder in sq. ins., s stroke of piston in ins., and n number of revolu- tions per minute. Injection Pipes. — One each Bottom and Side to* each condenser; area of each equal to supply 70 times volume of water evaporated when engine is working at a maximum ; and in Marine and River engines the addition of a Bilge, which is properly a branch of bottom pipe. Note i.— Proportions given wiil admit of a sufficient volume of water when en- gine is in operation in the Gulf Stream, where the water at times is at temperature of 84°, and volume required to give water of condensation a temperature of ioo° is 70 times that of volume evaporated. 2. Velocity of flow of water through cock or valve 20 feet per second in river or at shallow draught, and 30 feet in sea or deep draught service. Feed Pump* — (Single acting, Marine ), Volume, .006 to .01 steam cylinder. (River and Land), or when fresh water alone or a surface condenser is used, .004 to .007. NON-CONDENSING. For a Range of Pressures of from SO to ISO ll>s. (Mercu- rial Gauge) per Sq. Inch, Cat Off at Half Stroke. Piston-rod. — (Wrought Iron), Diam., .125 to .2 steam cylinder. (Steel), .8 diam. Steam and Exhaust Valves . — Area is determined by rules given for them in a condensing engine, using for divisors 30 000 and *22 750. A decrease in volume of cylinder is not attended with a proportionate decrease of their area, it being greater with less volume. Feed-pump.* — (Single acting, Marine), Volume, .008 to .016 steam cylin- der. (River and Land), or where fresh water alone is used, .005 to .011. O-eTieral IRrules. Engines. Cylinder. Thickness . — (Vertical), ^~ = t ; (Horizontal),^— =:t ; (In- clined), divide by 2000 in a ratio inversely as sine of angle of inclination. D representing diameter of cylinder, p extreme pressure in lbs. per sq. inch that it may be subjected to, and t thickness in ins. Shafts , Gudgeons, Journals, etc. To resist Torsion. See rules, pp. 790, 796. Coupling Bolts. ~\J n representing number of bolts, D diameter of shaft, d' distance of centre of bolts from centre of shaft, and d diameter of bolts, all in ins. Cross-head, Wrought-iron. (Cylinder), a ~- = S, and y/y = d, or ~ =. b. a representing area of cylinder in sq. ins., I length of cross-head between centres of its journals in feet, and S product of square of depth d, and breadth, b, of section, both in ins. (Air-pump), ^-= S, and as above for d and b. If section of either of them is cylindrical, for S put ^Sx 1.7. Diam. of boss twice, and of end journals same as that of piston-rod. Sec- tion at ends .5 that of centre. * See Formulas, pnge 736. STEAM-ENGINE. 729 Steam-pipe. — Its area should exceed that of steam-valve, proportionate to its length and exposure to the air. Connecting-rod. — Length, 2.25 times stroke of piston when it is at all practicable to afford the space ; when, however, it is imperative to reduce this proportion, it may be twice the stroke. Comparative friction of long and short connecting-rods is, for length of stroke of piston, 12 per cent, additional; twice, 3 per cent. ; and for thrice, 1.33. Neck. — Diam. 1 to 1.1 that of piston-rod. Centre of body ( Horizontal ), 1.25 ins. ; (Vertical), .06 inch per foot of length of rod. With two connecting-rods or links, area of necks .65 to .75 area of attached rod. When a second set of rods is used, as in some air-pump connections, area of necks, in a ratio, inversely as their lengths to that of first set. Straps of Connecting-rods , Links, etc. — (Strap), area at its least section .65 neck of attached rod ; (Gib and Keg), .3 diam. of neck, width, 1.25 times, (Slot) 1.35 times (Draft) of keys .6 to .8 inch per foot. Distance of Slot from end of rod .5 diam. of pin. /P 1 Pins (Cranks, Beams, etc.). 3 / — . 355 — d. P representing pressure or thrust of rod or beam , l length of journal in ins., and C, for Wrought iron = 6 40, Cast, 560. Puddled steel , 600, and Cast , 1200. Length, 1.3 to 1.5 times their diam., and pressure should not exceed 750 lbs. per sq. inch for propeller engine, and 1000 for side-wheel. Cranks (Wr ought-iron). — Hub, compared with neck of shaft, 1.75 diam., and 1 depth. Eye, compared with pin, 2 diam., and 1.5 depth. Web, at pe- riphery of hub, width, .7 width, and in depth .5 depth of hub ; and at periph- ery of eye, width, .8 width, and in depth, .6 depth of eye. (Cast-iron.) Diameters of Hub and Eye respectively, twice diam. of neck of shaft, and 2.25 times that of crank pin. Radii for fillets of sides of web .5 width of web at end for which fillet is designed; for fillets at back of web, .5 that at sides of their respective ends. Beams, Open or Trussed. — Length from centres 1.8 to 2 stroke of piston, and depth .5 length. If strapped, Strap at its least dimensions .9 area of piston-rod, its depth equal to .5 its breadth. End centre journals each 1, and main centre journals 2.5 times area of piston or driving-rod. This proportion for strap is when depth of beam is .5 length, as above; conse- quently, when its depth is less, area of strap must be increased; and when depth of strap is greater or less than .5 width, its area is determined by product of its bd 2 , being same as if its depth was .5 its width. (Cast-iron). Area of Section of Centre. = A. p representing extreme pressure upon piston in lbs., d depth in ins., and l length in feet. Depth at centre .5 to .75 diam. of cylinder, and, when of uniform thick- ness, a thickness of not less than .1 of depth. Vibration of End Centres. — l -r- 2 — V {l -r- 2) 2 — (s 4- 2) 2 = vibration at each end ; s representing stroke of piston, in feet. Plumber Blocks (Shaft). — Binder d \J C == depth, d representing diam. of bolts when two to binder , l distance between bolts, b breadth of binder, all in ins., and C for wrought iron 1, steel .85, and cast iron .2. Holding-down Bolts. P - 4 - 3 C = area at base of thread of each bolt. C for mild steel for small and large bolts 6000 and 7000, for wrought iron 4500 and 6000, if but two are used. 9 Binder (Brass) . y = depth. 730 STEAM-ENGINE. Codes . — Angles of sides of plug from 7 0 to 8° from plane of it. Pumps . — Velocity of water in pump openings should not exceed 500 feet per minute. Fly-wheels and Governors . — See Rules, pages 451 and 452. Water-wheels. Water-wheels {Arms '). — Number from .75 to .8 diam. of wheel in feet. ( Blades ) Wood . — For a distance of from 5 to 5.5 feet between arms, thick- ness from .09 to .1 inch for each foot of diam. of wheel. Area of blades, compared with area of immersed amidship section of a vessel, depends upon dip of wheels, their distance apart, model and rig of vessel. In River service , area of a single line of blade surface varies from .3 to .4 that of immersed section; in Bay or Sound service , it varies from .15 to .2 ; and in Sea service , it varies from .67 to .1. Note.— A wrought-iron blade .625 inch thick bent at a stress withstood by an oak blade 3.5 ins. thick. FLaclial and. Feathering. Radial . — Loss of effect is sum of loss by oblique action of wheel blades upon the water, their slip, and thrust and drag of arms and blades as they enter and leave the water. Loss by oblique action is computed by taking mean of square of sines of angles of blades when fully immersed in the water. Loss by oblique action of blades of wheel of steamer Arctic , when her wheels were immersed 7 feet 9 ins. and 5 feet 9 ins., was 25.5 and 18.5 per cent., which was the loss of useful effect of the portion of total power developed by engines, which was applied to wheels. Feathering . — Loss of effect is confined to thrust and drag of arms and blades as they enter and leave the water. Comparative Effects. — In two wheels of a like diameter (26 feet, and 6 feet immer- sion), like number and depth of blades, etc., the losses are as follows : Radial 26.6 per cent. | Feathering 15.4 per cent. Loss of effect by thrust and drag in a feathering wheel, having these elements and included in the above given loss, is computed at 2 per cent. Relative loss of effect of the two wheels is, approximately, for ordinary immer- sions, 20 and, 15 per cent, from circumference of wheel. — - ^ d=zc. d and d f representing depths of blades 3 d 2 — d ' 2 below surface of water, and c centre of pressure, all in like dimensions , from bottom Centre of Pressure, In the cases here given, centres of pressure are as follows : Radial 6.41ns. | Feathering 8.5 ins. IPropellers. Propellers (Screw). — Pitch should vary with area of circle described by screw to area of midship section of vessel. AREA, TWO-BLADED. Area of disk of propeller to mid- ) ship section being 1 to J 6 1 5 4-5 4 I 3-5 3 2-5 2 Ratio of pitch to the diameter of V propeller is 1 to } .8 1.02 1. 11 1.2 j 1.27 131 1.4 1.47 For Four-bladed screws, multiply ratio of pitch to diam. as given above, by 1.35. Length , .166 diam. STEAM-ENGINE. 731 Slip .— Slip of a screw propeller is directly as its pitch, and economical effect of a screw is inversely as its pitch ; greater the pitch less the effect. An expanding pitch has less slip than a uniform pitch, and, consequently, is more effective. To Compute Thrust of* a Eropeller. ]jp 2I 1 — T. S representing speed of vessel in knots per hour. SLIDE YALYES. All Dimensions in Inches. To Compute Lap required on. Steam End, to Cnt-ofF at any given Part of Stroke of Biston. Rule. — From length of stroke subtract length of stroke that is to be made before steam is cut off ; divide remainder by stroke, and extract square root of quotient. Multiply this root by half throw of valve, from product subtract half lead, and remainder will give lap required. Example.— Having stroke of piston 60 ins., stroke of valve 16 ins., lap upon ex- haust side .5 in. = one thirty-second of valve stroke, lap ppon steam side 3.25 ins., lead 2 ins., steam to be cut off at five sixths stroke; what is the lap? 60 — — of 60 = 10. = . 408. . 408 X — = 3- 264, and 3. 264 — — = 2. 264 ins. 6 V 6° 2 2 To Ascertain Lap reqnirecL on Steam End, to Cnt-ofF at various Portions of Stroke. Valve Distance of piston from end of its stroke when steam is cut off, in parts of length of its stroke. without Lead. 1 5 1 7 1 5 1 1 1 1 2 12 3 24 ¥ t» TS 2T Lap in parts of) stroke j •354 •323 .286 .27 •25 .228 .204 .177 .144 .102 Illustration. — Take elements of preceding case. Under i is .204, and .204 X 16 = 3.264 ins. lap. When Valve is to have Lead . — Subtract half proposed lead from lap as- certained by table, and remainder will give proper lap to give to valve. If, then, as last case, valve was to have 2 ins. lead, then 3.264 — 2 2 = 2. 264 ins. To Compute at what Part of Stroke any- given Lap on Steam Side will Cut off. Rule. — To lap on steam side, as determined above, add lead ; divide sum by half length of throw of valve. From a table of natural sines (pages 390- 402) find the arc, sine of which is equal to quotient ; to this arc add 90°, and from their sum subtract arc, cosine of which is equal to lap on steam side, divided by half throw of valve. Find cosine of remaining arc, add 1 to it, and multiply sum by half stroke, and product will give length of that part of stroke that will be made by piston before steam is cut off. Example. — Take elements of preceding case. Cos. ^sin. — + 90 0 — cos. -f 1 X—= cos. (32 0 13' -}- 90 0 — 73 0 34') 60 = 48° 39', and cos. 48° 39' -f- 1 x — = 1. 66 X 30 = 49. 8 ins. 2 To Ascertain Breadth, of Ports. Half throw of valve should be at least equal to lap on steam side, added to breadth of port. If this breadth does not give required area of port, throw of valve must be increased until required area is attained. 7 32 STEAM-ENGINE. IPortion of Stroke at -wliick Exhausting 3?ort is Closed, and Opened. Lap on Exhaust Side of Lap on Exhaust Side of Valve in Parts of its Throw. Portion of Stroke at which Steam is cut off. * tV Valve in Parts of its Throw. Portion of Stroke at which Steam is cut off. xV .001 .008 .013 .022 .125 .062 5 .031 25 109 071 •053 041 • 093-074 .058 .043 -043-033 .0331.022 B .125 .062 5 •031 25 .026 .052 .066 .082 .008 .022 •033 .044 .004 .015 •023 •°33 Units in columns of table A express distance of piston, in parts of its stroke, from end of stroke when exhaust port in advance of it is closed; and those in columns of table B express distance of piston, in parts of its stroke, from end of its stroke when exhaust port behind it is opened. Illustration.— A slide valve is to be cut off at one sixth from end of stroke, lap on exhaust side is one thirty-second of stroke of valve (16 ins.), and stroke of piston is 60 ins. At what point of stroke of piston will exhaust port in advance of it be closed and the one behind it open. Under one sixth in table A, opposite to one thirty-second, is .053, which x 60, length of stroke = 3. 18 ins. ; and under one sixth in table B, opposite to one thirty- second, is .033, which X 60 == 1.98 ins. If lap on exhaust side of this valve w T as increased, effect would be to cause port in advance of valve to be closed sooner and port behind it opened later. And if lap on exhaust side was removed entirely, the port in advance of piston would be shut, and the one behind it open, at same time. Lap on steam side should always be greater than that on exhaust side, and differ- ence greater the higher the velocity of piston. In fast-running engines, alike to locomotives, it is necessary to open exhaust valve before end of stroke of piston, in order to give more time for escape of the steam. To Compute Stroke of* a Slide Valve. Rule.— To twice lap add twice width of a steam port in ins., and sum will give stroke required. Expansion by lap, with a slide valve operated by an eccentric alone, cannot be extended beyond one third of stroke of a piston without interfering with efficient operation of valve; with a link motion, however, this distortion of the valve is somewhat compensated. When lap is increased, throw of eccentric should also be increased. When low expansion is required, a cut-off valve should be resorted to in addition to main valve. To Compute Distance of* a Diston from End of its Stroke, when Dead produces its Effect. Rule. — Divide lead by width of steam port, both in ins., and term the quotient sine ; multiply its corresponding versed sine by half stroke, and product will give distance of piston from end of its stroke, when steam is ad- mitted for return stroke and exhaustion ceases. Example.— Stroke of piston is 48 ins., width of port 2.5 ins., and lead .5 inch; what will be distance of piston from end of stroke when exhaustion commences? . 5 - 4 - 2. 5 == . 2 = sine , ver. sin. of . 2 = .0202, and .0202 x — = .4848 ins. 2 To Compute Dead, when Distance of a Diston from tlie End of Stroke is given. Rule. — Divide distance in ins. by half stroke in ins., and term quotient versed sine ; multiply corresponding sine by width of steam port, and prod- uct will give lead. Example. — Assume elements of preceding case. 48 4 8 4 8 — = - 0202 = ver. sin. , and sine of ver. sin. .0202 == . 2, and . 2 X 2. 5 = . 5 inch. STEAM-EJSTGINE. 733 To Compute Distance ofaPiston from End of its Strolze, wlxen Steam is admitted. for its Return Stroke. ’ Rule.— D ivide width of steam port, and also that width, less the lead, by .5 stroke of slide, and term quotients versed sines first and second. Ascer- tain their corresponding arcs, and multiply versed sine of difference between first and second by .5 stroke, aild product will give distance. Example.— Assume elements of preceding case, lap = .5 inch , and stroke of slide 6 ins. and = ■ 8333, and .667 and ver. sin. 8o° 24' a, 7 o° 33' x ^ = . 3528 inch. To Compute Dap and. Lead of Locomotive Valves. To cut off at .33, .25, and .125 of stroke of piston, lap = 289, .25, and .177 t, outside lead = .07 t, and inside lead = .3 t. t representing stroke of valve, all in ins. HORSE-POWER. Horse-power is designated as Nominal , Indicated, and Actual. Nominal, is adopted and referred to by Manufacturers of steam-engines, in order to express capacity of an engine, elements thereof being confined to dimensions of steam cylinder, and a conventional pressure of steam and speed of piston. . Indicated, designates full capacity in the cylinder, as developed in opera- tion, and without any deductions for friction. Actual, lefers to its actual power as developed by its operation, involving elements of mean pressure upon piston, its velocity, and a just deduction for friction of operation of the engine. To Compute Horse-power of aix Engine. 1ST ominal .— Non- condensing, —— , and Condensing. = B? D revre- 1000 u 1400 x senting diameter of cylinder in ins., and v velocity of piston in feet per minute. Non-condensing is based upon uniform steam-pressure of 60 lbs. per sq. inch (steam-gauge), cut off at .5 stroke, deducting one sixth for friction and losses, with a mean velocity of piston, ranging from 250 to 4=50 feet per minute. 1 Condensing is based upon uniform steam-pressure of 30 lbs. per sq. inch (steam-gauge), cut off at .5 stroke, deducting one fifth* for friction and losses, with a mean velocity of piston of 300 feet per minute for an engine of short stroke, and of 400 feet for one of long stroke. Actual.— Non-condensing. - Ft 2 8 r -- HP. A representing area of cylinder in sq. ins., P mean effective pressure upon cylinder piston, inclusive of atmosphere, f friction of engine in all its parts, added to friction of load, both in lbs. per sq. inch, s stroke of piston in feet, and r number of revolutions per minute. Sum of these resistances is from 12.5 to 20 per cent., according to pressure of steam, being least with highest pressure. , * This value may be safely estimated at 2.5 lbs. per sq. inch for friction of engine in all its parts, and friction of load may be taken at 7.5 per cent, of remaining pressure. t This value is best obtained by an Indicator', when one is not used, refer to rule and table, op. 710-12 In estimating value of P, add 14.7 lbs., for atmospheric pressure, to that indicated by steam gauge or safety-valve. Clearance of piston at each end of cylinder is included in this estimate. 1 This value may be safely estimated in engines of magnitude at 1.5 to 2 lbs. per sq. inch, for friction ©f engine in all its parts, and friction of load may be taken at 5 to 7.5 per cent, of remaining pressure. bum of these resistances in ordinary marine engines is from 10 to 20 per cent., according to pressure of steam, exclusive of power required to deliver water of condensation at level of discharge or load-line di£fer “ t desi 8" s a, „ti- 3 Q 734 STEAM-ENGINE. Illustration.— Diameter of cylinder of anon-condensing engine is loins., stroke of piston 4 feet, revolutions 45 per minute, and mean pressure of steam (steam gauge) 60 lbs. per sq. inch. A=78.54 sq. ins. P 60+14.7 = 74 7 lbs - /= 2 *S+(6o+ *447-^2.5) X .075=7.92 lbs. Then 78-54 X (60+ '4-7~7-9 2 +<4-7) Xi X 4 X 45 _ n w 3300° Note i. Power of a non-condensing engine is sensibly affected by character of its exhaust, as to whether it is into a heater, or through a contracted pipe, to afford a. blast to combustion. 2 .— If an indicator is not used to determine pressure of steam in a cylinder, a safe estimate of it, when acting expansively, is .9 of full pressure, and when at full stroke from .75 to .8. Condensing. A Pt ~f* g. a 33 000 Power required to work the air-pump of an engine varies from .7 to .9 lhs. per sq. inch upon cylinder piston. Illustration.— Diameter of cylinder of a marine steam-engine is 60 ins., stroke of piston 10 feet, revolutions 15 per minute, pressure of steam 50 lbs. per sq. inch, cut off at .25 stroke, and clearance 2 per cent. A = 2827. 4 sq. ins. P (per Ex., page 713) = 28.62 lbs. f— 1.5 + 28.62 — 1.5 X .05 == 2.467 tbs. •_ Then °827-4 X 28.66-2.856 X2X10X 15 = 662.2-5 TP. 33 000 From which is to be deducted in marine engines power necessary to discharge water of condensation at level of load-line, which is determined by pressure due to elevation of water, area of air-pump piston, and velocity of its discharge in feet per second. Indicated. AP2s>: =H?and 33cooip 33 000 P 2 s r British. Admiralty Buile. — Nominal. 7 Av D2 v 33 000 6000 = IP. French. — {Force de Cheval.) 1.695 D 2 Lr = IP- D and L in meters. Illustration.— Assume a diameter of cylinder of .254 meters, with a stroke of piston of . 3 meters and 250 revolutions per minute. 1.695 X -254 2 X .3 X 250 = 8.18 IP. A Force de Cheval — 4500 kilometers per minute = 32 549 foot-lbs. = .987 57 IP. One IP == 1. 0139 Force de Chevaux. Compound Indicated. ALr|ii hyp. log. R" — .000053 = IP. L representing length of stroke in feet , R ,/ combined ratio of both cylinders , and b> back pressure. j Illustration. — Assume area of cylinder 3 sq. ins., stroke 6 feet, one stroke of ( piston, and steam 60 lbs. per sq. inch, cut off at .25. A = 3 sq. ins . , L = 6 feet , n = 1 stroke, P = 60 lbs. , R" = 5- 9 6 9> b = 3 ' per sq. inch , and r = . 5, and 1 + hyp. log. R" = 1 + 1.7865. Then 3X6X.5X X 1 + 1.7865 — 3) X-oooo53 = 9X 10.052X2.7865 — 3 \5* 909 / _ , X .000053 = .on 93 IP, which, x 2 for I revolution, = .023 86 IP per revolution. To Compute Volume of Water required to "be Evapo- rated in an Engine. Rule.— Multiply volume of steam expended in cylinder and steam-chests by twice number of revolutions, and multiply product by density of steam at given pressure. t X For reference see 2d and 3d foot-note on previous paf?e. STEAM-ENGINE. 735 Example. — Wbat volume of water will an engine require to be evaporated per revolution, diam. of cylinder being 70 ins., stroke of piston 10 feet, and pressure of steam 34 lbs. per sq. inch, including atmosphere, cut off at .5 of stroke? Area of cylinder =2 3848.5 ins. 10X12-7-2 = 60 ins., 60 X 3848.5 =2 230910 cube ins. Add, for clearance at one end, volume of nozzle, steam-chest, etc., 17 317 cube ins. Then 230 910 -f- 17 317 -4- 1728 X 2 = 287.3 cube feet , which, x .001336, density of steam at 34 lbs. pressure (see Note 2), =2.3838 cube feet. Note r. — This refers to expenditure of steam alone; in practice, however, a large quantity of water “ foaming,” differing in different cases, is carried into cylinder in combination with the steam; to which is to be added loss by leaks, gauges, etc. 2. — Volume of steam is readily computed by aid of table, pp. 708-9. Thus, den- sity or weight of one cube foot of steam at above pressure =2 .0835 lbs. Hence, as 62.5 lbs. : 1 cube foot .0835 lbs. : .001 336 cube foot. To Compute Volume of Circulating Water required. Toy an Engine. 1114 + .3T — t „ m ^ — - p = v . T representing temperature of steam upon entering the con- denser , t, t\ and t" temperatures of feed water , of water of condensation discharged , and of circulating water , all in degrees. Illustration.— Assume exhaust steam at 8 lbs. per sq. inch, temperatures of dis- charge ioo°, feed water 120 0 , and sea-water 75 0 . 1114-f .3 X 183— 120 _ 41.05 times. 100 — 75 * y Temperature at 8 lbs. pressure = 183°. To Compute Volume ofFlow through an Injection IPipe. Rule.— M ultiply square root of product, of 64.33 and depth of centre of opening i to condenser, from surface of external water, added to height of a column of water due to vacuum, in condenser, all in feet, by area of opening in sq. ins. ; and .6 product, divided by 2.4 (144 -h 60) will give volume in cube feet per minute. Example. — Diameter of an injection pipe is 5.375 ins., height of external water above condenser 6.13 feet, and vacuum 24.45 ^s. ; what is volume of flow per min.? Area of 5.375 ins. = 22.69 ins., c = .6. Vacuum ^ - - - 45 inS ’ == J2 lbs. ; 12X2.24 feet (sea- water) = 26. 88 feet, and 26. 88 -f 6. 13 — 33 . 1 jeet. ° 4 Then ^64.33 X 33.1 X 22.69 x ,6 628. 15 2= 261.73 cube feet. To Compute Area of an Injection Eipe. Rule.— A scertain volume of water required by rule, page 706, in cube ins. per second, multiply it by number of volumes of water required for con- densation, by rule, page 707, divide it by velocity due to flow in feet per second, and again by 12, and quotient will give area in sq. ins. Example.— An engine having a cylinder 70 ins. diam., stroke of piston 10 feet per minute 15, and steam 19.3 lbs., mercurial gauge cut off at .5; what should be area of its injection pipe at its maximum operation? Volume of cylinder 267.25 cube feet, cut off at .5 = 133.625 ins. wot p r nm rm t ‘ lt 34 lbS ' I4 ’ 7) = - 001 336. Velocit y of flow of injected watei (computed from vacuum and elevation of condensing water) 33 feet per second. Then 133.625 X 15 X 2 X 1728 -h 60 = 115452 cube ins. steam per second, and 115 452 X .001 336 =2 154.24 cube ins. water per second. onh a a T^nn^lT e L • re q uired t0 condense steam is about 70 times volume is about 4o P times d ’ WhlCh ° U 7 ° CCUrS m the Gulf of Mexico 5 ordinary requirement 154 24 -f- 11.59 (= 7-5 Per cent, for leakage of valves, etc.) = 165.83, which, x 70 as above, = u 608. 1 cube ins., and n 608. 1 -4- 33 x 12 2= 29. 31 sq. ins. STEAM-ENGINE. 7 3 6 Coefficient of velocity for flow under like conditions == .6; hence, 29.31^.6 = 48.85 sq. ins. ■nj ote This is required capacity for one pipe. It is proper and customary that there should be two pipes, to meet contingency of operation of one being arrested. To Compute Area of* a Deed Dump. {Sea-water.) Rule. — Divide volume of water required in cube ins. by number of single strokes of piston, both per minute, and divide quotient by stroke of pump, in ins. ; multiply this quotient by 6 (for waste, leaks, “running up, etc.), and product will give area of pump in sq. ins. Example— Assume volume to be 5 cube feet and revolutions of engine 15 per minute, with a stroke of pump of 3.5 feet. 5 X i7 2 j _ ^ which -r- 3.5 X 12 = 13 72, and 13.72 X 6 = 82.32 sq. ins. 15 Note.— In fresh water, this proportion may be reduced one half. STEAM-IN JECTOlt. William Sellers & Co. Self-adjusting. Volume of* Water Discharged per Hour. Pressure of Steam in Lbs. No. No. 60 80 100 120 Cub. feet. Cub. feet. Cub. feet. Cub. feet. 3 28.12 31.66 35-2 38-75 7 4 52.16 58.44 64.72 7 1 8 5 82.18 92.02 101.86 111.7 9 6 119.09 133-33 147-57 161.82 10 Pressure of Steam in Lbs. 60 80 100 120 Cub. feet. Cub. feet. Cub. feet. Cub. feet. 162.65 182.1 201.55 221 213.2 238.8 264.4 290 269.97 302.28 334 - 59 366.9 333-64 373-57 4 * 3-49 453-41 Highest temperature admissible of feed water 135 0 . To Compute Size of Injector re which, X 60, = 313.2 lbs. 1728 To Compute Net Volume of Feed Water required per I DP per Hour. Operation.— Assume elements of formula, page 716, and illustration, page 717. Then .1154 X 2.76 X 60 = 19.11 lbs. Feed Pipes . — fv = diameter for small, and ^-Vv, for large pumps. d representing diameter of plunger in ins., and v its velocity in feet per minute. STEAM-ENGINE. 737 Itesvilts of Operations of Steam-engines. ( D . K. Clark.) Condensing Engine. Actual Ratio of Expan- sion. Steam per IIP as cut-off. Coal per IIP. Initial Pressure at cut-off. Steam per IIP per hour. SINGLE. 5-2 6.07 Lbs. I4-5I 14.27 12.92 Lbs. 2-5 2.2 Lbs. 34-5 46 23.25 50 Lbs. 17.4 18.7 20.72 19.6 18.62 3.62 Sulzer Corliss valves. 10 4. 122 3.3 Simpi-hpntpfl Dim 60 * ’ J COMPOUND ' Receiver I.85 I.852 4.OI 1-857 2.486 o. 221 14-45 14.85 10.94 13-34 13. 18 13-87 actual 22. 21 1. 61 56 85.5 J. Elder & Co J [ Marine, jacketed 1 Receiver J. & E. Wood j [ stationary 2.14 — Donkin j [ Woolf, stationary j jacketed 5o-5 22.51 15-37 American, Woolf j [ 1st cylinder [ both O' ** x 2.3I 5-63 3-77 Q. IQ _ 90 “ “ jacketed < f 1st cylinder iboth -j 20.71 — 90 14.1 NON-CONDENSING. Marshall, Sons, & Co 4.8 16.87 76 25-9 29.6 3 i -36 21.24 Davev, Paxman, & Co 5 T,nnnmnt,ive “Great Britain” l.AC 31-36 21.24 / 5 “ “ 2.94 — T 7 [Practical Efficiency of Steam-engines. Initial Volume =. i. Cylinders. Most Efficient Ratio of Ex- pansion. Steam * per IH? per hour. Cylinders. Most Efficient Ratio of Ex- pansion. Steam * per IH* per hour. CONDENSING. Single cylinder, jacketed. . . 6 Lbs. 19-5 Compound, jacketed, Woolf Compound, Woolf. IO 7 Lbs. 20.5 23 Single cylinder A “ “ superheated Compound, jacketed, Re- 1 ceiver j T 4 6 18.5 1 9 NON- CONDENSING. Single cylinder, + jacketed. . Single cylinder, $ 4 3 to to M * From boiler. t 70 lbs. pressure. \ 90 lbs. pressure. Standard Operation of a Portable Engine . Grate 5.5 sq. feet. Heating surface 220 “ “ Coal per EP per hour 6.25 lbs. “ “ sq. foot of grate. 9 “ “ “ hour 50 “ Water evaporated from 1 and at 212° per hour. j 45° “ “ per H? per hour 62.5 “ “ “ sq. foot of] grate ; lbs. 81.8 Ratio of heating surface of grate 40 to x. MIXTURE OF AIR AND STEAM. Water contains a portion of air or other uncondensable gaseous matter, and when it is converted into steam, this air is mixed with it, and when steam is condensed it is left in a gaseous state. If means were not taken to remove this air or gaseous matter from condenser of a steam-engine, it would fill it and cylinder, and obstruct their operation; but, notwithstanding the ordinary means of removing it (by air- pump), a certain quantity of it always remains in condenser. 20 volumes of water absorb 1 volume of air. 3 Q* 738 STEAM-ENGINE. £ o- i 5 ‘•sS I VO'O VO VO VO t-* t^OO CO CO O CO IN C) ft © ® & ■ 8*0 •° is b0« — o*S D • to to r/j bD j lONIN m 10X0 io xoio lO XOIOIO op 3 ' *MM«ejc5 - fa bO «f bfiS 2 S g-S S\5P 5a 5 ©^ 6 £ .2 <2 © ’2 bC 3 5 C3 3 2 .^i ft CJi |1 a_i 1 2 § +j o .«£ S.I ■p » bfl 8 S ! £ ►? gq © £ .a r “ a © S a ig ^ s a H ^ oa ^ § “2 © Is € s a tg o © o tS ° S M § ■» ^ 2 Diam. of Plunger in any single Cylinder Pump for like Volume and Speed. £ C c 10X0 10 10 10 . t"~. t'- 10X0 10 10 IO £ goo vo ro io iooo oo « w « n c oi "t- io iovd t^co oidiN w n n 4444NNNovtNH c MMMMHII-IMHHWMHHWO j. u- c c Volume delivered per Minute, at stated Number of Strokes. * m 1 m M w N t oOOOOOOOOOOOOOOOOOOOOOOO^ g i Single Strokes or Displacements per Minute of one Plunger. |S' 2 > 22 § 2 § 22 §I§§S 2 §§§§§ 8 '°' 8 '' gS © o O O O OOOOOOOOOOOOOOOOOO a s> & & & &&&&&&§,& Displace- ment per Stroke of one Plunger. §J«fl Sg' fe 8 £ 58 3 3 3 3 £ Sn W Sow . «T «T io H 3 $ ■ 3 MHHcicicieimrornMVV ^-vo m h o Length Stroke. 2 ro rj- xovo OOOOOOOOOOOOOOOOOjOiojo Diameter of Water- plungers. u-1 IO 10 10 10X0 ^ t^IlO IO © ° •° 2 a ^ stxB O c3 ca 3* ° £ m £ o ,o STEAM-ENGINE. — BOILER. 739 BOILER. Its efficiency is determined by proportional quantity of heat of com- bustion of fuel used, which it applies to the conversion of water into steam, or it may be determined by weight of water evaporated per lb. of fuel. In following results and computations, water is held to be evaporated from stand- ard temperature of 212 0 . Proportion of surplus air, in operation of a furnace, in excess of that which is chemically required for combustion of the fuel, is diminished as rate of combustion is increased; and this diminution is one of the causes why the temperature in a furnace is increased with rapidity of combustion. When combustion is rapid, some air should be introduced in a furnace above the grates, in order the better to consume the gases evolved. Natural Draught. Grate {Coal) should have a surface area of 1 sq. foot for a combustion of 15 lbs. of coal per hour, length not to exceed 1.5 times width of furnace, and set at an inclination toward bridge-wall of 1 to 1.5 ins. in every foot of length. When, however, rate of combustion is not high, in consequence Of low ve- locity of draught of furnace, or fuel being insufficient, this proportion of area must be increased to one sq. foot for every 12 lbs. of fuel. Width of bars the least practicable, spaces between them being from .5 to .75 of an inch, according to fuel used. Anthracite requiring less space than bituminous. Short grates are most economical in combustion, but generate steam less rapidly than long. Level of grate under a plain cylindrical boiler gives best effect with a fire 5 ins. deep, when grate is but 7.5 ins. from lowest point. Depth, Cast-iron, .6 square root of length in ins. {Wood), their area should be 1.25 to 1.4 that for coal. Automatic (Vicar’s). — Its operation effects increased rapidity in firing and more effective evaporation. Ash-pit. — Transverse area of it, for a combustion of 15 lbs. of cogl per hour, 2 to .25 area of grate surface for bituminous coal, and .25 to .3 for anthracite. Or 15 to 20 ins. in depth for a width of furnace of 42 ins. Furnace or Combustion Chamber. — {Coal) Volume of it from 2.75 to 3 cube feet per sq. foot of grate surface. {Wood) 4.6 to 5 cube feet. The higher the rate of combustion the greater the volume, bituminous coal requiring more than anthracite. Velocity of current of air entering an ash-pit may be estimated at 12 feet per second. Volume of air and smoke for each cube foot of water converted into steam is, from coal, 1780 to 1950 cube feet, and for wood, 3900. Rate of Combustion. — In lbs. of coal per sq. foot of grate per hour. Cornish Boilers, slowest, 4 ; ordinary, 10. Stationary , 12 to 16. Marine , 16 to 24. Quickest: complete combustion of dry coal, 20 to 23; of caking coal, 24 to 27 ; Blast or Fan and Locomotive , 40 to 120. Bridge-wall {Calorimeter).— Cross-section of an area of 1.2 to 1.6 sq. ins. for each lb. of bituminous coal consumed per hour, or from 18 to 24 sq. ins. for each sq. foot of grate, for a combustion of 15 lbs. of coal per hour. Temperature of a furnace is assumed to range from 1500° to 2000°, and volume of air required for combustion of 1 lb. of bituminous coal, together with products of combustion, is 154.81 cube feet, which, when exposed to above temperatures, makes volume of heated air at bridge- wall from 600 to 750 cube feet for each lb. of coal consumed upon grate. 740 STEAM-ENGINE. BOILER. Hence, at a velocity of draught of about 12 feet per second, area at bridge- wall, required to admit of this volume being passed off in an hour, is 2 to 2.5 sq. ins., and proportionately for increased velocity, but in practice it may be 1. a to 1.6 ins. When 20 lbs. of coal per hour are consumed upon a sq. foot of grate, 20 x 1.2 or 1.6 = 24 or 32 sq. ins. are required, and in a like proportion for other quantities. Or, When area of flues is determined upon, and area over bridge-wall is required, it should be taken at from .7 to .8 area of lower flues for a natural draught, and from .5 to .6 for a blast. When one half of tubes were closed in a fire-tubular marine boiler, the evapora- tion per lb. of coal was reduced but 1. 5 per cent. Firing. — Coal of a depth up to 12 ins. is more effective than at a less depth. Admission of air above the grate increases evaporative effect, but diminishes the rapidity of it. Air admitted at bridge-wall effects a better result than when admitted at door, and when in small volumes, and in streams or currents, it arrests or pre- vents smoke. It may be admitted by an area of 4 sq. ins. per sq. foot of grate. Combustion is the most complete with firings or charges at intervals of from 1 5 to 20 minutes. With a fuel economizer (Green’s) an increased evaporative effect of 9 per cent, has been obtained. When external flues of a Lancashire boiler were closed, evaporative power was slightly increased, but evaporative efficiency was decreased; and when 25 per cent, of like surface in setting of a plain cylindrical boiler was cut off, evaporation was reduced but 1.5 per cent. When temperature at base of chimney was 630°, with a fire 12 ins. in depth, it was decreased to 556° with one 9 ins. in depth, and to 539 0 with one 6 ins. High wind increases evaporative effect of a furnace;. Stationary or Land.— Set at an inclination downward of .5 inch in 10 feet. Smoke Preventing. —A test of C. Wye Williams's design of preventing smoke, at Newcastle, 1857, as reported by Messrs. Longridge, Armstrong, and Richardson, gave an increased evaporative effect with the “practical prevention of smoke.” Hence it was concluded, “ That by an easy method of firing, combined with a due admission of air in front of furnace, and a proper arrangement of grate, emission of smoke may be effectually prevented in ordinary marine multi-tubular boilers, with suitable coals. 2d. That prevention of smoke increases economic value of fuel and evaporative power of boiler. 3d. That coals from the Hartley district have an evaporative power fully equal to that of the best Welsh steam-coals.” Heating Surfaces. Marine ( Sea-water ). — Grate and heating surfaces should be increased about .07 over that for fresh water. Relative Value of Heating Surfaces. Horizontal surface above the flame = 1 I Horizontal beneath the flame = . 1 Vertical = -5 1 Tubes and flues. = .56 Minimum Volumes of* Fuel Consumed per Sq. Foot of* Grrate per Hour, for given Surface-ratios. (D. K. Clark.) Surface-ratios of Heating Surface to Grate. Stationary Marine Portable Locomotive (coal) (coke) 20 30 Lbs. 12. 1 11. 2 3-2 5-2 7 50 Lbs. 18.9 i7-5 5 Lbs. 26 24 n.7 16 18.3 25 26.3 36 32.5 44 At extreme consumption of fuel (120 lbs.) coke will withstand disturbing effect of a blast better than coal. STEAM-ENGINE. — BOILER. 741 A scale of sediment one sixteenth of an inch thick will effect a loss of 14.7 per cent, of fuel. One sq. toot' of fire surface is held to be as effective as three of heating. Relation of Grrate, Heating Surface, and Fuel. When Grate and Heating Surface are constant, greater the weight of fuel consumed per hour, greater the volume of water evaporated ; but the volume is in a decreased proportion to fuel consumed. In treating of relations of grate, surface, and fuel, D. K. Clark, in his valuable treatise, submits, that in 1852 he investigated the question of evaporative perform- ance of locomotive-boilers, using coke; and he deduced from them, that assuming a constant efficiency of fuel, or proportion of water evaporated to fuel, evaporative effect, or volume of water which a boiler evaporates per hour, decreases directly as grate-area is increased; that is to say, larger the grate, less the evaporation of water at same rate of efficiency of fuel, even with same heating surface. 2d. That evaporative effect increases directly as square of heating surface with same area of grate and efficiency of fuel. 3d. Necessary heating surface increases directly as square root of effect— viz for four times effect, with same efficiency, twice heating surface only is required. ’ 4th. Necessary heating surface increases directly as square root of grate, with same efficiency; that is, for instance, if grate is enlarged to four times its first’area twice heating surface would be required, and would be. sufficient, to evaporate same vol- ume of water per hour with same efficiency of fuel. Result of 40 experiments with a stationary boiler (fresh water), with an evaporation of 9 lbs. water, per lb. of fuel consumed, the coefficient .002 22 was deduced. ( ji \ 2 -) .00222 ==W. W representing volume of water in cube feet, and g and h areas of grate and heating surfaces in sq. feet. Illustration. — Assume a heating surface of 90 feet, and a grate of v what will be the evaporation 1 5 ’ To Compute Areas ofGrate and Heating Surfaces Volume of Water, and Weight of EPnel. For a Temperature of 281°, or Pressure of 50 lbs. per Sq. Inch. Eo Compute Ratio of Heating Surface to -Area of Grate and. to inflect a Given Evaporation. Then 90-^3 x .002 22 = 1.998 cube feet. When Water per Sq. Foot of Grate per Hour and Surface Ratio are Given. W — X R 2 To Compute Weight of Fuel. — x R 2 ^ == F, and x R 2 = (E — C) F. F, and x R 2 — (E — C) F. Illustration. — Assume elements as preceding. .02 .02 STEAM-ENGINE. — BOILER. 742 When Efficiency of Fuel and Fuel consumed per Sq. Foot of Grate per Hour are given . = E or efficiency of fuel or weight of water evaporated per lb. /(E — C)F Of fad. f— y— = R - To Compute Fuel tliat may fee consumed per Sq. Foot of Grate per Hour, corresponding to a driven Effi- ciency. When Efficiency of Fuel, that is, Weight of Water evaporated per Lb. of Fuel, and the Surface Ratio , are given. , R!± CF c + ^K ! = Eiand £*l = r . F ’ ' ' F Illustration. — Assume elements as preceding. E— C" .02 x 5 ° 2 ~f~ IQ X _^5 _ I3 33; ,0 + .02 X 5° - = i3-33, and .02 X 5° 2 = 15 lbs. is *^3 ; — Combustion of Coal per sq. foot grate. -Natural Draught , from 20 to 25 lbs. can be consumed per hour. — Steam-jet , 30 lbs., and Exhaust-blast 65 to 80 lbs. From Results of Experiments upon Marine Boilers, see Manual of D. K. Clark, page 808; he deduced the following formula, as applicable to all surface ratios in such boilers. Newcastle .021 56 R 2 -}~9-7 1 hnd for Wigan .01 R 2 -j- 10.75 F = W in lbs. And the general formulas he deduced from all the various experiments are as follows. , . From and at 212 0 . Portable 008 R 2 -}“8.6 F = W. Stationary... 0222 R 2 -j- 9.56 F == W. Locomotive, coke Marine 016 R 2 -f- 10.25 F = W. Locomotive, coal, .009 R 2 -f 9-7 F = W. .. 0178 R 2 -}- 7.94 F = W. As the maximum evaporative power of fuel is a fixed quantity, the preceding formulas are not fully applicable in minimum rates of its consumption and e\apo- rative quality. With coal and coke the limits of evaporative efficiency may be taken respectively at 12.5 and 12 lbs. water from and at 212 0 . Illustration l— Assume a marine fire-tubular boiler with a surface ratio of heat- ing surface to grate of 30 and a consumption of coal of 15 lbs. per sq. foot ot grate per hour, what will be its evaporation per sq. foot of grate? .016 X 3P 2 -f* 10.25 X 15 = 168.15 lbs. 2.— Assume a like boiler, using fresh water, to have a ratio of heating surface to grate of 30 and an evaporation of 165 lbs. water per sq. foot of grate per hour, what would be consumption of coal per sq. foot of grate per hour? 165— .016 xjp; m jo. 25 Tube Surface (Iron) per lb. of coal 1.58, per sq. foot of grate 32, and per IIP 4- 27 sq. feet. Locomotive Boiler has from 60 to 90 sq. feet per foot of grate, and consumes 65 lbs. coal per sq. foot per hour. Evaporative Capacity- of Tubes of Varying Length. Total Length of Tubes 12 Feet 3 ins. (M. Paul Hevrer, 1874.) Furnace and TUI 1 E S. Surface and Water. 3 ins. in Length of Tubes. 3.02 Feet. 3.02 Feet. 3.02 Feet. 3.02 Feet. Surface in sq. feet 76.43 ■ 179 179 179 179 Water evaporated per sq. ) foot per hour in lbs j 24- 5 8.72 4.42 2.52 1.68 STEAM-ENGINE. BOILER. 743 Results of Operation of Boilers under Varying Propor- tions of Grate, Area, and. Length, of Heating Surface, Draught of Furnace, and Rate of Combustion. Fire-tubular. Agricultural and Hoisting tl U U Locomotive English Marine 1 Stationary 4 . Area of Grate. Heating Surface. Grate to Heating Surface. Coal per Sq. Foot of Grate per Hour. Evapor Water fi per sq. ft. of grate. ation of rom 212 0 per lb. of Coal. Fuel. Sq. Feet. Sq. Feet. Ratio. Lbs. Lbs. Lbs. 4-7 158 34 13 1 19 9-33 Welsh. , 3-2 220 69 12.8 151 11.83 ( 26. 25 9 ^ 3-5 36-7 30.86 327 10.6 (16 8l8 51 38 375 10.47 10.5 788 75 45 419 11.04 10.6 IO56 100 i 57 1401 10.41 22 748 34 24- 3 265 10.7 18 749 41.6 23.6 264 11. 2 10.3 9 J 5 50 41.25 468 11.36 10.3 508 49-3 27.63 309.8 n -54 Lanc’r 10.8 151.2 14 27.76 205 7-39 Anth’e 3 . 1-5 945 30 28.87 293-7 10.17 Welsh. 3 i -5 767 24.4 14 141.4 10. 1 “ 2 and 4 Wigan. 3 Experimented at New York. i New Castle. * Effect of reducing the tube-surfaces was tried by stopping one half the number of tubes in alter- nate diagonal rows, so that the tube surface was reduced 206.5 sq. feet. The results with fires 12 ins. deep were as follows : Tubes open. Tubes half closed. Coal per sq. foot of grate per hour 25 lbs. 24 lbs. Water from 212 0 per lb. of coal 12.41 “ 12.23 “ Smoke per hour, very light 2.8 minutes. 8 minutes. Evaporative Effects of Boilers for Different Rates of Combustion, and Surface Ratios. {D. K. Clark.) Water from arid at 212 0 per Hoar. Surface Ratio 30. Fuel per Sq. Foot of Grate per Hour. Statio Wa per Sq. foot. NARY. ter per lb. of Coal . Mar Wa per Sq. foot. INE. ter per lb. of Coal . Port. Wa per Sq. foot. 1 .BLE. ter per lb. of Coal . C01 Wa per Sq. foot. Locos/ al. ter per lb. of Coal . IOTIVE. Co Wt per Sq. foot. ke. iter per lb. of Coal. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. 10 116 11. 6 117 n -7 93 9-3 io 5 10.5 95 9.5 15 163 10.9 168 11. 2 136 9 154 10.3 135 9 20 211 10.6 219 10.9 W 9 9 o 202 IO. 1 175 8.7 30 307 10.2 322 10.7 265 8.8 299 10 254 8-5 Surface Ratio SO. 15 187 12.5 187-5 12.5 149 9.9 168 rr.2 164 20 247 12.3 248 12.5 192 9.6 217 10.9 203 30 342 11.4 348 11. 6 278 9-3 3 i 4 10.4 283 40 438 10.9 450 *i -3 364 9.1 411 10.3 362 50 534 10.7 552 11 450 9 508 10. 1 442 10.9 10.2 9.4 9 8.8 Surface Ratio 75. Locomotive, coal. . “ coke . Water. Fuel per Sq. Foot 30 40 50 Per sq. foot. “ lb. coal. Lbs. Lbs. Lbs. 342 11.4 439 11 536 10.7 “ sq. foot. V lb. coal. 338 XI -3 -k p 00 A 497 9.9 Lbs. 633 10.7 576 9.6 Lbs. 77 8 10.4 695 9-3 90 Lbs. 927 l °. 3 8i5 9 Lbs. 1020 10.2 894 L „ y • ’ . 01 ieea-water is raised above that ob reduced condensin S engine, the proportions of surfaces may be correspondingly 744 STEAM-ENGINE. BOILEE. Results of Operation, of various Designs of Boiler, nu- clei’ varying Proportions of Grate, Calorimeter, Area and. Length of Beating Surface, Draught, Firing, and Bate of Combustion. Stationary. Area of Grate. Heat- ing Surface. Grate to Heating Surface. Circuit of Heating Surface. Temperature of Chimney. Coal per S(j. Foot of Grate per Hour. Water E from 212 0 per lb. of Coal. lyaporated per Sq. Foot of Grate per Hour. Sq. Ft. Sq. Ft. Ratio. Feet. 0 Lbs. Lbs. Lbs. Lancashire double 1 internal and ex- > 20.5 612 29.8 79 51 1 15-35 8.32 125 4 ternal Sued 1 . . . ) U U 2 21 767 36.5 80 505 21.5 10.88 204 4 Galloway vertical ) water-tubular 2 . ( 21 719 22.8 79 505 22.7 10.77 212 4 U .1 2 3 i -5 719 34-3 80 630 18.3 IQ. 17 162 4 Fairbairn 1 20.5. 1017 49-5 • \ 387 15.27 8.67 133 French 1 20.1 607 3°-3 : ■ — > 16.42 8.12 133 „ Cylindrical flued3. . . 14.2 377-5 26.8 56 292 7-43 9.08 59 5 Marine. At Pressure of Atmosphere. Horizontal fire-tub. 2 10.3 5°8 49-3 — _ 27-5 II.92 328 ? U U 2 10.3 508 49-3 — — 41.25 II.36 469 8 U U 2 10.3 3 ° 2 30 — — 24 12.23 268 9 a u i 9-3 749 39 — 21 IO 182 10 u u 28.5 749 26.3 — — 21.15 8.94 164 10 u u 28.5 749 26.3 — — 19 II. 13 335 11 , (C. Wye Williams) j i 5 - 5 , 749 48.3 — 600 37-4 IO.63 398 “ U } 22 749 34 - 600 - 17.27 11.7 202 12 :: } 42 749 17.6 — — 16 9- 6 5 154 13 1 n u 3 10.8 150 13-9 8.5 j 10.99 8-95 88 14 ' “ “ 3 4-32 i 47 34 8.5 — 27.58 7.24 40 J 4 1 Trial in France. 2 At Wigan, 1866-68, height of chimneys 100 feet. 3 Navy- yard, Washington, U. S., chimney 61 feet. 4 At pressure of atmosphere, fires 12 ins. ; deep, at 40 lbs. pressure^ evaporation was reduced 12 per cent. 5 Bituminous coals. , 6 Anthracite, at pressure of 6.5 lbs. above atmosphere. 7 Fires 14 ins. deep, air ad- mitted through furnace - doors. 8 Ditto do., jet blast. 9- Half tubes closed up. 10 Air through grate only. 11 Air through grate and door, no smoke. 12 One open- ing in door, temp. 625°, with two 633°, with four 638°, and with six 6oo°. *3 Long grates, air spaces fully open, no smoke. I 4 0ne furnace, anthracite coal, 5 ins. deep. Dranglit. Draught of Furnace. —Volume of gas varies directly as its absolute tem- perature, and draught is best when absolute temperature of gas in chimney is to that of external air as 25 to 12. r ~f~ 461-2 — _ — v". V, V', and V" representing absolute temperatures at T 32° + 461.2° V' ’ ’ 1 J or temperature given , and at 32 °, in degrees and volume of furnace gas at tempera- ture T in cube feet. Illustration. — Assume temperature of furnace or T = 1500 0 , and 12 lbs. air per lb. of fuel. -522 — ~h 4 ^ r ; 2 _— 3 ,q8 and as 150 cube feet is volume of gas per lb. of fuel at 12 32° + 461.2° lbs. supply of air , 150 X 3-98 = 597 cube feet. — c. W representing weight of fuel consumed in furnace per second in lbs., v volume of air at 32 0 supplied per lb. of fuel in cube feet, t absolute temperature of gas disch arged by chimney in degrees , a area of chimney in sq.feet, and C velocity of current in chimney in feet per second. STEAM-ENGINE. — BOILER. '45 .USTRATION.— Assume W = .16, V = 150, t .16 X 150 X 1000 24000 2466 1000 °, and a: = 9-73 feet. 5 X 493-2° .084 to .087 = D. D representing weight of a cube foot of gas discharged by ley, in lbs. Illustration. 493- 2 X -086 3= .0424 lb. (1 -(- G -}- — ^ = H. G representing a coefficient of resistance and friction of rough grate and fuel * f coefficient of friction of gas through flues and over surfaces, 1 1 length of flues and chimney , m hydraulic mean depth, % and H height oj chimney , all in feet. Illustration i. — Assume C = 9.73, l = 60, and m = .72, all in feet. 64-: g.73 2 / . .012 X 6o\ 94-67 ■ _ . C a V' + ) = X 14 = 20.6 feeL — = W. 64.33 V -72 / 64.33 vt '.—Assume preceding elements. ?-73 X 5 X 493- 2 _ _ # l6 ^ 150 X xooo When. H is given. 3 6 Duration of smoke in an hour, ) very light minutes 1.1 - Comparative K fleet of Draught and Blasts. By late experiments in England, with boilers of two steamers, to deter- mine relative effects of the different methods of combustion, the results were: Natural draught 1, Jet 1.25, and Blast 1.6. Flow of -Adr. (HawJcsley.) In Cylindrical Pipes. 3g 6ff = V, = 3 „ /i£ 6 =Q , If?L\ aud 135 21200 000 In Conduits of Various Sections. 796 /— — v, - v —h V Gl 633 000 a ’ , /a 3 h Yah Q h 3 V 3 C l = ^ ~^r = ^6’ and 6^000 000 = IP - In which x inch water is taken as equivalent to a pressure of 5.2 lbs. per sq. inch for any passage. V representing velocity in feet per second , h head of water in ins. , d diameter of pipe, t length , and C perimeter, all in feet, a area of section in sq.feet, Q (Va) volume of air discharged per second in cube feet, and BP horse-power. Safety Yalves. Up to a pressure of 100 lbs. per sq. inch, area in sq. ins. equal product of tveight of water evaporated in lbs. per hour by .006. Act of Congress (V. S .). — For boilers having flat or stayed surfaces, 30 ins. for every 500 sq. feet of effective heating surface; for cylindrical boilers, or cylindrical flued, 24 sq. ins. Board of Trade, Eng . — Two of .5 inch area per sq. foot of grate. Or, /— = V 452 diameter. G representing area of grate in sq. ins. Locked Safety-valves . — Effective heating surface, less than 700 sq. feet, valve 2 ins. in diameter; less than 1500, 3 ins. in diameter; less than 2000, 4 ins. in diameter; less than 2500, 5 ins. in diameter; and above 2500, 6 ins. in diameter. Or, (.05 G -j- .005 S) ~ = area of each of two valves. G representing sq. inch, per sq.foot of grate, and S sq. inch, per sq.foot of heating surface. STEAM-ENGINE. FLUES AND TUBES. 747 Illustration. — Assume G = 50 sq. feet, S = 1600 sq. feet, and P = 80 lbs. (m. g.) ’hen, (.05 X 50 -f-- 005 X 1600) X Vioo-r-8o = 2.5 + 8 X 1.118 = 11.73 sq. ins. IPipes. Area. .25 G -f- .01 S G representing area of grate and S area of heat- surface, both in sq. feet, and P pressure per mercurial gauge in lbs. Copper), Thickness. Steam, . 125 -f ; Feed, . 125 -f ; Blow ( Bottom Surface ), .125 — : Supply , .1 4 -— ; Discharge, .1 -f — ; Feed, Suction, 9OOO 300 2CO Bilge discharge, .09 and Steam Blow-off, .05-] . d representing 200 500 internal diam. of pipe, and p internal pressure per sq. inch in lbs. Flanges. — Of brass, thickness 4 times that of pipe; breadth, 2.25 times diam. of bolt ; bolls , diam. equal to and pitch 5 times thickness of flange. For lower pressure or stress, pitch of bolts 6 times. Fines and Tubes. Flues and Tubes.— Cross section, for 15 lbs. of coal consumed per hour, an area of from .18 to .2 area of grate, area being measurably inverse to diameter, and directly increased with length. Thus, in Horizontal Tubular Boilers, area .18 to .2 area per sq. foot of grate, and in Vertical Tubular .22 teo .25, area decreasing with their length, but not in proportion to reduction of temperature of the heated air, area at their termination being from .7 to .8 that of calorimeter or area immediately at bridge-wall. Large flues absorb more heat than small, as both volume and intensity of heat is greater with equal surfaces. Tubes. — Surface 1 sq. foot, if brass, and 1.33, if iron, for each lb. of coal consumed per hour ; or 20 of brass and 27 of iron for eacli sq. foot of grate, and 2.6 sq. feet of brass and 3.7 of iron per JH?. Set in vertical rows, and spaces between them increased in width with number of the rows. Temperature of base of Chimney or Smoke-pipe, or termination of the flues or tubes, is estimated at 500° ; and base of chimney, or its calorimeter , with natural draught, should have an area of 1.33 sq. ins. for every lb. of coal consumed per hour. With tubes of small diameter, compared to their length, this proportion may be reduced to 1 and 1.2 ins. When combustion in a furnace is very complete, the flues and tubes may be shorter than when it is incomplete. Evaporation. 1 sq. foot of grate surface, at a combustion of 15 lbs. coal per hour, will evaporate 2.3 cube feet of salt water per hour. A sq. foot of heating surface, at a like combustion of fuel, will evaporate from 5 to 6.2 lbs. of salt water per hour ; and at a combustion of 40 lbs. coal per hour (as upon Western rivers of U. S.), from 10 to 11 lbs. fresh water, exclusive of that lost by being blown out from boilers. 13.8 to 17.2 sq. feet of surface will evaporate 1 cube foot of salt water per hour, at a combustion of 15 lbs. coal per hour per sq. foot of grate. Relative evaporating powers of Iron, Brass, and Copper are as 1, 1.32, and 1.56. Note. — Boilers of Steamer Arctic, of N. Y., vertical tubular, having a surface of 33.5 to 1 of grate, consuming 13 lbs. of coal per sq. foot of grate per hour, evapo- rated 8.56 lbs. of salt water per lb. of coal, including that lost by blowing out of saturated water. 748 STEAM-ENGINE. SMOKE-PIPES AND CHIMNEYS. Water Surface . At low evaporations, 3 sq. feet are required for each sq. foot of grate sur- face, and at high evaporation 4 to 5 sq. feet. Steam liooin. From 15 to 18 times volume that there are cube feet of steam expended for each single stroke of piston for 25 revolutions per minute, increasing directly with their number. Or, .8 cube feet per IIP for a side-wheel engine, and .65 for an ordinary and .55 for a fast-running screw-propeller. Space is required proportionate to volume of steam per stroke of piston. Thus, with like boilers, the space may be inversely as the pressures. Steam-drums and steam-chimneys, by their height, add to the effect of their volume, by furnishing space for water that is drawn up mechanically by the current of steam, to gravitate before reaching the steam-pipe. Grate. — Area in sq. feet per lb. of coal per hour for following boilers. Width, 1.5 diameter of furnace: Cornish and Lancashire, slow I Portable, moderate forced . . .03 sq. foot. combustion 2 sq. foot. Locomotive and like, strong Marine, tubular 05 to .066 “ “ | blast 01 “ “ Thickness of Tubes per B W G. External diameter in ins 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 Thickness for pressure of 50 lbs., number. .12 12 n 11 mo 10 10 9 “ “ “ 100 u u ..11 10 99 98 88 7 Smoke-pipes and. Chimneys. Area at their base should exceed that of extremity of flue or flues, to which they are connected. In Marine service smoke-pipe should be from .16 to .2 area of grate. In Locomotive, it should be .1 to .083. - r Intensity of their draught is as square root of their height. Hence, rela- tive volumes of their draught is determined by formula : fh .ia — volume in sq. feet, h representing height of pipe or chimney in feet, and a its area in sq. feet. When wood is consumed their area should be 1.6 times that of coal. Chimneys {Masonry ). — Diameter at their base should not be less than from . 1 to .11 of their height. Batter or inclination of their external surface .35 inch to a foot, which is about equal to 1 brick (.5 brick each side) in 25 feet. Diameter of base should be determined by internal diameter at top, and necessary batter due to height. Thickness of walls should be determined by internal diameter at top ; thus, for a diameter of 4 feet and less, thickness may be 1 brick, but for a diameter in excess of that 1.5 bricks. A rea. 15 C fh — a. C representing weight of coal consumed per hour in lbs., and a area of ditto at top , in sq. ins. {Brick masom'y .) — 25 tons weight per sq. foot of brickwork in height is safe if laid in hydraulic mortar. Less the height of a smoke-pipe or chimney, the higher the temperature of its gases is required. ; t ; STEAM-ENGINE.— PUMPS.— PLATES AND BOLTS. 749 Velocities of Current of Heated Air in a Chimney 100 Feet in, Height. In Feet per Second. Air at Base of Chimney. , A ._ Air at Base of Chimney. 32 o 5 o° 250° | 350 ° Feet. 24 Feet. 30 28 27 Feet. 33 3i 3° 450 Feet. 35 34 33 6o° 7 °o 8o° Feet. 250 Feet. 26 25 24 35o Feet. 29 29 450 Feet. 33 32 32 hen Height of Chimney is less than ioo feet . — Multiply velocity as ob- d for temperature by .1 square root of height of chimney in feet. Draught consequent upon a steam -jet in a smoke-pipe or chimney is nearly equal to that of a moderate blast. The most effective draught is when absolute temperature of heated air or gas is to that of external air as 25 to 12, or nearly equal to temperature of melting lead. In chimneys of gas retorts, ovens, and like furnaces, the draught is more intense for a like height of chimney than in ordinary furnaces, in con- sequence of the great mass of brick masonry, which, becoming heated, adds to intensity of draught. Chimneys. Lawrence Manufacturing Co., Mass. Octagonal. Height above ground 21 1 feet. Diameters 15, and 10 feet 1.5 ins. Wall at base 23.5, and at top 11.5 ins. Shell at base 15 ins., at top 3.75 ins. Foundation 22 feet deep. England. —Square.— Height 190 feet. Diameter at base. kt “ 3°° “ “ “ Round. “ 312 “ “ “ “ “ 300 “ “ “ Diameter at base usually . 1 of height above ground. 20 feet. 29 “ 30 “ 20 “ Vacuum at base of chimney ranges from .375 to .43 ins. of water. Circulating Pumps. Single-acting. — .6 volume of single-acting air-pump and .32 of double- acting. Double-acting. — .53 volume of double-acting air-pump. Volume of Pump compared to Steam Cylinder or Cylinders. Engine. Pump. Volume. Expansive, 1.5 to 5 times Single-acting 08 to .045. Compound do. 04510.035. Expansive, 1.5 to 5 times Double r acting 045 to .025. Compound do. 025 to .02. Voices .— Area such as to restrict the mean velocity of the flow to 450 feet per minute. PLATES AND BOLTS. Wrought-iron. — Tensile strength ranges from 45500 to 70000 lbs. per sq. inch for plates, and 60000 to 65000 lbs. for bolts, being increased when subjected to a moderate temperature. English plates range from 45000 to 56000 lbs., and bolts from 55000 to 59000 lbs. D. K. Clark gives best quality of Yorkshire 56000 lbs., of Staffordshire 44 800 lbs. Test of IPlates. ( XJ . S.) — All plates to be stamped at diagonal corners at about four ins. from edge, and also in or near to their centre, with name of manu- facturer, his location, and tensile stress they will bear. Plates subjected to a tensile stress under 45000 lbs. per sq. inch, should contract in area of section 12 per cent., 45000 and under 50000, 15, and 50000 and over, 25, at point of rupture. 3 R* 750 STEAM-ENGINE. PLATES. Brands. (C No. i) Charcoal No. i. — Plates , will sustain a stress of 40000 lbs. per sq. inch ; hard and unsuited for flanging or bending. (C No. 1 R H) Reheated , hard and durable, suited for furnaces, unsuited for con- tinued bending. (C H No. 1 S) Shell, will sustain a stress of 50000 to 54 000 lbs. in direction of fibre, and 34000 to 44000, across it: hard and unsuited for flanging or even bending with a short radius. (C H No. 1 F) Flange, will sustain a stress of 50000 to 54000 lbs., soft and suited tor flanging. (C H No. 1 F B) Furnace and (C H No. 1 FFB) Flange Furnace. The first is hard, but capable of being flanged, the other is hard, and suited for flanging. The especial brands are Sligo, Eureka , Pine , etc. The best English plates known are the Yorkshire , as Low Moor , Bowling , Farnley, Monk Bridge, Cooper db Co., etc. (See Steam-boilers, W. H. Shock, U. S. N., 1880.) Steel.— Tensile strength ranges from 75000 to 96000 lbs. Mr. Kirkaldy gives 85 966 lbs. as a mean. When used in construction of boiler-plates should be mild in quality, containing but about .25 to .33 per cent, of carbon; for when it contains a greater proportion, although of greater tensile strength, it is unsuited for boilers, from its hardness and consequent shortness in its resistance to bending. Crucible steel may be used, but that obtained by the Bessemer or Siemens-Martin process is best adapted for boiler-plates. Its strength becomes impaired by the processes of punching and shearing, rendering it proper thereafter to submit it to annealing. Steel rivets, when of a very mild character and uniformly heated to a bright red, are superior to iron in their resistance to concussion and stress. Copper. — Tensile strength is 33000 lbs., being reduced when subjected to a temperature exceeding 120°. At 212 0 being 32 000, and at 550° 25 000 lbs. "W roiiglit-iron Shell IPlates. Pressure and. Thickness. (U. S. Law.) Based upon a Standard of One Sixth of Tensile Strength of Plates. Iron or Steel. Results with a Tensile Strength of 50000 Lbs. Thick- Diameters in Ins. ness. 36 38 40 42 44 46 48 54 60 66 72 | 1 78 Inch. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. •25 116 no 104 99 95 9 1 87 77 69 63 58 53 •3125 145 137 130 124 118 IX 3 109 96 8 7 79 72 67 •375 174 165 156 149 142 136 130 116 104 95 87 80 •5 232 220 208 198 190 182 174 154 138 126 116 106 1 84 90 96 102 108 | 114 120 126 132 i35 140 144 Inch. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. •375 74 69 65 61 58 55 52 49 47 46 44 43 •4375 86 80 76 71 68 64 61 57 55 53 5 i 50 •5 99 92 87 81 77 ' 73 69 65 *>3 61 39 57 •5625 hi 103 98 91 87 82 78 73 . 7* 69 67 64 •75 148 138 130 121 115 109 103 97 94 9 l 88 85 •875 172 160 152 142 136 128 122 no 106 102 100 1 198 184 174 162 i 54 146 138 130 126 122 118 114 To which 20 per c6nt. is to be added for double riveting and drilled holes. Iron plates .375 inch in thickness will bear, with stay bolts at 4, 5, and 6 ins. apart from centres, respectively 170, 150, and 120 lbs. per sq. inch. Iron plates, as tested by Mr. Phillips at Plymouth Dockyard, .4375 inch in thick- ness, with screw stay bolts 1.375 ins. in diameter riveted over heads, 15.75 and 15.25 ins. from centre = 240 sq. ins. of surface for each bolt; bulged between bolts and drew from bolts at a pressure of 105 lbs. per sq. inch of plate. Iron plates .5 inch in thickness, under like conditions with preceding case, bulged and drew from bolts at a pressure of 140 lbs. per sq. inch of plate. Hence, it ap- pears. resistances of plates are as squares of their thickness. When nuts were applied to ends of bolt through .4375 inch plate, its resistance in- creased to 165 lbs. per sq. inch of plate. STEAM-ENGINE. — SHELLS. — PLATES. 75i Cylindrical Shelly. (U. S. Law.) To Compute Pressure for a Given Thickness and. Diameter, or Thickness for a Given Pressure and Diameter. For Pressure . Rule.— M ultiply thickness of plate in ins. by one sixth of tensile strength of metal, and divide product by radius or half diameter of shell in ins. . . When rivet-holes are drilled, and longitudinal courses are double riveted, add one fifth to result as above attained. Example. — Assume boiler 8 feet in diam., and plates .5 inch thick; what work- ing pressure will it sustain, tensile strength of plates equal to a stress of 60000 lbs.? 8 X 12 5000 , ,, . c y 60 000 -r- one sixth -4 — = — Q - = 104. 16 lbs. 3 2 48 For Thickness. Rule.— M ultiply pressure by radius of shell, and divide product by one sixth of tensile strength of metal. Example.— A ssume pressure, radius, and tensile strength as preceding. 104. r6 X 96 -r- 2 = jooo^ = 5 inch . 60 000 -j- one sixth 10 000 For Evaporation of Salt Water.— Add one sixth to thickness of plates and sec- tional area of stay bolts. IPor Freight and. Fiver Steamboats. Standard. — 150 lbs. pressure for a boiler 42 ins. in diameter and plates .25 inch thick. For Pressure. Rule— M ultiply thickness of plate by 12 600, and divide result by radius of boiler in ins. Example.— A ssume a boiler 42 ins. in diameter, and plates .25 inch in thickness; what working pressure will it sustain ? _ .25 X 12 600-7- 42 -T- 2 = 150 lbs. Proof — All boilers by U. S. Law to be tested to a hydrostatic pressure of 50 per cent, above that of their working pressure. Relative Mean Strength of Riveted Joints compared to that of Dlates. Allowances being made for Imperfections of Rivets, etc. Plates , 100; Triple, .72 to .75; Double or Square, .68 to .72; Double with double abut straps, .7 to .75 ; Staggered, .65 ; Single, .56 to .6. Board of Trade , England. Coefficient or Factor of Safety. — When shells are of best material and workmanship, rivet-holes drilled when plates are in place, abut strapped, plates at least .625 inch in thickness and double riveted, with rivets com- puted at a resistance not to exceed 75 per cent, over the single shear,* the coefficient is taken at 5. Boilers must be tested by hydrostatic pressure to twice that of working pressure. Tensile strengths of plates are taken, with fibre 47000 lbs. per sq. inch, across it 40 000 lbs., and when in superheaters from 30 000 to 22 400 lbs. 47 000 ^ ^ f __ p an( j D P C — _ ^ p re p rege nting pressure that shell will sus- D G 47 000 B 2 tain per sq. inch in lbs . , B least per cent, of strength of rivet or plate (■ whichever is least) at lap , D diam. of shell and t thickness of plate, both in ins., and C coefficient of safety. Shearing or detrusive resistance of wrought iron is from 70 to 80 per cent, of its tensile strength. 752 STEAM-ENGINE. SHELLS. PLATES. Illustration.— A ssume T = 50000 lbs. tensile strength of plate, B — 75 per cen D — 120 ins., C = 5, and < = .5. What pressure will shell sustain, and what shoe be thickness of plates for such pressure and diameter? 50000 X .75 X .5 X 2 __ . 120X62.5 xs' • — = 62. 5 lbs. , and — — . e inch. 120 X 5 50000 X .75 X 2 5 For all practicable deficiencies in drilling, punching, and riveting in tran verse, courses, if existing, this coefficient is increased up to 6.75, and in lo gitudinal courses to 8.75, and when courses are not properly broken, ; addition is made to above of .4. Diameter of rivets should not be less than thickness of plates. Molesworth. P d_ cttG -Yd — Pj ancl — t. d representing diameter and t thickness metal, both in ins., P working pressure in lbs. per sq. inch, and C as follows : Single riveted. Double riveted. Best Yorkshire plates ) rC — 6200 and 2800 u Staffordshire plates i one ninth of tensile (« u 7 Ordinary plates. stren S th - j « = 3 5 ^ a Working stress not to exceed .2 tensile strength of joint or riveted plate. Then for a pressure of no lbs., and a diameter of 42 ins., as given for a standard U. S. boiler. Taking C as above for best single-riveted plate at 6200, — ° X 42 = . 272 4 - ins ..... ... 2 X 6200 m thickness, or . 122 inch m excess of U. S. Law for a plain cylindrical boiler, single r iveted. Lloyd's. Thickness of shells to be computed from strength of longitudinal joints. * Tri ^ < J C ^ — d n a ~~ p — — ~ — == x i an( t — z. t representing thick- ness of plate, D diameter of shell, p pitch and d diameter of rivets , all in ins. ; J per cent, of strength of joint or rivets, the least to be taken ; C a constant as per table ; Y working pressure in lbs. per sq. inch / n number and a area of rivet / x per cent, of strength of plate at joint compared with solid plate, and z per cent, of strength of tivets compared with solid plate. When plates are drilled, take .9 of z, and when rivets are in double shear, put 1.75 a for a. Constants. * J C_ p PD D “ ’ C J ~ Joint. Ik .5 inch and under. on Plate . 75 inch and under. s. Above .75 inch. .375 inch and under. Steel Plates. .3625 I '.‘75 inch inch and and under. | under. Above •75inch. t ( punched holes 155 170 165 1 80 170 200 21^ I 230 240 ljap j drilled do I9O I90 200 Double abut j punched holes strap (drilled do. I70 180 180 190 215 230 250 260 When plates, as in steam-chimneys, superheaters, etc., are exposed to direct ac- tion of the flame, these constants are to be reduced .33. Illustrations. — Assume pitch 4 ins., diam. of rivet 1.375 ins., and thickness of plate 1 inch, both single and double riveted. Area 1.375 — 1.48 sq. ins. - — * ° 75 = .656 per cent, strength of joint compared to solid plate. — *^’ 4 - — .37 per cent, strength of rivet to solid plate when single riveted, and — — . 647 4X1 per cent, when double riveted. Rivets at Joint. ^ X 100 with punched holes and by 90 with drilled. p t STEAM-ENGINE. PLATES. — ABUT STRAPS, ETC. 753 IPlates. To Compute Thickness of Flates for a Griven Pressure and. Fitch, and. Pressure and Fitch, for Griven Thick- ness. sixteenths of an inch , p pitch of stays or distance apart at centres in ins., P working pressure in lbs. per sq. inch, and C a constant, as follows : For a Tensile Strength of Metal of 50000 Lbs. per Sq. Inch. Screw Stay-bolts with Riveted Heads. — Plates up to .4375 inch in thickness C = 90, and above that 100. Screw Stay-bolts with Nuts. — Plates up to .4375 inch in thickness C = iio, and above that 120. Screw Stay-bolts with Double Nuts and Washers . — Up to 4.375 ins. in thickness C = 140, and above that 160. When stay-bolts are not exposed to corrosion, these constants may be reduced .2. Resistance of a flat surface decreases in a higher ratio than space between stays. Hence, C must be decreased in proportion to increase of pitch above that of ordinary boiler-plates. Illustration i.— Assume pressure no lbs. per sq. inch, and pitch of stays 5 ins. ; what should be thickness of plate for screw-bolts and riveted heads? 2. — Assume thickness of metal 5 sixteenths inch thick, stay-bolts screwed and riveted over its threads, and working pressure of steam 80 lbs. per sq. inch. Double Abuts should be at least .625 thickness of plate covered. Single, .125 thicker than plate covered, and Double , .625. Direct. — Tensile stress should not exceed 5000 lbs. per sq. inch for Iron, and 7000 for Steel. Diagonal or Oblique. — Ascertain area of direct stay required to sustain the surface ; multiply it by length of diagonal stay, and divide product by length of a line drawn at a right angle to surface stayed, to end of diagonal stay, and quotient will give area of stay increased to that which is required. Stress upon an oblique stay is also equal to stress which a perpendicular stay supporting a like surface would sustain, divided by cosine of angle which it forms with perpendicular to surface to be supported. Illustration. — Assume pressure no lbs. per sq. inch, area of supported surface 36 sq. ins., and angle of stay 45 0 ; what would be pressure or stress upon stay? t representing thickness of metal in f no X 5 2 95 =f -LA- — 5. 38— sixteenth. 95 >c 2 X Q5 — -A- = 5.45 ins. pitch. Abut Straps. Stays. Cosine 45 0 = .7O7 n. Then no X 36 - 4 -. 707 11 = 5600 lbs. 754 STEAM-ENGINE. — GIRDERS. — FLUES, ETC. .66 neck of rod. Stay-Dolts. — Iron, are not to be subjected to a greater stress than 6000 lbs. per sq. inch of section ; Steel, 8000 lbs., both areas computed from weakest part of rod, and when of steel they are not to be welded. To Compute Diameter and. IPitcli of Stay - Dolts, and Resistance they -will Sustain. Screwed, d 95 WP d 70 70 = d, — p, and Vp r-.p, and VP m Socket. — — a 95 P. d representing diameter in ins. Illustration. — A ssume pitch of stay bolts 6 ins., and working pressure 100 lbs. per sq. inch; what should be diameters of bolts, both screw and socket? 6 X V 100 o - 7 , o ^ , 6 X V 100 = . 857 inch Screwed , and 70 95 : 6 2, inch Socket. C d 2 t = P, Grirders. (Lloyd's.) P (L —p) DL_^ /P (L— p) D L = d. L representing (L — p) D L ’ C d 2 *’ V Ot length of girder, d its depth, t its thickness at centre or sum of its thicknesses , D its distance apart from centre to centre , and p pitch of stays, all in ins., and C a constant as per following : One stay to each girder, C = 6000. If two or three — 9000. If four = 10 200. Illustration. — Assume triple stayed girder, 24 ins. in length, 3 ins. in depth, 1 inch thick, and stayed at intervals of 6 ins. ; what working pressure will it sustain ? ^ ooooX6 2 X 1 324000 C = 9000. Then . w , = — — = >25 U>s. (24 — 6) X 6 X 24 2592 ITlm.es, -A^rclied. or Circmlai? Furnaces. U. S. Law. .3125 inch for each 16 ins. of diameter. English iron, being harder than American, is better constructed to resist compression, and consequently a less thickness of metal is required for like stress. 89 6< >o t 2 _ /PLD_ D “ ’ V 89 600 “ Lloyd's. 89 600 t 2 : D, and ^ — L. D representing L D “ ’ V 89 600 “ ’ PL external diameter of flue or furnace , and t thickness of plate, both in ins.. L length of flue or furnace between its ends or between its rings, in feet , and P working press- ure in lbs. per sq. inch. Illustration.— A ssume diameter of flue 16 ins., length 6 feet, and working press- use of steam 80 lbs. per sq. inch. Then 80 X 6 X 16 89 600 = V '°%57 — * 2 9 Furnace . — P not to exceed 8000 t STEAM-ENGINE. — RIVETING. 755 IllustratMn.-— Assume diameter ot ST ^ one 48 ins., working pressure of steam 80 lbs., ana iengm o ZO~ V A V ifi , ,, • . 7 ... Then / 80 X 6 X 48 __ / ' ^ __ __ 507 inch thickness \ 89600 RIVETING. -D, . The strength of a joint is determined by ascertaining which of *e Uvo"’the plate or fhe rivets, has the least resistance; the stress on the first being tensile and the latter detrusrve. *1 * in he taken from that of the article under consider- The for construction and location of the joint, and A^c^^ttr^rontress, as with or across the fibre of the metal, or exposed to high heat as in a superheater. IVith or Across the Fibre .— From experiments of Mr. D. Kirkaldy and othersfthe difference in strength of Iron plates is ascertained to be from 6.5 to 18 per cent., the average 10 per cent. a, ; ~oinioa The relative strength of plates with or across the fibre, as detenni^dby MnKiAaldy^for ‘Fagersta^’ is 9 per cent., and for “Siemens” it is without material difference. puSedlo™^ to be 15 per cent. In Riveted Joint exposed to a tensile stress, area of rivets should be to area of section of plates through line of rivets, running a little in i excess up to .5625 inch diameter of rivet, and somewhat less beyond that > d ing determined by relative shearing and tensile resistances of met and plate. Note.— For Riveting of Hulls of Vessels, see pp. 828-30. Essentially by Nelson Foley. Single Lap Riveting. S— A = b for plate, for rivets, ptb' = a, and P P L *-27 *>' f _ d p representing pitch , t thickness of plate, and d diameter oj rivets, being punched. Illustration.— A ssume j) = 3 in*-, d= i inch, a = . 7 8 54 «, and t=. S inch. 3 1 .66 per cent, strength of lap, =, - 5^3 P* '««**• °f rimt to ^ id ,P Me ’ i— .66' Z 3 ins.,and X .5 = 7 on Die Lap Riveting. Preceding formulas for single lap riveting apply to this, wit'll substitution of 2 a for a and .64 for 1.27. Illustration. — Assume p = 3 ins., t = .5 inch , and 6' = .589. 3X.5X.589 „ .64X.58Q 3 — <7 e = .4418 area ofd, 4 jtf x.sa. B d, 2— l” = £5 d •4418X2 . ,, ' 3 -=.5896'. 3X-5 Diameter of Rivets as Determined by IPlate. B Or Strength at Joint. Diam. = Thickness of Plate X B Or Strength at Joint. Diam. = Thickness of Plate X B Or Strength at Joint. Diam. = Thickness of Plate X Per Cent. T = S. .9 per cent, of Section Per Cent. T = S. .9 per cent, of Section Per Cent. T = S. .9 per cent, of Section 68 of Rivet. of Rivet. of Rivet. x -35 x -5 7 X 1.56 x -73 74 1. 81 2 69 1.42 x -57 72 1.64 1.82 75 1.91 2.12 70 1.48 1.65 73 1.72 1.91 76 •2 . 2.25 1 = .5 men ana h = 7 o per cent., tensile stren, to shearing being as 7 to 6. What should be diameter of the rivets? .5 X 1.48 X -|* = .863 inch. When rivets are in double shear, put 1.9 a for a. STEAM-ENGINE. — DUTY. — EVAPORATION. 757 Triple Lap Riveting. Preceding formulas for single lap riveting apply to this, with substitution of 3 a for a and .42 fof 1.27. lLLUSTRATiox.-Assumep = 3 ^^-5^,and& -.883. 4417 area ofd, X -5 = -74 . diam. , J ' “ and i44^X 3 = . 8 8 3 &'• 3 X -5 2 = -75 6, Diameter of Rivets as Determined by IPlate. B Or Strength at Joint. Per Cent. 70 7 1 7 2 Diam. = Thickness of Plate X •99 1.04 1.09 .9 per cent of Section of Rivet. 1. 1 i-i5 Or Strength at Joint. Per Cent. 73 74 75 Diam. = Thickness of Plate X Or Strength at Joint. t : 15 1. 21 1.27 9 per cent, of Section of Rivet. 1.27 1-34 1. 41 Per Cent. 76 77 78 Diam. = Thickness of Plate X T = S. 1-34 1.42 i-5 .9 per cent, of Section of Rivet. 1.67 Operation.— As shown by preceding tables. Greneral Formtilas and Illnstrations. 1.27 BT^ and y. i (1 — iB) S. ’ p tT i. 75 (1 — B) S P* T Tftrefc tw Single Shear. Rivets in Double Shear. Rivets in Triple Shear . Zigzag Riveting .27 B T 2-5 (1 , a 2.5 S _ , t — d, and — 77=- = 0 . B) S ’ ptT 7 - „„„ Riveting. Strength of plate between holes diagonally is equafto that horizontally between holes, when diagonal pitch _ .6 and or- izontal = diameter of rivet + .4. Thus, .6 p -}- .4 p — diagonal pitch. Duty of Steam-engines?. The conventional duty of an engine is the number of lbs. raised by it 1 foot in height by a bushel of bituminous coal (112 lbs.). Cornish Engine.— Axe rage duty, 70000000 lbs. ; the highest duty ranging from 47 000 000 to 101 900 000 lbs. ■ A condensing marine engine, working with steam at .75 lbs. (mercurial gauge), cut offat -5 stroke, will require from 1.75 to 2 lbs. bituminous coal per TP per hour. Relative Cost of Steam-engines for Eqnal Effects. In Lbs. of Coal per IP per Hour. Lbs. A theoretically perfect engine A Cornish condensing engine 2 A marine condensing engine J Evaporative Rower of Boilers. The Evaporative power of a boiler, in lbs. of water per lb. of fuel consumed, is ascertained approximately by formula x 8 33 ( S \ e = lbs. S representing total heating surface in sq. feet , F fuel consumed bilf .per hour, and e theoretical evaporative power of the fuel. Illustration. — Assume evaporative power of the fuel at 15, consumption per hour 800 lbs., and heating surface 1600. Then {u^TfsSo) x 15 = IO ' 448 lbs ' 3S STE AM-EN GINE. — WEIGHTS, Efficiency of boiler. 1.833 1600 : - 7 33 - \1600 X 2 -f 800/ . n T ^ e o e iT. ap0 .? Uve power of differ ent fuels, from and at 2I2 ° is for coals fr, to 16.8 lbs., the average of Newcastle being is., fornatent f , cohe I3 . 3 , Peat IO , 3 , and Woods, when INTotes on Horse-power horizontal zection r-ir cube r«, t of water, and , s sj. reel of graLara’a^r ? u b ^^Sing'siSS^nnd T^foot of grate-area! k° ri , SUf H h !! 0llerS WlH range from 3 4 times that of the nominal. Multitubular £oilers .-. 75 sq. foot of grate-area and 2.5 of heating surface. ^ iglits of 1 Steam-engines. Side-wheels.- American Marine Engine. Vertical beam. Oscillating. Inclined * Without frame. Frame. Water- wheels. ( No. Cylinders. Volume. Weight per Cube Foot. Wood.* Wood. * Wood.* Wood.* Wood.* Iron. Iron. Iron. Wood. Wood. Wood. Wood. Iron. Iron. Iron. Iron. 1 2 1 2 1 2 2 2 Cube Feet. 63 216 430 253 725 540 1502 535 Lbs. 1100 1040! 1225 1480$ io89f 850 55 °§ 1100 t With frame 1109. % Including boilers. Service. Screw Propellers . — American Marine ( Condensing ). River. Coast. Coast. Coast. Sea. Sea. Sea. Sea. § Single frame. Engine. Cylinders. Vertical direct, Jet Condens’g . “ “ Surface Cond’g “ Jet “ Horizontal back-action . “ direct... Vertical compound. No. “ direct . :i!l o l O “ Non-Condensing. Volume. CubeFeet. 4 12.5 12.5 33 506 68 67 4.8 24-3 425 3-6 35 1.86 2.77 | Weights. Engine. Lbs. 22040 59006 48 130 120450 1 523 o 6 q 289 680 201 000 24 7<>5 94196 1 022 400 30 534 172028 14 410 *4 759 , Boilers. Description. English Marine ( Condensing ). Weights. Trunk Horizontal direct. Vertical direct . Oscillating Vertical compound Horizontal compound. Engines. I Propeller and Shafting. Boilers and Water. Total. Tons. Tons. Tons. Tons. 121 47 257 425 223 85 303 611 165 48 144 357 11 7 43 135 295 4-25 , *75 7-25 12.25 497 167 656 1320 55 i 5 1 10 180 130 27 162 3 i 9 STEAM-ENGINE. — WEIGHT OF BOILERS. 759 Land-engines. — (Non- condensing . ) Engine. Volume of Cyl’r. Engine. Spur-wheel and Connections. Sugar-Mill Complete. Boilers, Grates, etc. Engine per Cube Foot of Cylinder. Lbs. Lbs. Lbs. Lbs. Lbs. Vertical! 1 18 ins. X4 feet 7 67 200 37800 89 600 26 880 9600 beam f 30 ins. X5 feet 24-5 105000 137 179 265 879 75060 4290 Horizon' 1 , 14 ins. X 2 feet .2.2 10 914 — — 8 200 5100 22 ins. X 4 feet 10.6 56 000 — 30140 5600 To Oompxite Weiglit of a, Vertical 13 earn and Side-wlieel Jet Condensing Engine. (T. F. Rowland, A.S.C.E.) Including all Metals, Boiler and A ffackmenfs , Smoke-pipe, Grates , Iron Floors , and Iron in Wooden Water-wheels, omitting Coal-biinJcers. For a Pressure per Mercurial Gauge of 40 tbs. per Sq. Inch. For surface condenser add 10 to 15 per cent. Rule. — Multiply volume of cylinder in cube feet by Coefficient in follow- ing table corresponding to length of stroke, and product will give rough weight in lbs. For finished weight deduct 6 per cent. Stroke. Coefficient. Stroke. Coefficient. Stroke. Coefficient. Stroke. Coefficient. Feet. 5 2467 Feet. 7 2213 Feet. 9 1865 Feet. 11 1619 6 2340 8 • 2000 10 W 30 12 1546 Example 1.— What are the rough and finished weights of a vertical beam engine cylinder 80 ins. in diameter and 12 feet stroke of piston? Area of 80 ins. = 5026. 56, which x 12 feet = 419 cube feet , and X 1546 for 12 feet stroke = 647 774 lbs. rough weight. Then 647 774 X -06 = 38 866, and 647 774 — 38 866 = 608 908 lbs. finished weight. WEIGHTS OF BOILERS. Weights of Iron Boilers (including Doors and Plates , and exclusive of Smoke- pipes and Grates) per Sq. Foot of Heating Surface. Surface Measured from Grates to Base of Smoke-pipe or Top of Steam Chimney. Boiler. For a Working Pressure of 40 Lbs . Weight. Single return, Flue* water bottom u it a “ “ ‘ | Multi-flue * ! water bottom Horizontal return, Tubular f water bottom “ “ “ t u “ “ * Vertical “ “ f Horizontal direct, Tubular *. .water bottom Lbs. 25.6 to 32.9 24 tO 30 27 to 45 25 tO 43 22.5 to 35 21 ' to 33 17.7 to 26.7 18.5 to 26.5 19.8 to 23.8 17 to 21 23.5 to 24 18. 1 to 18.6 16.3 to 17.3 24 to 26 Weight of Cylindrical Furnace .and Shell Boilers, all complete for Sea Service and for a pressure of 60 lbs. steam, 200 lbs. per IIP. * w^!?l? f : f ' ,rn L Ce i qUare -- S wu- 7 1 L n,irical - . , t Section of furnace and shell square. ♦ v\ rought-iron heads, .375 inch thick, flues, .25 inch, and surface computed to half diameter of shell. Cylindrical, external furnace, t 36 ins. in diam., .25’inch thick.. “ Flue “ 4361042 “ .25 “ “ Horizontal direct, Tubular Locomotive. Vertical Cylinder direct, Tubular Notes, i. The range in the units of weight arises from peculiarities of construc- tion, consequent upon proportionate number of furnaces, thicknesses of metal vol- ume of shell compared with heating surface, character of staying, etc. 2. If pressure is increased the above units must be proportionately increased. 76O STEAM-ENGINE— BOILER-BOW EE, COMBUSTION. Boiler-power. The power of a boiler is the volume or weight of steam alone (indepen- dent of l any water that it may hold in suspension) that it will generate at its operating pressure in a unit of time. \ r *» rinp boilers of the ordinary type and proportions, with natural draught, burn- cA produce 3-5 *5.5 Iff per s^oot of grate pertour^ w.th a iSAScS' hour. 8 to 10 ik. cr^pa^ expansion, under a pressure of 70 lbs., 20 lbs. steam. . With a blast draught and consuming 30 to 40 lbs. of a fair quality of coal pel sq. foot of grate per hour, 7 to 10 IP per hour can be attained, in locomotive boilers having from 50 to 90 sq. feet of heating surface per sq. foot and pressure. To Compute Volume of Air and Gas in a Furnace. When Volume at a Given Temperature is known. Rule— M ultiply given volume by its absolute temperature, and divide product by the given abso- lute temperature. Note. Absolute temperature is obtained by adding 461° to given or acquired temperature. Fvamptf —Assume volume of air entering a furnace at 1 cube foot, its tempera- ture 6o°, and temperature of furnace 1623°; what would be the increase of volume . 1 X 1623° -f- 461° 2084 ~ 521 = 4 times. 6o° 461° Volume of Furnace Gas per ITb. of Coal. ( Ranlane .] Tempera- ture. 12 Lbs. \ir Supplied. 18 Lbs. 24 Lbs. 32 0 150 225 300 68 161 241 322 104 172 258 344 212 205 3°7 409 572 3 H 47 i 628 Tempera- ture. 12 Lbs. k\r Supplied. 18 Lbs. 24 Lbs. 752 ° 369 553 738 1112 479 718 957 1472 588 882 1176 1832 697 1046 1395 2500 906 1 J 357 1812 Temperature of ordinary boiler furnaces ranges from 1500 0 to 2500° The opening of a furnace door to clean the fire involves a loss of from 4 to 7 per cent, of fuel. For other illustrations, see ante , page 744-6. Rate of Combustion. The rate of combustion in a furnace is computed by the lbs. of fuel consumed per sq. foot of grate per hour. ■ . Tn .rpnpnl practice the rate for a natural draught is, for anthracite coal from 7 to 16 lbs., for bituminous, from 10 to 25 lbs., and with artificial orfonced raugi ^ y a blower, exhaust-blast, or steam-jet, the rate may be increased from 30 to 120 lbs. The dimensions or size of coal must he reduced and the depth of the fire increased directly, as the intensity of the draught is increased. Temperature of gases at base of chimney or pipe resistance of surface of chimney is as square of velocity of current ot & ases. Ordinarily from 20 to 32 per cent, of total heat of combustion is 1 “S’be added ‘the production of the chimney draught m a mmebote, ^ f ^‘' m e \ «nd the di'utioS losses bv incomplete combustion of the gaseous portion of the tuel and tnc anuiiou of the gLes by an excess of air, making a total of fully 60 per cent. ( Steam-boilers, Wm. H. Shock, XI. S. N., i83i.) STRENGTH OF MATERIALS. — ELASTICITY. j 6 l STRENGTH OF MATERIALS. Strength of a material is measured by its resistance to alteration of form, when subjected to stress and to rupture, which is designated as Crushing, Detrusive, Tensile, Torsion, and Transverse, although trans- verse is a combination of tensile and crushing, and detrusive is a form of torsion at short lengths of application. ELASTICITY AND STRENGTH. Strength of a material is resistance which a body opposes to a per- manent separation of its parts, and is measured by its resistance to alteration of form, or to stress. Cohesion is force with which component parts of a rigid body adhere to each other. Elasticity is resistance which a body opposes to a change of form. Elasticity and Strength , according to manner in which a force is exerted upon a body, are distinguished as Crushing Strength , or Resistance to Com- pression ; Detrusive Strength , or Resistance to Shearing ; Tensile Strength, or Absolute Resistance ; Torsional Strength , or Resistance to Torsion ; and Transverse Strength , of Resistance to Flexure. Limit of Stiffness is flexure, and limit of Resistance is fracture. Neutral Axis , or Line of Equilibrium, is the line at which extension ter- minates and compression begins. Resilience , or toughness of bodies, is strength and flexibility combined ; hence, any material or body which bears greatest load, and bends most at time of fracture, is toughest. Stiffest bar or beam that can be cut out of a cylinder is that of which depth is to breadth as square root of 3 to i ; strongest , as square root of 2 to 1 ; and most resilient , that which has breadth and depth equal. Stress expresses condition of a material when it is loaded, or extended in excess of its elastic limit. General law regarding deflection is, that it increases, cceteris paribus, di- rectly as cube of length of beam, bar, etc., and inversely as breadth and cube of depth. Resistance of Flexure of a body at its cross-section is very nearly .9 of its tensile resistance. Coefficient of Elasticity. Elasticity of any material subjected to a tensile or compressive force, within its limits, is measured by a fraction of the length, per unit of force per unit of sectional area, termed a constant, and coefficient of elasticity is usually defined as the weight which would stretch a perfectly elastic bar of uniform section to double its length. Unit of force and area is usually taken at one lb. per sq. inch. E represent - ing denominator of fraction. Example.— I f a bar of iron is extended one 12000000th part of its length per lb. of stress per sq. inch of section, x r 12000000 E * The bar would, therefore, be stretched to double its normal length by a force of 12000000 lbs. per sq. inch, if the material were perfectly elastic. s* 762 STRENGTH OF MATERIALS. ELASTICITY. The same method of expressing coefficient of elasticity is applied to re- sistance to compression. That is, coefficient, in weight, is expressed by de- nominator of fraction of its length by which a bar is compressed per unit of weight per sq. inch of section. Ultimate extension of cast iron is 500th part of its length. Extension of Cast-iron Bars,when suspended V ertically. 1 Inch Square and 10 Feet in Length. Weight applied at one End. Weight. | Extension. Set. Weight. Extension. Set. Weight. Extension. | Set. Lba. Ins. Ins. Lbs. Ins. Ins. Lbs. Ins. Ins. 529 .0044 — 21x7 .0190 .000059 8468 .0871 .00855 1058 .0092 .000015 4234 • 0397 . .O0265 14820 . 1829 •02555 Woods. — MM. Chevaudier and Wertheim deduced that there was no limit of elasticity in woods, there being a permanent set for every extension. They, however, adopted a set of .00005 of length as limit of elasticity. This is empirical. MODULUS OF ELASTICITY. Modulus or Coefficient of Elasticity of any material is measure of its elastic reaction or force, and is height of a column of the material, pressing on its base, which is to the weight causing a certain degree of compression as length of material is to the diminution of its length. It is computed by this analogy : As extension or diminution of length of any given material is to its length in inches, so is the force that pro- duced that extension or diminution to the modulus of its elasticity. P Or x : P :: l : w = — . x representing length a substance 1 inch square and ifoot ’ ’ * * " x in length would be extended or diminished by force P, and w weight of modulus in lbs. To Compute "WeigHt of Modulus of Elasticity. Rule. — A s extension or compression of length of any material 1 inch square, is to its length, so is the weight that produced that extension or com- pression, to modulus of elasticity in lbs. Example.— I f a bar of cast iron, 1 inch square and 10 feet in length, is extended .008 inch, with a weight of 1000 lbs., what is the weight of its modulus of elasticity? .008 : 120.(10 X 12) :: 1000 : 15000000 lbs. To Compute Moduli of Elasticity. When a Bar or Beam is Supported at Both Ends and Loaded in Centre. Rule.— M ultiply weight or stress per sq. inch in lbs. by length of material in ins., and divide product by modulus of weight. I W = M; E M ~T~ = W. I representing length in ins., M modulus , W weight in lbs. per sq. inch , and E compression or extension , Example i. — If a wrought- Won rod, 60 feet in length and .2 inch in diameter, is subjected to a stress of 150 lbs., what will it be extended? Modulus of elasticity of iron wire is 28 230 500 lbs. (see following table), and area of it . 2 2 X .7854 = . 314 16. .31416 = 477-46 lbs. per sq. inch , and 60 X 12 = 720 ins. Then 477-46 X 720 . 34J 77 1 : 2 „ OJ2 jg inch. “ 28 230 500 28 230 500 2.— Take elements of preceding case under rule for weight of modulus. 120 X 'OOQ _ oc g frlc>l .008 X 15000000 = IOOO lbs 15 000000 120 STRENGTH OF MATERIALS. — COHESION. 763 Modulus of Elasticity- and "Weight of Various Materials. Substances. Height. Weight. Substances. Height. Feet. Lbs. Ash 4 970 000 1 656 670 Beech 4 600000 I 345 DOO Brass, yellow 2 460000 8 464 DOO “ wire 4 1 12 000 14632 720 Copper, cast 4 800000 18 240 DOO Elm 5 680 000 1499 500 Fir, red 8 330 000 2 Ol6 DOO Glass 4 440 000 5 550 DOO Gun-metal 2 790 000 8 844 300 Hempen fibres. . . . 5 000000 1 70 DOO Ice 6 000000 2 37O DOO Iron, cast 5750000 17 968 500 11 wrought 7 550000 25 820 DOO “ wire 8377000 28 230 500 Feet. Larch Lead, cast Lignum-vitse Limestone Mahogany Marble, white Oak Pine, pitch “ white., Steel, cast “ wire Stone, Portland . . . Tin, cast Zinc 4 415 000 146000 1 850000 2 400 000 6 570 000 2 150000 4 750 000 8 700 000 8 970 000 8 530000 9 000 000 1 672 coo 1 053 000 4 480 000 Weight. Lbs. 1 074 000 720000 1 080 400 3 300 000 2 071 060 2 508 000 1 710000 2 430 000 1 830 000 26 650000 28 689 000 1 718 800 3 510 000 13 440000 Weight a Material will hear per Sq. Inch, without Permanent Alteration of its Length. Material. | Lbs. Material. Lbs. Material. ] Lbs. Metals. Brass 6 00 Stones , etc. Marble Woods. Beech 2360 3240 4290 2060 Gun -metal IO OOO Limestone* 4QOO 2000 Elm Tron cast 15 000 17 800 I ^OO Portland 1500 Fir, red “ wrought... Lead Larch Woods. Mahogany 3000 3960 Steel 45 000 Ash 1 3540 Oak * Tensile strength 2800.. Comparative Resilience of Woods. Ash Chestnut | Larch .. .84 I | Spruce Beech 86 Elm Oak . . .63 Teak Cedar 66 Fir .. .4 | | Pitch Pine. . . •• -57 | Yellow Pine. MODULUS OF COHESION. To Compute Length of a Prism of a Miaterial which would he Severed hy its own "Weight when Suspended. Rule. — D ivide tensile resistance of material per sq. inch hy weight of a foot of it in length, and quotient will give length in feet. Illustration. — Assume tensile resistance of a wrought-iron rod to be 60000 lbs. per sq. inch. Weight of 1 foot = 3.4 lbs. Then 60000 - 4 - 3.4 = 17 647.06 feet. Length in Feet required to Tear Asunder the following Substances : Rawhide 15 375 feet. | Hemp twine. .. 75000 feet. | Catgut 25 000 feet. Elasticity of Ivory as compared with Glass is as .95 to i. When Height is given. Rule. — M ultiply weight of 1 foot in length and 1 inch square of material by height of its modulus in feet, and product will give weight. To Compute Height of Modulus of Elasticity. Rule. — D ivide weight of modulus of elasticity of material by weight of 1 foot of it, and quotient will give height in feet. " Example. — Take elements of preceding case (page 762I, weight of 1 foot being 3 lbs. ; what is height of its modulus of elasticity ? 15 000 000 -r- 3 = 5 000 000 feet. 7 6 4 STRENGTH OF MATERIALS. CRUSHING. From a series of elaborate experiments by Mr. E. Hodgkinson, for the Railway Structure Commission of England, he deduced following formulas for extension and compression of Cast Iron : Extension: 13934040 — 290743200 c c 2 Compression : 12 931 560 — — 522 979 200 — = W. e and c representing extension and compression , and l length in ins. Illustration. — What weight will extend a bar of cast iron, 4 ins. square and 10 feet in length, to extent of .2 inch? 2 2 ^ 13 934 040 X 290 743 200 - 1 — ^ == 23 223. 4 — 8076. 2 = 15 147. 2, which x 4 ins - 120 120 = 60 588. 8 lbs. CRUSHING STRENGTH. Crushing Strength of any body is in proportion to area of its section, and inversely as its height. In tapered columns, it is determined by the least diameter. When height of a column is not 5 times its side or diameter, crushing strength is at its maximum. Cast Iron. — Experiments upon bars give a mean crushing strength of 100000 lbs. per sq. inch of section, and 5000 lbs. per sq. inch as just sufficient to overcome elasticity of metal ; and when height exceeds 3 times diameter, the iron yields by flexure. When it is 10 times, it is reduced as 1 to 1.75 ; when it is 15 times, as 1 to 2 ; when it is 20 times, as 1 to 3 ; when it is 30 times, as 1 to 4 ; and when it is 40 times, as 1 to 6. Experiments of Mr. Hodgkinson have determined that an increase of strength of about one eighth of destructive weight is obtained by enlarging diameter of a column in its middle. In columns of same thickness, strength is inversely proportional to the 1 * fi 3 power of length nearly. A hollow column, having a greater diameter at one end^ than the other, has not any additional strength over that of an uniform cylinder. Wrought Iron— Experiments give a mean crushing stress of 47 000 lbs. per sq. inch, and it will yield to any extent with 27 000 lbs. per sq. inch, while cast iron will bear 80000 lbs. to produce same effect. Effects. — A wrought bar will bear a compression of g of its length, with- out its utility being destroyed. With cast iron, a pressure beyond 27000 lbs. per sq. inch is of little, if any, use in practice. j Glass and hard Slones have a crushing strength from 7 to 9 times greater than tensile ; hence an approximate value of their crushing strength may be ; obtained from their tensile, and contrariwise. Various experiments show that the capacity of stones, etc., to resist effects of freezing is a fair exponent of that to resist compression. Seasoning. — Seasoned woods have nearly twice crushing strength of un- seasoned. Elastic Limit compared to Crushing Resistance. Wrought-iron Commerce 545 Cast steel Bessemer steel 615 Fagersta stee i Cast steel 473 STRENGTH OF MATERIALS. CRUSHING. 765 Crvisliing Strength, of various Materials, deduced from Experiments ofMaj. Wade, Hodgkinson, Capt. Nleigs, TJ. S. .A.., Stevens Institute, and. lay Gr. JLi. Yose. Reduced to a Uniform Measure of One Sq. Inch. Cast Iron. Figures and Material. Gun-metal, American. “ mean Low Moor, No. 1, English. “ No. 2, “ Clyde, No. 3, “ Crushing Weight. Lbs. 174803 85 000 125 000 100 000 62 450 92330 106 039 Figures and Material. Clyde, average, English Stirling, mean of all, English . . “ extreme, English Extreme, English { Average (Hodgkinson), English Blaenavon No. 2 Crushing Weight. Lbs. 82 000 122393 134 400 53 760 153 200 84 240 109 700 Wrought Iron. Aluminium bronze, 95 cop. . Fine brass. Cast copper Steel, cast Fagersta . 1 127 720 1 83500 I English || 1 ,47040 1 1 ‘ ‘ average | Various Metals. 129 920 164 800 1 17 000 105 000 250000 154 5 oo Steel, Bessemer “ “ soft “ tempered “ Siemens Tin, cast Lead 65 200 40 000 37 850 50000 66 200 335 ooo 15500 7 730 Elastic Crushing Strength of Wrought Iron and Crucible Steel is equal to its ten* sile, of Bessemer Steel, 50 per cent, of its transverse strength. Woods. Ash Beech Birch Box Cedar, red “ seasoned. ... Chestnut.... ... Elm “ seasoned “ English Hickory, white Larch Locust Mahogany, Spanish. 6 663 6963 3300 7900 10513 5968 6500 5 350 6 831 10000 10300 8925 3 200 5 5 oo 9 ll 3 8198 Maple Oak, American white.. “ Canadian white. . “ “ live . . . “ English “ Dantzic, dry Pine, pitch “ white.... ;.. “ yellow “ Deal, Christiana. Spruce, white Teak Walnut Willow, seasoned O IOO IOOOO 7500 5982 6850 9500 6 484 7 700 8947 5 775 8 200 5850 5 950 12 100 6645 6 000 Oak. Crosswise of Fibre. Larch 1300 Pines . 55 o Increase in Strength of Cubes of Sandstone , per Sq. Inch {under Blocks of Wood), as Area of Surface is increased. (Gen' l Gillmore , U. S. A.) Inches. •5 I 1 Yellow Berea sandstone . . Blue “ “ Lbs. 6080 Lba. 6990 9500 Lbs. 8 226 10 730 Lbs. 8 955 12000 2.25 2.75 3 4 Lbs. Lbs. Lbs. Lbs. 9 r 3 ° 9838 10125 11 720 12 500 13 200 — — 766 STRENGTH OF MATERIALS.— CRUSHING. Stones, Cements, etc. (Per Sq. Inch.) Figures and Material. Basalt, Scotch “ Welsh Beton, N. Y. S. ConcretiDg Co. Brick, pressed “ Gloucester, Mass. . “ hard burned Crushing Weight. “ common | “ yellow-faced burned, Eng. “ Stourbridge fire-clay, “ Staffordshire blue, “ “ stock, English “ Fareham, English 44 red, English “ Sydney, N. S Caen, France Cement, Hydraulic, pure, Eng. | 44 Portland, sand i 44 44 sand 3 44 44 3 mos 44 44 i sand, 3 mos. . . . , 44 44 9 mos 44 44 i sand, 9 mos 44 44 12 inch cubes , ) 12 mos. > i sand and gravel ) « « 3 ^ “ Roman 44 “ pure, Eng 44 Rosendale “ Sheppey, Eng Concrete, lime i, gravel 3 j Freestone, Belleville, N. J “ Connecticut 44 Dorchester, Mass “ Little Falls, N. Y. .. . Glass, crown Gneiss Granite, Aberdeen, Eng 44 Cornish, “ “ Dublin, ’ “ 44 Newry, “ * Tested by author at Stevens’ Institute, N. J. r, • , 1 m n + O Lbs. 8300 16 800 800 1 400 6 222 10 219 14 216* 363° 800 4000 1440 1 650 7 200 2 250 5 boo 808 2 228 1543 17 000 32 000 1 280 600 3 800 2464 5980 2 33 ° 2650 1 800 342 750 3270 1 280 460 775 3 522 33*9 3069 2991 31 000 19600 10 760 6 339 10450 12 850 Figures and Material. Granite, Patapsco, Md. . ‘ Portland, Eng. * Quincy, Mass. . Greenstone, Irish Limestone 44 compact, Eng. .. . 44 Magnesian, “ .... 44 Anglesea “ 44 Irish “ Marble, Baltimore, Md ‘ 4 East Chester, N. Y. t 44 Hastings, N. Y 44 Irish “ Italian 44 44 white,..,. “ Lee, Mass 44 Montgomery Co., Pa.. . . 44 Statuary 44 Stockbridge, Mass.$. ... 41 Symington, large .. . 44 “ fine crystal . . 44 “ strata horizontal Masonry, brick, common. . . . 44 44 in cement. , Mortar, good 44 lime and sand 44 “ beaten... 44 common Oolite, Portland Pottery pipe, Chelsea Sandstone, Aquia Creek . 44 Arbroath, Eng. . . 44 Connecticut 44 Craigleth, Eng.. . “ Derby grit “ ... 44 Holyh’d quartz, Eng. 44 Seneca II.... 44 Yorkshire, Eng. . Slate, Irish Terra Cotta Whinstone, Scotch. + Post-office, Wash. $ City Hall, New York. \ WasMu^n, D. C. I Smitbsoaian Institute. Safe Load of Hollow, Cylindrical, and Solid Columns, Arches, CHords, etc-., of Cast Iron. Hollow Columns. Per Sq. Inch. (F. W. Shields, M. I. C. E.) Length. Thick- ness. Load. Length. Thic H Load, ness. Length. Thick- ness. Load. Length. Thick- ness. Load. 20 to 24 diam’s. Inch. •375 •5 Lbs. 2800 336 o 20 to 24 | diam’s. Inch. Lbs. .625 3920 •75 1 44 So 25 to 30 diam’s. Inch. •375 •5 Lbs. 2240 2800 25 to 30 diam’s. Inch. .625 •75 Lbs. 336 o 3920 Solid Columns, etc.- 336 ° lbs- per sq. inch. (Brunei.) Arclies.— 5600 lbs. per sq. inch. STRENGTH OF MATERIALS. CRUSHING. 767 Chords and Posts.— 1 inch diameter and not more than 15 diameters in Jength .2 of breaking weight of metal. .625 inch diameter and not more than 25 diameters in length .5 of breaking weight of metal, and when more than 25 diameters in length from .1 to .025 of breaking weight of metal. (. Baltimore Bridge Go.) Wrought-iron Cylinders and Rectangular Tubes. External Internal Thickness. Area. Crushing Weight Length. Diameter. Diameter. per Sq. Inch. Cylinders. Ins. Ins. Ins. Sq. Ins. Lbs. 10 feet 1-495 1.292 . 1 •444 14661 10 2.49 2.275 .107 .804 29 779 10 6.366 6. 106 •13 2.547 35886 Rectangular Tubes. 10 feet 4- 1 X 4.1 •03 •504 10980 5 “ a! 4.1 X 4.1 •°3 •504 n 514 10 “ 4- 1 X 4.1 .06 1.02 19 261 10 “ 0 4-25 X 4-25 •134 2-395 21585 7-5 “ 4-25 X 4-25 •134 2-395 23 202 10 “ ’ O. 8.4 X 4-25 J.26 i.126 6.89 29981 10 “ 8.1 X 8.1 .06 2.07 13276 7.66 “ 8.1 X 8.1 .06 2.07 133OO 10 “ ) internal 8.1 X 8.1 .0637 3-55i 19 732 5 “ . j diaphrag’s 8. 1 X 8.1 .0637 3-551 23 208 Strength per Sq. Inch of 2-Inch Cubes under Blochs of Wood. ( GenH Gillmore , U. S. A.) Surfaces Worked to a Clear Bed. Granite. Lbs. Staten Island blue 22250 Maine 15000 Quincy, dark 17750 “ light 14750 Westchester Co., N. Y 18250 Millstone Point, Conn 16 187 New London, Conn 12 500 Richmond, Ya 21 250 “ “gray 14 100 Cape Ann, Mass, Westerly, R. L, gray 14937 Fall RiVer, Mass., gray 1593 7 Garrisons, Hudson River, gray. . 13 370 Duluth, Minn., dark 17750 Keene, N. H., bluish gray 12 875 Used in Central Park, N. Y., red 17 500 Jersey City, N. J., soap 20750 Passaic Co. , “ gray 24 040 Limestone. Glen’s Falls, N. Y 11475 Lake Champlain, N. Y 25000 Canajoharie, N. Y 20700 Kingston, “ 13900 Garrisons, “ 18500 Marblehead, 0 ., white 12600 Joliet, 111 ., white 16900 Lime Island, Mich., drab \ 18000 1 2 1: 000 Sturgeon Bay, Wis., bluish drab 21 500 Limestone. Lbs. Bardstown, Ky., dark 16250 Cooper Co., Mo., dark drab 6650 Erie Co. , N. Y. , blue 12 250 Caen, France 3 650 Marble. East Chester, N. Y 13 504 Italian, common 13062 Dorset, Vt 7612 Mill Creek, 111 ., drab 9687 North Bay, Wis., drab 20025 Sandstone. Little Falls, N. Y., brown 9850 Belleville, N. J., gray. n 700 Middletown, Conn., brown 6950 Haverstraw, N. Y. , red 4 350 Medina, N. Y., pink 17725 Berea, 0 . , drab { 7250 ’ ’ l 10250 Vermillion, 0 ., drab 8850 Fond du Lac, Wis., purple 6250 Marquette, Mich., “ 7450 Seneca, 0 . , red brown 9 687 Cleveland, O. , olive green 6 800 Albion, N. Y., brown 13 500 Kasota, Minn., pink 10700 Fontenac, Minn., light buff. .... 6 250 Craigleth, Edinburgh 12000 Dorchester, N. B., freestone 9 150 Massillon, 0 ., yellow drab 8 750 AY arrensb u rg, Mo. , bl u i sh d rab . 5 000 CRUSHING. STRENGTH OF MATERIALS. To Compute Crushing Weight of Columns. Deduced by Mr. L. D. B. Gordon from Results of Experiments of various Authors. a representing area of metal in sq. ins., r ratio of length to least external diameter or side , and W crushing weight in tons. Illustration.— What is the crushing weight of a hollow cylindrical column of cast iron io ins. in diameter, 24 feet in length, and 1 inch in thickness? Length. Feet. 2 Ins. 3 Ins. 4 Ins. 5 Ins. 6 Ins. 7 Ins. 8 Ins. 9 Ins. 10 Ins. 11 Ins. 12 Ins. 13 Ins. 14 Ins. T r 5 Ins. 5 12.4 44 102 184 288 414 560 728 916 1126 1354 — — — 6 9.4 3 6 88 164 264 386 532 698 884 1082 1320 I 57° • — — 7 7.2 30 76 146 242 360 502 660 850 1056 1282 1530 1798 2086 8 24 66 130 218 332 470 630 812 1016 1240 i486 • 1754 2040 9 — 20 56 114 198 306 440 59 6 774 974 1196 1440 1706 1992 10 18 48 102 180 282 410 560 739 932 1152 1392 1656 1940 12 — — 38 80 136 238 354 494 658 846 1056 1292 1550 1828 14 28 64 122 200 304 432 586 774 966 1192 1440 1712 16 5 2 100 I70 262 378 520 686 878 1094 1332 i59 6 18 — 44 84 144 226 332 462 616 796 1000 1228 1482 20 — — — 72 124 196 292 410 552 720 912 1130 1372 Subtract weight that may be borne by a column, of diameter of internal diameter of tube from external diameter, and remainder will give weight that may be borne. Thickness of metal should not be less than one twelfth diameter of colufnn. Illustration. — Required the safe load of a solid cast-iron column 6 ins. in diam- eter and 20 feet in length. Under 6 and in a line with 20 is 72, which X 1000 = 72 000 lbs. Note. — This is about one sixth of destructive weight. Cast Iron. ( Hodgkinson .) Round Solid or Hollow. -- - a - — W. For rectangular put 500. r 2 Rectangular Solid or Hollow. - — = W. For L, T, U, etc., put — - — Wrought Iron. [Stoney.) Round Solid. — 16 a - =2 W. Rectangular Solid, r 2 2400 3000 Steel. [Baker.) Round Solid. —Strong steel, - 51 a - = W ; m ild steel, — ' - - 2 - =± W. 1400 Rectangular Solid . — Strong steel, — - — mild steel, 1 1600 Area of 10 ins. =78.54. — — 2 = 28, 8, and 28. 8 2 = 829.44. Area of 10 ins — Weight home with Safety by Solid Cast-iron Columns. In 1000 Lbs.— [New Jersey Steel and Iron Co.) Diameter. For Tubes or Hollow Columns. STRENGTH OF MATERIALS CRUSHING. S „ft L»* „ *»•««» b »„ « /«, ... .<* » «» “* •** “ " TFoods, one seventh to one tenth. WOODS. TO compute Destructive Weight of Colour. d4 .6 Cylinder. -75- o = than 30 diameters in length. . w. Rectangle. C = W. £/iort Columns , or less W r a S __ w. d representing diameter and s column of Wee dimensions in lbs. Ash 22000 “ Canadian 17000 Beech I 5 r 5°° Cedar *4 000 Elm Red Pine 17 5 °° Yellow pine 12000 White u 9 000 Spruce 14000 Walnut 12500 Coefficients Elm, rock 26000 Fir, Dantzic 22000 Oak, white 20000 “ Eng 23000 - ~ Pitch Pine 2000 Illustr \tioh. —What Is destructive weight of a column of yellow pine ,o ina souare and “feet in length or height? ^ x 12000 = X 12 000 = 833 333 tbs. RectangulI-oJ^ ^0y nniun~coCm Sti^ronen^h of .column, divided by this least dimension and by wfdth of column, all dimensions m ms. L ’ T i L | T .36 •35 •33 • 3 1 .29 •27 .26 .24 •23 19 .098 .097 093 089 084 08 077 073 •43 •43 • 4 2 • 4 Illustration.— Assume a'whiteiakLoJLn, secured at both ends, 12 by 8 ins., and 20 feet in length which = . Hence, 12 X 8 X .097 = 9 - 3 ™ tons - strerrsth axrd Staffiie. mUex - YeUow pine . . . Cast Iron 1000 1 Cast Steel. . . . 2518 | Oak 108.8 | Pine.. . . . . . 7 -5 .■■^^^l^irm^orVit-e^im^s^s^nd^f f^Alak^d^vid^by^amH br^Mim^^ie^ * All ton., except when otherwi.e designated, arc 2240 lbs. a T STRENGTH OP MATERIALS.— DEFLECTION. DEFLECTION". Deflection of Bars, Beams, GHrders, etc. and wilC lik « l '“ 1 "’ S^^®M fc -'srfssSSsi swwr** **"- - In experiments of Hodgkinson, it was further shown that sets from dp flections were very nearly as squares of deflections. 1 d In a rectanguiar bar beam, etc., position of neutral axis is in its centro V s 110t S f enS1 i y altere f b y variations in amount of strain applied In bars, beams, etc., oi cast and wrought iron, position of neutral axis P varies in rL beam ’t and ° nly fixed whiIe elasticity of beam is perfect. When a bar, beam, etc., is bent so as to injure its elasticity, neutral line changes and continues t ° change during loading of beam, until its elasticity is destroyed susnendlrt oi beamS ’ SS’’ are of same len S th , deflection of one, weight being is as R tn d , f nL° ri K eI1 K’ C °T are Supported at both Ends and Loaded in Middle , strain being same, is as 2 to 1 ; and when length and loads are same, deflection will be as 16 to 1, for strain will be four times fore all °other Xed h* ^ ^ ° n ° ne su PP orte d at both ends ; there- lore a n other things being same, element of deflection will be four times greater ; also, as deflection is as element of deflection into square of length then, as lengths at whi ch weights are borne in their cases are as 1 to 2 ^de- flection is as 1 : 2 2 x 4 = 1 to 16. ^ fleC ( J 101 ? of a bar ’ beam > etc., having section of a triangle, and supported at its ends, is .33 greater when edge of angle is up than when it is down? whfrp^f co . uatera ^ deflection of a beam, etc., under stress of its load, where a horizontal surface is required, it should be cambered on its unner surface, equal to computed deflection. upper STRENGTH OF MATERIALS. — DEFLECTION. Safe Deflection.— One fortieth of an inch for each foot of span, with a factor of safety for load of .33 of destructive weight = xfVch but for ordinary loads and purposes, Cast Iron , Wnr to Wo 0 i and Wrought Iron , WffTF to Woo or WiFT>> after beam, etc., has become set. When Length is uniform , with same weight, deflection is inversely as breadth and square of depth into element of deflection, which is inversely as depth. Hence, other things being equal, deflection will vary inversely as breadth and cube of depth. Illustration.— Deflections of two pine battens, of uniform breadth and depth, and equally loaded, but of lengths of 3 and 6 feet, were as 1 to 7.8. Deflection of different bars, beams, etc., arising from their own weight, having their several dimensions proportional, will be as square of either of their like dimensions. jjj construction of models on a scale Intended to be executed in full di- mensions, this result should be kept in view. When a continuous girder, uniformly loaded, is supported at three points by two equal spans, middle portion is deflected downwards over middle bear- ing and it sustains by suspension the extreme portions, which also have a bearing on outer bearings. Middle portion is, by deflection, convex up- wards and outer portions are concave upwards; and there is a point of “contrary flexure,” where curvature is reversed, being at junction of con- vex and concave curves, at each side of middle bearing. This point is dis- tant from middle bearing, on each side, one fourth of span. Of remaining three fourths of each span, a half is borne by suspension by middle portion, and a half is supported by abutment. Hence, distribution of load on bear- ings is easily computed, as given above. Deflection of each span is to that of an independent beam of same length of span as 2 to 5. In a beam of three equal spans, deflection at middle of either of side spans is to that of an independent beam as 13 to 25. . In a long continuous beam, supported at regular intervals, deflection ot each span is to that of an independent beam of one span as 1 to 5. Cylinder .— If a bar or beam is cylindrical, deflection is 1.7 times that of a square beam, other things being equal. Formulas for Deflection of Beams of Rectangular Sec- tion, etc. r Z 3 w &d 3 CD Fixed \ Loaded at One End. ^ - = D ; ana = W. at One End. Fixed \ Loaded in Middle, at -< Uniformly. bd 3 C ,7’ ” "■ l 3 3 Z 3 w 8 6 d 3 CD _ “ Uniformly. ^ = D ; and = V. Z 3 w Doth Ends. ' < Ends. I 24 b d 3 C 5 Z3W n . 2 4 H 3 CD _ — D ; and — — — = W. Z 3 = D;and 8X246 7 CD = W. Supported | at -( Both Ends. f Loaded in Middle. Uniformly. 8 X 24 Z> d 3 C ’ 5 I 3 Z 3 W _ , x6H 3 CD ___ = D;and — 75— = W. 5 Z 3 W _ 8 X 16 5 d s C D 2 — = D ; and =■ W. “ at any one Point. 8 X 16 b d 3 C n 2 W 5 I 3 Z 6 d 3 C D Z 6 d 3 C = W. Supported in Middle. 3 Z 3 W _ . 5 X 16 b d 3 C Ends Uniformly loaded. 5 x l6 6 d 3 q ~ D ’ and ^73 — w * l representing length , b breadth , and d depth , all in ins . , W weight or stress in lbs. or tons , m n distances of weight from supports , C a constant , and D deflection , m ms. 77 2 STRENGTH OF MATERIALS. DEFLECTION, Fixed at Both Ends. Deflection of Beams of Rectangular Section. , = D; Uniformly , = Fixed at f r . . . _ — . One End '[ Loaded at 0ne End - { Loaded in Middle. Z 3 w b cZ 3 C Z 3 W 24 6 ^3 (J Z 3 W Supported { Loaded in Middle. jo ^ ni I i6 6d3 C 's.| “ , = D ; Uniformly, == D ; Uniformly, 5W 8 X 24 b cZ 3 C 5 Z 3 w m 2 w 2 W Z 6 tZ 3 (j = D. 8 X 16 b rf 3 c W weight in tons. - = D. = D. i?oZ/i Ends. / a aZ any one Point. C a Constant as follows. Cast Iron. Rectangular Bars.— Loaded at One End 875. “ at the Middle 28000. Round Bars . — Loaded at One End 594. “ at the Middle 19000. Wrought Iron. Cast Iron. Rectangular.— For tons and l in ins. put C = 47 000. 28 000. “ “ l in feet “ C = 27. 16. Round. “ “ Zinins. “ 0 = 32000. 19000. “ “ l in feet “ C= 18. 10.7. Hence, in order to preserve same stiffness in bars, beams, etc., depth must be increased in same proportion as length, breadth remaining constant. Woods. Mean of LasleWs, Barlow, etc. ( D . K. Clark.) /3 W Supported at Both Ends. Loaded in Middle. Ash, Canadian 1476 u Eng 2722 Beech 2418 Blue Gum 2559 Elm 1227 Fir, Dantzic 2490 u Memel 3630 “ Riga 2920 Greenheart 1888 Iron Bark 4378 C Iron-wood 4228 Larch 2100 Mahogany, Honduras 2118 “ Mexican. 3608 “ Spanish. . 3360 Norway spar 2465 Oak, Baltimore 2761 “ Canadian 3445 “ Dantzic 2080 “ Eng 1848 bd 3 C = D. Oak, French 2656 “ white 2114 Pitch pine 2968 Bed “ 2434 Rock elm 2319 Spruce 3300 “ Amer 2669 “ Scotch 1583 Teak 1804 Yellow pine 2084 Application of Table : To Compute Deflection of a Rectangular Beam of Wood. Illustration. — What is the deflection of a floor beam of yellow pine, 3 by 12 ins., 12 feet between its supports, under a uniformly distributed load of 3000 lbs. ? 5 X 12 3 x 3000 15000 . — — . 299 inch. C = 2084. 8X3X128x2084 50016 Hence , To compute weight that may be borne by a given deflection of such a beam, 8 X 3 X 12 3 X 2084 X .299 14 955 t. 1 — - 3 • = =2991 lbs. 5X12 3 5 Deflection, of Continuous Grirders or Beams. Beams of Uniform Dimensions, Supported at Three or More Bearinqs, (D. K. Clark. ) 1. Two Equal Spans or 3 Bearings. Weight on 1st and 3d bearing = .375 W l 4k “ 2d bearing = 1.25 W l 2. Three Equal Spans or 4 Bearings. Weight on 1st and 4th bearing = .4 W l “ a 2d “> 3d = 1.1 W l 3. Four Equal Spans or 5 Bearings. Weight on 1st and 5th bearing = .39 W l j Weight on 2d and 4th bearing = 1.14 W l Weight on 3d bearing = .93 W l. Z 3 W Cylindrical Beam. -rz-r. = D ; d* C Z 3 STRENGTH OF MATERIALS. DEFLECTION. 773 To Compute Maximum Load, that may he Dome hy a Rectangular Beam. Deflection not to exceed Assigned Limit of one hundred and twentieth of an Inch for each Foot of Span. Supported at Both Ends. Loaded in Middle. d3 _ w b and d representing breadth and depth in ins. , l length in feet, C con- l 2 C ’ slant, and W weight or load in lbs. Constants Cast Iron 0003 Wrought Iron .0021 Hickory 018 Teak 024 Oak, red 039 Hemlock 039 Pine, white 039 Chestnut, horse. 051 Oak, white 027 Ash, white 03 Pine, pitch 033 ( “ yellow ... r . .036 Illustration.— What is maximum load that may be borne by a beam of white pine, 3 by 12 ins., 20 feet between its supports, and loaded in its middle? A> 1 Q -0 1 2 Feet. Ins. Ins. Lbs. Ins. Lbs. Ins. Lbs. Ins. c 1. American. |j| 1.83 I I 600 .06 266 .027 148 .015 I 2. English... “ 2-75 2 2 4480 .08 131° .022 1310 .022 1.29 3. “ .... j 2-75 i -5 2-5 8960 . 104 2128 .025 00 .022 1.25 4. * “ .... “ 2-75 i -5 3 8960 .q88 co 0 0 •O37 2259 .022 .88 To Compute Deflection of, and Weight Chat may he home hy, a Rectangular Bar or Beam of Wrought Iron. \VJ 3 W l 3 _ ^ 60000 b d 3 CD _y 60 000 b d ^ D 60 000 b d 3 C l^ OOOOO U U,~ ^ w ~ Illustration. — What weight will a beam 2 ins. in breadth, 5 ins. in depth, and 15 feet between its supports, bear with safe deflection of yiy of an inch for each foot of space, or T of its length ? C from table = .88. D = of 15 = .12 inch. 60000 X 2 X 5 3 X -88 X . 12 _ 1800000 _ 533 30 j bs 15 3 ~ 3375 “ *3 33/3 D. K. Clark gives for Elastic deflection, 47 000 for Rectangular bars, and 32 000 for Cylindrical. Note.— Deflection of to of the length may be allowed under special cir- cumstances; but under ordinary loads the deflection should not exceed one fourth of these, as yg 1 ^ to ^y 1 ^. Practice in U. S. is to allow y^ 1 ^ after girder has taken its permanent set. In small bridges there is a slight increase in deflection from high speeds, about .166 or .144 of the normal deflection, with the same load moving at slow speed. In large girders there is no perceptible difference between the deflection at high and low speeds. 3T* 774 STRENGTH OF MATERIALS. DEFLECTION. Deflection of Wronght-iron Rolled Beams. Supported at Both Ends. Weight applied in Middle. W n - C at Reduced Weight and Deflection. 70000 d 2 (4 a + 1.155 «') D No. Form. Length. Fla Width, nges. Mean Thick- ness. Web. Depth . Weight and by Actual Observation. Deflection at one sixth of Destructive Weight. C Feet. Ins. Inch. Inch. InS. Lbs. Ins. Lbs. Inch. *• X 10 3 *485 •5 7 12000 •4 3800 .127 1.05 2. “ 20 4.6 .8 •5 9-85 16000 i-i5 6300 •453 .92 3- 20 5-7 •643 .6 “•75 20 000 .85 8000 ■34 .98 To Compute Deflection of, and Weigh, t that may De Dome hy, a W r o vigli t— iron Rolled Ream of Uniform and Sym- metrical Section. Supported at Both Ends. W Z3 Weight applied in Middle. — D. (D. K. Clark.) 70000 d 2 (4 a -)- 1 . 155 a') D : W. 70000 d 2 (4 a- f- 1. 155 a') l 3 l representing span in feet, d reputed depth , or depth less thickness of lower flange in ins., a area of section of lower flange, a ' area of section of web for reputed depth of beam, both in sq. ins., and W weight or stress in lbs. Illustration'. — What is deflection of a wrought-iron rolled beam of New Jersey Steel and Iron Co., 10.5 ins. in depth, flanges 5 by .5 ins., and width of web .47 inch, when loaded in its middle with 8000 lbs., and supported over a span of 20 feet? q. ins., and a' = 10 X .47 = 4.7 sq. ins . 64000000 d — 10. 5 - Then — .5 = 10 ms., a = 5 X- 5 = 2.5 8000 X 20 3 107 899 500 '±= . 59 inch. 70000 X io 2 X (4 X 2.5 -f 1. 155 X 4-7) If weight is uniformly distributed, divide by 112 500 instead of 70000. A like beam 6 ins. in depth, loaded with 2608 lbs., and supported over a span of 12 feet, gave by actual test a deflection of .3 inch, and by above formula it is also .3 inch. Note. — Deflection for such a beam, for a statical weight or stress of 17 100 lbs., uniformly distributed , by rules of N. J. Steel and Iron Co., would be .54 inch, which, with difference in weights, will make deflections alike. Deflection of Wrought-iron Riveted. Reams. Supported at Both Ends. Weight applied in Middle. W 1 3 - = C at Reduced Weight and Deflection. 168000 ( - (~+f) d 2 D No. Form. Length. Flanges. Angles. Web. Depth. Weight and by Actual Observation. Deflection at one sixth of Destructive Weight. c Feet. Ins. IllS. Iuch. Ins. Lbs. Inch. Lbs. ! Inch. 2.125X2 I '• T 7 i \ “ X-28 2.125X2 [•25 , 7 4 216 . 1 4062 .096 • 6 3 JL 1 l X.29 J i 1 [ 4 - 5 X 2X2 I 2 T n.66-^ •5 4 - 5 X X.3125 2X2 1 f .25 12.5 77 280 .46 12 880 •075 1.96 t •375 X.3125 J 1 4 - 5 X 2X2 I 3 . “ 22.5 \ •5 7 X X -375 3X3 1 ^•375 16.5 115584 •875 19265 4 * OO 3-86 \ c •5 X -4375 J STRENGTH OF MATERIALS. DEFLECTION. 775 To Compnte Deflection of, and. Weight tLat may "be "borne by, a Riveted. Beam of Wrought Iron. w * 3 - = D. ^ooo /£+£ + £) a. CD ld.0 ■ 2 4/ W. (il -f- (l' . & \ 168 OOO ^ — J C a, a', and a" representing areas of upper and lower flanges with their angle pieces, and of web for its entire depth , all in sq. ins. Note.— I f there are not any flanges, as in No. 1, angle pieces alone are to be computed for flange area. Illustration.— What weight will a riveted and flanged beam of following dimen- sions sustain, at a distance between its supports of 25 feet, and at a safe deflection of . 2 inch or of its length ? Top flange 6 X - 5 ins - I Web 5 Bottom flange 6 X -5 “ I Depth 17 “ Angles 2.25 X 2.25 X .5 ins. a and a' each = 6 X .5 — 3 + 2.25 -f 2.25 — .5 X .5 X 2 = 7 sq. ins. a" — . 5 x 17 = 8. 5 sq. ins. C, as per No. 2 .43, but inasmuch as flanges in this case are much heavier, assume .5. 168000 Then - 17 2 x 2 x .5 = 44 303120 15625 Strength of a Riveted beam compared to a Solid beam is as 1 to 1.5, while for equal weights its deflection is 1.5 to 1. Tubular Girders. Wrought Iron. Supported at Both Ends. Weight applied in Middle. No. Section. Length of Bearing. Breadth. Dej Inter- nal. >th. Ex- ternal. Weight. Deflection. Deflection at .008 inch for each Foot of j Span. 1 1 £ 1 ft •o VO Feet. Ins. Ins. Ins. Lbs. Ins. Inch. ( cT I. □ Thickness .03 inch 3*75 1.9 2.94 3 448 .1 •°3 288 2. “ “ .525 “ 3° i5-5 22.95 24 33 685 •56 .24 473 top .372 “ ) 3- “ bottom .244 “ / 30 16 23.28 24 32 53 8 1. 11 .24 224 sides .125 “ ) 4- “ Thickness .75 “ 45 24 34-25 35-75 128 850 1.85 •36 362 O Thickness .0375“ 17 12 11.925 12 2 755 •65 .136 62.8 7 • 0 “ .0416“ *7 9-25 13-535 13.62 2 262 .62* .136 47-9 “ -i43 “ 17 9- 2 5 14.714 15 l6 800 1.39* .136 119 * Destructive weight. To Compute Deflection of, and Weiglit tLat may L>e "borne by, a "Wronglit-iron Tubular Girder. 16 b d* C D — W. - =D. 1 3 16 b d* C~ Illustration.— What weight may be safely borne by a wrought-iron tube, alike to No. 3 in preceding table, for a length of 40 feet, and a deflection of .32 inch? 16 X 16 X 24 s X 224 X .24 __ 190253629 _ 6 lbs 30 3 27000 Flanged Rivets. Deflection of Iron and Steel Flanged Rails within their elastic limit, compared with their transverse strength, is as 17 to 20, and with double-headed it is as n to 23. STRENGTH OF MATERIALS. DEFLECTION, RAILS. Supported at Both Ends. Weight applied in Middle. Iron . No. Fokm. Length of Bearing. Head. Bottom. Weight per Yard. | Depth. Area. Observed Weight and Deflection. Destructive Weight and Deflection. Feet. Ins. Ins. Lbs. Ins. Sq. Ins. Lbs. Ins. Lbs. In3. 1. 1 2-75 2.25X1 2.25X1 60 4-5 6.166 13 440 •°34 26 680 .065 2. u 4-5 2 3 XI 2.3 Xi 65 4-5 6.68 11 200 .11 24640 .204 3 - u 5 2.3 Xi 2.3 Xi 82 5-4 8.25 25 760 .2 51520 •378 4 - T 2-75 3-5 X .8 2.25X1 60 4 6.7 11 200 •035 26 680 .065 5 - A 2.58 2.23X1 3-5 X -6 57 3-5 5.85 II 200 •°97 20160 .128 Steel. -d ti Depth. Observed Destructive No. Form. tr<~ .5 e 0 ^ n a Bottom. Weig] per Yard Web. Centres of Heads. Total. Area. Weight and Deflection. Weight and Deflection. Feet. Ins. Ins. Lbs. Inch. Ins. Ins. S.IU 3 . Lbs. In. Lbs. Inch. «• I 5 - 78 •75 4.2 5-4 7.67 36 086 •25 80 192 •55 7- “ Bessemer. 3.62 - 86 - - 5-5 8-43 22 400 .14 26 680 .165 8 -X 5 2-5 6 -375X;|7 1 84 .65 3-37 4-5 8. 24 27 290 .24 27 290 .24 To Compute Deflection of Double-headed Hails within. Elastic Limit. (D. K. Clarlc.) t Supported at Both Ends. Weight applied at Middle. IRON. W 1 3 — -- = D. a representing area of one head , less portion per - 57000 (4 ad ' 2 - f- 1. 155 td 3 ) taining to web , d whole depth of rail , d' vertical distance between centres of heads f t thickness of web, all in ins., I length in feet, and W weight in lbs. STEEL. \ For 57 000 put 67 400. Illustration. — Take case No. 3 (Iron), in preceding table, with a weight of 26000 lbs. ; what will be its deflection between bearings 5 feet apart? a = 1.911. ^' = 4.2. d = 5.4. t~. 82. 26000 X 5 3 3 250 000 Then 57000 (4 X 1-911 X 4-2 2 + 1. 155 X .82 X 5-4 3 ) 57 000 X 284 .2 inch. To Compute Deflection of Iron and Steel Hails of* TU n- symmetrical Section within Elastic Limits. Elastic Deflection of Steel Flanged Rails of Metropolitan Railway of London, as determined by Mr. Kirkaldy, at a span of 5 feet, and loaded in middle, was ,02 inch per ton. ( See Manual of D. K. Clark, pp. 667-670.) STRENGTH OF MATERIALS. DEFLECTION. 777 CAST IRON. Deflection of Bectangnlar Bars and Beams of various Sections, etc., by TJ. S. Ordnance Corps, Barlow, Hodgkinson, and CnLitt. Supported at Both Ends. Weight applied in Middle. . 6 ©*gl 2 x. American. 2. English. . . 3. “ ... 4. “ ... 5 - “ ••• Length of Bear- ing. Breadth. Depth. Wei* By Actual Observation. ;ht and Deflect At one sixth of Breaking Weight. don. At 1th of an inch for each foot of span. Feet. Ins. Ins. Lbs. Ins. Lbs. Ins. Lbs. Ins. 1.66 2 2 5000 .036 1666 .012 1805 .013 4 I 1 212 •32 80 . 12 22 •033 16 4 4 1008 •4 5333 2 . II. 337 ° x -33 4-5 3 1 1120 1.42 2x5 .27 30 •037 4-5 1 l 2 : 5 5 2231 •51 422 .1 156 •037 3.81 4. 11 3- 8 9 2-37 2-33 To Compute Deflection of, and "Weiglit tliat may t>e ■borne V>y, a Rectangular Bar or Beam of Cast Iron. 10 400 b d 3 C D W 1 3 - = C. W l 3 - — D. zW. 10400 b cZ 3 D 10400 6 d 3 C l 3 Illustration. — What weight will a beam 2 ins. in breadth, 5 ins. in depth, and 16 feet between its supports, bear with safe deflection of y^ of an inch for each foot of span, or yy 1 ^ of its length ? C from table = 3. 89. D = yU of 16 = 1. 33 ins. 10400 X 2 X 5 s X 3.89 X t.3 S _ 13 451 620 _ ^ 16 3 — " 4096 Clark gives C uniform for Rectangular bars of 2.69, and 1.85 for Cylindrical. FLANGED BEAMS. Cast Iron. Supported at Both Ends. Weight applied in Middle. To Compute Deflection of, and W eiglit tliat. may "be "borne Toy, a Flanged Beam of Cast Iron of "Uniform and Symmetrical Section. Wl 3 :I). 27 000 d 2 (4 a -{- 1. 155 a' 2 ) D = W. 27 000 d 2 (4 a -f- 1. 155 a' 2 ) l 3 Illustration. — What is deflection of a cast-iron beam (Hodgkinson’s) 7.15 ins., flanges 2.6 X -86 ins. and 5X1-6 ins., and width of web 1 inch, when loaded in its middle with n 200 lbs., over a span of 15 feet? ^ = 7.15 — 1-6 = 5.55 ins., <1 = 5 X 1.6 = 8 ins., and a' = 7.15 — 1.6 = 5.55 ins. 11200X 15 3 __ 37800000 Then : 1.48 ins. 27000 5.55^ (4 X 8 + ..155 X 5-55=) 27000 3a8 (32 + 35-57) Note i. — The observed deflection of this beam was 1.28 ins., at one sixth of its de- structive weight it was .3, and at y|-^ of an inch for each foot of span it was .125 inch. 2.— The mean ratio of elastic to destructive stress is 73 per cent. Formulas for value of deflection signify that deflection varies directly as weight, and as cube of length; and inversely as breadth, cube of depth, and coefficient of elasticity. 77 8 STRENGTH OF MATERIALS. DEFLECTION. Elastic Strength of Beams of Unsymmetrical Section.— Elastic strength is approximately Reducible from ultimate strength, according to ordinary ratio of one to the other, ascertained experimentally. Elastic strength and de- flection of a homogeneous beam of any section is same, whether in its nor- mal position or turned upside down. Comparative Strength, a nd Deflection of Cast-iron Flanged. Beams. Description of Beam. Comp. Strength. Description of Beam. Comp. Strength, Beam of equal flanges .58 Beam with flanges as 1 to 4.5. . .78 “ with only bottom flange. .72 “ “ u 1 to 5.5. . .82 “ “ flanges as 1 to 2 .b 3 “ “ “ 1 to 6 . . “ “ “ 1 to 4 •73 “ “ “ 1 to 6 . 73 •' .92 SHAFTS. To Compute Deflection and Distributed Weight for Limit of Deflection. Wrought Iron. Deflection. Supported at Ends. Fixed at Ends. Weight. Supported at Ends. Fixed at Ends. = w. = w. Round. W Z 3 and W Z 3 = D. 664 cZ 4 and 1330 STRENGTH OF MATERIALS. — DETRUSIVE. 783 Sh.ea.x*ing. Wrought Iron. Resistance to shearing of American is about 75 per cent., and of English 80 per cent., of its tensile strength. Resistance to shearing of plates and bolts is not in a direct ratio. It ap- proximates to that of square of depth of former, and to square of diameter of latter. Results of Experiments upon. Shearing Strength of "Various Metals Toy Barallel Cutters. Wrought Iron .— Thickness from .5 to 1 inch, 50000 lbs. per sq. inch. Made by Inclined Cutters , angle = 7 0 . Plates. Thickness. Power. Bolts. Diam. Power. Ins. •05 .297 .24 •51 I Lbs. 54 ° 11 196 14 93 ° 39 1 5 ° 44 800 Brass Ins. 1. 11 •775 •775 .32 1. 142 Lbs. 29 700 ji 310 28 720 3°93 35 4 i° Copper Cf np] Steel Wrought iron j Wrought iron j Result of Experiments in Shearing, made at \J . S. NTavy Yard, Washington, on Wrought-iron Bolts. Diam. Minimum. Stress. Maximum. Per Sq.Inch. Diam. Minimum. Stress. Maximum. Per Sq.Inch. Inch. •5 •75 Lbs. 8900 18 400 Lbs. 9400 19650 Lbs. 44149 39 553 Inch. •875 1 Lbs. 25 500 32 900 Lbs. 27 600 35 8oo Lbs. 4 i 5°3 40 708 Mean 41 033 lbs. Result of Experiments on .875 Inch. Wrought-iron Bolts. (E. Clark.) Lbs. I Tons. II Lbs. I Tons. Single shear 54096 24.15 Double shear of two .625-inch plates Double “ 46904 I 22.1 II riveted together (one section) 45 696 | 20.4 Tensile strength 50 176 lbs. Riveted Joints. Experiments on strength of riveted joints showed that while the plates were destroyed with a stress of 43 546 lbs., the rivets were strained by a stress of 39 088 lbs. Cast Iron. Resistance to shearing is very nearly equal to its tensile strength. An average of English being 24 000 lbs. per sq. inch. Steel. Shearing strength of steel of all kinds (including Fagersta) is about 72 per cent, of its tensile strength. Treenails. Oak treenails, 1 to 1.75 ins. in diameter, have an average shearing strength of 1.8 tons per sq. inch, and in order to fully develop their strength, the planks into w r hich they are driven should be 3 times their diameter. Woods. When a beam or any piece of w r ood is let in (not mortised) at an inclina- tion to another piece, so that thrust will bear in direction of fibres of beam that is cut, depth of cut at right angles to fibres should not be more than .2 of length of piece, fibres of which, by their cohesion, resist thrust. Ash 650 lbs. I Deal 625 lbs. I Pine 650 lbs. Chestnut 600 “ | Oak 2300 “ | Spruce 625 “ STRENGTH OF MATERIALS. TENSILE. TENSILE STRENGTH. Tensile Strength is resistance of the fibres or particles of a body to separation. It is therefore proportional to their number, or to area of its transverse section, and in metals it varies with their temperature, generally decreasing as temperature is increased. In silver, tenacity decreases more rapidly than temperature ; and in copper, gold, and plat- inum less rapidly. Cast Iron- Experiments on Cast-iron bars give a tensile strength of from 4000 to 5000 lbs. per sq. inch of its section, as just sufficient to balance elasticity of the metal; and as a bar of it is extended the 12300th part of its length for every 1000 lbs. of direct strain, or one sixteenth of an inch in 64.06 feet per sq. inch of its section, it is deduced that its elasticity is fully excited when it is extended less than the 2400th part of its length, and extension of it at its limit of elasticity, which is about .5 of its destructive weight, is esti- mated at 1500th part of its length. Average ultimate extension is 500th part of its length. A bar will contract or expand .000006173 inch, or the 162000th of its length, for each degree of heat ; and assuming extreme of moderate range of temperature in this country 140° (— 20° + 120°), it will contract or ex- pand with this change .0008642 inch, or the 1157th part of its length. It follows, then, that as 1000 lbs. will extend a bar the 12300th part of its length, contraction or extension for 1157th part will be equivalent to a force of 10648 lbs. (4.75 tons) per sq. inch of section. It shrinks in cooling from one eighty-fifth to one ninety-eighth of its length. Mean tensile strength of American, as determined by Maj. Wade for U. S. Ordnance Corps, is 31 829 lbs. (14.21 tons) per sq. inch of section; mean of English, as determined by Mr. E. Hodgkinson for Commission on Applica- tion of Iron to Railway Structures, 1849, is 19484 lbs. (8.7 tons) ; and by Col, Wilmot, at Woolwich, in 1858, for gun-metal, is 23257 lbs. (10.35 tons), varying from 12320 lbs. (5.5 tons) to 25 520 lbs. (10.5 tons). Mean ultimate extension of four descriptions of English, as determined for Commission above referred to, was, for lengths of 10 feet, .1997 inch, being 600th part of its length ; and this weight would compress a bar the 775th part of its length. Tensile strength of strongest piece ever tested — 45 97° lbs. ( 2 °*5 2 tons). This was a mixture of grades 1, 2, and 3 from furnace of Robert P. Parrott at Greenwood, N. Y., and at 3d fusion. At 2.5 tons per sq. inch it will extend same as wrought iron at 5.6 tons. From experiments of Maj. Wade he deduced the following mean results : Density. I Tensile. I Transverse. I Torsion. 1 Crushing. I Hardness. 7.225 |1 31829 1 8182 I 8614 I 144916 I 22.34 Tensile per sq. inch of section ; Transverse per sq. inch, one end fixed, load applied at other end at a distance of i foot ; and Torsion per sq. inch, stress applied at end of a lever i foot in length. Green sand castings are 6 per cent, stronger than dry, and 30 per cent, stronger than chilled ; but when castings are chilled and annealed, a gain of 1 15 per cent, is attained over those made in green sand. Resistance to crushing and tensile stress is for American as 4.55 to 1, and for English as 5.6 to 7 to 1. Strength increasing with density. STRENGTH OF MATERIALS. TENSILE. 785 Remelting. — Strength, as well as density, are increased by repeated re- meltings. The increase is the result of the gradual abstraction of the con- stituent carbon of the iron, and the consequent approximation of the metal to wrought iron. Result of the 4th melting of pig iron, as determined by Major Wade, was to increase its strength from 12880 lbs. (5.75 tons) to 27888 lbs. (12.45 tons), and its specific gravity from 6.9 to 7.4. Three successive meltings of Greenwood iron, N. Y., gave tensile strength of 21 300, 30 100, and 35 700 lbs. Result of 5th melting by Mr. Bramwell was to increase strength of Acadian iron from 16800 lbs. (7.5 tons) to 41 440 lbs. (18.5 tons). Remelting increases its resistance to a crushing stress from 70 to 80 tons (14 per cent.) per sq. inch of section. Hot and. Cold Blast. Mr. Hodgkinson deduced from experiments that relative strength of 1.2 and 3 ins. square was as 100, 80, and 77, and that hot blast had less tensile strength than cold blast, but greater resistance to a crushing stress. Captain James ascertained that tensile strength of .75 inch bars, cut out of 2 and 3 inch bars, had only half strength of a bar cast 1 inch square. Mr. Robert Stephenson concluded, from experiments of recent date, that average strength of hot blast was not much less than that of cold blast ; but that cold blast, or mixtures of cold blast, were more regular, and that mixt- ures of cold blast and hot blast were better than either separate. Stirling’s IVLixed or Toughened Iron. By mixture of a portion of malleable iron with cast iron, carefully fused in a crucible, a tensile strain of 25 764 lbs. has been attained. This mixt- ure, when judiciously managed and duly proportioned, increases resistance of cast iron about one third ; greatest effect being obtained with a propor- tion of about 30 per cent, of malleable iron. NXallea'ble Cast Iron. Tensile strength of annealed malleable is guaranteed by some Manufact- urers of it at 56000 lbs. ; it is capable of sustaining 22400 lbs. without per- manent set. AWr 011 glit Iron. Experiments on English bars gave a tensile strength of from 22 000 lbs. to 26400 lbs. per sq. inch of its section, as just sufficient to balance elasticity of the metal ; and as a bar of it is extended the 28 oooth part of its length for every 1000 lbs. of direct strain, or one sixteenth of an inch m 116.66 teet per sq. inch of its section, it is deduced that its elasticity is fully excited when it is extended the 1000th part of its length, and extension of it at its limit of elasticity, which is from .45 to .5 of its destructive weight, is esti- mated at 1520th part of its length. A bar will expand or contract .000 006 614 inch, or 151 200 part of its length for each degree of heat ; and assuming, as before stated for cast iron, that extreme range of temperature in air in this country is 140°, it will contract or expand with this change .000926, or 1080th of its length, which is equn a- lent to a force of 20 740 lbs. (9.25 tons) per sq. inch of section. Mean tensile strength of American bars and plates (45000 to 76000), 60 500 lbs. (27 tons) per sq. inch of section ; as determined by Prof. Johnson in 1836, is 55900 lbs. ; and mean of English, as determined ly Capt. Brown, Barlow, Brunei, and Fairbairn, is 53 900 lbs. ; and by Mr. Kirkaldy, bars and plates (47040 to 55910) 51 475 lbs. (22.97 tons). - U* 786 STRENGTH OF MATERIALS. TENSILE. Greatest strength observed 73449 lbs. (32.79 tons). Ultimate strength, as given by Mr. D. K. Clark, 59 732 lbs. (26.66 tons). Average ultimate extension is 600th part of its length. Strength of plates, as determined by Sir William Fairbairn, is fully 9 per cent, greater with fibre than across it. Resistance of wrought iron to crushing and tensile strains is, as a mean, as 1.5 to 1 for American; and for English 1.2 to 1. Reheating. — Experiments to determine results from repeated heating and laminating, furnished following : From 1 to 6 reheatings and rollings, tensile stress increased from 43 904 lbs. to 61 824 lbs., and from 6 to 12 it was reduced to 43904 again. Effect of Temperature . — Tensile strength at different temperatures is as follows: 6o°, 1; 114 0 , i-Hi 2I2 °, 1.2; 250°, 1.32; 270°, 1.35; 325 0 , 1.41 ; 435 °) 1 - 4 - Experiments of Franklin Institute gave at 8o° 56000 lbs. I 720 0 . 55 000 lbs. I 1240 0 22 000 lbs. 570 0 66500 44 I 1050 0 32000 u I 1317 0 9000 “ Annealing. — Tensile strength is reduced fully 1 ton per sq. inch by an- nealing. Cold Rolling. — Bars are materially stronger than when hot rolled, strength being increased from one fifth to one half, and elongation reduced from 21 to 8 per cent. Hammering increases strength in some cases to one fifth. Welding. — Strength is reduced from a range of 3 to 44 per cent. 20 per cent., or one fifth, is held to be a fair mean. Temperature. — From o° to 400° strength is not essentially affected, but at high temperature it is reduced. When heated to redness its strength is re- duced fully 25 per cent. Tensile strength at 23 0 was found to be .024 per cent, less than at 64°. Cutting Screw Threads reduces strength from n to 33 per cent. Hardening in water, oil, etc., reduces elongation, but does not essentially increase the strength. Case Hardening reduces strength fully 10 per cent. Galvanizing does not affect strength of plates. Angled Bars , etc. — Their strength is fully 10 per cent, less than for bolts and plates. Elements connected, with. Tensile Resistance of various Substances. Substances. Stress per Sq. Inch for limit of Elasticity. Ratio of Stress to that causing Rupture. Substances. Stress per Sq. Inch for limit of Elasticity. Ratio of Stress to that causing Rupture. Lbs. Lbs. Beech 3 355 4 000 . 'I Wrought iron, ordinary “ “ Swedish.... 17 600 24 400 (18850 (22 4OO 15 OOO .3 Cast iron, English . 22 •34 •35 •35 .26 “ “ American Oak 5000 2 856 52 000 .2 •23 .62 44 44 English Steel plates, .5 inch “ “ American. . . “ wire . 757°° • 5 44 wire, No. 9, unannealed 47 532 .46 Yellow pine 3 332 •23 44 “ 44 annealed.. 36300 •45 Turning . — Removing outer surface does not reduce the strength of bolts. STRENGTH OF MATERIALS. TENSILE. 787 TIE-RODS. Results of Experiments on Tensile Strength ofWrought- iron Tie-rods. Common English Iron, 1.1875 Ins. in Diameter. Description of Connection. 1 Breaking Weight. Semicircular hook fitted to a circular and welded eye Lbs. Two semicircular hooks hooked together 16220 29 120 48 160 56 000 Right-angled hook or goose-neck fitted into a cylindrical eye '■ Two links or welded eyes connected together Straight rod without any connective articulation Ratio of Ductility and Malleability of Metals. In order of Wire-drawing Ductility. In order of Laminable Ductility. In order of Wire-drawing Ductility. In order of Laminable Ductility. In order of Wire-drawing Ductility. In order of Laminable Ductility. Gold. Silver. Platinum. Iron. Copper. Zinc. Tin. Lead. Nickel. Gold. Silver. Copper. Tin. Platinum. Lead. Zinc. Iron. Nickel. Relative resistance of Wrought Iron and Copper to tension and compres- sion is as 100 to 54.5. Steel. Experiments of Mr. Kirkaldy, 1858-61, give an average tensile strength for bars of 134400 lbs. (60 tons) per sq. inch for tool-steel, and 62720 lbs. (28 tons) for puddled. Greatest observed strength being 148 288 lbs. (66.2 tons). Plates, mean, 86800 lbs. (32 to 45.5 tons) with fibre, and 81 760 lbs. (36.5 tons) across it. Its resistance to crushing compared to tension is as 2.1 to 1. Hardening .— Its strength is very materially increased by being cooled in oil, ranging from 12 to 55 per cent. Crucible.— Experiments by the Steel Committee of Society of Civil Engineers, England, 1868-70, give a tensile strength of 91 571 lbs. per sq. inch (40.88 tons), with an elongation of .163 per cent., or 1 part in 613, and an elastic extension of .000 034 7th part for every 1000 lbs. per sq. inch, or 1 part in 28818. Bessemer.— Experiments by same Committee give a tensile strength of 76653 lbs. per sq. inch (34.22 tons) with an elongation of .144 per cent., or 1 part in 695, and an elastic extension of .oooo34 82d part for every 1000 lbs. per sq. inch, or 1 part in 28 719. Result of Experiments by Committe of Society of Civil Engineers of England, 1868-70, and Mr. Daniel Kir- kaldy, 1875. Per Sq. Inch. Steel. Elastic Strength. Elastic E in Parts of Length. ixtension per iooo^ Lbs. Ratio of Elastic to Ultimate Strength. Destructive Weight. Crucible Lbs. 49 840 44 800 48 608 39 200 32 080 28784 Tons. Per Cent. .225 .204 In Length. .0005 .00045 Per Cent. 58.2 59 59 * 2 5i-5 46.4 44.4 Lbs. 86 464 75 757 78 176 7 2 576 69888 64512 Tons. 38-6 Bessemer 20 Fagersta, unannealed . “ annealed Siemens, unannealed. . “ annealed.... 21.7 17-5 14.56 12.85 33- 82 34 - 9 3 2 -4 31-2 28.8 STRENGTH OF MATERIALS. TENSILE, Average Tensile Elasticity- of Steel Bars and 3 ?lates. (Com. of Civil Engineers , 1870.) Description. lasticity per Sq. Inch. Elastic Exten- sion in Parts of Length. Ratio of Elas- tic to Destruc- tive Strength. Lbs. Parts. Per Cent. 50 557 1 in 485 58.2 43 814 i in 675 55 56 5 6 o — 64.8 34048 — 55-6 55 574 — 64-7 40 858 — 54 30710 1 in 980 59 - 2 26940 1 in 1020 56-5 32 5 00 — 46.4 28780 — 44.4 40174 — 58.8 42 112 1 in 185 — Bars. Crucible, hammered and rolled Bessemer, “ “ ... .. Fagersta, rolled “ unannealed u hammered and rolled “ “ “ annealed., “ plates, unannealed. “ “ annealed unannealed... annealed , “ tires Krupp’s shaft Siemens, Tensile strength of steel increases by reheating and rolling up to second operation, but decreases after that. Tensile Strength, of Various Materials, deduced from Experiments of TJ. S. Ordnance Department, Eair- hairn, Hodgkinson, KLirkaldy-, and "by- the Author. Power or Weight required tq, tear asunder One Sq. Inch , in Lbs. Metals. Lbs. Steel, Pittsburgh, mean 94450 Bessemer, rolled - - - f 76 650 “ hammered Metals. Lbs. Antimony, cast 1 053 Bismuth, cast 3248 Cast Iron, Greenwood 45 97 ° mean, Major Wade. . . 31 829 gun-metal, mean 37 232 malleable, annealed. . 56000 Eng., strong 29000 “ weak 13400 , , ( 1 c 600 “ averages { 2 J 28o u gun-metal 23257 “ mean* 19484 11 Low Moor, No. 2 14076 “ Clyde, No. 1.. . . 16125 “ “ No. 3.... 23468 u Stirling, mean.. 25764 Copper, wrought 34000 rolled 36 000 cast 24350 bolt 36800 wire 61 200 Gold 20384 Lead, cast 1 800 “ pipe.....:. ....... ........ 2240 “ “ encased 3 759 “ rolled sheet 3320 Platinum wire 53000 Silver, cast 40000 Steel, cast, maximum 142000 “ mean 88560 puddled, maximum 173817 Amer. Tool Co 179980 „.i vo i 2IOOOO ( 3OOOOO plates, lengthwise 96 300 u crosswise .... 93 700 Chrome bar 180000 125 000 152900 “ Eng., cast 134000 “ u “ plates, mean 93500 “ “ plates 86800 “ “ puddled plates 62720 u u crubible 91 570 “ “ homogeneous 96280 “ “ blistered, bars 104000 “ “ Fagersta bars 89600 “ “ “ plates 98560 “ “ Whitworth’s { 89600 ( 152000 “ “ Siemens’s plates. . j ^ |g° “ “ Krupp’s shaft 92243 Tin, cast. .... f; 5000 u Banca.. 2100 Wire rope, per lb. w’t per fathom 4480 “ “ galvanized steel, u 6720 Wrought Iron, boiler plates. . . j J^ooo “■ rivets....... 65000 “ bolts, mean “ “ inferior 30000 “ hammered 54000 “ shaft 44 750 u wire 73600 tc u No. 9 100000 “ “ No. 20 120000 “ “ diam. .0069 inch 301 168 “ “ galv’ized .058 “ 64960 “ Eng., heavy forging. 33600 “ “ plates, lengthw’e 53800 “ “ “ crosswise 48800 * By Coram’fl on application of Iron to Railway Structure. STRENGTH OF MATERIALS. TENSILE. 789 Metals. Lbs. Wrought Iron, Eng., mean 51000 Eng., Low Moor 57600 “ Lancashire 48800 “ Thames 65921 “ armor-plates .... 40000 “ bar...... (31300 ( 50000 “ “ charcoal 63000 “ rivet, scrap 51 760 Russian, bar, best 59500 “ “ 49000 Swedish, “ best 72000 “ “ 48900 Zinc 3 500 sheet. Alloys or Compositions. Alloy, Cop. 60, Iron 2, Zinc 35, Tin 2. 85 120 “ Tin 10, Antimony 1 noco Aluminium, Cop. 90 71 6co “ maximum 96320 Bell-metal 3670 Brass, cast 18000 “ wire 49000 Bronze, Phosphor., extreme 50915 “ mean 34464 “ ordinary . 23500 “ Cop. 10, Tin 1 33000 “ “ 9, “ 1 38080 “ “ 8, “ 1 36000 “ “ 2, Zinc 1 29000 Gun-metal, ordinary 18000 mean , . 33600 “ bars 42040 Speculum metal 7000 Yellow metal 48 700 Woods. Ash, white 14000 “ American 9500 “ English 16000 Bamboo 6 300 Bay 14000 Beech, English 11500 Birch 15000 “ Amer., black 7000 Box, African 23000 Bullet 19000 Cedar, Lebanon 11400 West Indian 7500 “ American 600 Chestnut 500 “ horse 10000 Cypress 6 000 Deal, Christiana 12400 Ebony 27000 Elm \ 6000 l 13000 Gum, blue 18000 “ Alabama 15860 Hackmatack 12000 Hickory 11000 Holly .. . 16000 Lance ( x 7 35° ( 23000 Woods. Larch 1 ... .. Lignum vitae Locust Mahogany, Honduras “• Spanish.. Oak, Pa. , seasoned “ Va., “ ........ “ white ‘ ‘ live, Ala “ red u African “ English “ Dantzic Pear Pine, Ya “ Riga “ yellow' “ white “ red Poon Poplar Redwood, Cal.. Spruce, white Sycamore. Teak, India “ African Walnut, Eng “ black “ Mich Willow Yew r Across Fibre. Oak Pine Lbs. ( 4 200 i 9500 ..II 800 I 16000 ( 20500 . . 21 OOO I 8000 ( 12 000 •• 20333 . . 25 222 . . l6 500 . . l6 380 . . IO 250 . . 9 500 J 4 500 l 7 571 . • 4 200 . . 9 860 . . 19 200 . . 14 OOO . . 13 OC O ..11 8co . . 13 000 . . 13300 . . 7 000 .. 10833 | 10290 \ 12400 ( 9 660 l *3 . . 15000 . . 2 1 000 . . 7 800 . . 16633 ■ • 17 580 . . 13 000 . . 8 000 . . 2 300 • 550 Miscellaneous. Basalt, Scotch Beton, N. Y. Stone Con’g Co. . . . j Blue stone Brick, extreme inferior. Cement, Portland, 7 days. “ pure, 1 mo “ sand 2, 320 days “ U J u “ pure, •“ “ sand 1, in water 1 mo “ “ 1 “ 1 y’r u “ 3? 1 year. . “ “ 5,. 1 “ “ “ 7, 1 “ Hydraulic Rosedale, Ulst. Co., 7 days “ sand i, 30 “ 9 mos 1469 300 500 77 75o 100 290 400 860 393 713 948 1 152 201 3X9 310 214 163 284 104 102 560 700 79 o STRENGTH OP MATERIALS. TORSION. Miscellaneous. Cement, Roman, in water 7 days “ “ 1 mo., “ “ 1 year “ sand 1, 42 days. Flax Glass, crown . . . . Glue Granite Gutta Percha’. . . , Hemp rope Ivory Leather belting. . Limestone Marble, statuary. , “ Italian.., Marble, white “ Irish Lbs. 90 «5 286 284 1 99 160 25 000 2546 4000 578 3 5 oo 12000 16000 1 000 330 670 2 800 3 200 5 200 9000 17 600 Miscellaneous. Lbs. Mortar, 1 year j “ hydraulic j “ ordinary 35 Oxhide 6 300 Rope, Manila 9000 “ tarredhemp 15000 Sandstone 150 “ fine green 1260 “ Arbroath j i ^3 “ Caithness { 473 l 1054 “ Portland . { 857 ( 1 000 “ Craigleth 453 Silk fibre . . * . . 52 000 Slate * {,;& Whalebone 7000 TORSIONAL STRENGTH. SHAFTS AND GUDGEONS. Shafts are divided into Shafts and Spindles , according to their mag- nitude, and are subjected to Torsion and Lateral Stress combined, or to Lateral Stress alone. A Gudgeon is the metal journal or Arbor upon which a wooden shaft revolves. Lateral Stiffness and Strength. — Shafts of equal length have lateral stiff- ness as their breadth and cube of their depth, and have lateral strength as their breadth and square of their depths. Shafts of different lengths have lateral stiffness directly as tlieir breadth and cube of their depth, and inversely as cube of their length ; and have lateral strength directly as their breadth and as square of their depth, and inversely as their length. Hollow Shafts having equal lengths and equal quantities of material have lateral stiffness as square of their diameter, and have lateral strength as their diameters. Hence, in hollow shafts, one having twice the diameter of an- other will have four times the stiffness, and but double the strength ; and when having equal lengths, by an increase in diameter they increase in stiff- ness in a greater proportion than in strength. When a solid shaft is subjected to torsional stress, its centre is a neutral axis, about which both intensity and leverage of resistance increase as radius or side ; and the two in combination, or moment of resistance per sq. inch, t, increase as square of radius or side. Round Shaft. — Radius of ring of resistance is radius of gyration of sec- f tion, being alike to that of a circular plate revolving on its axis, viz., .7071 radius. The ultimate moment of resistance then is expressed by product of sectional area of shaft, by ultimate shearing resistance per sq. inch of material by radius, and by .7071. Or, .7854 d 2 r S X -7071 = .278 d 3 S = Rf. (D. K. Clark.) d representing diameter of shaft and r radius, S ultimate shearing stress of mate- rial in lbs. per sq. inch, R radius through which stress is applied , in ins. , and W moment of load or destructive stress, in lbs. .278 d 3 S _ R W _ _ , /R W — 5 — =W; ^5 = S; an,J v / ^ _X 1,534 = - Hence, STRENGTH OF MATERIALS. — TORSION. 791 Round Shaft. — Strength, compared to a square of equal sectional, area, is about as 1 to .85. Diameter of a round section, compared to side of square section of equal resistance, is as 1 to .96. Square Shaft. — Moment of torsional resistance of a square shaft exceeds that of a round of same sectional area, in consequence of projection of cor- ners of square ; but inasmuch as material is less disposed to resist torsional stress, the resistance of a square shaft, compared to a round one of like area of section, is as 1 to 1.18, and of like side and diameter, as 1.08 to 1. Hence, -^ 7 8 X n°8 S = w HoUwo Round Shafts. ' ■ *? ■ S = W. When Section is comparatively Thin. x ' 57 ^ = W. s representing side , d and c V external and internal diameters , and t thickness of metal in ins. Torsional Angle of a bar, etc., under equal stress, will vary as its length. Hence, torsional strength of bars of like diameters is inversely as their lengths. Stress upon a shaft from a weight upon it is proportional to product of the parts of shaft multiplied into each other. Thus, if a shaft is 10 feet in length, and a weight upon centre of gravity of the stress is at a point 2 feet from one end, the parts 2 and 8, multiplied together, are equal to 16; but if weight or stress were applied in middle of the shaft, parts 5 and 5, multiplied together, would produce 25. When load upon a shaft is uniformly distributed over any part of it, it is consid- ered as united in middle of that part; and if load is not uniformly distributed, it is considered as united at its centre of gravity. Deflection of a shaft produced by a load which is uniformly distributed over its length is same as when .625 of load is applied at middle of its length. Resistance of body of a shaft to lateral stress is as its breadth and square of its depth; hence diameter will be as product of length of it , and length of it on one side of a given point , less square of that length. Illustration. — Length of a shaft between centres of its journals is 10 feet; what should be relative cubes of its diameters when load is applied at 1, 2, and 5 feet from one end? and what when load is uniformly distributed over length of it? lX l 1 — I 3 = d 3 ] and when uniformly distributed, d 3 -r -2 — d x . 10 X 1 = 10 — 1 2 — 9 — cube of diameter at 1 foot ; 10X2 = 20 — 2 2 = 16 == cube of diameter at 2 feet ; 10 X 5 = 50 — 5 2 = 25 = cube of diameter at 5 feet. When a load is uniformly distributed, stress is greatest at middle of length, and is equal to half of it; 25 -f- 2 = 12. 5 — cube of diameter at 5 feet. Torsional Strength of any square bar or beam is as cube of its side, and of a cylinder as cube of its diameter. Hollow cylinders or shafts have great- er torsional strength than solid ones containing same volume of material. To Compute Diameter of* a Solid. Shaft of* Cast or Wrought Iron to Tfcesist Lateral Stress alone. When Stress is in or near Middle. Rule. — Multiply weight by length of shaft in feet ; divide product by 500 for cast iron and 560 for wrought iron, and cube root of quotient will give diameter in ins. Example. — Weight of a water-wheel upon a cast-iron shaft is 50000 lbs., its length 30 feet, and centre of stress of wheel 7 feet from one end ; what should be diameter of its body ? 3J ^ 5 ° 000 X 3 ° ^ _ I4 42 i ns } weight was in middle of its length. Hence diameter at 7 feet from one end will be, as by preceding Buie, 30X7 — 7 2 — 161 = relative cube of diameter at 7 feet ; 30 X 15 — if 2 — 22s — relative cube of diameter at 15 feet, or at middle of its length. Then, as ^225 : 14.42 :: -^161 : 12.91 ins., diameter of shaft at 7 feet from one end. 79 2 STRENGTH OF MATERIALS. TORSION. For Bronze, 420; Cast steel, 1000 to 1500; and Puddled steel, 500. When Stress is uniformly laid along Length of Shaft. Rule. — Divide cube root of product of weight and length by 9.3 for Cast iron and 10.6 for Wrought iron, and quotient will give diameter in ins. For Bronze, 8.5 ; Cast steel, 18.6 to 27.9 ; and Puddled steel, 9.3. When Diameter for Stress applied in Middle is given . Rule. — Take cube root of .625 of cube of diameter, and this root will give diameter required. Example. — Diameter of a shaft when stress is uniformly applied along its length is 14.42 ins. ; what should be its diameter, stress being applied in middle? To Compute Diameter of* a, Solid. Shaft of Cast Iron to Pfcesist its 'Weight alone. Rule. — Multiply cube of its length by .007, and square root of product will give diameter in ins. Example. — Length of a shaft is 30 feet; what should be its diameter in body? V(3° 3 X .007) = ^189 — , 3 . 75 i n s To Compute Diameter of a Hollow Shaft of Cast Iron to Sustain its Load in Addition to its Weight. When Stress is in or near Middle. Rule. — Divide continued product of .012 times cube of length, and number of times weight of shaft in lbs., by square of internal diameter added to 1, and twice square root of quotient added to internal diameter will give whole diameter in ins. Example. — Weight of a water-wheel upon a hollow shaft 30 feet in length is 2.5 times its own weight, and internal diameter is 9 ins. ; what should be whole diam- eter of shaft? To Compute Diameter of a Ffcoxxnd or Square Shaft to Ptesist Combined Stress of Torsion and 'Weight. Rule. — Multiply extreme of pressure upon crank-pin, or at pitch-line of pinion, or at centre of effect upon the blades of a water-wheel, etc., that a shaft may at any time be subjected to ; by length of crank or radius of wheel, etc., in feet ; divide the product by Coefficient in following Table, and cube root of quotient will give diameter of shaft or its journal in ins. Example. — What should be diameter for journal of a wrought-iron water-wheel shaft, extreme pressure upon crank-pin being 59 400 lbs., and crank 5 feet in length ? When Two Shafts are used , as in Steam-vessels , etc., with One Engine . Rule. — Divide three times cube of diameter for one shaft by four, and cube root of quotient will give diameter of shaft in ins. Example. — Area of journal of a shaft is 113 ins. ; what should be diameter, two shafts being used? Example. —Apply rule to preceding case. v 50 000 x 30 9-3 — = 12.31 ins. V.625 x 14.42 3 = -^.625 X 3000 — 12.33 ins. HOLLOW SHAFTS. 2 A '.012 X 30 3 X 2. 5' V * + 9 2 > : )+ 9=:2 \/1 6.28 ms. , and 6. 28 -f- 9 = 15. 28 ins. C = 120. ' 59 4oo X 5 12 ° 2475, and V 2 475 = *3-53 STRENGTH OF MATERIALS. TORSION. 793 Torsional Strength, of Various NLetals. [Maj. Wm. Wade , U. S. Ordnance Corps , 1851, Steel Committee [ England , 1868], and Stevens Institute , N. J., 1878.) Reduced to a Uniform Measure of One Inch in Diameter or Side- Stress applied at One Foot from Axis of Body and at Face of Axis. „ „ . t C d.a Destructive Stress Bars and Metals. Tensile Strength. at 25 Ins. Computed at 12 Ins. Cast Iron. Lbs. Lbs. Lbs. I Area 1 sq. inch ) 45 000 520 1082 Area 2. 97 sq. ins. ) ll 3800 7904 *7^. Diam. 1 Least . . . IP 1 =1.9 [Mean... ins. ) Greatest. §111 Side 1 inch ) 111 Area 1 sq. inch j 9000 31 829 45000 u 1550 2145 2840 350 3664 4462 59°7 728 Wrought Iron. Diam. f Least . . . Wm =1.9 'Mean ins. ( Greatest. Area 2. 83 sq. ins. 38027 56 300 74 592 1250 1375 1500 2600 2860 3120 Bronze. u Diam.= ( Least 1.9 ins. (Greatest. Area 2.83 sq. ins. 17 698 56 786 500 650 1040 1352 Cast Steel. tt Diam.= (Least 1.9 ins. (Greatest. Area 2.83 sq. ins. 42 000 128000 2600 7760 5408 16140 Bessemer Steel. u Diam. = 1.382 ins. ) Area 1.5 sq. ins. j 36 960 1568 3261 49 2 530 650 850 728 152 197 788 2353 1236 245 115 235 230 80 105 To Compute Diameter of Shafts of Oak and. Pine. Multiply diameter ascertained for Cast Iron as follows: Oak by 1.83, Yellow Pine by 1.716. Metals and Woods. Ultimate Torsional Strength . — Of Cast Iron may be taken as equal to its transverse strength for American and .9 for English, or as .26 of its tensile strength for American and .23 for English. Of Wrought Iron, as .7 to .8 of its transverse strength for American and .7 to 1 for English, and of Steel, as .72 of its tensile strength. Elastic Torsional Strength . — Of Cast Iron may be taken as equal to its transverse strength, of Wrought Iron 40 per cent, of its ultimate torsional strength, of Steel 44 per cent, of its tensile strength, and 45 per cent, of its ultimate torsional strength. # Bessemer Steel . — Has a torsional strength of 6670 lbs. per sq. inch at a ra- dius of one foot, being somewhat less than that of Cast Iron, Fagersta has 50 per cent, of its ultimate transverse strength, and Siemens 44.5 per cent, of its ultimate tensile. _ 3X 794 STRENGTH OF MATERIALS.— TORSION. Note.— E xamples here given are deduced from instances of successful practice* where diameter has been less, fracture has almost uuiversally taken place stress being increased beyond ordinary limit. * ’ 2.— When shafts of less diameter than 12 ins. are required, Coefficients here given may be slightly reduced or increased, according to quality of the metal and diame- ter of shaft; but when they exceed this diameter, Coefficients may not be increased as strength of a shaft decreases very materially as its diameter increases. Order of shafts, with reference to degree of torsional stress to which they may be subjected, is as follows : J 1. Fly-wheel. | 2. Water-wheel. | 3. Secondary shaft. J 4. Tertiary, etc. Hence, diameters of their journals may be reduced in this order. To Compute Diameter of a Wrought-iron Centre Shaft for connecting Two Engines at a Right Angle. Conditions of such a shaft are as follows : Greatest stress that it is subjected to is when leading engine is at ,75 of its stroke, and following engine .25 of its stroke ; hence, position of each crank is as sin. 22 0 30 x 2 = .7 0 ? 1 of length of crank or radius of power. Consequently, ■ = d. P representing extreme pressure on piston. Note.— I n computing P it is necessary to take very extreme pressure that piston may be subjected to, however short the period of time. Average pressure does not meet requirement of case. Illustration. — E xtreme pressure upon each piston of two engines connected at a right angle was hi 592 lbs., and stroke of pistons 10 feet; what should have been diameter of centre shaft ? and what of each wheel or driving shaft? 3 / fill 592 X 2 x .707 „ / 7 88q55 „ _ . V \ ^ — 7 = y — 125 =18.48 ms. centre shaft. For ordinary mill purposes, driving shafts should be as cube roots of 2^ of ? times cube of centre shaft. , 0 O o w J Thus ^ 11 ^ = ,6.79 » s . To Compute Torsional Strength of Hollow Shafts and Cylinders. Rule.— F rom fourth power of exterior diameter subtract fourth power of interior diameter, and multiply remainder by Coefficient of material ; divide this product by product of exterior diameter and length or distance from axis at which stress is applied in feet, and quotient will give resistance in lbs. d±—d'*C n 0r ’ - = R - Example. —What torsional stress may be borne by a hollow cast-iron shaft, hav- ing diameters of 3 and 2 ins., power being applied at one foot from its axis? C = 130. 3 4 — 2* X 130 = 8450, which -f- 3 X 1 = = 2816.6 lbs. To Compute Torsional Strength of Round and Square Shafts. Rule.— Multiply Coefficient in preceding Table by cube of side -or of diameter of shaft, etc., and divide product by distance from axis at which stress is applied in feet ; quotient will give resistance in lbs. Illustration; — What torsional stress may be borne by a cast-iron shaft of best material, 2 ins. in diameter, power applied at 2 feet from its axis. C from table = 130. 130 X ? 3 1040 = 520 lbs. For steamers, when from heeling of vessel or roughness of sea the stress may be confined to one wheel alone, diameter of journal of its shaft should be equal to that of centre shaft. STRENGTH OF MATERIALS. — TORSION. 795 GUDGEONS. 'Po Compute Diameter of a Single Gudgeon of Cast Iron, to Support a given Weiglit or Stress. Rule.— Divide square root of weight in lbs. by 25 for Cast iron, and 26 for Wrought iron, and quotient will give diameter in ins. Example.— Weight upon a gudgeon of a cast-iron water-wheel shaft is 62 500 lbs. ; what should be its diameter ? To Compute Diameter of Two Gudgeons of Cast Iron, to Support a given Stress or "W eight. Rule.— Proceed as for two shafts, page 792. To Compute XJltimate Torsional Strength, of Hound, and Square Shafts. (D. K. Clark.) S representing ultimate shearing strength , and W moment of load, both in lbs., s side of square shaft, and R radius of stress, both in ins. Illustration. — What is ultimate torsional strength of a round cast-iron shaft 4 ins. in diameter, stress applied at 5 feet from its axis ? By experiments of Major Wade, ordinary foundry iron has a torsional strength of 7725 lbs., or 644 lbs. per sq. inch at radius of one foot. When Torsional Strength per sq. inch for radius of 1 inch is ascertained, substitute C for .278, .4, .2224, or .32. n . v Stress which will give a bar a permanent set of .5° is about .7 of that which will break it, and this proportion is quite uniform, even when strength of material mav vary essentially. Wrought Iron, compared with Cast Iron, has equal strength under a stress which does not produce a permanent set, but this set commences under a less force in wrought iron than cast, and progresses more rapidly thereafter. Strongest bar of wrought iron acquired a permanent set under a less strain than a cast-iron bar of lowest grade. Strongest bars give longest fractures. .2 d 3 S When S is not known, substitute for Steel. Round. ^ — w. g 72 s=z y 2 p er cen t. of tensile strength. Torsional Strength of Cast Steel is from 2 to 3 times that of Cast Iron. Following rules are purposed to apply in all instances to diameters of journals of°shafts, or to diameter or side of bearings of beams, etc., where length of journal or distance upon which strain bears does not greatly ex- ceed diameter of journal or side of beam, etc. ; hence, when length or distance is greatly increased, diameter or side must be correspondingly increased. Coefficients for torsional breaking stress of Iron, Bronze, and Steel, as de- termined by Major Wade, are : Wrought Iron, 640 ; Cast Iron, 560 ; Bronze, 460 ; Cast Steel, 1 120 to 1680. Puddled Steel does not differ essentially from that of cast iron. ^62 500 250 25 — 25 — “ 10 ins. Cast Iron. Round. Square. — = W, and 1. Hollow. .278 ( d 4 — d'*) S R d Assume S = 20000 lbs. Thus, take preceding illustration. Then ? 7 2 3 x 4 _ __ g 24Q v 5 x 12 •30 S 3 s = W. Square. — = W. STRENGTH OF MATERIALS. TORSION. F jrmulas for Minimum and Maximum Diam. of Wrouglit-iron Shafts. (A. E. Seaton , London , 1883, and Board of Trade, Eng.) Compound Engines . \J^ 1 d 2 ] S — diameter. D and d representing diam- eter of low and high pressure cylinders , and S half stroke, all in ins., p pressure of steam m boiler , in lbs. per so. inch, and C a rnpifirieni. ns • Angle of Crank. Shai Crank. fts. Pro- peller. Angle of Crank. Shafts. I Crank. I Pro- peller. Angle Crank. Shai Crank. ? t8. I Pro- peller. Angle of Crank. Shai Crank. fts. Pro- peller. O O OS (2468 (4000 2880 5400 IOO° | 2279 2659 ( 4000 1 5400 | IlO° i 2I 3 I , 1 4000 2487 5400 1 120° | 2016 (4000 2352 5400 /iH? y — C = diameter. A. E. Seaton, London, 1883. Side-wheel Engines, Sea Service.— One cylinder crank journal, C = 8o; outboard 100; Two cylinder crank journal 50; outboard 65 ; and centre shaft 58. Propeller Engines.— One cylinder crank journal 150: Tunnel 130: Two cylinder compound crank 130; Tunnel no; Two cranks, crank 100; Tunnel 85; Three cranks, crank 90; and Tunnel 78. ’ River Service.— C may be reduced one fifth. Illustration. — With a compound propeller engine, steam cylinders 20 and 40 ins. in diameter, by 40 ins. stroke, operating under a pressure of 80 lbs. steam (mercurial gauge), what should be the diameter of the shafts of wrought iron? a /20 2 X 80 —J— 40 2 X 15 . . „ A6000 _ V ^ X 40 = y X 40 = 8.24 ms. crank shaft ; 4000 and , /56000 V ^5400" X 4 ° = 7- 46 ms. propeller shaft. «T ournals of Shafts, etc. Journals or bearings of shafts should be proportioned with reference to pressure or load to be sustained by the journal. Simplest measure of bear- ing capacity of a journal is product of its length by its diameter, in sq. ins. • and axial area or section thus obtained, multiplied by a coefficient of pressure per sq. inch, will give bearing capacity. Sir William Fairbairn and Mr. Box give instances of weights on bearings of shafts etc. from which following deductions are made, showing pressure per sq inch of axial section of journal : . F H Crank pins, 687 to 1150 lbs. per sq. inch. Link bearings, 456 to 690 lbs. per sq. inch. axTa? Sea 6 ^ bearings ’ as a & eneral ™le, should not exceed 750 lbs. per sq. inch of Length of Journals should be 1.12 to 1.5 times diameter. Journals of Locomotives or Like Axles are usually made twice diameter, and to sustain a pressure of 300 lbs. per sq. inch of axial area, or 10 sq. ins. per ton of load. Solid. Cylindrical Couplings or Sleeves. *?’ 3 7^7“ hi = ^ ’ .25 d-}-.i2 — 1 c. d representing diameter , ZnhnYjffX ° f l ! e ^ °S top °r scarf of shaft, k breadth of key, its depth be- ing half its bieadth, and D diameter of coupling or sleeve , all in ins. Flanged Couplings. d +V 3;5 d=D; 3 d-}- 1 = F; . 3 df-. 4 z=l; d- fi=L; Z-h 4 — s. D repre- senting diameter of body of coupling F diameter of flanges, l thickness of both flanges, L length of each coupling , s projection of end of one shaft and retrocession of other from centre of coupling, and d diameter of shaft, all in ins. Supports for Shafts. ( Molesworth .) - L- L representing distance of supports apart, in feet. STRENGTH OF MATERIALS. TORSION, 797 To Resist Lateral Stress. = D- W representing weight or pressure at centre of length in lbs., and D diameter or side , if square, in ins. Value of C. Wrought Iron, 560; Cast Iron, 500; Cast Steel, 1000 to 1500; Bronze, 420; and Wood, 40. When Weight is distributed put 2 C. Values of C for Shafting of Various Metals , as observed by different 16 W r Authorities , and deduced from Formulas of Navier. Ultimate Resistance. - = C. Metal. ! c Metal. c Metal. C Wrought Iron. Cast Iron. Steel. American, Pemb e , Me. 61673 ( 36846 American, Conn. . 82 926 Ulster 61 815 American, mean { 3830° “ Spindle 102 131 “ mean 66 436 l 42 821 “ Nash. I. Co. 95213 English, refined Swedish 49! 4 8 “ 18 trials 44 957 English, Shear hi 191 54 585 61 909 English, mean. . | 22 132 38217 Bessemer j 73060 79662 16 W R 7T <23 HVIill and. Factory Shafts. ( J . B. Francis.) Cylindrical. Square. 3V2WR = T. - d 3 T 16 R = W. /s 3 T \ = T - hr ^ 3 )^ 2 _W. Mean value of T. Cast Iron { 22000 65000 mean 35000 “ Eng. 30000 Wrought Iron { 49 000 ° ( 94000 “ mean 50000 “ “ Eng. 45000 Steel \ 76000 ( in 000 mean 86000 “ Bessemer 78000 Illustration.— What is the ultimate or destructive weights that may be borne by a Round Cast-iron shaft 2 ins. in diameter, and by a Square shaft 1.75 ins. side, stress applied at 25 ins. from axis? Assume T = 36000. Round. Square. 3.1416X23X36000 /1.75 3 X 36000 . 16x25 y:> V 25 Their lengths should be reduced, and diameter increased, in following cases : 1st. At high velocities, to admit of increased diameter of journals, thereby rendering them less liable to heating. 2d. As they approach extremity of a line of shafting. 3d. Attachment of intermediate pulleys or gearing. ■4-3^ 4 - ^2 = 1837.8 lbs. Prime Movers of Power. Wrought Iron, Cast Iron, ght 3 j* ;. V > JIP Transmitters of Power. - d, and .02 nd 3 = IEP. = d, and .032 nd* = IIP. = d, and .01 wd 3 = IIP. Steel. ^/ 6 “-„ IIP = <*, and .016 n d» = IBP. 3 /167IIP _ , and oo6nd 3 = up. , / 8 3.5 IEP = d and 0,2 n d* = jip . V n V n IIP representing horse-power transmitted , n number of revolutions, and d diameter of shaft in ins. Illustration 1.— What should be diameter of a wrought-irom shaft, to simply transmit 128 IP at 100 revolutions per minute? 3 /50XJ28 _ 3 /6400 _ {ns V 100 100 2.— What BP will a steel shaft of 4 ins. diameter transmit at 100 revolutions per minute? .032 X 100 X 4 3 = 204. 8 horses , 79 $ STRENGTH OF MATERIALS. — TRANSVERSE. TRANSVERSE STRENGTH. Transverse or Lateral Strength of any Bar , Beam , Rod, etc., is in propor- tion to product of its breadth and square of its depth; in like-sided bars, beams, etc., it is as cube of side, and in cylinders as cube of diameter of section. When One End is Fixed and the Other Projecting , strength is inversely as distance of weight from section acted upon ; and stress upon any section is directly as distance of weight from that section. When Both Ends are Supported only, strength is 4 times greater for an equal length, when weight is applied in middle between supports, than if one end only is fixed. When Both Ends are Fixed , strength is 6 times greater for an equal length, when weight is applied in middle, than if one end only is fixed. When Ends Rest merely upon Two Supports, compared to one When Ends are Fixed , strength of any bar, beam, etc., to support a weight in centre of it, is as 2 to 3. When Weight or Stress is Uniformly Distributed, weight or stress that can be supported, compared with that when weight or stress is applied at one end or in middle between supports, is as 2 to 1. HVTetals. In Metals, less dimension of side of a beam, etc., or diameter of a cylinder, greater its proportionate transverse strength, in consequence of their having a greater proportion of chilled or hammered surface, compared to their ele- ments of strength, resulting from dimensions alone. Strength of a Cylinder , compared to a Square of like diameter or sides, is as 5.5 to 8. Strength of a Hollow Cylinder to that of a Solid Cylinder, of same area of section, is about as 1.65 to 1, depending essentially upon the proportionate thickness of metal compared to diameter. Strength of an Equilateral Triangle, Fixed at One End and Loaded at the Other, having an edge up, compared to a Square of the same area, is as 22 to 27 ; and strength of one, having an edge down, compared to one with an edge up, is as 10 to 7. Note.— In Barlow and other authors the comparison in this case is made when the beam, etc., rested upon supports. Hence the stress is contrariwise. Strongest rectangular bar or beam that can be cut out of a cylinder is one of which the squares of breadth and depth of it, and diameter of the cylinder, are as 1, 2, and 3 respectively. , Oast Iron. Mean transverse strength of American, as determined by Major Wade, is 681 lbs. per. sq. inch, suspended from a bar fixed at one end and loaded at the other ; and mean of English, as determined by Eairbairn, Barlow, and others, is 500 lbs. Experiments upon bars of cast iron, 1, 2, and 3 ins. square, give a result of transverse strength of 447, 348, and 338 lbs. respectively ; being in the ratio of 1, .78, and .756. Woods. Beams of wood, when laid with their annular layers vertical, are stronger than when they are laid horizontal, in the proportion of 8 to 7. Relative Stiffness of ^Materials to Resist a Transverse Stress. Ash 089 I Cast Iron 1 I Oak 095 I Wrought iron 1.3 Beech 073 | Elm 073 | White pine. .. .1 | Yellow pine.. .087 STRENGTH OF MATERIALS. — TRANSVERSE. 799 Strength of a Rectangular Beam in an Inclined position, to resist a vertical stress, is to its strength in a horizontal position, as square of radius to square of cosine of elevation ; that is, as square of length of beam to square of dis- tance between its points of support, measured upon a horizontal plane. Transverse Strength, of Various [Materials. [U S. Ordnance Department , Hodglcinson , Fairbairn , KirJcaldy, and by the Author.) Power reduced to uniform, Measure of One Inch Square , and One Foot in Length ; Weight suspended from one End. Metals. Brass Cast Iron, mean of 4 grades “ “ (Maj. Wade) “ ordinary u extreme, West P’t F’dry “ gun-metal,* “ “ “ Eng., Low Moor, cold blast. “ u Ponkey, “ “ “ Ystalyfera \ l “ “ mean, 65 kinds “ “ “ 15 kinds, coldblast “ u planed bar “ “ rough bar Copper Steel, hammered, mean “ cast, soft “ “ hard “ hematite, hammered “ Krupp’s shaft “ Fagersta, hammered Wrought Iron, mean “ u English “ 11 Swedish t 260 660 681 575 740 472 581 770 500 641 5*8 534 244 1500 1540 4200 1620 2096 1200 606 475 665 Woods. Ash “ English “ Canada Balsam, Canada Beech “ white Birch Cedar, white f‘ Cuba Chestnut Elm “ Canada, red Fir, Baltic, mean “ Canada, yellow “ red “ Norway “ Dantzic Riga “ Memel “ “ red Greenheart, Guiana Gum, blue Hackmatack Hemlock 168 160 120 87 130 112 160 115 160 63 105 160 125 170 i53 58 117 120 123 163 112 161 75 160 136 102 100 Woods. Hickory. Iron wood, Burmah Larch, Russian Lignumvitse Locust Mahogany Mangrove Maple ... Oak, white “ live “ red, black “ African “ English “ French “ Dantzic “ Canada “ Sardinia “ Spanish Pine, white “ pitch “ yellow “ Georgia Poon Poplar Spruce, Canada “ black Sycamore Tamarack Teak V. . . . Walnut Willow White wood 170 250 240 118 162 295 112 162 202 150 160 135 207 105 88 146 142 105 125 !37 130 200 184 112 125 87 125 ICO 165 1 12 87 Il6 Stones, Bricks, etc. Brick, common, mean “ pressed, “ . ..* u English, stock “ “ fine Brick arch Cement, mean “ “ Portland. j “ 11 Sheppey “ “ hydraulic, Portland. 11 “ Roman u Puzzuolana “ Portland, 1 year “ Roman, “ Concrete, Eng., fire-brick beam, ) cement j 20 40 11 . 8 14 15 15 10.2 37-5 5 5 2 2.5 3-i * This was with a tensile strength of 27000 lbs. t With 840 lbs. the deflection was 1 inch, and the elasticity of the metal destroyed. 800 STRENGTH OF MATERIALS. TRANSVERSE. Stones, Bricks, etc. Concrete, Eng. , fire-brick, sand 3 , 1 lime 1 j '7 “ Eng. , clay and chalk 5.4 Flagging, blue, New York 31*25 Freestone, Conn 13 “ Dorchester 10.8 New Jersey, mean 19 “ New York 24 “ Eng., Craigleth 10.7 “ “ Darby, Victoria. . . 1.3 11 “ Park Spring 4.3 Glass, flooring 42.5 Granite, blue, coarse 18 “ Quincy 26 “ mean 25 “ Eng., Cornish 22 Limestone “ English 11 Marble Stones, Bricks, etc. Marble, Adelaide “ Italian... Mortar, lime, 60 days “ 1 lime, 1 sand Oolite, English, Portland Paving, Scotch, Caithness “ Ireland, Valentia “ Welsh “ English, Yorkshire, blue. . “ “ Arbroath Slate “ Bangor “ English, Llangollen Stones, English, Bath “ “ Kentish, Rag “ “ Yorkshire, landing “ Caen 4*5 2*5 2 *•75 1*25 21.2 68 68.5 57 10.4 1 7 81 90 43 5*2 35*8 22.5 12.5 Elastic Transverse Strength of Woods, compared with their Breaking Weight , is as follows : Ash . Elm... Larch.. Per Cent. .. 29 Norway Spruce.. Oak, Dantzic . . . . ‘ k English . . . . Per Cent. .. 30 .. 36 Red Pine.. . . , Riga Fir Teak .. 38 Pitch Pine Yellow Pine, Increase in. Strength, of several Woods "by Seasoning. Per Cent. Ash 44.7 ] Beech ..61.9 | Elm 12.3 | Oak 26.1 | White pine. .. .9 Concretes, Cements, etc. Materials. Breaking Weight. Materials. Breaking Weight. concretes (English). Fire-brick beam, Portl’d cement Lbs. 3. 1 bricks (English). Best stock Lbs. 11 . 8 I A “ sand 3 parts, lime 1 part cements (English). Blue clay and chalk . 7 Fire-brick New brick IO. 7 5*4 37*5 10. 2 5 Old brick AW. / 0. 1 Portland J Sheppey Stock-brick, well burned 5 * 8 “ inferior, burned. . . 2*5 Transverse Strength of Various Figures of* Cast Iron. Reduced to Uniform Measure of Sectional Area of One Inch Square and One Foot in Length. Fixed at one End ; Weight suspended from the other. Form of Bar or Beam. Breaking Weight. Form of Bar or Beam. Breaking Weight. Lbs. Lbs. Square Square, diagonal vertical. . . Cylinder. 673 568 573 Rectangular prism. 2 X .5 ins. in depth. . . 3 X .33 “ in depth. . . 4X.25 “ in depth... Equilateral triangle, an edge up Equilateral triangle, an edge down 1456 2392 2652 560 958 O Hollow cylinder; greater diameter twice that of lesser 794 T 2 ins. in depth X 2 X ) .268 inch in width. . . j A 2 ins. in depth X 2 X ) .268 inch in width . . ) 2068 555 STRENGTH OF MATERIALS. — TRANSVERSE. 801 Solid, and. Hollow Cylinders of* various Materials. One Foot in Length. Fixed at one End ; Weight suspended from the other. Materials. [ External Diam. Internal Diam. Breaking Weight. Materials. External Diam. ; Internal | Diam. Breaking Weight. WOODS. Ash Ins, 2 Inch. Lbs. 685 604 772 75 610 METAL. Cast iron, cold) Ins. Ins. Lbs. 2 1 blast ) 3 — 12 COO Fir* 2 STONEWARE. White pine. . u a 1 2 1 1 Rolled pipe of ) fine clay ) 2.87 1.928 190 * An inch-square batten, from same plank as this specimen, broke at 139 lbs. Formulas for Transverse Stress of Rectangular Bars, Beams, Cylinders, etc. Fixed at One End. Loaded at the Other. Bars, Beams, etc. IW Vd 2 = S; Sbd 2 l = W IW Sbd 2 IW SO 2 jiw /lT a “ Bars, Beams, etc. n ~\y and Cylinder 3/ — b and d. Fixed at Both Ends. Loaded in Middle . IW • „ . 6 S b d* _ 6 Sbd 2 6 bd 2 — S; zW: W — l: \ Sb IW 6 Sd 2 ~~ / - = d ; and Cylinder 3 = b and d. V 6 S b ’ J V 6 S Fixed at Both Ends. Loaded at any Other Point than in Middle . Bars, Beams, etc. 2 m n W 2,1b d mnW ■S' 3 lbd 2 S 2 mnW d 2 ~ ’ V }2mnW , ....... „ /2mnW , , V7sTF = d; and Cylinder 3^^- = 6 and d '2 mn W ’ 3$b 2 m n W 3 S Id- Bars, Beams, etc. 3 s i 0 ' * v 3 SI Supported at Both Ends. Loaded in Middle, t W _ o 4S6 4S&rf 2 ffd 2 ~ S] Z = W; 'IW w \ / l ^ -z=d: and Cylinder 3 b and d. / 4 S b ’ V 4 s Z W 4 S d 2 Supported at Both Ends. Loaded at any Other Point than in Middle . ™ . mnW Bars, Beams, etc. ^ — = S ; Imn W Slbd 2 = W; mnW m n W S b d 2 ~ 1 Sld 2 ~ V mnW , , _ , . , „ A -g IF = d; and Cylinder 'm w W S « - — 6 and d. In Square Beams, etc., for b and d put = d. In Cylinders, for b d 2 put d 3 as above. When weight is uniformly distributed , same formulas will apply , W repre- senting only half required or given weight. S representing stress in a Bar , fleam, or Cylinder , one foot in length, and one inch square, side , or in diameter; and W weight , m Z&s. ; 6 breadth, and d depth , in ins.; ‘ length, m distance of weight from one end, and nfrom the other, all in feet. B rick-work. A brick arch, having a rise of 2 feet, and a span of 15 feet 9 ins., and 2 feet in width, with a depth at its crown of 4 ins., bore 358400 lbs. laid. along its centre. 0 802 STRENGTH OF MATERIALS. TRANSVERSE. Coefficient or Factor of Safety. Coefficient or factor of safety of different materials must be taken in view of importance of structure, or instrument, probable or required period of du- ration of it, and if it is to bear a quiescent, vibratory, gradual, or percussive stress, and to meet these varied conditions, it will range from .125 to .3 of the maximum or ultimate strength here given or ascertained. To Compute Transverse Strength, of a Rectangular Bar or Beam. When a Bar or Beam, is Fixed at One End , and Loaded at the Other. Rule. — Multiply Coefficient of material in preceding Tables,, or, as may be ascertained, by breadth and square of depth in ins., and divide product by length in feet. Note. —When a beam, etc., is loaded uniformly throughout its length, result must be doubled. Example.— What weight will a cast-iron bar, 2 ins. square and projecting 30 ins. in length, bear without permanent injury? Assume strength of material at 660, and its elasticity at one fifth or .2 of its strength. Then 660 X - 2 X = lbs . 2.5 2.5 If Dimensions of a Beam or Bar are Required to Support a Given Weight at its End. Rule. — D ivide product of weight and length in feet by Coeffi- cient of material, and quotient will give product of breadth and square of depth. Example. — What is the depth of a wrought- iron beam, 2 ins. broad, necessary to support 576 lbs. suspended at 30 ins. from fixed end ? Assume strength of iron at 150. Then = 9.6, and /— = 2.19 ins. depth. 150 V 2 When a Beam or Bar is Fixed at Both Ends, and Loaded in the Middle. Rule. — Multiply Coefficient of material by 6 times breadth and square of depth in ins., and divide product by length in feet. Note.— When beam is loaded uniformly throughout its length, result must be doubled. Example.— What weight will a bar of cast iron, 2 ins. square and 5 feet in length, support in middle, without permanent injury? Assume strength of material as in a previous case at .2 of 660. Then 660 X-2X2X6X2 2 6336 = ~ - - — 1267.2 lbs. If Dimensions of a Beam or Bar are Required to Support a Given Weight in Middle, between Fixed Ends. Rule.— Divide product of weight and length in feet by 6 times Coefficient of material, and quotient will give prod- uct of breadth and square of depth. Example.— What dimensions will a square cast-iron bar, 5 feet in length, require to support without permanent injury a stress of 2160 lbs. ? Assume strength of material at .2 of 660 or 132, as preceding. Then 2I ^°— — =. 10 — 13.64, which , divided by 2 for assumed breadth — 6.82, 132 X 6 792 and -\J 6. 82 = 2.61 ins. depth. When Breadth or Depth is Required. Rule.— Divide product obtained by preceding rules by square of depth, and quotient is breadth ; or by breadth, and square root of quotient is depth. Example. — If 128 is the product, and depth is 8; then 128 -r- 8 2 = 2, breadth. Also, 128 2 = 64, and V 6 4 = 8 ? depth. STRENGTH OF MATERIALS. TRANSVERSE. 803 When Weight is not in Middle between Ends. Rule:.— -Multiply Coefficient of material by 3 times length in feet, and breadth and square of depth in ins., and divide product by twice product of distances of weight, or stress from either end. Example.— What weight will a cast-iron bar, fixed at both ends, 2 ins. square and 5 feet in length, bear without permanent injury, 2 feet Irom one end? Assume strength of material at .2 of 660 or 132, as preceding. When a Beam or Bar is Supported at Both Ends , and Loaded in Middle . Rule:.— M ultiply Coefficient of material by 4 times breadth and square of depth in ins., and divide product by length in feet. Note. When beam is loaded uniformly throughout its length, result must be doubled. Example.— What weight will a cast-iron bar, 5 feet between the supports, and 2 ins. square, bear in middle, without permanent injury? Assume strength of iron at 132, as preceding. If Dimensions are Required to Support a Given Weight. Rule.— D ivide product of weight and length in feet by 4 times Coefficient of material, and quotient will give product of breadth, and square of depth. When Weight is not in Middle between Supports. Rule.— M ultiply Coef- ficient of material by length in feet, and breadth and square of depth in ins., and divide product by product of distances of weight, or stress from either support. Example.— What weight will a cast-iron bar, 2 ins. square and 5 feet in length, support without permanent injury, at a distance of 2 feet from one end, or support ? Assume strength of iron at 132, as preceding. To Compute Pressure upon Ends or upon Supports. Rule i. — D ivide product of weight and its distance from nearest end or support, by whole length, and quotient will give pressure upon end or sup- port farthest from weight. 2.— Divide product of weight and its distance from farthest end, or sup- port, by whole length, and quotient will give pressure upon end or support nearest weight. Example.— W hat is pressure upon supports in case of preceding example? 880 X 2 — 352 lbs. upon support farthest from the weight ; 880 X - = 528 lbs. upon 5 _ 5 support nearest to weight. When a Bar or Beam , Fixed or Supported at Both Ends , bears Two Weights at Unequal Distances from Ends. m W l w A T . n w . V W , „ r , 1 = pressure at w end , and — — | — - — — pressure at W end . L L L L m and n representing distances of greatest and least weights from their nearest end , W and w greatest and least weights , L whole length , l distance from least weight to farthest end , and V distance of greatest weight from, farthest end. Illustration. — A beam 10 feet in length, having both ends fixed in a wall, bears two weights— viz., one of 1000 lbs., at 4 feet from one of its ends, and the other of 2000 lbs., at 4 feet from the other end; what is pressure upon each end? Then 132 X 2 X 4 X 2 2 = 4224 -r- 5 — 844. 8 lbs. Then 132 x 5 X 2 X 2 2 2 X (5 — 2) 5^ = 880 lbs. 6 ... 4 X 1000 . 6 X 2000 _ 1400 lbs. at w ; - = 1600 lbs. at W. ^ 10 10 804 STRENGTH OF MATERIALS. TRANSVERSE. When Plane of Bar or Beam Projects Obliquely Upward or Downward. When Fixed at One End and Loaded at the Other. Rule. — Multiply Co- efficient of material by breadth and square of depth in ins., and divide product by product of length in feet and cosine of angle of elevation or depression. Note.— When beam is loaded uniformly along its length, result must be doubled. Example. — What is weight an ash beam, 5 feet in length, 3 ins. square, and pro- jecting upward at an angle of 7 0 15', will bear without permanent injury? Assume breaking weight of ash at 160, and its elasticity at .25 of its strength, and cosine of 7 0 15' = .992. Then 160 X -25 X 3 X 3 2 5 X -99 2 = 1080 ■= 217.74 lbs. To Compute Transverse Strength, of an Eqnilateral Tri- angle or T Beam. Rule. — Proceed as for a rectangular beam, taking following proportions of Coefficient of material : Fixed at One or Both Ends. Supported at Both Ends. Equilateral triangle, edge up. . . Equilateral triangle, edge down X beam, flange up Equilateral triangle, edge up. . . Equilateral triangle, edge down X beam, flange up bxd 2 X. 2 C b X d 2 X -34 “ b X d 2 X .42 “ b X d 2 X .34 “ bXd 2 X- 2 “ b X d 2 X .42 “ To Compute Transverse Strength of a Solid. Cylinder. Rule. — Proceed as for a rectangular beam, and take .6 of Coefficient or of product. A mean of 18 results with cold blast gun-metal, gave a coefficient for 740 lbs. When Fixed at One End , and Loaded at the Other. Rule. — Multiply weight to be supported in lbs. by length of cylinder in feet ; divide product by .6 of Coefficient of material, and cube root of quotient will give diameter. Note.— When cylinder is loaded uniformly throughout its length, cube root of half quotient will give diameter. Example.— What should be diameter of a cast-iron cylindrical beam of gun-metal, i 8 ins. in length, to break at 15 000 lbs. ? t 15 000 X t > „ /10000 - = 3 / = 2 . V 444 61 ins. .6 X 74° When Fixed at Both Ends , and Loaded in Middle. Rule.-— M ultiply weight to be supported in lbs. by length of cylinder between supports in feet; divide product by .6 of Coefficient of material, and cube root of one sixth of quotient will give diameter. Note.— W hen cylinder is loaded uniformly along its length, cube root of half the quotient will give diameter. Example. — What is the diameter of a cast-iron cylinder of gun-metal, 2 feet be- tween supports, that will break at 35 964 lbs. ? 3 j^ 64>C2 = i6 3/^ = 3 ins. .6x74° V 6 Mean results of cylinder and square bars gave 444 and 740 lbs. Hence, strength of a cylinder compared to a square is as 444 to 740 or .6 to 1. Then 4 X 3 3 X 444 = 47 952 lbs. To Compute Diameter of* a. Solid. Cylinder to Support a given Weight. When Supported at Both Ends , and Loaded in Middle. Rule. — Multiply weight to be supported in lbs. by length of cylinder between supports in feet; divide product by .6 of Coefficient of material, and cube root of one fourth of quotient will give diameter. STRENGTH OF MATERIALS. TRANSVERSE. 805 Note.— When cylinder is loaded uniformly along its length, cube root of half the quotient will give diameter. Example.— What is diameter of a cast-iron gun-metal cylinder, 1 foot between its supports, that will break at 48000 lbs. ? 48 000 X 1 0.0 A 08 ,, . 2 = 108, and 3 / — — 3.61 ms. .6X740 V 4 Ttecta.iagu.lar, ( D . K. Clark.) (1) Loaded at Middle, —j— — W. (2) Loaded at One End. —j— — W. Cylindrical. (3) Loaded at Middle. 5 ' 5 ^ ■ — = W. (4) Loaded at One End. I ’ 375 ^ --- =: W. W representing ultimate stress in tons. Above Coefficients are for iron of a tensile strength of 7 tons per sq. inch. (1) (2) (3) ( 4 ) (1) (2) ( 3 ) ( 4 ) 9.2 2-3 6.3 1.6 For 12 tons put 13 “ 13-8 3-4 9.4 2.4 10.4 2.6 71 1.8 14-5 3-6 10.2 2.6 ii -5 2.9 7-9 2 14 16 4 11 2.8 12.7 3-2 8.6 2.2 15 “ 17.2 4-3 11. 8 3 To Compute Destructive W eiglit, or Loads tliat may "be borne "by Wrought-iron Rolled. Beams and Grirders, or Riveted Tubes of* various Figures and Sections. Supported at Both Ends. Load applied in Middle. When Section of Beam or Girder is that of any of the Figures in follow- ing Table. Rule. — Divide product of area of section, depth, and Coefficient for girder, etc., from following Table, by length between supports in feet, and quotient will give destructive weight in lbs. Note. — The Coefficients given are based upon experiments with English iron. Solid Beams. Illustration. — What load will destroy a wro ight-iron grooved beam of following dimensions, 10 feet in length between supports, and loaded in its middle? Flanges, 5.7 X .64 inch; Web, .6 inch; Depth, n.75 ins. ; Area, 13.34 sq. ins. Assume Coefficient 4638 as for like case (12) in following table, page 806. 13.34X11.75X4638 726821 = = 72 682. i lbs. 10 10 Ultimate stress for such a beam by experiment was estimated at 97997 lbs. Formulas of Various Authors give following Results: D. K. Clark. d (4 a + 1. 1555 a') .6 l = W. a representing area of section of lower flange , a area of section of web, less one flange, d depth of beam., less average depth of one flange, all in ins., I length in feet , and W ultimate destructive weight in tons. This formula is based upon the assumption that the beam has lateral support. 1T -75 — -6 (4 X 5-7 X .6 -f- 1. 155 X ix.75 — .6 X .6) _ 238.69 .6 X 10 : 39.78, which X 2240 = 88 107 lbs. Molesworth. - - b -- _ W. C = 7616 lbs . , and for b d 2 put b d' — 2 b' d' 2 . b and d representing exterior and b' and d' interior dimensions, and l length in ins. ’)] = 786.6 — 558. - = 57 805.4 lbs. 5.7 X ii- 75 2 — [5-7 — *6 X 11.75 — (.64 X 2 2 )] = 786.6 — 558.9 = 227.7. Then 4 X 7616 x 227.7 ^ 693665 _ Fairbairn’s formula would give a result less than half of the first, and Hodgkin- son’s alike to that of Molesworth. 3 v 806 STRENGTH OF MATERIALS. TRANSVERSE. WROUGHT IRON. Transverse Strength, of Wrought-iron Rolled Beams and Girders. {Barlow, Fairbairn , Hughes , KirTcaldy , dc.) Reduced to Uniform Measure of One Foot in Length. Supported at Both Ends ; Stress or Weight applied in Middle. Section. Area (A). Destructive Weieht. l w Flanges. Web. Depth {d). Distance. For Length of Distance. One Foot(/). , A d u Ins. Ins. Ins. Feet. Ins. Sq. Ins. Lbs. Lbs. (W). — 1 1 1 1 2 500 2500 2 500 2 2 2 9 4 6 600 18 150 2 266 — i -5 3 2 9 4-5 10080 27 720 2053 — 1 3 3 3 7050 21 150 2 350 — . 1 1 5 •78 474 2370 2370 3-5 X .6 .8 3-5 2 7 5-65 20 160 52480 2654 25 Xi 4 X .38 } -325 7 j'v > f i - > . : 2 9 5-9 - : 44000 121 000 2930 2.6 X1.25 .85 5 4 6 7-44 19000 85 500 2 298 3 X -49 •5 7-°7 10 5.87 24 200 242 000 5830 4.6 X .8 •5 9- 8 5 20 n -5 38 080 761 600 6724 5-7 X .64 .6 ii-75 10 x 3-34 72 688 726 880 4 638 2.85X .38 • 3 i 2.5 4 x -75 3150 12 600 2 880 7 X -5 4 X -5 } - 38 16.5 22 6 18.9 49 280 1 108 800 3 556 4 - 5 X -375 2X2X.3125 } .38 14.25 1 6 5 10.5 47 000 775 500 5 183 4- 5X28 4 - 5 X 3 } - 25 7 7 6-35 24380 170660 3840 3-9 3-9 }■* 6 7 6 2.62 9976 74 820 4766 i5-5 I 5-5 } - 53 24 30 41.4 128 885 3 866 550 3 8 9 6 ' 2 4 24 •75 •75 } 35-75 45 87.38 257 080 11 568600 3 703 * - .131 {12.4 (12.138 ■h 5-05 17885 178850 2 856 j - • x 43 I 15 X 9-75 I 10 5-56 26 250 262 500 3 x 47 $ J •75 5-2 5 7.72 102 480 512400 12 760. Steel. These results are very conclusive of the correctness of above formula, as will be seen in cases given, and they are deduced from beams and girders varying from i to 45 feet in length ; hence, when length qf a beam or girder of any of the sections given is less, relative breaking weight may be in- creased, in consequence of increased stability of beam or girder. For full experiments on Tubes and Tubular Girders, etc., see Rep. of Commas on Railway Structures , London , 1849. Tensile strength of iron assumed at 45 000 lbs. per sq. inch. STRENGTH OF MATERIALS. TRANSVERSE. 807 Elements of Rolled Wrought - iron Beams and Channel Bars. With Safe Load Uniformly Distributed. F or Length of One Foot. Tlie ISTew Jersey Steel and Iroix Co., Trenton, IN'. J. ( Beams Supported Sidewise.) Width. Designation. inch. X Beams. Extra Light. Light. Heavy. Light. Heavy. Light. Heavy. go lbs. 120 “ 55 “ Light. Heavy. Light. Heavy. Extra Heavy. Extra Light. Light. Heavy. Light. Heavy. Light. Heavy. Deck Beams. Channels. Extra Light. Light. Heavy. Extra Light. Light. Extra Light. Light. % “ Heavy. Light. Heavy. Light. Heavy. Weight Web. Flange. Web. Flange. Total. Foot. Inch. Ins. Sq. Ins. Sq. Ins. Sq. Ins. Lbs. •1875 2 •75 1.02 1.77 6 •25 2-75 1 1.91 2.91 10 •3125 3 1.25 2.41 3.66 12.3 •25 2-75 1.2 1.79 2.99 10 •3125 3 1.56 2.34 3-9 13-3 •25 3 i -5 2.51 4.01 13-3 •3 3-5 1.8 3 -n 4.91 16.7 •5 5 3 5-7 8.7 30 .625 5-25 3-75 8.09 11.84 40 •3 3-75 2. 1 3-4 5-5 18.3 • 3 4 2.4 3-97 6-37 21.7 •375 4-5 2.96 5-07 8.03 26.7 .3 4 2.7 4-3 7 23- 3 •375 4-5 3 - 3 8 5.12 8-5 28.3 •57 4-5 5 -i 3 7.2 12.33 41.7 •3125 4-5 3.28 5.62 8.9 30 •375 4-5 3-94 6.5 10.44 35 •47 5 4-93 8-43 13-36 45 •47 4.8 5-75 6.58 12.33 41.7 .6, 5-5 7-39 9-38 16.77 56.7 • 5 5 7-59 7-45 15.04 50 .6 5-75 9.07 10.95 20.02 66.7 • 3 1 4-5 2.17 3.18 5-35 18.3 •38 4-5 3-04 3-25 6.29 21.7 .2 1.5 .6 .85 i -45 5 .2 i -5 .8 .85 1.65 5-5 .2 1.625 1 .92 1.92 6-3 .18 1-875 1.08 1.17 2.25 7-5 .28 2.25 1.68 1.52 3-2 11 •4 2-5 2.4 1.92 4-32 15 .2 2 1.4 1. 14 2-54 8-5 •25 2.5 « i -75 1.85 3-6 12 .2 2.2 1.6 i -7 3-3 11 .26 2-5 2.08 2.4 4.48 i 5 •33 2-5 2.97 2. 11 5.08 16.7 •43 3-125 3-87 3 -i 5 7.02 23-3 •375 2-75 3-94 2.06 6 20 •33 3 4.04 2.96 7 23-3 .68 4 8-33 5 - 77 . 14. 1 46.7 •5 4 7-5 4-5 12 40 •75 1 4-75 1125 7.6 18.85 63-3 Load. 18 000 30 100 36 800 38 700 49 100 62 600 7 6 800 132000 172 000 1 01 000 135000 168 000 167 000 199 000 268 000 250000 286 000 360000 377 000 51 1 000 551000 748 000 63 500 91 800 10 500 15700 22 800 33 680 45 7 00 583°° 39 5°o 62 000 65 800 88950 104 000 146 000 134 750 200 100 381 000 401 000 625000 The loads given in the table are such as will effect a maximum strain upon the metal of 12000 lbs. per sq. inch. For permanent stress, absolutely free from vibration, a greater strain would be allowable, and, contrariwise, if the stress is mainly that of a live load the loads here given should be re- A difference of 25 per cent, in either direction should be made, according to the character of the load to be supported or stress to be borne. Steel beams have greater estimated strength than iron, but their stiffness is not materially greater. 808 STRENGTH OF MATERIALS. TRANSVERSE. Elastic Transverse Strength of W rough t-iron Bars is about 45 per cent, of their transverse strength, and of Plates 55 per cent., or 48 per cent, of their tensile strength ; of solid rolled beams, 50 per cent. ; and of double-headed rails, 46 per cent, of their transverse strength ; of Fagersta Steel, 56 per cent, of its transverse strength ; of double-headed Steel rails, 47 per cent. ; of Bes- semer Steel, 37.5 to 48 per cent. ; of Steel flanged, 68 per cent. ; and of Wrought-iron Steel flanged, 62 per cent, of its transverse strength. Transverse strength of Solid Cast-iron Beams or Girders is about 50 per cent, of ultimate strength; of double-headed or flanged rails, 46 per cent. ; and of single- flanged rails, 62 per cent, of its tensile strength. Note. — The actual breaking weight of a 10.5 ins. beam of New Jersey Steel and Iron Co., weight 35 lbs. per foot, for a length of span of 20 feet, is 60000 lbs. Channel and Deck 13 earns and Strut Bars. With Safe Load Uniformly Distributed for Length of One Foot. ( Beam supported Sidewise.) Depth. Designation. Wi Web. dth. Flange. Area. Section. Weight per Foot. Load. Strength Sidewise. as Strut. Edgewise. Ins. Channel. Inch. Ins. Sq. Ins. Lbs. Lbs. Lbs. Lbs. 3 Extra Light u tt .2 i -5 i -45 5 10500 51 341 4 .2 i -5 1.65 5-5 15 700 49 597 5 it It .2 1.625 1.92 6-33 22 800 57 930 6 u u .18 i -875 2.25 7-5 33680 77 1403 6 Light .28 2.25 3-2 11 45700 IOI 1343 6 Heavy •4 2.5 4-32 i 5 583°° 123 1257 7 Extra Light li it .2 2 2.54 8-33 39 5oo 82 1700 7 •25 2-5 3-6 12 62 000 136 1883 8 it it .2 2.2 3-3 11 65 800 109 2493 8 Light U .26 2-5 4.48 15 88 950 142 2480 9 •33 2-5 5.08 16-33 104 000 124 2892 9 Heavy •43 3-125 7.02 23-33 146000 190 2925 10.5 Light •375 2-75 6 20 134 750 160 3685 12.25 .42 3 8.62 28.33 238 000 172 5275 12.25 Heavy .68 4 14. 1 46.66 381 000 3 i 7 5170 15 Light •5 4 12 40 401 000 3 01 7833 15 Heavy •75 4-75 18.85 63-33 625 000 428 7762 7 ) Deck ( • 3 i 4-5 5-35 21.66 63 500 35 i 36 8 j Beams. ( Strut Bars. .38 4-5 6.29 18.38 91 800 547 37 5 Light, Single Heavy, u — — i -55 5-33 9 100 44 457 5 — — 2.15 7-33 11 900 48 433 Operation! of Table. To Compute Depth of a Beam, to Support a Uniformly Distri.bn.ted. Load. Rule. — M ultiply load in lbs. by length of span in feet, and take from table the beam, the load of which is nearest to and in excess of the product thus obtained. Example.— What should be depth of a beam to sustain with safety a uniformly distributed load of 30000 lbs., over a span of 15 feet? 30000 X 15 = 450000, which is load for a heavy beam 12.25 ins. in depth. Weight of beam should be added to load. Inversely .— If the load is required, divide load in table by span of beam in feet, and subtract weight of beam. To Compute Deflection of Like Beams. Rule. — D ivide square of span in feet by 70 times depth of beam in ins. Example.— Assume beam as preceding. 15 2 225 , STRENGTH OF MATERIALS. TRANSVERSE. 8O9 Comparative Strength, and Deflection of Cast-iron Description of Beam. Beam of equal flanges “ with only bottom flange. “ with flanges as i to 2. . . with flanges as 1 to 4. . . Comp. Strength. •58 .72 .63 •73 Description of Beam. Comp. Strength. Beam with flanges as 1 to 4. 5 . . . .78 “ with flanges as 1 105.5... .82 “ with flanges as 1 to 6 1 with flanges as 1 to 6.73. . .92 Dimensions and Proportions of Wronght-iron Flanged 13 earns. (D. K. Clark.) Depth. 25 Ins 3 3 3' 4 4 4- 75 5 5 5- 5 6 6.25 6.25 6.25 7 7 7 7 8 8 8 8 8 9- 25 9-5 10 10 14 14 16 Breadth of Flanges. Thickness. Web. Flanges. 3 1.625 3 4-5 2.25 3-25 2.25 2.25 3- 625 3.625 2-375 2- 5 4 5 5-125 3- 75 4- 5 4-5 4-75 4- 75 5 6 5- 5 6 5.625 Inch. 1875 5 875 5 5 5 3 i2 5 j7S 375 4375 3125 3 i2 5 3 i2 5 281 3 i2 5 3i 2 5 3 i2 5 3i 2 5 375 375 375 4375 4375 375 4375 4375 75 5 6 25 5625 5625 5625 75 Weight per Lineal Foot. Inch. ,2187 25 187 125 75 3 I2 5 4375 4375 5625 4375 375 (.062 375 4375 4375 5 4375 375 5 5 6 25 5625 6875 5625 5625 625 8i75 9375 875 8x75 8175 Lbs. 5-5 5-5 8 11 18 12.5 14 14 19 *9 15 15 21 29 29 24 30 32 32 36 42 56 60 60 62 Ultimate Strength. Loaded in Middle. Lbs. 2 800 5600 2 490 5 49° 8510 6 940 13 440 19 270 11 880 23 830 13 440 13000 17470 14790 17 020 23300 25980 20830 21 280 34 5oo 44 800 47 040 41 560 59 3 6 o 56 000 58 240 76 160 100 800 136 640 150020 1 52 260 188 160 Safe Stress Uniformly Distributed. Lbs. 910 I 860 830 1 830 2 830 2 310 4480 642O 3960 794O 4 440 4 330 5 820 4 930 5670 7760 8660 6940 7 090 11 500 14 930 15 680 13850 19 750 18660 19 410 25390 33 600 45 530 50000 50 750 62 720 ■Wrought-iron Rectangular Girders or Tubes. (Sit'd.) Supported at Both Ends. Loaded in Middle. AdG — w. A representing area of section in sq. ins., d depth in ins., I length be- tween supports in feet, and W destructive weight in lbs. Ti lustration —What is the destructive weight of a rectangular girder, 35.75 ins^ in depth by 24 in breadth, metal .75 inch thick, and length between supports 45 feet? Assume C or coefficient = 37 coo, as per case (r 7 ) in preceding table, page 806, and area = 87. 375 ins.. Then 8 7-375 X 35-375 X 3?oo _ 1 1 557 5 2 3 _ ^ 8 45 45 W l , By experiment it was 257 080 lbs. By Inversion — A, and A c — , result of 259373 lbs., and Molesworth’s Hodgkinson’s formula would give 303 907 lbs. 8lO STRENGTH OF MATERIALS. TRANSVERSE. I 2 W Unequally Loaded Beams, etc. m n — w- l representing length between supports , and m and n distances from points of support, all in like denomination , and W and w destructive and safe weights also in like denomination. To Compute Destructive Weight and. Area of Bottom Blate. AdC = W; W l : = A; and W m n l “ " ’ c d ~~ 25 c d l = A ‘ A re P reseniin 9 Mea of plate in sq. ins., d and l depth and length, m and n distances of load at other points than in middle , all in feet, and W weight in lbs. Note. — Sufficient metal should be provided in sides to resist transverse and shearing stress, and in upper flange to resist crushing. Illustration.— What area of wrought iron is necessary in bottom plate of a rec- tangular tubular girder, 3 feet in depth, supported at both ends, and loaded in middle with 130000 lbs. ? C, ascertained by experiment for destructive stress, 180000 lbs., and area 7. x sq.ms. 130000 X 30 —5 3 — — 7. 22 sq. ins. 180000 X 3 Wronght-iron Cylindrical Beams or Tubes. — W. Illustration. — What is destructive weight of a cylindrical tube, 12.4 ins. in diameter, .131 inch in thickness, and 10 feet between its supports? Area of metal = 5.05 sq. ins., and C = 2856, as in the 19th case of table, page 806. Then 5 - ° 5Xl2 - 4X 2856 = i 7 = 17 884.2 lbs. D. K. Clark. ^ ^ = W. d representing diameter , t thickness of metal, and l length, all in ins., S tensile strength of metal per sq. inch, and W weight, both in lbs. 3.14 X 12. 4 2 X .131 X 45000 2 846 250 S = 45 000 lbs. - = 23 718.7 lbs. 10 X 12 Molesworth’s formula gives a result of 23286.1 lbs. "Wr onght-ir on Elliptical Beams or Tribes. A d'C l - = W. Illustration. — Assume diameter of tube 9.75 and 15 ins., metal .143 inch n thickness, and distance between supports 10 feet. A = 5.56 sq. ins. C = 3147, as per case (20) in preceding table, page 806. Then 5. 56 X 15 X 3M7 _ 262459.8 = 26 245.9 lbs. D. K. Clark. 1. 57 (&2 + d 2 )<*3 — L- = M. Q *7854 ( r * — r'*)' = M. ^ - = M - “f! * 11 r 2 = M. t representing rcidius , £ transverse and c conjugate diameters, and s side. To Compute Common Centre of Gravity and Vertical Distance between Centres of Crushing and Tensile Stress of a. Grirder or Beam. Rule. — Multiply surface of section of each part, or figure composing whole, by distance of its centre from centre of one of the two extreme parts or figures, as . ; divide sum of their products by sum of surfaces of sec- tion, and result will give distance of common centre of gravity from centres of each extreme part or figure. Example.— Take annexed figure. C 2.5 X 1 X o =2.5 Xo = .0 Above .325 X .325x3.31=1.076 .38 X 4 X (^ + 5-62 + 1) = 1.52 X 6.31 = 9-59 1 4.345 10.667 Dividing 10.667 by 4.345 = 2. 455 = distance of common centre from centre of upper part. 1.52 Xo 1=1.52 Xo = .0 Below •3 2 5 X 5*62 X ( 5 | ? + v) = 1-826 X 3 =5-478 2.5 X (| + 5 -6 2 + '-f )= 2 -5 X 6 . 3 i= 25 - v 5.846 21.253 Dividing 21.225 by 5.846 = 3-631 = distance of common centre from centre of lower part. Hence, 3.631 + — = 3.821 = distance of common centre from bottom , and 3.631 + 2.652 = 6.283 =s distance between centres of gravity. 820 STRENGTH OF MATERIALS. TRANSVERSE. To Compute Neutral Axis of a Beam of Unsymmetrical Section.— Figs. 3 , 4 , 5 , 6, 7 , 8, and 9. (D. K. Clark.) Operation.— Divide section as reduced into its simple elements and assume a datum-line from which moments of elements are to be computed. Multiply area of each element by distance of its own centre of gravity from datum-line, to ascertain its moment. Divide sum of these moments bv to- tal reduced area ; and quotient is distance of centre of gravity of reduced section, or of neutral axis of whole section, from datum-line. Illustration.— F ig. 8 annexed is i 2 ins. deep, i 2 ins. wide, and i inch thick. Extend web, c d, to the lower surface at d' and d", leaving 5.5 ins! of web, a d' and d" b , on each side. Reduce this width in the ratio of 1.73 to 1, or to (5.5-1- 1. 73 =) 3. 2 ins., and set off d' a' and d" b' each equal to 3. 2 ms. Then reduced flange, a' b', is (3. 2 x 2 = 6. 4 -f 1 =) 7-4 ins - wide, and reduced section consists of two rectangles a ' b' and c d. Assume any datum-line, as ef at upper end of sec- tion, and bisect depths of rectangles, or take intersections of their diagonals at g and o, for their centres of gravity. Distances of these from datum-line are 5.5 and 11.5 ins. respectively, and areas of the 1 rectangles are nXi^nsq. ins., and 7.4 x 1 = 7.4 sq. ins. Then, cd—n x 5.5=60.5 a ' ~ 7-4 X 11.5= 85.1 18.4 145.6 = 7.91 ins . Showing that centre of gravity of reduced section, being neutral axis of whole section, is 7.91 ins. below upper edge, in line it. Centre of gravity of entire section at . , it may be added, is 8.65 ins. below upper edge, or .74 inch lower than that of reduced section. Neutral axes of other sections, Figs. 3 to 7, found by same process, are marked on the figures. Section of a flange rail, No. 7, which is very various in breadth, may be treated in two ways: either by preparatorily averaging projections of head and flange into rectangular forms; or, by taking it as it is, and dividing it into a con- siderable number of strips parallel to base, for each of which the moment, with re- STlftPit to ASRIIITIPH Hntnm.Hno 1 C? in ooonrfoinA/1 F i ^ a • u - e-' / f -9 d 1 u -~rr TT~ f 0 a' d' d" b' b paicwiei tu u u&u, iui eauu ui wiiicTi ine moment, witn re- spect to assumed datum-line, is to be ascertained. First mode of treatment is ap- proximate; second is more nearly exact. To Compute Ultimate Strength of Homogeneous Beams of Unsymmetrical Section. Operation. — Resuming section, Fig. 9, for which neutral axis has been ascertained, To Compute Tensile Resistance , Divide portion below neutral axis i 1, Fig. 9, with reduced width of flange, a' b\ into parallel strips, say .5 inch deep, as shown, and multiply area of eacli strip by its mean distance from neutral axis for proportional quantity of resistance at strip. Divide sum of products, amounting in this case to 31.3, by extreme depth below neutral axis = 4.09 ins., and multiply quotient by 1.73 S (ultimate tensile resist- r TF -"^ T~~1 ance at lower surface). ‘The final product is total tensile of v resistance of section ; or, 31.3 X i • 73 S — — r — == 13- 24 S total tensile resistance. 4 .° 9 S representing ultimate tensile strength of material per sq. inch. Again, multiply area of each strip by square of its mean distance from neu- tral axis, and divide sum of these new products, amounting to 104.64, by sum of first products. The quotient is distance of resultant centre of tensile stress, d', from neutral axis. Or, resultant centre is, 104 ^ 4 = 3. 34 ins. below neutral axis. 3 i -3 This process is that of ascertaining centre of gravity of all the tensile resistances. — vr 1 y: x-t — — i X STRENGTH OF MATERIALS. TRANSVERSE. 821 By a similar process for upper portion in compression, sum of first products is ascertained to be same as for lower portion = 3 i- 3- as ,„SX ^ = 3-34 8, and 3£ -3 X 3-3 4_g ^ S total compressive resistance, wbich is 4 same as total tensile resistance, in conformity with general law of equal- ity of tensile and compressive stress in a section. Sum of products of areas of stress, divided by squares of tlieir distances respec- tively from neutral axis, is 164.9, and resultant centre c, Fig. 9, is -—— = 5.27 ins. above neutral axis. o nm nf distances of centres of stress or of resistance from neutral axis, 3.34 + , 27 = 8 6 ^‘“Ssmnceapart of these centres as represented by central line.c d. Abbreviated Computation. -As upper part of section is a rectangle, its resultant centre =4 of height, or 7.91 X f = 5 - 27 ins. above neutral axis Average resist- ance is half maximum stress, viz., that at upper portion, which is 3.34 S per sq. ^ nC k' ! 7.9IX3-34S O Area of rectangle therefore =17.91 X 1 = 7.91 sq. ins., and — p ~ — 13.21 compressive resistance , as before determined. ^ ^ 4 S cZ Moment of tensile resistance = 13. 21 X 8.61 ins. = 113. 76 S, also = — , or —j~ — W. S representing total resistance of section in lbs., d vertical drawee apart of centres of tension and compression , and l length between supports , all m in . Strength- of Beam Inverted.— When inverted, maximum tensional resistance of beam at its lower surface c, Fig. 8, is 1.73 S. „ t „ ari : nQ nnf i 7 9 1 X t. 73 o g i 0 f a i tensile re- Area of rectangle nc — 7.91 sq. ins., ana - . , , ^ — °-79 0 sistance, or about one half of beam in its normal position. Note.— F or other rule for computation of centre of gravity, see Strengt Loudon, 1870. Metals. Density. ( Least Cast Iron.... 'Greatest. ( Mean Wrought Iron { Neatest! Cast Steel. .. . j Greatest. Bronze (Greatest. 6.9 7-4 7.225 7- 7°4 7.858 7.729 8- 953 7.978 8-953 Compres- TenBile . sion. 1 Sq. Ins. 84529 174 120 144916 40 OOO 127 720 198944 39 1 9 8 5 Sq. Tns. 9000 45 97° 31 829 38027 74 59 2 128 OOO 17 698 56786 Torsion. Trans- verse. Sq. Ins. 8 614 2913 3 643 28 280 1916 1 852 — 2656 | — Sq. Ins. 416 958 680 542 mis, etc. B. Baker , Major Wade. Tensile Hard- to Com- ness. pression. I to 9.4 4-57 1 “ 3.8 33-51 1 “ 4.6 22.34 I “ I 10.45 I “ 1.7 12. 14 1 to 3.1 — ; — 4-57 — 5-94 IT actors of Safety. Girders , Beams, etc., of cast iron should not be subjected to a greater stress than one sixth of their destructive weight, and they should not be subjected to an impulsive stress greater than one eighth. The following are submitted by English Board of Trade, Commission- ers, etc. Structure. Cast Iron. Girders Columns Tanks Machinery I Stress. Factor. j Structure. Stress. Factor. - — 1 Dead 3 to 6 I Wrought Iron. Girders Dead 3 Live 6 • 0 Bridges Mixed 4 Live . ! Shock 4 8 10 Steel, S Bridges Mixed 4 822 STRENGTH OP MATERIALS. TRANSVERSE. Girders, Beams, Lintels, etc. Transverse or Lateral Strength of any Girder, Beam, Breast-summer Lintel, etc is in proportion to product of its breadth and square of its depth, and area of its cross-section. Best form of section for Cast-iron girders or beams, etc., is deduced from experiments of Mr. E. Hodgkinson, and such as have this form of section m T_ are known as Hodgkinson’s. 7 R ^7 e deduc . ed fr0 7 m his experiments directs, that area, of bottom flanqe should be 6 times that of top flange— flanges connected by a thin ver- tical web, sufficiently rigid, however, to give the requisite lateral stiff- ness tapering both upward and downward from the neutral axis - and in order to set aside risk of an imperfect casting, bv any great dispro- portmn between web and flanges, it should be tapered so as to connect with them, with a thickness corresponding to that of flange. As both Cast and Wrought iron resist compression or crushing with a grea er force than extension, it follows that the flange of a girder'or beam of +1 2 • °, f thes ? meta . ls * which is subjected to a crushing strain, according as the girder or beam is supported at both ends , or fixed at one end , should be of less area than the other flange, which is subjected to extension or a ten- siiB stress. When girders are subjected to impulses, and sustain vibrating loads, as in budges, etc., best proportion between top and bottom flange is as i to 4 • as a general rule, they should be as narrow and deep as practicable, and should never be deflected to more than .002 of their length. ^ ub,lc Halls, Churches, and Buildings where weight of people alone t p t r “ vided . f , or > an estimate of 175 lbs. per sq. foot of floor surface is sufficient to provide for weight of flooring and load upon it. In comput- ing other weight to be provided for it should be that which may at any time bear upon any portion of their floors; usual allowance, however, is for a weight of 280 lbs. per sq. foot of floor surface for stores and factories. noJed a to 3X!!’- h as , in baildin S? and bridges, where the structure 'is ex- posed to sudden impulses the load or stress to be sustained should not ex- ceed horn .2 to .16 of breaking weight of material employed ; but when load weight*™ ° r Str6SS C1U,eScent: 11 may be increased to .3 and .25 of breaking An open-web girder or beam, etc., is to be estimated in its resistance on he mad 1 ® P nncl P le as lf had a solid web. In cast metals, allowance is to and flanges ° f Strength due to unec l ual contraction in cooling of web In Cast Iron, the mean resistances to Crushing and Extension are, for American as 4.55 to 1, and for English as 5.6 to 7 to 1 ; and in Wrought Iron 't; American as 1.5 to 1 and for English as ,.2 to 1 ; hence the inass of metal below neutral axis will be greatest in these proportions when stress is intei mediate between ends or supports of girders, etc. „ r ] V ™ t d ™. Girder * or Beams, w hen sawed in two or more pieces, and slips ® Ijctw ® en them, and whole bolted together, are made stifler by the operation, and are rendered less liable to decay. nr^o r rt e ion C of T W rn h ‘A®? are stron S er tba " "'hen cast on a side, in the flange up f * aad 1 iey are stron S est also when cast with bottom Most economical construction of a Girder or Beam, with reference to at- taining greatest strength with least material, is as follows : The outline of STRENGTH OF MATERIALS. TRANSVERSE. 823 top bottom, and sides should be a curve of various forms, according as breadth or deptli throughout is equal, and as girder or beam is loaded only at one end, or in middle, or uniformly throughout. Breaking Weights of Similar Beams are to each other as Squares of their like Linear Dimensions. By Board of Trade regulations in England, iron may be strained to 5 tons per sq. inch in tension and compression, and by regulation of the Ponts et Chaussees, France, 3.81 tons. Rivets .75 and 1 inch in diameter, and set 3 ins. from centre in top of girder, and 4 ins. at bottom. Character of fracture, as to whether it is crystalline or fibrous, depends upon character of blows ; thus, sharp blows will render it crystalline, and slow will not disturb its fibrous structure. For spans exceeding 40 feet, wrought iron is held to be preferable to cast iron. Riveting, when well executed, is not liable to be affected by impact or velocity of load. A Coupled Girder or Beam is one composed of two, fastened together, and set one over the other. Trussed Beams or Grirclers. Wrought and Cast Iron possess different powers of resistance to tension and com- pression- and when a beam is so constructed that these two materials act in uni- son with each other at stress due to load required to be borne , their combination will effect an essential economy of material. In consequence of the difficulty of adjust- ing a tension-rod to the stress required to be borne, it is held to be impracticable to construct a perfect truss beam. Fairbairn declares that it is better for tension of truss-rod to be low than high, which position is fully supported by following elements of the two metals : Wrought Iron has great tensile strength, and, having great ductility, it undergoes much elongation when acted upon by a tensile force. On the contrary, Cast Iron has great crushing strength, and, having but little ductility, it undergoes hut little elongation when acted upon by a tensile stress-, and, when these metals are re- leased from the action of a high tensile stress, the set of one differs widely from that of the other, that of wrought iron being the greatest. Under same increase of temperature, expansion of wrought is considerably great- er than that of cast iron; 1.81* tons per sq. inch is required to produce in wrought iron same extension as in cast iron by 1 ton. Fairbairn, in his experiments upon English metals, deduced that within limits of stress of 13440 lbs. per sq. inch for cast iron, and 30240 lbs. per sq. inch for wrought iron, tensile force applied to wrought iron must be 2.25 times tensile force applied to cast iron, to produce equal elongations. Relative tensile strengths of cast and wrought iron being as 1 to 1.35, and their resistance to extension as 1 to 2.25, therefore, where no initial tension is applied to a truss-rod, cast iron must be ruptured before wrought iron is sensibly extended. Resistance of cast iron in a trussed beam or girder is not wholly that of tensile strength, but it is a combination of both tensile and crushing strengths, or a trans- verse strength; hence, in estimating resistance of a trussed beam or girder, trans- verse strength of it is to be used in connection with tensile strength of truss. Mean transverse strength of a cast-iron bar, one inch square and one foot in length, supported at both ends, stress applied in the middle, without set , is about 900 lbs. ; and as mean tensile strength of wrought iron, also without set , is about 20000 lbs. per sq. inch, ratio between sections of beams and of truss should be in ratio of transverse strength per sq. inch of beam and of tensile strength of truss. Girders under consideration are those alone in which truss is attached to beam at its lower flange, in which case it presents following conditions: * Elongation of cast and wrought iron being 5500 and 10000, hence 10 000 -r- 5500 = 1.81. 824 STRENGTH OF MATERIALS. TRANSVERSE. 1. When truss runs parallel to lower flange. 2. When truv ? nt ™ ,• to lower flange, being depressed below its centre. 3. When beam is archeTunwnvd and truss runs as a chord to curve. arched upward , Consequently in all these cases section of beam is that of an onen one with a cast-iron upper flange and web, and a wrought-iron lower flange increased fnTt^ re sistance over a wholly cast-iron beam in proportion to the increased tensile strength of wrought iron over cast iron for equal sections of metals. strength From various experiments made upon trussed beams, it is shown : 1. That their rigidity far exceeds that of simple beams: in some cases it ?nl°h 8 rnr es h greater ” 2 '- That u when truss resilts ruptur^, uppe“ fiSS of beam T en - by com P r ® sslon ) there is a great gain in strength. 3. ThaUheir strength s gieatly increased by upper flange being made larger than lower one , Th , t arei of mefah ' S Sreater tha “ tUat ofa wrought-iron tubular beam containing' same Comparative Value of Wrought-iron Bars, Hollow Grirders, or Tubes ol 'Various Figures {English). Circular tubes, riveted Flanged beams x 2 Elliptic tubes, riveted *.!!.!. n Rectangular tubes, riveted * * * * * x 5 Circular, uniform thickness Plate beams * Elliptic, uniform thickness!.*.’.*..*.**” jg Rectangular, uniform thickness General Deductions from Experiments of Stephenson , Fairbaim Cubitt Hughes , etc. in “ irn ® h . ows in his experiments that with a stress of about 12 320 lbs ner sn ibs - 0,1 wrought iron ’ the sets and asara A cast-iron beam may be bent to .3 of its breaking weight if load is laid on ufi y ’ 1? Dd ' l6 i - f 1 laid on at once ’ wi]1 P r °duce same effect, if weight of beam compared with weight laid on. Hence, beams of cast ironXuld be S capable of bearing more than 6 times greatest weight which will be laid upon them ^ams of cast or wrought iron, if fixed or supported at both ends flanffp«? should be m proportion to relative resistances of material to crushing or extension Breaking weights in similar beams are to each other as squares of their like linear dimensions; that is, breaking weights of beams are computed by multiplying to- gether area of their section, depth, and a Constant , determined from experiments on be a wee^si?pports.' CU ar f ° r “ UDder illvestigation > and dividing product by distance 2 4 C 4 1‘ and wrought ' iron learns, having similar resistances, have weights nearly as r i .^ 0 < X fl beam /, r girder ' eeestructed of plates of wrought-iron, compared to a single rib and flanged beam X, of equal weights, has a resistance as 100 to 93. g Resistance of beams or girders, where depth is greater than their breadth when supported at top, is much increased. In some cases the difference is fully one third. When a beam is of equal thickness throughout its length, its curve of equilibrium shmdd hP u! • SUPP °h Py Unif ° rm StreSS with ec P ,al ^(stance in every part! should be an Ellipse, and if beam is an open one, its curve of equilibrium for a uni! foi m load should be that of a Parabola. Hence, when middle portion is not wholly removed, its curve should be a compound of an ellipse and a parabola approaching nearer to the latter as the middle part is decreased. ? 1 P S iron ° f CaSt ir0D ’ UP t0 a SI>an ° f 40 feet ’ involve a less cost than of wrought Cast-iron beams and girders should not be loaded to exceed .2, or subjected to a greater stress than 166 of their destructive weight ; and when the stress is attended with concussion and vibration, this proportion must be increased. WnJ^Ll^ t ' ,I !? n ’ gir l® r8 * n ? y be ™ ade 50 feet in length > and best f°rm is that of a fixed ioadj flanges sbouid be as 1 to 6 ’ whei Forms of girders for spaces exceeding limit of those of simple cast iron are vari- inc - ipa 0nB !' adopted are those of straight or arched cast-iron girders in and Tubidar CeS ’ and b ° ted together — Trussed > Bowstring, and wroughtdron Box STRENGTH OF MATERIALS. — TRANSVERSE. 825 Straight or Arched Girder , formed of separate castings, is entirely dependent upon bolts of connection for its strength. Trussed or Bowstring Girder is made of one or more castings to a single piece, and its strength depends, other than upon the depth or area of it, upon the proper abstinent of the tension, or the initial strain, upon the wrought-iron truss. Box or Tubular Girder is made of wrought iron, and is best constructed with cast-iron tops, in order to resist compression: this form of girder is best adapted to afford lateral stiffness. When a girder has four or more supports, its condition as regards a stress upon its middle is essentially that of a beam fixed at both ends. The following results of the resistances of materials will show how they should be distributed in order to obtain maximum of strength with minimum of dimensions : To Tension, j Cast iron t 21 OOO { 32 OOO “ English.. ( 13000 ) 23 OOO Granite 57 8 Limestone f 670 \ 2 800 To Tension. Oak, white, mean. “ English “ . Wrought iron “ English Yellow pine 11 000 6 500 S 45 000 1 59 000 ( 31 000 i 53 000 16000 To Crush’g. 7500 3 100 47 000 83 000 40 000 65000 4000 The best iron has greatest tensile strength, and least compressive or crushing. Conditions of Forms and Dimensions of a Symmetrical Beam or Grirder. When Fixed at One End, and Loaded at the Other . 1. When Depth is uniform throughout entire Length , section at every point must be in proportion to product of length, breadth, and square of depth, and as square of depth is in every point the same, breadth must vary directly as length ; consequently, each side of beam must be a vertical plane, tapering gradually to end. 2 . When Breadth is uniform throughout entire Length , depth must vary as square root of length ; hence upper or lower sides, or both, must be deter- mined by a parabolic curve. 3 . When Section at every point is similar , that is , a Circle , an Ellipse , a Square , or a Rectangle , Sides of which bear a fixed Proportion to each other , the section at every point being a regular figure, for a circle, the diameter at every point must be as cube root of length ; and for an ellipse or a rec- tangle, breadth and depth must vary as cube root of length. Illustration.— A rectangular beam as above, 6 ins. wide and 1 foot in depth at its extreme end, and 4 feet in length, is capable of bearing 6480 lbs. ; what should be its dimension at 3 feet ? x 5 g 7? an d .^/ 3 — 1. 442 . Then 1.587 : 1.442 :: 1 : .9086, and 6 and 12 X .9086 == 5.452 and 10.9. TT 5.452 X 10.9 2 , . 6 X 12 2 Hence — 2 i6, and = 216. 3 4 When Fixed at One End , and Loaded uniformly throughout its Length. 1. When Depth is uniform throughout its entire length , breadth must in- crease as the square of length. 2 . When Breadth is uniform throughout its entire length, depth will vary directly as length. 3 . When Section at every point is similar , as a Circle , Ellipse , Square , and Rectangle , section at every point being a regular figure, cube of depth must be in ratio of square of length. 826 STRENGTH OF MATERIALS. TRANSVERSE. Illustration. —Take preceding case. Then 4 2 : 32 :: I2 3 ; 9?2j and = ^ in depth When Supported at Both Ends. 1. When Loaded in the Middle, Coefficient or Factor of Safety of the beam or product of breadth and square of depth, must be in proportion to distance from nearest support ; consequently, whether the lines forming the beam are straight or curved, they meet in the centre, and of course the two halves are 2. When Depth is Uniform throughout, breadth must be in ratio of length, of' len^tlr Breadih is Uni f orm throughout, depth will vary as square root 4. When Section at every point is similar , as a Circle, Ellipse, Square , and Rectangle, section at every point being a regular figure, cube of depth will be as square of distance from supported end. 1 When Supported at Both Ends, and Loaded uniformly throughout its Length. 1. When Depth is Uniform, breadth will be as product of length of beam and length of it on one side of given point, less square of length on one side of given point. 0 2. When Breadth is Uniform, depth will be as square root of product of length of beam and length of it on one side of given point, less square of length on one side of given point. 4 3. When Section at every point is similar , as a Circle, Ellipse, Square , and Rectangle, section at every point being a regular figure, cube of depth will be as product of length of beam and length of it on one side of given point less square of length on one side of given point. ’ DRIlliptical-sided. Beams. To Determine Side or Curve of an Elliptical-sided Beam J LI L re P resentm 9 load in lbs., I length in feet , C coefficient, and b breadth in ins. Illustration.— W hat should be depth in centre of abeam of white pine 10 feet in length between its supports, and 5 ins. in breadth, to support a load of ioioo lbs.? Assume C = 100. Then 7 ™°°°*™^ A^oooo _ V 2 X 100 x 5 V 1000 Hence, outline of beam is that of a semi-ellipse, having 10 feet for its transverse diameter, and 9 ins. for its semi-conjugate. Note.— W eight of Girder, Beam, etc., should in all cases be added to stress or load. Miscellaneous Illustrations. 1.— What should be side of a rectangular white oak beam, 2 ins. in width and 6 feet between its supports, to sustain a load of 360 lbs. ? ’ Assume stress at .2 of breaking weight of 150 lbs. = 30. / 6 X 360 _ /2160 v 4 X 2 X 30 \ 240 — * = 3 ins. v 4 X 2 X 3® v 2.— What should be breadth and depth of such a beam if square? 3 / 6 x 36° ? /'2160 V 4X30 ~v . 4 X 30 V 120 3* — What should be diameter of a cylinder? 360 X 6 = 2. 62 ins. .6 X 3 ° = 120, and ins - STRENGTH OF MATERIALS. TRANSVERSE. 827 STEEL. To Compute Transverse Strength, of Steel Bars. Supported at Both Ends. Weight applied in Middle. x.155 S bd* =w g represen ti n g tensile strength in lbs . , l length between supports in ins . , and W weight in lbs. Illustration —What is ultimate destructive stress of a bar of Crucible steel, 2 ins. square, and 2 feet between supports ? s = 9° 000 lbs. 1.155 X 90000 X 2 3 = 831 600 6 J 6s . 2 X 12 24 Elastic Transverse Strength is 50 per cent, of its ultimate strength. Hardening in oil increases its strength from 12 to 56 per cent. Thus, Soft steel, 121 520 lbs. ; soft steel, cooled in water, 90 160 lbs. ; soft steel, cooled in oil, 215 120 lbs. Knipps is about .45 of its tensile breaking weight, .24 of its compressive or crushing strength, .38 of its transverse, and .39 of its torsional. Friction of a steel shaft compared to one of wrought iron is as .625 to 1. Capacity of steel to resist a transverse stress is much less than to resist torsion. Relative diameters of steel and wrought-iron shafts, to resist equal trans- verse stress, are as .98 to 1, and weight of such a proportion of steel shaft compared with one of wrought iron will be about 4 per cent, less, and friction of bearing will be 6 per cent. less. CYLINDERS, FLUES, AND TUBES. Hollow Cylinders. Cast Iron. To Compute Elements of* Hollow Cylinders ■vvithin Limits of* Elastic Strength. (D. K. Clark.) P P S X hyp. log. R = P. r y— ^ = S. ~ = hyp. log. R. S representing ** * hyp. log. R o elastic tensile strength of metal in lbs. per sq. inch , R ratio of external diameter to in- ternal = — = — , and P internal pressure in lbs. per sq. inch, d and d' representing ’ dr internal and external diameter , and r and r' internal and external radii , all in ins. Note.— Hyperbolic Logarithm of a number is equal to product of its common logarithm and 2.3026. Illustration i. — Diameters of a hydrostatic cylinder 5.3 by 13.125 ins. ; what pressure within its elastic strength will it sustain per sq. inch? Assume S = 10000 lbs. Hyp. log. R = - 3 ‘ -- - X 2.3026 = log. 2.5 X 2.3026 = .92. 5-3 Then 10 000 X .92 = 9200 lbs. per sq. inch. Note.— F or Bursting Strength take maximum strength of metal. 2. —A water-pipe .75 inch thick has an internal diameter of 10 ins., what is its bursting pressure ? S = 30 000 lbs. Hyp. log . 10 — = . 1398. Then 30000 X *1398 = 4194 Ibd. 3. — If it were required of a hydrostatic press to sustain a pressure of 589050 lbs. upon a ram of 5 ins. in diameter, what would be pressure on ram, and what should be thickness of metal, assuming it equal to an elastic tensile stress of 15000 lbs. per sq. inch ? Area of 5 ins. = 19.635. 589 05 0 _ ooQ —p ress ure per sq. inch on ram. i 9- 6 35 Then 30000 = 2, which = hyp. log. R — 7. 39, and 7. 39 X 5 = 3 6 -95 = external di< 15000 ameter. 36.95 — 5 = 31.95, which -f- 2 = 15-975 ins. thickness of metal. 828 STRENGTH OF MATERIALS. TRANSVERSE. Wrought Iron. and. Steel. R + hyp. log. •— — 1 d Q -p 2P „ 2 P , 2 S “ P * tT. 7, 7 d ' ~ S ' ^+ I = ( p + h ypiog.R). Illustration i.— If diameters of a wrought-iron cylinder are 5 and u ins and ingp^ssure^? StmCtlVe strengt ^ of : metal is 4° 000 lb s. per sq. inch, what is 5 its break- ~ = 3- Hyp. log. 3 = .477 12 X 2.3026 = 1.0986. tviqv. 3'f _I -°986 — 1 men X 40000 = 61 972 lbs. per sq. inch = 61 972 X 5 -r- 15 — 5 = 30 986. 2 lbs. per sq. inch of section of metal. steam-boiler 6 feet in internal diameter, of wrought-iron plates .375 inch thick and double riveted longitudinally, burst at a joint bv a pressure of mi lbs per sq. inch ; what was resistance of joint per sq. inch of its section ? J y 72 + . 375 X 2 — 1. 0104. Hyp. log. 1. 0104 = .010345. Then - 2 x 300 _ 600 1. 0104 -J-. 010 345 — 1 “ .020745 = 2 9 405 Ms. per sq. inch of section of joint. SHIP AND BOILER PLATES. (- Seepages 751-757 /or Boiler Riveting.) Ultimate Tensile Strength, of Riveted. and. "Welded Joints of Wrought-iron IPlates. (D. K. Clark.) Entire Plate = 100. Joints. •5 Plate. •375 •4375 Aver- age. Joints. •5 Plate. •375 •4375 Aver- age. Scarf- welded Lap-welded Single hand riveted. “ “ snap-1 headed f 50 40 50 102 66 60 56 106 69 50 52 104 62 50 53 Double riv’d, snap- headed “ “counter-] sunk and snap- headed j I 1 59 53 72 69 70 72 67 65 “ “by machine “ “ counter-) sunk head . . . J 40 44 52 S 2 54 50 1 49 49 “ “with single] welt, counters’k and snap-headed) 1 52 65 60 59 Strength of Riveted Joints per Sq. Inch of Single Plate . ( Wm. Fairbairn.) Single Lapped— Machine riveted. Pitch 3 times, 25 000 lbs. Hand riveted. Pitch 3 times, 24 000 lbs. Rivets “staggered,” and equidistant from centres, 3050Q lbs. Abut Joints .— Hand riveted. Rivets not “ staggered,” and equidistant from centres, single cover or strap, 30 000 lbs. Rivets “square,” single cover or strap, 42 000 lbs. ; double covers or straps, 55 000 lbs. Comparative Strength of Riveted Joints. Entire Plate .375 ins. thick = 100. Double riveted, double strap, or fish- ) Q plated joint j 80 Double riveted lap joint 72 Double riveted, single strap, or fish- 1 , plated joint j Single riveted lap joint 60 For all joints of plates over .5 inch, other than double welded, these proportions are too high. ^A closer pitch of rivets should be adopted in single than in double riveted abuts, STRENGTH OF MATERIALS. TRANSVERSE, 829 Dimensions of R,ivets, Pitcli, Lap, etc, Plate. Thickness. Diam. of Rivet. Length from Head. Pitch. Single. ** a P- Double. Staggered. Inch. •25 • 3 I2 5 •375 •5625 .625 •75 •875 1 Ins. •5 .625 •75 .8125 •9375 1 1. 125 1.25 i -5 Ins. 1. 125 1-375 1.625 2.25 2.75 3 3-25 4 ' 4-5 Ins. i -5 1.625 1- 75 2.125 2 - 375 2.625 3 3 - 375 4 - 375 Ins. 1.5625 2 2.4375 2.625 3 3- 2 5 3.625 4 4.875 Ins. 2.75 3 - 4375 4.125 4 - 4375 5.1875 5 - 5 6.1875 6.875 8.25 Ins. 2 - 4375 3 3.625 3 - 9375 4 - 5625 4.8125 5 - 4375 6.0625 7- 2 5 1 1.5 1 4-5 4-375 4-»75 °* 2 5 7-^3 Straps. — Single, .125 thicker than the plate; Double, each .625 of thickness of plate. To Compute Diameter of Trivet. Ordinarily , T 1.25 + . 1875 — d. T representing thickness of plate , and d diameter of rivet. Fitcli of Rivets. {Nelson Foley.) Plates. Metal between the Holes. Diam. of Rivets. 1 Plates. Metal between the Holes. Diam. of Rivets. Single Staggered. 52 to 62 per cent. 68 to 75 “ “ 1.4 to 2.3 1.4 tO 2. 1 I Square . . (Triple. . . 70 to 78 per cent. 76 to 80 “ “ .99 to 1.7 .77 to 1 Proportions of Single Rivet Wrought-iron Joints. {French. ) Thickness of Plate. Diameter of Rivets. Pitch of Rivets. Width of Lap. Mil’s Inch. Mil’s Inch. Mil’s Ins. Mil’s Ins. 3 .118 8 •315 27 1.06 3 ° 1. 18 4 .158 10 •394 32 1.26 34 i -34 5 .197 12 .472 37 1.46 40 1.58 6 .236 *4 •551 43 1.69 44 I -73 7 .276 16 •63 48 1.89 50 1.97 8 • 3 I 5 17 .669 5 i 2.01 54 2.13 9 •354 19 .748 54 2.13 56 2.2 Thickness 1 of Plate. Diameter 1 of Rivets. Pitch of Rivets. Width of Lap. Mil’s Inch. Mil’s Ins. Mil’s Ins. Mil’s Ins. 10 -394 20 .787 56 2.2 58 2.28 11 •433 21 .827 57 2. 24 60 2.36 12 .472 22 .866 58 2.28 60 2.36 13 .512 23 .906 60 2.36 62 2.44 14 -551 24 -945 62 2.44 64 2.52 15 • 59 1 25 .984 63 2.48 66 2.6 16 .63 26 1.024 65 2.56 68 2.68 Result of Experiments on Dorible Riveted, and Double Strapped Rlate Joints. [Mr. Brunei.) Plates 20 ins. in width, 5 inch thick , Abut jointed, with a Strap or Fish-plate on each side , 10 ins. in width. Holes Punched. For Boiler Riveting see pp. 755-57* 4 a 830 STRENGTH OF MATERIALS. TRANSVERSE. Hi.il Is of Vessels. Diameter of Rivets. Plate. U. S. and British Lloyds. Liverpool Reg f y. Admiralty, Eng. Mill wall, Eng. ■■.’I Pitch of Rivets. Length ( Counter- sunk. >f Ri 7eta. Snap- headed. Inch. Inch. Ins. Ins. Inch. Ins. Ins. Ins. •3125 .625 •5 •5 .625 *•75 1-125 1.5 •375 .625 .625 .625 .625 2 1.25 1.625 •4375 .625 .625 •75 .625 2.125 1-375 1-75 •5 •75 •75 •75 •75 2.25 i -5 2 •5625 •75 •75 •875 •75 2-437 1.6875 2.1875 .625 •75 .8125 •875 ^875 2.56 1-9375 2-375 .6875 •875 •875 •875 •875 2.812 2.1875 2.625 •75 •875 •875 1 •875 3- 125 2-375 2-75 .8125 •875 •9375 1 .875 3-375 2-5 2.875 •875 1 1 1. 125 1 3-625 2.625 3 •9375 1 1.0625 1.125 i 3-875 ' 2.75 3-125 1 1 1.125 1. 125 1 4.125 2.875 3-25 Lap of Joint or Course should be .5 pitch of rivets added to .3 diam. of rivet. Note.— Lloyd’s requires a spacing of 4.5 diameter. Liverpool Registry, 4. Ad- miralty, 4.5 to 5 in edges and abuts of bottom and bulkhead plates, and 5 to 6 in other water-tight work. Bureau Veritas , 4 diameters for single riveting, and 4 5 for double. STEEL PLATES. Steel Plates, according to M. Barba, .354 inch thick are equal to wrought iron .472 inch thick, or as 3 to 4; consequently, when iron rivets are used, their diameter should be in proportion to an iron plate. It is ascertained also that they are best united by iron rivets. A steel plate .3125 inch thick requires an iron rivet .5625 inch in diam- eter, and 1.375 ins. apart. Bridge Blates and Rivets. Plates .25 to .5 inch thick. Rivets .75 to 1 inch diameter, and 3 ins. apart from centres in upper flange or girder, and 4 ins. in lower Rivet Heads. Ellipsoidal , Fig. 1. — D diameter , R radius of head = D, r radius of flange = .4 D, c depth at centre p ,5 D. Segmental , Fig. 2. — D diameter , c depth at centre = .625 /'ITX 2 ' ~P Tiortrl nr - IV n rl I n /'i /I IV \ y 1 D, R radius of head — .75 D, 0 depth below head = . 125 D, Countersunk . — Head 1.52 D, angle 6o°. Countersink .45 diam. of plate. Cheesehead or heads, section of which is a parallelogram. Head .45 D, diameter 1.5 D. Rivets. Shearing strength of a Lowmoor rivet = 40 320 d 2 or 18 d 2 in tons. d representing diameter of rivet in ins. Memoranda. Punching holes for riveting weakens plates, varying from 10 to 20 per cent., ac- cording to their temper, hardest losing most. Countersunk riveting does not impair strength of joint, as compared with ex- ternal head. Diagonal abut joints are stronger than square. Shearing strength of rivets should not exceed that of plates. Maximum strength of joint is attained at 90 to 100 per cent, of net section of plate. Shearing strength of English wrought iron is taken at 80 per cent, of its tensile strength. STRENGTH OF MATERIALS. — TRANSVERSE. 83 I LEAD PIPE. Resistance of Lead Bipe to Internal Pressure. ( Kirkaldy , Jar dine, and Fairbairn.) Diam. Thick- ness. Weight Foot. Bursting Pressure. Diam. Thick- ness. Weight per Foot. Bursting Pressure, Diam. Thick- ness. Weight per Foot. Bursting Pressure. Inch. •5 .625 •75 X Inch. .2 .2 .22 .2 Lbs. 2- 3 2.6 3- 8 4. 1 Lbs. 1579 1349 1191 9t 1 Ins. 1.25 1-5 1-5 i-5 Inch. .21 .24 . 2 • 2 Lbs. 5-3 7 - 1 Lbs. 6$3 734 528 62 $ Ins. 2 2 3 3 Inch. .21 .2 •25 •25 Lbs. 9.2 Lbs. 49 8 448 364 374 Tensile strength of metal = 2240 lbs. per sq. inch. To Compute Thickness of a Lead. Bipe when Diameter and Pressure in Lbs. per Sq. Inch is given. j> ULE Multiply pressure in lbs. per sq. inch by internal diameter of pipe in ins., and divide product by twice tensile resistance of metal in lbs. per sq. inch. Illustration —Diameter of a lead pipe is 3 ins., and pressure to which it is to be submitted is 370 lbs. per sq. inch; what should be thickness of metal? 370 X 4 mo . ' • AL — tLA = — . 248 ins. 2240 X 2 4480 Difference in Weight between Pipes of “Common,” “Middling,” and “Strong” is 12 per cent. To Compute Weiglit of Lead Bipe. D 2 — d p 3. 86 == W. D and d representing external and internal diameters in ins., and W weight of a lineal foot in lbs. To Compute Maximum or Bursting Pressure tliat may be Lome by a Lead Bipe. Rule.— Multiply tensile resistance of metal in lbs. per sq. inch by twice thickness of pipe, and divide product by internal diameter, both in ms. Illustration.— What is bursting pressure of a lead pipe 3 ins. in diameter and .5 inch thick? 2 ^° X -5 XJ _ ^12 == 6 6 lbs 3 3 Assume a column of water 34 feet in height to weigh 15 lbs. per sq. inch; what head of water would such a pipe sustain at point of rupture? 15 • 34 •• 746.6 : 1692.3 feet. Resistance of Grlass GHLoLes and Cylinders to Internal Pressure and Collapse. ( Flint Glass.) Bursting Pressure. Diameter. GLOBES. Thickness. Per Sq. Inch. Diameter. cylin: Length. DER. Thickness. Per Sq. Inch. * Inch. Lbs. Ins. Ins. Inch. Lbs. 4 .024 84 4 7 .079 282 5 .022 90 Elliptical (Crown Glass). 6 •059 152 Colla 4 - 1 1 7 1 psing Pressure. [ *019 1 109 5 1 - OI 4 292 II 3 1 14 I * OI 4 1 85 4 .025 1000* 4 7 •034 202 6 1 -°59 j 900* ll 4 * Unbroken. 1 *4 1 .064 1 297 832 STRENGTH OF MATERIALS. — TRANSVERSE. Manganese Bronze. Manganese Bronze , No. 2, has a Tensile strength of 72 000 to 78 600 lbs. per sq. inch, its elastic limit is from 35000 to 50000 lbs., its ultimate elon- gation 12 to 22 per cent., and its hardness alike to that of mild steel. Transverse Strength— Destructive stress of a bar 1 inch square, supported at both ends at a distance of 1 foot = 4200 lbs., bending to a right angle be- fore breaking, and requiring 1700 lbs. to give it a permanent set. MEMORANDA. Cast Iron. Beams cast horizontally are stronger than when cast vertically. Relative strength of columns of like material and of equal weights is : Cylindrical, 100; Square, 93; Cruciform, 98; Triangular, no. ( Hodgkinson .) If strength of a cylindrical column is 100, one of a square, a side of which is equal to diameter of the cylinder, is as 150. Repetition of Stress . — A piece submitted to transverse stress broke at 1956th strain, with a stress .75 of that of its original ultimate resistance. Resistance to Bursting of Thick Cylinders . — Mean resistance to bursting, of chambers of cast-iron guns is as follows ( Major Rodman , U.S.A.) : Thickness of metal — 1 calibre, length = 3 calibres, 52 217 lbs. per sq. inch. Thickness of metal — .5 calibre, length == 3 calibres, 49 100 lbs. per sq. inch. The tensile strength of the iron being 18 820 lbs. Diam. of cylinder 2 ins., length 12 ins., metal 2 ins., 80229 lbs - P er sq. inch. Diam. of cylinder 3 ins., length 12 ins., metal 3 ins., 93702 lbs. per sq. inch. Tensile strength of iron being 26 866 lbs. Sudden Applications of Stress.— 'Loss of strength by sudden application of load was, by experiment, 18.6 per cent, in excess of load applied gradually, and its elongation 20 per cent, greater. Low Temperature. — Tensile strength at 23 0 under sudden application of load, was reduced 3.6 per cent., and elongation 18 per cent. "W i* outfit Iron. Increased Hammering gives 20 per cent, greater strength with decreased elongation. Hardening. — Water increases strength more than oil or tar. A bar .87 inch in diameter, forged and hardened in water, attained a tensile strength of 73448 lbs. {Mr. Kirkaldy.) Case Hardening. — Loss of tensile strength 4950 lbs. per sq. inch. Cold Rolling added 18.5 per cent, to tensile strength, and when plates were reduced .33 in thickness, strength was nearly doubled, with but .1 per cent, elongation. Specific gravity was reduced. Fibre. — Plates are about 12 per cent, stronger with fibre than across it. Angles , Tees, etc., have from 2200 to 4500 lbs. less tensile strength than rectangular bars. Galvanizing does not perceptibly affect strength. W elding . — Strength as affected by welding varies by experiment from 2.6 to 43.8 per cent, less, average being 19.4. Elastic Strength is about .45 of its tensile breaking weight, .15 of its com- pressive or crushing strength, and .5 of its transverse strength. Effect of Screw Threads. — 1 inch bolts lose by dies 6.1 1 per cent., and by chasing 28 per cent. Steel. Steel can be hardened in water at a temperature of 310°. STRENGTH OF MATERIALS. — TRANSVERSE. 833 WOODS. To Compute Transverse Strength, of Large Timber. Destructive Stress. . 3 S b d 2 Fixed at One End , and Loaded at the Othei . — ^ . x.8S^ 2 Fixed at Both Ends , and Loaded in Middle. ■ zW. 4 Supported at Both Ends , arcd Loaded in Middle. 1.2 S be? 2 = W. iVzed aZ j&rfA Lrcds, arad Loaded at any other point than | -45 s 6 d2 — w, the Middle. 3 l Supported at Both Ends , cmd Loaded at any other point ) • 3 B ^ the Middle. T w ; ^ m n i S. * Hence, W Z „ , Wi 5 tZ, and l representing breadth , tZepto, and length to or between supports allin ins S m^an of tensile and crushing strengths of material at two thirds of its Value, as determined: by experiments , W ultimate, weight or stress m lbs., and m and n dis- tances of load from nearest supports in ins. When a beam is uniformly loaded, the stress is twice that if applied in its middle or at one end. Values of 1.3 S. Hence, for other coefficients, as .3, 1.8, etc., the values will be proportional. Woods. Ash, white “ ' Canadian “ English Beech Birch Cedar “ Cuban Chestnut Cypress Elm, English.. “ Rock, Canada. . Fir, Dantzic Greenheart Gum, blue Hackmatack Iron wood Larch Woods. Locust Mahogany, Honduras. Oak, Pa “ Va u white “ English u Dantzic “ French Pine, Va “ pitch “ white “ yellow “ Canada.. Redwood, Cal Spruce Teak Walnut, black Illustration i. — What is destructive stress of a beam of English oak, 2 ins. square, and 6 feet between its supports? 1.2 from table — 1.7, and S = .66 of 5700 (mean of tensile and crushing strength) = 3762 lbs. 1.7 X 2X^X3762 _ 5 ii «3 = Jlo6 lbSi 6 X 12 72 By experiment of Mr. Laslett it was 688 lbs. 2. — What is destructive stress of a beam of yellow pine, 3 ins. by 12, and 14 feet between its supports? 1. 2 from table = 3. 87, and S = .66 of 10 200 (mean of tensile and crushing strength) = 6732 lbs. 3.87 X 3 X 12 2 X 6732 ___ 11 65482 7 14 X 12 168 - = 69 374 lbs. If the beam was fixed at both ends then 3.87 would be 5.8. Or, s 1.2 : 1.8 :: 3.87 : 5.8. Safe Statical Loads for Rectangular Beams of Various ^Materials, One Inch in Breadth, and One Boot in Length. Supported at Both Ends and Loaded in Middle. A American, Af African, B Baltic, B’k Black, C Canadian, D Dantzic, HZ English, G Georgia, M Memel, P Pitch, R Riga, W White, Y Yellow. — Figures at Head of Columns denote Destructive Weight of Material in Lbs. 834 STRENGTH OF MATERIALS. TRANSVERSE. Locust. 1 180 O -vf »i-0 O O -vhvo O VO ^ -^O 0 vO a* m Tj- oi c->. O ON vO 0 m 0 vo 00 00 10 O it CN ON hi (V Oi V in H m O m on 00 01 hi rh h3 n ro 1000 h to ov moo m ovvo m 0 ' mmciNNCOCOtJ- 10 VO Hickory. Maple. Af Oak. G Pine. 800 Lbs. 160 640 1440 2560 4 000 5760 7840 10240 12 960 16 000 19360 23040 27040 3 1 3 6 ° 36 000 40 960 Ash. D Fir. Rock Elm. 680 Lbs. 136 544 1 224 2 176 3400 4 896 6 664 8704 11 016 13 600 16456 19584 22 984 26 656 31 600 34816 M FJr. Chestnut. E Ash 640 Lbs. 128 512 1 152 2 048 3 200 4608 6 272 8 192 10 368 12 800 15488 18432 21 632 25 088 28 800 32768 W Oak. B Fir. 600 .0000000000000000 n NOOOO M 0 CICOCO M 0 NOOOO NON h vO CNO moo vo no ion m mo H H roviONON -v c-. 0 m n- 0 H H M M d M m gie S ° £0 g *r> ^ n. 0 PhOm 0000000000 000000 » h v 0 vo mvo on h 0 >-i -v- onvo mvo ,c m voNNNO\foo ovo moo m m * •-I h n m m c^oo h m mco m tj-oo Hi H M H d Cl 01 i-ii m « so « 10 momomomomomomomo n * 0 M ”^00 NOO VN O 0 O N -VOO 01 OO ,0 m •<+■ ovo 'O nh mo mo m m mo 00 h oi m mo 00 0 n m no mo H HI H M 01 N 01 Spruce. Sycamore. Elm. W Pine. 500 Lbs. 100 400 900 1 600 2 500 3600 4 900 6 400 8 100 10000 12 100 14400 16 900 19 600 22 500 25 600 A . . « g-Sr 0 $ Lbs. 90 360 810 1440 2 250 3240 4410 5760 7 290 9000 10 890 12 960 15 210 17 640 20 250 23.040 Birch. Hack- matack. Hemlock. 400 Lbs. 80 320 720 1 280 2 000 2 880 3920 5 120 6 480 8 000 9680 11 520 13520 15 680 1 8 000 20480 E N 0 « -to 0^0 Tf-vo 0+0 'S-O O 0 't- aJOmt-^NOOm ovoo 0 >1 h v 0 00 .0 oi moo m h 0 « Ti-r^oioo m + m ^ H h n m '+■ mo n On 0 N -v-o 4j » 0 ^0 0 T 3 °° & (2 ' .OOOOOOOOOOOOOOOO « O ■+ VO OO '+-'+0 OO -cj- -+-0 OO •g oi m On m m ovoo ooooiOHt^mm 1-5 h 01 01 m tj-o Choo 0 hi m m •£. « h n ro mo r^oo o O h n m ij- mo G IHMHHHHH « l 2 ^ O a ^ *S 5 *S ^ % A, o 3 o a, & 3 s N & S v .w > O .5 3 © g , © y 2 ca a m rS3 03 "o ^ S3 c3 .3 n't a £ »c £ £ 33 Ctf „ STRENGTH OF MATERIALS. TRANSVERSE. 835 Following Coefficients or Factors of Safety are for .125 of average de- structive weight : Coefficients Ash “ English “ Canada Beech Chestnut Elm, Canada “ English Fir, Riga Hemlock for Various Woods. 85 80 60 58 65 80 42 Hickory Larch Locust Maple Oak, white . . , “ English “ Dantzic “ Adriatic, 100 130 40 150 80 60 62 55 ( Hatfield and others.) Ogk, Canada “ French Pine, pitch “ yellow “ red Georgia • o white......... “ Canada red. . . . 70 85 68 65 55 100 120 62 60 Spruce 65 Illustration. — What safe weight will a beam of white pine sustain, 4 ins. in breadth. 12 in depth, and 15 feet between its supports, when loaded in its middle? and what when uniformly loaded? Coefficient as above, 62. Then 4 X 12 X 62 _ 2 ^g a g ^ loaded in its middle , and 2380.8 X 2 = 4761.6 lbs. if uniformly loaded. Floor Beams of Wood. Condition of stress borne by a Floor beam is that of a beam supported at both ends and uniformly loaded ; but from irregularity in its loading and unloading, and from necessity of its possessing great rigidity, it is proper to estimate its capacity as a beam loaded at middle of its length. To Compute Capacity of Floor Beams, Grirdei-s, etc. Supported at Both Ends. Rule.— Divide product of breadth and square of depth, in ins., and Coef- cient for material, by length in feet, and result will give weight in lbs. Or, & = W. Fixed at Both Ends. C = W. I t Example.— The dimensions of a white-pine floor timber are 4 by 12 ins., and its length between supports 15 feet; what weight will it sustain in its centre? A X I2 2 X 62 C as per preceding table == 62. Then — = 2380.8 lbs. When Uniformly Loaded. Multiply the results by 2. To Compute Deptti of a Floo : 1 Beam. Supported at Both Ends. When Length between Supports , Breadth and Distance between Supports , for One Foot , between Centres of Beams are Given. Rule.— Divide product of length in feet, and weight to be borne in lbs., by product of breadth in ins., and Coefficient for material, and square root of quotient will give depth in ins., for distance between centres of one foot. Or, A-77 = d. Fixed at Both Ends. . / r— ; = d. ’ V bG V i-5 b c When Uniformly Loaded , W represents but half required or given weight. Example. — T ake elements of preceding case, distance between centres of beam 15 ins. 0 = 62. /i 5 X 2380. 8 /357 12 Then / — ^ A — = 12 ms. V 4 X 62 V 2 4 8 When Distance between Centres of Beams is greater or less than one Foot. Rule. — Divide product of square of depth for a beam, When distance between centres is one foot , by distance given, by 12, and square root of quotient will give depth of beam. 836 STRENGTH OF MATERIALS. — TRANSVERSE. Example.— Assume beam in preceding case to be set 15 ins. from centres of ad joining beams; what should be its depth? 2 X 15 — ' 3.42 ins . Then /12 2 X 15 /2160 V“^— = v/^ = I3 ' 4 To Compute Breadth, of' a Floor Beam or Grirder. /Supported at Both Ends. When Length and Depth are given. Rule.— D ivide product of length in feet, and weight to be borne in -lbs.$ by product of square of depth in i and Coefficient for material, and quotient will give breadth in ins. „ l W _. ' 7 w - = &: Fixed at Both Ends. — — b. ’ d 2 C ' 1.5 d 2 C ~ When Uniformly Loaded, W represents but half required or given weight. Example.— Take elements of preceding cases. Then *SXH*o-*± 35712 _ . i2 2 X 6 2 8928 4 mS ' When Distance between Centres of Beams is greater or less than One Foot. Rule. Divide product of breadth for a beam, When distance between centres is one foot, and distance given, by 12, and result will give breadth. Example.— Assume beam, as in preceding case, to be set 15 ins. from centre of adjoining beams; what should be its breadth? Then When Weight is Suspended or Stress borne at any other point than the Middle , See Formulas, page 801. Header and Trimmer or Carriage Beams. Conditions of stress borne or to be provided for by them are as follows : Header supports .5 of weight of and upon tail beams inserted into or at- tached to it, and stress upon it is due directlv to its length and weight of and upon tail beams it supports, alike to a girder loaded at different points. . Trimmer or Carriage beams support, in addition to that borne by them directly as floor beams, each .5 weight on headers. Note. — In consequence, of effect of mortising (when bridles or stirrups are not used), a reduction of tully one inch should be made iu computing the capacity of depth of headers and trimmers. To Compute Breadth of a Header Beam. When Uniformly Loaded. Rule.— C ompute weight to be borne in lbs. by tail beams, divide it by two (one half only being supported by header), mul- tiply result by length of beam in feet, and divide product by product of twice Coefficient of material and square of depth, and result will give breadth in ins. ■ - . W-r-zl 2 (j ^2 = W representing weight per sq. foot. Example. — What should be breadth of a Georgia pine header, 13 ins. in depth. 10 feet in length, supporting tail beams 12 feet in length, bearing 200 lbs. per sq. foot of area supported? C, as per preceding table, 100, and depth = 13 — 1 = 12 ins. Then 12 X 10 X 2oo-f-2 x 10 ^toS' = 4 ' I7; <2 = 15 — 1 = 14; 19X4X 19 X 10 = 2 X 540 • • 5 X 450 800 = 4. 32 ms. 23 X 100 X 14 2 Note 1.— Depth of trimmer beams is usually determined by depth of floor beams; when not, proceed to determine it as tor a header. 2 when a trimmer beam is mortised to receive headers, it is proper to deduct 1 inch from its depth, as in preceding illustrations. When bridle or stirrup irons are used to suspend headers, a deduction of the thickness of the iron only is neces- sary, usually .5 inch. With Two Headers and One Set of Tail Beams— Fig. 1. Operation. — Proceed for each weight or load as for a beam, when weights are sustained or stress borne at other point than the middle. a L - — W and w. a representing area of floor in sq. feet , L load per sq. foot , •5X5 and W and w weights or loads at points of rest on trimmers. Note. —Hatfield and some other authors give complex and extended formulas, to deduce the di- mensions of a Girder or Beam, under a like stress. Upon consideration, however, it will readily be recognized that a beam loaded at more than one point is simply two or more beams, as the case may be, loaded at different points, and connected together. Illustration. — What should be breadth of a trimmer beam of Yellow or Georgia pine, 25 feet in length, 12 ins. in depth, sustaining two headers 12 feet in length, set at 15 feet from one wall and 5 feet from the other, to support with safety 300 lbs. per sq. foot of floor? I — 25, m — 15, n= 10, 5 = 5, r = 20, C= 100, and d= 12 — 1 = 11 for loss by mortising. ..Xs Xjg = I«2S2 = 4500 lbs. at W, and 12X5X - ^ = = 4500 lbs. at w. .5X5 - 2 5 5X5 - 2 5 Then X 10 X 4500 __ 675000 _ g im breadth for load on header at is feet, 25 X 11 2 X 100 302 500 an( j 5 X 20 X 45 go, _ 45 000 g _ 1-4 g i nSt breadth for load on header at ant * 25 X 11 2 X 100 302500 2.23 + 1.48 = 3.71 ins. combined breadth. <—m — * r-- ■ l 1 W OD 838 STRENGTH OF MATERIALS. TRANSVERSE. With Two Headers and Two Sets of Tail Beams. — Fig. 2. Fig 2t Operation.— Pi oceed as directed for Fig. 1. Illustration. — What should be breadth of a trimmer beam of yellow pine 25 feet in length, 15 ins. in depth, sustaining two headers 12 feet’ in length, set at 15 feet from one wall and 5 feet from the other, to support with safety 300 lbs. per sq foot of floor? ’ 1 — 25, m = 15, n = 10, 5 = 5, r — 20, C = 100, and d— 15 — 1 = 14 for loss by mortising. 12X15X300 54000 — ■ - 13 500 lbs. at W, and K— dr 4 II — 72 — i J _ : 1 tl 71 < :~ 4 1 3 1 — -5X-5 = 4500 lbs. at w. 12X5X 300 _ 18000 •5X.5 _ -25 Then 15 x IQ X 1 3 5QQ _ 2025000: _ 25 X 14 2 X 100 490000 " T ins . , and 4.14 .92 = 5.06 ins . . combined oreadth. 4- »4 ins . , ami A_ X 20 X 45°o = 45£°°o = 25 X 14 X 100 490000 Fig. 3- With Three Headers and Two Sets oj Tail Beams.— Fig. 3. — 1 HM 1 — T l — f— - M *| —rri— l-H> 1 1 1 1 15 X 7 — 3 x 200 Then 5 X -5 .25 7 X 13 X 5250 _ 47 7 750 Operation.— Proceed as directed for Fig. 1. Illustration. —What should be breadth of a trimmer beam of yellow pine, 20 feet in length 13 ins. in depth, sustaining 3 headers 15 feet in length, set at 3, 7j and 13 feet from one wall, to sustain a load of 200 lbs. per sq. foot of floor? 1 = 20, m = 7 , n = i 3 , s = 7 , 0 = 3, d = i 3 — 1 = 12 ins., and C = 100. 15X7 X 200 21 000 .5X.5 “ -25 15X7 — 3X20 , = 3000 Lbs. at w . - 5250 lbs. at W ; 20 X 12 2 X 100 288000 3 X 17 X 3000 __ i53°oo 288 000 = 3000 lbs. at w ; and :.66 ins . ; 5 X .5 7 X 13 X 3000 20 X 12 2 X 100 " 273000 — .95 ins.; and 20 X 12 2 X 100 bined breadth. = .53 ins. 288 OOQ Hence, 1.66 + .95 + .53 = 3., 14 ins. com - Stirrmps ox- Bridles. Stirrups are resorted to in flooring designed for heavy loads, in order to avoid the weakening of the trimmers by mortising. Average wrought iron will sustain from 40000 to 50000 lbs. per sq. inch. Hence 45 000 lbs. as a mean, which -4- 5 for a factor of safety, = 9000 lbs. A stirrup supports one half weight of header, and being doubled (looped), the stress on it is but .5^-2 = .25 of load on header. To Compute Dimensions of Stirrups or Bridles. W-f-2 2 X 9000 = area. Hence area thickness — = width. Illustration. — What should be area and width of .75 inch wrought-iron stirrup irons for a weight on a header beam of 240000 lbs. ? 240 000 — 2 2 X 9000 120000 „ „ ' . 6.66 = — ^ = 6.66 sq. ms . , and = 8. 8 ins. = width. 10000 .75 STRENGTH OF MATERIALS. TRANSVERSE. 839 Condition of stress borne by a Girder is that of a beam fixed or supported at both ends, as the case may be, supporting weight borne by all beams resting thereon, at the points at which they rest. Rule.— Multiply length in feet by weight to be borne in lbs., divide product "by twice* the Coefficient, and quotient will give product of breadth and square of depth in ins. Example.— It is required to determine dimensions of a yellow-pine girder, 15 feet between its supports, to sustain ends of two lengths of beams, each resting upon it and adjoining walls, 15 feet in length, having a superincumbent weight, including that of beams, of 200 lbs. per sq. foot. Condition of stress upon such a girder is that of a number of beams, 30 feet in length (15X2), supported at their ends, and sustaining a uniform stress along their length, of 200 lbs. upon every superficial foot of their area. Coefficient . 2 of 500 =* 100. To Compute Greatest Load upon a, Grirder, and. Dimen- sions thereof.— Fig. 1. Then, for weight and dimensions, same formulas will apply. Illustration — Assume weight of 8000 lbs. at 3 feet from one end of a white-pine beam 12 feet in length between its bearings, and another weight of 3000 lbs. at 5 feet from other end. C . 2 of 500 — 100. 8000 X 3 X 12 — 3 = 216 000 effect of weight at location 1, and 3000 X 5 X 12 — 5 — io5 000 effect of weight at location 2. Hence 1, being greatest, = W, and 2 = w. Grirder. To Compute Dimensions of a Grirder. ins. Or, if 15 ins., then Then 30 X 15 X 200 ~T 1 5 X 200 - 4 - 2, for half support on their walls = 45 000 lbs. — == 3375 = b and d 2 . Assuming b = 12 ins., then =16.77 3., then = 15 ins. then ins. , then When a Beam is Loaded at Two Points. Fig. 1. < - 1 —m— ni ^ == effect of weight at 1, ^ = effect of weight at 2 , y ( W X n + w s) t= the two effects at 1 , and j {tb r -f W m) == two effects at 2 . Then, 3 — X 8000 = 18 000 at W, and x 8750 = 3750 at w ; and ’to 12 12 (8000 X 9 + 3000 X 5) = 21 750 = total effect at W, and ~ (3000 X 7 + 8000 X 3) = 18 750 = total effect at tv. 5 Hence,. to ascertain dimensions at greatest stress, Hence,. to ascertain dimensions at greatest stress, 12 x 300 * For being uniformly loaded. STRENGTH OF MATERIALS. TRANSVERSE. Verification . — Assume a beam as above loaded with 21 750 lbs. at 3 feet from end. Then, by formula for 801, — X g 2 * 750 = ** 1 ^. _ 8 ins 12 X 10 2 X 100 120000 y Fig. 2. < Equivalent Weight at Middle, — Fig. 2. 1 > w' o ;= A ; in y H*trt Z-f- W n 1^2 = B; = E ; and ^ - 4 - 2* = D = ic $ equivalent load at middle. Illustration.— What should bo breadth of a beam of Georgia pine, font in 1 rr! it , 7 w i |d l w A. w- E ....... i® 20 feet in length, 15 ins. in depth' uniformly loaded with 4000 lbs., and sustaining 3 headers or concentrated loads of 6000 lbs., at respective distances of 4 and 9 feet from one end and 7000 lbs. at 6 feet from other end ? 4, r — 16, m — 9, n = 6000 x 4 C = 850 X .2 = 170. 4000 X 20 20 - = 2400; = 6, ^^15 — 1 = 14, L = 4000, and 6coo X 9 7000 X 6 ■ = 54 °° 5 : = 4200; and ■ Hence, Then 20 X 70000 20-^ 2 2 = 2000. 2400 -j- 5400 -j- 42CO -j- 2000 = 14 000 lbs. 14000 X 10 X 10 1 400 000 - = 70 000 lbs ., effect at middle. ■=z 10. 5-f- ins. ' 4 X 14 2 X 170 133 280 Operation deduced by Graphic Delineation of Greatest Stress without uni- form Load. Fig. 3. < 1 > Moments of weights = A r W m n and , t , j 19200, 29700, and 29400, and let fall perpendiculars 1, 2, and 3 proportionate thereto. Connect w', W, and w with A B, and sum of distances of in- tersections of these lines upon perpendiculars, from 1, 2, and 3, respectively, will give stress upon A B at these points. Whence, greatest stress at greatest load will be ascertained to be 61 800 lbs. When Loaded at Three Points, m no as in Fig. 2. J (W n + w s) + w T = Gnatest Stms • Illustration. — Take elements of above case, omitting uniformly distributed load, — (6000 X 11 X 7000 X 6) -f- 6000 11 ^ ^ = — X 108 000 -f- 13 200 =. 61 800 lbs. Deflection of G-irders and Beams. W 13 __ d< Cbd* ' CM*-*" ~7^ W; = aad l/—lr^ = l 1 represent- ing length in feet , b and d breadth and depth, and D defection in ins. Values of C for Various Woods. {Hatfield.) Ash 4000 Chestnut 2550 Hemlock 2800 Hickory 3850 Larch 2093 Pine, Georgia 5900 Oak, white 3100 “ pitch 2836 “ English, mean. . 2686 “ white 2900 Spruce 3500 “ red 4259 Illustration. — What would be deflection of a floor beam of white pine, 10 feet in length, 4 ins. in breadth, and 8 in depth, with 4000 lbs. loaded in its middle? _ 4000 X 10 3 4000000 . . . C = 2900. — -r = = .674 inch. 2900 X 4 X 8 3 5 939 200 * Load uniformly distributed. STRENGTH OF MATERIALS. TRANSVERSE. 84 1 When Weight is Uniformly Distributed . A fair allowance for deflection of floor beams, etc., is .03 inch per foot of length; 04 inch may be safely resorted to. AW eights of Floors and. of Loads. Dwellings .— Weight of ordinary floor plank of white pine or spruce, 3 lbs. per sq. foot, and of Georgia pine, 4.5 lbs. Plastering, Lathing, and Furring will average 9 lbs. per sq. foot. Clay Blocks ( Flat Arch) 5.25 X 7-25 ins. in depth and 1 foot in length, 21 lbs. = 80 lbs. per cube foot of volume. Floors of dwellings will average 5 lbs. per sq. foot for white pine or spruce, and on iron girders will average from 17 to 20 lbs. per sq. foot. Weight of men, women, and children over 5 years of age, 105.5 ^s., and one third of each will occupy an average area of 12 X 16 ins. = 192 sq. ins. = 78.5 lbs. per sq. foot. Of men alone 15 X 20 ins. = 300 sq. ins. =48 m 100 sq. ieet. Bridges , etc .— Weight of a body of men, as of infantry closely packed, = 138 lbs. each, and they will occupy an area of 20 X 15 ins. = 300 sq. ms.— 66.24 lbs. per sq. foot of floor of bridge, and as a live or walking load, 80 lbs. per sq. foot. „ „ „ . Weight of a dense and stationary crowd of men, 120 lbs. per sq. toot. Bridging of Floor Beams increases their resistance to deflection in a very essential degree, depending upon the rigidity and frequency of the bridges. Weight on. Floors, etc., in addition to Weight ofStruct- ure, per Sq.. Foot. Relative resistance of scarfs in Oak and Pine, 2 ins. square, and 4 feet in length, by experiments of Col. Beaufoy. Scarf 12 ins . in Length and 13 ins. from End , or 1 inch from Fulcrum . Vertical . — no lbs. gave away in scarf. Horizontal , large end uppermost and towards fulcrum .— 101 lbs. fastenings drew through small end of scarf; small end uppermost , etc., 87 lbs. gave away in thick part of scarf. Statical or Dead Load at .2 of destructive stress, but for ordinary pur- poses it may be increased to .25, and in some cases with good materials to .3. Live Load at .1 to .125 of destructive stress. See also page 802. Ball rooms, Brick or stone walls Churches and Theatres. . . 1x5 to 150 “ 80 “ 85 lbs. Roofs, wind and snow 30 to 35 lbs. Slate roofs Dwellings Factories. Grain 200 to 400 “ 100 “ 4 ° “ Snow, per inch Street bridges. , Warehouses Wind 250 to 500 50 Scarfs, Factors of Safety. 842 SUSPENSION BRIDGE. SUSPENSION BRIDGE. Compute Elements. = stress at • . C representing chord or span , a half chord , and v versed sine of chord or curve of deflection, in feet, L distributed load inclusive of suspended struct- ure, Q load per lineal foot, and S stress at centre, all in tons, x distance of any point from centre of curve, and h height of chain at x above centre of it, both in feet, 5 stress on chain at any point, as x, from centre of span, s stress on any tension-rod, and t stress at abutments, all in tons, n number of tension-rods, o angle of tangent of chain with horizon at any point,, as x, r angle of chain with vertical at abutments l length of chain, infect, and z angle of direction of chain. Assume C = 300 feet, L == 1000 tons, v = 25 feet, x = 100 feet, n = 30, r = 71° 34', and 0 = 12 0 32'. Then, ^ 300 X 1000 8X25 = 1500 tons - 1500 -J- 1 = 1536.56 tons = s', 25 X 100 2 * * S * * * (. 5 X 300) 2 4 X hi 2 X 100 = 11. 11 feet — h) — . 2222 = 12° 32* — tan. 0 ; 4 X 25 3°° 1500 2 \/ ( ' = -3333 = 7 l0 34'— = cot. angle r; 5 X 3°°) 2 “h — 25 2 = 305. 5 feet =. I ; 300 X 1000 8X1 500 : 25 = v ; yj + 1 = 1428.6 tons — t) and 30 2 X 25 1000 = 34. 48 tons — s ; - = .3162 = 18 0 26'. V(2 X 25) 2 (300 -4- 2) 2 For a deflection of .125 of span, horizontal stress is equal to total load. To Construct curve, see Geometry, page 230. To Compute Batio wliicli Stress 01a Oliains or Cables at either Boint of Suspension Bears to whole Suspended "Weight of Structure and. Load. == R. R representing ratio. 2 X sin. z Illustration. — Assume elements of preceding case. - ■ x * 6g == i- 58 ratio. By a preceding formula it would be 1. 536. Stress on Back Stays . — The cables being led over rollers, having free mo- tion, tension upon them is same, whether angle z is same as that of r or not. Stress on Piers . — When angles r and i are alike, stress on piers will be vertical, but when angle of i is greater or less than r, stress will be oblique. To Compute Horizontal Stress and Vertical Pressure on Biers. S cos. z = Si, S cos. n r= S 0, S sin. « = and S sin. w = Po. St and S 0 representing stress , and P i and P 0 pressure, inward and outward. No-te —Span of New York and Brooklyn Bridge 1505.5 feet, deflection 128 feet, angle of deflection at piers from horizontal 15 0 10'. TRACTION. 843 TRACTION*. JR, e suits of' Experiments on Traction of Roads and. Pavements. ( M . Morin.) 1 st. Traction is directly proportional to load, and inversely proportional to diameter of wheel. 2d. Upon a paved or Macadamized road resistance is independent of width of tire, when it exceeds from 3 to 4 ins. 3d. At a walking pace traction is same, under same circumstances, for carriages with or without springs. 4th. Upon hard Macadamized, and upon paved roads, traction increases with velocity: increments of traction being directly proportional to incre- ments of velocity above velocity qf 3.28 feet per second, or about 2.25 miles per hour. The equal increment of traction thus due to each equal increment of velocity is less as road is more smooth, and carriage less rigid or better hung. 5th. Upon soft roads of earth, sand, or turf, or roads thickly gravelled, traction is independent of velocity. 6th. Upon a well-made and compact pavement of dressed stones, traction at a walking pace is not more than .75 of that upon best Macadamized roads under similar circumstances ; at a trotting pace it is equal to it. 7th. Destruction of a road is in all cases greater as diameters of wheels are less, and it is greater in carriages without springs than with them. Experiments made with the carriage of a siege train on a solid gravel road and on a good sand road gave following deductions : 1. That at a walk traction on a good sand road is less than that on a good firm gravel road. 2. That at high speeds traction on a good sand road increases very rapidly with velocity. Thus, a vehicle without springs, on a good sand road, gave a traction 2.64! times greater than with a similar vehicle on same road with springs. Ttesxilts -witli a Dynamometer. Wagon and Load 2240 lbs. * Roadway. Relat’e num- ber of horses for like effect. Roadway. Relat’e num- ber of horses for like effect. t>n railway R ]hs, t . . , , Telford road, 46 lbs C,7C On best stone tracks, 12.5 lbs. Good plank road, 32 to 50 lbs. Stone block pavement, 32.5 “ Macadamized road, 65 lbs. . . . 1.56 4 to 6.25 4.06 8. 12 Broken stone or con’te, 46 lbs. Gravel or earth, 140-147 lbs. j Common earth road, 200 lbs. . 3/3 5-75 17- 5 18 - 37 25 Note. — B y recent experiments of M. Dupuit, he deduced that traction is inversely proportional to square root of diameter of wheel. Relation of force or draught to weight of vehicle and load over 6 different con- structions of road, gave for different speeds as follows: Walk. Trot. Walk. Trot. Stage coach, 5 tons. . 1.3 1 | Carriage, seats only, on springs. . 1.29 1 Resistance to Traction on Common JRoacls. On Macadamized or Uniform Surfaces. {M. Dupuit.) 1. Resistance is directly proportional to pressure. 2. It is independent of width of tire. 3. It is inversely as square root of diameter of wheel. 4. It is independent of speed. * See Treatise on Roads , Streets , and Pavements , by Brev. Maj.-Gcn'l Q. A. Gillmore, U. S. A. t Telford estimated it at 3.5. 844 TRACTION. On Paved and Rough Roads. Resistance increases with speed, and is diminished by an enlargement of tire up to a moderate limit. Traction on Various Roads . — Traction of a wheeled vehicle is to its weight upon various roads as follows : Per Ton. Stone track, best 12.5 to 15 “ “ .... 28 to 39 “ pavement. 14 Asphalted. ...... 22 Plank 22 Block stone ( pavement. . . . J 32 to 36 to 28 to 45 to 35 Per 100 lbs. Per Ton. • 55 to .58 Telford road 46 to 78 x.25 to 1.3 Macadamized. . . 46 to 90 .5 to 1.5 “ loose 67 tO I 12 I to 1.25 Gravel 134 to 180 .98 to 2 Sandy 140 to 3*3 1.4 to 1.6 Earth 4 200 to 29c Per 100 lbs. 2. 1 to 3. 5 2 to 4 3 to s 6 to 8 6.3 to 14 9 to 13 Hence, a horse that can draw 140 lbs. at a walk, can draw upon a gravel road 6 -f - 8 140 X 100 = 2000 lbs. 2 Resistance on Common Roads or Fields. ( Bedford Experiments , 1874.*) Gravelled Road. (Hard and dry , rising 1 in 430.) Maxi- mum Draft. Average Draft. Average Speed per Hour. H> de- veloped per Minute. Draft per Ton on Level. Work per IP per Horse. Lbs. Lbs. Miles. tP. Lbs. IP. 2 horse wagon without springs. 320 159 2-5 1.06 43.5 or. 0192 •53 4 “ “ “ “ 400 251 2.6 1.74 44-5 “ -02 .87 2 “ “ with “ 300 133 2.47 .88 34-7 “ * OI 5 •44 1 “ cart without “ 180 49.4 2.65 •35 28 “ .Q125 •35 Arable Field. (Hard and dry , rising 1 in 1000.) 2 horse wagon without springs. 1000 700 2-35 4.36 210 or. 099 2. 18 4 !! “ - u !! 1200 997 2.52 6.7 194 “ .0^3 3-35 2 “ “ with “ 1000 710 ^•35 4-45 210 “ .099 1.22 1 “ cart without “ 400 212 2.61 1.48 140 -0625 1.48 Fore wheels of wagons were 39 ins., and hind 57 ins. in diam. ; tires varying from 2.25 to 4 ins. ; and wheels of cart were 54 ins. in diam., and tires 3.5 and 4 ins. Springs reduced resistance on road 20 per cent., but did not lessen it in the field. From these data it appears, that on a hard road, resistance is only from .25 to .16 of resistance in field. Lowest resistance is that of cart on road = 28 lbs. per ton; due, no doubt, to absence of small wheels alike to those of the wagons. Assuming average power without springs to be .6 IP on road, as average for a day’s work, it represents .6 X 33000 = 19800 foot-lbs. per minute for power of a horse on such a road. Resistance of a smooth and well-laid granite track (tramway), alike to those in London and on Commercial Road, is from 12.5 to 13 lbs. per ton. Omnibus. f (Weight 5758 lbs.) Average Speed per Hour. Per Ton. Total. Granite pavement (courses 3 to 4 ins.) 2.87 miles. 17.41 lbs. 44.75 lbs. Asphalt roadway 3.56 “ 27.14 “ 69-75 “ Wood pavement. . 3.34 “ 41.6 “ 106.88 “ Macadam road, gravelly 3.45 “ 44-48 “ 114.32 “ “ “ granite, new 3.51 101.09 “ 259.8 “ Note. — The resistance noted for an asphalt roadway is apparently inconsistent with that for a granite pavement, for when it is properly constructed it is least resistant of all pavements. * See report in Engineering , July io, 1874, page 23. f Report Soc. Arts , London , 1875. TRACTION, 845 Per Ton. Total. 31.2 lbs. 33 J bs. 44 u 46 “ 62 u 65 “ 140 “ 147 “ W agon. ( Sir John Macneil. ) Weight 2342 lbs. Speed 2.5 Miles per Hour. Well-made stone pavement Road made with 6 ins. of broken hard stone, on a foundation) of stones in pavement, or upon a bottom of concrete | [ Old flint road, or a road made with a thick coating of broken | Road made with a thick coating of gravel, on earth Stage Coach.. {Sir John Macneil) Weight 3192 lbs. Gradients 1 to 20 to 600. Metalled Road. At 6 mfie's per hour 62 lbs. per ton. :: ■* :: « W » « TJnnr —It was found that, from some unexplained cause, the net frictional resistance at equal speeds . . prmRiderablv according to gradient, resistances being a maximum for steepest gradient , and a varied cons^erably , accormng . g , e less than x i n 600. Mode of action of To Compute Resistance to Traction on Various Roads. (Sir John Macneil.) ON A LEVEL. Rule —Divide weight of vehicle and load in lbs. by its unit in following table, and to quotient add .025 of load; add sum to product of velocity of vehicle in feet per second, and Coefficient in following table for the particular road, and result will give power required in lbs. Or W + w 1 w .025 + Cv = T. W and w repi'esenting weights of vehicle and load ’ unit ' Coefficients for Traction of Various Vehicles. Stage coach I 2 horse wagon without springs 54 TTpuvv wiip-nn cn 2 “ u with ‘ 42 Heavy wagon 93 4 horse wagon without springs 55 I 1 cart without 4-3 Coefficients for Roads of Various Construction. Macadamized road Gravel, clean 13 u muddy 3 2 Stone tramway • • 1.2 .... 12. 1 Pavement 2 Broken stone, dry and clean 5 “ “ covered with dust. .. . 8 “ “ muddy 10 Sand and Gravel Illustration. — What is the traction or resistance of a stage coach weighing 2200 lbs., with a load of 1600 lbs., when driven at a velocity of 9 feet per second over a dry and clean broken stone road ? 2200 + 1600 i6qq x -025 .j. 5 x 9 = 123 lbs. 100 To Compute Rower necessary to Sustain a "Vehicle -upon aix Inclined. Road, and also its Rressure thereon, omit- ting Efi'ect of Friction. AT AN INCLINATION. W : A C : : 0 : B C, and W : A C : : p : A B. Or, r e : e 0 :: A B : B C; W : c o :: l : h: whence, W j = eo - * A Assume A B of such a length that vertical rise, B BC = i foot; then, W _ W ac_ Vab 2 4-i = W sin. A = 0. and W A B WAB AG VAB 2 4-i B = W cos. A — p. TRACTION. 846 W W V W ■ W Z' 2 Or, — = P: — — =P] or, - - == P, and — =p. W representing 1 1 VZ' 2 + 1 Vz' 2 +i weights of vehicle and load 0, and ¥ power or force necessary to sustain load on road , p pressure of load on surface , all in lbs ., h height of plane , Z inclined length of road or plane, and l' horizontal length, all in feet. Illustration. — W hat is power required to sustain a carriage and its load, weigh- ing 3800 lbs., upon a road, inclination of which is 1 in 35, and what is its pressure upon road ? Sin. A = .028 56. Cos. A = .99994. Z = 35.014. Then 3800 X • 028 56 = 108. 53 lbs. = power, and 3800 X • 999 94 = 3799- 77 Z&s. pressure. To Compute Resistance of a Load, on an Inclined. Road. Rule. — A scertain the tractive power required, and add to it the power necessary to sustain load upon inclination, if load is to ascend, and subtract it if to descend. Example 1.— In preceding example tractive power required is 123 lbs., and sus- taining power for that inclination 108.53; hence 123-1-108.53 = 231.53 lbs. 2. — If this load was to be drawn down a like elevation. Then 123 — 108.53 = 14.47 ^ s • To Compute Rower necessary to Move and. Sustain a "Vehicle either Ascending or Descending an Elevation, and at a given Velocity, omitting Effect of Friction. ( W -4- w w\ — — — h 1- — cos. L + (W -4- w) sin. L. 4- v c = R. L. representing angle of t 40/ elevation for a stage wagon and a stage coach, and t units as preceding ; upper sign taken when vehicle descends the plane , and lower when it ascends. Illustration. — A ssume a stage coach to weigh 2060 lbs., added to which is a load of 1100 lbs., running at a speed of 9 feet per second over a broken stone road covered with dust, and having an inclination of 1 in 30; what is power necessary to move and sustain it up the inclination, and what down it ? v = g , c = 8, sin. of L. = sin. of i° 54'+ = .0333, and cos. L- =.9995. /2060 + 1100 , noo\ . — : 7 . . Then ( \- - ) x .9995 + ( 2060+ 1100) X -0333 + 8 X 9 = 59-°7 + \ 100 40 / 105.23 + 72 = 236.3 lbs. up inclination. . , /2060 + 1 100 . 1 roo\ . - — — — - — : — ‘ . And ^ — b J X .9995 + 8X9 — (2060 + 1100) X .0333 = 59.07 + 72 — 105. 23 = 25. 84 lbs. down inclination. Tractive and Statical Resistance of Elevations. (Gillmore.) T ■■ - —g. T representing traction in lbs. per ton , W weight of load in lbs., VW 2 - T2 and g' grade of road. Illustration.— Assume traction as per preceding table, page 844, 200, and weight of vehicle 2 tons ; what should be least grade of road ? 200 X 2 \J 4480 2 — - = .0897 = - + . 200 X 2 Showing that, for a road upon which traction is 200 lbs. per ton, the grade should not exceed one in height to one eleventh fall of base; hence, generally, the proper grade of any description of road will be equal to force necessary to draw load upon like road when level. Practically, greatest grade of a Telford or Macadamized road in good condition = .05, and a horse can attain at a walk a required height upon this grade, without more fatigue and in nearly same time that he would require to attain a like height over a longer road with a grade of .033, that he could ascend at a trot. For passenger traffic, grades should not exceed .033. traction. 847 Resistance of Gravity at Different Inclinations of Grrade. For a Load of 100 Los. 1 in 5 1 in 10 1 in 15 1 in 20 Lbs. 19.61 9-95 6.65 4.99 1 in 25 1 in 30 1 in 35 in 40 1 in 45 1 in 50 1 in 55 1 in 60 Lbs. 2.22 1 in 70 1 in 80 1 in 90 1 in 100 Inclination of Roads .- Power of draught at different inclinations and velocities is as follows ( Sir John Macneil) : Inclination. Angle. 1 in 20 2° 52' i in 26 2 ° X2' 1 in 30 1° 55' 1 in 40 1° 26' 1 in 60 57 - S' Traction at Speeds of per Feet Hour of per Mile. 6 Miles. 8 Miles. 10 Miles. 264 268 296 203.4 213 219 225 176 165 196 200 132 160 166 172 88 in 120 128 Frictional Resistance per Ton at Speeds of per Hour ol 6 Miles. 8 Miles. 10 Miles. 76 96 112 63 68 4 1 63 66 56 61 65 72 78 81 Grrade. Grade of a road should be reduced to least of practicable attainment, and as a general rule should be as low as 1 in 33, and steepest grade that is ad- missible on a broken stone road is 1 in 20. The condition of traction is /+sin. a L, which should not exceed P, and sin. a should not exceed j- —f or/, f representing coefficient of friction , a angle of in- clination, L load , and P power in lbs. Illustration.- In case, page 846, weight or load = 2060 + 1x00 = 3160 ’ efficient of friction for such a road = .042 per 100 lbs., and sin. 1 54 - -°33 1 • Then .042 + .033 16 X 3160 — 237.5 lbs. Traction of a Vehicle compared to its Weight on Different Roads. (F. Robertson , F. R. A. S.) 1 in 68 I Flint foundation 1 hj 34 1 “ 49 I Gravel road 1 ‘ J 5 Sandy road 1 in 7. Assuming a horse to have a tractive force of 140 lbs. continuously and steadily at a walk, he can draw at a walk on a gravel road 15 X 140 = 2100 lbs. Friction of* Ptoacls. Friction of Roads— According to Babbage and others, a wagon and load weighing 1000 lbs. requires a traction as follows . Of Load. of Load * Macadamized °33 Dry high road 025 Well paved road 014 ( .0035 Railroad | .0059 033 of load. Stone pavement — Macadamized road Gravelled road in Gravel road, new 083 Common road, bad order. . .07 Sand road 063 Broken stone, rutted 052 “ “ fair order... .028 “ “ perfect order .015 Macadamized, new 045 “ 033 “ gravelly 02 Earth, good order 025 Sled , hard snow, iron shod.,,. . Coefficients of Friction in Proportion to Load. Per 100 lbs. Per Ton. Per 100 lbs. Wood pavement 019 Asphalt roadway 012 Stone pavement 015 Granite “ 008 Stone “ very smooth .006 Plank road 01 1 -0035 Railway j .0059 Stone track 05 848 TRACTION. To Compute Frictional Resistance to Traction of* a Stage Coach on a NXetalled Road in Good Condition. 3° + 4 v + V 10 v — R - v representing speed in miles per hour , and R frictional resistance to traction per ton. Note.— F ormula is applicable to wagons at low speeds. Canal, Slackwater, and. River. On a canal and water, resistance to traction varies as square of velocity from that of 2 feet per second to that of 11.5 feet. When velocity is less than .33 miles per hour, resistance varies in a less degree. In towing, velocity is ordinarily 1 to 2.5 miles per hour. Resistance of a boat in a canal depends very much upon the comparative areas of transverse sections of it and boat, it being reduced as difference increases. In a mixed navigation of canal and slack-water, 3 horses or strong mules will tow a full-built, rough-bottomed canal boat, with an immersed sectional area of 94.5 sq. feet, and a displacement of 240 tons, 1.75 to 2 miles per hour for periods of 12 hours. With a section of but 24.5 sq. feet, or a displacement of 65 tons, an aver- age speed of 2.5 miles is attained for a like period. By the observations of Mr. J. F. Smith, Engineer of the Schuylkill Navigation Co., a canal boat, with an immersed section alike to that above given, can be towed for 10 hours per day as follows: Per Hour. By 1 horse or mule. I By 2 horses 'or mules. By 3 horses or mules. By 4 horses or mules. By 8 horses or mules. 1 mile. 1.5 miles. 1.75 miles. 1.875 miles. 2.5 miles. Assuming then, the tractive power of a horse as given in table, page 437, the above elements determine results as follows: Horses. Miles. Tractive Power divided by Load. in Lbs. per Ton. rraction in Lbs. per Sq. Foot of immersed Section. 1 1 2504-240 1.04 2.65 2 i-5 165 X 24-240 1.38 3-49 3 I -75 140 X 3 -S- 240 i -75 4.44 3 i - 8 75 132 X 3 4 - 240 1-65 4.19 3 2 125 x 34-240 1.56 3-9 8 3 (hght) 2-5 100 X 3 -4- 65 4.61 * | 12.24 Upon a canal of less section and depth, a displacement of 105 tons, with an im- mersed section of 43 sq. feet, a speed of 2 miles with 2 horses was readily obtained, which would give a traction of 2.38 lbs. per ton, and of 5.71 lbs. per sq. foot of im- mersed section. Maximum Power of a. Horse 011 a Canal. (Molesworth.) Miles per hour 2.5 3 3.5 45678 9 10 Duration of work in ) 0 hours } II- 5 8 5-9 4-5 2.9 2 1.5 1.125 -9 -75 Load drawn in tons.. 520 243 153 102 52 30 19 13 9 6.5 Street Railroads or Tramways. ( GenH Gillmore.*) Upon a level road, and at a speed of 5 miles per hour, the power required to draw a car and its load is from to of total weight, varying with condition of rails and dryness or moisture of their surface. * Treatise on Roads, Streets, and Pavements. D. Van Nostrand, 1876, N. Y. TRACTION. WATER. 849 To Compute Resistance of* a Car. Ty6 _f. Txv — c - v l*L?L — r ; and /-f-c-f-r = R. T representing weight A A u J ) } 400 _ in tons, f f notion in lbs., v speed in miles per hour , a area of front or section of car in sq. feet , c concussion , r resistance of atmosphere , and R total resistance , aii in tos. Illustration. — Assume a car and load of 8960 lbs., with an area of section of 56 sq. feet, and a speed of 5 miles per hour. Then 5 2 X 56 400 896° __ ^ £ 0ns . ^ x 6 = 24 lbs. friction ; - = 6.66 Z&s. concussion ; 2240 3 = 3. 5 Z&s. resistance of air ; and 24 -j- 6. 66 + 3. 5 = 34- 16 In average condition of a road, the resistance of a car may be taken at which, in preceding case, would be 74.66 lbs. On a descending grade, therefore, of 1 in 74.66, the application of a brake would not be required. WATER. Fresh Water. Constitution of it by weight and measure is By Weight. By Measure. I By Weight. By Measure. Oxygen... 88.9 1 I Hydrogen.. 11.1 2 Cube inch of distilled water at its maximum density of 39.1 0 , barom- eter at 30 ins., weighs 252.879 grains, and it is 772.708 times heavier than atmospheric air. Cube foot (at 39. i°) weighs 998.8 ounces, or 62.425 lbs. N oxe< For facility of computation, weight of a cube foot of water is usually taken at 1000 ounces and 62.5 lbs. At a temperature of 32 0 it weighs 62.418 lbs., at 62° (standard tem- perature) 62.355 lbs., and at 212 0 59- 6 4 lbs. Below 39.1 0 its density decreases, at first very slow, but progressing rapidly to point of conge- lation, weight of a cube foot of ice being but 57.5 lbs. Its weight as compared with sea-water is nearly as 39 to 40. It expands .085 53 its volume in freezing. From 40° to 12 0 it ex- pands .00236 its volume, and from 40° to 212 0 it expands .0467- times = .000271 5 for each degree, giving an increase in volume of 1 cube foot in 21.41 feet. "Volumes of Pure Water. At 32 0 27.684 cube ins. = 1 cube foot. “ 39-i° 27.68 u “ = 1 “ 62° 27.712 “ “ = 1 “ 212° 28.978 “ “ =1 U * At 62° 1 Ton =35-923 cube feet. “ “ 1 Lb. =27.71 “ ins. “ 39.1 0 1 Tonneau = 35.3156 “ feet. “ “ 1 Kilogr. = .0353 “ “ Height of a Column of Water at 62° or 62.355 lbs. 1 lb. per sq. inch = 2. 3093 feet, and at pressure of atmosphere = 33.947 feet = 10.347 meters. Ice and. Snow. Cube foot of Ice at 32 0 weighs 57.5 lbs., and 1 lb. has a volume of 30.067 cube ins. Volume of water at 32 0 , compared with ice at 32 0 , is as 1 to 1.085 53 > ex- pansion being 8.553 P er cent * Cube foot of new fallen snow weighs 5.2 lbs., and it has 12 times bulk of water. 850 WATER. Rainfall. Annual Fall at different Places . Location. Alabama Albany Algiers Alleghany Antigua Archangel Auburn Bahamas Baltimore Barbadoes Bath, Me Belfast Biskra Bombay Bordeaux Boston Brussels Buffalo Burlington, Vt Calcutta Cape St. Franpois. . Cape Town Charleston Cherbourg Cologne Copenhagen Cracow Demerara 150 23- 31 54 39-7 24 23 *3-33 91.2 132.21 37- 52 30.87 36.92 38- 52 24- 5 v 1 2 9 Fairfield .......... | 32.93 Dover (Engl.) . . Dublin Dumfries East Hampton . Edinburgh Ins. 30 -I 7 4i-35 7-75 46.66 45 14.52 30.17 42.19 39-9 55-87 34-58 39-46 IIO 29.7 39-23 29 27.27 32 Location. Ft. Crawford, Wis.. Ft. Gibson, Ark Ft. Snelling, Iowa.. Fortr. Monroe, Va. Florence Frankfort, Oder. . “ Main. . Geneva Gibraltar Glasgow Gordon Castle, Sc’d Granada Great Britain . Greenock Halifax Hanover Havana Hong-kong Hudson India. Jamaica Jerusalem Key West Khassaya, India. . . Lewiston Liverpool London Louisiana Madeira . . . . Manchester . Marseilles. . Ins. 29-54 30.64 30.32 52-53 35-9 21.3 16.4 32.6 47.29 21.3 3 i 2 9-3 105 126 32 61.8 33 22.4 52 8i.35 39-32 60 130 34-31 65 31.39 610 23.15 34.12 25.2 51-85 Location. 22 4 l 36.14 43 18.2 Michigan Mississippi Mobile, 1842 Naples Newburg New York Ohio Palermo Paris Philadelphia Plymouth (Engl.).. Port Philip Poughkeepsie Providence Rochester Rome Santa Fe Savannah Schenectady Siberia . . . .' Sierra Leone Sitka St. Bernard St. Domingo St. Petersburg State of N.Y Sydney Tasmania Trieste Ultra Mullay, India Utica Venice Vera Cruz Vienna Washington West Point Ins. 33-5 45 54-94 41.8 40.5 36 36 22.8 23.1 49 44 29. 16 32.06 36.74 29 39 74.8 55 47-77 7-75 85-79 48 120 17.6 33- 79 49 35 46.4 263.21 39-3 34- i 62 19.6 34.62 48.7 Average rainfaH in England for a number of years was, South and East, 34 ins • W est and hilly, 43.02 to 50 ins., and percolation of it was estimated at 30 per cent. ’ Mean volume of water in a cube foot of air in England is 3.789 grains. Globe, mean depth ,5 ins Cape of Good Hope in 1846 in 3 hours, 6.2 “ At Khassaya, in 6 rainy months 550 ins. ; in 1 day, 25.5 “ Evaporation.— Mean daily evaporation, in India .22 inch; greatest .56* in Eng- land .08. At Dijon, when mean depth of rainfall was 26.9 ins. in 7 years’ evapora- tion was for a like period 26.1 ins., and in Lancashire, Eng., when fall was as 06 ins., evaporation was 25.65. Volume of* Rainfall. Rainfall, depth in ins. , x 2 323 200 = cube feet per sq. mile. X 17-37874 = millions of gallons per sq. mile. X 3630 = cube feet per acre. X 27 154. 3 = gallons per acre. Mineral Waters are divided into 5 groups, viz. : _. Carbonated, containing pure Carbonic acid — as, Seltzer, Germany; Spa, Bel- gium; Pyrmont, Westphalia; Seidlitz, Bohemia; and Sweet Springs, Virginia.’ 2. Sulphurous, containing Sulphuretted hydrogen— as, Harrowgat’e and Chelten- ham, England; Aix-la-Chapelle, Prussia; Blue Lick, Ky. ; Sulphur Springs, Va., etc. 3. Alkaline, containing Carbonate of soda— these are rare, as, Vichy, Ems. WATER. 851 a. Chalybeate, containing Carbonate of iron— as, Hampstead, Tunbridge, Chelten- ham, and Brighton, England; Spa, Belgium; Ballston and Saratoga, N. Y. ; and Bedford, Penn. c. Saline, containing salts— as, Epsom, Cheltenham, and Bath, England; Baden- Baden and Seltzer, Germany; Kissingen, Bavaria; Plombieres, France; Seidlitz, Bohemia ; Lucca, Italy ; Yellow Springs, Ohio ; Warm Springs, N. C. ; Congress Springs, N. Y. ; and Grenville, Ky. Brief Rules for Qualitative Analysis of Mineral Waters. First point to be determined, in examination of a mineral water, is to which of above classes does water in question belong. 1. If water reddens blue litmus paper before boiling, but not afterwards, and blue color of reddened paper is restored upon warming, it is Carbonated. 2. If it possesses a nauseous odor, and gives a black precipitate, with acetate of lead, it is Sulphurous. 3. If, after addition of a few drops of hydrochloric acid, it gives a blue precipitate, with yellow or red prussiate of potash, water is a Chalybeate. 4. If it restores blue color to litmus paper after boiling, it is Alkaline. 5. If it possesses neither of above properties in a marked degree, and leaves a large residue upon evaporation, it is a Saline water. River or canal water contains .05 \ 0 £ vo j ume 0 f gaseous matter. Spring or well water .07) Tie-agents. When water is pure it will not become turbid, or produce a precipitate with any of following Re-agents : Baryta Water, if a precipitate or opaqueness appear, Carbonic Acid is present. Chloride of Barium indicates Sulphates, Nitrate of Silver, Chlorides, and Oxalate of Ammonia, Lime salts. Sulphide of Hydrogen, slightly acid, Antimony, Arsenic, Tin, Copper, Gold, Platinum, Mercury, Silver, Lead, Bismuth, and Cadmium; Sul- phide of Ammonium , solution alkaloid by ammonia, Nickel, Cobalt, Manganese, Iron, Zinc, Alumina, and Chromium. Chloride of Mercury or Gold and Sulphate of Zinc, organic matter. Filter Beds. Fine sand, 2 feet 6 ins. ; Coarse sand, 6 ins. ; Clean shells, 6 ins., and Clean gravel 2 feet, will filter 700 gallons water in 24 hours by gravitation. Sea Water. Composition of it per volume: Chloride of Sodium (common salt). .2.51 Sulphuret of Magnesium 53 Chloride of “ 33 Carbonate of Lime “ of Magnesia Sulphate of Lime Water .01 96.6 By analysis of Dr. Murray, at specific gravity of 1.029, it contains Muriate of Soda 220.01 I Muriate of Magnesia 42.08 Sulphate of Soda 33. 16 | Muriate of Lime 7.84 303.09 Or, 1 part contains .030309 parts of salt = 3V P ar t °f its weight. Mean volume of solid matter in solution is 3.4 per cent., .75 of which is common salt. Boiling Points at Different Degrees of Saturation. Salt, by Weight, Boiling Salt, by Weight, Boiling Salt, by Weight, Boiling in 100 Parts. Point. in 100 Parts. Point. in 100 Parts. Point. 3-°3 = it 213. 2° I S - 1 5 = WS 217. 9 0 27.28=,^ 222. 5° 6-°6 = Ts 214. 4 0 l8.l8 = -^ 219 0 30 - 31 = if 223.7° 9.09= A 2 iS- 5 ° 2 I .22 = -3 7 3 220.2° 33-34 = ii 224.9° I2 - 12 — -h 216. 7 0 2 4-25 = A 221 .4° *36.37 =i| 226° * Saturated. WATER. WAVES OF THE SEA. 852 Deposits at Different Degrees of Saturation and. Tem- perature. When 1000 Parts are reduced by Evaporation. Volume of Water. Boiling Point. Salt in 100 Parts. Nature of Deposit. 1000 214 0 3 None. 299 217 0 10 Sulphate of Lime. 102 228° 29*5 Common Salt. It contains from 4 to 5.3 ounces of salt in a gallon of water. Saline Contents of Water from several Localities. Baltic 6.6 I Black Sea 21.6 Arctic 28.3 I British Channel. .. . 35.5 Mediterranean 39.4 Equator 39-42 South Atlantic. ..... 41.2 North Atlantic 42.6 Dead Sea 385 There are 62 volumes of carbonic acid in 1000 of sea- water. Cube foot at 62° weighs 64 lbs. Its weight compared with fresh water being very nearly as 40 to 39. Height of a Column of Water at 6o° or 64.3125 lbs. At 62°, 1 Ton = 35 cube feet. 1 Lb. per sq. inch = 2.239 f eet > an d at pressure of atmosphere = 32.966 feet = 10.048 meters. 'Weights. A ton of fresh water is taken at 36, and one of salt at 35 cube feet. WAVES OF THE SEA. Arnott estimated extreme height of the waves of an ocean, at a distance from land sufficiently great to be freed from any influence of it upon their culmination, to be 20 feet. French Exploring Expedition computed waves of the Pacific to be 22 feet in height. By observations of Mr. Douglass in 1853, he deduced that when waves had heights of 8 feet, there were 35 in number in one mile, and 8 per minute. 15 “ “ 5 and 6 “ “ 5 “ 20 “ “ 3 “ “ 4 “ J. Scott Russell divides waves into 2 classes — viz. : Waves of Translation, or of 1st order ; of Oscillation, or of 2d order. Waves of* the Dirst Order. 1. Velocity not affected by intensity of the generating impulse. 2. Motion of the particles always forward in same direction as wave, and same at bottom as at surface. 3. Character of wave, a prolate cycloid, in long waves, approaching a true cycloid. When height is more than one third of length, the wave breaks. Waves of* the Second Order. 1. Ordinary sea waves are waves of second order, but become waves of the first order as they enter shallow water. 2. Power of destruction directly proportional to height of wave, and great- est when crest breaks. 3. A wave of 10 feet in height and 32 feet in length would only agitate the water 6 ins. at 10 feet below surface; a wave of like height and too feet in length would only disturb the water 18 ins. at same depth. Average force of waves of Atlantic Ocean during summer months, as de- termined by Thomas Stevenson , was 61 1 lbs. per sq. foot; and for winter months 2086 lbs. During a heavy gale a force of 6983 lbs. was observed. WAVES OF THE SEA. 853 J. Scott Russell deduced that a wave 30 feet in height exerts a force of 1 ton per sq. foot, and that, in an exposed position in deep water, 1.75 tons may be exerted upon a vertical surface. At Cassis, France, when the water is deep outside, blocks of 15 cube me- ters were found insufficient to resist the action of waves. Breakwaters with vertical walls, or faces of an angle less than 1 to 1, will reflect waves without breaking them. Waves of oscillation have no effect on small stones at 22 feet below the surface, or on stones from 1.5 to 2 feet, 12 feet below surface. A roller 20 feet high will exert a force of about 1 ton per sq. foot. Greatest force observed at Skerry vore, 3 tons per sq. foot ; at Bell Rock, 1.5 tons per sq. foot. Waves of the second order, when reflected, will produce no effect at a depth of 12 feet below surface. Action of waves is most destructive at low-water line. Waves of first order are nearly as powerful at a great depth as at surface. To Compute 'Velocity'. When l is less than d. .55 yjl or 1.818 y/l — V. When l exceeds 1000 d. V 32.17 d = V, and When Height of Wave becomes a sen- sible Proportion to Depth , ^32. *7 y + 3 = V. To Compute Height of Waves in Reservoirs, etc. x. 5 ^/l + (2. 5 — V L ) = height in feet. L representing length of Reservoir, Pond, etc., exposed to direction of wind, in miles. Tidal Waves. Wave produced by action of sun and moon is termed Free Tide Wave. K Semi-diurnal tide wave is this, and has a period of 12 hours 24-f- minutes. Professor A iry declared that when length of a wave was not greater than depth of the water, its velocity depended only upon its length, and was pro- portionate to square root of its length. When length of a wave is not less than 1000 times depth of water, velocity of It depends only upon depth, and is proportionate to square root of it; velocity being same that a body falling free would acquire by falling through a height equal to half depth of water. For intermediate proportions, velocity can be obtained by a general equation. Under no circumstances does an unbroken wave exceed 30 or 40 feet in height. A wave breaks when its height above general level of water is equal to general depth of it. Diurnal and other tidal waves, so far as they arc free, may be all considered as running with the same velocity, but the column of the length of wave must be doubled for diurnal wave. Length of Wave. Depth of Water. Feet. 1 j Feet. 10 I Feet. I Feet. 100 1 1000 1 Velocity per Second. Feet. 10 000 1 Feet. ; IOO OOO Feet. Feet. Feet. Feet. Feet. Feet. Feet. 1 2.26 5-34 5-67 — — — 10 2.26 7 -i 5 16.88 17.92 17-93 — 100 — 7- I 5 22.62 53-19 56.67 56.71 1 000 — 22.62 7 i -54 168.83 179.21 10000 — — — 7 i -54 226.24 533-9 4 c WHEEL GEARING. 854 WHEEL GEARING. Pitch Line of a wheel is circle upon which pitch is measured, and it is circumference by which diameter, or velocity of wheel, is measured. Pitch is arc of circle of pitch line, is determined by number of teeth in wheel, and necessarily an aliquot part of pitch line. True or Chordial Pitch , or that by which dimensions of tooth of a wheel are alone determined, is a straight line drawn from centres of two contiguous teeth upon pitch line. Line of Centres is line between centres of two wheels. Radius of a wheel is semi - diameter bounded by periphery of the teeth. Pitch Radius is semi-diameter bounded by pitch line. Length of a Tooth is distance from its base to its extremity. Breadth of a Tooth is length of face of wheel. Depth of a Tooth is thickness from face to face at pitch line. Pace of a Tooth , or Addendum , is that part of its side which extends from its pitch line to its top or Addendum line. Flank of a Tooth is that part of its side which extends from pitch line to line of space at base of and between adjacent teeth ; its length, as well as that of face of tooth, is measured in direction of radius of wheel, and is a little greater than face of tooth, to admit of clearance between end of tooth and periphery of rim of wheel or rack. Cog Wheel is general term for a wheel having a number of cogs or teeth set in or upon, or radiating from, its circumference. Mortice Wheel is a wheel constructed for reception of teeth or cogs, which are fitted into recesses or sockets upon face of the wheel. Plate Wheels are wheels without arms. Rack is a series of teeth set in a plane. Sector is a wheel which reciprocates without forming a full revolution. Spur Wheel is a wheel having its teeth perpendicular to its axis. Bevel Wheel is a wheel having its teeth at an angle with its axis. Crown Wheel is a wheel having its teeth at a right angle with its axis. Mitre Wheel is a wheel having its teeth at an angle of 45 0 with its axis. Face Wheel is a wheel having its teeth set upon one of its sides. Annular or Internal Wheel is a wheel having its teeth convergent to its centre. Spur Gear. — Wheels which act upon each other in same plane. Bevel Gear —Wheels which act upon each other at an angle. Inside Gear or Pin Gearing.— Form of acting surfaces of teeth for a pitch-circle m inside gearing is exactly same with those suited for same pitch-circle in outside gearing, but relative position of teeth, spaces, and flanks are reversed, and adden- dum-circle is of less radius than pitch-circle. A Train is a series of wheels in connection with each other, and consists of a series of axles, each having on it two wheels, one is driven by a wheel on a preced- ing axis and other drives a wheel on following axis. Idle Wheel. — A wheel revolving upon an axis, which receives motion from a pre- ceding wheel and gives motion to a following ivheel, used only to affect direction of motion. Trundle. Lantern, or Wallower is when teeth of a pinion are constructed of round bars or solid cylinders set into two disks. Trundle with less than eight staves can- not be operated uniformly by a wheel with any number of teeth. Spur, Driver, or Leader is term for a wheel that impels another: one impelled is Pinion , Driven , or Follower. WHEEL GEARING. 855 Teeth of wheels should be as small and numerous as is consistent with strength. When a Pinion is driven by a wheel , number of teeth in pinion should not be less than 8. When a Wheel is driven by a pinion , number of teeth in pinion should not be less than jo. When 2 wheels act upon one another, greater is termed Wheel and lesser Pinion. When the tooth of a wheel is made of a material different from that of wheel it is termed a Cog ; in a pinion it is termed a Leap in a trundle a Stave, and on a disk a Pin. Material of which cogs are made is about one fourth strength of cast iron. Hence, product of their b d 2 should be 4 times that of iron teeth. Number of teeth in a wheel’ should always be prime to number of pmion ; that is, number of teeth in wheel should not be divisible by number of teeth in pinion without a remainder. This is in order to prevent the same teeth coming together so often and uniformly as to cause an irregular wear ot their faces. An odd tooth introduced into a wheel is termed a Hunting tooth or Cog. The least number of teeth that it is practicable to give to a wheel is regu- lated bv necessity of having at least one pair always in action, in order to provide for the contingency of a tooth breaking; and least number that can be employed in pinions having teeth of following classes is : Involute, 25 , Epicycloidal, 12 ; Staves or Pins, 6. Velocity Ratio in a Train of Wheels. — lo attain it with least number of teeth, it should, in each elementary combination, approximate as near as practicable to 3.59. A convenient practical rule is a range from 3 to 6. Illustration. i 6 36 216 1296 velocity ratio. j 2 3 4 elementary combination. To increase or diminish velocity in a given proportion, and with least quantity of wheel-work, number of teeth in each pinion should be to number of teeth in its wheel as 1 : 3-59- Even to save space and expense, ratio should never exceed 1 : 6. (Buchanan.) To Compute IPitcli. Rule —Divide circumference at pitch-line by number of teeth. Example.— A wheel 40 ins. in diameter requires 75 teeth; what is its pitch? 3.1416 X 40-^75 — 1.6755 ins. To Compute True or CLordial iPitcli. r ule —Divide 180° by number of teeth, ascertain sine of quotient, and multiply it by diameter of wheel. Example. — Number of teeth is 75, and diameter 40 ins. ; what is true pitch? l8o -5- 75 = 2 0 24', and sin. of 2 0 24' = .041 88, which X 40 = 1-6752 ins. To Compute Diameter. Rule.— M ultiply number of teeth by pitch, and divide product by 3.1416. Example.— Number of teeth in a wheel is 75, and pitch 1.6755 ins. ; what is di- ameter of it ? 75 x 1.6755 -5- 3. 1416 = 40 ins. When the True Pitch is given. Rule.— M ultiply number of teeth in wheel by true pitch, and again by .3184. " Example. — Take elements of preceding case. 75 x 1.6752 X .3184 = 40 ins • Or, Divide 180° by number of teeth, and multiply cosecant of quotient by pitch. l8o ^- 75 = 2 o 2 4 / , and cos. 2 0 24' = 23.88, which X 1.6752 = 40 ins. 856 WHEEL GEARING. To Compute INTurribei' of* Teeth. Rule. — Divide circumference by pitch. To Compute INAxrxiber of* Teeth in a Finion or Follower to have a given Velocity. Rule.— M ultiply velocity of driver by its number of teeth, and divide product by velocity of driven. Example i.— Velocity of a driver is 16 revolutions, number of its teeth and velocity of pinion is 48; what is number of its teeth? 16 X 54-4-48 = 18 teeth. 2.— A vvheel having 75 teeth is making 16 revolutions per minute: what is num- ber of teeth required in pinion to make 24 revolutions in same time? 16 X 75 -7- 24 = 50 teeth. To Compute Proportional Ptadins of* a Wheel or Finion. Rule.— Multiply length of line of centres by number of teeth in wheel for vvheel, and in pinion, for pinion, and divide by number of teeth in both vvheel and pinion. Example.— L ine of centres of a wheel and pinion is 36 ins., and number of teeth in wheel is 60, and in pinion 18; what are their radii? 3 3 * * 6 X 60 _ 7 36 X 18 0 . - = 27.69 ins. wheel. ^ ^ = 8.3 ms. pinion. 60 -f- 18 To Compute Diameter of* a Fin ion. . When Diameter of Wheel and Number of Teeth in Wheel and Pinion are gwen. Rule. Multiply diameter of vvheel by number of teeth in pinion, and divide product by number of teeth in wheel. Example.— D iameter of a wheel is 25 ins., number of its teeth 210, and number of teeth in pinion 30; what is diameter of pinion? 25 X 30-f- 210 = 3. 57 iws. To Compute Number of* Teeth required in a Train of Wheels to produce a given Velocity. Rule. Multiply number of teeth in driver by its number of revolutions and divide product by number of revolutions of each pinion, for each driver and pinion. inH^i MPLE i~ I , f a ? river in . a train of three wheels has 90 teeth, and makes 2 revo- of^thertwo J elocitles re( l uire d are 2, 10, and 18, what are number of teeth in each 10 : 90 : : 2 : 18 = teeth in 2 d wheel. 18 : 90 : : 2 : 10 = teeth in 3 d wheel. To Compute Velocity of* a Pinion. Rule.— Divide diameter, circumference, or number of teeth in driver, as case may be, by diameter, etc., of pinion. When there are a Series or Train of Wheels and Pinions. Rule. — Divide continued product of diameter, circumference, or number of teeth in wheels by continued product of diameter, etc., of pinions. Example 1.— Tf a vvheel of 32 teeth drives a pinion of 10, upon axis of which there is one 01 30 teeth, driving a pinion of 8, what are revolutions of last? 3 2 ^ 30 960 — X -y — = 12 revolutions. 10 8 80 2. Diameters of a train of wheels are 6, 9, 9. 10, and 12 ins. ; of pinions, 6, 6, 6, 6, and 6 ins. ; and number of revolutions of driving shaft or prime mover is 10 • what are revolutions of last pinion ? 6 X 9 X 9 X 10 X 12 X 10 583 200 , . ~6X 6X6 X6 X 6 = ~ 777 T = 75 reMms - WHEEL GEARING. 857 To Compute Proportion that Velocities of "Wheels in a Train should, "bear to one another. Rule. — Subtract less velocity from greater, and divide remainder by one less than number of wheels in train ; quotient is number, rising in arithmet- ical progression from less to greater velocity. Example. — What should be velocities of 3 wheels to produce 18 revolutions, the driver making 3? 1 ^ 3 — — 7.5 = number to be added to velocity of driver — 7. 5 -f- 3 = 10. 5, and 3 — 1—2 10.5 _j_ 7.5 = !8 revolutions. Hence 3, 10.5, and 18 are velocities of three wheels. Pitcli of* "Wlieels. To Compute Diameter of a "Wheel for a given Pitch, or Pitch for a given Diameter. From 8 to 192 Teeth. No. of Teeth. Diame- ter. No. of Teeth. Diame- ter. No. of Teeth. Diame- ter. No. of Teeth. Diame- ter. No. of Teeth. Diame- ter. 8 2.6l 45 14.33 82 26.II 1 1 9 37.88 156 49.66 9 2-93 46 14.65 83 26.43 120 38.2 157 49.98 10 3-24 47 14.97 84 26.74 121 38.52 158 50.3 11 3*55 48 15.29 85 27.06 122 38.84 159 50.61 12 3.86 49 15.61 86 27.38 123 39.16 160 50.93 13 4.18 50 15-93 87 27.7 I2 4 39-47 l6l 51.25 14 4.49 5i 16.24 88 28.02 125 39*79 162 51-57 15 4.81 52 16.56 89 28.33 126 40.II 163 51.89 16 5.12 53 16.88 90 28.65 127 40-43 164 52.21 1 7 1 5-44 54 17.2 9 1 28.97 128 40.75 165 52.52 18 5-7 6 55 17.52 92 29.29 I29 4I.O7 166 52.84 19 6.07 56 17.8 93 29.61 130 41.38 167 53-16 20 6-39 57 18.15 94 29.93 131 4I.7 168 53-48 21 6.71 58 18.47 95 30.24 132 42.02 169 53-8 22 7-03 59 18.79 96 30.56 133 42.34 170 54-12 23 7-34 60 i9.II 97 30.88 134 42.66 171 54-43 24 7.66 61 I9.42 98 3 1 * 2 135 42.98 172 54-75 25 7.98 62 19.74 99 3!‘52 I36 43 29 173 55-07 26 8.3 63 20.06 ! 100 31.84 137 43.61 174 55-39 27 8.61 64 20.38 101 32.15 138 4393 J 75 55-71 28 8-93 65 2O.7 102 3247 139 44-25 176 56.02 29 9-25 66 21.02 103 32.79 140 44-57 177 56.34 30 9 57 67 21-33 104 33.11 141 44.88 178 56.66 3 1 9.88 68 21.65 105 33.43 I42 45-2 179 56.98 32 10.2 69 2I.97 106 33.74 143 45-52 180 57 23 33 10.52 70 22.29 107 34.06 144 45-84 181 57.62 34 10.84 7i 22.6l 108 34.38 145 46.16 182 57 93 35 11. 16 72 22.92 109 34-7 I46 46.48 183 58-25 36 n.47 73 23.24 no 35*02 147 46.79 184 58-57 37 n.79 74 23.56 III 35-34 I48 47.11 185 58.89 38 12. 11 75 23.88 1 12 35.65 149 47-43 186 59.21 39 12.43 76 24.2 113 35*97 150 47*75 187 59*53 40 12.74 77 24.52 114 36.29 151 48.07 188 59*84 4i 13.06 78 24.83 “5 36.61 152 4839 189 60.16 42 13.38 79 25.15 Il6 36.93 153 48.7 190 60.48 43 13-7 80 2547 117 37-25 154 49.02 191 60.81 44 14.02 81 25.79 Il8 37*56 155 49-34 192 61.13 Pitch in this table is true pitch, as before described. To Compute Circumference of a "Wheel. Rule. — M ultiply number of teeth by their pitch, 858 WHEEL GEARING. To Compute Tfcevolntions of a Wheel or IPinion. Rule. — Multiply diameter or circumference of wheel or number of its teeth in ins., as case may be, by number of its revolutions, and divide prod- uct by diameter, circumference, or number of teeth in pinion. Example.— A pinion io ins. in diameter is driven by a wheel 2 feet in diameter, making 46 revolutions per minute; what is number of revolutions of pinion ? 2 X 12 X 46 -f- 10 = 110.4 revolutions. To Compute Number of Teeth, of a Wheel for a given. Diameter and. 3?itch. Rule. — D ivide diameter by pitch, and opposite to quotient in preceding table is given number of teeth. Example. — Diam. of wheel is 40 ins., and pitch 1.675; what is number of its teeth? 40 H- 1.675 = 23.88, and opposite thereto in table is 75 = number of teeth. To Compute Diameter of a Wheel for a given JPitch. and Number of Teeth. Rule. — M ultiply diameter in preceding table for number of teeth by pitch, and product will give diameter at pitch circle. Example. — What is diameter of a wheel to contain 48 teeth of 2.5 ins. pitch? 1 5. 29 X 2. 5 = 38. 225 ins. To Compute flitch of a Wheel for a given Diameter and IV n m her of Teeth. Rule. — D ivide diameter of wheel by diameter in table for number of teeth, and quotient will give pitch. Example. — What is pitch of a wheel when diameter of it is 50.94 ins., and num- 50.94 -T- 25.47 — 2 i ns - ber of its teeth 80? G-eneral Illustrations. 1. — A wheel 96 ins. in diameter, making 42 revolutions per minute, is to drive a shaft 75 revolutions per minute; what should be diameter of pinion? 96 X 4 2 “J“ 75 = 53- 7 6 ins - 2. — If a pinion is to make 20 revolutions per minute, required diameter of an- other to make 58 -revolutions in same time. 58 -4- 20 = 2.9 = ratio of their diameters. Hence, if one to make 20 revolutions is given a diameter of 30 ins., other will be 30 -f- 2.9 = 10.345 ins. 3. — Required diameter of a pinion to make 12.5 revolutions in same time as one of 32 ins. diameter making 26. 32 X 26 -4- 12. 5 = 66. 56 ins. 4. — A shaft, having 22 revolutions per minute, is to drive another shaft at rate of 15, distance between two shafts upon line of centres is 45 ins. ; what should be diameter of wheels? Then, 1 st. 22 -f- 15 : 22 : : 45 : 26.75 = ins. in radius of pinion. 2d. 22 -j— 1 5 : 15 : : 45 : 18. 24 = ins. in radius of spur. 5. — A driving shaft, having 16 revolutions per minute, is to drive a shaft 81 revo- lutions per minute, motion to be communicated by two geared wheels and two pul- leys, with an intermediate shaft; driving wheel is to contain 54 teeth, and driving pulley upon driven shaft is to be 25 ins. in diameter; required number of teeth in driven wheel, and diameter of driven pulley. Let driven wheel have a velocity of V16 X 81 = 36, a mean proportional between extreme velocities 16 and 81. Then, 1st. 36 : 16 :: 54 : 24 — teeth in driven wheel. 2d. 81 : 36 :: 25 : 11. 11 =in5. diameter of driven pulley. 6- — If, as in preceding case, whole number of revolutions of driving shaft, num- ber of teeth in its wheel, and diameters of pulleys are given, what are revolutions of shafts? Then, 1st. 18 : 16 : : 54 : 48 = revolutions of intermediate shaft. 2d. 15 : 48 :: 25 : 80 = revolutions of driven shaft. WHEEL GEARING. TEETH OE WHEELS. 859 TeetPL of Wheels. Tr , r>ic'v'cloidal. — In order that teeth of wheels and pinions should work evenly and without unnecessary rubbing friction, the face (from pitch line to top) of the outline should be determined by an epicycloidal curve (see page 228), and that of the flank (from pitch line to base ) by an hypocycloidal (see also page 228). When generating circle is equal to half diameter of pitch circle, hypocy- cloidal described by it is a straight diametrical line, and consequently out- line of a flank is a right line, and radial to centre of wheel. If a like generating circle is used to describe face of a tootli of other wheel or pinion respectively, the wheel and pinion will operate evenly. Illustration. — Determine all elements of wheel —viz. , Pitch circle, Number of teeth, Pitch, Length, Face, and Flank. Cut a template A to pitch circle c c of wheel, and secure it temporarily to a board. Having determined depth of tooth, set it off on pitch line, as a 0, Fig. 1, and above it apply a sec- , ond template, a; radius of wheel is equal to half radius of ninion- insert into, or attach exactly at its edge, a tracer . roll template a alon" a! and tracer will describe an epicycloidal curve, a r, and by m\erting a describe 0 and faces of a tooth are delineated. To describe flanks, define pitch line c c, Fig. 2, and arc n w, drawn at base of teeth or board A (as in Fig. 1), secure a strip of wood, w, equal in length to radius of wheel, and locate centre of it, *, draw radii x a and x 0, and they will define flanks, which should be filleted, as shown at s s. Define arc ^ ^ zz, and length of tooth is determined. \ ! ; 1 Proceed in like manner conversely for teeth of pinion, and • ! ! wheel and pinion thus constructed will operate truly. :l. I In construction of the teeth of a wheel or pinion in " iLjil the pattern-shop, it is customary to construct the wheel or pinion complete, out to face of wheel at base of teeth, and then to insert the teeth in rough, approximately shaped blocks, by a dovetail at their base, fitting into face of wheel, and then the outline of a tooth is described thereon ; the block is then remo , \ ed, fin- ished as a tooth, replaced, fastened, and filleted. Involute. Teeth of two wheels will work truly together when their face is that of an involute (see page 229), and that two such wheels should work truly, the circles from which the involute lines for each wheel are generated must be concentric with the wheels, with diameters in same ratio as those of the wheels. Assume Ac, Be, Fig. 3. pitch radii of two wheels designed to work together, through c, draw a right line, e i, and with perpendiculars c c, i c, describe arcs n 0, r. s, and involutes n c o and res define a face of each of the teeth. To describe teeth of a pair of B 4 * wheels of which A c, B c, Fig. 4, are pitch radii, draw c i, c e, per- pendicular to radials B i and A c, and they are to be taken as the radials of the involute arcs from which the faces of the teeth are to be defined; then fillet flanks at ' ^ base, as before described, Fig. 2. \J Involute teeth will work with truth, even at varying A ‘ distances apart of the centres of the wheels, and any wheels of a like pitch will woik in union, however varied their diameters. WHEEL GEARING. — TEETH OF WHEELS. „ Circular teeth are defined as follows : Assume A A, Fig. 5, pitch-line, b b line of base of teeth, and t i face-line. Set off on pitch-line divisions both of pitch and depth of teeth then 1-^ . Wltb a radius of 1.25 pitch describe arcs as o s 4 p!tch line for faces of teeth ' then draw ra- " dial hnes ov, ru, to centre of wheel for flanks strike arc 1 1 to define length of tooth, and fillet flanks at base as before described. Proportions of Teeth. In computing dimensions of a tooth, it is to be considered as a beam fixed at one end , .. . . , weight suspended from other, or face of beam • and it is essential to consider the element of velocity, as its stress in opera- tion, at high velocity with irregular action, is increased therebv. Dimensions of a tooth should be much greater than is necessary to resist direct stress upon it, as but one tooth is proportioned to bear whole stress upon wheel, although two or more are actually in contact at all times • but this requirement is in consequence of the great wear to which a tooth is’ sub- jected, shocks it is liable to from lost motion, when so worn as to reduce its depth and uniformity of bearing, and risk of the loss of a tooth from a defect. . A t00th running at a low velocity may be materially reduced in its dimen- sions, compared with one running at a high velocity and with a like stress. Result of operations with toothed wheels, for a long period of time has determined that a cast-iron (Eng.) tooth with a pitch of 3 ins. and a breadth ot 7.5 ms. will transmit, at a velocity of 6.66 feet per second, power of 59 16 To Compute Dimensions of* a Tooth, to Desist a given Stress. Rule.— M ultiply extreme pressure at pitch-line of wheel by length of 00 1 in decimal of a foot, divide product by Coefficient of material of tooth, and quotient will give product of breadth and square of depth. S l 0r ’ ~ b d2 - s representing stress in lbs. \ and l length in feet The Coefficient of cast iron for this or like purposes may be taken at from 50 to 70. Pitch A B = 1. Depth rs = . 45. Length co = . 75. Space 5 v = . 55. Working length ce = . 7. Play sv—rs£.i [-- Clearance e to 0 — .05. Face B c = .35. Note. — It is necessary first to determine pitch, in order to obtain either length or depth of a tooth. of tooth, pitch being 3 ins. ? 3 X .75 = 2.25 length of tooth , which H- Coefficient of material is taken at 60. 4886 x -1875 Example — Pressure at pitch-line of a cast- iron wheel (at a velocity of 6.66 feet per sec- ond) is 4886 lbs. ; what 'should be dimensions 1875 = length in decimal of a foot 60 — 15.27. If length = 2.25, pitch ==, 3, and depth = 1.35 ins. Pitches of Equivalent Strength for Cast Iron and Wood Iron 1. Hard wood *. 26. Then = 8.39 ins. breadth. I - 35 2 When Product of b d 2 is obtained , and it is required to ascertain either dimension, ff - depth , and ~ = breadth. ■ WHEEL GEARING. TEETH OF WHEELS. 86 1 To Compute Depth of a Tooth. 1. When Stress is given. Rule.— Extract square root of stress, and mul- tiply it by .02 for cast iron, and .027 for hard wood. 2. When IP is given. Rule.— E xtract square root of quotient of EP di- vided by velocity in feet per second, and multiply it by .466 for cast iron, and .637 for hard wood. Example.— IP to be transmitted by a tooth of cast iron is 60, and velocity of it at its pitch line is 6.66 feet per second; what should be depth of tooth? \/ 6^ X - 466 = , ' 398 , '" S - To Compute LP of a Tooth. Rule.— Multiply pressure at pitch-line by its velocity in feet per minute, and divide product by 33 000. Example. —What is IP of a tooth of dimensions and at velocity given in preced- ing example. 4886 X 6. 66 X 60" -r- 33 000 = 59. 16 horses. To Com pate Stress that may be borne by a Tooth. Rule— M ultiply Coefficient of material of tooth to resist a transverse strain, as estimated for this character of stress, by breadth and square of its depth, and divide product by extreme length of it in decimal of a foot. Example. — Dimensions of a cast-iron tooth in a wheel are 1.38 ins. in depth, 2.1 ins. in length, and 7.5 ins. in breadth; what is the stress it will bear? Coefficient assumed at 60. * 7 — - — = 4886 lbs. 2. 1 — 12 Following deductions by the rules of different authors for like elements are sub- mitted for a cast-iron tooth: Pitch 3 ins. 1 Depth 1.38 ins. | Breadth... 7.5 ins. | Length 2.1 ins. Actual Power in Stress Exerted Depth of | Actual Power in Stress Exerted Depth of at a velocity of 400 feet per min., 4886 lbs. Tooth, at a velocity of 400 feet per min., 4886 lbs. Tooth. By Above Rule ^ X -446- ■ “ Fairbairn .025 y/W Imperial Journal ^ - W 576 ' Ins. 1.398* *■75 1.76 / V By Ranlcine , / — V 1500 “ TredgoU A . “ Buchanan • Ins. 1.8 2.25 2.24 H representing horsepower (60), W stress in lbs., and v velocity in feet per second. Depth, Ditch, and Breadth. (M. Morin.) Cast iron j. ^ 028 y/W — d. .057 y/W = p - Hard w r ood 038 -fW = d. .079 y/W = P. W representing weight or stress upon tooth in lbs . , d depth of tooth , and P pitch in ins. When velocity of pitch-circle does not exceed 5 feet per second 5 = 4 c?, when it exceeds 5 feet b = 5 d, and if wheels are exposed to wet 6 = 6 d. b representing breadth. Illustration.— Assume pressure at pitch-line of a cast-iron wheel upon a tooth equal 6000 lbs., and velocity 5 feet per second. Then .028 -y/6ooo=: 2. 17 ins. Depth , and .057 y/6000 — 4. 46 ins. Fitch. Note. —F or further Illustrations of Formation of Teeth, Bevel Gearing, Willis’s Odontograph, Staves, Trundles, etc., see Mosely’s Engineering, Shelton’s Mechanic’s Guide, Fairbairn’s Mechanism and Machinery of Construction, etc. * This depth, with a breadth of 7.5 ins., is .1 of ultimate strength of average strength of American Cast Iron. 862 TEETH OF WHEELS. — WINDING ENGINES. PROPORTIONS OF WHEELS. With six flat A rms and Ribs upon one side of them , as ; or a Web in centre , as Rim. — Depth, measured from base of teeth, .45 to .5 of pitch of teeth, hav- ing a web upon its inner surface .4 of pitch in depth and .25 to .3 of it in width. Note.— W hen face of wheel is mortised, depth of rim should be 1.5 times pitch, and breadth of it 1.5 times breadth of tooth or cog. Hub . — When eye is proportionate to stress upon wheel, hub should be twice diameter of eye. In other cases depth around eye should be .75 to .8 of pitch. Arm . — Depth .4 to .45 of pitch. Breadth at rim 1.5 times pitch, increas- ing .5 inch per foot of length toward hub. Rib upon one edge of arm, or Web in its centre, should be from .25 to .3 pitch in width, and .4 to .45 of it in depth. When section of an arm differs from those above given, as with one with a plane section, as msmm , or with a double rib, as |sgaBsj , its dimensions should be proportioned to form of section. In a wheel of greater relative diameter, length of hub and breadth of arms, or of the rib or web, according as plane of arm is in that of wheel, or con- trariwise, should be made to exceed breadth of face of wheel (at the hub) in order to give it resistance to lateral strain. Number of arms in wheels should be as follows*. 1.5 to 3.25 feet in diameter 4 I 5 to 8.5 feet in diameter 6 3 25 “ 5 “ “ 5 I 8.5 “ 16 “ “ 8 16 to 24 feet in diameter 10 With light wheels, number of arms should be increased, in order better to sustain rigidity of rim. Mortise Wheels . — Their rim or face should be .9 pitch of tooth, and twice depth of rim of a solid wheel. WINDING ENGINES. With Winding Engines, for drawing coals, etc., out of a Pit, where it is required to give a certain number of revolutions, it is necessary to have given diameter of Drum and thickness of rope, which is flat made, and contrariwise. To Compute Diameter of -a Drum. Where Flat Ropes are used, and are wound one part over the other. Rule. — Divide depth of pit in ins. by product of number of revolutions and 3.1416, and from quotient subtract product of thickness of rope and number of rev- olutions ; remainder is diameter in ins. Example. — If an engine makes 20 revolutions, depth of pit being 600 feet, and rope 1 inch, what should be diameter of drum ? 600 X 12 20 X 3 - I 4 l6 — — 7200 - ! X 2° = - 20 = 94-59 MU- To Compute Diameter of Roll. Rule. — To area of drum add area or edge surface of rope ; then ascertain by inspection in table of areas, or by calculation, diameter that gives this area, and it is the diameter of Roll. WINDING ENGINES.' — WINDMILLS. 863 Example.— What is diameter of roll in preceding example? Area of 94.59 = 7027.2 + (area of 7200 X i) = 7 20O:=:l 4 227.2, and V^ 22 ?- 2 " .7854 = 134. 59 ins. Or, Radius of drum is increased number of revolutions multiplied by thickness of rope; as | 2 oXi= 67.295 ins. To Compute TSTuniDer of* Bevolntions. Rule.— T o area of drum add area of edge surface of rope ; from diameter of the" circle having that area subtract diameter of drum, and divide re- mainder by twice thickness of rope ; quotient will give number of revolutions. Example. Length of a rope is 2600 ins., its thickness 1 inch, and diametei of drum 20 ins. ; what is number of revolutions? Area of 2 o-|- area of rope = 314. 16-1-2600 = 2914. 16, diameter of which is 60.91, and 60.91— 2 q_ 2 5 revolutions. 1X2 Or, subtract diameter of drum from diameter of roll, and divide remainder by twice thickness of rope; as 60.91 — 20 = 40.91, and 40.91 -r- 1 X 2 = 20.45 revolutions. To Compute Boint of Meeting of Ascending and De- scending Buckets when two or more are used. To Compute Point of Meeting of Buckets. Rule. — D ivide sum of length of turns of rope by 2, and to quotient add length of last turn ; divide sum bv 2, multiply quotient by half number of revolutions, and product will give distance from centre of drum at which buckets will meet. Note 1.— Meetings will always be below half depth of pit. 2.— At half number of revolutions buckets will meet. Example —Diameter of a drum is 9 feet, thickness of rope 1 inch, and revolu- tions 20; what is depth of pit, and at what distance from top will buckets meet? 28.54 + 384? ^ 8 8 ^ 2 x .gg_ZL 22 — 35.995 x 10 = 359-95 fa *- 2 2 2 To Compute this Depth. Rule.— T o diameter of drum add thickness of rope in feet, and ascertain its circumference ; to diameter of drum add quo- tient of product of twice thickness of rope and number of revolutions less 1, divided by 12 for a diameter, and circumference of this diameter is length of last turn, also in feet ; add these two lengths together, multiply their sum by half number of revolutions, and product will give depth of pit. 9 -f thickness of rope = 9 + jj of * = 9- 083, which X 3- J 4 l6 = 28. 54 feet = length of first turn. 9.0833 + I >< 2 >< 26 ~ 1 x 3. I4i6 _ 38.48 feet = length of last turn. Then 28.54 + 38.48 X ^ = 67.02 X 10 = 670.2 feet, depth of pit. WINDMILLS. Driving Shaft of a vertical windmill should be set at an elevating angle with horizon when set upon low ground, and at a depressing angle when set upon elevated ground. Range of these angles is from 3 0 to 15V A velocity of wind of 10 feet per second is not generally sufficient to drive a loaded mill, and if velocity exceeds 35 feet per second the force is generally too great for ordinary structures. Angle of Sails should be from 18° to 30° at their least radius, and from 7 0 to 1 7 0 at their greatest radius, mean angle being from 15 0 to 17 0 to plane of motion of sails. Length of a whip (arm) is divided into 7 parts, sails ex- tending over 6 parts. 86 4 WIND-MILLS. ™ !p ^u l \ 0( i( f lellffth : B . readth -°33, at top .016 ; Depth .025, at ton ;° I2 3> Width of sail .33, at axis .2. Distance of sail from axi s 5 .oi4 of length of whip, ancl cross-bars 16 to 18 ins. from centres. To Compute Angles of Sails. 18 d 2 2 3 r 2 angle of sail with plane of its motion at any part of it. d repre- senting distance of part of sail from its axis , and r extreme radius of sail, both in feet. Illustration.— A ssume r = i 4 , and^ength of sail I2 feet, d = . 5 of 12 or three Then 23 0 — - = 23 — 5.88°= 17. 12° ?° — 72.88°. sixths of sail — 5 X 12 -j- (14 — 12) = 2 = 8 feet. 18X8 Hence, angle of sail with axis = 90° — 17. 12* _ # be as r tollowsf Sa ' 1S ' S divided into 6 equal P arts > an « les at ea ch of these parts will Distance from Axis. Angle of sail with axis 6 * s * 6 *o 3 o *o 7Q 5 , with plane of motion 22.5° 21 0 18.5° 150 85° 5° -~n 3.16 v r' sin. x IP X 1 080 000 v 3 To Compute Elements of Windmills. 5 ” A v 3 = A: y .1047 iiz=av ; ' R2 + r 2 _ 1 080 000 v representing velocity of wind per sec- °/ centre of percussion of sails, and R and r outer and inner radii of sails, all in feet , x mean angle of sail to plane of motion n number of revolution? of arms per minute, a v angular velocity, A area of sails in sq.feet, and IP horse-power. iLLUSTRATmN.-If a windmill has 4 arms of 28 feet, with a mean angle (x) of 160 area °f sai of sq. feet each, having an inner radius of 4 feet and is on’ eiated by wind at a velocity of 40 feet per second; what are its elements? P Then 11. 5 x 40 4 X 150 X 6 7 4°°o__ ^ 1 080 000 = 23 : 35 - 55 J /28 s 4 - 4 2 ; yJ —. 2 -20 feet; 35-55 X 1 080000 64 000 3. 16 X 40 20 X. 275 64 = A = 599_9 sq.feet. — n~ 22.95; Deductions from Velocities varying from 4 to 9 Feet per Second. {Mr. Smeaton.) . /• Vel , oc! . t / of windmill sails, so as to produce a maximum effect, is near- ly as velocity of wind, their shape and position being same. 2. Load at maximum is nearly, but somewhat less than, as square of ve- locity of wind, shape and position of sails being same. 3. Effects of same sails, at a maximum, are nearly, but somewhat less than, as cubes of velocity of wind. 4. Load of same sails, at maximum, is nearly as squares, and their effect as cubes of their number of turns in a given time. 5. In sails where figure and position are similar, and velocity of wind the lTi^gth^sail 1, ° f revolutions in a given time will be reciprocally as radius or 6. Load, at a maximum, which sails of a similar figure and position will overcome at a given distance from centre of motion, will be as cube of radius. 7. Effects of sails of similar figure and position are as square of radius. Velocity of extremities of Dutch sails, as well as of enlarged sails, in all their usual positions when unloaded, or even loaded to a maximum, is considerably greater than that of wind. WINDMILLS. WOOD AND TIMBER. 865 Results of Experiments on Effect of Windmill Sails. When a vertical windmill is employed to grind corn, the millstone usu- ally makes 5 revolutions to 1 of the sail. 1. When velocity of wind is 19 feet per second, sails make from n to 12 revolutions in a minute, and a mill will grind from 880 to 990 lbs. in an hour, or about 22 440 lbs. in 24 hours. 2. When velocity of wind is 30 feet per second, a mill will carry all sail, and make 22 revolutions in a minute, grinding 1984 lbs. of flour in an hour, or 47 616 lbs. in 24 hours. Reaxxlts of Operation of Windmills. {A. R. Woolf M. E.) Velocity of Wind 15 to 20 Miles per Hour. Revolutions of Wheel and Gallons of Water raised per Minute. Desig- Revolutions Water raised to an Elevation of Power Cost per Hour. nation of Mill. of Wheel. 25 Feet. 50 Feet. 100 Feet. 200 Feet. developed. Actual.* Per HP. Feet. No. Gallons. Gallons. Gallons. Gallons. HP Cents. Cents. 8.5 70 to 75 6.16 3-C2 — — .04 .60 15 IO 60 to 65 19.18 9 - 5 6 4-75 — .12 .70 5-8 14 50 to 55 45-14 22.57 11.25 5 .28 1.63 5-8 18 40 to 45 97.68 52.16 24.42 12.21 .61 2.83 4.6 20 35 to 40 124.95 63-75 3 i - 2 5 15-94 .78 3-56 4-5 25 30 to 35 212.38 106.96 49-73 26.74 i -34 4.26 3-2 * Including interest at 5 per cent, per annum. WOOD AND TIMBER. Selection of Standing Trees . — Wood grown in a moist soil is lighter, and decays sooner, than that grown in dry, sandy soil. Best Timber is that grown in a dark soil, intermixed with gravel. Poplar, Cypress, Willow, and all others which grow best in a wet soil, are exceptions. Hardest and densest woods, and least subject to decay, grow in warm climates ; but they are more liable to split and warp in seasoning. Trees grown upon plains or in centre of forests are less dense than those from edge of a forest, from side of a hill, or from open ground. Trees (in U. S.) should be selected in latter part of July or first part of August ; for at this season leaves of sound, healthy trees are fresh and green, while those of unsound are beginning to turn yellow. A sound, healthy tree is recognized by its top branches being well leaved, bark even and of a uniform color. A rounded top, few leaves, some of them turned yellow, a rougher bark than common, covered with parasitic plants, and with streaks or spots upon it, indicate a tree upon the de- cline. Decay of branches, and separation of bark from the wood, are infallible indications that the wood is impaired. Green timber contains 37 to 48 per cent, of liquids. By exposure to air in seasoning one year, it loses from 17 to 25 per cent., and when seasoned it retains from 10 to 15 per cent. According to M. Leplay, green wood contains about 45 per cent, of its weight of moisture. In Central Europe, wood cut in winter holds, at end of following summer, fully 40 per cent, of water, and when kept dry for sev- eral years retains from 15 to 20 per cent, of water. Felling Timber — Most suitable time for felling timber is in midwinter and in midsummer. Recent experiments indicate latter season and month of J uly. 1 4 D 866 WOOD AND TIMBER. A tree should be allowed to attain full maturity before being felled. Oak matures at 75 to 100 years and upward, according to circumstances ; Ash Larch, and Elm at 75 ; and Spruce and Fir at 80. Age and rate of growth of a tree are indicated by number and width of the rings of annual increase which are exhibited in a cross-section of its body. A tree should be cut as near to the ground as practicable, as the lower part furnishes best timber. Dressing Timber— As soon as a tree is felled, it should be stripped of its bark, raised from the ground, reduced to its required dimensions, and its sap-wood removed. Inspection of Timber— Quality of wood is in some degree indicated by its color, which should be nearly uniform, and a little deeper towards its cen- tre, and free from sudden transitions of color. White spots indicate decay. Sap-wood is known by its white color ; it is next to the bark, and soon rots. Defects of* Timber. Wind-shakes are serious defects, being circular cracks separating the con- centric layers of wood from each other. Splits , Checks , and Cracks , extending toward centre, if deep and strongly marked, render timber unfit for use, unless purpose for which it is intended will admit of its being split through them. Brash is when wood is porous, of a reddish color, and breaks short, with- out splinters. It is generally consequent upon decline of tree from age. Belted is that which has been killed before being felled, or which has died from other causes. It is objectionable. Knotty is that containing many knots, though sound ; usually of stinted growth. Twisted is when grain of it winds spirally ; it is unfit for long pieces. Dry-rot is indicated by yellow stains. Elm and Beech are soon affected, if left with the bark on. Large or decayed knots injuriously affect strength of timber. Heart-shake— Split or cleft in centre of tree, dividing it into segments. Star-shake . — Several splits radiating from centre of timber. Cup-shake . — Curved splits separating the rings wholly or in part. Rind-gall .— Curved swelling, usually caused by growth of layers over spot where a branch has been removed. Upset . — Fibres injured by crushing. Foxiness . — Yellow or red tinge, indicating incipient decay. Doatiness . — A speckled stain. Seasoning and Preserving Timber. Seasoning is extraction or dissipation of the vegetable juices and moisture or solidification of the albumen. When wood is exposed to currents of air at a high temperature, the moisture evaporates too rapidly, and it cracks ; / and when temperature is high and sap remains, it ferments, and drv-rot ensues. Wood requires time in which to season, very much in proportion to density of its fibres. Water Seasoning is total immersion of timber in water, for purpose of dissolving the sap, and when thus seasoned it is less liable to warp and crack, but is rendered more brittle. WOOD AND TIMBER. 867 For nurDose of seasoning, it should be piled under shelter and kept dry , ctrlim nl iced between each layer, one near each end of pile, and others at ho £ distances in order to keep the timber from winding. These strips should be one over the other, and in large piles she ^ ^ t tob» tVnVlr T io’ht timber may be piled in upper portion ot shelter, neavj timuer upon'°Tound floor. Each pile should contain but one description o± timber, and they should be at least 2.5 feet apart. . . . It should be repiled at interval* and all pieces indicating decay should be removed, to prevent their affecting those which are still sound. It reouires from 2 to 8 years to be seasoned thoroughly, according to its dimensions, and it should be worked as soon as it is thoroughly dry, for deteriorates after that time. Gradual seasoning is most favorable to strength and durability of timber. Various methods have been proposed for hastening the process, as Steaming, whicMias been applied with success; and results of experiments of various processes of saturating it with a solution of Corrosive sublimate and An \er)tic fluids are very satisfactory. Such process hardens and seasons wood, Se same tTme thlt it securest from dry-rot and from attacks of worms. Woods are densest and strongest at the roots and at their centres. Their strength decreasing with the decrease of their density. Oak timber loses one fifth of its might in seasoning, and about one third in becoming perfectly dry. Pitch pine, from the presence of pitch, requires time in excess ot that due to the density of its fibre. Mahogany should he seasoned slowly, Pine quickly. Whitewood should not be dried artificially, as the effect of heat is to twist it. Salt water renders wood harder, heavier, and more durable than fresh. Condition of timber, as to its soundness or decay, is readily recognized when struck with a quick blow. Timber that has been for a long time immersed in water, when brought into the air and dried, becomes brashy and useless. When trees are barked in the spring, they should not be felled until the foliage is dead. Timber cannot be seasoned by either smoking or charring ; but when it is exposed to worms or to the production of fungi , it is proper to smoke or char it, and it may be partially seasoned by being boiled or steamed. Timber houses are best provided with blinds which keep out rain and snow but which can be turned to admit air in fine weather, and the houses should be kept entirely free from any pieces of decayed wood. Kiln-drying is suited only for boards and pieces of small dimensions, as it is apt to cause cracks and to impair the strength, unless performed very slowly. Charring , Painting , or covering the surface is highly injurious to any but seasoned wood , as it effectually prevents drying of the inner part of the wood, in consequence of which fermentation and decay soon take place. Timber is subject to Common or Dry-rot , former occasioned by alternate exposure to moisture and dryness, and as progress of it is from the exterior, covering of the surface, if seasoned, with paint, tar, etc., is a preservative. 868 WOOD AND TIMBER. Common-rot is the consequence of its being piled in badly-ventilated slieds. Outward indications are yellow spots upon ends of pieces, and a yellowish dust in the checks and cracks, particularly where the pieces rest upon pil- i’ g-strips. Dry or Sap-rot is inherent in timber, and it is the putrefaction of the veg- etable albumen. Sap wood contains a large proportion of fermentable ele- ments. Insects attack wood for the sugar or gum contained in it, and f ungi subsist upon the albumen of wood ; hence, to arrest dry-rot, the albumen must be either extracted or solidified. Most effective method of preserving timber is that of expelling or ex- hausting its fluids, solidifying its albumen, and introducing an antiseptic liquid. Strength of impregnated timber is not reduced, and its resilience is improved. In desiccating timber by expelling its fluids by heat and air, its strength is increased fully 15 per cent. The saturation of wood with creosote, tar, antiseptics, etc., preserves it from the attack of worms. Jarrow wood, from Australia, is not subjected to their attack. In a perfectly dry atmosphere durability of woods is almost unlimited. Rafters of roofs are known to have existed 1000 years, and piles submerged in fresh water have been found perfectly sound" 800 years from period of their being driven. Resistance of woods to extension is greater than that of compression. Impregnation of Wood. Several of the successful processes are as follows : # Kyan , 1832.— Saturated with corrosive sublimate. Solution 1 lb. of chlo- ride of mercury to 4 gallons of water. Burnett ( Sir Wm .), 1838. — Impregnation with chloride of zinc by sub- mitting the wood endwise to a pressure of 150 lbs. per sq. inch. Solution, 1 lb. of the chloride to 4 gallons of water. Boucheri. — Impregnation by submitting the wood endwise to a pressure of about 15 lbs. per sq. inch. Solution, 1 lb. of sulphate of copper to 12.5 gallons of water. Bethel. — Impregnation by submitting the wood endwise to a pressure of 150 to 200 lbs. per sq. inch, with oil of creosote mixed with bituminous matter. Robbins , 1865. — Aqueous vapor dissipated by the wood being heated in a chamber, the albumen solidified, then submitted to vapor of coal tar, resin, or bituminous oils, which, being at a temperature not less than 325 0 , readily takes the place of the vapor expelled by a temperature of 212 0 . Hay ford, 187-. — Aqueous vapor dissipated by the wood being heated in a chamber to a temperature of from 250° to 270°, the albumen solidified, then air introduced to assist the splitting of the outer surfaces. When vapor is dissipated, dead oils are introduced under a pressure of 75 lbs. per sq. inch. Planks , Deals , and Battens. — When cut from Northern pine (Pinus Sylve- stris) are termed yellow or red deal, and when cut from spruce (Abies, alba , etc.) they are termed white deal. Desiccated wood, when exposed to air under ordinary circumstances, ab- sorbs 5 per cent, of water in the first three days ; and will continue to absorb it until it reaches from 14 to 16 per cent., the amount varying according to condition of the atmosphere. WOOD AND TIMBER. 869 Durability of Various Woods. Pieces 2 feet in Length ,. 1.5 ins. Square, driven 28.5 ins. into the Earth. After 2.5 Years. Acacia Ash, Amer Cedar, Va “ Lebanon.. Elm, Eng 44 Can Fir Larch Oak, Can “ Memel “ Dantzic . . . “ Chestnut.. Pine, pitch Teak . yellow . white . . Good Much decayed Very good Good Much decayed “ attacked Surface only attacked. , Very much decayed.., Very good Surface only attacked. , Attacked Very much decayed. . Very good After 5 Years, ( Externally decayed, rest per- ( fectly sound. Decayed. Sound as when driven. Tolerable. Entirely decayed. Decayed. Much decayed. ( Attacked in part only, rest fair ( condition. Very rotten. ( Some moderately, most very { much, decayed. ( Attacked in part only, rest fair ( condition. Much decayed. Very rotten. Somewhat soft, but good. Effect of Creosoting. Results of Experiments with Various Woods {E. R. Andrews). Spruce Oak . . . Water absorbed. Per cent. •2543 .0261 Hard pine. . . . ( dried \ creosoted. Gum, black.. ]*™soted. Birch, white. j^‘“‘ 0 V e ' d ; Per cent. . 16 ( dried | creosoted . f dried "j creosoted . Cotton- wood | cre0 soted. Sesquoia Gigantea of California, dried, .4722 ; creosoted, .0. Fluids will pass with the grain of w^ood with great facility, but will not enter it except to a very limited extent when applied externally. Absorption of Preserving Solution by different Woods for a Period of 7 Pays. Average Lbs. per Cube Foot. Black Oak 3.6 Chestnut 3 Hemlock. Red Oak. I Rock Oak 3.9 White Oak 3.1 Proportion of W a ter in various Woods. Alder (Betula alnus) 416 Ash (Fraxinus excelsior) 28.7 Beech (Fagus sylvatica) 33 Birch ( Betula alba) 30.8 Elm ( Ulmus campestris) 44-5 Horse-chestnut ( AEsculus hippocast. ) 38. 2 Larch ( Pinus larix) 48.6 Mountain Ash (Sorbus aucuparia). . 28.3 Oak ( Quercus robur) 34*7 Pine (Pinus Sylvestris L.) 39.7 Red Beech (Fagus sylvatica) 39.7 Red Pine (Pinus picea dur) 45.2 Spruce (Abies, alba , nigra, rubra , ' excelsa) Sycamore (Acer pseudo -platanus) . . White Oak (Quercus alba) White Pine (Pinus abies dur) White Poplar (Populus alba) 35 Willow (Salis caprea) 26 Pecrease in Pimensions of Timber by Seasoning Ins. Ins. 13-25 Woods. Cedar, Canada 14 to Elm 11 to 10.75 Oak, English 12 to 11.625 Pitch Pine, North. . . xoXioto 9.75X9-75 Weight of a beam of English oak, when wet, was reduced by seasoning from 972.25 to 630.5 lbs. Woods. Ins. Ins. Pitch Pine, South 18.375 to 18.25 Spruce 8.5 to 8.375 White Pine, American.. 12 to 11.875 Yellow Pine, North 18 to 17.875 870 WOOD AND TIMBER. Weight of a Cube Foot of Oalt and Yellow Fine. White Oak, Va. Yellow Pine. Va. Age. Round. Square. Round. Square. Live Oak. Green 64.7 67.7 47.8 39- 2 78.7 1 Year 53-6 53-5 39- 8 34-2 2 Years 46 49.9 34-3 33-5 66.7 In England, Timber sawed into boards is classed as follows : 6.5 to 7 ins. in width, Battens ; 8.5 to 10 ins., Deals ; and 11 to 12 ins., Planks. (/See also page 62.) Distillation. — From a single cord of pitch pine distilled by chemical ap- paratus, following substances and in quantities stated have been obtained : Charcoal 50 bushels. Illuminating Gas about 1000 cu. feet. Illuminating Oil and Tar. . . 50 gallons. Pitch or Besin 1.5 barrels. Pyroligneous Acid 100 gallons. Spirits of Turpentine 20 “ Tar . 1 barrel. Wood Spirit 5 gallons. Strength of Timber. Results of experiments have satisfactorily proved: That deflection was sensibly proportional to load ; That extension and compression were nearly the same, though former being the greater ; That, to produce equal deflection, load, when placed in the centre, was to a load uniformly distributed, as .638 to 1 ; That deflection under equal loads is inversely as breadths and cubes of the depths, and directly as cubes of the spans. (M. Morin.) It has also been shown, that density of wood varies very little with its age. That coefficient of elasticity diminishes after a certain age, and that it de- pends also on the dryness and the exposure of the ground where the wood is grown. Woods from a northerly exposure, on dry ground, have a high coefficient, while those from swamps or low moist ground have a low one. That tensile strength is influenced by age and exposure. The coefficient of elasticity of a tree cut down in full vigor, or before it arrives at this condition, does not present any sensible difference. That there is no limit of elasticity in wood, there being a permanent set for every extension. Average Result of Experiments on Tensile Strength of Wood in Various Positions per Sq. Inch. {MM. Chevandier and Werlheim.) With the fibre, 6900 lbs. Radially, 683 lbs., and Tangentially, 723 lbs. To Compute Volume of an Irregular Body'. By “ Simpson's Rule." Operation. —Take a right line in the figure for a base line, as A B, divide the fig- ure into any number of equal parts, and compute the areas of their plane sections as 1, 2, 3, etc., at the points of division , by rules applicable to area of a plane. Then, operate these areas as if they were the ordinates of a plane curve or figure of same length as the figure, and result will give volume required. Illustration. — Assume a figure having areas as follows, and A B = 24 feet. Sections, 1 c 4 3 2 1 2 Areas, 3 feet 5 “ Multiplier, 1 4 2 4 Products, 3 20 14 36 iz 84 and 84 x 24 -r- 4 - 4 - 3 = 168 cube feet. MISCELLANEOUS MIXTURES. 871 MISCELLANEOUS MIXTURES. Cements. Much depends upon manner in which a cement is applied as upon the cement itself, as best cement will prove worthless if improperly applied. Following rules must be rigorously adhered to to attain success : 1. Bring cement into intimate contact with surfaces to be united. This is best done by heating pieces to be joined in cases where cement is melted by heat, as with resin, shellac, marine glue, etc. Where solutions are used, cement must be well rubbed into surfaces, either with a brush (as in case of porcelain or glass), or by rubbing the two surfaces together (as in making a glue joint between pieces of wood). 2. As little cement as practicable should be allowed to remain between the united surfaces. To secure this, cement should be as liquid as practicable (thoroughly melted if used with heat), and surfaces should be pressed closely into contact until cement has hardened. 3. Time should be allowed for cement to dry or harden, and this is particularly the case in oil cements, such as copal varnish, boiled oil, white lead, etc. When two surfaces, each .5 inch across, are joined by means of a layer of white lead placed between them, 6 months may elapse before cement in middle of joint be- comes hard. At the end of a month the joint will be weak and easily separated; at end of 2 or 3 years it may be so firm that the material will part anywhere else than at joint. Hence, when article is to be used immediately, the only safe cements are those which are liquefied by heat and which become hard when cold. A joint made with marine glue is firm an hour after it has been made. Next to cements that are liquefied by heat are those which consist of substances dissolved in water or alcohol. A glue joint sets firmly in 24 hours; a joint made with shellac varnish becomes dry in 2 or 3 days. Oil cements, which do not dry by evaporation, but harden by oxidation (boiled oil, white lead, red lead, etc.) are slowest of all. Stone.— Resin, Yellow Wax, and Venetian Red, each 1 oz. ; melt and mix. Aquarium. Litharge, fine white dry Sand, and Plaster of Paris, each 1 gill; finely pulverized Resin, .33 gill. Mix thoroughly and make into a paste with boiled linseed oil to which drier has been added. Beat well, and let stand 4 or 5 hours before using it. After it has stood for 15 hours, however, it loses its strength. Glass cemented into a frame with this cement will resist percolation for either salt or fresh water. Adhesive for Fractures of all Kinds. White Lead ground with Linseed-oil Varnish, and kept from contact with the air. Requires a few weeks to harden. Stone or Iron. Compound equal parts of Sulphur and Pitch. Brass to Glass. Electrical. — Resin, 5 ozs. ; Beeswax, 1 oz. ; Red Ochre or Venetian Red, in pow- der, 1 oz. Dry earth thoroughly on a stove at above 212° Melt Wax and Resin together and stir in powder by degrees. Stir until cold lest earthy matter settle to bottom. Used for fastening brass-work to glass tubes, flasks, etc. Chinese Waterproof*. Schio-liao . — To 3 parts of Fresh Beaten Blood add 4 parts of Slaked Lime and a little Alum; a thin, pasty mass is produced, which can be used immediately. Materials which, are to be made specially waterproof are painted twice, or at most three times. Mr ooden public buildings of China are painted with schio-liao , vtfhich gives them an unpleasant red- dish appearance, but adds to their durability. Pasteboard treated with it receives anpearanee and strength of wood. China. Curd of milk, dried and powdered, 10 ozs. ; Quicklime, 1 oz. ; Camphor, 2 drachms. Mix, and keep air-tight. When used, a portion is to be mixed with a little water into a paste. Cisterns and Water-casks. Melted Glue, 8 parts; Linseed oil, boiled into a varnish with Litharge, 4 parts. Tbi3 cement hardens in about 48 hours, and renders the joints of wooden cisterns and casks air and 872 MISCELLANEOUS MIXTURES. Clotli or Leather. Shellac, 1 part; Pitch, 2 parts; India Rubber, 4 parts; and Gutta Percha, 10 parts; cut small; Linseed oil, 2 parts; melted together and mixed. Earthen and. Grlass Ware. Heat article to be mended a little above 212 0 , then apply a thin coating of gum Shellac upon both surfaces of broken vessel. Or, dissolve gum Shellac in alcohol, apply solution, and bind the parts firmly to- gether until cement is dry. Or, dilute white of egg with its bulk of water and beat up thoroughly. Mix to consistence of thin paste with powdered Quicklime. Use immediately. Entomologists’. Thick Mastic Varnish and Isinglass size, equal parts. Grixtta IPercha. Melt together, in an iron pan, 2 parts Common Pitch and 1 part Gutta Percha. Stir well together until thoroughly incorporated, and then pour liquid into cold water. When cold it is black, solid, and elastic ; but it softens with heat, and at ioo° is a thin fluid. It may be used as a soft paste, or in liquid state, and answers an excellent purpose in cementing metal, glass, porcelain ivory, etc. It may be used instead of putty for glazing. ’ Grlass. Sorer s.— Mix commercial Zinc White with half its bulk of fine Sand, add a solu- tion of Chloride of Zinc of 1.26 spec, grav., and mix thoroughly in a mortar. Apply immediately, as it hardens very quickly. Holes iix Castings. Sulphur in powder, 1 part; Sal-ammoniac, 2 parts; powdered Iron turnings, 80 parts. Make into a thick paste. Make only as required for immediate use. ILydranlic JPaint. Hydraulic cement mixed with oil forms an incombustible and waterproof paint for roofs of buildings, outhouses, walls, etc. Iron Ware. Sulphur. 2 parts; fine Black-lead, 1 part. Heat sulphur in an iron pan until it melts, then add the lead; stir well, and remove. When cool, break into pieces as required. Place upon opening of the ware to be mended, and solder with an iron. Kerosene Lamps, etc. Resin, 3 parts; Caustic Soda, 1; Water, 5, mixed with half its weight of Plaster of Paris. It sets firmly in about three quarters of an honr. Is of great adhesive power, not permeable to kero- sene, a low conductor of heat, and but superficially attacked by hot water. Ijeatlier to Iron, Steel, or Grlass. 1. — Glue, 1 quart, dissolved in Cider Vinegar; Venice Turpentine, 1 oz. ; boil very gently or simmer for 12 hours. Or, Glue and Isinglass equal parts, soak in water 10 hours, boil and add tannin until mixture becomes “ropy;” apply warm. Remove surface of leather where it is to be applied. 2. — Steep leather in an infusion of Nutgall, spread a layer of hot Glue on sur- face of metal, and apply flesh side of leather under pressure. Leather Eelting. Common Glue and Isinglass, equal parts, soaked for 10 hours in enough water to cover them. Bring gradually to a boiling heat and add pure Tannin until whole be- comes ropy or appears alike to white of eggs. Clean and rub surfaces to be joined, apply warm, and clamp firmly. ^Molding and Temporary Adhesion. Soft . — Melt Yellow Beeswax with its weight of Turpentine, and color with finely powdered Venetian red. When cold it has the hardness of soap, but is easily softened and molded with the fingers. MISCELLANEOUS MIXTURES. 873 Alai tli a, or Grreeli Alastic. Lime and Sand mixed in manner of mortar, and made into a proper consistency with milk or size without water. ALarUle. Plaster of Paris, in a saturated solution of Alum, baked in an oven, and reduced to powder. Mixed with water, and color if required. Aletal to Glass. Copal Varnish, 15 parts; Drying Oil, 5; Turpentine, 3. Melt in a water bath and add 10 of Slaked Lime. Alending Shells, etc. Gum Arabic, 5 parts; Rock Candy, 2; and White Lead, enough to color. Large 01>j ects. Wollaston's White.— Beeswax, 1 oz. ; Resin, 4 ozs. ; powdered Plaster of Paris, 5 oz. Melt together. Warm the edges of the object and apply warm. By means of this cement a piece of wood may be fastened to a chuck, which will hold when cool ; and when work is finished it may be removed by a smart stroke with tool. Any traces of cement may be removed by Benzine. ALarble Workers and. Coppersmiths. White of egg, mixed with finely-sifted Quicklime, will unite objects which are not submitted to moisture. Porcelain. Add Plaster of Paris to a strong solution of Alum till mixture is of consistency of cream. It sets readily, and is suited for cases in which large rather than small surfaces are to be united. Rust Joint. {Quick Setting.) — Sal-ammoniac in powder, 1 lb. ; Flour of Sulphur, 2 lbs. ; Iron borings, 80 lbs. Made to a paste with water. (Slow Setting.)— Sal-ammoniac, 2 lbs. ; Sulphur, 1 lb. ; Iron borings, 200 lbs. The latter cement is best if joint is not required for immediate use. Steam Boilers, Steam-pipes, etc. Finely powdered Litharge, 2 parts; very fine Sand, 1; and Quicklime slaked by exposure to air, 1. This mixture may be kept for any length of time without injuring. In using it, a portion is mixed into paste with linseed oil, boiled or crude. Apply quickly, as it soon becomes hard. Soft . — Red or White Lead in oil, 4 parts; Iron borings, 2 to 3 parts. Hard. — Iron borings and salt water, and a small quantity of Sal-ammoniac with fresh water. Transparent— Glass. India-rubber, 1 part in 64 of chloroform; gum Mastic in powder, 16 to 24 parts. Digest for two days, with frequent shaking. Or, pulverized Glass, 10 parts; powdered Fluor-spar, 20; soluble Silicate of Soda, 60. Both glass and fluor-spar must be in finest practicable condition, which is best done by shaking each in fine powder, with water, allowing coarser particles to de- posit, and then by pouring off remainder, which holds finest particles in suspension. The mixture must be made very rapidly, by quick stirring, and applied immediately. Uniting Leather and. ATetal. Wash metal with hot Gelatine; steep leather in an infusion of Nutgalls, hot, and bring the two together. Waterproof Alastic. Red Lead, 1 part ; ground Lime, 4 parts; sharp Sand and boiled Oil, 5 parts. Or, Red Lead, 1 part; Whiting, 5; and sharp Sand and boiled Oil, 10. "Wood to Iron. Litharge and Glycerine.— Finely powdered Oxide of Lead (litharge) and Concen- trated Glycerine. The composition is insoluble in most acids, is unaffected by action of moderate heat, sets rapidly, and acquires an extraordinary hardness. Turner's. — Melt 1 lb. of Resin, and add .25 lb. of Pitch. While boiling add Brick dust to give required consistency. In winter it may be necessary to add a little Tallow. 874 MISCELLANEOUS MIXTURES, GLUES. Marine. Dissolve India Rubber, 4 parts, in 34 parts of Coal-tar Naphtha; add powdered Shellac, 64 parts. While mixture is hot pour it upon metal plates in sheets. When required for use, heat it, and apply with a brush. Or, India Rubber, 1 part; Coal Tar, 12 parts; heat gently, mix, and add powdered Shellac, 20 parts. Cool. When used, heat to about 250 0 Or, Glue, 12 parts; Water, sufficient to dissolve; add Yellow Resin, 3 parts; and, when melted, add Turpentine, 4 parts. Strong Glue . — Add Powdered Chalk to common Glue. Mix thoroughly. HVHucilage. Curd of Skim Milk (carefully freed from Cream or Oil), washed thoroughly, and dissolved to saturation in a cold concentrated solution of Borax. This mucilage keeps well, and, as regards adhesive power, far surpasses gum Arabic. Or, Oxide of Lead, 4 lbs. ; Lamp-black, 2 lbs. ; Sulphur, 5 ozs. ; and India Rubber dissolved in Turpentine, 10 lbs. Boil together until they are thoroughly combined. Preservation of Mucilage . — A small quantity of Oil of Cloves poured into a bottle containing Gum Mucilage prevents it from becoming sour. To Resist IVIoistnre. Glue, 5 parts; Resin, 4 parts; Red Ochre, 2 parts; mixed with least practicable quantity of water. Or, Glue, 4 parts; Boiled Oil, 1 part, by weight. Oxide of Iron, 1 part. Or, Glue, 1 lb., melted in 2 quarts of skimmed Milk. Rarclim en t. Parchment Shavings, 1 lb. ; Water, 6 quarts. Boil until dissolved, then strain and evaporate slowly to proper consistence. Rice, or Japanese. Rice Flour; Water, sufficient quantity. Mix together cold, then boil, stirring it during the time. Liquid. Glue, Water, and Vinegar, each 2 parts. Dissolve in a water-bath, then add Al- cohol, 1 part. Or, Cologne or strong Glue, 2.2 lbs. ; Water, 1 quart; dissolve over a gentle heat; add Nitric Acid 36°, 7 ozs., in small quantities. Remove from over fire, and cool. Or, White Glue, 16 ozs. f White Lead, dry, 4 ozs. ; Rain Water, 2 pints. Adc[ Al- cohol, 4 ozs., and continue heat for a few minutes. Elastic and. Sweet.— Stamps or Rolls. Elastic . — Dissolve good Glue in water by a water-bath. Evaporate to a thick con- sistence, and add equal weight of Glycerine to Glue^ submit to heat until all water is evaporated, and pour into molds or on plates. Sweet— Substitute Sugar for the Glycerine. To Adhere Engravings or Litliograplis -upon Wood. Sandarach, 250 parts; Mastic in tears, 64 parts; Resin, 125 parts; Venice Tur pentine, 250 parts; and Alcohol, 1000 parts by measure. BROWNING, OR BRONZING, LIQUID. Sulphate of Copper, 1 oz. ; Sweet Spirit of Nitre, 1 oz. ; Water, 1 pint. Mix. Let stand a few days before use. MISCELLANEOUS MIXTURES. 875 Grim Barrels. Tincture of Muriate of Iron, i oz. ; Nitric Ether, i oz. ; Sulphate of Copper, 4 scruples; rain water, 1 pint. If the process is to be hurried, add 2 or 3 grains of Oxymuriate of Mercury. When barrel is finished, let it remain a short time in lime-water, to neutralize any acid which may have penetrated , then rub it well with an iron wire scratch-brush. After Browning. — Shellac, 1 oz. ; Dragon’s-blood, .25 oz. ; rectified Spirit, 1 qt. Dissolve and filter. . , , , . . . _ T , Or Nitric Acid spec. grav. 1.2; Nitric Ether, Alcohol, and Muriate of Iron, each 1 part.’ Mix, then add Sulphate of Copper 2 parts, dissolved in Water 10 parts. LACQUERS. Small Ax* iris, or Waterproof Paper. Beeswax, 13 lbs.; Spirits Turpentine, 13 gallons; Boiled Linseed Oil, 1 gallon. All ingredients should be pure and of best quality. Heat them together in a copper or earthen ves- sel over a gentle fire, in a water-bath, until they are well mixed. Bright Iron. Work. Linseed Oil, boiled, 80.5 parts; Litharge, 5.5 parts; White Lead, in oil, n.25 parts; Resin, pulverized, 2.75 parts. Add litharge to oil ; simmer over a slow fire 3 hours ; strain, and add resin and white lea<| -, keep it gently warmed, and stir until resin is dissolved. Or, Amber, 6 parts; Turpentine, 6 parts; Resin, 1 part; Asphaltum, 1 part; and Drying Oil, 3 parts; heat and mix well. Or, Shellac, 1 lb. ; Asphaltum, 6 lbs. ; and Turpentine, 1 gallon. Iron and. Steel. Clear Mastic, 10 parts; Camphor, 5 parts; Sandarac, 15 parts; and Elimi Gum, 5 parts. Dissolve in Alcohol, filter, and apply cold. Brass. Shellac, 8 ozs. ; Sandarac, 2 ozs. ; Annatto, 2 ozs. ; and Dragon’s-blood Resin, .25 oz. ; and Alcohol, 1 gallon. Or, Shellac, 8 ozs. ; and Alcohol, 1 gallon. Heat article slightly, and apply lacquer with a soft brush. Wood, Iron, or Walls, and rendering Cloth, Paper, etc., W aterproof. Heat 120 lbs. Oil Varnish in one vessel, 33 lbs. Quicklime in 22 lbs. water in an- other. Soon as lime effervesces, add 55 lbs. melted India Rubber. Stir mixture, and pour into vessel of hot Varnish. Stir, strain, and cool. When used, thin with Varnish and apply, preferably hot. To Clean Soiled Engravings. Ozone Bleach, 1 part; Water, 10; well mixed. INKS. Indelible, for Marking Linen, etc. 1.— Juice of Sloes, 1 pint; Gum, .5 oz. This requires no “ preparation ” or mordant, and is very durable. 2 — Nitrate of Silver, 1 part , Water, 6 parts, Gum, 1 part; Dissolve. 3.— Lunar Caustic, 2 parts; Sap Green and Gum Arabic, each 1 part; dissolve with distilled water. “Preparation.”— Soda, 1 oz. ; Water, 1 pint; Sap Green, .5 drachm. Dissolve, and wet article to be marked, then dry and apply the ink. Perpetual, for Tomb-stones , Marble, etc. — Pitch, n parts; Lamp-black, 1 part; Turpentine sufficient. Warm and mix. Copying Ink. — Add 1 oz. Sugar to a pint of ordinary Ink. SOLDERING. Base for Soldering. Strips of Zinc in diluted Muriatic, Nitric, or Sulphuric Acid, until as much is de- composed as acid will effect. Add Mercury, let it stand for a day; pour off the Water, and bottle the Mercury. When required, rub surface to be soldered with a cloth dipped in the Mercury. 8/6 MISCELLANEOUS MIXTURES. VARNISHES. W aterproof. combined* Sulphur ’ 1 lb ' ’ Linseed 0il > 1 S al1 - ; boil ^em until they are thoroughly Good for waterproof textile fabrics. Harness. India Rubber, .5 lb. ; Spirits of Turpentine, 1 gall. ; dissolve into a jelly ; then mix hot Linseed Oil, equal parts with the mass, and incorporate them well over a slow fire. Fastening Reatker on Top Rollers. Gum Arabic, 2.75 ozs., and a like volume of Isinglass, dissolved in Water. To Preserve Gr lass from tlie Sun. Reduce a quantity of Gum Tragacanth to fine powder, and dissolve it for 24 hours in white of egg well beat up. Water-color Drawings. Canada Balsam, 1 part; Oil of Turpentine, 2 parts. Mix and size drawing before applying. Objects of Natural History, Sliells, Risk, etc. Mucilage of Gum Tragacanth and of Gum Arabic, each 1 oz. Mix, and add spirit with Corrosive Sublimate, to precipitate the more stringy por- tion of the Gum. Iron and Steel. Mercury, 120 parts; Tin, 10 parts; Green Vitriol, 20 parts; Hydrochloric Acid of 1.2 sp. gr., 15 parts, and pure Water, 120 parts. Blackboards. Shellac Varnish, 5 gallons; Lamp-black, 5 ozs.; fine Emery, 3 ozs.; thin with Alcohol, and lay in 3 coats. Black. Heat, to boiling, Linseed Oil Varnish, 10 parts, with Burnt Umber, 2 parts, and powdered Asphaltum, 1 part. When cooled, dilute with Spirits of Turpentine as may be required. Balloon. Melt India Rubber in small pieces with its weight of boiled Linseed Oil. Thin with Oil of Turpentine. Transfer. Alcohol, 5 ozs. ; pure Venice Turpentine, 4 ozs. ; Mastic, 1 oz. To render Canvas Waterproof' and Bliable. Yellow Soap, 1 lb , boiled in 6 pints of Water, add, while hot, to 112 lbs. of oil Paint. Waterproof Bags. Pitch, 8 parts, Wax and Tallow, each 1 part. To Clean Varnish. Mix a lye of Potash or Soda, with a little powdered Chalk. STAINING. ? Wood and Ivory. Yellow. — Dilute Nitric Acid will produce it on wood. Red. — An infusion of Brazil Wood in Stale Urine, in the proportion of 1 lb. to a gallon, for wood, to be laid on when boiling hot, also Alum water before it dries. Or, a solution of Dragon’s-blood in Spirits of Wine. Black. — Strong solution of Nitric Acid. Blue. — For Ivory: soak it in a solution of Verdigris in Nitric Acid, which will turn it green ; then dip it into a solution of Pearlash boiling hot. Purple. — Soak Ivory in a solution of Sal-ammoniac into four times its weight of Nitrous Acid. Mahogany. — Brazil, Madder, and Logwood, dissolved in water and put on hot. MISCELLANEOUS MIXTURES. 8 77 MISCELL ANEOU S. Blacking for Harness. Beeswax, .5 lb. ; Ivory Black, a ozs. - Spirits of Turpentine, x oz. ; Pruss.an Blue ground in oil, i oz ; Copal „ g f mixture is quite cold; make it harness, then polish lightly with silk. To Clean Brass Ornaments. Brass ornaments that have not b-n gilt or lackered “^ n "’L?e,tn S£ «>- wHh swons Tripoii - To Harden Brills, Chisels, etc. Temper them in Mercury. To Clean Coral. -r^^SS^^e. If much discolored, let it remain in solution for a few hours. Blacking, withovit [Polishing. Molasses, 4 ozs. ; Lamp-black, .5 oz. ; Yeast, a table-spoonful; Eggs, 2; Olive 0,1, a teaspoonful ; Turpentine, a teaspoonful. Mix well. To be applied with a sponge, without brushing. Dubhing. Resin, 2 lbs. ; Tallow, 1 lb. ; Train-oil, 1 gallon. Anti-friction G-rease. Tallow 100 lbs • Palm-oil, 70 lbs. Boiled together, and when cooled to 80°, strain th" k'sTevefa'nd mix will 28 lbs. of Soda, and x. 5 gallons of Water, for Winter, take 25 lbs. more oil in place of the Tallow. Or, Black Lead, 1 part; Lard, 4 parts. To Attach Hair Belt to Boilers. Red Lead, x lb. ; White Lead, 3 lbs. ; and Whiting, 8 lbs. Mixed with boiled Lin- seed Oil to consistency of paint. [Pastils for Fumigating. Gum Arabic 2 ozs. ; Charcoal Powder, 5 ozs. ; Cascarilla Bark, powdered, .75 oz. ; Saltpetre, .25 drachm. Mix together with water, and make into shape. Bor Writing upon Zinc Labeis. -Horticultural Dissolve too grains of Chloride of Platinum in a pint of water; add a little Mu- Cila 0 g “ Sabammon^^'dr. ; Verdigris, . dr. ; Lamp-black, .5 dr. ; Water, xo drs. Mix. To [Remove old Ironmold. Remoisten part stained with ink, remove this by use of Muriatic Acid diluted by 5 or 6 times ifs weight of water, when old and new stain will be removed. To Cut India Rubber. Keep blade of knife w T et with water or a strong solution of Potash. Adhesive for Rubber Belts. Coat driving surface with Boiled Oil or Cold Tallow, and then apply powdered Chalk. X-jiard. 50 parts of finest Rape-oil, and 1 part of Caoutchouc, cut small. Apply heat until it is nearly all dissolved. To [Preserve Heather Belting or Hose. Apply warm Castor Oil. For hose, force it through it. To Oil Heather Belting. Apply a solution of India Rubber and Linseed Oil. 4 E miscellaneous mixtures. *-^Mow 1 I P bT’ T warm. Beef Tallow, 3 lbs. ; Beeswax, , lb. Heated and applied warm to both sidea 0 s Lay dull files in diluted Sulphuric Acid until they are bitten deep enough Apply A q ua.ammonia Rera ° Ve ° U fr ° m leather. Wash with a solution of Pearlash in water Or, Extract of Litherium diluted with from 200^3^ pa^f witen' , To Remove IPaint Apply hot^tmd ?et 2 rentai'n n for > i° day ’ 4 ° ZS ' ’ boiling Water, with Quicklime, .5 lb r, Extract of Litherium, thinly brushed over the surface ? or 3 times. r , . To Clean ALarTble Mix with water. n .. ' Past e for Cleaning Metals Spirits of Turpentfne.’ Rottenstone > 6 P arts - Mix with equal parts of Train Oil and Watchmaker’s Oil, whieh Place coils of thin Sheet Lead in a bottle with Olive ° r Thicl3:eils * a few weeks, and pour off the clear oil. th ° V 0lL Ex P°se it to the sun for .. , durable IPaste. wH h a mtle cold being added to it); add a little BrownSnrarm r,rL o 1 l e boding water is prevent fermentation, and a few drops of (hi of Lavender Th? u , b: which will coming moldy. When dried, dissolve in water. Lavender > whlch will prevent it be- lt will keep for two or three years in a covered vessel. Stains. lo Remove . — Stains of Iodine are removed bv reotififvi «n,v ( T 7 alic or Superoxalate of Potash; Ironmolds bv w d -r Sp i * t; Ink stams b ^ Ox- W] th Ink, then remove them in the usual way ’ bUt lf obstlDate > moisten them ot“SSS? fr0m 'removed by Spirits of Hartshorn, or Ch with fresh solution of part in solution of Ammonia or of Hyposulphite™? Soda^T^r” 6 White ’ dip the with clean water. p 01 ^ )OC ‘ a - In a few minutes wasli tu?e r oS’V he Stai “ ed Iinen ° Ver a baSin of water, and wet mark with Tine- Rod T°a rreserve Bottoms of Iron Steam-boilers r-5 parteby w 7 e 5 ig p ht ns; Wtian Red ’ 17 part3 i ™ tin «> &S Parts; and LUharge, , . T . To Preserve Sails. Aatfa e „VwUh% 2 lu b e“vitHo,, and il "*«> -° gallons MISCELLANEOUS OPERATIONS AND ILLUSTRATIONS. 879 Whitewash. For outside exposure, slack Lime, .5 bushel, iu a barrel; add common Salt, 1 lb. ; Sulphate of Zinc, .5 lb. ; and Sweet Milk, 1 gallon. Boiled Oil and finely powdered Charcoal, each 1 part; mix to the consistence of paint. Apply 2 or 3 coats. This composition is well adapted for. casks, water-spouts, etc. Rub surface with Pumice Stone am iter until the rising of the grain is removed. Then, with powdered Tripoli and boi] Linseed Oil, polish to a bright surface. Chrome Green, .25 oz. ; Sugar of Lead, 1 lb. ; ground fine, in sufficient Linseed Oil to moisten it. Mix to the consistency of cream, and apply with a soft brush. The glass should be well cleansed before the paint is applied. The above quantity is sufficient for about 200 feet of glass. To Make Drain Tiles Porous. Mix sawdust with the clay before burning. MISCELLANEOUS OPERATIONS AND ILLUSTRATIONS. I# it is required to lay out a tract of land in form of a square, to be en- closed with a post and rail fence, 5 rails high, and each rod of fence to con- tain 10 rails. What must be side of this square to contain just as many acres as there are rails in fence ? Operation, i mile =- 320 rods. Then 320 X 320 = 160, sq. rods in an acre = 640 acres y and 320 X 4 sides and X 10 rails = 12 800 rails per mile. Then as 646 acres : 12800 rails 12800 acres : 256000 rails, which will enclose 256000 ’acres, and -1/256000 X 69.5701 = number of yards in side of a sq. acre , and -f- 1760, yards in a mile = 20 miles. 2.— How many fifteens can be counted with four fives? 3. — What are the chances in favor of throwing one point with three dice? Operation. — Assume a bet to be upon the ace. Then there will be 6 X 6 X 6 = 216 different ways which the dice may present themselves , that is , with and without an ace . Then, if the ace side of the die is excluded, there will he 5 sides lefty and 5X5X5 = 125 ways without the ace. Therefore, there will remain only 216 — 125 = 91 ways in which there could he an ace. The chance, then, in favor of the ace is as 91 to 125 ; that is, out of 216 throws, the probability is that it will come up 91 times , and lose 125 times. 4. — The hour and minute hand of a clock are exactly together at 12; when are they next together ? Operation. — As the minute hand runs 11 times faster than the hour hand, then, as 11 : 60 :: 1 : 5 min. 27^ sec. — time past 1 o'clock. 5. — Assume a cube inch of glass to weigh 1.49 ounces troy, the same of sea-water .59, and of brandy .53. A gallon of this liquor in a glass bottle, which 'weighs 3.84 lbs., is thrown into sea-water. It is proposed to deter- mine if it will sink, and, if so, how much force will just buoy it up? Operation. 3.84 X 12 -f- 1.49 — 30.92 cube ins. of glass in bottle. 231 cube ins. in a gallon X .53 = 122.43 ounces of brandy. Then, bottle and brandy weigh 3.84 X 12-1-122.43 = 168.51 ounces , and contain 261.92 cube ins., which X -59 = 154-53 ounces, weight of an equal bulk of sea-water. And, 168.51 — 154. 53 = 13 98 ounces , weight necessary to support it in the water. To Preserve Woodwork. To IP sh Wood. Faint for Window Glass. 880 MISCELLANEOUS OPERATIONS AND ILLUSTRATIONS. 6.— A fountain has 4 supply cocks, A, B, C, and D, and under it is a ci«« tern, which can be filled by the cock A in 6 hours, by B in 8 hours, by C in 10 hours, and by D in 12 hours ; now, the cistern has 4 holes, designated E F, G, and II, and it can be emptied through E in 6 hours, F in 5 hours, G in 4 hours, and H in 3 hours. Suppose the cistern to be full of water, and that all the cocks and holes were opened together, in what time would the cistern be emntied? be emptied? Operation.— A ssume the cistern to hold 120 gallons. hrs. gall. : 1 : 20 at A. '.1 : 15 af B. : : 1 : 12 at C. 1 : 10 at D. [ hour , 57 gallons. hrs. gall. If 6 : 120 : 8 : 120 ; 10 : 120 ; 12 : 120 : Run in in 1 hr3. If 6 5 4 3 Run out in 1 hour , gall. 120 :: 1 120 :: 1 120 :: 1 120 :: 1 hrs. gall. 20 at E. 24 at F. 30 at G. 40 at H. 1 14 gallons. 57 Run out in 1 hour more than run in, 57 gallons. Then, as 57 gallons : 1 hour :: 120 gallons : 2.158-}- hours. 7.— A cistern, containing 60 gallons of water, has 3 cocks for discharging it ; one will empty it in 1 hour, a second in 2 hours, and a third in 3 hours ; in what time will it be emptied if they are all opened together? Operation.— 1st, .5 would run out in 1 hour by the 2d cock, and .333 by the 3d- consequently, by the 3 would the reservoir be emptied in 1 hour. .5 -f- .333 -}- 1 = §d~ § + f, being reduced to a common denominator , the sum of these 3 — JL 1 - . whence the proportion, 1 1 : 60 6 : 32^- minutes. A reservoir has 2 cocks, through which it is supplied ; by one of them it will fill in 40 minutes, and by the other in 50 minutes ; it has also a dis- charging cock, by which, when full, it may be emptied in 25 minutes. If the 3 cocks are left open, in what time would the cistern be tilled, assuming the velocity of the water to be uniform ? Operation.— T he least common multiple of 40, 50, and 25, is 200. Then, the 1st cock will fill it 5 times in 200 minutes , and the 2d, 4 times in 200 minutes , or both, 9 times in 200 minutes ; and, as the discharge cock will empty it 8 times in 200 minutes , hence 9 — 8 = 1, or once in 200 minutes = 3. 2 hours. 9- The time of the day is between 4 and 5, and the hour and minute hands are exactly together ; what is the time ? Operation.— Difference of speed of the hands is as 1 to 12 = 11. 4 hours X 60 = 240, which - 4 - n = 21 min. 49.09 sec., which is to be added to 4 hours. 10.— Out of a pipe of wine containing 84 gallons, 10 were drawn off, and the vessel refilled with water, after which 10 gallons of the mixture were drawn off, and then 10 more of water were poured in, and so on for a third and fourth time. It is required to compute how much pure wine remained in the vessel, supposing the two fluids to have been thoroughly mixed. Operation. 84 — 10 = 74, quantity after the 1st draught. Then, 84 : 10 74 : 8.8095, and 74 — 8.8095 = 65.1905, quantity after 2 d draught. 84:10: : 65. 1905 : 7. 7608, and 65. 1905 — 7. 7608 = 5 7. 4297, quantity after 3 d draught. 84: 10: ‘.57. 4297: 6. 8367, and 57.4297 — 6.8367 = 50.593, quantity after 4th draught , = result required. I 1 ’ — A. reservoir having a capacity of 10000 cube feet, has an influx of 750 and a discharge of 1000 cube feet per day. In what time will it be emptied ? Operation. — = 40 days. Contrariwise In what time will it be filled ? 1000 — 750 The discharge being 1000 and the influx 1250 cube feet per hour. 1250 — 1000 Operation. = 40 hours = 1 day 16 hours. MISCELLANEOUS OPERATIONS AND ILLUSTRATIONS. 88 1 12.— A son asked his father how old he was. His father answered him thus’: If you take away 5 from my years, and divide the remainder by 8, the quotient will be one third of your age ; but if you add 2 to your age, and multiply the whole bv 3, and then subtract 7 from the product, you will have the number of years of my age. What were the ages of father and son i Operation.— Assume father’s age 37. Then - 5 = 32, and 32 = 8 = 4, and 4 X 3 = 12, son’s age. Again : 12 + 2 = 14, and 14 X 3 = 42, and 42 - 7 = 35- Therefore 37 - 35 - 2, error too little. Again: Assume father’s age 45; then 45 — 5 = 4°, and 4° = 8 = 5- The ™^ e 5 x 3 — 15, son's age. Again: 15 + 2= 17, and 17 X 3 = 5L and 51 — 7 — 44- There- fore 45 — 44 = 1, error too little. Hence (45 sup. X 2 error) — (37 sup. X 1 error) = 90 - 37 = 53, and 2 — 1 = 1. Consequently, 53 is father's age. Then 53 — 5 = 48, and 48 -4- 8 = 6 = .333 of son's age , and 6 X 3 = 18 years , son's age. —Two companions have a parcel of guineas. Said A to B, if y 011 will give me one of your guineas I shall have as many as you have left, B re- plied, if you will give me one of your guineas I shall have twice as many as you will have left. How many guineas had each of them ? Operation.— A ssume B had 6. Then A would have had 4, for 6 — 1 = 4 + 1 = 5- Again : 4 (A’s parcel) — 1 = 3, and 6 + 1 = 7, and 3 X 2 = 6. Therefore 7 — 6 = 1, error too tittle. Again : Assume B had 8. Then A would have 6, for 8 — 1 = 6+ 1 =7. Again : 6 (A’s parcel) — 1 = 5, and 8 -f- 1 = 9, and 5X2 = 10. Therefore 10 — 9 == 1, error too great. Hence 8X1 = 8, and 6X1=6. Then 8 + 6 = 14, and 1 + 1 = 2. Whence, di- viding products by sum of errors, 14 = 2 = 7 = B’s parcel , and 7 — 1 — 5 + 1— 6 for A when he had received x of B ; also 5 -iX2 = 7 + x= 8 = B’s parcel when he had received 1 of A. 1 4 —If a traveller leaves New York at 8 o’clock in the morning, and walks towards New London at the rate of 3 miles per hour, without intermission ; and another traveller starts from New London at 4 o clock in the evening, and walks towards New York at the rate of 4 miles per hour continuously ; assuming distance between the two cities to be 130 miles, whereabouts upon the road will they meet? Operation. — From 8 to 4 o’clock is 8 hours; therefore, 8X3 = 24 miles, per- formed by A before B set out from New London; and, consequently, 130 — 24=106 are the miles to be travelled between them after that. Hence, as (3 + 4) 7 : 3 ’. I 106 : All — 45 a more miles travelled by A at the meeting; consequently? 24 + 45 f- = 69-f miles from New York is place of their meeting. I5 _If from a cask of wine a tenth part is drawn out and then it is filled with water ; after which a tenth part of the mixture is drawn out; again is filled, and again a tenth part of the mixture is drawn out: now, assume the fluids to mix uniformly at each time the cask is replenished, what frac- tional part of wine will remain after the process of drawing out and replen- ishing has been repeated four times ? Operation.— Since .1 of the wine is drawn out at first drawing, there must remain q After cask is filled with water, .1 of whole being drawn out, there will remain *.9 of mixture; but .9 of this mixture is wine; therefore, after second drawing, there will remain .9 of. 9 of wine, or ; and after third drawing, there will remain .9 9 3 Of. 9 of. 9 Of wine, or — . Hence, the part of wine remaining is expressed by the ratio .9, raised to a power exponent of which is number of times cask has been drawn from. q4 Therefore, fractional part of wine is — .6561. 4°E* 882 MISCELLANEOUS OPERATIONS AND ILLUSTRATIONS. 16. — There is a fish, the head of which is 9 ins. long, the tail as long as the head and half the body, and the body as long as both the head and tail. Required the length of the fish. of°ta ^ RATIOX '~ Assume b0(J y t0 be 2 4 ins - in length. Then 24-4- 2 -j- 9 = 21, length Hence 21 -f 9 = 30, length of body, which is 6 ins. too great. Again : assume the body to be 26 ins. in length. Then 26 - 4 - 2 -f 9 = 22 length of tail. Hence 22 -|- 9 = 31, length of body, which is 5 ins. too great. Therefore, by Double Position, divide difference of products (see rule, page 99) by difference of errors (the errors being alike), 26 X 6 — 24 X 5 = 36 = difference of products , and 6 — 5 = 1 = difference of errors. J Consequently, 36 = 1 = 36, length of body, and 36 -4- 2 -}- 9 = 27, length of tail , and 30 4-27 + 9 = 72 ins . , length required. 17. — A hare, 50 leaps before a greyhound, takes 4 leaps to the greyhound's 3, but 2 leaps of the hound are equal to 3 of the hare’s. How many leaps must the greyhound take before he can catch the hare? Operation.— As 2 leaps of the greyhound equal 3 of the hare, it follows that 6 of the greyhound equal 9 of the hare. While the greyhound takes 6 leaps, the hare takes 8; therefore, while the hare takes 8, the greyhound gains upon her 1. Hence, to gain 50 leaps, she must take 50 X 8 = 400 leaps; but, while hare takes 400 leaps, greyhound takes 300, since number of leaps taken by them are as 4 to 3. 18. — If a basket and 1000 eggs were laid in a right line 6 feet apart, and 10 men (designated from A to J) were to start from basket and to run alter- nately, collect the eggs singly, and place them in basket as collected, and each man to collect but 10 eggs in his turn, how many yards would each man run over, and what would be entire distance run over*? Operation. — A’s course would be 6X2 feet {first term) -f- 10 X 6 X 2 feet ( last term) = 132 = sum of first and last terms of progression. Then 1324-2X10=: 660 feet = number of times X half sum of extremes = sum of all the terms, or the distance run by A in his first turn. B’s course would be 11X6X2 = 132 feet { first term) 4-20X6X2 =' 240 feet {last term) = 372 = sum of first and last terms. Then 372 4-2X10 = i860 = sum of all the times, or B 's first turn. A’s last course would be 901 x 6 X 2 = 10812 feet for the first term and = 10920 feet for the last term of his last turn. 910X6X2 Then 10 812 -f- 10 920 - 4 - 2 X 10 = 108 660 = sum of the terms, or distance run. B’s last course would be 911 x 6 X 2 = 10932 feet for the first term and 920X6X2 — 11 040 feet for the last term of his last turn. Then 10 932 -f- 1 1 040 -4-2X10 = 109 860 = sum of the terms or distance run. Therefore, if A’s first and last runs — 660 and 108 660 feet, and the number of terms 10, then, by Progression, the sum of all the terms == 546 600 feet. And if B’s first and last runs= i860 and 109 860 feet, and the number of terms 10, then the sum of all the terms = 558600 feet. Consequently, 558 600 — 546 600 = 12 000 = common difference of runs, which, be- ing added to each man’s run = sum of all runs, or entire distance run over. A’s run, 546 600 = 182 200 yds. F’s B’s “ 558600 = 186200 “ G’s C’s “ 570600 = 190200 u H’s D’s “ 582600 = 194200 “ I’s E’s “ 594600 = 198200 “ J’s run, 606 600 = 202 200 yds. “ 618600 = 206200 u “ 630600 = 210200 11 u 642600 = 214200 u “ 654600 = 218200 “ 6 006 000 feet, which - 4 - 5280 = 1137.5 miles. I 9- i 11 a pair of scales, a body weighs 90 lbs. in one scale, and but 40 lbs. in the other, what is the true weight? V (40 x 90) = 60 lbs. MISCELLANEOUS OPERATIONS AND ILLUSTRATIONS. 883 20. If a steamboat, running uniformly at the rate of 15 miles per hour through the water, were to run for 1 hour with a current of 5 miles per hour, then to return against that current, what length of time would she require to reach the place from whence she started ? Operation. 15 — j- 5 — 20 miles, the distance vun during the houi . Then 15 — 5 = 10 miles is her effective velocity per hour when returning , and 20^-10 — 2 hours , the time of returning , and 2 + 1 = 3 hours , or the whole time oc- cupied. Or, Let d represent distance in one direction , t and t' greater and less times of run- ning in hours , and c current or tide. a , v X ? — d Then = velocity of boat through the water , and = c. ’ t X t' 1 21. — Flood-tide wave in a given river runs 20 miles per hour, current of it is 3 miles per hour. Assume the air to be quiescent, and a floating body set free at commencement of flow of the tide ; how long will it drift in one direction, the tide flowing for 6 hours from each point of river ? Operation. — Let x be the time required; 202: = distance the tide has run up, to- gether with the distance which the floating body has moved; 33 — whole distance which the body has floated. Then 20 x- - 3 x : 6 X 20, or the length in miles of a tide. c = — x6 = 7 hours. 3 minutes , 31.765 seconds. 20 — 3 ^ 3 22. — A steamboat, running at the rate of 10 miles per hour through the water, descends a river, the velocity of which is 4 miles per hour, and re- turns in 10 hours ; how far did she proceed? Operation.— Let x = distance required, returning. Then, f- -- — 10; 6 x + 143 1 14 0 — - — — time of going. = time of 10 + 4 10 — 4 - 840 ; 20X — 840 ; 840 - 4 - 20 == 42 miles. 23. — From Caldwell’s to Newburgh (Hudson River) is 18 miles ; the cur- rent of the river is such as to accelerate a boat descending, or letard one ascending, 1.5 miles per hour. Suppose two boats, running uniformly at the rate of 15 miles per hour through the water, were to start one from each place at the same time, where will they meet? Operation.— Let 3 = the distance from N. to the place of meeting; its distance from C., then , will be 18 — x. Speed of descending boat, 15 + 1.5 = 16.5 miles per hour ; of ascending boat, 15 x 18 — x 1.3 — 13. ^ miles per hour. — - — = time of boat descending to point of meeting. ^ ^ = time of boat ascending to point of meeting. x 18 — x These times are of course equal; therefore, -7—= — • 16.5 i 3-5 16. 53, and 13. $x -j- 16. s x = 297, or 30X = 297. Then, 13. 53 = 297 — Hence x — — 9.9 miles, the distance from Newburgh. 3 ° 24. — There is an island 73 miles in circumference; 3 men start together to walk around it and in the same direction : A walks 5 miles per day, B 8, and C 10 ; when will they all come aside of each other again ? Operation.— It is evident that A and C will be together every round gone by A ; hence it remains to ascertain when A and B will be in conjunction at an even round, as 3 miles are gained every day by B. Therefore, as 3 : x 73 : 24.33+; but, as the conjunction is a fractional number, it is necessary to ascertain what number of a multiplier will make the division a whole number. 7^-f-24.'n+ = s, the number of days required in which A will go round 5 times, B 8, and C 10 times. 884 MISCELLANEOUS OPERATIONS AND ILLUSTRATIONS. 25.— Assume a cow, at age of 2 years, to bring forth a cow-calf, and then to continue yearly to do the same, and every one of her produce to bring forth a cow-calf at age of 2 years, and yearly afterward in like manner ; how many would spring from the cow and her produce in 40 years ? Operation. — The increase in 1st year would be o, in 2d year 1, in 3d 1, in 4th 2 in 5O1 3, in 6th 5, and so on to 40 years or terms, each term being — sum of the two preceding ones. The last term, then, will be 165 580 141, from which is to be sub- tracted 1 for the parent cow, and the remainder, 165 580140, will represent increase required. 26— The interior dimensions of a box are required to be in the propor- tions of 2, 3, and 5, and to contain a volume of 1000 cube ins. ; what should be the dimensions ? Operation. — 3 / IO °° X 2 3 = 6. 43 ; 3 00X38 J j,6 5 ; and 3 A° ooX 5 3 =l6 { v 2X3X5 V 2x3x5 y V 2x3X5 And what for a box of one half the volume, or 500 cube ins., and retaining same proportionate dimensions ? Operation.— 2 x 3 X 5 = 30, and — = 15. Then /15X 6.433 /15 X 9-65 3 ^ / V 30 —S' 1 ! y ~ = 7. 66 ; and 3 y /15 X i 6 3 = 12 ms. 30 * V 30 27. — The chances of events or games being equal, what are the odds for or against the following results ? Five Events. Four Events. Odds. Against. In favor. Odds. Against. In favor. 31 to 1 4.33 to I All the 5 4 out of 5 1 out of 5 2 out of 5 15 to 1 2.2 to 1 All the 4 3 out of 4 1 out of 4 2 OUt Of 4 5 to 3 in favor of the 5 events result- ing 3 and 2. 5 to 3 against 2 events the 4 events do not result only, or that 2 and 2. Three Events. T'wo Events. Odds. Against. In favor. Odds. Against. In favor. 7 to 1 Even All the 3 ( 2 or all out t of 3 1 out of 3 1 2 or all out l of 3 3 to 1 Even Both events ( 1 only out \ of 2 1 out of 2 ( 1 only out \ of 2 3 to 1 in favor of the 3 events result- ing 2 and 1. Even that the events result 1 and 1. 28. — Required the chances or probabilities in events or games, when the chances or probabilities of the results, or the players, are equal. Events Games. That a named event occurs a majority or more of times. Against a named event occurring an exact majority of times. Against each event occur- ring an equal number of times. Events or Games. That a named event occurs a majority or more of times. Against a named event occurring an exact majority of times. Against each event occur- ring an equal number of times. 21 Even 5 to 1 _ ii Even 3.4 to 1 20 1.33 to 1 — 4.66 to 1 10 1.7 to I 3.06 to I *9 Even 4.5 to I — 9 Even 3 to 1 — 18 1.55 to I — 4. 4 to 1 8 1.75 to I 2.66 to 1 17 Even 4.4 to I — 7 Even 2.7 to 1 — 16 1.5 to 1 — 4. 1 to 1 6 2 to I — 2.2 to I 15 Even 4 to i — 5 Even 2.2 tO I — 14 1.5 to I — 3.8 to I 4 2.2 to I — 1.66 to I 13 Even 3.7 to I — 3 Even 1.66 to 1 — 12 1.6 to I 3.44 to I 2 3 to I — ' Even. 29. — The chances of consecutive events or results are as follows : 11. — 2047 to 1. | 10.— 1023 to 1. | 9. — 5ntoi. | 8. — 255 to 1. | 7. — 127 to 1. | 6. — 63 to 1. Hence it will be observed that the chances increase with the number of events very nearly in a duplicate ratio. Illustration. — The chances of n consecutive events compared with 10, are as 2047 to 1023, or 2 to 1. MISCELLANEOUS OPERATIONS AND ILLUSTRATIONS. 885 30.— Required the chances or probabilities of events or results in a given number of times. ... The numerator of a fraction expresses the chance or probability either for the re- sult or event to occur or fail, and the denominator all the chances or probabilities both for it to occur or fail. Thus, in a given number of events or games, if the chances are even, the proba- bility of any particular result is as -±- = ~ 5 , etc., being 1 out of 2, 2 out of 4, etc. , or even. If the number of events or games are 3, then the probability of any par- ticular result, as 2 and 1, or 1 and 2, is determined as follows : Number of permutations of 3 events are 1 X 2 X 3 = 6, which represents number of times that number of events can occur, 2 and 1, or 1 and 2, to which is to be added the 2 times or chances they can occur all in one way or the reverse thereto. HenC e 6 = 1= . 3 — = l,or 3 to 1 in favor of result; and probability ot 1 2 d - 6 4 4 3 1 . * one Dartv naming or winning tw’O precise events or results, as winning 2 out 01 3, is determined as follows: Number of permutations and chances, as before shown, 3 ’ ’ ' ' 8 are 8. Hence, number of his chances being 3, 3 — 3 ? — = — , or 3 to 5 in o 1 j ~ - 3 5 favor of result; and probability of one party naming or winning all or 3 events or results is determined as follows: Number of permutations and chances being also, as before shown, 8. Hence, as there is but one chance of such a lesult, 1 + 7" _ — A , or 1 to 7 in favor of result. ’ 8 8—1 7 If number of events, etc., are 4, then probability of any particular result, as 2 and 2, or of winning 2 or more of them, is determined as follows : Number of permutations and chances of 4 events are 16. Hence, as number of chances of such a result are : of the result, and that the results do not occur precisely 2 and 2. The number of = — or as 11 to 5 in favor 16 16 — 11 5 chances of such a result being 10, 6 -j- 10 — , or 5 to 3 against it. 5 3 If number of events, etc., are 5, then probability of any particular result, as 3 and 2, is determined as follows : Number of permutations and chances being 32, and number of chances of such a result being 20, ^ ^ ^ f , or as 5 to 3 in favor of the result; and that it may occur precisely 3 out of 5, the number of chances are 10 JO-f-22 32 l6 = — } or 11 to 5 against it. 3I . What is the dilatation of the iron in a railway track per mile, be- tween the temperatures of — 20° and +130°? Operation. 20 0 -f 130 0 = 150 0 . The dilatation of wrought iron (as per table, page 519) is, from 32 0 to 212 0 = i8o° = .ooi 257 5 times its length. 1 ° 47 9 . of 5280 (feet in a mile) = Hence, as 180 : 150 :: .001 257 5 : .001047 9 5-53 f ee t P er m ^ e - 32 — A steamer having an immersed amidship section of 125 sq. feet, has a speed of 15 miles per hour with 300 H>. What power would be required for one of like model, having a section of 150 sq. feet for a speed of 20 miles . As power required for like models is as cube of speeds, Then — 1.2 relative sections , and 125 Hence, 1 : 1.2 2.37 : 2.844 times IP. 20 a = 8000 i 5 3 = 3375 — 2 . 37 relative powers. 886 MARINE STEAMERS AND ENGINES. MARINE STEAMERS AND ENGINES. Iron Cruiser (IPropeller). “Zabiaca,’’ I. R N.— Vertical Direct Engine {Compound).— Length between perpendwulai'S, 2 2 8 feet ; at water-line of 12 feet, 220 feet ; beam, 30 feet ; hold, 17.5 Displacement at load draught of 12.58 and 14.58 feet, 1202 tons. Per inch at load- line, 11.58 tons. Areas.— Of Load-line, 4867 sq. feet; of Sails, 12 312 sq. feet. Coefficients. Of Total Displacement, . 5 ; of Surface, . 74 ; of Cylindroid from cyl- inder, .61; of Cylindroid from parallelopipedon, .475. Cylinders. — 34 and 59 ins. in diam. by 36 ins. stroke of piston. Pressure of Steam.— 78 lbs. per sq. inch, cut off at 23 ins. full throttle. Revolu- tions 89.4 per minute IIP, 1400. Pitch of Propeller, 19 feet. Speed, 14 knots pei hour. Fuel. — Anthracite coal, 1.6 lbs. per IIP per hour. * 4 Centres of Gravity. —Forward of after perpendicular, 100 feet; below meta-centre at draught of 10.46 and 12.21 feet, 2.81 feet, and at load-line 3.12 feet. Of Buoyan- cy, below load-line, 4. 97 feet. Of Engines , Boilers, Water, etc. , aft of centre of length 25.25 feet; do. above top of keel, 9.17 feet. b ’ Meta-centre. — Above centre of buoyancy for mean draught of u. 3 feet = 4 feet. Iron Freight and Passenger (Propellers). “ Orient. ’’-Vertical Direct Engine {Compound).— Length upon deck , 460 feet : beam, 46. 35 feet ; depth to main deck, 27. i feet ; to spar deck, 35. 1 feet. Immersed section at load-line , 1094 sq.feet. Displacement at load draught of 26 q feet, 9500 tons ; per inch, 40 tons. Tons, 3440-5380. Cylinders.— 1 of 60 ins. in diam., and 2 of 85 ins., by 5 feet stroke of piston. Con- denser. — Surface, 12 000 sq. feet. Propeller. — 4 blades, 22 feet in diam. Pitch, 30 teet. Shaft, 20 ins. in diam. ’ J Boilers.— 4 (cylindrical tubular), 15.5 feet in diam. by 17.5 feet in length; 6 fur- naces, 4 feet in diam. by 6 feet in length. Pressure of Steam, 75 lbs. per sq. inch. Revolutions, 60 per minute. IIP, 5400. Bulkheads, 12. Decks', 3 of iron. Capacity — 30 oo tons coal, 3600 tons (measurement) cargo, 120 1st class passen- gers, 130 2d, and 300 3d class, or 3000 troops and 406 horses. Water Ballast.— Aft, 82 feet in length. Rig 4-masted bark. Passage , 35 days Plymouth to Australia. Weights.- Hull, Engines, and Boilers, 4940 tons. “Arizona.’’— Vertical Direct Engine {Compound). — Length between verven- diculars and for tonnage, 450 feet; breadth, 45.5 feet; depth , 35.7 9 feet; Tons, 514T55. Cylinders.— 1 of 62 ins. and 2 of 90 ins. in diam., by 5.5 feet stroke of Diston Condenser.— Surface, i2 5 4 osq. feet. ’ J b 01 P lsl °n. Propeller (Cast Steel).— Diam., 23 feet; weight, 27 tons. Boilers.— 6 of 13.5 feet in diam., 3 of 10 feet in length, and 3 of 18 feet Heatina RewZtioZ 50 - 0 , p q er m nufe ai TTpV q 6 fee ^ of Steam, 86 lbs. per sf inch, revolutions, ^5 per minute. IIP, 6306. Speed, 17 knots per hour. “Normandie ’’—Vertical Direct Engines ( Compound ) -Length ,qo feet n ms. , beam, 49 feet 1 1 ms. Hold , 37 feet 5 ins. Mean draught at trial, 20 %{t % s f0 aCement ' 7656 tons • Irnnersed Section at load -draught of 24.25 feet, io 6o 6, mf" ^Hn^f l™ 3 £t 3 - 7 i inS ” and 3 of 74-875 ins. in diam. ; stroke of pistons, 67 ins. , ratio ot low to high pressure, i to 4.46. 1 strokeofiriston 3 ; T/f V f -P single acting, 34 ins. diam. ; 11 by 11 inch engines Centrifugal Pum 2>*—3, 12.5 ms. in diam., driven by three . P ,Sf^V CyI lf driCal tubular), 4 double end, 13.5 feet in diam., 18.5 feet in X Hefuna wL ’ I3 ' 75 T 9-5 feet in length. Grates, 808.5 sq. feet. Heating Surface, 21 405 sq. feet. Steam Room , 3950 cube feet. XrD, €SS %hn?f Ste ™ n \ 8 5 l bs - ; cut off at • 75 stroke. Revolutions, 59 per minute. IIP, 8006. Shaft, 23.625 ms. in diam. Propeller , 22 feet in diam. Pitch, 31 feet. complet e) 1 3,6 tons. S ^ h °" r ' WeigM of En g‘nes, Boilers, and Water in boilers, MARINE STEAMERS AND ENGINES. 887 “City of San Francisco.” — Vertical Direct Engine [Compound). — Length over all , 352 feet; for tonnage , 339 feet; beam, 40. 2 feet; hold, 28 feet 10 ins.; Load draught, 22 feet. Cylinders , 2.— 51 and 88 ins. in diam. by 5 feet stroke of piston. Condenser . — Surface, 6425 sq. feet. Pressure of Steam, 80 lbs. per sq. inch. Revolutions, 55. Speed, 14 knots per hour. Propeller, 4 blades, 20 feet in diam. by 25 feet pitch. Boilers.— 6 (cylindrical tubular), 13 feet in diam. Heating Surface , 10650 sq. feet. Grates , 378 sq. feet. Ratio of Grate to heating surface, 1 to 28,; to tube area, 9 to 1 ; to smoke-pipe area, 6.66 to 1. Iron -A.xixili.ary Freight. Vertical Direct Engine [Compound). — Length on deck, 135 feet ; beam, 22.5 feet; hold, 11 feet. Load-draught , 4 feet 10 ins. and 10 feet 6 ins. Free board, 1.5 feet. Cylinders . — 21 and 40 ins. in diam. by 27 ins. stroke of piston. Condenser. — Sur- face, 617 sq. feet. Boiler (cylindrical tubular). — 12 feet in diam. by 9.5 feet in length. Heating surface, 1205 sq. feet. Grates, 38.5 sq. feet. Pressure of Steam, 80 lbs. per sq. inch. Speed, 10.8 knots per hour. IIP, 370. Consumption of coal, 8.5 tons in 24 hours. Rig. — Schooner. “Isle of Dursey.” — Vertical Direct Engines [Compound Triple Expansion). — Length on deck, 210 feet ; beam, 31.25 feet ; hold, 14. 1 f$et. Tons, 620.963. Cylinders. — 2, each 15.75 and 22 ins., and 44.33 ins. in diam. ; stroke of piston, 2.75 feet. Condenser . — 466 .75 inch tubes, No. 18 B W G. Surface , 792 sq. feet. Propeller . — 4 blades, 12.5 feet in diam. Pitch, 14.5 feet. Surface, 38.5 sq. feet. Pressure of Steam, 150 lbs. per sq. inch. Revolutions , 73 per minute. Boiler . — 1 (horizontal tubular). Heating surface, 1650 sq. feet. Grate, 42 sq. feet. IIP per sq foot of grate, 12.3 ; of heating surface, .374. Total, 500. Fuel.— Bituminous, 1.5 lbs. per IIP per hour. Rig.— Fore-topsail schooner. Iron Fire-boat. “Zophar Mills.” — Vertical Direct Engine . — Length on load-line, n 5 feet ; beam, molded , 24 feet; hold at side, 8 feet 8 ins. Immersed section at load-line of 7 feet, 150 sq.feet. Cylinder. — 30 ins. in diam. by 30 ins. stroke of piston; volume of piston space, 12.25 cube feet. Condenser. — Surface , 900 sq. feet. Boilers (return tubular). — Two, 8 feet in width by 14 feet in length. Heating sur- face, 2120 sq. feet. Grates, 80 sq. feet. Pressure . — 70 lbs. per sq. inch, cut offat . 5 stroke. Revolutions , 84 at 45 lbs. press- ure, cut off at .5. Speed.— 12.5 miles per hour. Propeller. — 4-bladed, 8 feet 9 ins. in diam. Pumps, Vertical Duplex. Steam cylinders , 4. — 16 ins. in diam. by 9 ins. stroke. Pumps , 4. — 7.5 ins. in diam. by 9 ins. stroke. Receiving pipes , 8.5 ins. in diam. Revolutions , no per minute. Discharge . — 2200 gallons per minute; or, 8 streams, 2.5 ins. to 3.25 ins. hose, average 75 feet in length of hose each, and 1.5 ins. nozzles, 160 feet. Or, 4 streams, 3.25 ins. hose, 100 feet in length of hose each, connected to one length of 16 feet of 4-inch hose, and 3.25 ins. nozzle, 280 feet. Steel Launch. Inverted Direct Engine [Non- condensing).— Length, 25 feet ; beam , 5 feet; hold , 2. 5 feet. Cylinder.— 5 ins. in diam. by 5 ins. stroke of piston. Hull,— Frame, .75 x .75 inch, No. 12 W G. Keel, stem and stern-post, each, 1.5 X 1.25 ins. 888 MARINE STEAM VESSELS AND ENGINES. Steel Yaclits. (Propellers). “Lady Torfrida.”— Vertical Direct Engines (Compound). — Length, 200 feet 8 ins.; beam , 25 feet 7 ins. ; hold , 15 feet 7 ins. Tons, 611. Cylinders.— 3, one of 24 ins. in diam. , and two of 34 ins. by 30 ins. stroke of piston. Condenser. — Surface , 1978 sq. feet. Circulating Pump, double, 12 by 17 ins. Air- pump, single, 20 by 17 ins. Boilers (return tubular).— 14. 5 feet in diam. by 9 feet in length. Heating Sur- face, 1887 sq. feet. Grate, 77 sq. feet. Pressure of Steam, no lbs. Vacuum, 28.5 ins. IIP, 1020. Propeller, Manganese bronze, n feet in diam. Pitch, 14.5 feet. Speed.— is knots per hour. Iron. “ Isa.” — Vertical Direct Engines (Compound). — Length of keel, 118.66 feet; beam , 18.75 feet ; hold, 10 feet. Tons, 248. Cylinders, 3.— 10, 15, and 28 ins. in diam. by 2 feet stroke of piston. Condenser.— Surface, 350 sq. feet. Circulating Pump, 6 ins. in diam. by 12 ins. stroke. Pressure of Steam .— 120 lbs. per sq. inch full stroke. Revolutions, 112 per minute. Speed, 12 knots per hour. Propeller .— 2 blades, 8.5 feet in diam. Pitch, 12.25 feet. Composite. “Radha.” — Vertical Direct Engines (Compound). — Length for tonnage, 142 feet ; beam, 20 feet ; depth of hold, 8 feet 8.5 ins. Tons, 77.04 and 149.15. ’ Immersed section at load-draught of 8.25 feet, 115 sq.feet. Cylinders , 3. — 1 of 20 ins. in diam., and 2 of 26 by 2 feet stroke of piston. Condenser. — Surface , 800 sq. feet. Boiler (flue and return tubular). — 9 feet 8 ins. wide, and 15 feet in length. Heat- ing surface, 1947 sq. feet. Grate, 48 sq. feet. Propeller , 7.5 feet in diam. Pitch, 12 feet. Revolutions, 135 per minute. Pressure of Steam . — 100 lbs., cut off at .5. Blast draught. “Siesta.” — Vertical Direct Engine (Compound), Herreshoff . — Length on deck over all, 98 feet ; at water-line, 90.3 feet; beam at deck, 1 7 feet ; at water-line , 15.16 feet; depth of hull from rabbet of keel to top of shear plank, 8.33 feet ; draught of water at load-line, 5.66 feet. Immersed section at load-line, 43 sq. feet. Displacement at load-draught, 63.83 tons. Area of water section, 878.7 sq.feet, and of immersed surface of hull, 1438 sq.feet. Ratio of water surface to its circumscribing parallelogram, .64 ; of immersed trans- verse section to its do. do. , . 584 ; and of displacement of immersed hull above lower edge of rabbet of keel to its circumscribing parallelopiped, .5677. Cylinders, 2.— 10.5 and 18 ins. in diam. by 18 ins. stroke of piston. Volume of piston space, 3.45 cube feet. Relative volumes of displacement of cylinders, 1 to 2.96. Air-pump , single acting, 6 ins. in diam. by 6.25 ins. stroke. Circulating and Feed Pumps, single acting, 1.125 ins. in diam. by 18 ins. stroke. Condenser, External.— Surface, 731, 5 ins. by 29.5 ins. tubes; condensing surface, 235 sq. feet. Propeller, 4 blades, 4 feet 7 ins. in diam. Pitch , 8 feet. Helicoidal area of blades, 9.46 sq. feet. Transverse area, 6.59 sq. feet. Shaft. — Journal, 3.875 X 8 ins. ; stress, 3.75 ins. Engine space , 3 feet by 5.5 feet in length. Boiler (vertical double coil). — Diam. outside of casing, 6.66 feet; height, 8 feet 10 ins. Heating surface, 558 sq. feet. Grates , 26 sq. feet. Smoke-pipe, 23.5 ins. in diam. by 25 feet above grates. Steam room, 5.7 cube feet. Heating surface to Grate, 21.5 to 1. Pressure of Steam. 60.7 lbs. per sq. inch, cut off in small cylinder at .88 of stroke, and in large at .3 stroke. In small cylinder at end of stroke, 55.2 lbs. ; in large cylinder at commencement of stroke, 50.6; and at end of stroke, 15.6 lbs. Mean back pressure in small cylinder, 47.36 lbs. ; and in large, 5.77. Revolutions , 193 per minute. Speed', 12.75 miles (11.06 knots) per hour. Slip of Propeller, 27.3 per cent. MARINE STEAM VESSELS AND ENGINES. 889 Herreshoff.— Vertical Direct Engine {Compound).— Length on deck, 100 feet; beam, 12.5 feet. Cylinder.— 1 of 12.5 and 21.5 ins. in diam. by 16 ins. stroke of piston. Pressure of steam , 120 lbs. per sq. inch. Revolutions, 480 per minute. Speed , 22.5 knots per hour. Thrust of Propeller at 15.73 knots, 4080 lbs. Torpedo Boats. (^Propellers.) Iron. Vertical Direct Engine (Compound).— Length, no feet ; beam, 12.5 feet. Displacement , 52 tons. Cylinders, 1.— 12.5 ins. and 21.5 ins. ; stroke of piston, 16 ins. Boiler. — (Horizontal tubular.) Diam., 4.75 feet. Tubes, 125 of 2 ins. in diam. Heating surface, 1016 sq. feet. Speed, 20.3 knots per hour. Steel. Composite. “Torpedo Boat,” R. N. — Vertical Direct Engine (Compound), Herreshoff Mfg. Co.— Length, 59.5 feet; beam, ;.^feet. Cylinders. — 6 and 10.5 ins. in diam. by 10 ins. stroke of piston. Condenser , External.— Surface. Boiler (vertical coil).— Tubes 2 ins. in diam. and 300 feet in length. Propeller. — 4 blades, 38 ins. in diam. by 5 feet pitch. Weight at load-draught of hull of 1.5 feet; armament and stores, 8 tons. Iron. Side Wheels. “Princess Marie and Elizabeth. ’’—Oscillating Engine (Compound).— Length on load-line, 274.8 ins.; beam, 34.75 feet; hold, 24.25 feet. Tons, 1606. Cylinders, 2. — 60 and 104 ins. in diam. by 3.5 feet stroke of piston. Pressure of Steam.— 70 lbs. per sq. inch, cut off at .6 stroke. Revolutions, 32.75 per minute. Speed, 17. 12 knots per hour. IIP, 3543. Cjnsumption of fuel, 1.92 lbs. per IIP per hour. Cost, £54900 sterling. Cutter (Corrugated). “La Bonita.” — Inclined Engine (Non-condensing).— Length upon deck, 42 feet ; beam, 9 feet ; hold , 3 feet. Immersed section at load-line , 8.75 sq. feet. Displacement at load-draught of 1.3 feet, 8386 lbs. Tons, 9.65, 0 . M. Cylinder. — 8 ins. in diam.; stroke of piston, 1 foot; volume of piston space, .35 cube foot. Water-wheels. — Diam. 5.66 feet. Blades, 7 ; breadth, 2.3 feet; depth, 7 ins. Boiler.— (Horizontal tubular). Heating surface , 95 sq. feet. Grates, 6 do. Fuel , coal or wood. Exhaust draught. Pressure of steam. — 65 lbs. per sq. inch. Revolutions, 54 per minute. IIP, 9. Hull.— Corrugated and galvanized plates, .0625 inch thick. Weights.— Hull, 2876 lbs. ; Engine and wheels, 2400 lbs. ; Boiler, 2260 lbs. ; pipes, grates, etc., 750 lbs. Steel. Ferry Boat.— Inclined Engine (Surface Condensing). — Length, 78 feet; beam, 15 feet; hold, 8 feet amidships and 5 feet at ends. Load-draught, 2. 25 feet. Cylinder , 15.5 ins. by 24 ins. stroke of piston. Boiler (cylindrical tubular).— Steel; heating surface, 220 sq. feet. Pressure of Steam, 80 lbs. per sq. inch. Plates, .125, .1875, and .25 ins. Light Draught. “Ho-nam.” — Vertical Beam Engine (Compound). — Length upon deck, 280 feet; beam, Tsfeet ; depth from hold to upper deck, 30 feet. Tons, 2364. Cylinders , 1 — 40 and 72 ins. in diam. ; stroke of piston, 10 feet. IIP, 3000. Speed , 16.14 knots per hour. Passengers, 2000. Decks , 3: Main, Saloon, and Promenade. Rig.— Schooner. F 89O MARINE STEAM VESSELS AND ENGINES, Wood Side "Wheels. Passenger and Deck Cargo. “City of Fall River,” New York to Fall River, Mass., Vertical Beam En- gine (Compound). — From notes of James E. Sague and John B. Adger , Jr. Length on water-line , 260 feet; over all , 273 feet; beam, 42 feet; over guards , 73 feet; hold 18 feet ; 1723 tons N. M. Immersed section at load-line of 9 feet 3 ins. (1750 tons), 365 sq.feet; and at load- draught of 12 feet, 480 sq.feet. Displacement at load-draught , 2350 tons. Cylinders , 2. — 1 of 44 ins. in diam. by 8 feet stroke of piston, and 1 of 68 ins. in diam. by 12 feet stroke. Clearance at each end of high pressure, 4.6 per cent, and at low pressure, 3 per cent. Volumes, 85 and 303 cube feet. Receiver, 89.13 cube feet. Air-pump , 37 ins. in diam. by 4.75 feet stroke of pis- ton. Condenser. — Surface, 4067 sq. feet. Water-wheels , 25.5 feet in diam. Blades , feathering, 12 of 40 ins. in depth by 10 feet in length. Centre of Pressure on Blades, n. 22 feet from axis of shaft. Boilers.— 2 (flue and return tubular), 17.5 feet in width by 15 feet in length, 220 tubes 3.5 ins. in diam. and 12 feet in length. Grates, 230 sq. feet. Fuel. —Anthracite. Natural Draught. Consumption, 1463 lbs. per hour; refuse, 281 lbs. = 19-23 per cent, per sq. foot of grate, and 12.73 lbs. Pressure of Steam.— High-pressure cylinder, throttle open and cut off at .445, mean in boiler per guage, 70 lbs. ; in receiver, n lbs. ; mean effective pressure, 41.8 lbs. per sq. inch. Low-pressure cylinder, at point of cutting off of .45, 17.42 lbs. above zero; mean effective pressure, 12.4 lbs. per sq. inch. Expansion of steam, 6.99 times. Vacuum, 28.4 ins. IP.— High-pressure cylinder, 783; low-pressure, 840. Revolutions, 25.8 per minute. Feed Water , 27 854 lbs. per hour; per IEP, 17.17 lbs. Temperatures. — Feed water, 97 0 ; sea water, 49. 4 0 ; water of condensation, 90 0 ; heat units per hour per IIP, 19 090. Stress of wheels, 20.4 per cent. Condensing water, per IIP per hour, 407 lbs. Consumption.— Compound engine, 2.03 lbs. per IIP; and as a simple condensing engine, without high-pressure cylinder, 2.84 lbs. Evaporation per hour , 1208 lbs. water; per lb. of combustible from 212 0 , 11.75 lbs.; from temperature of feed (97 0 ), 10.22 lbs. ; from feed per lb. of coal, 826 lbs. Temperature of gases in chimney 485°. Heating Surface , 29 sq. feet to 1 of grate. Weights, Engine, and Frame, 250 tons; boilers complete, 120 tons; water, 50 tons. Speed, 14.14 knots per hour; and IIP, 1623. Draft of water , 10.65 feet; Displacement, 1938 tons. “City of Boston,” New York and Norwich. —Vertical Beam Engine (Con- densing). — Length upon load-line, 320 feet ; beam, 39 feet ; hold, 12.6 feet. Immersed section at load-line, 288 sq.feet. Displacement 1450 tons, at load-draught of 8. 25 feet. Cylinder. — 80 ins. in diam. by 12 feet stroke of piston. Volume, 419 cube feet. Water-wheels. — Diam. 37 feet 8 ins. Arms , 36. Blades, 37; breadth of do., 10 feet; depth of do., 30.5 ins. Dip at load-line, 4.25 feet. Boilers.— 2 (flue and return tubular), Shells, 12.5 feet in diam., and in length 26.5 feet. Heating Surface, 10 120 sq. feet. Grates, 184 sq. feet. Pressure of Steam. — 35 lbs. per sq. inch, cut off at .5 stroke. Revolutions (maxi- mum), 19.75 per minute. IIP, 2500. Fuel. — Anthracite; Blast. Consumption, at ordinary speed, 5200 lbs. per hour. Weights of Engine, Boilers, etc., 263 tons. Hull.—^ Weight, 800 tons. Light draught of hull without fuel, water, or furni- ture, 7 feet. MARINE STEAM VESSELS AND ENGINES. 89 1 Wood Propellers. Herreshoff, R. N., Vertical Direct Engine [Compound).— Length on deck , 46 feet ; over all, 48 feet; beam, gfeet; hold , 5 feet. Displacement at load-line, 7 -44 tons. Area of section at load-lime , 217.8 sq.feet. Area of wetted surface, 365. 5 sq. feet. Coefficient of fineness, . 396. Cylinder.— 8 and 14 ins. in diam. by 9 ins. stroke of piston. Condenser, External. — Surface. Propeller .— 4 blades, 3 feet in diam. by 4 feet 1 inch pitch. Blower, 42 ins. in diam. Boiler (vertical coil). Heating surface , 174 sq. feet. Grates , 12. 5 sq. feet. Pressure of Steam, 53 lbs. per sq. inch. Revolutions, 333 per minute. IIP, 68.4. Speed 10 18 knots per hour. With 129 lbs. and 466 revolutions, 14.26 knots. IIP, 169.5.’ Weight of Engines, Boiler , and Water, 5300 lbs. Herreshoff, Vertical Direct Engine [Compound). — Length over all , 8 6 feet; beam , n feet. Displacement, 27 tons. Cylinder .— 13 and 22 ins. in diam. by 12 ins. stroke of piston. Surface Condensing. Pressure , 130 lbs. per sq. inch. Revolutions, 460 per minute. Speed, 20 knots per hour. IIP, 425. Propeller , 3 blades. Pitch, 5 feet. Herreshoff R I N.— Vertical Direct Engine [Compound).— Length over all, 60 feet ; beam , 7 feet ; hold , 5. 5 feet. Displacement at load-draught of 32 ms . , 7 tows (2240 lbs. ). Cylinders.— 8 and 14 ins. in diam. by 9 ins. stroke of piston. Surface condenser. Pressure of Steam .— 140 lbs. per sq. inch, cut off at .5. Revolutions , 600 per minute. Speed, 19.875 knots per hour. Cable or Rope Towing. “ Nyitra. Horizontal Direct Engines [Condensing).— Length of boat, xf&feet; beam, 24. 5 feet ; hold, 7. 5 feet. Immersed section, 74.4 sq.feet. Displacement, 200 tons at load-line of 3.7 5 feet. Immersed section , 263.7 sq. feet. Displacement, 949 tons. Tow. 3 barges. Cylinders . — 2 of 14.18 ins. in diam. by 23.625 ins. stroke of piston. IIP, net effective , 100. Speed, 7.73 miles per hour. Propellers. — Twin, 4 feet 2 ins. in diam. Stress. — Cable, 7485 lbs. Per ton of displacement, 6.5 lbs. ; per sq. foot of im- mersed section, 22 lbs. Fuel. — Per mile and ton of displacement (1149), .078 lbs. Towing. Wood Side VWlieels. u Wm h . Webb.”— Harbor and Coast.— Vertical Beam Engines [Condensing). —Length upon deck, 185.5 feet; beam, 30. 25 feet; hold, 10.8 feet. Immersed Section at load-line, 194 sq.feet. Displacement 498.25 tons, at load- draught of 7. 25 feet. Cylinders.— 2, of 44 ins. in diam. by 10 feet stroke of piston ; volume, 21 1 cube feet. Condensers.— Jet, 2, volume 105 cube feet. Air-pumps.— 2, volume 45 cube feet. Water-wheels . — Diam., 30 feet. Blades (divided), 21; breadth of do., 4.6 feet; depth of do., 2.33 feet. Dip at load-line, 3.75 feet. Boilers. — 2 (return flue). Heating surface, 3280 sq. feet. Grates, 147.5 sq. feet. Smolce-pipe.— Area, 11.6 sq. feet, and 35 feet in height above the grate level. Pressure of Steam .— 35 lbs. per sq. inch, cut off at .5 stroke. Revolutions, 22 pel minute. IIP, 1500. Fuel . — Anthracite or Bituminous. Consumption, 1680 lbs. per hour. Speed .— 20 miles per hour. Weights. — Engines, Wheels, Frame, and Boilers, 310579 lbs. 892 RIVER STEAMBOATS AND ENGINES. Wood Side ‘W'h.eels. Passenger. “Daniel Drew,” New York to Albany.— Vertical Beam Engine ( Condensing ). — Length upon deck , 251.66 feet ; at load-line , 244 feet ; beam , 31 feet ; hold , 9. 25 feet. Immersed section at load-line , 136 sq.feet. Displacement 380 tons , load-draught of 4.83 feet. Cylinder. — 60 ins. in diam. by 10 feet stroke of piston; volume , 196 cube feet. Condenser. — «/e£, volume 68 cube feet. Air-pump, volume 26 cube feet. Water-wheels. — Diam. 29 feet. Arms , 24. Blades , 24; breadth of do., 9 feet; depth of do., 26 ins. Dip at load-line, 2.33 feet. Boilers . — 2 (return flue), 29 feet in length by 9 feet in width at furnace. Shell, diam. 8 feet. Heating surface , 3350 sq. feet. Grates, 105 sq. feet. Cross area of lower flues, 15.5 sq. feet; of upper, 13 sq. feet. Weight , 80650 lbs. Smoke-pipes. — 2, area 25.13 sq. feet, and 32 feet in height above the grate level. Pressure of Steam . — 35 lbs. per sq. inch, cut off at .5 stroke. Revolutions ( maxi- mum ), 26 per minute. IIP, 1720. Fuel. — Anthracite; Blast. Consumption , 3800 lbs. per hour. Speed , 22.3 miles per hour. Slip of Wheels from Centre of Pressure, 12.5 per cent. Frames. — Molded, 15.75 ins. ; sided, 4 ins. ; and 20 ins. apart at centres. “ Mary Powell,” Hudson River.— Vertical Beam Engine {Condensing).— Length on water-line, 286 feet ; over all , 294 feet ; beam , 34 feet 3 ins. ; over all, 64 feet ; hold , 9 feet. Deck to promenade deck, 10 feet. Immersed section at load - line of 6 feet , 200 sq. feet. Displacement , 800 tons at mean load-draught of 6 feet. Area of transverse head surface of hull above water, 2000 sq.feet. Cylinder . — 72 ins. in diam. by 12 feet stroke of piston; volume, 338 cube feet. Clearance at each end, 12.5 cube feet. Steam and Exhaust Valves , 14.75 ins. in diam. Air-pump, 40 ins. in diam. by 5 feet 2 ins. stroke of piston. Condenser. — Jet , 128 cube feet. Crank-pin , 8.75 ins. in diam. x 10.75 ins. Beam, 22.5 feet in length; centre, 9.75 in diam. Water-wheels — Diam. 31 feet; blades (divided), 26; breadth of do., 10 feet 6 ins. ; width, 1 foot 6 ins. ; immersion, 3 feet 6 ins. Shafts.— Journal, 15.625 ins. by 17 ins.’ Boilers . — 2 (flue and return tubular), of steel, n feet front by 26 feet in length; shell, 10 feet in diam. and 16 feet 1 inch in length. Furnaces, 2 in each, of 4 feet 10 ins. by 8 feet in length. Heating Surface, 2660 sq. feet; and Superheating, 340 sq. feet in each. Grates , 152 sq. feet. Flues, 10 in each, transverse area, n feet 7 ins. Tubes, 80 in each, 4.5 ins. in diam., 6 feet 6 ins. in length, and 8 feet 7 ins. in transverse area. Steam Chimneys . 8 feet in diam. x 12 feet in height. Smoke-pipe , 4 feet 6 ins. in diam. and 68 feet in height from grates. Combustion , Blast. Blowers , 4 feet in diam. and 3 feet in width. Revolutions, 78 per minute. Fuel (anthracite), 6280 lbs. per hour, or 40 lbs. per sq. foot of grate per hour. Per sq. foot of heating surface, 2.25 lbs. Speed , 23.65 miles per hour. Pressure of Steam, 28 lbs. per sq. inch, cut ofT at .47 stroke; terminal pressure, 16.4 lbs. ; throttle, .625 open. Vacuum, 25 ins. Revolutions , 22.75 per minute. Temperatures.— Reservoir, 120°. Feed water, 120 0 . Chimney, 740 0 . IP.— Total, 1900. IIP, 1560. Net, 1450. Evaporation.— Water per lb. of coal, from 120 0 , 7 lbs. ; per lb. of combustible, from 120 0 , 8. 2 lbs. Steam per total IP per hour, 21. 1 lbs. Coal per do. do., 3. 14 lbs. Weights. Engine. — Frame, keelson, out -board wheel -frames donkey engine, and boiler, blower engines and blowers, all complete, 360000 lbs. Boilers.— Iron return flue. 120000 lbs. Steel return tubular, 116000 lbs. Water, 128000 lbs. Capacity.— 2000 passengers and their baggage. Memoranda. — Th i s vessel was originally but 266 feet in length, and when length- ened the cylinder of 62 ins. in diam. was removed and replaced with one of 72 ins. Engine designed throughout for original cylinder and a pressure of from 50 to 55 lbs., cutting olf at .625 of stroke, with throttle wide open. Engines and Boilers built by Fletcher, Harrison, & Co., New York, 1861 and 1875. RIVER STEAMBOATS AND ENGINES. 893 ‘‘Solano ” Ferry Boat.— Vertical Beam Engines [Condensing).— Length over all , 424 feet; on keel, 406 feet; beam [molded), 64 feet; hold at 18.5 feet; at ends, 15 feet 10 ins. ; width over guards, 116 feet. Light draught, 5 feet; loaded, 6.5 feet. Tons , 3541. Cylinders.— 2 of 60 ins. diam. by 11 feet stroke of piston. Wheels , 34 feet in diam. by 17 feet face. Blades, 24. Boilers, 8.— Steel; 7 feet in diam. by 28 feet in length. Heating surface, 19640 sq. feet. ’ Grates, 288 sq. feet. IIP, 4000. Passenger and. Liglit IPreiglrt. “Seth Grosvenor.”— Steeple Engine [Condensing).— Length upon deck, 9 $feet ; beam , 17.2 feet; hold, 5 feet. Immersed section at load-line , 43 sq.feet. Displacement 73 tons, at load-draught of 3-25 fat. Cylinder .— 28 ins. in diam. by 3 feet stroke of piston; volume, 12.8 cube feet. Water-wheels. — Diam. 13.5 feet. Blades, 14; breadth of do., 3 feet; depth of do., 1.25 feet. Boiler (flue and return tubular ). — Heating surface, 540 sq. feet. Grates, 22.5 sq. feet. Area of tubes, 367 sq. ins. IIP, 90. Weights.— Engine, Wheels, Frame, and Boiler, 61 556 lbs. = 27-4 tons. The operation of this vessel was in every way successful, being: very fast, economical in fuel, etc., and she would have been improved if the hull had had 15 feet additional length, all other dimensions and capacities remaining the same. "Wood. Stern Wlieels. Passenger and Deck Dreiglit. “Montana.” — Horizontal Engines [Non-condensing). — Length upon deck [over all), 248 feet; at water-line , 245 feet; beam, 48 feet 8 ins. [over all, 50 feet 4 ins.); hold, 6 feet; draught of water at load-line , 5.5 feet. Immersed section at load-line, 244 sq. feet. Displacement at mean light draught of 22 ins . , 594 tons (2000 lbs. ) Cylinders.— Two, 18 ins. in diam. by 7 feet stroke of piston. Valves, 4.5 and 5 ins. in diam. Piston-rod , 4 ins. Steam-pipe, 4.5 ins. Connect- ing-rod, 30 feet in length. Water-wheel , 19 feet in diam. by 35 feet face; blades, 3 feet in depth. Shaft, 10.25 ins. in diam. Boilers. Four (horizontal tubular), 42 ins. in diam. by 26 feet in length. Two flues in each, 15 ins. in diam. Heating surface, 0 ffective, 1023, total 1431 sq. feet. Furnace, 6.5 X 17 feet. Grates , 4.16 X 17 feet; surface, 70.8 sq. feet. Smoke-pipes. Two, 3 feet in diam. by 55 feet 3 ins. in height. Exhaust or Blower draught. Calorimeter. — Of Bridge, 15.27; of Flues, 9.82; and of Chimneys, 14.14 sq. feet. Areas of grate, compared to calorimeter of flues, 7.2; to ditto, of chimneys, 5; and of bridge, 4.6 sq. feet. Steam-room, 562 ; and water space, 294 cube feet. Hull.— Frames, 4X6 ins. and 15 ins. apart at centres. Intermediate do., 4X6 ins., and running for 7.5 feet each side of keelson. Planking. — Bottom, oak, 4 ins. ; side do., 2.5 to 4 ins. Deck beams, pine, 3X6 ins. Deck plank, 2.5 ins. Keelson, oak; side do., eight each side, one each 7, 8.75, and 9 ins., and five 6.75 ins. Wales, one each side, 9 and 7 ins. by 3, and one 10 X 2.5 ins. Deck posts, 3.5 X 3 ins. ar *d 4 feet apart. Deck beams, 5. 5 X 3 ins. Knuckles , oak, 6 X 12 ins. Bulkheads, one longitudinal and one athwartship at shear of stern. Sheathing of wrought iron, .0625 to .125 inch from just below light to load-line. Hog Posts .— White pine, 8.5 and u ins. square. Chains , 1.5 ins. in diam. Weights. — Boilers, 29 264; water, 18 351 ; and boilers, chimneys, grates, and water, 55672 lbs. Hull, oak, 520560; Pine, 91 437; Bolts, spikes, etc., 8000, and Deck and guards, 76000 lbs. ; Hull alone, 310 tons. Weight of hull compared to one of iron as 8 to 5, effecting a difference of about 100 tons. 894 RIVER STEAMBOATS. SAILING VESSELS. Passenger and. Peck PreigLt. “Pittsburgh.” — Horizontal Engines {Non- condensing).— Length on deck , 252 feet; beam, 39 feet ; hold, 6 feet; draught of water at load-line, 2 feet. Immersed section at load-line, 75 sq.feet. Displacement at load-draught of 2 feet, 380 tons (2000 lbs.). Cylinders.— Two, 21 ins. in diam. by 7 feet stroke of piston. Water-wheel .— 21 feet in diam. by 28 feet face. Boilers.— 2 (horizontal tubular), 47 ins. in diam. by 28 feet in length. Two fires in each. Iron. Stern Wheels. Horizontal Engines ( Non-condensing ). — Length upon deck , no feet ; beam, 14 feet ( deck projecting over, 4 feet ) ; hold, 3.5 feet. Immersed section at load-line , 10.25 sq.feet. Displacement at load-draught of 1.1 feet, 33 tons. Cylinders.— Two, of 10 ins. in diam. by 3 feet stroke of piston; volume of piston space, 1.6 cube feet. Wheel. — Diam. 13 feet. Blades , 13; breadth of do., 8.5 feet; depth of do., 8 ins. Revolutions, 33 per minute. Boiler .— One (horizontal tubular). Tubes, 100 of 2 ins. in diam. Fuel . — Bituminous coal. Consumption , 4480 lbs. in 24 hours. Hull.— Plates, keel, No. 3; bilges, No. 4; bottom, No. 5; sides, Nos. 6 and 7. Frames , 2.5 X - 5 ins., and 20 ins. apart from centres. Steel. “Chattahoochee.”— Inclined Engines {Non- condensing).— Length on deck , 157 feet; beam, 31. 5 feet; hold, 5 feet. Immersed section at load-line, 153 sq.feet. Freight capacity , 400 tons (2000 lbs.). Cylinders.— Two, 15 ins. in diam. by 5 feet stroke; volume of piston space, 12.26 cube feet. Wheel.— One, 18 feet in diam. ; blades, 2 feet in depth. Boilers . — Three (cylindrical flued). Diam. 42 ins. ; length, 22 feet; 2 flues of 10 ins. in each. Heating surface, 690 sq. feet. Grates , 48 sq. feet. Pressure of Steam, 160 lbs. per sq. inch, cut off at .375. Revolutions, 22 per min. Consumption of Fuel, 12 tons (2000 lbs.) in 24 hours. Plating of Hull , .1875 to .25 inch. Light draught , 21 ins. Iron Propellers. Vertical Direct Engines {Non-condensing). — Length on deck , 70 feet; beam, 10.5 feet ; draught, 12 ins. Propellers, 2.-2 blades, 16. ins. in diam., set n ins. below water-line. Boiler (tubular coil). Revolutions , 480 per minute. Speed, 10.49 miles P er hour. Water led to propellers through tunnels in bottom at sides. “Louise.”— Vertical Tandem Engines {Compound).— Length, 60 feet; beam , 12 feet ; hold, 4.25 feet. Displacement at load-draught of 2. 5 feet , 8 tons. Cylinders , 5 and 10 ins. in diam. by 8 ins. stroke of piston. Surface Condenser.— Boiler (vertical tubular), 4 feet in diam. by 8.5 in length. Iron Sailing "Vessels. Passenger and Preiglxt. English. — Ship . — Length upon deck , 178 feet ; do. at mean load-line of 19 . 16 feet . ,177 feet ; keel, 171 feet; beam, 32.88 feet ; depth of hold, 21.75 feet ; keel (mean), i.'j^feet. Immersed section at load-line, 387 sq. feet. Displacement at load-draught of 19. 16 feet, 1385 tons; at deep load-draught of 20 feet, 1495 tons; and, in proportion to its circumscribing parallelopipedon, . 524. Load-line.— Area at load-draught, 4557 sq. feet. Angle of entrance 57°; of clear- ance, 64°. Area in proportion to its eircumscrib ng parallelogram, .784. YACHTS. CUTTERS. PILOT BOAT. 895 rprttre of Gravity , 6.416 feet below mean load-line. Centre of Displacement (grav- ity of b 6- 25 feet below load-line; and 4.33 feet before middle of length of load-line. Immersed Surface.— Bottom, 7370 sq. feet. Keel , 1130 sq. feet. Sails , 13 282 sq.feet. Meta-centre , 6.66 feet above centre of gravity of displacement Centre of Effort before centre of displacement, 3. 5 feet ; height of do. above mean load-line, 55. 5 ^et. Launch.. 'Wood. Steam Launch “ Herreshoff. ’’—Vertical Engine {Compound).— Length, 33 feet 1 inch ; beam, 8.7 5 feet. Displacement at mean load-draught of {to rabbet of keel) 19 ins., 8929 lbs. Weights.— Hull and Machinery, 6555 lbs. Coal, 1120 lbs. Yachts. Wood. “America ” Schooner.— Length over all, 98 feet ; upon deck, 94 feet; at load-line, qo 5 feet; beam, 22.5 feet ; at load-line, 22 feet ; depth of hold 9. 25 feet. Height at side from under side of garboard sir alee, 1 1 feet. Sheer , forward, 3 feet ; aft, 1 . 5 feet, Tmmersed section at load-line , 121.8 sq. feet. Displacement at load-draught of 8. 5 feet from under side of garboard strake and of 11 feet aft, 191 tons; and, m pro- portion to Volume of circumscribing par allelopipedon, .375. Displacement at 4 feet {from garboard strake ), 43 tons ; at 5 feet, 66 tons ; at 6 feet, 93 tons; at 7 feet, 127 tons; and at 8 feet , 167 tons. Centre of Gravity.— Longitudinally, 1.75 feet aft of centre of length upon load- line Sectional, 2.58 feet below load-line. Of Fore body, 14.25 feet forward; and of After body, 19 feet aft. Meta-centre , 6.72 feet above centre of gravity. Centre of Effort , 31 17 feet from load-line. Centre of Lateral Resistance , 6.33 feet abaft of centre of gravity. Area of Load-line , 1280 sq. feet. Mean girths of im- mersed section to load-line, 25 feet. Load-draught.— Forward, 4.91 feet; aft, 11.5 feet. Rake of Stem, 17 feet. Soars — Mainmast , 81 feet in length by 22 ins. in diam. Foremast , 79.5 feet in lenath by 24 ins. in diam. Main boom , 58 feet in length. Main gaff , 28 feet. Fore gaff ; 24 feet. Rake, 2.7 ins. per foot. Drag of Keel, 3 feet. Tons, 170.56. “Julia,” Sloop .—Length for tonnage , 72.25 feet; on water-line, qo feet 7 ins.; beam, 19 feet 8 ins.; hold, 6 feet 8 ins. Tons, 0 . M. 83.4; N. M. 43.98. Load-draught, 6.25 feet. Sails.— Mainsail, hoist, 49.75 feet, foot 54.25, and gaff 27.66; Jib , hoist, 49.75 feet, foot 39. 5, and stay 63. 5. Gaff topsail, hoist, 24. 5 feet. Areas. — Mainsail, 2322 sq. feet. Jib, 986, and Topsail, 454. Cutters. “Tara” {English) Sloop.— Length on load-line, 66 feet; beam, 11.5 feet. Immersed section at load-line, 11.5 sq.feet. Displacement, 75 tons. Spars.-Mast, deck to hounds, 42 feet. Boom, 58 feet. Gaff, 39 feet Bowsprit outside of stem, 30 feet. Mast to stem, 26 feet. Topmast, foot to hounds, 25 feet. Balloon topsail yard, 46 feet. Canvas, area, 3450 sq. feet. Tons, t. H., 90. Ballast .— At Keel , 38.5 tons. Hull, 1.5 tons. “Mischief” {English), Sloop .— Length on load-line, 61 feet ; beam, 19.9 feet. Immersed section at load-line , 60 sq. feet. Displacement , 55 tons. Pilot Boat. “Wm. H. Aspinwall,” Schooner. —Length of keel , 74 feet; upon deck, 80 feet; beam, 19 feet ; hold , 7.6 feet. Draught of water. 6 feet forward ; aft , 9. $ feet. Keel, 22 ins. in depth. False keel, 12 ins. in depth at centre. Spars.— Mainmast. 77 feet in length. Foremast, 76 feet. Main boom, 46 feet Main gaff, 21 feet. Fore gaff, 20 feet. Tons.— N. M., 46.32. 896 PASSAGES OF STEAMBOATS. ICE-BOATS. PASSAGES OF STEAMBOATS. Distances in Statute Miles. 1807, Clermont , of N. Y., New York to per hour, neglecting effect of the tide. Albany, 145 miles, in 32 hours = 4. 53 miles 1811, New Orleans , of Pittsburgh, Penn, (non-condensing and stern-wheel) Pitts- burgh to Louisville, Ky., 650 miles, in 2 days 22 hours. 1849, Alida, of N. Y., Caldwell’s, N. Y., to Pier 1, North River, 43.25 miles, in 1 hour 42 min . , ebb tide = 2. 75 miles per hour. Speed = 22. 19 miles per hour i860 30th Street, N. Y., to Cozzens’s Pier, West Point, 50.5 miles, in 2 hours 4 min., and to Poughkeepsie, 74.25 miles, in 3 hours 27 min., 5 landings, flood tide. And 1853 Robinson Street to Kingston Light, 90.375 miles, in 4 hours, making 6 landings' tiond t.idp ' ° ’ 1850, Buckeye State, of Pittsburgh, Penn, (non - condensing), Cincinnati to Pitts- burgh, 500 miles (200 passengers), 53 landings, in 1 day 19 hours; 4 miles per hour adverse current. Speed == 15.63 miles and 1.23 landings per hour. Average depth of water in channel 7 feet. 1852, Reindeer , of N. Y., New York to Hudson, 116.5 miles, in 4 hours 57 min making 5 landings. Flood tide. ^ 5/ ’ 1853, Shotwell , of Louisville, Ky. (non-condensing), New Orleans to Louisville 1450 miles, 8 landings, in 4 days 9 hours; 4.5 to 5.5 miles per hour adverse cur- rent. Speed = 18.81 miles per hour. Note.— In 1817-18 the average duration of a passage from New Orleans to Louisville was 27 days. 12 hours; the shortest, 25 days. * 1855, New Princess, of New Orleans (non-condensing), New Orleans, La., to Natchez Miss., 310 miles, in 17 hours 30 min.; 3.5 to 4 miles per hour adverse current! Speed == 20.98 miles per hour. 1864, Daniel Drew, of N. Y., Jay Street. N. Y., to Albany, 148 miles, in 6 hours 51 min . , 9 landings. Flood tide. Speed of boat = 2?. 6 miles per hour. 1867, Mary Powell, of N. Y., Desbrosses Street, N. Y., to Newburgh, 60.5 miles, in 2 hours 50 min. , 3 landings; from Poughkeepsie to Rondout Light, 15.375 miles, in 39 min., flood tide. 1873, Milton to Poughkeepsie, light draught and flood tide, 4 miles, in 9 min. ; and 1874, Desbrosses Street to Piermont, 24 miles, in 1 hour; to Caldwell’s, 43.25 miles, in 1 hour 50 min. Speed = 22. 77 to 23 miles per hour. Runs from New York to Albany, 146 miles, by different Boats. 1826, Sun. 12 hours 16 min. 182 6, North America* . 10 “ 20 “ 1841, Troy f 8 “ 10 “ 1841, South America t. 7 “ 28 “ 1852, Fr. Skiddy § 6 hours 24 min. i860, Armenia n 7 “ 22 “ 1864, Daniel Drew } 6 “ 51 “ 1864 , Ch’ncey Vibbard X. 6 “ 42 “ * 7 landings. + 4 landings. % 9 landings. § 6 landings. || 11 landings. Timing Distance .— From 14th St., Hudson River, N. Y., to College at Mount St. Vincent, 13 miles. Note. — Where landings have been made, and the river crossed, the distance between the points given is correspondingly increased. 1870, R. E. Lee , of St. Louis (non-condensing), New Orleans to St. Louis, Mo., 1180 miles (without passengers or freight), 4 to 5 miles per hour adverse current; Vicks- burg, 1 day 38 min.; Memphis, 2 days 6 hours 9 min.; Cairo, 3 days 1 hour.; and to St. Louis, 3 days 18 hours 14 min., inclusive of all stoppages. 1870, Natchez, of Cincinnati, Ohio, from New Orleans to Baton Rouge, 120 miles, in 7 hours 40 min. 42 sec. Runs from Neio Orleans to Natchez, 295 miles , by different Boats. 1814, Orleans, 6 days 6 hours 40 min. I 1856, New Princess , 17 hours 30 min. 1840, Edward Shippen, 1 day 8 hours. | 1870, R. E. Lee , 16 hours 36 min. 47 sec. Ice-L>oats. Distances in Statute Miles. 1872, Haze, of Poughkeepsie, N. Y., to buoy off Milton, 4 miles, in 4 min. 1872, Whiz, of Poughkeepsie, N. Y., to New Hamburg, 8.375 miles, in 8 min. i i PASSAGES OF STEAMERS AND SAILING VESSELS. 897 PASSAGES OF STEAMERS AND SAILING VESSELS. Distances in Geographical Miles or Knots. Steamers. Side-wlxeels. 1807. Phoenix , of Hoboken, N. J. (John Stevens), New York, N. Y., to Philadelphia Penn. First passage of a steam vessel at sea. 1814, Morning Star , of Eng., River Clyde to London, Eng. First passage of an English steamer at sea. 1817, Caledonia , of Eng. , Margate, Eng., to Cassel, Germ., 180 miles, in 24 hours. 1810 Savannah , of N.Y., about 340 tons 0 . M., Tybee Light, Savannah River, Ga to Rock Light, Liverpool, Eng., 3640 miles, in 25 dags 14 hours; 6 days 21 hours of which were under steam. 1825 Enterprise , of Eng., 500 tons, Falmouth, Eng., to Table Ray, Africa, in 57 days ; and to Calcutta, India, in 113 days. First passage of a steamer to India. 1830, Hugh Lindsay , 41 1 tons, 80 HP, Bombay, India, to Suez, Egypt, 3103 miles, in 31 days running time. 1837, Atlanta , of Eng., 650 tons, Falmouth, Eng., to Calcutta, in 91 days. 1839, Great Western , of Eng., Liverpool to New York, N. Y., 3017 miles, in 12 days 18 hours. 1870 Scotia , of Eng., Queenstown, Ireland, to Sandy Hook, N. J., 2780 miles, in 8 days 7 hours 31 min. 1866, New York to Queenstown, 2798 miles, in 8 days 2 hours 48 win.; thence to Liverpool, Eng., 270 miles, in 14 hours 59 min.; total, 8 days 17 hours 47 min. Screw. 1874. India Government Boat , Steel, length 87 feet, beam 12 feet, draught of water 3.75 feet, mean speed for one mile 20.77 miles per hour, and maintained a speed of 18.92 miles in 1 hour. 1877, Lusitania, of Eng., London to Melbourne, Australia, via Cape, 11 445 miles, in 38 days 23 hours 40 min. Sailing Vessels. 1851, Chrysolite (clipper ship), of Eng., Liverpool, Eng., to Anjer, Java, 13000 miles, in 88 days. The Oriental , of N. Y., ran the same course in 89 days. 1853, Trade Wind (clipper ship), of N. Y., San Francisco, Cal., to New York, N. Y., 13610 miles, in 75 days. 1854, Lightning (clipper ship), of Boston, Mass., Melbourne, Australia, to Liver- pool, Eng., 12 190 miles, in 64 days. 1854, Comet (clipper ship), of N. Y., Liverpool, Eng., to Hong Kong, China, 13040 miles, in 84 days. 1854, Sierra Nevada (schooner), of N. H., Hong Kong, China, to San Francisco, Cal., 6000 miles, in 34. days. 1854, Red Jacket (clipper ship), of N. Y., Sandy Hook, N. J., to Melbourne, Aus- tralia, 12720 miles, in 69 days 11 hours 1 min. 1855, Euterpe (half-clipper ship) of Rockland, Me., New York to Calcutta, India, 12 500 miles, in 78 days. i860, Andrew Jackson (clipper ship), of Boston, New York, N. Y., to San Fran- cisco, Cal., 13 610 miles, in 80 days 4 hours. 1865, Dreadnought (clipper ship), of Boston, Honolulu, Sandwich Islands, to New Bedford, Mass., 13470 miles, in 82 days ; and 1859, Sandy Hook, N. J., to Rock Light, Liverpool, Eng., 3000 miles, in 13 days 8 hours. 1865, Sovereign of the Seas (medium ship), of Boston, Mass., in 22 days sailed 5391 miles = 245 miles per day. For 4 days sailed 341.78 miles per day, and for 1 day 375 miles. 1866, Henrietta (schooner yacht), of N. Y., Sandy Hook, N. J., to the Needles, Eng., 3053 miles, in 13 days 21 hours 55 min. 16 sec. 1866, Ariel and Serica (clipper ships), of England, Foo-chou-foo Bar, China, to the Downs, Eng., 13 500 miles, in 98 days. 1869, Sappho (schooner yacht), of N. Y., Light-ship off Sandy Hook, N. J., to Queenstown, Ireland, 2857 nvles, in 12 days 9 hours 34 min. 898 ELEMENTS OP MACHINES AND ENGINES, ELEMENTS OF MACHINES AND ENGINES. BLOWING ENGINES. Furnaces. — Two. Fineries. — Two. {England.) 240 Tons Forge Pig Iron per Week . Engine (non-condensing).— Cylinder, 20 ins. in diam. by 8 feet stroke of pistoa Boilers. — Six (plain cylindrical), 36 ins. in diam. and 28 feet* in length. Grates , 100 sq. feet. Blowing Cylinders. — Two, 62 ins. in diam. by 8 feet stroke of piston. Pressure , 2.17 lbs. per sq. inch. Revolutions , 22 per minute. Pipes , 3 feet in diam.= 168 area of cylinder. Tuyeres. — Each Furnace, 2 of 3 ins. in diam. ; 1 of 3.25 ins. ; and 1, 3 of 3 ins. Each Finery, 6 of 1.33 ins. ; and 1, 4 of 1. 125 ins. Temperature of Blast , 6oo°. Ore, 40 to 45 per cent, of iron. Furnaces. — Eight , diam. 16 to 18 feet. Dowlais Iron Works {England). 1300 Tons Forge Iron per Week; discharging 44000 Cube Feet of Air per Minute. Engine (non-condensing). — Cylinder , 55 ins. in diam. by 13 feet stroke of piston. Pressure of Steam.— 60 lbs. per sq. inch, cut off at .33 the stroke of piston. Valves , 120 ins. in area. Boilers. — Eight (cylindrical flued, internal furnace), 7 feet in diam. and 42 feet in length ; one flue 4 feet in diam. Grates , 288 sq. feet. Fly Wheel. — Diam., 22 feet; weight, 25 tons. Blowing Cylinder , 144 ins. in diam. by 12 feet stroke of piston. Revolutions , 20 per minute. Blast , 3.25 lbs. per sq. inch. Discharge pipe, diam. 5 feet, and 420 feet in length. Valves. — Exhaust, 56 sq. feet; Delivery, 16 sq. feet. Furnaces . — Lackenby {England). 800 Tons Iron per Week. Engine (horizontal, compound condensing). — 32 and 60 ins. in diam. by 4.5 feet stroke of piston. Blowing Cylinders. — Two, 80 ins. in diam. by 4.5 feet stroke of piston. Pressure, 4. 5 lbs. per sq. inch. Revolutions , 24 per minute. Pipe, 30 ins. in diam. ; volume, 12.25 times that of blowing cylinders. IP. — Engine, 290 lbs. ; Blowing cylinders, 258; efficiency, 89 per cent. Valves. — Area of admission, .16 of area of piston; of exit, .125. Volume. — 190000 cube feet of air are supplied per ton of air. Blower and. Exhausting Fan. (Sturtevant’s.) Blower. Grate Surface. Inlet. Outlet. Diam. of Pulley. Face of Pulley. Revolu- tions. Air per Minute. IP. No. Sq. Feet. Diam. Ins. Diam. Ins. Ins. Ins. Per Min. Cube Feet. 00 5 5 4 2.75 2 3000 500 — 0 6 5-75 4-75 3 2.25 2600 6oo — X 8 6-5 5-75 3-5 3 2200 764 .6 2 10 7-5 7-5 3-75 3-5 1928 1 019 •79 3 14 9 9 4-25 4 1638 1427 1 .11 4 20 10.5 10.5 5 5 1410 1936 x -5i 5 27 12 12 6 5- 25 1194 2 701 2.1 6 36 14 14 7 6-5 1018 3669 2.86 7 48 16 16 9 7-5 878 4847 3-77 8 62 18 18 10 8.5 7 66 6115 4.76 9 80 21 21 12 10.5 671 8i54 6.35 10 100 24 24 14 12 59 8 10702 8-34 * 40 feet would have afforded economy in fuel. ELEMENTS OF MACHINES AND ENGINES. 899 COTTON FACTORIES. (English . ) For driving 22060 Hand-mule Spindles , with Preparation, and 260 Looms , with common Sizing. Engine (condensing)— Cylinder, 37 ins. in diam. by 7 feet stroke of piston; volume of piston space, 53.6 cube feet. Pressure of Steam.— ( Indicated average) 16.73 lbs. per sq. inch. Revolutions , 17 per minute. , . _ . . Friction of Engine and Shafting.-( Indicated) 4.75 lbs. per sq. inch of piston. IIP 125. Total power = i. Available, deducting friction 717. f 305 hand-mule spindles, with preparation , , . or 230 self-acting “ “ Notes.— Each IIP will drive j or £ throstle “ “ t or 10. 5 looms, with common sizing. Including ^ == 3 hand -mule, or 2.25 self-acting spindles. 1 self-acting spindle = 1.2 hand-mule spindles. DREDGING MACHINES. Dr edging 20 Feet from Water-line, or 180 Tons of Mud or Silt per Hour 11 Feet from Water-line. Length upon deck , 123 feet; beam, 26 feet Breadth over all , 41 feet. Immersed section at load-line , 60 sq.feet. Displacement , 141 tons, at load-draught 0/2.83 feet. Engine (non-condensing ).-Cylinders, two, 12.125 ins. in diam. by 4 feet stroke of piston. Boilers,— Two (cylindrical flue), diam. 40.5 ins., and length, 20 feet 3 ins.; two flues, 14.625 ins. in diam. Heating surface , 617 sq. feet. Grates , 37 sq. teet. Pressure of Steam, 25 lbs. per sq. inch; throttle .25 open, cut off at .5 the stroke of piston. Revolutions , 42 per minute. Buckets. — Two sets of 12, 2.5 feet in length by 15 ins. at top and 2 feet deep; vol- ume, 6.25 cube feet. Chain Links , 8 ins. in length by .5 inch diam. Scows or Camels. — Four, of 40 tons capacity each. STEAM HOPPER DREDGER* ( Wm . Simons Sf Co.) Iron. “Neptune” (English). — Length, 150 feet; breadth, yifeet. Dredge from 6 Ins. to 2 5 Feet. Capacity of Hopper, 500 to 600 Tons. Engines. — Two (compound), 375 IP, for dredging and propulsion, and one for raising bucket-frame and anchor-posts. A like designed dredger of 1000 tons capacity has dredged 10000 tons silt per week and transported it 7.5 miles. Dredging 400 Tons of Mud or Silt per Hour , 5 to 35 Feet in Depth Capacity of Hopper, 1300 Tons. Engines.— Two (compound), IP 700. Speed.— 7.5 knots per hour. Steam Dredging Crane. (English.) Lift , 30 Feet per Hour. Lbs. 21 280 24 640 Lifting Power. Volume of Bucket. Mud or Silt. Coal and Sand. Excava- tion Ground. Weight of Crane. Lifting Power. Volume of Bucket. Mud or Silt. Coal and Sand. Excava- tion Ground. Tons. Lbs. Tons. Tons. C. Yds. Lbs. Tons. Lbs. Ton®. Tons. C. Yds. 2. 5 1x20 25 20 20 18000 5 2240 50 40 3 ° 3 1680 37-5 32 25 33480 7 3360 60 54 40 gOO ELEMENTS OF MACHINES AND ENGINES, Iron. Dredger and Hopper Barge (Compound; English). — Length, extreme 120 feet' beam, 32 feet ; hold, 10. 5 feet. Breadth of bucket well, 6. 75 feet. Load-draught, 6 feet. Cylinders.— 21.5 and 40 ins. in diam. by 2.5 feet stroke of piston. Condenser .— Surface , 600 sq. feet. Circulating Pump , single acting.— 15 ins. in diam. by xc ins stroke of piston. Boiler. — Heating surface, 1150 sq. feet. Grates , 40 sq. feet. Steam -room 300 cube feet. ’ 0 Shaft. — 7.75 ins. in diam. Bucket ladder . — Of wrought iron, 74 feet in length, 5 feet in depth at centre, and 2 feet 2 ins. at ends. Buckets. — 34; volume, 15 cube feet each. Excavation and Delivery .— For a transit of 7.5 miles, 3000 tons per day. Hopper Barge . — Length between perpendiculars, 115 feet; beam, 32 feet; hold, 9 feet 11 ins. Load-line with 400 Tons dredge, 8 feet. Hoppers — Length, 50 feet ; breadth at top , 22 feet; at bottom, 9 feet. Cost. — Dredge , $ 90000; Hoppers, $ 18 000 each. Maintenance.— Dredger, 1.75 cents; Hopper, 1.7 cents; Towing, 1.2 cents per ton of dredge excavated and delivered. “Hercules,” Panama Canal. — Length on deck , 100 feet ; beams, 40, 60, and 4 c feet ; depth of hold, 12 feet. Slot , 36 feet in length by 6 feet 7 ins. in width. Ways.— Two, one 40 feet and one 60 feet, by 5 feet in width. Buckets. — 38 ; volume, 1.33 cube yards. Spuds, 2 feet in diam. and 60 in length. Engines .— Two of 100 IP each, and two of 40 IP each. Boilers .— Three (horizontal tubular), 16 feet in length. Elevator and Discharge.— Maximum, 24 cube yards per minute. Crane. (“W ood.) Hull. — Length on deck, 100 feet ; beam, ^\feet ; load-draught, 4.5 feet. Radius of crane, 46 feet; height, 70 feet; counter -balance, 70 tons. Boiler. — Heating surface, 500 sq. feet. Pressure of Steam, 80 lbs. per sq. inch. HP, 150. Propellers. — Two, 4.25 feet in diam. Speed, 5 miles per hour. Engine to operate crane. Cylinder . — 10 ms. in diam. by 12 ins. stroke of piston. FLOUE MILLS. 30 Barrels of Flour per Hour. Water-wheels, Overshot . — 5, diam. 18 feet by 14.5 feet face. Buckets , 15 ; ns. ih depth. Water. — Head, 2.5 feet. Opening, 2.5 ins. by 14 feet in length over each wheel. 5 Barrels of Flour per Hour , and Elevating 400 Bushels of Grain 36 Feet. Water-wheel, Overshot . — Diam. 22 feet by 8 feet face. Buckets, 52 of 1 foot in depth. Water. — Head , from centre of opening, 25 ins. Opening, 1.75 ins. by 80 ins. in length. Revolutions, 3.5 per minute. Stones, three of 4.5 feet; revolutions, 130, Three Run of Stones , Diameter 4 Feet. Water-wheel, Overshot. — Diam. 19 feet by 8 feet face. Buckets , 14 ins. in depth. Or, Steam-engine (non-condensing). — Cylinder, 13 ins. in diam. by 4 feet stroke. Boiler (cylindrical flued).— Diam. 5 feet by 30 in length; two flues 20 ins. in diam. ELEMENTS OF MACHINES AND ENGINES. QOI IP No. 4 6 io 15 20 25 HOISTING ENGINES. Eor 3 ?ile Driving, Hoisting, NTining, etc. Lidgerwood Manuf’g Co., New York. Single Cylinders. Cylinder. Ins. 5X5 6X8 7 X 10 8 X 10 9 X 12 10 X 12 Capacity. Lbs. 1000 1250 1800 2800 4000 5000 Cost, with Boiler.* $ 600 675 825 1050 1275 1375 12 20 30 40 50 Complete. Double Cylinders. Cylinder. Ins. 5 X 8 6X8 7 X 10 8 X 10 9 X 12 10 X 12 Capacity. Lbs. 2000 2500 3500 6000 8000 9000 Cost, with Boiler.* 950 1050 1350 1550 2000 2350 Details and Operation. Engine. Dram. Be Dimen- sions. >iler. Tubes. Ram. Leaders. Hoist. Lift. Ram. Blows per Minute. Piles per 10 Hours. Fuel per Hour. H? 10* 20 Ins. 12 X 24 14 X 26 Ins. 32 X 75 40 X 84 No. 48 of 2 in. 80 of 2 in. Lbs. 1953 2700 Feet. 40 75 Feet. 8 to 12 8 to ii No. 25 1 29 No. 50 IOO Lbs. 70 80 * Weight complete, 8500 lbs. Mining Engines and. Boilers. {Various Capacities.) Engine , Boiler , etc., as given for Pile Driving, page 902. Operation. — 250 to 300 tons of coal in 10 hours. Fuel , 40 lbs. coal per hour. Water, 20 gallons per hour. Weight of Engine and Boiler, 4500 lbs. Hancock Inspirator. F01 ' a Lift of Water of 25 Feet. No. Diam Steam-pipe. ieter. Suction. Discharge at Pressure of 60 Lbs. No. Diam Steam-pipe. eter. Suction. Discharge at Pressure of 60 Lbs. Ins. Ins. G’lls.per h’r. Ins. Ins. G’lls.per h’r. 10 •375 •5 120 30 1.25 i-5 1260 12.5 •5 •75 220 35 1.25 i-5 1740 15 •5 •75 300 40 1-5 2 2230 20 •75 1 * 540 45 i-5 2 2820 25 1 1.25 900 50 2 2-5 3480 Temperature of water not over 145 0 for a low lift, and ioo° for a high lift. HYDROSTATIC PRESS. ( Cotton) 30 Bales of Cotton per Hour. Engine (non-condensing). — Cylinder, 10 ins. in diam. by 3 feet stroke of piston. Pressure of Steam,, 50 lbs. per sq. inch, full stroke. Revolutions, 45 to 60 per minute. Presses. — Two, with 12-inch rams; stroke, 4.5 feet. Pumps. — Two, diam. 2 ins. ; stroke, 6 ins. For 83 Bales per Hour. Engine (non-condensing).— Cylinder, 14 ins. in diam. by 4 feet stroke of piston. Boilers. — Three (plain cylindrical), 30 ins. in diam. and 26 feet in length. Grates , 32 sq. feet. Pressure of Steam , 40 lbs. per sq. inch. Revolutions , 60 per minute. Presses. — Four, geared 6 to 1, with two screws, each of 7.5 ins. in diam. by 1.625 in pitch. Shaft (wrought iron). — Journal, 8.5 ins. Fly Wheel, 16 feet in diam. ; weight, 8960 lbs. 902 ELEMENTS OF MACHINES AND ENGINES. LOCOMOTIVE. “Experiment” [Compound).— Cylinders, one each, 12 and 26 ins. in diam., and one 26 ins. by 2 feet stroke of piston. Boiler.— Heating surface , 1083.5 sq. feet. Grate , 17. 1 sq. feet. Pressure of Steam , 150 lbs. per sq. inch, cut off at .35. Speed , 50 miles per hour. Weight.— Empty, 34-75 tons. Street Railroad or Tramway Engine. Cylinder , 7 ins. in diam. by n ins. stroke of piston. Boiler 78 tubes 1.75 ins. in diam. by 4 feet in length. Heating surface , 160 sq. feet. Grate , 4.25 sq. feet. Wheels, 2.33 feet in diam. Base , 4.5 feet. Gauge , 4 feet 8. 5 ins. — Average per mile in England, 2.52 pence sterling = 4.48 cents. PILE-DRIVING. Driving One Pile. Engine (non-condensing).— Cylinder, 6 ins. in diam. by 1 foot stroke of piston. Boiler (vertical tubular).— 32 ins. in diam., and 6.166 feet in height. Grates, 3.7 sq. feet. Furnace, 20 ins. in height. Tubes, 35, 2 ins. in diam., 4.5 feet in length. Revolutions, 150 per minute. Drum, 12 ins. in diam., geared 4 to 1. Leader , 40 feet in height. Ram.— 2000 lbs., 2 blows per minute. Fuel, 30 lbs. coal per hour. Driving Two Piles. Engine (non-condensing).— Cylinders, two, 6 ins. in diam. by 18 ins. stroke of piston. Boiler (horizontal tubular).— Shell, diam. 3 feet, and 6 feet in length. Furnace end 3.75 feet in width, 3.5 feet in length, and 6 feet in height. Pressure of Steam, 60 lbs. per sq. inch. Revolutions, 60 to 80 per minute. Frame 8.5 feet in width by 26 feet in length. Leaders , 3 feet in width by 24 feet • in height. Rams.— Two, 1000 lbs. each, 5 blows per minute. PUMPING ENGINES. ' Corliss Steam-engine Co., Providence, R. I . — Vertical -Beam Engine (Com- , pound).— Cylinders.— 18 and 36 ins. in diam. by 6 feet stroke of piston. Pumps. — Four plunger, 19 ins. in diam. by 3 feet stroke of piston. Displacement per revolution of engine, 84.96 cube feet. Boilers. — Three, vertical fire tubular. Grate.— 93 sq. feet. Heating surface, 1680 sn feet. Pressure of Steam. 125 lbs. per sq. inch,. cut off at .22 feet. Revolutions, / , 36 per minute. IIP 313. Fly-wheel.— 25 feet in diam., weight 62000 lbs. Fuel —Cumberland coal, 486 lbs. per hour, inclusive of kindling and raising steam, i Ash and Clinkers. 9.4 per cent. Duty for one week, 113 271 000 foot-lbs. Water delivered, 17621 gallons per minute, against head of 180 feet. Duty, average for 1883, per 100 lbs. anthracite coal, 106 048 000 foot-lbs. For Elevating 200000 Gallons of Water per Hour. Lynn Mass . — Engine (Compound). — Cylinders, 17.5 and 36 ins. in diam. by 7 feet stroke of piston; volume of piston space, 61.2 cube feet. Air Pump (double act- ing), 11.25 i ns - diam. by 49.5 ins. stroke of piston. Pump Plunger , 18.5 ins. in diam. by 7 feet stroke. Boilers .— Two (return Sued), horizontal tubular; diam. of shell, 5 feet; drum, 3 feet; tubes, 3 ins. Length of shell, 16 feet. Grates, 27.5 sq. feet. Pressure of Steam, 90.5 lbs. ; average in high-pressure cylinder 86 lbs., cut off at 1 foot, or to an average of 44.5 lbs. ; average in low-pressure cylinder, 27 lbs., cut off at 6 ins., or to an average of 10.8 lbs. Revolutions, 18.3 per minute. Fly Wheel. — Weight, 24000 lbs. Evaporation of Water , 4644 lbs. per hour. Loss of action by Pump, 4 per cent. Consumption of Coal— Lackawanna, 291 lbs. per hour. Duty 205772 gallons of water per hour, under a load and frictional resistance of 73.41 ibs. per square inch, equal to 103 923 217 foot-lbs. for each 100 lbs. of coal. ELEMENTS OF MACHINES, MILLS, ETC. 903 “ Gashill at Saratoga , N. Y. Engine ( Horizontal Compound). Cylinders.— High pressure, 2 of 21 ins. diam. Low pressure, 2 of 42 ins. diam., all 3 feet stroke of piston. Pumps.— Two of 20 ins. diam. by 3 feet stroke of piston. Fly Wheel, 12.33 feet in diam. ; weight, 12000 lbs. Boilers (horizontal tubular).— Two of 5.5 feet in diam. by 18 feet in length. Heat- ing surface, 2957 sq. feet. Grates , 51 sq. feet of grate; to heating surface, 1 to 58, and to transverse section of tubes, 1 to 7. Chimneys , 75 feet. Pressure of Steam.— Mean of 20 hours, 74.25 lbs. per sq. inch. Revolutions , 17.87 per minute. IIP. High-pressure cylinders, 109.2; low-pressure, 76.65. Total, 185.8. Fuel. Anthracite, 6.9 lbs. per sq. foot of grate per hour. Evaporation , per sq. foot of heating surface per hour, 1.175 lbs. ; per lb. of coal, 9.25 lbs. ; per cent, of non-combustible, 3.2. Duty, 112899993 foot-lbs. per 100 lbs. coal. Heating surface per IIP, 14.9. Steam per sq. foot of surface per hour, 1. 19 lbs. ; per sq. foot of surface per lb. of coal per hour from 212°, 11.28 lbs. Ericsson’s Caloric. For an Elevation of 50 Feet. Dimen- sions. Space occupied. Floor. | Height. Volume per Hour. Pipes, Suction and Dis- 1 charge, j Fuel per Hour. Nut Anthr. I Gas. Furnace. Gas. | Coal. COST Deep Pump. Well Pi Extra. Pipes pe Plain. imp. sr Foot. Galvan Ins. Ins. Ins. Gall. Ins. Lbs. Cub. ft. % $ $ $ % 5 34 Xi 8 48 150 •75 — 15 150 — ' — — — 6 39X20 51 200 •75 2-5 18 200 210 — — — 8 48X21 63 350 1 3-3 25 235 250 10 .64 .86 12 54 X 27 63 800 i -5 6 — — 320 15 .80 *•*5 12* 42X52 65 1600 2 12 — — ‘ 450 25 t .92 1.25 * Over 90 feet, 92 cents. t Duplex. Including engine and pump, oil-can and wrench, complete in all but suction and discharge-pipe. SUGAR MILLS. Expressing 40000 lbs. Cane-juice per day, or for a Crop of 5000 Boxes of 450 lbs. each in four Months ’ Grinding. Engine (non-condensing).— Cylinder, 18 ins. in diam. by 4 feet stroke of piston. Boiler (cylindrical flued).— 64 ins. in diam. and 36 feet in length; two return dues, 20 ins. in diam. Heating surface, 660 sq. feet. Grates, 30 sq. feet. Pressure of Steam, 60 lbs. per sq. inch, cut off at . 5 the stroke of piston. Revolu- tions, 40 per minute. Rolls. — One set of 3, 28 ins. in diam. by 6 feet in length; geared 1 to 14. Shafts, 11 and 12 ins. in diam. Spur Wheel , 20 feet in diam. by 1 foot in width. Fly Wheel. 18 feet in diam. ; weight, 17 400 lbs. Weights.— Engine, 61 460 lbs. ; Sugar Mill, 65 730 lbs. ; Spur Wheel and Connect- ing Machinery to Mill, 28680 lbs. ; Boiler, 18 520 lbs. ; Appendages, 6730 lbs. Total, 181 120 lbs. STONE AND ORE BREAKERS. (Blake’s.) No. Re- ceiver. Pul D’m. ley. Face. 45 > % Power re- quired. Weight. No. Re- ceiver. Pul D’m. ley. Face. | V’locitj per 1 Minute, Power quired. Weight. Ins. Feet. Ins. Feet. PP. Lbs. Ins. Feet. Ins. Feet. HP. Lbs. A 4X10 1.66 6 250 4 4 000 5 9Xi5 2-5 9 250 9 13360 1 5XlO 2-75 6 180 5 6 700 6 11X15 2-33 6 180 9 11 6co 2 7X10 2 7-5 250 6 8000 7 13X15 2-33 8 180 9 11 760 3 5Xi5 2-33 8 180 9 9 100 8 15X20 3-5 10 150 12 32 600 4 7X15 ! 2.33 9 180 9 10490 9 18X24 6 12 125 12 37 5oo Note. — Amount of product depends on distance jaws are set apart, and speed. Product given in Table is due when jaws are set 1.5 ins. open at bottom, and ma- chine is run at its proper speed and diligently fed. It will also vary somewhat with character of stone. Hard stone or ore will crush faster than sandstone. A cube yard of stone is about one and one third tons. 904 ELEMENTS OF MACHINES. CHIMNEYS. STEAM FIRE-ENGINE. Ainoskeag, IN'. FI . ls^t Class. Steam Cylinder.— Two of 7.625 ins. in diam. by 8 ins. stroke of piston. Water Cylinder.— Two of 4.5 ins. in diam. Boiler (vertical tubular).— Heating surface , 175 sq. feet. Grates , 4.75 sq. feet. Pressure of Steam.— 100 lbs. per sq. inch. Revolutions , 200 per minute. Discharges.— Two gates of 2.5 ins., through hose, one of 1.25 ins. and two of 1 inch. Projection. — Horizontal, 1.25 ins. stream, 311 feet; two 1 inch streams, 256 feet. Vertical, 1.25 ins. stream, 200 feet. Water Pressure. — With 1.125 ins. nozzle, 200 lbs. Time of Raising Steam,— From cold water, 25 lbs., 4 min. 45 sec. Weights. — Engine complete, 6000 lbs. ; water, 300 lbs. SAW-MILL. Two Vertical Saws, 34 Ins. Stroke, Lathes, etc. Engine (non-condensing). Cylinder. — 10 ins. in diam. by 4 feet stroke of piston. Boilers. — Three (plain cylindrical), 30 ins. in diam. by 20 feet in length. Pressure of Steam.— go lbs. per sq. inch. Revolutions , 35 per minute. Note.— This engine has cut, of yellow-pine timber, 30 feet by 18 ins. in 1 minute. STONE SAWING. Emerson Stone Saw Co. (Diamond Stone Saw, Pittsburgh, Penn.).— 20 IP, 150 sq. feet of Berea sandstone, inclusive of both sides of cut, in 1 hour. CHIMNEYS. Lawrence, Mass . Octagonal, 222 Feet above Ground, and 19 Feet below. Foundation, 35 Feet square and of Concrete 7 Feet deep. ( Hiram F. Mills.) Shaft.— 234 feet in height, 20 feet at base, and 11.5 at top; 28 ins. thick at base and 8 at top. Core.— 2 feet thick for 27 feet, and 1 foot for 154. Horizontal Flues. — 7.5 feet square, and Vertical flue or cylinder of 8.5 feet, 234 high, with walls 20 ins. thick for 20 feet, 16 for 17 feet, 12 for 52 feet, and 8 for 145 feet. Purpose. — For 700 sq. feet grate surface. Weight. — 2250 tons. Bricks , 550000. New York Steam Heating Co. Quadrilateral, 220 Feet above Ground and 1 Foot below. {Chas. E. Emery , Ph.D .) Shaft. 220 feet in height, and 27 feet 10 ins. by 8 feet 4 ins. in the clear inside. Foundation.— 1 foot below high water. Capacity.— Boilers of 16000 IP. Cost of Steam-Engines and Boilers complete, and. of Operation per Bay of IO Honrs, inclusive of Labor, Enel, and llepairs. {Chas. E Emery, Ph.D.) HP. Engine. Water orated IIP per Hour. Evap- i per Lb. of Coal. Coa IIP. 1 per Day. Labor. Sup- plies and Re- pairs. Cost of Coal.* Total Cost of Operat’n, including Coal. 6.25 12.5 29 1 1 2 276 552 Portable Vertical^ Horizontal ( ^ 8 Single Condensing. . . Lbs. 42 3 8 32 23 22.2 22.2 Lbs. 7-5 7-5 8 8.8 8.8 8.8 Lbs. 56 51 40 26. 1 25.2 25.2 Lbs. 394 7 i 7 1 308 3 3 oo 7 8 3 i 15 663 $ i -75 i -75 2.25 3 - 75 4 - 25 6 $ •33 .41 .60 1.17 2. 12 4.02 •73 1- 33 2 - 43 6.14 14.58 29. 16 2.86 3-56 5-45 11.66 22.27 4 1 - S 2 * $ 4.42 per ton (2240 lbs.), including cartage. GRAPHIC OPERATION. 905 GRAPHIC OPERATION. Solutions of Questions toy- a G-rapliic Operation. 1. If a man walks 5 miles in 1 hour, how far will he walk in 4 hours? continue the time to Operation. — Draw horizontal line, divide it into equal parts, as 1, 2, 3, and 4, representing hours. From each of these points let fall vertical lines A C, 1 1, etc., and divide A C into miles, as 5, 10, 15, and 20, and from these points draw equi- distant lines parallel to the horizontal. Hence, the horizontal lines represent time or hours, and the vertical, distance or miles. Therefore, as any inclined line in diagram represents both time and distance, course of man walking 5 miles in an hour is represented by diagonal Ac; and if he walks for 4 hours, 4, and read off from vertical line A C the distance = 20 miles. 2. How far will a man walk in 2 hours at rate of 10 miles in 1 hour ? His course is shown by the line A o, representing 20 miles. 3. If two men start from a point at the same time, one walking at the rate of 5 miles in an hour and the other at 10 miles, how far apart will they be at the end of 2 hours ? Their courses being shown by the lines A r and A 0, the distance r 0 represents the difference of their distances, 10 20 = 10 miles. 4. How long have they been walking? Their courses are now shown by the lines A 0 and A 4, the distance 2 4 represents the difference of their times, or 2 ^ 4 = 2 hours. 5. When they are 10 miles apart, how long have they been walking? Their courses are again shown by the lines A r and A 0, the distance r o repre- sents the difference of their distances of 10 miles, and A 2, 2 hours. 6. If a man walks a given distance at rate of 3.5 miles per hour, and then runs part of distance back at rate of 7 miles, and walks remainder of dis- tance in 5 minutes, occupying 25 minutes of time in all, how far did he run? Operation. — Draw horizontal line, as A C, representing whole time of 25 minutes; set off point e representing a convenient fraction of an hour (as 10 minutes), and a i equal to corresponding fraction of 3.5 miles (or .5833); draw diagonal A n, produced indefinitely to 0 , and it will represent the rate of 3.5 miles per hour. Set off C r equal to 5 minutes, upon same scale as that of A C; let fall vertical r s , and draw diagonal C u at same angle of inclination as that of An; then from point u draw diagonal u 0 , inclined at such a rate as to represent 7 miles per hour; thus, if i n represents rate of 3.5 miles, s 0, being one half of the distance, will represent 7 miles. The whole distance between the two points is thus determined by C x, and dis- tance ran by u s, measured by scale of miles employed. Verification.— The distances A e and A i are respectively 10 minutes = .166 of an hour, and .5833 mile = .166 of 3.5 miles. Hence, C x~. 875 mile, and u s = . 5833 mile. Consequently, the man walked A 0 = .875 mile = 15 minutes, ran Om = .5833 mile = 5 minutes, and walked u C = .2916 mile. 7. If a second man were to set out from C at same time the man referred to in preceding question started from A, and to walk to A and return to C, at a uniform rate of speed and occupying same time of 25 minutes, at which points and times will he meet the first man? Operation . — As A C represents whole time, and C x distance between the two points, v z and 0 x will represent course of second man walking at a uniform rate, and he will meet the first man, on his outward course, at a distance from his start- ing-point of A, represented by A o, and at the time A a; and on his return course at distance Av.xm. and at the time A c. 4 g* go6 MISCELLANEOUS. MISCELLANEOUS. NTo., Diameter, and. Number of Sliot. {American Standard.) Compressed Back Shot. No. Diam. Shot per Lb. No. Diam. Shot per Lb. No. Diam. Shot per Lb. Inch. NO. Inch. No. Inch. No. 3 •25 284 1 •3 173 00 •34 115 2 .27 232 . 0 •32 140 000 •36 98 Balls, .38 Inch, 85 No. per lb. ; .44 Inch, 50 No. per lb. Chilled Shot. No. Diam, Shot per Oz. No. Diam. Shot per Oz. No. Diam. Shot per Oz. No. Diam. Shot per Oz. Inch. No, Inch. No. Inch. No. Inch. No. 12 •05 23 8 5 , 9 .08 585 6 .11 223 1 .16 73 11 .06 , 1380 8 Trap 495 5 .12 172 B •17 61 10 Trap 1130 8 .09 4°9 4 •13 136 BB .18 52 10 •°7 868 7 Trap 345 3 .14 109 BBB .19 43 9 Trap I 7 i 6 7 .1 299 2 15 88 Drop Shot. No, Diam. Pellets per, Oz. No. Diam. Pellets per Oz. No. Diam. Pellets per Oz. No. Diam . Pellets perOz. Inch.. No. Inch. No. Inch. No. Inch. No. Extra Fine Dust • OI 5 . 84021 9 Trap 688 5 . 12 168 BBB .19 42 Fine Dust •03 10784 9 .08 568 4 •13 132 T .2 36 Dust .04 4 565 ' 8 Trap 472 3 .14 106 12 -05 2 326 8 .09 399 2 •15 86 TT .21 31 11 .06 I 346 7 Trap 338 1 . 16 71 F .22 27 10 Trap I 056 7 . 1 291 B •17 59 so .07 848 6 . 1 1 218 BB . 18 50 FF •23 24 The scale of the Le Roy standard (adopted by the Sportsman’s Convention) com- mences with .21 inch for TT shot, and reduces .01 inch for each size to .05 inch for No. 12. The number of pellets per oz. being the actual number in perfect shot. The number of pellets by this standard is nearly identical with that of the Amer- ican Standard. Tatham’s scale is same as Le- Boy’s; but number of pellets is deduced mathemat- ically,, by computing them from the specific gravity of the lead. Drains. Diameter and G-rade of, to.. Discharge Rainfall. Diam. Grade ' 1 Inch. , Acres. Diam. Grade 1 Inch. Acres. Ins. Ins. 4 3 ° •5 40 1.2 20 .6 20 1-5 5 80 •5 7 20 1.2 60 .6 60 *■5 20 1. 8 120 . L 5 6 60 IL 8q i.8, Diam. Grade 1 Inch. Acres. Diam. Grade 1 Inch. Acres. Ins. Ins. 5-8 60 2. 1 80 9 120 2. 1 15 240 7 - f 80 2.5 120 7.8 60 2-75 80 9 12 , 120 4-5 60 10 80 5-3 18 240 10 British and Nletrio Measures, Commercial Equivalents of. (tr. Johnstons Stones , F. R. S.) Length. Millimeters. Inch 9 T 4-4 Foot 3°4-^ Yard.. 25.4 Weight. Grammes. Pound 453-6 Ounce 28.35 Grain .0648 Volume. Cube Centimeter. Gallon.. 4554 Quart 1136 Ounce 28.4 MEMORANDA, 907 MEMORANDA. Kiysical and. MecTianical Elements, Constructions, and Results. Belting. Double. —600 HP (to be transmitted) -f- velocity of belt in feet per minute, or 191 IP 4- number of revolutions per minute X diameter of pulley in feet — width in ins. Machine Belts . — 1500 to 2000 BP -r- velocity of belt in feet per minute = width in ins. ( Edward Sawyer.) Blast Pipe of a Locomotive. Best height is from 6 to 8 diameters of pipe, and best effect when expanded to full diam. of pipe at 2 diameters from base. Boiler Riveting. A riveting gang (2 riveters and 1 boy) will drive in shell, furnace, etc., a mean of 12.5 rivets per hour. Brick or Compressed Fuel is composed of coal dust agglomerated by pitchy matter, compressed in molds, and subjected to a high temperature in an oven, in order to expel the moisture or volatile portion of the pitch and any fire- damp that may exist in the cells of the coal. Bridge, Highest. At Garabil, France, 413 feet from floor to surface of water, and 1800 feet in length. Bronze, JVTalleakle. P. Dronier, in Paris, makes alloys of copper and tin malleable by adding from .5 per cent, to 2 per cent, quicksilver. Building Department, Requirements of. (New Yorlc.) Furnace Flues of Dwelling Houses hereafter constructed at least 8-inch walls on each side. The inner 4 ins. of which, from bottom of flue to a point two feet above 2d story floor, built of fire-brick laid with fire-clay mortar; and least dimensions of furnace flue 8 ins. square, or 4 ins. wide and 16 ins. long, inside measure; and when furnace flues are located in the usual stacks, side of flue inside of house to which it belongs may be 4 ins. thick. If preferred, furnace flues may be made of fire-clay pipe of proper size, built in the walls, with an air space of 1 inch between them, and 4 ins. of brick wall on outside. Boiler Flues to be lined with fire-brick at least 25 feet in height from bottom, and in no case walls of said flues to be less than 8 ins. thick. All flues not built for furnaces or boilers must be altered to conform to the above requirements before they are used as such. Buildings, Proteetion of, from Lightning. A wire rope of 4 lbs. per yard is held to be the most efficient. Single Conductors, weighing 8 lbs. per yard and 4 lbs. for duplicated and all others, may be located 50 feet apart, thus bringing every portion of the building to which they are applied within 25 feet of their protection. Iron is the be^st material for a conductor; it should be continuous, and all joints soldered. Several points are preferable to one. and greater surface should be given to connections w’ith the earth than usually practised. (Sir W. Thompson.) For othe^ information, see Van Nostrand's Magazine , 1 V. F, Aug. 1882 ,page 154. Cement. Iron to Stone.— Fine iron filings, 20 parts, Plaster of Paris, 60, and Sal Ammoniac, 1 ; mixed fluid w’ith vinegar, and applied forthwith. Chimney Braviglit. W — w h = D. W and w representing weights of a cube foot of air at external and internal temperatures , h height of chimney or pipe in feet , and D value of draught. See Weight of Air, page 521. Chinese or India Ink improves with age, should be kept in dry air, and in rubbing it down, the movement should be in a right line and with very little pressure. MEMORANDA. 908 Coal, Effective Value off Theoretical quantity of heat per IP is 2564 units per hour, and average quantity of heat in a lb. of coal that is utilized in the generation of steam in a boiler is 8500 units; hencej theoretical quantity of coal required per IP per hour= = .3 lbs., after the water has been heated into atmospheric steam, being theoretically nearly 7.5 per cent, of total heat re- quired to change 30 lbs. water at 6o° into steam of 60 lbs. effective pressure. The total heat developed by the combustion of coal, when utilized evaporatively, ranges from .55 to .8, but in practice it does not exceed 65 per cent. Coast and. Bay Service. A velocity of current of 2.5 feet per second will scour and transport silt, and 5 to 6.5 feet sand. For river scour the velocities are very much less. Cold, Greatest. — 220 0 , produced by a bath of Carbon, Bisulphide, and liquid Nitrous Acid. Corrosion of* Iron, and Steel. The corrosion of steel over iron is, as a mean, fully one third greater. Cost of Family of Mechanics in Erance ranges from $220 to $600 per annum, of which clothing costs 16 parts, food 61, rent 15, and mis- cellaneous 8. Crushing Resistance of Brich. A pressed brick of Philadelphia clay withstood a pressure of 500000 lbs. for a period of 5 minutes. Earthwork. Shovelling. — Horizontal, 12 feet. Vertical, 6 feet. When thrown horizontal, 12 to 20 feet, 1 stage is required, and from 20 to 30, 2 stages. When vertical, 6 to 10 feet, 1 stage is required. Wheelbarrow. — Proper distance up to 200 feet. Number of Loads and Volume of Earth, per Lay. One Laborer . (C. Herschell , C. E.) Distance. Trips. Volume. Distance. Trips. Volume. Distance. Trips. Volume. Feet. No. Cub. Yds. Feet. No. Cub. Yds. Feet. No. Cub. Yds. 20 120 23-5 150 96 13-3 350 88 11. 6 5 ° no 16.9 200 94 12.8 400 86 11. 2 7 ° 100 14.4 250 92 12.4 450 84 10.9 100 98 13.8 300 90 12 500 82 10.5 Volume of a barrow load, 2.5 cube feet. Portable Railroad and Hand Cars.— For a distance of 550 feet, 60 cube yards can be transported per day. Horse Cart.—'V olume of Earth transported per Lay. One Laborer. Distance. Trips. Volume. Distance. Trips. Volume. I | Distance. Trips. Volume. Feet. No. Cub. Yds. Feet. No. Cub. Yds. Feet. No. Cub. Yds. 300 86 17. 1 1000 43 8.6 2000 25 5 500 67 13.6 I5°° ; 3 i 6.4 1 2500 21 4*3 Volume of each load, 8 cube feet. Ox Cart is less in cost at expense of time. / Electinc Light, Candle Lower off Maxim Incandescent Lamp.— Current with 30 Faure cells, 74 volts, 1.81 Amperes, 16 standard candles. With 50 like cells, 124 volts, and 3.2 Amperes, 333 candles. ( Paget Hills , LL. D.) The elavated electric lights at Los Angeles, Cal., are distinctly visible at sea for a distance of 80 miles. Engine and Sugar Mill, Weights off Engine (non- condensing). —Cylinder .— 30 ins. in diam. by 5 feet stroke of piston. Boilers (cylindrical flue).— 70 ins in diam. by 40 feet in length. Weights. — Engine, 105000 lbs. ; Boilers , com- plete, 75 000 lbs. ; Sugar-mill , 40 ins. by 8 feet, 220050 lbs. ; Connecting Machinery , 137 1 79 lbs. Cane carriers, etc., 46 787 lbs. MEMORANDA. 909 Filtering Stone. Artificial— Clay, 15 parts; Levigated Chalk, 1.5; and Glass Sand, coarse, 83.5. Mixed in water, molded, and hard burned. Fire-engine, Steam. Relative effect for equal cost compared with a hand engine, as 1 to 1 13. Each IH> requires about 112 weight of engine. Floating Bodies, Velocities of. At low speeds resistance increases somewhat less than square of velocity. In a Canal, at a speed of 5 miles per hour, a large wave is raised, which at a speed of 9 miles disappears, and when speed is superior to that of the wave, resistance of boat is less in proportion to velocity, and immersion is reduced. Length of Vessel. — The proper length for a vessel in feet (upon the wave-line theory) is fifteen sixteenths of square of her speed in knots per hour. Flow of ^Air. 67 y/h = Velocity per second X C. h representing column of water in ins., and C a coefficient ranging from 56 to 100. Circular orifices, thin plate 5^ to -79 Cylindrical mouth-pieces, short 81 “ .84 do. do. rounded at inner end 92 ‘ .93 Conical converging mouth pieces 9 “ 1 Conoidal mouth piece, alike to contracted vein 97 ‘ 1 Fines, Corrugated. ( Wm. Parker.) ~ - — Working stress in lbs. per sq. inch. T representing thickness in 16 ths of an inch , and D diameter in ins. Steel, corrugations 1.5 ins. deep. Experiments upon a furnace 31.875 ins. in diarn., 6.75 feet in length, and with 13 corrugations. Foundation Files. When piles are driven to a solid foundation, they act as columns of support, and are designated Columns , and when they deriv° their supporting power from the friction of the soil alone, they are termed Piles. Authorities differ greatly as to the factor of safety for Piles, varying .1 to .01 of impact of ram. ( Weisbach. ) As columns, their safe load may be taken at from 750 to 900 lbs. per sq. inch. Authorities give a higher value (Rankine and Mahon, 1000); but it is to be borne in mind that when piles are driven to a solid resistance, they are frequently split, and consequently their resistance is much decreased. As a rule, the following coefficients for ordinary structures are submitted: When the piles are wholly free from vibration consequent upon external impulse, .35 to .4. and when the structures are heavy and exposed to irregular loading, as storehouses, etc. , .15 to .2. Ordinarily, the bearing of a properly driven pile not less than 10 ins. in diam. may be taken at 10 tons. Friction of Bottoms of Vessels. At a velocity of 7 knots per hour, a foul bottom requires 2.42 LP over that for a clean bottom. Friction of Planed Brass Surfaces in muddy water is .4 pressure. Gras, Steam, and Hot-air Engines. Relative costs of gas, steam, and air engines per IP: Otto Gas engine, 8.75; Steam engine, 3.5; and Hot-air engine, 4. Heat. Available heat) *6431535 upended per IIP per hour) Total heat of combustion X Coefficient for fuel consumption of coal per IIP. Coal 14000X772 units = 10808000. Theoretical evaporative power — 15 lbs. 16 431 535 water. Efficiency of furnace = .5; then 10 808 000 X • 5 = 5 404 000, and — 5 404 000 = 3.04 lbs. per IIP per hour. Ice Boats, Speed of. Maj.-Gen. Z. B. Tower. U. S. A., assigns the speed of Ice boats at twice that of the wind, and the angle of sail, to attain greatest speed, to be less than 90 0 . Japan Coal. Analysis of Bituminous. — Specific Gravity, 1.231. Carbon, 77.59. Hydrogen, 5.28. Oxygen, 3.26. Nitrogen, 2.75. Sulphur, 1.65 Ash, 8.49. and loss, .98. Its evaporative effect = 4. 16 lbs. water per lb. of coal. 910 MEMORANDA. Lee-way. A full modelled vessel, with an immersed section of i to 6 of her longitudinal section, and with an area of 36 sq. feet of sails to 1 of immersed sec- tion, will drift to leeward 1 mile in 6. A medium modelled vessel, with an im- mersed section of 1 to 8, and with like areas of sail and section, will drift 1 in 9. LigRt, Standard, of. Photometric , English. — Spermaceti candles, 6 per lb. ; 120 grains per hour. Carcel burner = 9.5 candles. Locomotive .A.xles, Friction of. .016 of weight. Hence, if radius of wheel = .1, axle friction at periphery -~ 10 == 3.73 at periphery. Mercurial Grange. To prevent freezing, apply or introduce Glycerine on top of column. Metal Products of TJ. S., 18 SS. Value, $222000000. Mississippi Diver, Silt in. Near St, Charles the volume of silt borne per day in 1879 was 475457 cube yards, and on one day, July 3. it was 4 113600. At times the volume equals 3 ozs. per cube foot of water. Motive Power. A sailing vessel having a length 6 times that of her breadth, requires, for a speed of 10 knots per hour, an impelling force of 48 lbs. per sq. foot of immersed section. Mowing Machine. Kirby's (Auburn, N. Y.) — 670 lbs., 2 horses, 1 acre heavy clover in 46 min. Ordnance, Energy of. In a competitive test ot a 9-incli Woolwich gun, and a 5.75-inch Krupp, the energy per inch of circumference of bore was re- spectively 1 18 and 123 foot-tons; their penetration therefore by the wrought-iron standard being about the same, but their total energies were respectively 16400 and 5800 foot-tons. At Mepper a shot of no lbs., with a velocity of 1749 feet per second, and a strik- ing energy of 2300 foot-tons, passed through a target composed of two plates of soft wrought iron 7 ins. thick, with 10 ins. of wood between them, and passed 800 yards beyond. Petroleum. One lb. crude oil heated 1 lb. water 315. 75 0 = 28.21 lbs. water at 6o° converted to steam at 212 0 . Relative evaporative effects of Oil and Anthra- cite coal as 1 to 3.45. Dopnlation, Comparative Density of, and jNT umber of Persons living in a House in different Cities. Chicago, 4; Baltimore and Naples, 4.5; Philadelphia, 6; London, Boston, and Cairo, 8; Marseilles, 9; Pekin, 10; Amsterdam, n ; New York, 13.5; Hamburg, 17.07; Rome and Munich, 27; Paris, 29; Buda Pesth, 34.2; Madrid, 40; St. Peters- burg, 43.9; Vienna, 60.5; and in Berlin, 63. Dower of a Volcano. An eruption of that of Cotopaxi has projected a mass of rock of a volume of 100 cube yards a distance of 9 miles. Dower Required to Dx*aw a "Vessel or Load up an In- clined Hydrostatic Dail or Slip "Way. (Wm. Boyd, Eng.) WI = R; Cc?W-f-D = F; and P d' c =f W representing weight of vessel , or load and cradle , I inclination of ways , as length rise , R resistance of vessel or load , F friction of cradle and rollers , and f friction of plunger in stuffing-box , all in tons, C and c coefficients of friction of cradle and stuffing-box , d diameter of axle of rollers, d' product of circumference of plunger and depth of collar or stuffing , all in ins., and P pressure per sq. inch on plunger , in lbs. Hence, W = I, and R 4 - F - 4 -/ = power in tons. ’ length ’ Illustration. — Assume weight of a vessel and cradle 2000 tons, pressure on plunger 2500 lbs. per sq. inch, inclination of w*ays 1 in 20, diameters of axle of roll- ers and of rollers 3 and 10 ins., depth of collar 2 ins., and circumference of plunger 50; what would be the power required? C = .2, and c = .6. 2000 . .2 X 3 X 2000 . 2500 X 2 X 5° X - 6 Then = 100 tons ; - = 120 tons ; - = 67 tons ; 20 10 2240 and 100 -j- i2o-j- 67 = 287 tons. 2240 MEMORANDA. 91 1 Qt „„ TTlfir Ordinary OistriDntion of Power If P P0wer developed by engine, 88 IH> ; Power expended in its operation, Friction of load .^ys"*' I Power expended by slip of propeller. ... 14 Friction ; ” | “ “ m propulsion 71 -C3 _ Contrifuaal has lifted water 28 to 29 feet, drawn it horizontally 800 feeYanTt Also drawn it 24 feet, and projected it 50 feet. ’ Trains. Power and Resistance. — A railway train running at rat^f^omi^s per hour = 88 feet per second, and velocity a body would acquire • * qq fppt — 88 — 8 o 2 2 = 120.3 feet. Consequently, in addition to power m m resistance to train, as much power must be expended to "n motion at lids speed, as would lift it in mass to a beigbt of 1 21 feet in a second. Tf thP train weiehed 100 tons = 224000 lbs., then 224000 X 120.3 = 26747200 foot lbs and if this result was obtained in a period of 5 minutes, it would require 1^ 3 -J 5 X224000- 33 000 = 163. 3 H> in addition to that required for frictional rG To raise the speed of a train from 40 (58.66 feet per second) to 45 (66 feet per sec- ond^ miles per hour, the power required^in addition to that of friction would be as c8 66^8^02 = 53-44 feet is to 66 -=-8.02 = 67. 57 feet = 67.57 — 53-44 = 1 4- 1 3 f eet. Assume a train of 100 tons, running at rate of 60 miles per hour, and total retard- in a nower at 1 its weight 100-4-10 = 10. Then 224000 X 10 X 120.3 — 26947 200 . 22^400 = 1203 feet, which train would run before stopping. If, however, tram was ascending a grade of i in ioo, the retarding force — .11 (n • IO °) ® ^ 640 distance in which tram would come to rest would be 26947200^24640 = 1093.6 feet. Relative Non-condixctibility of Materials. Material.. Per cent. Material. Per cent. Material. Percent. Hair fait. 100. 83.2 7 i -5 68 Mineral wool, No. 1 Charcon 1 67-5 63.2 55-3 55 Lime, slacked Asbestos 48 36*3 34-5 13.6 Mineral Wool, No. 2 “ “ and tar Sawdust Pine wood Coal ashes Loam Air space, 2 ins. . . Resistance to a S team -vessel in au- mm w mer. In air 10 per cent, of IIP, and in waiter, at a speed of 20 miles per hour, 90 per cent., or 8 IIP per sq. foot of immersed amidship section. Saws, Circular. 30 ins. in diameter, are run at 2000 revolutions per minute = 3-57 rail ’ es - Sp Ur Gf ear has been driven at a velocity of 1 mile per minute. Sugar Mill Rollers. 5 feet by 28 ins., at 2.5 revolutions per minute, requires 20 IP, and 18 feet per minute is proper speed of such rolls. Surface Condensation, Experiments on. (B. G. Nichol.) Tube of Brass , .75 Inch External Diameter. No. 18 B W G, Surface = 1.0656 sa. feet. Duration o f Experiment , 20 Minutes. Horizontal. Steam. Vertical. Temperature 2 55° 256° Pressure per sq. inch per gauge. . . 17.75 lbs. 18.25 lbs. Condensation by tube surface 18.5835 “ 29.9585 “ “ per sq. ft. of “ per hour 52-32 “ 84.34 Condensed during experiment 19.0625 “ 1 30-4375 “ 253° 2 54 ° 16.75 lbs. 17.25 lbs. 24.0835 “ 43.0835 “ 67.8 121.29 “ 24.5625 “ 43-5625“ lbs. Steamers^ icngmes, w eignts ox. cmymx,, m Fittings ready for Service per IIP. Mercantile steamer 480 lbs. I Light draught 280 English Naval “ 360 “ | Torpedoes 60 Ordinary Marine Boiler with Water 196 lbs. "Wind, Pressure of. Estimate of upon Structures. — 30 lbs. per sq. foot. Per lineal foot of a locomotive train = 10 feet in height, 300 lbs. per sq. foot, A Tornado has developed a pressure of 93 lbs. per sq. foot. 912 MEMORANDA. "Via, Suez Canal. Passages by Steamers.— 1882, “ Stirling Castle,” Shang hai to Gravesend, in 29 days 22 hours and 15 min., including 1 day 22 hours and ~>q min. in coaling and detentions. ' 0 “ Glenare ,” Amoy to New York, N. Y., in 44 days and 12 hours , including deten- tion at Suez. From Gibraltar in n days. Zinc Foil in Steam-boilers. Zinc in an iron steam-boiler consti- tutes a voltaic element, which decomposes the water, liberating oxygen and hydro- gen. The oxygen combines with fatty acids and makes soap, which, coating the tubes, prevents the adhesion of the salts left by evaporation. The mealy deposit can then be readily removed. Files. To Compute Extreme Load a Foundation Pile will Sustain. R 2 h p „ “ - = L. R representing weight of ram , P weight of pile , and L extreme * T~ tv X S load , all in lbs.; h height of fall of ram, and s distance of depression of pile with last blows , both in feet. Illustration. — Assume a ram 1000 lbs. to fall 20 feet upon a pile of 400 lbs., what resistance will the earth bear, or what weight will the pile sustain when driven by the last blow, from a height of 20 feet, .5 inch? s ==.5 of 12 ins. =.0416. Then 1000 2 X 20 400 -J- 1000 X .0416 20 000 000 58.24 = 343 406 lbs. Perimeter. The limits or bounds of a figure, or sum of all its sides. Of a canal it is the length of the bottom and wet sides of its transverse section. Flood Wave. The flood wave of the Ohio River in March (1884) was 71 feet 1 inch at Cincinnati, being higher than that of any previous record. Ice. Crushing Strength of, as determined by U. S. testing machine, ranged from 327 to 1000 lbs. per sq. inch. Atmosphere. If pure air is exhausted of 2.5 per cent, of its oxygen, it will not support the combustion of a candle. Blasting Paper. Unsized paper coated with a hot mixture of yellow prussiate of potash and charcoal, each 17 parts; refined saltpetre, 35; potassium chlorate, 70; wheat starch, 10, and water, 1500. Dry, cut into strips, and roll into cartridges. Circular Saws. Speed, 9000 feet per minute. Thus, for an 8 ins., 4500 revolutions, and progressively up to a 72 ins., 500 revolutions. (Emerson.) Foods. Relative Value of, compared witL. IOO Lbs. of Good Bay. Additional to page 203. Lbs. Lbs. Lbs. Acorns 68 Barley and Rye, mix’d 179 Barley straw 180 Buckwheat 64 Buckwheat straw 200 Linseed 59 Mangel-wurzel 339 Pease and Beans 45 Pea-straw 153 Potatoes 175 Rye 54 Turnips 504 Wheat 46 Wheat, Pea, and Oat- chaff 167 Depth of tlie Ooean. Mean depth is estimated by Dr. Krummel at 1877 fathoms = .4624 geographical mile. Gas-engine. A gas-engine 1.5 actual IP will cost, with gas at 8 cents per hour, 10 cents per hour for 10 hours. (Am. Engineer.) Locomotive. Average daily run 100 miles at a cost of $ 12.80 for driver, fireman, fuel, and repairs. (A'. J. Central R. R. Co.) Consumption of Fuel per Mile. Passenger , 25 to 30 lbs. coal. Freight , 45 to 55 Ibs. 3 or one cord wood per 40 miles. MEMORANDA. 9 13 Masonry. In laying stones in mortar or cement, they should rest upon the course beneath them, more than upon the material of joint. Steel Gr un (Krupp’s). Bore, 15.75 ins.; length of bore, 28.5 feet- of gun 32.66 feet. Weight, 72 tons. Charge, 385 lbs. prismatic powder; projectile chilled iron, 1660 lbs., with an explosive charge of 22 lbs. of powder. ’ Moment of shot at muzzle, estimated at 31 000 foot-tons, and range 15 miles. Saw-Mill. 7722 feet of 1 inch Poplar boards in One Hour. Engine (Non-condensing). Cylinder . — 12 by 24 ins. stroke of piston. Boilers. Two (cylindrical flued), 38 ins. in diam. by 26 feet in length, two 14 ins return flues m each Heating Surface .— 7 80 sq. feet. Grates .-42. 5 sq. feet. Pressure of Steam .— 125 lbs. per sq. inch, cut off at 16.5 ins. ^Revolutions.— 2 so to 350 per minute. Saws. — Two circular, 60 and 66 ins. in Note.— Grates set 28 ins. from under side of boilers, without bridge-wall and a combustion chamber under boilers, 4 feet in depth. Fuel, sawdust. ’ Steam Heating. 62 500 cube feet of space requires 6000 sq. feet of heat- ing surface to attain a temperature of 70 0 in the vicinity of the city of New York in its coldest weather. J u Or, One sq. foot of iron pipe will heat 10. 5 cube feet of space in an ordinary build- ing, temperature of exterior air 70°. (Felix Campbell.) * o ° r Steam at a Pressure of 60 lbs. + atmosphere has a \ elocity of efflux of 890 feet per second, and as expanded, a velocity of 1445 feet. Blasting. In small blasts 1 lb. powder will detach 4.5 tons material and in large blasts 2.75 tons. (See page 443. ) dl> na m Delta, Metal (Iron and Bronze). Specific gravity 8 4 (See page 384). b J 4 Jarrah Wood of Australia. A avails. Melting point 1800°. Impervious to insects and the Teredo Relative water evaporat- G as anr>, and to maintain it in motion 133.22 H>. Average work of a car-horse 5. 75 hours per day for a term of service of 6 years. Strong (fraught-horses will exert a power of 143 lb& @ 2.75 miles per hour for 22 milesfand an ordinary one 121 lbs. per 25 miles. (Gayffier.) Cable Railway. Mr. Wright gives the power required per ton* at 1.92 IP. * All tons here and elsewhere are given at 2240 lbs. 'T gi 6 APPENDIX. APPENDIX. River Steamboat. Wood Side "Wheels. ITreiglit and Passenger. “ Bostona. ” — Horizontal Lever Engines {Non-condensing).— Length on deck 302 feet 10 ins. ; beam , 43 feet 4 ins.; hold , 6 feet. Tons , 993.52. Immersed section of light draught of 26 ws., 83 sg./ee*. Capacity for freight 1200 tons (2000 lbs.). ’ Cylinders.— Two of 25 ins. in diam. by 8 feet stroke of piston. Boilers. — Four of steel, 47 ins. in diam. by 30 feet in length, 6 flues in each Heating surface , 903 sq. feet. Grate surface , 98 sq. feet. £ Pressure of Steam, 154 lbs. per sq. inch, cut off at .625. Revolutions , per minute. Speed , 10 miles per hour against current of upper Ohio, 3 to 5 miles. To Compute N£eta-cen tre of Hall of a Vessel. Operation of Formula in Naval Architecture, page 660. Assume a sharp-modelled yacht, 45 feet in length, 13.5 feet beam, and 9.5 feet hold, with an immersed amidship section of 42 sq. feet, and a displacement of goo cube feet at a mean draught of water of 6 feet. 2 /» y 3 d x - f - — — — = Meta-centre. See pages 650, 659. Ordinates (dec) taken at intervals of 2.5 feet are as follows: .216 2 - x 97 y = o = y 1 3 = .63 = y 23 = i. 3 3 = y 3 3 — 2 3 = 8 y 4 3 = 2.8 s = 21.952 y5 3 — 3 .63 _ 46.656 y 63 — 5 3 =125 y 7 = 5 8 3 ~ 195. II2 y 8 =6.53 = 287.496 y 9 3 = 6.7 s = 300.763 yio 3 — 6.75 = 307.547 y 11 =6.5 =287.496 y 12 = 6.25 = 244.14 y I 3 3 = 5 .8 =195.112 y ,43 3 = 5 = 125 y 15 =4.2 = 74.088 y l6 = 3*25 = 34. 328 y I 7 3 = 2> 4 =13.824 y l8 g — 1, 5 = 3-375 y 19 = .8 = .512 ya ° 3 = o = .0 2272.814 2-5 5682.035 Summation of function of cubes of ordinates for value of / y 3 dx = 5682.035. And - 0 f 5682.035 0 f 6. 3 i = 4.21 feet. 3 9 °° 3 Note. — The other elements of this vessel are: Area of load-line , 401.12 sq. feet ; Displacement in weight , 27.974 tons ; do. at load- draught, .955 tons per inch; Depth of centre of gravity of displacement below load- line, 1.49 feet; Volume of displacement, to volume of immersed dimensions, 26.8 per cent. To Compute Height of Jet in a Ooxidnit IPipe from a Constant Head. (Weisbach.) h v 2 h' TV : — t v . . = ■ — ‘ — h, and — = h". h, h', and h" representing heights I + ( C + C 'd)(v) 2 " due to velocity of efflux, loss of head and of ascent, l length o f pipe or conduit, and d and d' diameters of pipe and jet, all in feet, v velocity of efflux in feet per second, C and C' coefficients of friction of inlet of pipe and outlet , and z a divisor determined by experiment with diameters of . 5 to 1.25 ins. , ranging from 1.06 to 1.08. Illustration. — If conduit pipe for a fountain is 350 feet in length, and 2 ins. in diameter, to what height will a jet of .5 inch ascend under a head of 40 feet? Assume C andC" .8 and .5, h — 25 feet, d — 2 ins. = .166, and .5 = .5-4-12 = .0416. Then 25 1 -f ^.8-f .5 --4.9 feet. 66 / APPENDIX. 917 To Compute Head and Discharge of Water in Dipes of Grreat Length. It becomes necessary first to determine the velocity of the flow, which is = 4 v — v — !. 27 o — independent of friction. V representing volume of water 3. 1416 d 2 0 d~ ... in cube feet, and d diameter of pipe in ms. When head, length, and diameter of pipe are given V2 g h J i+C + c- Coefficients of friction C, for velocity of flow, range from .0234 to .0191 for veloci- ties from 3 to 13 feet per second, and c that for the pipe as a mean at . 5. See Weis- bach’s Mechanics, Vol. i., page 431. Illustration.— What head must be given to a pipe 150 feet in length and 5 ins in diameter, to discharge 25 cube feet of water per minute, and what velocity will it attain at that head? - - J C^.024 and c — . 5* Then 1.273 ^ = *- 2 73 X 2.4 = 3.055 feet velocity per second , and (1+.5 + .024 150 X -- 2 ) |^ = i. 5 + 8;6 4 X. 14 = 142/^ head. \ 5 / 64.33 _ •x/ds c !l V 2 _ . . Or, 4. 72 — - — V in cube feet per minute , and . 538 P / — r— — “ 171 ins - y/ 1 -r-h v Illustration.— A ssume elements of preceding case. Then 4.72 ^ 3I25 — = 4 ? 2 * = 25 ' 67 cube f eet > aDd -538 $ J lS ° * **— V 150 -‘r- 1.42 10.25 = . 538 X ^69 607 = . 538 X 9- 3 01 — 5 i ns - To Compute Fall of a Canal or Open ConcUxit to Con- duct and Discharge a. Given Volume of Water per Coefficient of friction in such case is assumed by Du Buat and others at .007 565. C l JL x — = h. h representing height of fall, l length of canal, and p net perime - ter, all in % ; A area of section of canal in sq. feet, and v velocity of flow in feet per second. Illustration i -What fall should he given to a canal with a section of 3 feet at botlom 7 at top, and 3 in depth, and a length of 2600 feet, to conduct 40 cube feet of water per second ? C = .oo 7 6, p = 3 -f ( V3 2 + 2 2 X 2) = 10. 21 feet, A = 7 + ^ ~ = 1 5 sq. feet, and v — — 2. 66 feet. Then .00,6 -600XKX2Z x 2A6= = I3 45 x = M 8/eef. ' 15 64.33 2.— What is volume of water conducted by a canal, with a section of 4 feet at bottom, 12 at top, and 5 in depth, with a fall of 3 feet, and a length of 5800 feet? Jf—X^gh-v. A = U+UX5 _ 40 sq . feet, and p = 4 + (V5 2 + 4"' X 2} = 16.8 feet. f 40 ^3 X 64.33 x 3 = J X >93 = 3-3 feet, and V .0076 X 5800 X 16. i 40 X 3. 23 feet velocity = 129. 2 cube feet. For Dimensions of transverse profile of a canal, see Weisbach, page 492, vol. i. 918 APPENDIX. STEAM, VACUUM, AND HYDROSTATIC GAUGES. (Crosby’s.) INCHES. Diameter of Dial. . 12 10 8.5 6-75 6 5.5 4-5 3-5 2-5 Brass No 00 0 1 2 3 4 5 6 7 Iron No — _ i -5 2.5 3-5 4-5 5-5 6-5 7-3 ADJUSTABLE-POP SAFETY-VALVES. (Crosby’s.) INCHES. Diameter of Valve. 1 | 1-25 1 1,5 1 2 ! 2 - 5 I 3 | 3-5 1 4 1 5 Capacity in IP 10 1 20 1 3 ° 1 50 1 .&> 100 150 | 200 1 3 °° STEAM SIPHON. An Independent Lifting Pump. Capacity for a Discharge Pipe 2 Ins . in Diameter, per Minute. Water raised. j Pressure. Discharge. Water raised. Pressure. Discharge. Feet. Ins. Lbs. Gallons. Feet. Ins. Lbs. Gallons. 14 6 30 63.54 13 2 60 119.68 13 2 40 85-71 13 2 70 138.44 f 13 2 50 IOO 13 2 80 157-57 DISTANCES, VELOCITIES, AND ACCELERATION. To Compute Velocities of an Accelerated Body. fv 2 -j- (2 v' S), Or, v -f t v' = V. v and v' representing original and accelerated velocities, and V final velocity , all in feet per second ; S distance or space passed over in feet , and t time in seconds . ~ — = v '- v ' representing average velocity in feet per second. V' t == S, arid 2 V' — V = v. Illustration i. A body moving with a velocity of 10 feet per second, is acceler- ated at rate of 4 feet per second, per second, for a period of 6 seconds; what are its different velocities? v = 10, v' = 4, t — 6. Then, 10 -f 6l<4 = 34 feet final velocity. 10 = 22 feel average velocity. 22X6 = 132 feet distance passed over. Vio 2 -f (2 X 4 X 13 2 ) = V IJ & — 34 f eet > and 2 X 22 — 34 = 10 feet original velocity. k , V — v And, — = v , V + t> X « = s, -Xt = v' S, r 2 -j- 2 v’ S = V 2 , V — v — — t, and \/V 2 — 2 V S = v. 2.- A body is projected vertically with a velocity of 200 fee' t per se ^ d , and is retarded at the rate of 30 feet per second, per second; what t J passed through when its velocity is reduced to 80 feet per second, and in what time . v — 200, v' = 30, and V = 80. v = 30 , Then 200 — 80 30 = 4 seconds. 80 4 - 200 , — — X 4 = 560 feet. A vehicle being drawn with a velocity of 25 feet per second is accelerated 5 feet per second, per second; what is its velocity and time of operation at the end of 100 feet? v = 25 , Then 5, and V = 100. 25 — iK seconds. 3 + 25 , 5 APPENDIX. . A stream of water after flowing a distance of 120 feet, is ascertained to have a vetocIty ofTo fteT per’second, with an accelerating velocity of a feet per second, per second; what was its primitive velocity and time of flow . S = X20, V = 4 P, ^ = 2. 40- 33-47 Then V40 2 — 2 X 2 X 120 = 33.4 7 f eet - - — 3. 26 seconds. j Delivery" and Friction in Dose. (R. F. Hartford , Am. Soc. C. E.) Hose 2. 5 ins. in diameter. Nozzles not exceeding 1. 5 ins. J 2 ^.$T- G Rubber or Leather . .0408 v d 2 and .497 c d 2 V p .= G ) 4.0484 G 2 d '2.012 G_ d . 12>i8 cy/P and = and P; .012 857 b G 2 ?; c /p v a* ■ c 2 d* * ' .003 175 6 c 2 d 4 P Z and .000021 4 6 l v* d* =J>; P-p = P'; P* = r » = . , 3I4-9 6 < 32^6 Ot* — V 2 and — u' 2 * v representing velocity of air m feet per second , and ’ 400 200 in miles per hour. Wolff . 1 . Wiley X„te. For u.ofal table, and formula, aee “ Windmill, a, aPnme Mover, by A. R. M , & Sons, New York, 1885. 922 APPENDIX. To Compute Head in. JL,L>s. per Sq. Incli to Resist Fric- tion of An* in Long and Rectilineal Pipes, etc. V 1728 V2 L a 60" V) (3. 7 d ) 5 83. ] = H : H (3.7 d) 5 83. ya r -* 7 ^(3.7^ 5 83.3 == V; 5 / L ^ a 60" v P-{-H y 83.1 H ' 3 ‘ 7 ~ a ’ ana I2 " x 33 000 ~ ^ V representing volume discharged ^ cube feet per minute, L length of pipe in feet, d diameter of pipe in ins., H head and n nlln Ur /ri ™ and P er *?• inch i v velocity of discharge in feet per sec- ond, a area of discharge m sq. ms., and HP horse-power of friction of air alone. h Jmo3 RA ? I ^ )N 'T' AsSUme volume of air discharged 44000 cube feet per minute diameter of discharge pipe 40.5+ ms. (say 1280 sq. ins. net), length of pipe 1000 feet, and pressure at discharge 3.5 lbs. per sq. inch. Then 44° °°X r 728 1280 X 60 44 000 2 X 1000 6 310406 250000 74i- 5 IP. r i 936090 000 X 1000 83.1 X .3068 99° feet, and (3.7 x 4°- 5~H 5 X 83.1 =6310406250000. 3068 lbs. ; cu ^ feet . -4- 3-7 = 40.5 ins., and 1280 x 60 X 990 X 3. 5 + -3068 12 X 33000 Ice or Cold Producing Machines witliont Rower or Pressure. Dry- Agent System. (E. Gillet, New York.) The Operation of this Machine is Automatic. Ice produced in 1 hour; water or other liquids rapidly cooled, from ioo° and above to freezing-point and below. ’ produce simultaneously 35 lbs. water per hour at a tempera peraturJ 06 ^ -maohines produce in like manner 70 lbs. water per hour at a like tem- Ihe capacity of these machines can be increased to any size as required, and they ar £u P ™ e to t ^ ie P r .°duction of ice, refrigerating of brine, meats, etc. the Refrigerator, which is or can be applied to the machine, is cooled by the ex- cess of low temperature of the solution, after making the ice, cooling the water etc. Ihe chemical element cannot crystalize while in process of evaporation 5 but when allowed to coo it will crystalize at once, and be readily granulated if allowed to drop from a low elevation. Owing to the use of one or more Economizers embraced in the system, 1. 5 to 2 lbs. of water is required per lb. of ice. 7 5 This system consists simply in mixing a special salt with water, and reducing it by evaporation, or evaporating a few hundredweight of water per hour, the water decreasing in temperature while passing from one Economizer to another, and in- versely on returning to Evaporator. ’ Friction, of Water in Pipes. (Weisbach.) . 1865 l V 2 fi V c = h. I representing length of pipes in feet , v = , or velocity in feet per second, V volume of water in cube feet elevated per second, d diameter of pipes m ms., and C a coefficient, ranging from .069 when velocity — .1 foot, .0387 for ’ -° 375 f° r l f™t, .0265 for 2 feet, .023 for 4 feet, .0214 for 6 feet, .0205 fir 8 feet, .0193 /or 12 feet, and .01 82 /or 20 feet. Illustration.— Assume volume 125 cube feet, raised 25 feet per hour, through a pipe 2 ins. in diameter and 500 feet in length; how many feet of vertical head will the friction m the pipe be equal to ? Then ^ 36^ x 2 2 " = 1,59 velocit > r > and c = -° 28 - Hence > " ^ 5 g X 1 59 X .028 = 3.3 feet, and 25-1-3.3 = 28.3 feet. ORTHOGRAPHY OF TECHNICAL AVORDS AND TERMS. 923 ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. a uniformity of expression. Abut. To meet, to adjoin to at the end to border upon. Abut end of a log, etc. , is that having the greatest diameter or side. But and Butt end, when applied in this manner, are corruptions. Adit In Mining , the opening into a mine. ‘ • Tho middle or centre of a vessel, either fore and aft or athwartships. Tht:“p tome m of a d veSat 0 , and is termed dead flat. rod tn nainted and carved or sculptured ornaments of imaginary foliage* a^d^ani mals, 'In which there are no perfect figures of either. Synonymous with Moresque. . Arbor The principal axis or spindle of a machine of revolution. facesTof a bodyl" forming^n'e^terhir angle^me^ea^o^er.^T^e edges of V a bodyj as a brick, are arrises. Ashlar In Masonry , stones roughly squared, or when faced. Hart. Across, from side to side, transverse, across the line of a vessel's course. Athwartships , reaching across a vessel, from side to si e. Bagasse. Sugar-cane in its crushed state, as delivered from the rollers of a mill. Balk. In Carpentry, a piece of timber from 4 to 10 ins. square. Baluster. A small column or pilaster; a collection of them, joined by a rail, forms a balustrade. Banister is a corruption of balustrade. Bark A ship without a mizzen-topsail, and formerly a small ship. Bateau. A light boat, with great length proportionate to its beam, and wider at its centre than at its ends. Batten In Carpentry , a piece of wood from i to 2 5 ins. thick ^ [ rom 1 7 infCbread?h When less than 6 feet in length, it is termed a deal-end. Berme. In Fortifications and Engineering , a space emba P nk- and a moat or fosse, to arrest the ruins of a rampart The level top oi ment of a canal, opposite to and alike to the towpath. Bevel. A term for a plane having any other angle than 45° or 9° • Binnacle. The case in which the compass, or compasses (when two are used), set on board of a vessel. a Bit. The part of a bridle which is put into an animal’s mouth. In Caipentry, a boring instrument. Bitter End. The inboard end of a vessel’s cable abaft the bitts. Bills. A vertical frame upon a deck of a vessel, around or upon which is secured cables, hawsers, sheets, etc. Bogie. Pivoted truck, to ease the running of an engine or car around a curve^ Boomkin. A short spar projecting from the bow or quarter of a vessel, to exten the tack of a sail to windward. Bowlder. A stone rounded by natural attrition; a rounded mass of rock ported from its original bed. Breast-summer. A lintel beam in the exterior wall of a building. Buhr-stone. A stone which is nearly pure silex, full of pores and cavities, a used for Mills. Bunting. Woolen texture of which colors and flags are made. Burden. A load. The quantity that a ship will carry. Hence burdensome. Cag. A small cask, differing from a barrel only in size. Commonly written Keg. 924 ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. Caliber. An instrument with semi-circular legs, to measure diameters of snheres or exterior and interior diameters of cylinders, bores, etc. ’ A pair of Calibers is superfluous and improper. Calk. To stop seams and pay them with pitch, etc. To point an iron shoe so as to prevent its slipping. 0 Cam An irregular curved instrument, having its axis eccentric to the shaft upon which it is fixed. 0 “ u Camber. To camber is to cut a beam or mold a structure archwise as deck beams of a vessel. » ueLK ' Camboose. The stove or range in which the cooking in a vessel is effected Tho cooking-room of a vessel; this term is usually confined to merchant vessels n vessels of war it is termed Galley. in CW In Engineering, a decked vessel, having great stability, designed for use weight or bufk SUn V6SSelS ° r structures - A| s° to transport loads of great A Scow is open decked. Cantle. A fragment; a piece; the raised portion of the hind part of a saddle. Cantime. The space between the sides of two casks stowed aside of each other TantUne. “ “ th ° CaDtline ° f tW ° 0tliers > il is said to be stowed bilge and Capstan. A vertical windlass. inferring fisheries!' VCSSel ( ° f 25 ° r 3 ° t0nS ’ bllrdeD ) used u P on the coast of Prance reeelveX 2^?.^ V n J ber se ‘ fore . and aft from the deck beams of a vessel, to leceive the ends of the ledges in framing a deck. Carvel built.— A term applied to the manner of construction of small boats to signify that the edges of their bottom planks are laid to each other like to the man- ner of planking vessels. Opposed to the term Clincher. Caster. A small phial or bottle for the table. Casters. Small wheels placed upon the legs of tables, etc., to allow them to be moved with facility. Catamaran. A small raft of logs, usually consisting of three, the centre one be- ing longer and wider than the others, and designed for use in an open roadstead and upon a sea-coast. Chamfer. A slope, groove, or small gutter cut in wood, metal, or stone. Chapelling. Wearing a ship around without bracing her fore yards. Chimney. The flue of a fireplace or furnace, constructed of masonry in houses and furnaces, and of metal, as in a steam boiler. See Pipe. Chinse. To chinse is to calk slightly with a knife or chisel. Chock. In Naval Architecture , small pieces of wood used to make good any de- ficiency in a piece of timber, frame, etc. See Furrings. Choke. To stop, to obstruct, to block up, to hinder, etc. Cleats. Pieces of wood or metal of various shapes, according to their uses, either to belay ropes upon, to resist or support weights or strains, as sheet, shoar, beam cleats, etc. Clincher built. A term applied to the construction of vessels’ bottoms, when the lower edges of the planks overlay the next under them. Coak. A cylinder, cube, or triangle of hard wood let into the ends or faces of two pieces of timber to be secured together. The metallic eyes in a sheave through which the pin runs. In Naval Architecture , the oblong ridges banded on the masts of ships. Coamings. Raised borders around the edges of hatches. Coble. A small fishing-boat. Cocoon. The case which certain insects make for a covering during the period of their metamorphosis to the pupa state. ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. 925 Cog. In Mechanics, a short piece of wood or other material let into the faces of a bodv to impart motion to another. A term applied to a tooth in a wheel when it is made of a different material than that of the wheel. In Mining, an intrusion of matter into fissures of rocks, as when a mass of unstratifled rocks appears to be in- jected into a rent in the stratified rocks. Coaaina In Carpentry , the cutting of a piece of timber so as to leave a part alike^to a^cog, and the notching of the upper piece so as to conform to and receive it. Alike to indenting or tabling. Colter. The fore iron of a plough that cuts earth or sod. Compass. In Geometry , an instrument for describing circles, measuring figures, etc. A pair of Compasses is superfluous and improper. Connectina Rod In Mechanics , the connection between a prime and secondary mover, as between the piston-rod of a steam-engine and the crank of a water-wheel or fly-wheel shaft. The term Pitman is local, and altogether inapplicable. Contrariwise. Conversely, opposite. Crossways is a corruption. Corridor. A gallery or passage in or around a building, connected with various departments, sometimes running within a quadrangle ; it may be opened or enclosed. In Fortifications , a covert way. Cyma. A molding in a cornice. Damasquinerie. Inlaying in metal. Davit. A short boom fitted to hoist an anchor or boat. Deals. In Carpentry , the pieces of timber into which a log is cut or sawed up. Their usual thickness is 3 by 9 ins. and exceeding 6 feet in length. Improperly restricted to the wood of fir-trees. Dike. In Engineering , an embankment of greater length than breadth, imper- vious to water, and designed as a wall to a reservoir, a drain, or to resist the influx of a river or sea. Dingay (Nautical). A ship or vessel’s small boat. Dock. In Marine Architecture , an. enclosure in a harbor or shore of a river, for the reception, repair, or security of vessels or timber. It may be wholly or only partially enclosed. See Pier. When applied to a single pier or jetty, it is a misapplication. Dowel. A pin of wood or metal inserted in the edge or face of two boards or pieces, so as to secure them together. This is very similar to coaking, but is used in a diminutive sense. An illustration of it is had in the manner a cooper secures two or more pieces in the head of a cask. Draught. A representation by delineation. The depth which a vessel or any floating body sinks into water. The act of drawing. A detachment of men from the main body, etc. Ordinarily written draft. Dutchman. In Mechanics , a piece of like material with the structure, let into a slack place, to cover slack or bad work. See Shim. Edgewise. An edge put into a particular direction. Hence endwise and sidewise have similar significations with reference to an end and a side. Edgeways is a corruption. Euphroe. A piece of wood by which the crowfoot of an awning is extended. Fault. In Mining , a break of strata, with displacement, which interrupts opera- tions. Also, fissures traversing the strata. Felloe , Felloes. The pieces of wood which form the rim of a wheel. Fetch. Length of a reservoir, pond, etc., along which the wind may blow towards the embankment or dam. Flange. A projection from an end or from the body of an instrument, or any part composing it, for the purpose of receiving, confining, or of securing it to a sup- port or to a second piece. Flier. In Carpentry , a straight line of steps in a stairway. Frap. To bind together with a rope, as to frap a fall, etc. 9 26 ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. Frieze In Architecture, the part of the entablature of a column which is between f "" TeVt “id next the base, left by the removal of the top or « *». “ — » br fn., tbeir faces to the required shape or level. Gliding. Putting galets into pointing-mortar or cement. Galete Pieces of stone chipped off by the stroke of a chisel. See Spall : Galiot. A small galley built for speed, having one mast, and from .6 to =o thwarts for rowers A Dutch-constructed brigantine, j Gate. In Mechanics, the hole through which molten metal is poured into a mold , for casting. " “"^^^. wheete for transmlttitigmbtlon. To gear* ) by passins around his belly. In Printing, the bands ol a press. Gnarled. Knotty. ii nvnvtfi To clean a vessel’s bottom by burning. Tig Z tin, off grass, shells, etc, from a ship’s bottom. Synonymous r< with Breaming. u, and aSSenSlotheSem 0 Cat harping*; ropes which brace in the shrouds of in to the huU of a vessel when ter ends drop below h4r aD C ®«i»r InNaral Architecture, calking with a large maul or beetle. "to press, to crowd, to wedge in. In Nautical language, to squeeze tight. ", Pe ^ d - /o from one tack to another ; hence JiUng, the shifting of a boom. Jigging. Washing minerals in a sieve of tbe floor timbers, anf e eSy W M l" When Rcated on & floors or at the sides, it is termed a sisters or a side keelson. Si S«,. 'wm> «*»»"•»“ »•"■«*■ “CSX * ™ » .> « — »' - — « ■» " will not float or sit upright. ^ rin^ or securing her to a SI < ^-'anting edge of a sail when not secured to a spar or stay. ( fici ( ( to 1 clei C the C pie( whi of s C C( Cl of tt ORTHOGRAPHY OP TECHNICAL WORDS AND TERMS. g 2 f Luf The fullest part of the bow of a vessel. Mall. A large double-headed wooden hammer. Mantle. To expand, to spread. Mantelpiece. The shelf ovef a fireplace in front of a chimney. Marquetry. Checkered or inlaid work in wood. Matrass. A chemical vessel with a body alike to an egg, and a tapering neck. Mattress A quilted bed ; a bed stuffed with hair, moss, etc., and quilted. Mitred. In Mechanics, cut to an angle of 45 °, or two pieces joined so as to make a right angle. Mizzen-mast. The aftermost mast in a three-masted vessel. Mold In Mechanics , a matrix in which a casting is formed. A number of pieces of vellum or like substance, between which gold and silver are laid for ^the P^rpos q of bein°- beaten. Thin pieces of materials cut to curves or any required figure In Naval 'Architecture pieces of thin board cut to the lines of a vessel s timbers etc. Fine earth, such as constitutes soil. A substance which forms upon bodies in warm and confined damp air. This orthography is by analogy, as gold, sold, old, bold, cold, fold, etc. Molding. In Architecture, a projection beyond a wall, from a column, wainscot, etc Moresque. See Arabesque. Mortise. A hole cut in any material to receive the end or tenon of another piece. Muck. A mass of dung in a moist state, or of dung and putrefied vegetable matter. Mullion. A vertical bar dividing the lights in a window ; the horizontal are termed transoms. Net. Clear of deductions, as net weight. Newel. An upright post, around which winding stairs turn. Nigged. Stone hewed with a pick or pointed hammer instead of a chisel. Ogee. A molding with a concave and convex outline, like to an S. See Cyma and Talon. Paillasse. Masonry raised upon a floor. A bed. Pargeting. In Architecture , rough plastering, alike to that upon chimneys. Parquetry. Inlaying of wood in figures. See Marquetry. Parral. The rope by which a yard is secured to a mast at its centre. Pawl. The catch which stops, or holds, or falls on to a ratchet wheel. Peek The upper or pointed corner of a sail extended by a gaff, or a yard set ob- liquely to a mast To peek a yard is to point it perpendicularly to a mast. Pendant. A short rope over the head of a mast for the attachment of tackles thereto ; a tackle, etc. Pennant. A small pointed flag. Pier. In Marine Architecture, a mole or jetty, projecting into a river or sea, to protect vessels from the sea, or for convenience of their lading. See Dock. Erroneously termed a Dock. Pile. In Engineering , spars pointed at one end and driven into soil to support a superstructure or holdfast. Spile is a corruption. Pipe. In Mechanics , a metallic tube. The flue of a fireplace or furnace when constructed of metal; usually of a cylindrical form. The term or application of Stack (which refers solely to masonry) to a metallic pipe is a misappli- cation. Piragua. A small vessel with two masts and two boom -sails. Commonly termed Perry-augur. Pirogue. A canoe formed from a single log, propelled by paddles or by a sail, with the aid of an outrigger. Plastering. Jl dill uuu * 65 ^* • In Architecture , covering with plaster cement or mortar upon walls , England, termed laying , if in one or two coat work; and priclcing up, a t or laths. In — 0 if in three-coat work. Plumber block. A bearing to receive and support the journal of a shaft. Polctcre. Masts of one piece, without tops. 928 ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. Poppets. In Naval Architecture , pieces of timber set perpendicular to a vessel’s bilge-ways, and extending to her bottom, to support her in launching. Porch. An arched vestibule at the entrance of a building. A vestibule supported by columns. A portico. Portico. A gallery near to the ground, the sides being open. A piazza encom- passed with arches supported by columns, where persons may walk; the roof may be flat or vaulted. J Pozzuolana. A loose, porous, volcanic substance, composed of silicious argilla- ceous, and calcareous earths and iron. Prize. In Mechanics , to raise with a lever. To pry and a pry are corruptions. Proa, Flying. A narrow canoe, the outer or lee side being nearly flat. A frame- work, projecting several feet to the windward side, supports a solid bearing in the form of a canoe. Used in the Ladrcne Islands. Purlin. In Carpentry , a piece of timber laid horizontal upon the rafters of a roof, to support the covering. Ramp. In Architecture , a flight of steps on a line tangential to the steps. A concave sweep connecting a higher and lower portion of a railing wall etc. A sloping line of a surface, as an inclined platform. Rarefaction. The act or process of distending bodies, by separating their parts and rendering them more rare or porous. It is opposed to Condensation. Rebate. In Mechanics , to pare down an edge of a board or a plate for the purpose of receiving another board or plate by lapping. To lap and unite edges of boards and plates. In Naval Architecture , the grooves in the side of the keel tor receiving the garboard strake of plank. Commonly written Rabbet. Remou. Eddy water without progressive action, in bed of a river* a return of water against direction of flow of a river. Rendering. In Architecture , laying plaster or mortar upon mortar or walls. Rendered and Set refers to two coats or layers, and Rendered , Floated, and Set to three coats or layers. Reniform. Kidney-shaped. Resin. The residuum of the distillation of turpentine. R 0S i n is a corruption. Riband. In Naval Architecture , a long, narrow, flexible piece of timber. Rimer. A bit or boring tool for making a tapering hole. In Mechanics , to Rim is to bevel out a hole. Riming. The opening of the seams between the planks of vessel for the purpose of calking them. Rotary. Turning upon an axis, as a wheel. Rynd. The metallic collar in the upper mill-stone by which it is connected the spindle. Sagging. A term applied to the hull of a vessel when her centre drops below ends. The converse of Hogging. Scallop. To mark or cut an edge into segments of circles. Scarcement. A set back in the face of a wall or in a bank of earth. A footii Scarf. To join ; to piece ; to unite two pieces of timber at their ends by rui the end of one over and upon the other, and bolting or securing them togethei Scend. The settling of a vessel below the level of her keel. Selvagee. A strap made of rope-yarns, without being twisted or laid up, a tained in form by knotting it at intervals. Sennit. Braided cordage. Sewage. The matter borne off by a sewer. Sewed. In nautical language , the condition of a vessel aground ; she is s sewed by as much as the difference in depth of water around her and her depth. Sewerage. The system of sewers. Shaky. Cracked or split, or as timber loosely put together. Shammy. Leather prepared from the skin of a chamois goat. ORTHOGRAPHY OF TECHNICAL WORDS AND sheer In Naval Architecture, the curve or bend of a ship's deck To at the upper cuts, and used to elevate heavy bodies, as masts, e “. piec e of wood or iron lot into a slack place in a fra »a h Shoal, A great multitude; a crowd; a multitude of fish. 77'ar. ‘ IToblique brace, the upper end resting against the substance to be sup- P Shales. Pieces of plank under The heels *^ n which wood; coa l, etc. , are thfown' or tlTToKw Aural contraction of a river. A young p,g. Sidewise. See Edgewise. Sianalled. Communicated by signals. Zi‘ the horizontal piece of tim ‘ be^sto A nf at the bottom ^ t0 draw duids out of vessels. sfr'Tht extreme after-part of the keel of a vessel; the portion that supports the rudder-post. Slantwise. Oblique ; not perpendicular. Sleek To make smooth. Refuse; small coal. . , JZker. A spherical-shaped, curved, or plane-surfaced instrument with which to smooth surfaces. Slue The turning of a substance upon an axis within its figu . J , 1 term applied to planks when their edges at their ends are curved or rofS'upwa^wsSe .{ the ends of a full-modelled vessel. r ctene etc chipped off by the stroke of a hammer or the force Spall. A p.ece of stone, etc. chippea pjeces of a blow. Spoiling, break g P triangnlar spaC e between the outer lines or^SLf an IrcCa horizontal line drawn from its apex, and a vertical hue i) 0 <=e of shielding the deck-beams from the shock of a sea. , alike to alme, M e, r with leafhgo. ■vulcanized rubber, used to facilitate the drying of wet floors ; 0r “f S ° ber Th( Stack In Masonry, a number of chimneys or p.pes standing together. ^The^ppli^ the smoke-pipe of a steam-boiler is wholly erroneous. Stape. Ill Engineering, the interval or distance between two e.evalions, in sho ling, throwing, or lifting. Sleeving. The elevation of a vessel’s bowsprit, cathead, etc. Stroke. A breadth of plank. Strut. An oblique brace to support a rafter. Style. The gnomon of a sun-dial. o ^ , rAr t s-T;r“r„=rrr,- ... • blow from a hammer. 930 ORTHOGRAPHY OP TECHNICAL WORDS AND TERMS. Talus. In Architecture , tlie slope or batter of a wall naraoet etc Tn a sloping heap of rubble at foot of a cliff. ’ parapet > etc - Iu Geology, distffints weight*^®''’ “ W °° deB beariB ® t0 receive the end « f a girder to Templet. A mold cut to an exact section of any piece or structure izB> - Terring. The earth overlying a quarry. Tester. The top covering of a bedstead. Tioles. The pins in the gunwale of a boat which are used as rowlocks Thwarts. The athwartship seats in a boat. eurrenUnstead of tbe^wind. ° f & VeSSe ‘ ^ aBCh ° r ’ Wben shc rides in dire <=tiott of the Tire. The metal hoop that binds the felloes of a wheel. shoS^ecureA St0PPCr ° f * PieC ° ° f ordBanco ' The iron bottom which grape- frames Mai ’ k ' W °° deB PiDS CmpIoyed secure the planking of a vessel to the boring at greaVfepff ** instr,!ment in the comminution of rock in earth- piecesof timbe/set'horLomAti 6 ’ a movabIe form °f support. In Mast-making , two pieces oi timber set horizontally upon opposite sides of a mast-head. u 2 nee. In Seamanship, to haul or tie up by means of a rope or tricing-line. furaace ° n ° r Tuyere ’ The nozzle of a bellows or blast-pipe in a forge or smelting- Vice - In Mechanics, a press to hold fast anything to be worked upon. Voyal. In Seamanship, a purchase applied to the weighing of an anchor lead to a capstan. Wagon. An open or partially enclosed four-wheeled vehicle, adapted for transportation of persons, goods, etc. Wear. In nautical language, to put a vessel upon a contrary tack bv turi her around stem to the wind. y Weir. A dam across a river or stream to arrest the water; a fence of twiff stakes in a stream to divert the run offish. Whipple-tree. The bar to which the traces of harness are fastened. Wind-rode. The situation of a vessel at anchor, when she rides in directio- the wind instead of the current. Windrow. A row or line of hay, etc., raked together. TFtMe An instrument fitted to the end of a boom or mast, with a ring thr 1 which a boom is rigged out or mast set up. Woold. To wind ; particularly to bind a rope around a spar, etc. Roil. To render turbid, to stir or mix. THE EJSTB. I <