*^'i v^'^ ct-, i. -J- ,^x -' --"^■^- x^^. .> ^^/ ^b ^- ^81^ ^ , -/^^ ■■^^^. "... t: aV -r^ - '^^.. xO°<. Cf^ o 0^ ^.^TT.^^V -^7* s '^ * / "^^ A^ wr?9^ o 0^ ^^1^^^%^^ ^ ,^^.% .'■^" .-^^V.o 3f^ .-^ H -r. 0^ ^ * * , ^^"\ V?' o 9 * > Ax 47 (li \ n- "e ~- 5^ li 7 7 •. "i r I' so XGivEEi: \X19S'^^CKRT-B00K, \"E.S Of / (. IT-'^ ,f H A S W E L L'S ENGINEERS' AND MECHANICS' POCKET-BOOK, CONTAINING UNITED STATES AND FOREIGN WEIGHTS AND MEASURES ; TABLES OF AREAS AND CIRCUMFERENCES OF CIRCLES, CIRCULAR SEG- MENTS, AND ZONES OF A CIRCLE ; SQUARES AND CUBES, SQUARE AND CUBE ROOTS ; LENGTHS OF CIRCULAR AND SEMI-ELLIPTIC ARCS ; AND RULES OF ARITHMETIC. MENSURATION OF SURFACES AND SOLIDS; THE MECHANICAL POWERS ; GEOMETRY, TRIGONOMETRY, GRAVITY, STRENGTH OF MATERIALS, WATER WHEELS, HYDRAULICS, HYDROSTATICS, PNEU- MATICS, STATICS, DYNAMICS, GUNNERY, HEAT, WINDING EN- GINES, TONNAGE, SHOT, SHELLS, &C. STEAM AND THE STE AM-ENGINE j COMBUSTION, WATER, CABLES AND ANCHORS, FUEL, AIR, GUNS, &C., &C, TABLES OF THE WEIGHTS OF METALS, PIPES, &C. MISCELLANEOUS NOTES, AND EXERCISES, &c., &:c. / BY CHARLES H. HASWELL. CHIEF ENGINEER U. S. NAVY. All examination of facts is the foundation of science. NEW-YORK: HARPER 6c BROTHERS, 82 CLIFF- STREET. 1844. ]m-^ f^' Kntfired, accordii^ig to Act of Con-zrps^, in the year 1844, by • IIaki'kr & Bkothkrs, In thp Clerk's Office of the Southern District of New-York. ERRATA. Page 140, 27th line, for " common," read cannon. " 143, 12th line, for *y 10x64.33," read v/lOxbISC " 142, 16th and 17th lines, omit *'and that product, attain, by the velocity in feet per second." "^ Page 142, 20th line, omit *' x5=:7500." " 158, 15th line, for " 594,000," read 59,400. " 179, 26th line, for " 5}" read 4,V. \ " 218, last line, after "horses' power," read, witU plain cyli»^ drical boilers. I Page 226, 17th line, for *' Picton," read Pictou. *' 240, 19th line, for " brushes," read bushes. i " 248, in cohimn 2d, insert i opposite to '* 0.211." •* 251, 3d line from bottom, for '* 176.7149," read 1767 145 1^ and in bottom line, for "460.6957," read 460.6947. j Page 260, last line, for " 92000," read 9200. ^y^2Z~ 4^::^^^^;^. TO CAPTAIN ROBERT F.STOCKTON, U. S. NAVY, AS A TRIBUTE TO THE LIBERALITY AND ENTERPRISE HE HAS EVINCED IN HIS PATRONAGE OF MECHANICAL SCIENCE, THIS EDITION IS, WITH PERxMISSION, BY THE AUTHOR. Washington^ Aug. 1, 1843. PREFACE. The following work is submitted to the Engineers and Mechanics of the United States by one of their number, who trusts that it will be found a convenient summary for reference to Tables, Results, and Rules connected with the discharge of their various duties. The Tables have been selected from the latest and best publications, and information has been sought from various sources, to render it useful to the Operative Engineer, Me- jchanic, or Student. The want of a work of this description in this country has long been felt, and this is peculiarly fitted to supply that want, in consequence of the adaptation of its rules to the metals, woods, and manufactures of the United States. Having for many years experienced inconvenience for the want of a compilation of tables and rules by a Prac- tical Mechanic, together with the total absence of units for the weights and strengths of American materials, I was in- duced to attempt the labour of a compilation and the neces- sary experiments to furnish this work. The proportions of the parts of the steam-engine and boilers will be found to differ in most instances materially from the English authorities ; but as they are based upon the actual results of the most successful experience, I do not hesitate to put them forth, being well assured that an adherence to them will ensure both success and satisfaction. A2 Vi PREFACE. The sources of information from which I have principally- compiled are Adcock, Grier, Gregory, the Library of Use- ful Knowledge, and the Ordnance Manual ; and to the la- bours of the authors of these valuable works I freely ac- knowledge my indebtedness. In my own efforts, I have been materially assisted by the officers of the West Point Foundry Association, who liber- ally furnished me wdth the means of making such experi- ments as were considered necessary, and to the Engineer of that establishment, Mr. B. H. Bartol, I am indebted for much valuable information and assistance. To the Young Engineer I would say, cultivate a knowl- edge of physical laws, without which, eminence in his pro- fession can never be securely attained ; and if this volume should assist him in the attainment of so desirable a result, the object of the author will be fully accomplished. We have seen a proof copy of Has well's Engi- neers' AND Mechanics' Pocket Book, and approve of its design and the subjects treated of: a work of this description has long been wanted, and we confidently express a conviction of its usefulness and appHcation, which in extent exceeds that of any work of its class with which we are acquainted. GOUVERNEUR KeMBLE, William Kemble, Robe'rt p. Parrott, B. H. Bartol, James P. Allaire, B. R. M'Ilvaine, Horatio Allen, C. E. Detmold, Charles W. Hackley, Hogg & Delamater, Stillman & Co., T. F. Secor & Co., Brown & Bell, Smith & Dimon, Merrick & Towne, I We5^ Point Foundry I Association, N. Y. I Allaire Works, N. Y, > Civil Engineers. { Professor of Mathematics, \ Columbia College, N. Y. Phxnix Foundry, N. Y. Novelty Works, N. Y. S Steam- Engine Manufac- turers, N. Y. \ Shipbuilders, N. Y. SSouthwark Foundry f Philadelphia. CONTENTS. Notation 11 Explanations OF Characters. • . 12 U. S. WEIGHTS AND MEASURES. Measures of Length 13 Measures of Surface 14 Measures of Capacity 14 Measures of Solidity 15 Measures of Weight 15 Miscellaneous 16 Measures of Value 16 Mint Value of Foreign Coins 16 FOREIGN WEIGHTS AND MEASURES. Measures of Length 17 Measures of Surface 18 Measures of Capacity 18 Measures of Solidity 19 Measures of Weight 20 Scripture and Ancient Meas- ures 21 Table for finding the Distances of ODjects at Sea 21 Reduction of Longitude 22 Vulgar Fractions 23 Decimal Fractions 25 Diwdecimals 30 Rule of Three 31 Compound Proportion 32 Involution 32 Evolution 32 Arithmetical Progression 34 Geometrical Progression 35 Permutation 36 Combination 36 Position 36 Fellowship 37 Alligation 38 Compound Interest 38 Discount 39 Equation of Payments 39 Annuities • • • • 40 Perpetuities 41 Chronological Problems 41 To find the Moon's Age 42 Table of Epacts, Dominical Letters, &c 42 Promiscuous Questions 43 GEOMETRY. Definitions 46 CONIC SECTIONS. p^gg Construction of Figures 48 Definitions 54 To construct a Parabola 54 To construct a Hyperbola 54 Ellipse 55 Parabolas 56 Hyperbolas 57 MENSURATION OF SURFACES. Triangles, Trapeziums, and Trape- zoids 59 Regular Polygons 60 Regular Bodies, Irregular Figures, and Circles 61 Arcs of a Circle 62 Sectors, Segments, and Zones 63 Ungulas and Ellipses 64 Parabolas and Hyperbolas 65 Cylindrical Rings and Cycloids 66 Cylinders, Cones, Pyramids, Spheres, and Circular Spindles 67 By Mathematical Formulce. Lines of Circle, Ellipse, and Para- bola 67 Areas of Quadrilaterals, Circle, Cyl- inder, Spherical Zone or Segment, Circular Spindle, Spherical Tri- angle, or any Surface of Revolu- tion 68 Capillary Tube 69 Useful Factors 69 Examples in Mensuration 70 Areas of the Segments of a Circle 72 Lengths of Circular Arcs 75 To find the Length of an Elliptic Curve 77 Lengths op Semi-elliptic Arcs . . 78 Areas of THE Zones OF A Circle. 80 MENSURATION OF SOLIDS. Of Cubes and Parallelopipedons .... 81 " Regular Bodies 81 " Cylinders, Prisms, and Ungulas 81 " Cones and Pyramids 82 *' Wedges and Prismoids 83 " Spheres 83 " Spheroids 84 " Circular Spindles 85 " Elliptic Spindles 85 " Parabolic Conoids and Spindles . 86 CONTENTS. IX Page Of Hyperholoids and Hyperbolic Co- noids 86 " Cylindrical Rings 87 By Mathematical FormulcB 87 Cask Gauging- 88 Examples in Mensuration 89 Areas OF Circles 91 Circumferences OF Circles 95 Squares, Cubes, AND Roots 99 To find the Square of a J^umber above 1000 116 To find the Cube or Square Root of a higher JsTumber than is contain- ed in the Table 118 To find the Cube of a JSTumber above 1000 118 To find the Sixth Root of a JVumber 118 To find the Cube or Square Root of a JSTumber consisting of Integers and Decimals 119 Sides of Equal Squares 120 Plane Trigonometry 123 Oblique-angled Triangles 124 Natural Sines, Cosines, and Tan- gents 125 Sines and Secants 127 MECHANICAL POWERS. Lever 128 Wheel and Axle 130 Inclined Plane 131 Wedge 133 Screw 133 Pulley 135 CENTRES OF GRAVITY. Surfaces 137 Solids 138 Gravitation 139 Promiscuous Examples 142 Gravities of Bodies . . .' 143 Specific Gravities 143 Proof of Spirituous Liquors 144 Table of Specific Gravities 145 STRENGTH OF MATERIALS. ■Cohesion 148 Transverse Strength 149 Deflexion 156 Journals of Shafts 158 Gudgeons and Shafts 160 Teeth of Wheels 161 Velocity of Wheels 162 Strength of Wheels 163 General Explanations concerning Wheels 164 Horse Power 165 ANIMAL STRENGTH. Men 165 Horses 166 HYDROSTATICS. Pagft Of Pressure 167 Construction of Banks 168 Flood Gates 168 Pipes 168 Hydrostatic Press 169 HYDRAULICS AND HYDRODYNAMICS. Of Sluices 170 Of Vertical Jipertures or Slits 170 Of Streams and Jets 171 Velocity of Streams 171 To find the Velocity of Water run- 7iing through Pipes 1 72 Waves 172 Table showing the Head necessary to overcome the Friction of Water in Horizontal Pipes 173 General Rules 174 Table of the Rise of Watar in Rivers 175 Water Wheels 176 To find the Power of a Stream 176 Barker's Mill 178 To find the Centre of Gyration of a Water Wheel 179 JVotes 179 PNEUMATICS. Weighty Elasticity, and Rarity of Air 180 Measurement of Heights by means of the Barometer 181 Velocity and Force of Wind 181 STATICS. Pressure of Earth against Walls . . 182 Dynamics 183 Pendulums 185 Centre of Gyration 186 Centres of Percussion and Os- cillation 188 Central Forces 190 Fly Wheels 192 Governors 192 Gunnery 193 Friction 194 HEAT. Communication of Caloric 196 Radiation of Caloric 197 Specific Caloric 197 Evaporation 200 Congelation and Liquefaction 200 Distillation 200 Miscellaneous 201 Melting Points of Alloys 202 Gunpowder 203 Dimensions of Powder Barrels .... 203 Light 204 Tonnage 204 CONTENTS. Page Piling op Balls and Shells 206 Weights and Dimensions of Balls and Shells 207 Winding Engines 208 Fraudulent Balances 209 Measuring of Timber 210 Steam 211 Steam acting- Expansively 214 STEAM-ENGINE. Condensing' Engines 216 Boilfirs 217 JVon- condensing Engines 218 Boilers 218 General Rules 219 Saturation in Marine Boilers 219 Sjnoke Pipes, or Chimneys 220 Belts 220 Power of Engines 221 To find the Volume the Steam, of a Cubic Foot of Water occupies 222 To find the Power of an Engine ne- cessary to raise Water 222 To find the Velocity necessary to Discharge a Given Quantity of Water 223 COMBUSTION. Fuel 224 .Analysis of Fuels 225 .Anthracite 226 Charcoal 226 Coke 226 Water 227 Motion of Bodies in Fluids 229 Air 232 Dimensions and Weight of Guns, Shot, and Shells, U. S. Army. 233 Weia-ht and Dimensions of Leaden Bills 234 Expansion of Shot 234 Weight and Dimensions of Shot 234 Dimensions and Weight of Guns, Shot, and Shells, U. S. Navy. . 236 Penetration of Shot and Shells 237 Penetration of Shells 238 MISCELLANEOUS. Recapitulation of Weights of Vari- ous Substances 239 Weights of a Cubic Foot of Various Substances 239 Slating 239 Capacity of Cisterns 240 Page Compositions 240 Sizes ofJ^uts 240 Screws 241 Strength of Copper 241 Digging 241 Coal Gas 241 Mcohol 242 Composition Sheathing J^ails 242 Cement 242 Brown Mortar 242 Bricks, Laths, &c. 243 Hai/ 243 Hills in an ^cre of Ground 243 Displacement of Vessels 243 Weight of Lead Pipe 244 TiJ...-- 244 Relative Prices of American Wrought Iron 345 Power required to Punch Iron and Copperplates 245 IRON. Weight of Square Rolled- Iron 246 Weight of Round Rolled Iron 247 Weight of Flat Rolled Iron 248 Weight of a Square Foot of Cast and Wrought Iron, Coppery and Lead....:. 2,51 Cast Iron 251 Weight of Cast Iron Pipes 252 Weight of a Square Foot 254 Weight of Cast Iron and Lead Balls 255 Weight of Copper Rods or Bolts • . • • 256 Weight of Riveted Copper Pipes .... 256 Copper 257 Lead 257 Brass 257 Cables and Anchors 258 Cables 259 ' Tables of Hemp Cables and Ropes . . 259 ' To ascertain the Strength of Cables. 259 To ascertain the Weight of Manilla Ropes and Hawsers • • 259 To ascertain the Strength of Ropes. 260 To ascertain the Weight of Cable-laid Ropes 260 To ascertain the Weight of Tarred Ropes and Cables 260 Blowing Engines 261: Dimensions of a Furnace, Engine, &c 2611 MISCELLANEOUS NOTES. On Materials 262 Solders, Cements, and Paints 263 Paints, Lackers, and Sta,ining 26'* NOTATION. 1 = 2 = 3 = 4 = 5 = 1- S-. 9 = 10: 20: 30: 40: 50: 60: 70: 80: 90 100 1 1. rll. = 111. = IV. zV. = VI. = VIL = VIII. zIX. = X. = XX. = XXX. = XL. = L. = LX. = LXX. =:LXXX. = xc. 500 = D, or 10. 1,000 = M, or CIO. MM. i As often as a character is repeated, \ so many times is its value repeated. ( A less character before a greater } diminishes its value, as IV = I from ( V, or 1 subtracted from 5 = 4. c A less character after a greater in- \ creases its value, as XI = X+I, or ho+i = ii. c For every annexed, this be- \ comes 10 times as many. I For every C and 0, placed one at \ each end, it becomes 10 times as ( many. 2,000 __ 5,000=:V, or 100. 6,000 = VI. 10,000 =% or CCIOO. 50,000 ^T, or 1000. 60,000 :==LX. 100,000 = 0, or CCCIOOO. 1,000,000 =M>r CCCCIOOOO. 2,000,000 = MM. Examples.— 1840, MDCCCXL. 18560, XVIIIDLX. A bar, thus — , over any > number, increases it 1000 times. 12 EXPLANATION OF CHARACTERS. EXPLANATIONS OF THE CHARACTERS USED IN THE FOLLOWING TABLES AND CALCULATIONS. = Equal to, as, 12 inches = 1 foot, or 8x8 = 16x4. + Plus, or more, signifies addition ; as, 4+6+5 = 15. — Minus, or less, signifies subtraction ; as, 15—5 — 10. X Multiplied hy, or into, signifies multiplication ; as, 8x9 r=: 72 ~ Divided hy, signifies division ; as, 72-:-9 = 8. •'• so is I Proportion ; as, 2 : 4 : : 8 : 16 ; that is, as 2 is to 4 so is 8 :'-to S ^o\Q. V Prefixed to any number signifies that the square root of that number is required ; as, ^16=:4; that is, 4x4=r 16. ^ Signifies that the cube root of that number is required • as ^64 1=4; that is, 4X4X4 = 64. 2 added to a number signifies that that number is to be squared • thus, 42 means that 4 is to be multiplied by 4. 3 added to a number signifies that that number is to be cuhed • thus, 43 IS = 4 X 4 X 4 == 64. The power or number of times a number is to be multiplied by itself is shown by the num- ber added ; as, 2 3 4 5^ &c. ~~ The har si gnifies that the numbers are to be taken together ; as, 8— 2+6 r= 12, or 3x5+3 = 24. . Decimal point, signifies when prefixed to a number, that that nunaber has a unit (1) for its denominator ; as, .1 is-i, .155 . ^^ ToVo ' ^^• CO Signifies difference, and is placed between two quantities when it is not evident which of them is the greater. ° Degrees, ' minutes, " seconds, '" thirds. < Signifies angle. ^~^ ^~-^' ^— -J <^c., denote inverse powers of a, and are equal a^ a^ a^ 7 Is put between two quantities to express that the former is greater than the latter ; as, alb, reads a greater than b. L Signifies the reverse ; as, a Z, b, reads a less than b. ( ) Parentheses are used to show that all the figures within them are to be operated upon as if they were only one : thus (3+2)x5 = 25. p is used to express the ratio of the circumference of a circle to its diameter = 3.1415926, &c. A A' a:' K!" signifies A, A prime, A second, A third, &c. dXd, a.d, or ad, signifies that a is to be multiplied by d. To ascertain the value of a decimal of a unit, see Reduction of Decimals, page 28. Note. The degrees of temperature used in this work are those of Fahrenheit. WEIGHTS AND MEASURES. 13 UNITED STATES' WEIGHTS AND MEASURES. 12 inches — 1 foot. 3 feet = 1 yard. 5i yards = 1 rod. 40 rods = 1 furlong 8 furlongs = 1 mile. Measures of Length. 36= 3. 198= 16^= 5i 7920= 660 = 220 = 40. 63360 = 5280 =1760 =320 = 8. The inch is sometimes divided into 3 larley corns, or 12 lines. A hair's breadth is the 48th of an inch. Gunter^s Chain. 7.92 inches =: 1 link. 100 links = 4 rods, or 22 yards. Ropes and Cables. 6 feet = 1 fathom. 120 fathoms = 1 cable's length. Geographical and Nautical Measure. 1 degree of a great circle of the earth =: 69.77 Statute miles. 1 mile = 2046.5 yards. Log Lines. 1 knot = 51.1625 feet, or 51 feet lj+ inches. 1 fathom = 5.11625 feet, or 5 feet 1^+ inches. Estimating a mile at 61391 feet, and using a 30" glass. If a 28" glass is used, and eight divisions, then 1 knot = 47 feet 9 + inches. 1 fathom =z 5 feet llf inches. The line should be about 150 fathoms long, having 10 fathoms between the chip and first knot for stray line. 'Note.— Bowditch gives Q\'20 feet in a sea miUy whichy if taken as the lengthy will make the divisions 51 feet and 5 I-IO feet. Cloth. 1 nail = 2i inches = ^^th of a yard. 1 quarter = 4 nails. 5 quarters = 1 ell English Pendulums. 6 points =: 1 line. 12 lines =: 1 inch. Shoemakers\ No. 1 is 4i inches in length, and erery succeeding number is I of an inch. ^ There are 28 divisions, in tw^o series of numbers, viz., from 1 to "13, and 1 to 15. B 14 WEIGHTS AND MEASURES. Circles, 60 seconds = 1 minute. 60 minutes = I degree. 360 degrees = 1 circle. 1 day is ... . 1 minute is . 3600 1296000 : : 21600. .002739 of a year. .000694 of a day. Miscellaneous, 1 palm = 3 inches. 1 hand = 4 inches. 1 span : 1 metre : : 9 inches. : 3.28174 feet. The standard of measure is a brass rod, which, at the tempera- ture of 32° Fahrenheit, is the standard yard. The standard yard of the State of New-York bears, to a pendulum vibrating seconds in vacuo, at Columbia College, the relation of 1.000000 to 1.086141 at a temperature of 32° Fahrenheit. 1 yard is 000568 of a mile. 1 inch is 0000158 of a mile. Measures of Surface, 144 square inches = 1 square foot. I 9 square feet =z \ square yard. | Inches. 1296 Rods. Land. 30J square yards r=: 1 square rod. 40 square rods = 1 square rood, 4 square roods ) _ ^ ^^^^ 10 square chains 5 640 acres = 1 square mile, E.— 208.710321 /ee^, 69.5701 yards, or 220 by 198 /eei square = 1 acre. Paper, 24 sheets = 1 quire. I sheets. 20 quires = 1 ream. | 480. Roods. Yards. 1210. 4840 = 160. 3097600 = 102400 = 2560. Note.— 2 Cap Demy . Medium Royal . Super-royal Imperial Elephant 13 X16 inches 19ixi5i- " 22 X18 " 24 X19 " 27 X19 " 29 X21i " 27ix22i *' Draiving Paper. Columbier . 33} X 23 inches. Atlas . . 33 X 26 Theorem . 34 x 28 Doub. Elephant, 40 x 26 Antiquarian . 52 x 31 Emperor . 40 x 60 Uncle Sam . 48 x 120 Measures of Capacity, Liquid. 4 gills = 1 pint. 2 pints =z 1 quart. 4 quarts = 1 gallon. The standard gallon measures 231 cubic inches, and contains Pints. Gills. 8. 32 = 8 WEIGHTS AND MEASURES. 15 8 3388822 avoirdupois pounds, or 58372.1754 troy grains of distilled water at 39° 83 Fahrenheit ; the barometer at 30 inches. The gallon of the State of New-York contains 221.184 cubic inch- es, or 8 pounds of pure water at its maximum density. The Imperial gallon (British) contains 277.274 cubic inches. Pints. Quarts. Gallons. 8. 16 = 8. 64 — 32 = 8. which contains 2150.42 Dry. 2 pints = 1 quart. 4 quarts = 1 gallon. 2 gallons = 1 peck. 4 pecks = 1 bushel. The standard bushel is the Wincheste\ cubic inches, or 77.627413 lbs. avoirdupois of distilled water at its maximum density. Its dimensions are 18i inches diameter inside, 19^ inches out- side, and 8 inches deep ; and when heaped, the cone must not be less than 6 inches high. The bushel of the State of New- York contains 80 lbs. of pure water at its maximum density, or 2211.84 cubic inches. Measures of Solidity. 1728 cubic inches r=z l foot. 27 cubic feet =. 1 yard. Inches. 46656. Miscellaneous, 1 chaldron = 36 bushels, or . 57.25 cubic feet. Dry gallon of New- York . . 276.48 cubic inches. 1 perch of stone . . . . 24.75 cubic feet. Measures of Weight Avoirdupois. 16 drachms = 1 ounce. 16 ounces •=: 1 pound. 112 pounds r= 1 cwt. 20 cwt. = 1 ton. 1 lb. = 14 oz. 11 dwt. 16 gr. troy. The standard avoirdupois pound is the weight of 21^7015 cubic inches of distilled water weighed in air, at the tempdfature of the maximum density (3^^.83), the barometer being at 30 inches. Troy. Ounces. Pounds. Drachms. 256. 28672 = 1792. 573440 = 35840 = 2240. 24 grains = 1 dwt 20 dwt. = 1 ounce. 12 ounces = 1 pound. ^ Apothecaries. 20 grains ' = 1 scruple. 3 scruples = 1 drachm. 8 drachms = 1 ounce. 12 ounces = 1 p?^nd. Grains. 480. 5760 :240. Grains. Scruples. Drachms. 60. 480 = 24. 5760 — 288 = 96. 16 WEIGHTS AND MEASURES. Diamond. 16 parts — 1 grain = 0.8 troy grains. 4 grains = 1 carat = 3.2 " 7000 troy grains = 1 lb. avoirdupois. 175 troy pounds = 144 lbs. " 175 troy ounces = 192 oz. " 437i troy grains = 1 oz. " 1 troy pound = .8228+ lb. " Miscellaneous, 1 cubic foot of anthracite coal from 50 to 55 lbs. 1 cubic foot of bituminous coal from 42 to 55 lbs. 1 cubic foot Cumberland coal = 53 lbs. 1 cubic foot charcoal . — 18.5 '' (hard wood). 1 cubic foot charcoal . = 18. " (pine wood). 1 bushel bituminous coal — 80 " 1 stone . . . . = 14 " Coals are usually purchased at the conventional rate of 28 bush- els (5 pecks) to a ton. Measures of Value. 1 eagle = 258 troy grains. 1 dollar = 412.5 " 1 cent — 168 " The standard of gold and silver is 900 parts of pure metal, and 100 of alloy, in 1000 parts of coin. Relative Mint Value of Foreign Gold Coins, By Laiv of Congress^ August, 1834. Brazil. 1 Johannes 1 Dobraon . 1 Dobra 1 Moidore . 'k Crusado . England. ^Guinea . ^Sovereign France. 1 Double Louis (1786) 1 Double Napoleon . Colombia. 1 Doubloon Mexico. 1 Doubloon Portugal. 1 Dobraon . 1 Dobra 1 Johannes 1 Moidore . 1 Milrea . Spain. 1 Doubloon (1772) 1 Doubloon (1801) 1 Pistole . Weight. Dwt. Gr. 18 34 12 18 06 6 22 16i 5 9i 5 3^ 10 11 8 7 17 8^ 17 8i 34 12 18 6 18 6 22 19^ 17 8i 17 9 4 3i Value. $17,068 32.714 17.305 6.560 .638 5.116 4.875 9.694 7.713 15.538 15.538 32.714 17.305 17.068 6.560 .780 16.030 * 15.538 3.883 23.2 grains of pure gold = $1.00. United States Eagle preceding 1834,' $10,668. FOREIGN WEIGHTS AND MEASURES. 17 Mint Value of Foreign Coins. England. France. Austria. Prussia. Russia. Sweden. 1 Shilling . 5 Francs 1 Sous .... 1 Crown, or rix dollar . 1 Ducat 1 Ducat 1 Ducat = 10 roubles . 1 Rouble 1 Ducat 1 Rix dollar . $0,244 0.935 0.0093 0.97 2.22 2.202 7.724 0.743 2.19 1.08 The relative value of gold and silver is as 1 to 15 Ji. r.A y/ Measures, of Length. ^ Yardis^the length of a pendulum vibrating seconds in vacuo in Lon donfaf the level of the sea ; measured on a brass rod, at the tem perature of 62° Fahrenheit, =39.1393 Imperial inches. # French. Old System.— 1 Line =: 1 1 1 1 1 1 JVew System. — 1 1 1 1 1 1 Austrian Prussian Swedish Spanish 12 points . Inch = 12 lines . Foot = 12 inches . Toise = 6 feet . League =: 2280 toises . League = 2000 toises . Fathom =: 5 feet. Millimetre Centimetre Decimetre . Metre .... Decametre . Hecatometre Foot .... Foot . . . Foot .... Foot .... League (common) 0.08884 U. S. inches. = 1.06604 = 12.7925 = 76.7550 " (common), (post). = .03938 U. S. inches. = .39380 = 3.93809 = 39.38091 = 393.80917 = 3938.09171 " = 12.448 " = 12.361 = 11.690 = 11.034 = 3.448 U. S. miles. Table showing the relative length of Foreign Measures compared with British. Plcuies. Measures. Inches. Places. . Measures. Inches. Amsterdam . Foot 11.14 Malta . . . Foot 11.17 Antwerp . . " 11.24 Moscow " 13.17 Bavaria " 11.42 Naples . Palmo 10.38 Berlin . u 12.19 Prussia Foot 12.35 Bremen . " 11.38 Persia . Arish 38.27 Brussels u 11.45 Rhineland Foot . 12.35 China . " Mathematic. " Builder's " Tradesman's 13.12 12.71 13.32 Riga . . Rome . Russia . 10.79 11.60 13.75 " " Surveyor's 12.58 Sardinia Palmo 9.78 Copenhage a . " 12.35 Sicily . 9.53 Dresden ' " 11.14 Spaiij . Foot 11.12 England " 12.00 " Toesas 66.72 Florence Braccio 21.60 " Palmo 8.34 France . Pi^d de Roi Metre 12.79 39.381 Strasburgh Sweden Foot 11.39 11.69 Geneva . Foot 19.20 Turin . «' 12.72 Genoa . Palmo 9.72 Venice . (( 13.40 Hamburgh Foot 11.29 Vienna . u 12.45 Hanover (( 11.45 Zurich . it 11.81 Leipsic . " 11.11 Utrecht " 10.74 Lisbon . " 12.96 Warsaw *' 14.03 " , Palmo 8.64 18 FOREIGN WEIGHTS AND MEASURES. Table showing the relative length of Foreign Road Measures com- pared with British. Places. Measures. Yards. Placet. Measures. Tarda. Arabia . Mile 2148 Hungary . . Mile 9113 Bohemia " 10137 Ireland . . . " 3038 China . Li 629 Netherlands . " 1093 Denmark Mile 8244 Persia . . . Parasang 6086 England . " Statute 1760 Poland . . . Mile, long 8101 " GeogTaph. 2025 Portugal . . League 6760 Flanders " 6869 Prussia . . . Mile 8468 France . League, marine 6075 Rome . . . " 2025 " common 4861 Russia . . . Verst 1167 ii " post 4264 Scotland . . Mile 1984 .Germany Mile, long 10126 Spain . . . League, com. 7416 11700 9153 Hamj3,ttrgh Hanover a * 8244 1T559 Sweden . . •Switzerland . Mile Holland . " 6395 Turkey . . . Berri 1826 Measures of Surface, French. Old System.— 1 Square Inch 1 Arpent (Paris) . 1 Arpent (woodland) JSTew System. — 1 Jire . 1 Decare 1 Hecatare . 1 Square Metre . 1 Are . = 1.1364 U. S. inches. = 900 square toises. = 100 square royal perches. = 100 square metres. = 10 ares. = 100 ares. = 1550.85 square inches, or 10.7698 square feet. = 1076.98 Table showing the relation of Foreigv. 3feasures of Surface compa- red with British. Amsterdam Berlin. Canary Isles England . Geneva . Hamburgh Hanover Ireland . Naples . Sq. yards. Acre Moggia Places. Measures. Sq. yards. Portugal . . Geira 6970 Prussia . . . Morgen 3053 Rome .... Pezza 3158 Russia . . . Dessetina 13066.6 Scotland . . Acre 6150 Spain . . . Fanegada 5500 Sweden . . Tunneland 5900 Switzerland . Faux 7855 Vienna . . . Joch 6889 Zurich . . . Common acre 3875.6 Measures of Capacity. British. The Imperial gallon measures 277.274 cubic inches, containing 10 lbs. avoirdupois of distilled water, weighed in air, at the temperature ot 62^, the barometer at 30 inches. For Grain. 8 bushels = 1 quarter. 1 quarter = 10.2694 cubic feet. CkfaL or heaped measure. 3 bushels = 1 sack. 12 sacks = 1 chaldron. Imperial bushel = 2218.192 cubic inches. Heaped bushel, 19^ ins. diam., cone 6 ms. high = 2815.4872 cub. ins. 1 chaldron = 58.658 cubic feet, and weighs 3136 pounds. 1 chaldron (Newcastle) = 5936 pounds. tt c , u ;«« French. J^ew System.-l Litre = 1 cub. decimetre, or 61.074 U. S. cub- ms. Old System. - 1 Boisseau = 13 litres = 793.964 cub. ms., or 3.43 galls. 1 Pinte = 0.931 litres, or 56.817 cubic inches. Spanish. 1 Wine Arroba = 4.2455 gallons. 1 Fanega (common measure) = 1.593 bushels. FOREIGN WEIGHTS AND MEASURES. 19 Table showing the relative Capacity of Foreign Liquid Measures compared with British. Places. Measnres. Cub. Inch. Places. Measures. Cub.Inch. Amsterdam Anker 2331 Naples . . . Wine Barille 2544 " . . Stoop 146 " Oil Stajo 1133 Antwerp . . " 194 Oporto Almude 1555 Bordeaux . . Barrique 14033 Rome Wine Barille 2560 Bremen . . . Stubgens 194.5 " Oil 2240 Canaries . . Arrobas 949 u Boccali 80 Constantino pie Almud 319 Russia Weddras 752 Copenhagei I . Anker 2355 Kunkas 94 Florence . Oil Barille 1946 Scotland Pint 103.5 . Wine " 2427 Sicily Oil Caffiri 662 France . . Litre 61.07 Spain Azumbres 22.5 Geneva . . Setier 2760 Quartillos 30.5 Genoa . . Wine Barille 4530 Sweden Eimer 4794 " . Pinte 90.5 Trieste Orne 4007 Hamburgh . Stubgen 221 Tripoli Mattari 1376 Hanover " 231 Tunis Oil " 1157 Hungary Ehner 4474 Venice Secchio 628 Leghorn . . Oil Barille 1942 Vienna Eimer 3452 Lisbon . Almude 1040 '* Maas 86.33 Malta . . . Caffiri 1270 Table showing the relative Capacity of Foreign Dry Measures com- pared with British. Places. Measures. Cub.Inch. Places. Measures. Cub.Inch. Alexandria Rebele 9587 Malta . . . Salme 16930 " . . Kislos 10418 Marseilles Charge 9411 Algiers . . . Tarrie 1219 Milan . Moggi 8444 Amsterdam . Mudde 6596 Naples . Tomoli 3122 " . . Sack 4947 Oporto . Alquiere 1051 Antwerp . . Viertel 4705 Persia . Artaba 4013 Azores . . . Alquiere 731 Poland . Zorzec 3120 Berlin . . . Scheffel 3180 Riga . . Loop 3978 Bremen . . . " 4339 Rome . Rubbio 16904 Candia . . . Charge 9288 " auarti 4226 Constantinople Kislos 2023 Rotterdam Sach 6361 Copenhagen . Toende 8489 Russia . Chetwert 12448 Corsica . . . Stajo 6014 Sardinia Starelli 2988 Florence . . Stari 1449 Scotland Firlot 2197 Geneva . . . Coupes 4739 Sicily . Salme gros 21014 Genoa . . . Mina 7382 " " generale 16886 Greece . . . Medimni 2390 Smyrna. Kislos 2141 Hamburgh . . Scheffel 6426 Spain . Catrize 41269 Hanover . . Malter 6868 Sweden . Tunnar 8940 Leghorn . . . Stajo 1501 Trieste . Stari 4521 " ... Sacco 4503 Tripoli . Caffiri 19780 Lisbon . . . Alquiere 817 Tunis . u 21855 " ... Fanega 3268 Venice . Stajo 4945 Madeira . . . Alquiere 684 Vienna . Metzen 3753 Malaga . . . Fanega 3783 Measures of Solidity. French. 1 Cubic Foot = 2093.470 U. S. inches. Decistre = 3.5375 cubic feet. Steve (a cubic metre) . . . . = 35.375 " Decastere = 353.75 " 1 Stere = 61074.664 cubic inches. For the Square and Cubic Measures of other countries, take the length of the measure in table, page 17, and square or cube it as required. 20 FOREIGN WEIGHTS AND BIEASURES. British. French, Measures of Weight, 1 troy Grain = .003961 cubic inches of distilled water. 1 trov Pound = 22.815689 cubic inches of water. Old System.— 1 Grain 1 Gros 1 Once 1 Livre jVew System. — Milligramme Centigramme Decigramme Gramme Decagramme Hecatogramme Spanish . Swedish Austrian . Prussian . 0.8188 grains troy. = 58.9548 = 1.0780 oz. avoirdupois. = 1.0780 lbs. = .01543 troy grains. = .15433 = 1.54331 = 15.43315 = 154.33159 == 1543.3159 1 Millier = 1000 Kilogrammes = 1 ton sea weight. 1 Kilogramme . = 2.204737 lbs. avoirdupois. 1 Pound avoirdupois — 0.4535685 Kilogramme. 1 Pound troy . = 0.3732223 " 1 " . = 1.0152 lbs. avoirdupois. 1 " . =0.9376 1 " . =1.2351 1 » . =1.0333 Note.— /n the new French system, the values of the base of each measure, viz^ Metre, Litre, Stere, Arc, and Gramme, are decreased or increased by the following words prefixed to them. Thus, Milli expresses the 1000th part. Centi " 100th " Deci " 10th " Deca '* 10 times the value. Hecato expresses 100 times the value. Chilio " 1000 Myrio " 10000 Table showing the relative value oi Foreign Weights compared with British. Number Nnmber equal to equal to Places. Weights. lOU avoir- dupois pounds. Plaxxi. ._[ Weights. lOO avoir- dupois pounds. Aleppo . . . Rottoli 20.46 Hanover . . Pound 93.20 Oke 35.80 Japan . . Catty 76.92 Alexandria Rottoli 107. Leghorn . Pound 133.56 Algiers . . . Amsterdam . 84. Leipsic . . "■ (common) 97.14 Pound 91.8 Lyons . " (silk) 98.81 Antwerp . . Barcelona . . " 96.75 Madeira " 143.20 u 112.6 Mocha . Maund 33.33 Batavia . . . Catty 76.78 Morea . Pound 90.79 Bengal . . . Berlin . . . Seer 53.57 Naples . Rottoli 50.91 Pound 96.8 Rome . Pound 133.69 Bologna . . . Bremen . . . 125.3 Rotterdam " 91.80 u 90.93 Russia . " 110.86 Brunswick . . " 97.14 Sicily . " 142.85 Cairo .... Rottoli 105. Smyrna Oke 36.51 Candia . . . 85.9 Sumatra Catty 35.56 China . . . Catty 75.45 Sweden Pound 106.67 Constantinople Oke 35.55 " " 120.68 Copenhagen . Corsica . . . Pound 90.80 Tangiers " (miner's) 94.27 131.72 Tripoli . Rottoli 89.28 Cyprus . . . Damascus . . Rottoli 19.07 Tunis . " 90.09 25.28 Venice . Pound (heavy) 94.74 Florence . . Pound 133.56 " " (light) 150. Geneva . . . " (heavy) 82.35 Vienna . (i 81. Genoa . . . :; :; 92.86 Warsaw 4< 112.25 Hamburgh . . 93.63 SCRIPTURE AND ANCIEJMT MEASURES. 21 Scripture Long Measures. A digit . A palm . A span Feet. Inches. . =0 0.912 . =0 3.648 . =0 10.944 A cubit . • A fathom . Feet. Inches. . =1 9.888 . =7 3.552 Grecian Long Measures. A digit . A pous (foot) A cubit . Feet. Inches. . =0 0.7554 . =1 0.0875 . =1 1.5984 A stadium A mile Feet. Inches. . = 604 4.5 . =4835 A Greek or Olympic foot = 12.108 inches. A Pythic or natural foot = 9.768 " Jewish Long Measures. A cubit A Sabbath day's journey . Feet. , = 1.824 . =3648. Feet. A mile . . . = 7296 A day's journey . = 175104 (or 33 miles 864 feet). Roman Long Measures. A digit . An uncia (inch) A pes (foot) Feet. Inches. = .72575 = .967 = 11.604 Feet. Inches. A cubit . . = 1 5.406 A passus . = 4 10.02 A mile . . = 4835 Miscellaneous. Arabian foot . Babylonian foot Feet. . = 1.095 . = 1.140 Feet. Hebrew foot . . = 1.212 " cubit . . — 1.817 Egyptian . . = 1.421 sacred cubit = 2.002 Note. — The above dimensions are British. Table for finding the Distance of Objects at Sea, in Statute Miles. Height in feet. Distance in miles. Height in feet. Distance in miles. Height in feet. Distance in miles. Height in feet. Distance in miles. *.582 1. 11 4.39 30 7.25 200. 18.72 1 1.31 12 4.58 35 7.83 300 22.91 2 1.87 13 4.77 40 8.37 400 26.46 3 2.29 14 4.95 45 8.87 500 29.58 4 2.63 15 5.12 50 9.35 1000 32.41 5 2.96 16 5.29 60 10.25 2000 59.20 6 3.24 17 5.45 70 11.07 3000 72.50 7 3.49 18 5.61 80 11.83 4000 83.7 8 3.73 19 5.77 90 12.55 5000 93.5 9 3.96 20 5.92 100 13.23 1 mile. 96.1 10 4.18 25 6.61 150 16.20 The difference in two levels is as the square of the distance. Thus, if the height is required for 2 miles, P :22 :: 6.99: 27.96 inches; and if for 100 miles, P : lOO^ : : 6.99 : 1.103+ miles. For Geographical miles, the distance for one mile is 7.962 inches. 22 DISTANCES. Example. — If a man at the foretop-gallant-mast-head of a ship, 100 feet from the water, sees another and a large ship (hull to), how far are the ships apart '? A large ship's bulwarks are, say 20 feet from the water. Then, by table, 100 feet . . . . r= 13.23 20 " . . . . = 5.92 Distance . . 19.15 miles. Note. — 1-13 should be added for horizontal refraction. To Reduce Longitude into Time. Multiply the number of degrees, minutes, and seconds by 4, and the product is the time. Example. — Required the time corresponding to 50° dl\ oOo 31' 4 h.3 22' 4:' Ans. If time is to be reduced, then 4)3 22 50 31 Ans. thus, 13x15=66° 48' 15^'. Degrees of longitude are to each other in length, as the cosines of their latitudes. Or, multiplying by 15 ; h. ■m. 4 27 For every 5^ they are as follows Miles. 60. 59.77 Equator 5° 10° 15° 20° 25° 30° 35° 40° 45° 59.09 57.96 56.38 54.38 51.96 49.15 45.96 42.43 50° 55° 60° 65° 70° 75° 80° 85° 90° Miles. 38.57 34.41 30. 25.36 20.52 15.53 10.42 5.23 0.00 VULGAR FRACTIONS. 23 VULGAR FRACTIONS. A Fraction, or broken number, is one or more parts of a Unit. Example. — 12 inches are 1 foot. Here, 1 foot is the unit, and 12 inches its parts ; 3 inches^ therefore, are the one fourth of a foot, for 3 is the quarter or fourth of 12. A Vulgar Fraction is a fraction expressed by two numbers placed one above the other, with a line between them ; as, 50 cents is the ^ of a dollar. The upper number is called the J^umerator^ because it shows the number of parts used. The lower number is called the Denominator, because it denominates, or gives name to the fraction. The Terms of a fraction express both numerator and denominator ; as, 6 and 9 are the terms of ^. A Proper fraction has the numerator equal to, or less than the denominator ; as, An Improper fraction is the reverse of a proper one ; as, ^, &c. A Mixed fraction is a compound of a whole number and a fraction ; as, 5|-, &c. A Compound fraction is the fraction of a fraction ; as, J of ^, &c. A Complex fraction is one that has a fraction for its numerator or denominator, or I 5 — 3i both ; as, "o, or 3^, or #, or -5, &c. t 4 f 6 A Fraction denotes division, and its value is equal to the quotient obtained by di- viding- the numerator by the denominator ; thus, ^^ is equal to 3, and ^-^ is equal to ^. REDUCTION OF VULGAR FRACTIONS. To find the greatest Number that ivill divide Two or more Numbers without a Remainder. Rule.— Divide the greater number by the less; then divide the divisor by the remainder ; and so on, dividing always the last divisor by the last remainder, until nothing remains. Example.-— What is the greatest common measure of 1908 and 936 1 936) 1908 (2 1872 36) 936 (26 72 216 216 So 36 is the greatest common measure. To find the least Common Multiple of Two or more Numbers, Rule. — Divide by any number that will divide two or more of the given numbers without a remainder, and set the quotients with the undivided numbers in a line beneath. Divide the second line as before, and so on, until there are no two nimibers that can be divided ; then the continued product of the divisors and quotients will give the multiple required. Example.— What is the least common multiple of 40, 50, and 25 ? 5) 40 . 50 . 25 5) 8 . 10 . 5 2) 8. 2. 1 4.1.1 Then 5X5X2X4 = 200 Aria, 24 VULGAR FRACTIONS. To reduce Fractions to their lowest Terms, Rule. — Divide the terms by any number that wi.ll divide them without a re- mainder, or by their greatest common measure at once. Example. — Reduce J|4 of ^ foot to its lowest terms. 9Fo il^-lO = -9 f -^ = l%-3 = f , or 9 inches. To reduce a Mixed Fraction to its equivalent^ an Improper Fraction. Note. — Mixed and improper fractions are the same; thus^ 5^= y. For illus- tration^ see folloicing- examples : Rule. — Multiply the whole number by the denominator of the fraction, and to the product add the numerator; then set that sum above the denominator. Example. — Reduce 23j to a fraction. 23x6+2 =140 6 6 Example. — Reduce -^-|- inches to its value in feet. 6 123-i-6 = 20| ; that is, 20 feet and -J or | of a foot. To reduce a Whole Numler to an equivalent Fraction having a given Denominator. Rule.— Multiply the whole number by the given denominator, and set the prod uct over the said denominator. Example.— Reduce 8 to a fraction vrhose denominator shall be 9. 8X9 = 72; then '^-^ the answer. To reduce a Compound Fraction to an equivalent Simple one. Rule.— Multiply all the numerators together for a numerator, and al^the de- nominators together for a denominator. Note.— ^FAen there are terms that are common^ they may he omitted. Example.— Reduce § of | of § to a simple fraction. 13 2 6 1 ^ 2><4><3 = 24 = 4'^"^- 1 3- "Si 1 Or, — X— X^= 7, by cancelling the 2's and 3's. Example.— Reduce J of | of a pound to a simple fraction. sXf = 1 ^ns. To reduce Fractions of different Denominations to equivalent ones having a common Denominator. Rule.— Multiply each numerator by all the denominators except its own for the new numerators ; and multiply all the denominators together for a common de- nominator. Note. — In this, as in all other operations, whole numbers, mixed, or compound fractions, must first be reduced to the form of simple fractions. Example. — Reduce i, §, and ^ to a common denominator. 1X3X4=12^ 2X2X4 = 16 V =lf = If = -J:| 3X2X3 = 18S -* -* - 2X3X4 = 24 The operation may be performed mentally ; thus, Reduce i, ^, f , and 4 ^o a common denominator. ♦ 3 1.12 " 1__1 6_6 1— 2J> 2 — ^- '5—'B' I—I- 2 — 1- VULGAR FRACTIONS. 25 To reduce Complex Fractions to Simple ones «.?rrr7f^eLtbVllf^ then multiply the nu- Example.— Simplify the complex fraction ?f. 2§= f 8X 5L40 5 4f=V 3X24=^ =9 *^''"- ADDITION OF VULGAR FRACTIONS. to^yw^nVf^K^ fractions have a common denominator, add all the numerators together, and then place the sum over the denominators. "uiiieraiors reducld'^Zf'' ^mT'''^ /rflc^207i5 have not a common denominator, they must he reduced to one. Also, compound and complex must be reduced to simple fractimis Example.— Add ^ and | together. |-+| = |=1 Ans. Example.— Add ^ of ^ of ^^ to 2| of ^. 2^off =yx|=|i. Then, ^8+11^1X31^^^^ SUBTRACTION OF VULGAR FRACTIONS. th?n''«ni;^fjtT'® ^^® fractions the same as for other operations, when necessarv- SmZ^dro^aTor"""^'"'"" *^^ ^^^^^' ^-^ -^ '^^ remainderovT^^e Example.— What is the difference between | and | ? Example.— Subtract | from J. 6X9 = 54) 3X8=:24>=:54_24_30 - ^ MULTIPLICATION OF VULGAR FRACTIONS. tor^''to£e'iherfoT«?.L^^^ previously required ; multiply all the numera- denoSor. numerator, and all the denominators together for a new Example.— What is the product of J and f 7 3w3 9 1 /,„„ T^Q — 3^— 4 Ans. Example.— What is the product of 6 and § of 5 ? i Axf of 5 = ^X y = V = 20 Ans, c 26 APPLICATION OF KEDrCTION OF VULGAR FRACTIONS. DIVISION OF VULGAR FRACTIONS. Rule.— Prepare the fractions as before ; then dMde the numerator by the nu- merator, and the denominator by the denominator, if they will exactly divide ; but if not, invert the terms of the divisor, and multiply the dividend by it, as in multi- plication. Example.— Divide ^-^ by |. To find the Value of a Fraction in Parts of the whole Number. Rule.— Multiply the whole number bv the numerator, and divide by the denom- inator ; then, if anything remains, multiply it by the parts in the next inferior de- nomination, and divide bv the denominator, as before, and so on as far as necessa- ry ; so shall the quotients placed in order be the value of the fraction required. Example.— What is the value of ^ of § of $9 1 ^of| = |X-^= ^-i=^^Jins. Example.— Reduce | of a pound to avoirdupois ounces. 3 1 4) 3(0 lbs. 16 ounces in a lb. 4)l8_ 12 ounces, Ans. Example.— Reduce ^^ of a day to hours. r'o X^ = ft = 7^ hours, .3/t.. To reduce a Fraction from one Denomination to another. Rule —Multiply the number of parts in the next less denominator by the nu- merator if the reduction is to be to a less name, but multiply by the denominator if to a greater. Example.— Reduce ^ of a dollar to the fraction of a penny. 1 100 100 ot;^i Iv -V- = — r- = ^-T, the answer. 4 "^ 1 4 1 ' Example.— Reduce | of an avoirdupois pound to the fraction of an ounce. ^-X y^ = ^^ = f , the answer. Example. — Reduce ? of a cwt. to the fraction of a lb. 2y^±y^23 ^ 224 __ 3_2^ ^^le answer. Example.— Reduce § of | of a mile to the fraction of a foot. ^ of ^ - -«-X ^^^ - ^^-^^ = ^^^, the answer. 3"*4~12'^l ■" 12 1' Example.— Reduce ^ of a square foot to the fraction of an inch. 1^ 144 _ 144 4^'~T~— ~T" For Rule of Three in Vulgar Fractions^ see page 29. 4 -V •«'■*• DECIMAL FRACTIONS. A Decimal Fraction is that which has for its denominator a unit (1), with as many ciphers annexed as the numerator has places ; it is usually expressed by set- ting down the numerator only, with a point on the left of it. Thus, -^ is .4, y^^ is .85, -^-^ is .0075, and y oV¥oo ^^ •^^^^- ^^^^" ^^^'® ^^ ^ deficiency of fig- ures in the numerator, prefix ciphers to make up as many places as there are piphers in the denominator. DECIMAL FRACTIOA^S. 2% Mixed niunbers consist of a whole number and a fraction ; as, 3.25, which is the same as 3.f-^, or ff^. Ciphers on the right hand make no alteration in their value ; for .4, .40, .400 are decimals of the same value, each being ^, or |. ADDITION OF DECIMALS. • Rule.— Set the numbers under each other accordmg to the value of their places, as in whole numbers, in which state the decimal points will stand directly under each other. Then, beginning at the right hand, add up all the columns of numbers as in integers, and place the point directly below all the other points. Example.— Add together 25.125, 56.19, 1.875, and 293.7325. 25.125 56.19 1.875 293.7325 376.9225 the sum. SUBTRACTION OF DECIMAL FRACTIONS. Rule. — Place the numbers under each other as in addition ; then subtract as in whole numbers, and point off the decimals as in the last rule. Example.— Subtract 15.150 from 89.1759. 89.1759 15.150 ^ 74!0259 Rem. MULTIPLICATION OF DECIMALS. Rule. — Place the factors, and multiply them together the same as if they were whole numbers ; then poi*:: off in the product just as many places of decimals as there are decimals in both the factors. But if there be not so many figures in the product, supply the deficiency by prefixing ciphers. Example. — Multiply 1.56 by .75. 1.56 .75 780 1092 1.1700 Prod. BY CONTRACTION. To contract the Operation so as to retain only as many Decimal places in the Product as may he thought necessary. Rule.— Set the unit's place of the multiplier under the figure of the multipli- cand whose place is the same as is to be retained for the last in the product, and dispose of the rest of the figures in the contrary order to what they are usually placed in. Then, in multiplying, reject all the figures that are more to the right hand than each multiplying figure, and set down the products, so that their right- hand figures may fall in a column straight below each other ; and observe to in- crease the first figure in every line with what would arise from the figures omit- ted ; thus, add 1 for every result from 5 to 14, 2 from 15 to 24, 3 from 25 to 34, 4 from 35 to 44, &c., &c., and the sum of all the lines will be the product as required. Example.— Multiply 13.57493 by 46.20517, and retain only four places of deci- mals in the product. 13.574 93 71 502.64 54 299 72 8 144 96+2 for 18 27150+2 " 18 6 79+4 " 35 14+1 " 5 9+2 " 21 627.2320 28 DECIMAL FRACTIONS. Example.— Multiply 27.14986 by 92.41035, and retain only five places of deci mals. ^ns. 2508.92806. DIVISION OF DECIMALS. Rule. — Divide as in whole numbers, and point off in tlie quotient as many places for decimals as the decimal places in the dividend exceed those in the divisor; but if there are not so many places, supply the deficiency by prefixing ciphers. Example. — Divide 53.00 by 6.75. 6.75) 53.00 ( = 7.851+. Here 3 ciphers were annexed to carry out the division. BY CONTRACTION. Rule. — ^Take only as many figures of the divisor as will be equal to the number of figures, both integers and decimals, to be in the quotient, and find how many times they may be contained in the first figures of the dividend, as usual. Let each remainder be a new dividend ; and for every such dividend leave out one figure more on the right-hand side of the di\asor, carrying for the figures cut off as in Contraction of Multiplication. Note. — When there are not so many figures in the divisor as are required to be in the quotient, continue the first operation till the number of figures in the divisor be equal to those remaining to be found in the quotient, after ichich begin the contraction. Example. — Di\dde 2508.92806 by 92.41035, so as to have only four places of deci- mals in the quotient. 92.410315) 2508.928106 (27.1498 1848 207 +1 660 721 646 872 +2 13 849 9 241 4608 3 69 6 912 832+4 80 74 1 2 6 Example. — Divide 4109.2351 by 230.409, retaining only four decimals in the quo- tient. Ans. 17.8345. REDUCTION OF DECIMALS. To reduce a Vulgar Fraction to its equivalent Decimal. Rule. — Divide the numerator by the denominator, annexing ciphers to the nu- merator as far as necessary. Example.— Reduce 4 to a decimal. 5)4^ .8 Ans. To find the Value of a Decimal in Terms of an Inferior Denomi- nation. Rule. — Multiply the decimal by the number of parts in the next lower denomi- nation, and cut off as many places for a remainder, to the right hand, as there are places in the given decimal. Multiply that remainder by the parts in the next lower denomination, again cut- ting off for a remainder, and so on through all the parts of the integer. Example.— What is the value of .875 dollars ? .875 100 Cents, 87,500 2? Mills, 5.000 Ans. 87 cents 5 mills. DECIMAL FRACTIONS. 29 Example.— What is the content of .140 cubic feet in inches ? .140 1728 cubic inches in a cubic foot. ^^•^^ ^715. 241.3-9^2^ cubic inches. Example.— What is the value of .00129 of a foot 7 Ans. .01548 inches. Example.— What is the value of 1.075 tons in pounds 1 Ans. 2408. To reduce Decimals to equivalent Decimals of higher Denomina- tions. Rule.— Divide by the number of parts in the next higher denomination, contin- uing the operation as far as required. Example.— Reduce 1 inch to the decimal of a foot. 12 j 1.00000 I .08333, &c., Ans. Example. — Reduce 14 minutes to the decimal of a day. 601 14.00000 24 1 .23333 .00972, &c., Ans. Example.— Reduce 14" 12"' to the decimal of a minute. 14" 12"' 60 852."' 14.2" .23066', &c., Atis. Note. — When there are several numberSy to be reduced all to the decimal of the highest. Reduce them all to the lowest denomination, and proceed as for one denomi nation. .—Reduce 5 feet 10 inches and 3 barleycorns to the decimal of a yard Feet. Inches. Be. 5 10 3 12 70 3 3 213. 12 71. 3 5.9166 1.9722, &c., yards, Ans. RULE OF THREE IN DECIMALS. Rule. — Prepare the terms by reducing the vulgar fractions to decimals, com- pound numbers to decimals of the highest denomination, the first and third terms to the same name ; then proceed as in whole numbers. See Rule, page 31. Example.— If i a ton of iron cost ^ of a dollar, what will .625 of a ton cost? ^=.5 I .5:. 75: -..625 | = .75 \ .625 .5) .46875 .9375 dollars, Ans. C2 90 DUODECIMALS. DUODECIMALS. In Duodecimals, or Cross Multiplication, the dimensions are taken in feet, inch- es, and twelfths of an inch. Rule.— Set down the dimensions to be multiplied together, one under the other, so that feet may stand under feet, inches under inches, &c. Multiply each term of the multiplicand, beginning at the lowest, by the feet in the multiplier, and set the result of each immediately under its corresponding term, carrying 1 for every 12, from one term to the other. In like manner, multi- ply all the multiplicand by the inches of the multiplier, and then by the twelfth parts, setting the result of each term one place farther to the right hand for every multiplier. The sum of the products is the answer. Example.— Multiply 1 foot 3 inches by 1 foot one inch. Feet. Inches. 1 3 1 1 1 3 1 3 14 3 Proof. — 1 foot 3 inches is 15 inches, and 1 foot 1 inch is 13 inches ; and 15X13 = 195 square inches. Now the above product reads 1 foot 4 inches and 3 twelfths of an inch, and 1 foot = 144 square inches. 4 inches =48 " 3 twelfths = _ 3 195 Example.— How many square inches are there in a board 35 feet 4^ inches long and 12 feet 3^ inches wide 1 Feet. Inches. Twelfths. 35 4 6 12 3 4 424 6 8 10 1 6 11 9 6 434 3 11 Example.— Multiply 20 feet 6^ inches by 40 feet 6 inches. By duodecimals, Ans. 831 feet 11 inches 3 twelfths equal 831 square feet and 135 square inches. By decimals . . 40 feet 6 inches = 40.5 20 " 6^ " = 20.541666, &c. Feet . . . 831.937499 144 Square inches . 134.999856 Table showing the value of Duodecimals in Square Feet, and Decimals of an Inch. Sq. feet. Sq. inches. 1 Foot . . . . . . = 1 or 144. 1 Inch = ^ " 12. 1 Twelfth = j^j " 1. ^2 of 1 twelfth = j^-g " .083333, &c. .jLofJ^ofdo = 20 W .006944. &c. Application of this Table. What number of square inches are there m a floor 100^ feet broad and 25 feet 6 Inches and 6 twelfths long 1 Ans. 2566 feet 11 mches 3 twelfths equal 2566 feet 135 inches. RULE OF THREE. 31 RULE OF THREE. The Rule of Three teaches how to find a fourth proportional to three given numbers. It is either Direct or Inverse. It is Direct when more requires more, or less requires less. Thus, if 3 barrels of flour cost $18, what will 10 barrels cost 1 Or, if 300 lbs. of lead cost $25.50, what will 10 lbs. cost 1 In both of these cases the Proportion is Direct, and the stating must be, As 3:18 ::10: ^ns. 60. 300 : 25.50 : : 10 : ^ns. .85. It is Inverse when more requires less, or less requires more. Thus, if 6 men build a certain quantity of wall in 10 days, in how many days will 8 men build the like quantity 1 Or, if 3 men dig 100 feet of trench in 7 days, in how many days will 2 men perform the same work 7 Here the Proportion is Inverse, and the stating must be. As 8 : 10 : : 6 : ^ns. 7^. 2: 7::3: ^715. 10^. The fourth term is always found by multiplying the 2d and 3d terms together, and dividing the product by the 1st term. Of the three given numbers necessary for the stating, two of them contain the supposition, and the third a demand. Rule. — State the question by setting down in a straight line the three neces- sary numbers in the following manner : Let the 2d term be that number of supposition which is of the same denomina- tion as that the answer, or 4th term, is to be, making the demanding- number the 3d term, and the other number the 1st term when the question is in Direct Propor- tion, but contrariwise if in Inverse Proportion, that is, let the demanding number be the 1st term. Then multiply the 2d and 3d terms together, and divide by the 1st, and the prod- uct will be the answer, or 4th term sought, of the same denomination as the 2d term. Note. — If the first and third terms are of different denominations, reduce them to the same. If, after division, there be any remainder, reduce it to the next lower de- nomination, and divide by the same divisor as before, and the quotient will be of this last denomination. Sometimes two or more statings are necessary, which may always be known by the nature of the question. Example 1.— K20 tons of u-on cost $225, what will 500 tons costl Tons. Dolls. Tons. 20 : 225 : : 500 500 210) 11250i0 5625 dollars, Ans. Example 2.— If 15 men raise 100 tons of iron ore in 12 days, how many men will raise a like quantity in 5 days 1 Days. Men. Days. As 5 : 15 : : 12 12 5) 180 36 men, Jlns. Example 3.— A wall that is to be built to the height of 36 feet was raised 9 feet high by 16 men in 6 days : how many men -could finish it in 4 days at the same rate of working ? Days. Men. Days. Men. 4 : 16 : : 6 : 24 .^ns. Then, if 9 feet require 24 men, what will 27 feet require 1 9 : 24 : : 27 : 72 Ans. Example 4.— If the third of six be three, what will the fourth of twenty be ? ^ns. 7J. 32 INVOLUTION ^EVOLUTION. COMPOUND PROPORTION. Compound Proportion is the rule by means of which such questions as would require two or more statings in simple proportion (Role of Three) can be resolved in one. . -, . . j j As the rule, however, is but little used, and not easily acquired, it is deemed preferable to omit it here, and to show the operation by two or more statings. Example.— How many men can dig a trench 135 feet long in 8 days, when 16 men can dig 54 feet in 6 "days 1 Feet. Men. Feet. Men. First . . . As 54 : 16 : : 135 : 40 Days Men. Days. Men. Second . . . As 8 : 40 : : 6 : 30 ^ns. Example.— If a man travel 130 miles ih 3 days of twelve hours each, in how many days of 10 hours each would he require to travel 360 miles 1 Miles, Davs. Miles. Days. First . . . As 130 : 3 : : 360 : 8.307 Hours. Days. Hours. Days. Second . . . As 10 : 8.307 : : 12 : 9.9684 ^ns. Example.— If 12 men in 15 davs of 12 hours build a wall 30 feet long, 6 wide, and 3 deep, in how many days of 8 hours will 60 men build a wall 300 feet long, 8 wide, and 6 deep 1 *^^^- ^20 days. INVOLUTION. Involution is the multiplying any number into itself a certain number of times. The products obtained are called Powers. The number is called the Root, or "^When^a number is multiplied by itself once, the product is the square of that number ; twice, the cube ; three times, the biquadrate, &c. Thus, of the number 5. 5 is the Root, or 1st power. 5X5= 25 " Square, or 2d power, and is expressed 52. 5X5X5 = 125 " Cube, or 3d power, and is expressed 5^ 5X5X5X5 = 625 " Biquadrate, or 4th power, and is expressed 5*. The little figure denoting the power is called the Index or Exponent. Example.— What is the cube of 9 7 -^ns. 729. Example.— What is the 9th power of 2 1 -^ns. 512. Example.— What is the cube of 1 7 -^ns. |J. Example.— What is the 4th power of 1.5 1 -^ns. 5.0625. EVOLUTION. Evolution is finding the Root of any number. The sign y/ placed before any number, indicates the square root of that number is required or shown. , 1. . j u •♦ The same character expresses any other root by placing the mdex above it. , , Thus, y/25 = 5, and 4+2 = ^36. 3/ And, ^27='9vand 3/64= 4. Roots which only approximate are called Surd Roots. Rule.— Point oflf the given number from units' place into periods of two figures Find the greatest square in the left-hand period, and place its root in the quo- tient; subtract the square number from the left-hand period, and to the remainder bring down the next period for a dividend. , ,. . . Double the root already found for a divisor ; find how many times the dmsor is contained in the dividend, exclusive of the right-hand figure, place the result in the quotient, and at the right hand of the divisor. EVOLUTION. 33 Multiply the divisor by the last quotient figure, and subtract the product from the dividend ; bring down the next period, and proceed as before. Note. — Mixed decimals must be pointed off both ways from units. Example.— What is the square root of 2 ? 11 2.060606 (1.414, &c. l| 1 100 96 ^ll 400 281 2824 11900 4 11296 2828 1 604 Example. — What is the square root of 144 1 11 144 (12 Atis. 22 044 44 00 Example.— What is the square root of 12 ? Ans. 3.464101. SQUARE ROOTS OF VULGAR FRACTIONS. Rule. — Reduce the fractions to their lowest terms, and that fraction to a decimal, and proceed as in whole numbers and decimals. Note. — When the terms of the fractions are squares, take the root of each and set one above the other ; as, | is the square root of 5-g-. Example.— What is the square root of ^1 Ans. 0.86602540. To find the 4th root of a number, extract the square root twice, and for the 8th root thrice, &c., &c. TO EXTRACT THE CUBE ROOT. Rule.— From the table of Roots (page 99) take the nearest cube to the given number, and call it the assumed cube. Then say, as the given number added to twice the assumed cube is to the assu- med cube added to twice the given number, so is the root of the assumed cube to the required root, nearly. And, by using in like manner the root thus found as an assumed cube, and pro- ceeding as above, another root will be found still nearer , and in the same manner as far as may be deemed necessary. Example.— What is the cube root of 10517.9 ? Nearest cube, page 99 , 10648, root 22. 10648. 10517.9 21296 21035.8 10517.9 10648. 31813.9 : 31683.8 : : 22 : 21.9+ Ans. To extract any Root whatever. Let P represent the number, n " the index of the power, A " the assumed power, r its root, R " the required root of P. Then say, as the sum of w+lxA and n— IXP is to the sum of n+lXP and n — IXA, so is the assumed root r to the required root R. Example.— What is the cube root of 1500 1 The nearest cube, page 99, is 1331, root 11. 34 ARITHMETICAL PROGRESSION. P = 1500, 71 = 3, A = 1331, r=ll; Uien, n+lXA = 53^, Ti+lXP =6000 n— IXP = 3000, 7i— IX A = 2662 8324 : 8662 : : 11 : 11.446+ Ans, ARITHMETICAL PROGRESSION. Arithmetical Progression is a series of numbers increasing or decreasing by a constant number or diiference; as, 1, 3, 5, 7, 9, 15, 12, 9, 6, 3. Tiie numbers which form the series are called Terms; the first and last are called the Ex- tr ernes, and the others the Means. When any three of the following parts are given, the remaining two can be found, viz. : The First term, the Last term, the JSTumber of terms, the Common Difference, and the Sum of all the terms. When the First Term, the Common Difference, and the Number of Terms are given, to find the Last Term. Rule.— Multiply the number of terms less one, by the common difference, and to the product add the first term. Example.— A man travelled for 12 days, going 3 miles the first day, 8 the second, and so on ; how far did he travel the last day 7 12—1x5-1-3 = 58 Ans. When the Number of Terms and the Extremes are given, to find the Common Difference. Rule.— Di\ide the difference of the extremes, by one less than the ntimber of terms. Example.— The extremes are 3 and 15, and the number of terms 7 ; what is the ccsnmon difference % 15— 3-i-(7— 1) = 2 Ans. When the Extremes and Number of Temis are given, to find the Sum of all the Terms. Rule.— Multiply the number of terms by half the sum of the extremes. Example.— How many times does the hammer of a clock strike in 12 hours 1 12X(13-r-2)=78 Ans. When the Common Difference and the Extremes are given, to find the Number of Terms. Rule.— Divide the difference of the extremes by the common difference, and add one to the quotient. Example.— A man travelled 3 miles the first day, 5 the second, 7 the third, and so on, till he went 57 miles in one day. How many days had he travelled at the close of the last day 1 57— 3-r-2+l = 28 Ans. To find two Arithmetical Means betiveen two given Extremes. Rule.— Subtract the less extreme from the greater, and divide the difference by 3, and the quotient will be the common difference, which, being added to the less extreme, or taken from the greater, will give the means. Example.— Find two arithmetical means between 4 and 16. ]fi — 1-7-3= 4 com. dif. 4-|-4 = 8 one mean. 16 — i = 12 second mean. To find any Number of Arithmetical Means between two Extremes, Rule.— Subtract the less extreme from the greater, and divide the difference by one more than the number of means required to be found, and then proceed as in the foregoing rule. GEOMETRICAL PROGRESSION. 35 GEOMETRICAL PROGRESSION. Geometrical Progression is any series of numbers continually increasing by a constant multiplier, or decreasing by a constant divisor. As, 1, 2, 4, 8, 16, and 15, U, 3%. The constant multiplier or divisor is the Ratio. When any three of the following parts are given, the remaming two can be found, viz. : The First term, the Last term, the Number of terms, the Ratio, and the Sum of all the Terms. When the Ratio, Number of Terms, and the First Term are given, to find the Last Term. Rule.— Write a few of the leading terms of the series, and place their indices over them, beginning with a cipher. Add together the most convenient indices, to make an index less by one than the number of the term sought. Multiply together the terms of the series or powers belonging to those indices, and the product, multiplied by the first term, will be the answer. Note.— W^/tcTi the first term is equal to the ratio, the indices must begin with a unit. Example.— The first term is 1, the ratio 2, and the number of terms 23 ; what is the last term 1 Indices. 01234 5 6 7 Terms. 1, 2, 4, 8, 16, 32, 64, 128. 1^2+3+4+54-7 = 22. 128X32X16X8X4X2 = 4194304X1 = 4194304 Ans. Example.— If one cent had been put out at interest in 1630, what would it have amounted to in 1834 if it had doubled itself every 12 years 1 1834—1630 = 204-i-12 = 17. 12 3 4 5 6 1, 2, 4, 8, 16, 32, 64, 1+2+3+4+6 = 16. 1X2X4X8X16X64 = 65538X2 = $1,310.72 Ans When the First Term, the Ratio, and the Number of Teivns are given, to find the Sum of the Series. Rule.— Raise the ratio to a power whose index is equal to the number of terms, from which subtract 1 ; then divide the remainder by the ratio less 1, and multiply the quotient by the first term. Example.— If a man were to buy 12 horses, giving 2 cents for the first horse, 6 cents for the second, and so on, what would they cost him 1 312 = 531441—1 = 531440-r.(3— 1) = 2 = 265720X2 = $5,314.40 Ans. By another Method, the greater Extreme being known. (Greater extreme X ratio) -less extreme ^ g^^ ^^ ^^^ g^^.^^^ Ratio —1 354294x3—2 = 1062880 Thus 3—1 5.314.40, ^715., as above. A TABLE OF GEOMETRICAL PROGRESSION, Whereby any questions of Geometrical Progression proceeding from 1, and of double ratio J may he solved by inspection, if the number of terms ex- ceed 7wt 50. 1 8 128 15 1 2 9 256 16 4 10 512 17 8 11 1024 18 16 12 204S 19 32 13 4096 20 64 14 8192 21 16384 32768 65536 131072 262144 524288 1048576 22 23 24 25 26 27 28 2097152 4194304 16777216 33554432 67108864 134217728 36 30 31 32 33 34 35 36 PERMUTATION- -COMBINATION — -POSITION. Table— (Continued.) 268435456 37 68719476736 44 8796093022208 536870912 38 137438953472 45 17592186044416 1073741824 39 274877906944 46 35184372088832 2147483648 40 549755813888 47 70368744177664 4294967296 41 1099511627776 48 140737488355328 8589934592 42 2199023255552 49 281474976710656 17179869184 43 4398046511104 50 562949953421312 34359738368 100 i 633825300114114700748351602688 PERMUTATION. Permutation is a rule for finding how many different ways, any given number of things may be varied in their^ position. Rule.— Multiply all the terras continually together, and the last product will be the answer. Example. — How many variations will the nine digits admit of 7 1X2X3X4X5X6X7X8X9 = 362880 .ans. COMBINATION. Combination is a rule for finding how often a less number of things, can be chosen from a greater. Rule.— Multiply together the natural series, 1, 2, 3, &c., up to the number to be taken at a time. Take a series of as many terms, decreasing by 1, from the nuna- ber out of which the choice is to be made, and find their continued product. Di- vide this last product by the former, and the quotient is the answer. Example. — How many combinations may be made of 7 letters out of 12? IX 2X 3X4X5X6X7 := 5040. 12X11X10X9X8X7X6 = 3991680-J-5040izi: 792 Ans. Example. — How many combinations can be madje of 5 letters out of 10? 10X9X8X7X6 ^^^ ^ — — - — - — - — r=2o2 Ans. 1X2X3X4XO POSITION. Position is of two kinds. Single and Double, and is determined by the number of Suppositions. SINGLE POSITION. Rule.— Take any number, and proceed with it as though it were the correct one ; then say, as the result is to the given sum, so is the supposed number to the number required. Example.— A commander of a vessel, after sending away in boats i, |, and ^ of his crew, had left 300 ; what number had he in conunand 1 Suppose he had . 600. ^ of 600 is 200 |of600isl00 I of 600 is 150 450 150 : 300 : : 600 : 1200 Ans. Example.— A person being asked his age, replied, iff 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 ? -^w^- 37^ years. DOUBLE POSITION. Rule.— Take any two numbers, and proceed with each according to the condi- FELLOWSHIP DOUBLE FELLOWSHIP. 37 tions of the question ; multiply the results or errors by the contrary supposition ; that is, the first position by the last error, and the last position by the first error. If the errors be too great, mark them + ; and if too little, — . Then, if the errors are alike, divide the difference of the products by the differ- ence of the errors ; but if they are unlike, divide the sum of the products by the sum of the errors. Example.— F asked G how much his boat cost; he replied that if it cost him 6 mes as much as it did, and $30 more, it would stand him in S300. What Was the ice of the boat 1 Suppose it cost . . 60 . . or 30 G times. 6 times. 360 180 and 30 more, and 30 more. 390 210 300 300 90+ 90— 30 2d position. 60 1st position. 90 2700 5400 90 5400 180) 8100 (45 Ans. 720* 900 900 Example. — Wliat is the length of a fish when the head is 9 inches long, the tail as long as its head and half its body, and the body as long as both the head and tail 1 Ans. 6 feet. FELLOWSHIP. Fellowship is a method of ascertaining gains or losses of individuals engaged in joint operations. 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 $9,500. A paid $3, and B paid $2 for the ticket ; how much is each one's share "? 5 : 9.500 : : 3 : 5.700, A's share. 5 : 9.500 : : 2 : 3.800, B's share. DOUBLE FELLOWSHIP, Or Fellowship with Time. Rule.— Multiply each share by the time of its interest in the Fellowship ; then, as the sum of the products is to the product of each interest, so is the whole gain or loss to each share of the gain or loss. Example.— A ship's company take a prize of $10,000, which they divide accord- ing to their rate of pay and time of service on board. The officers have been on board 6 months, and the men 3 months ; the pay of the lieutenants is $100 ; mid- shipmen $50, and men $10 per month ; and there are 2 lieutenants, 4 midshipmen, and 50 men. What is each one's share 1 2 lieutenants $100 = 200X6=1200 4 midshipmen 50 = 200x6 = 1200 somen 10 = 500x3 = 1500 Lieutenants Midshipmen Men . 3900 : 1200 3900 : 1200 3900 : 1500 3900 10.000 : 3.076.92-r- 2 = $1,538.46 10.000 : 3.076.92-^ 4 = $769.23 10.000 : 3.846.16-J-50= $76.92 38 ALLIGATION COMPOUND INTEREST. ALLIGATION. Alligation is a method of finding the mean rate or quality of different materials when mixed together. When it is required to find the mean price of the mixture, observe the following Rule. — Multiply each quantity by its rate, then divide the sum of these products by the sum of the quantities, and the quotient will be the rate of the composition. Example. — If 10 lbs. of copper at 20 cents per lb., 1 lb. of tin at 5 cents, and 1 lb of lead at 4 cents, be mixed together, what is the value of the composition 1 10X20 I3Z 200 IX 5= 5 _1X 4= 4 12 ) 209 (17.3^ Ans. When the Prices and Mean Price are given, to find lohat Quantity of mch Article must be taken. Rule 1. — Connect with a line each price that is less than the mean rate with one or more that is greater. Write the difference between the mixture rate and that of each of the simples opposite the price with which it is connected ; then the sum of the differences against any price will express the quantity to be taken of that price. Example. — How much gunpowder, at 72, 54, and 48 cents per pound, will com- pose a mixtiue worth 60 cents a pound 7 ( 48 \ 12, at 48 cents ) m{5\\) 12, at 54 cents V Ans. 1 12 J 12+5 = 18, at 72 cents ) Proof.— 12X48+12X54+18X72 = 2520-r-12+12+12+6 = 60. Should it be required to mix a definite quantity of any one article, the quantities of each, determined by the above rule, must be increased or decreased in the proportion they bear to the defined quantity. Thus, had it been required to mix 18 pounds at 48 cents, the result would be 18 at 48, 18 at 54, and 27 at 72 cents per pound. Again, when the whole composition is limited, say, As the sum of the relative quantities, as found by the above rule, is to the whole quantity required, so is each quantity so found to the required quantity of each. Example. — Were 100 pounds of the above mLxture wanting, the result would be obtained thus : As 42 : 100 : : 12 : 28 f. 42 : 100 : : 12 : 284. 42 : 100 : : 18 : 421. COMPOUND INTEREST. If any principal be multiplied by the amount (in the following table) opposite the years, and under the rate per cent., the sum will be the amount of that princi-' pal at compound interest for the time and rate taken. Example.— What is the amount of $500 for 10 years, at 6 per cent. ? Tabular number . 1.79084X500 = $895.42 Ans. DISCOUNT EQUATION OF PAYMENTS. 39 Table showing tlie amount of £\ or%\^ c^c.^for any number of years not exceeding 24, at the rates of 5 and 6 per cent, compound interest. Years. 5 per cent. 1 1.05 2 1.1025 3 1.15762 4 1.21550 5 1.27G28 6 1.34009 7 1.40710 8 1.47745 9 1.55152 10 1.62889 11 1.71033 12 1.79585 6 per cent. Years. 5 per cent. 6 per cent. 1.06 13 1.88564 2.13292 1.1236 14 1.97993 2.26090 1.19101 15 2.07892 2.39655 1.26247 16 2.18287 2.54035 1.33322 17 2.29201 2.69277 1.41851 18 2.40661 2.85433 1.50363 19 2.52695 3.02559 1.59384 20 2.65329 3.20713 1.68947 21 2.78596 3.39956 1.79084 22 2.92526 3.60353 1.89829 23 3.07152 3.81974 2.01219 24 3.22509 4.04893 DISCOUNT. The Time, Rate per Cent., and Interest being given, to find the Principal. Rule.— Divide the given interest by the interest of $1, for the given rate and time. Example.— What sum of money at 6 p§r cent, will in 14 months gain ^14 1 As .07-i-$14 = $200 Ans. The Principal, Interest, and Time being given, to find the Rate per Cent. Rule.— Divide the given interest by the interest of ihe given sum, for the time, at 1 per cent. Example.— A broker received $32.66 interest for the use of $400, 14 months; what was that per cent. ? The interest on $32.66 for 14 months, is 4.66. Then, as 4.66H-32.66 = 7 per cent., Ans. The Principal, Rate per Cent., and Interest being given, to find the Time. Rule.— Divide the given interest by the interest of the sum at the rate per cent, for one year. Example.— In what time will $108 gain 11.34, at 7 per cent.1 The interest on $108 for one year is 7.56. Then, as 7.56-j-11.34 = 1.5 years, Ans. EQUATION OF PAYMENTS. Multiply each sum by its time of paynnent in days, and divide the sum of the products by the sum of the payments. Example.— A owes B $300 in 15 days, $60 in 12 days, and $350 in 20 days ; when is the whole due "? 300X15 = 4500 60X12= 720 350X20 = 7000 710 ) 12220 (17+ days, Ans. 40 ANNUITIES. ANNUITIES. The Annuity, Time, and Rate of Interest given, to find the Amount. Rule.— Raise the ratio to a power denoted by the time, from which subtract 1 ; divide the remainder by the ratio less 1, and the quotient, multipUed by the annui- ty, will give the amount. Note. — $1 or £1 added to the given rate per cent, is the ratio, and the preceding- table in Compound Interest is a table of ratios. Example.— What is the amount of an annual pension of $100, interest 5 per cent., which has remained unpaid for four vears? 1.05 ratio ; then 1.05^ — 1.215506:25—1 — .:21o50625-i-(1.0^—l).05= 4.310125X100 = 431.0125 dollars. The Annuity, Time, and Rate given, to find the Present Worth. Rule.— Divide the annuity by the ratio involved to the time, gubtraot-tbe'^fto- .tien U r om th e ana uiui^ and the remainder will be the present worth. Example.— What is the present worth of a pension or salary of $500, to continue 10 years at 6 per cent, compound interest ] §500, by the last rule, is worth $0590.3975, which, divided by 1.0610 (bv table page 39, is 1.79084) = $36c0.05 ^ns. ' Or, by the following table, multiply the tabular number by the given annuity and the product will be the present worth : * Table showing the present worth of ^1 or £\ anmiitij, at 5 and 6 per cent, compound interest Jor any nuniber of years under 34. Years. 5 per cent. 6 per cent. Years. 5 per cent. , 6 per cent. 1 0.95238 0.94339 18 11.68958 10.8276 2 1.85941 1.83339 19 12.08.532 11.15811 3 2.72325 2.67301 20 12.46221 11.46992 4 3.54595 3.4651 21 12.82115 11.76407 5 4.32948 4.21236 22 13.163 12.04158 6 5.07569 4.91732 23 13.48807 12.30338 7 5.78637 5.58238 24 13.79864 12.5.5035 8 6.46321 6.20979 25 14.09394 12.78335 9 7.10782 6.80169 26 14.37518 13.00316 10 7.72173 7.36008 27 14.64303 13.21053 11 8.30641 7.88687 28 14.89813 13.40616 12 8.86325 8.38384 29 15.14107 33.59072 13 9.39357 8.85268 30 15.37245 13.76483 14 9.89864 9.29498 31 15.59281 13.92908 15 10.37966 9.71225 32 15.80268 14.08398 16 10.83777 10.10589 33 36.00255 14.22917 17 10.27407 10.47726 34 16.1929 14.36613 Example. — Same as above ; 10 years at 6 per cent, gives 7.30008X500 =: $3680.04 Jlns. ^ When annuities do not commence till a certain period of time, they are said to be in Reversion. To find the Present Worth of an Annuity in Reversion. Rule.— Take two numbers under the rate in the above table, viz., that oppo- site the sum of the two given times and that of the time of reversion, and multiply their difference by the annuity, and the product is the present worth. Example.— What is the present worth of a reversion of a lease of $40 per an- num, to continue for six years, but not to commence until tlie end of 2 years, al- lowing 6 j)er cent, to the purchaser 7 By table, 8 years .... 6.20979 " 2 " .... 1.83339 4.37640X40= $175.05 ^7w< PERPETUITIES CHRONOLOGICAL PROBLEMS . 41 For half yearly and quarterly payments, the amount for the given time, multi- plied by the number in the following table, will be the true amount : Rate per ct. Half yearly. Quarterly. Rate per ct. 3 3^ 4 4i 5 Half yearly. 1.007445 1.008675 1.009902 1.011126 1.012348 Quarterly. 1.011181 1.013031 1.014877 1.016720 1.018559 5i 6 7 1.013567 1.014781 1.015993 1.017204 1.020395 1.022257 1.024055 1.025880 Example.— What will an annuity of $50, payable yearly, amount to in 4 years at 5 per cent., and what if payable half yearly 1 By table, page 39, 1.21550— 1~(1.05—1) =4.310X50 = $2J5.50 Jlns., for yearly payment, and . . 215.50X1.012348 = $218.16 " half yearly do. PEEPETUITIES. Perpetuities are such annuities as continue forever. Rule.— Divide the annuity by the rate per cent., multiply by the tabular num- ber above, and the quotient will be the answer. Example.— What is the present worth of a $100 annuity, payable semi-annually, at 5 per cent. 1 100-r.05 = 2000X1.012348 (from preceding table) = $2,024.70 ^ns. For Perpetuities in Reversion, subtract the present worth of the annuity for the time of reversion from the worth of the annuity, to commence immediately. Example.— What is the present worth of an estate of $50 per annum, at 5 per cent., to commence in 4 years 1 50-f-.05 = 1000. $50, for 4 years, at 5 per cent. = 3.54595 (from table) X50 = 177.29 $822.71 ^ns.y which in 4 years, at 5 per cent, compound interest, would produce $1000. CHRONOLOGICAL PROBLEMS. The Golden Number is a period of 19 years, in which the changes of the moon fell on the same days of the month as before. ' To find the Golden Number, or Lunar Cycle. Rule. — Add one to the given year ; divide the sum by 19, and the remainder is the golden number. Note. — IfQ remain^ it will be 19. Example.— What is the golden number for 1830 1 1830+1-M9 = 96 rem. : 7 Ans. To find the Epact. Rule. — Divide the centuries of the given year by 4 ; multiply the remainder by 17, and to this product add the quotient, multiplied by 43; divide this sum plus 86 by 25, multiplying the golden number by 11, from wliich subtract the last quotient, and, rejecting the 30's, the remainder will be the answer. Example. — Required the epact for 1830. Centuries. 18-M = 4|. 2X17 = 34. 4X43= 172+34 = 206+86 = 292-^25 = 11, last quotient. Golden number, as ascertained above, 7X11 =: 77 — 11 (last quotient) = 66, rejecting 30's=6 Ans. Example.— What is the epact for 1839 1 Ans. 15. D2 42 TABLE OF EPACTS, DOMINICAL LETTERS, ETC. TO FIND THE MOOX'S AGE ON ANY GIVEN DAY. Rule.— To the day of the month add the epact and number of the month, then reject the 30's, and the answer will be the moon's age. January 0, February'- 2, March 1, Example.— For 5th February, 1841. Given day Epact Number of month . Numbers of the Month. April 2, I July 5, May 3, August 6, June 4, I Septembers, October 8, November 10, December 10. ¥ age of the moon. The Cycle of the Sun is the 28 years before the days of the week return to the same days of the month. Table of Epacts^ Dominical Letters ^ and an Almanac^ from 1776 to 1875. February, March, November, February,* August. May. January. October. January,* April, July. September, December. June. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 33 24 25 2o 27 28 29 30 31 N. B. — In leap-: marked *. -ear, January and February must be taken in the columns Years Days. Dom. Let- ters. *: % a Years IDom. Days. Let- 1 ters. % Years Dom. Days. Let- ters. § a Years Days Dom. Let- ters. 1 1776 Friday* GF 9 1801 [Sunday. 1 D 15 1826 1 Wedn'y. A 22 1851iSat'y. E 28 1777 Saturd'y E 20 1802 Monday. C 26 1827 Thursd. G 3 1852 Mon.* DC 9 1778 Sunday. D 1 1803 Tuesd'y. B 7 1828Saturd.* FE 14 1853, Tues. B 20 1779 Monday. C 12 1804 Thurs.* AG 18 1829 Sunday. D 25 1854 Wedn. A ] 1780 Wedn.* BA 23 1805 Friday. F 29 1830 Monday. C 6 1855iThur. G 12 1781 Thiirsd. G 4 1806 Saturd'y E 11 1831 Tuesd'y. B 17 1856 Sat' y* FE 23 1782 JFriday. F 15 1807 Sunday. D 22 1832 Thurs.* AG 28 1857 Sund. D 4 1783 Saturd'y E 26 1808 Tuesd.* CB 3 1833 Friday F 9 1858 Mond. C 15 1784 Mond.* DC 7 1809 Wedn'y. A 14 1834 Saturd'y E 20 1859 Tues. B 26 1785 Tuesd'y B 18 1810 Thursd. G 25 1835; Sunday. D 1 1860 iThu.* AG 7 1786 Wedn'y. A 29 1811 Friday. F 6 1836 Tuesd * CB 12 1861 ! Friday F 18 1787 Thursd. G 11 1812 Sunday* ED 17 1837j Wedn'y. A 23 1862 Satur. E 29 1788 Saturd.* FE 22 1813 Monday. C 28 1838; Thursd. G 4 1863 Sund. D 11 1789 Sunday. D 3 1814 Tuesd'y. B 9 1839 Friday. F 15 1864 |Tue.* CB 22 1790 Monday. C 14 1815 Wedn'y. A 20 1840Sund'y.* ED 26 1865 Wedn.' A 3 1791 Tuesd'y. B 25 1816 Friday.* GF 1 1841 Monday. C 7 1866 Thur. G 14 1792 Thurs.* AG 6 1817 Saturd'y E 12 1842|Tuesd'y. B 18 1867 Friday F 25 1793 Friday.' F 17 1818 Simday. D 23 1843 Wedn'y. A 29 1868 Sun.* ED 6 1794 Saturd'y E 28 1819 Monday. C 4 1844 j Friday.* GF 11 1869 Mond. C 17 1795 Sunday. D 9 1820 Wedn.* BA 15 1845 Saturd'y E 22 1870 Tues B 28 1796 Tuesd.* CB 20 1821 Thursd. G 26 1846 Sunday. D 3 1871 Wedn. A 9 1797 Wedn'y. A 1 1822 Friday. F 7 1847 Monday. C 14 1872 Frid.* GF 20 1798 Thursd. G 12 1823 Saturd'y E 18 1848 Wed'y.* BA 25 1873 Satur. E 1 1799 Friday. F 23 18$M Mond'y* DC 29 184»Thursd. G 6 1874 Sund. D 12 1800 Saturd'y E 4 1825 Tuesd'y. B 11 1850 Friday. F 17 1875 Mond. C 23 * Distinguishes the leap-years. PROMISCUOUS QUESTIONS. 43 Use of the above Table— To find the day of the week on which any given day of the month falls in any year from 1776 to 1875. Example.— The great fire occurred in New- York on the 16th December, 1835; what was the day of the week 1 Against 1835 we find Sunday, and at top, under December, we find that the 13th ^ was Sunday ; consequently, the 16th was Wednesday. PEOMISCUOUS QUESTIONS. 1. If SlOO principal gain $5 interest in one year, what amount will gain $20 in 8 months 1 As 12 months : 5 : : 8 months : 3.33, the interest for 8 months. And, as 3.33 : : 100 : : 20 : 600 the answer. 2. A reservoir has two cocks, through which it is supplied ; by one of them it will fill in 40 minutes, and by the other in 50 min- utes ; it has also a discharging cock, by which, when fall, it may be emptied in 25 minutes. If the three cocks are left open, in what time would the cistern be filled, assuming the velocity of the water to be uniform ] The least common multiple of 40, 50, and 25 is 200. Then . . the 1st cock will fill it 5 times in 200 minutes, the 2d *' 4 " 200 or both 9 times in 200 minutes ; and, as the discharge-cock will empty it 8 times in 200 minutes, then 9—8 = 1, or once in 3.20 hours, Ans. 3. Out of a pipe of wine, containing 84 gallons, 10 gallons were drawn off, and the vessel replenished with water ; after which 10 gallons of the mixture was likewise drawn off, and then 10 gallons more of water were poured in, and so on for a third and fourth time. It is required to find how much pure wine remained in the vessel, supposing the two fluids to have been thoroughly mixed ] 84—10 = 74 As 84 : 10 : 84 : 10 : 84 I 10 : 74 : 8.80952 65.19048 : 7.76077 57.42971 : 6.83687 6.83687 50.59284 Ans. 4. A traveller leaves New- York at 8 o'clock in the morning, and walks towards New-London at the rate of 3 miles an hour, without intermission ; another traveller sets out from New- London at 4 o'clock the same evening, and walks for New-York at the rate of 4 miles an hour, constantly ; now, supposing the distance between the two cities to be 130 miles, whereabout on the road will they meetl From 8 o'clock till 4 o'clock is 8 hours ; therefore, 8x3=24 miles, performed by A before B set out from New-London ; and, con- sequently, 130—24 = 106 are the miles to travel between them after 44 PROMISCUOUS QUESTIONS. that. Hence, as 7 = 3+4 : 3 : : 106 : -^^^ 45f more miles trav- elled by A at the meeting ; consequently, 24+45^ z=z 69^ miles from New-York is the place of their meeting. 5. What part of ^3 is a third part of $2 1 loff ofl = lxfxJ = f Ans. 6. The hour and minute hand of a clock are exactly together at 12 ; when are they next together ] As the minute hand runs 11 times as fast as the hour hand ; then, 11 ; 60 : : 1 : 5 mm. 5^ sec. The time, then, is 5 min. 5^^ sec. past 1 o'clock. 7. The time of the day is betw^een 4 and 5, and the hour and min- ute hands are exactly together ; what is the time ? The speed of the hands is as 1 to 11. 4 hours X60 =240, which -f-11 = 21^^ min. added to 4 hours, Ans. 8. A can do a piece of work in 3 weeks, B can do thrice as much in 8 weeks, and C five times as much in 12 weeks ; in what time can they finish it jointly ] Week. Week. Week. As 3 : 1 : : 1 : 1 work done by A in one week. 8 : 3 : : 1 : f " B 12 : 5 : : 1 :j% " C " Then, by addition, ^+|4-t2 ^^'i^^ ^^ the work done by them all in one week ; these, reduced to a common denominator, become yj ^^-\-^==:^=:l', whence, 9 : 6 : : 8 : 5| Ans. 9. A cistern, containing 60 gallons of water, has 3 unequal cocks for discharging it ; one cock 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 all run together '? First, i w^ould run out in 1 hour by the second cock, and J by the third ; consequently, by the 3 was the reservoir supplied one hour, "l+^+l = f +f +f being reduced to a common denominator, the sum of these 3 = V ; whence the proportion, 1 1 : 60 : : 6 : 32 j^ minutes, the time required. 10. What will a body, weighing 10 lbs. troy, lose by being carried to the height of 7 miles above the surface of the earth 1 As the gravitation or iveight of a body above the earth is inversely as the square of its distance, and the earth's diameter being, say 3993 milesy then 3993+7 — 4000. And, as 4000^ : 3993^ : : 10 : 9.965 lbs., Ans. 11. Suppose a cubic inch of common glass weighs 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 ll^s., is thrown into the water. It is proposed to determine if it will sink ; and if so, how much force will just buoy it up 1 3.84X12-M.49 = 30.92 cubic inches of glass in the bottle. 231 cubic inches in a gallon x.53 = 122.43 ounces of brandy. PROBIISCUOUS QUESTIONS. 45' Then, bottle and brandy weigh 3. 84x 12+122.43 = 168.51 ounces, and contain 261.92 cubic inches, which, X-59 == 154.53 ounces, the weight of an equal bulk of salt water. And, 168.51 — 154.53= 13.98 ounces, the weight necessary to sup- port it in the water. 12. How many fifteens can be counted with four fives 1 Ans, 4. 13. What is the radius of a circular acre '? (Side of a square x 1.128 r^ diameter of an equal circle.) 208.710321, the side of a square acre, x 1.128 =:= 235.50-1-2 (for radius) = 117.75 feet, Ans. 14. From Caldwell's to Newburg is 18 miles ; the current of the river is such as to accelerate a boat descending, or retard one as- cending U miles per hour. Suppose two boats, driven 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 1 Call X the distance from N to the place of meeting ; its distance from C, then, will be 18— a:. Speed of descending boat, 15+1.5 ~ 16.5 miles per hour. ' Speed of ascending boat, 15—1.5 = 13.5 miles per hour. — - = time of boat descending to point of meeting. 16.5 18 X = time of boat ascending to point of meeting. 13.5 These times are, of course, equal ; therefore, - - - = - •♦ Then, 13.5x = 297— 16.5a;, and 13.52;+16.5a; = 297, or 30a; = 297. 297 Hence, x = -— - = 9.9 miles, the distance from Newburgh, Ans. 15. A steamboat, going at the rate of 10 miles per hour through the water, descends a river, the velocity of which is 4 miles per hour, and returns in 10 hours ; how far did she proceed] Let X = distance required, — — = time of going, — -— = time of returning. -^ - = 10 ; 6z+14x = 840 ; 20x = 840 ; 840-^-20 = 42, ^ns. 14^6 ' ^ 16. The flood tide wave of a river runs 20 miles per hour, the current of it is 3 miles per hour. Assume the air to be quiescent, and a floating body set in motion at the commencement of the flow of the tide ; how long will the body drift in one direction, the tide flowing six hours from each point of the river 1 Let X be the time required ; 20x = distance the tide has run up, together with the distance which the floating body has moved ; 3x= whole distance which the body has floated. Then 20x— 3z = 6X20, or the length in miles of a tide. x= -^r— X6 = 7 hours, 3 minutes, 31^^^ seconds, ^ns. 17. If a steamboat, going uniformly at the rate of 15 miles in an 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] 15+5 = 20 miles, tbe distance gone during the hour. Then 15—5 = 10 miles, is her etfective velocity per hour when returning, and 20^10 = 2 hours, the time of returning, and 2+1 = 3 hours, or the whole time occupied, Jins. 46 GEOMETRY. GEOMETRY. Definitions. A Foint has position, but not magnitude. A Line is length without breadth, and is either Right, Curved, or Mixed. A Right Line is the shortest distance between two points. A Mixed Line is composed of a right and a curved line. A Superficies has length and breadth only, and is plane or curved. A Solid has length, breadth, and thickness. Kxi Angle is the opening of two lines having different directions, and is either Right, Acute, or Obtuse. A Right Angle is made by a line perpendicular to another, falling upon it. An Acute Angle is less than a right angle. An Obtuse Angle is greater than a right angle. A Triangle is a figure of three sides. An Equilateral Triangle has all its sides equal. An Isosceles Triangle has two of its sides equal A Scalene Triangle has all its sides unequal. A Right-angled Triangle has one right angle. An Obtuse-angled Triangle has one obtuse angle. An Acute-angled Triangle has all its angles acute. A Quadrangle or Quadrilateral is a figure of four sides, and has the following par- ticular names, viz. : A Parallelogram, having its opposite sides parallel. A Square, having length and breadth equal. A Rectangle, a parallelogram having a right angle. A Rhombus (or Lozenge), having equal sides, but its ang es not right angles. A Rhomboid, a parallelogram, its angles not being right angles. A Trapezium, having unequal sides. A Trapezoid, having only one pair of opposite sides parallel. Note. — A Triangle is sometimes called a Trigon, and a Square a Tetragon. Polygons are plane figures having more than four sides, and are either Regular or Irregular, according as their sides and angles are equal or unequal, and they are named from the number of their sides or angles. Thus : A Pentagon has five sides. 'A Hexagon A Heptagon An Octagon A Nonagon A Decagon An Undecagon A Dodecagon SIX seven eight nine ten eleven twelve A Circle is a plane figure bounded by a curve line, called the Circumference (or "Periphery). An Arc is any part of the circumference of a circle. A Chord is a right line joining the extremities of an arc. A Segment of a circle is any part bounded by an arc and its chord. The Radius of a circle is a line drawn from the centre to the circumference. A Sector is any part of a circle bounded by an arc and its two radii. * A Semicircle is half a circle. A Quadrant is a quarter of a circle. A Zone is a part of a circle included between two parallel chords and their arcs. A Lune is the space between the intersecting arcs of two eccentric circles. A Gnomon is the space included between the lines forming two similar parallelo- grams, of which the smaller is inscribed within the larger, so as to have one angle in each common to both. A Secant is a line that cuts a circle, lying partly within and partly without it. A Cosecant is the secant of the complement of an arc. A Sine of an arc is a line running from one extremity of an arc perpendicular to a diameter passing through the other extremity, and the sine of an angle is the sine of the arc that measures that angle. The Versed Sine of an arc or angle is the part of the di'imeter intercp^^ted betwee>, the sine and the arc. GEOMETRY. 47 The Cosine of an arc or angle is the part of the diameter intercepted between the sine and the centre. A Tangent is a right line that touches a circle without cutting it. A Cotangent is the tangent of the complement of the arc. The Circumference of every circle is supposed to be divided into 360 equal parts called Degrees ; each degree into 60 Minutes, and each minute into 60 Seconds, and so on. The Complement of an angle is what remains after subtracting the angle from 90 degrees. The Supplement of an angle is what remains after subtracting the angle from 180 degrees. To exemplify these definitions, let A c 6, in the following diagram, be an assumed arc of a circle described with the radius A B. B k, the Cosine of the arc A c b. A g, the Tangent of do. CB b, the Complement, and 6 B E, the Supplement of the arc A c b. C g, the Cotangent of the arc, written coH B^, the Cosecant of the arc, written cosec. m b, the Coversed sine of the arc, or, by convention, of the angle A B 6 ; written coversin. A c &, an Arc of the circle AGED. A b, the Chord of that arc. e D rf, a Segment of the circle. A B, the Radius. A B 6 c, a Sector. A D E B, a Semicircle. C B E, a Quadrant. A c « E2 54 CONIC SECTIONS. CONIC SECTIONS. Definitions, A Cone is a solid figure having a circle for its base, and termina- ted in a vertex. Conic Sections are the figures made by a plane cutting a cone. An Ellipse is the section of a cone when cut by a plane obhquel through both sides. A Parabola is the section of a cone when cut by a plane paralle; to its side. , , , . ^ A Hyperbola is the section of a cone when cut by a plane, making a greater angle with the base than the side of the cone makes The Transverse Axis is the longest straight line that can be drawn ^^ The Conjugate Axis is a line drawn through the centre, at right angles to the transverse axis. ^ , An Ordinate is a right line drawn from any point of the curve perpendicular to either of the diameters. An Abscissa is a part of any diameter contained between its ver- tex and an ordinate. , w *u 4. ^; The Parameter of any diameter is a third proportional to that di- ameter and its conjugate. J. . • 14-^ The Focus is the point in the axis where the ordinate is equal to half the parameter. u i^ ^,. A ConoziZ is a solid generated by the revolving of a parabola or hyperbola around its axis. A ,S;?/iero2(^ is a solid generated in like manner to a conoid by an ellipse. To construct a Parabola— fig. 1. B CONIC SECTIONS, 55 Draw an isosceles triangle, A B D, whose base shall be equal to that of the proposed parabola, and its altitude, C B, twice that of it. Divide each side, A B, D B, into 8, or any number of equal parts ; then draw lines 1 1, 2 2, 3 3, &c., and their intersection will define the curve of a parabola. Note. — The following figures are drawn to a scale of 100 parts to an inch. To construct an Hyperbola,* the Transverse and Conjugate Diameters being given— fig. 2. Make A B the transverse diameter, and C D, perpendicular to it, the conjugate. Bisect A B in 0, and from 0, with the radius O C or D, de- scribe the circle D / c F, cutting A B produced in F and /, which points will be the foci. In A B produced take any number of points, n n, &c., and from F and /, as centres, with B 72, A tz, as radii, describe arcs, cutting each other in s s, &c. Through s 5, &c., draw the curve s B s, and it will be the hyper- bola required. ^ To describe hyperbolas by another metliod, see Gregory's Mathematics, p. 160. 56 CONIC SECTIONS. To find the length of the Ordinate, E F, of an Ellipse, the Transverse, A B, Conjugate, C D, and Abscissa, A F and F B, heing known— fig. 3. Rule. — As the transverse diameter is to the conjugate, so is the square root of the product of the abscissae to the ordinate which divides them. Example.— The transverse axis, A B, is 100 ; the conjugate, C D, is 60; one abscissa, B F, is 20 ; the other, A F, is (100— 20) = 80. 100 : 60 : : v'^OxSO : 24 Ans. The Transverse and Conjugate diameters, and an Ordinate heing known, to find the Ahscissce — fig. 3. Rule. — As the conjugate diameter is to the transverse, so is the square root of the difference of the squares of the ordinate and semi- conjugate to the distance between the ordinate and centre ; and this distance, being added to and subtracted from the semi-trans- verse, will give the abscissas required. Example.— The transverse diameter, A B, is 100 ; the conjugate, C D, is 60 ; and the ordinate, F E, is 24. 60 : 100 : : ^ 24^—302 . 39^ distance between the ordinate and centre ; then lOOH-2— 30 =:20, one abscissa ; 100-^2+30 = 80, the other abscissa. When the Conjugate, Ordinate, and Ahscissce are known, to find the Transverse— fig. 3. Rule. — To or from the semi-conjugate, according as the greater or less abscissa is used, add, or subtract the square root of the dif- ference of the squares of the ordinate and semi-conjugate. Then, as this sum or difference is to the abscissa, so is the conjugate to the transverse. Example. — The ordinate, F E, is 24 ; the less abscissa, F B, is 20 ; and the conjugate, C D, is 60. 30—^242-302 — 12 ; then 12 : 20 : : 60 : 100 Ans, The Transverse^ Ordinate, and Ahscissce heing given, to find the Con- jugate—fig. 3. Rule. — As the square root of the product of the abscissae is to the ordinate, so is the transverse diameter to the conjugate. Example. — The transverse is 100, the ordinate 24, one abscissa 20, the other 80. -v/80X20 : 24 : : 100 : 60 Ans. PARABOLAS. Any three of the four following terms heing given, viz., any two Or- dinates and their Abscissce, to find the fourth — fig. 4. Rule. — As any abscissa is to the square of its ordinate, so i^ ■^R" other abscissa to the square of its ordinate. CONIC SECTIONS. 4. e 57 Example. — The abscissa, e g, is 50, its ordinate, c g, 35.35 ; re- quired the ordinate A F, whose abscissa, e F, is 100. 50 : 35.35=^ : : 100 : ^2500 r=r 50 Ans. HYPERBOLAS. B When the Transverse, the Con jit gale, and the less Abscissa, B n, are given, to find an Ordinate, e n — fig. 5. Note. — In hyperbolas, the less abscissa, added to the axis, gives the greater. Rule. — As the transverse diameter is to the conjugate, so is the square root of the product of the abscissae to the ordinate required. When the Transverse, the Conjugate, and an Ordinate are given, to find the AhscisscB — fiig. 5. Rule. — To the square of half the conjugate add the square of the ordinate, and extract the square root of that sum. Then, as the conjugate diameter is to the transverse, so is the square root to half the sum of the abscissae. To this half sum add half the transverse diameter for the greater abscissa, and subtract it for the less. When the Transverse^ the Abscissce, and Ordinate are given, to find the Conjugate — fig. 5. Rule. — As the square root of the product of the abscissae is to the ordinate, so is the transverse diameter to the conjugate. When the Conjugate, the Ordinate, and the Abscissce are given, to find the Transverse — fiig. 5. Rule. — Add the square of the ordinate to the square of half the conjugate, and extract the square root of that sum. To this root add half the conjugate when the less abcissa is used, and subtract it when the greater is used, reserving the difference or sum. Then, as the square of the ordinate is to the product of the ab- scissa and conjugate, so is the sum, or difference above found, to the transverse diameter. 58 CONIC SECTIONS. Examples. — In the hyperbola, /^5. 2 and 5, the transverse dian eter is 100, the conjugate 60, and the abscissa, B n, is 40 ; required the ordinate e n. 100 : 60 : : v^(40+100x40) = 74.8 : 44.8 Ans. The transverse is 100, the conjugate 60, and ordinate e n, 44.8 what are the abscissas'? Ans. 40 and 140. The transverse is 100, the ordinate 44.8, the abscissas 140 and 40 what is the conjugate 1 Ans. 60. The conjugate is 60, the ordinate 44.8, and the less abscissa 40 what is the transverse 1 A7is. 100. MENSURATION OF SURFACES. 59 MENSURATION OF SURFACES. OF FOUR-SIDED FIGURES 2. a '■ To find the Area of a four-sided Figure, v)hether it be a Square, Paral- lelogram, Rhombus, or a Rhomboid. Rule. — Multiply the length by the breadth or perpendicular height, and the product will be the area. OF TRIANGLES. To find the Area of a Triangle^figs. 5 and 6, r c 6. Rule. — Multiply the base a b hy the perpendicular height c d^ and half the product will be the area. To find the Area of a Triangle by the length of its sides. Rule. — From half the sum of the three sides subtract each side separately ; then multiply the half sum and the three remainders continually together, and the square root of the product will be the area. To find the Length of one side of a Right-angled Triangle, wtien the Length of the other two sides are given — fig. 7. Rule. — To find the hypothenuse a c. Add together the square of the two legs a b and a c, and extract the square root of that sum. To find one of the legs. Subtract the square of the leg, of which the length is known, from the square of the hypothenuse, and the square root of the difference will be the answer. Note. — For Spherical Triangles, see page 68. OF TRAPEZIUMS AND TRAPEZOIDS. To' find the Area of a Trapezium — fig. 8. Rule. — Multiply the diagonal a c by the sum of the two perpen- diculars falling upon it from the opposite angles, and half the product will be the area. 60 MENSURATION OF SURFACES. To find the Area of a Trapezoid— fig. 9. Rule. — Multiply the sura of the parallel sides a h, d c, by a h, the perpendicular distance between them, and half the product will be the area. OF REGULAR POLYGONS. Rule.— Multiply half the perimeter of the figure by the perpen- dicular, falling from its centre upon one of the sides, and the prod- uct will be the area. To find the Area of a Regular Polygon, when the side only is given. Rule. — ^^lultiply the square of the side by the multiplier opposite to the name of the polygon in the following table, and the product will be the area. No. of Sides. Name of Polygon. Angle. Angle of Folygnn. Area. A B c 3 Trigon 120° 60^ 0.433012 2. 1.732 .5773 4 Tetragon 90 90 1.000000 1.41 1.414 .7071 5 Pentagon 72 108 1.720477 1.238 1.175 .8506 6 Hexagon > 60 120 2.598076 1.156 —Radius C =:l'gth ( of side 7 Heptagon 51^ 128i 3.633912 1.11 .8677 1.152 8 Octagon 45 135 4.828427 1.08 .7653 1.3065 9 No n agon 40 140 6.181824 1.06 .6840 1.4619 10 Decagon 36 144 7.694208 1.05 .6180 1.6180 11 Qndecagon 32fy 147TT 9.365640 1.04 .5634 1.7747 12 Dodecagon 1 30 150 11.196152 1.037 .5176 1.9318 Additional uses of the foregoing Table. The third and fourth columns nf the table will greatly facilitate the construction of these figures, with the aid of the sector. Thus, if it is required to describe an oc- tagon, opposite to it, in column third, is 45 ; then, with the chord of 60 on the sector as radius, describe a circle, taking the length 45 on the same line of the sector ; mark this distance off on the circumference, which, being repeated around the circle, will give the points of the sides. The fourth column gives the angle which any two adjoining sides of the respective figures make with each other. Take the length of a perpendicular drawn from the centre to one of the sides of a polygon, and multiply this by the numbers in column A, the product will be the ra- dius of the circle that contains the figure. The radius of a circle multiplied by the number in column B, will give the length of the side of the corresponding figure which that circle will contain. The length of the side of a pohgon multiplied by the corresponding number in the column C, will give the radius of the circumscribing circle. MENSURATION OF SURFACES 61 OF REGULAR BODIES. To find the Superficies of any Regular Body. Rule. — Multiply the tabular surface in the following table by the square of the linear edge, and the product will be the superficies. Number of Sides. Names. Surfaces. 4 Tetrahedron 1.73205 6 Hexahedron. 6.00000 8 Octahedron. 3.46410 12 Dodecahedron. 20.64573 20 Icosahedron. 8.66025 OF IRREGULAR FIGURES. Tofmd the Area of an Irregular Polygon, abcdefg—fig. 10. Rule. — Draw diagonals to divide the ligure into trapeziums and triangles ; find the area of each separately, and the sum of the whole will give the area required. To find the Area of a Long Irregular Figure, bdca—fig. 11. Rule. — Take the breadth in several places, and at equal distan- ces apart ; add them together, and divide the sum by the number of breadths for the mean breadth ; then multiply that by the length of the figure, and the product will be the area. OF CIRCLES. 14. 6 A/' rr^ ' 1 ^^ ■^^ / o \ \^ q\ '^\ hi. To find the Diameter and Circumference of any Circle. Rule 1.— Multiply the diameter by 3.1416, and the product will be the circumference. F 62 MENSURATION OF SURFACES. Rule 2.— Divide the circumference by 3.1416, and the quotient will be the diameter. Rule 3.— Or, as 7 is to 22, so is the diameter to the circumfer- ence. Or, as 22 is to 7, so is the circumference to the diameter. Or, as 1 13 is to 355, so is the diameter to the circumference, &c To find the Area of a Circle. Rule 1.— Multiply the square of the diameter by .7854, or the square of the circumference by .07958, and the product will be the area. Rule 2.— Multiply half the circumference by half the diameter. Rule 3.— As 14 is to 11, %o is the square of the diameter to the area , or, as 88 is to 7, so is the square of the circumference to the area. To find the Length of any Arc of a Circle— fig. 12. Rule 1.— From 8 times the chord of half the arc a.c, subtract the chord ab of the whole arc ; one third of the remainder will be the length nearly. Rule 2.— Multiply the radius ao of the circle by .0174533, and that product by the degrees in the arc. Rule 3.— As 180 is to the number of degrees in the arc, so is 3.1416 times the radius to its length. 1. When the Chord of the Arc and the Versed Sine of half the Arc are given. Rule 4.— To 15 times the square of the chord ab, add 33 times the square of the versed sine c d, and reserve the number. To the square of the chord add 4 times the square of the versed sine, and the square root of the sum will be twice the chord of half the arc. Multiply twice the chord of half the arc by 10 times the square of the versed sine, divide the product by the reserved number, and add the quotient to twice the chord of half the arc : the sum will be the length of the arc very nearly. Note.— 1. diameter X .8862 = side of an equal square. 2. circumference X .2821= " " " 3. diameter X .7071 = " of the inscribed square. 4. circumference X .2251 = " " " 5. area X .9003= " " " 6. side of a square XI. 4142 = diam. of its circums. circle. 7. " " X4.443 =circum. " " 8. " " Xl.128 = diam. of an equal circle. 9. <' " X 3.545 =circum. " " 10. square inches X 1-273 = round inches. When the Chord of the Arc, and the Chord of half the Arc are given. Rule 5.— From the square of the chord of half the arc subtract Note.— If the length for any number of degrees, minutes, &c., is required (see page 67 for the units, radius being 1), multiply them by the number of degrees, &c. in the arc, and the answer is the length. MENSURATION OF SUEFACES. 63 the square of half the chord of the arc, and the remainder will be the square of the versed sine : then proceed as above. Note. — The chord of half the arc is equal to the square root of the sum of the squares of the versed sine or height, and half the chord of the entire arc. When the Diameter and the Versed Sine of half the Arc are given. Rule 6. — From 60 times the diameter co, subtract 27 times cd the versed sine, and reserve the number. Multiply the diameter by the versed sine, and the square root of the product will be the chord of half the arc. Multiply twice the chord of half the arc by 10 times the versed sine, divide the product by the reserved number, and add the quo- tient to twice the chord of half the arc : the sum will be the length of the arc very nearly. • Note. — When the diameter and chord of the arc are given, the versed sine may be found thus : From the square of the diameter subtract the square of the chord, and extract the square root of the remainder. Subtract this root from the diameter, and half the remainder will give the versed sine of half the arc. The square of the chord of half the arc being divided by the diameter, vi^ill give the versed sine ; or, being divided by the versed sine, vv^iU give th.e diameter. To find the Area of a Sector of a Circle — fig. 13. Rule 1. — Multiply the length of the arc adb by half the length of the radius ao. Rule 2. — As 360 is to the degrees in the arc of the sector, so is the area of the circle to the area of the sector. Note. — If tlie diameter or radius is not given, add the square of half the chord of the arc to the square of the versed sine of half the arc ; this sum being divided by the versed sine, will give the diameter. To find the Area of a Segment of a Circle— fig. 12. (See table of Areas, page 72.) Rule 1. — Find the area of the sector having the same arc with the segment, then find the area of the triangle formed by the chord of the segment and the radii of the sector, and the difference of these areas, according as the segment is greater or less than a semicircle, will be the area required. Rule 2.— To the chord ah oi the whole arc, add the chord ac of half the arc, and I of it more ; then multiply the sum by the versed sine c d, and y\ of the product will be the area. Rule 3.— Multiply the chord of the segment by the versed sine, divide the product by 3, and multiply the remainder by 2. Cube the height, find how often twice the length of the chord is contained in it, and add the quotient to the former product, and it will give the area nearly. To find the Area of a Circular Zone— fig. 14. (See table of Areas, page 80.) Rule 1. — When the zone is less than a semicircle. To the area of the trapezoid ahcd add the area of the segments ab, cd; their sum is the area. 64 MENSURATION OF SURFACES. Rule 2. — When the zone is greater than a semicircle. To the area of the parallelogram bgdh, add the area of the segments big, dkh; their sum is the area. To find the Convex Surface of any Zone or Segment— figs. 38 and 39. Rule.— Multiply the height c b, or b d, of the zone or segment by the circumference of the sphere, and the product is the surface. OF UNGULAS. To find the Convex Surface of the Ungulas—figs. 27, 28, 29, and 30. Rules.— For fig, 27, multiply the length of the arc line abc of the base by the height ad. ^ For fig. 28, multiply the circumference of the base of the cylinder efg by half the sum of the greater and less lengths a e, cf For fig. 29, multiply the sine ad, of half the arc ag, of the base a eg, by the diameter eg of the cylmder, and from this product sub- tract the product* of the arc age and cosine df Multiply the dif- ference thus found by the quotient of the height g b, divided by the versed sine e d. For fig. 30 (conceive the section to be continued till it meets the side of the cylinder produced), then find the surface of each of the ungulas thus formed, and their difference is the surface required. Note. — For rules to ascertain the surface of conical ungulas, see Ryan's Bonny- costless Mensuration, page 136 (1639). To find the Area of a Circular Ring or Space included between two Concentric Circles— fig. 54. Rule.— Find the areas of the two circles ad, be separately, and their difference will be the area of the ring. OF ELLIPSES. 16. To find the Circumference of an Ellipse— fig. 15. I^uLE. — Square the two axes ah and cd, and multiply the square root of half their sum by 3.1416 ; the product will be the circumfer- ence. To find the Area of an Ellipse— fig. 15. Rule. — Multiply the two diameters together, and the product by .7854. * When this product exceeds the other, add them together, and when the cosine is 0, the product is 0. MENSURATION OF SURFACES. 65 To find the Area of an Elliptic Segment, <^^g—fig- 16. Rule. — Divide the height of the segment a;? by the axis ah, of which it is a part, and find in the table of circular segments, page 72, a segment having the same versed sine as this quotient ; then multiply the segment thus found and the two axes of the ellipse to- gether, and the product will give the area. 17. Uh OF PARABOLAS 18. r To find the Area of a Parabola — fig. 17. Rule. — Multiply the base dfhy the height ^e, and | of the prod- uct will be the area. To find the Area of a Frustrum of a Parabola — fig. 17. Rule. — Multiply the difference of the cubes of the two ends of the frustrum acdf by twice its altitude b e, and divide the product by three times the difference of the squares of the ends. To find the Length of a Parabolic Curve cut off by a Double Ordinate — fig. 18. Rule. — To the square of the ordinate a b add A of the square of the abscissa c b ; the square root of that sum, multiplied by 2, will give the length of the curve nearly. OF HYPERBOLAS 20. To find the Area of a Hyperbola — fig. 19. Rule. — To the product of the transverse and abscissa add | of the square of the abscissa a b, and multiply the square root of the sum by 21. Add 4 times the square root of tHe product of the transverse and abscissa to the product last found, and divide the sum by 75. Divide 4 times the product of the conjugate and abscissa by the transverse, and this last quotient, multiplied by the former, will give the area nearly. F2 QQ MENSURATION OF SURFACES. To find the Length of a Hyperbolic Curve— fig. 20. Rule.— As the transverse is to the conjugate, so is the conjugate to the parameter. To 21 times the parameter of the axis add 19 times the transverse, and to 21 times the parameter add 9 times the transverse, and multiply each of these sums by the quotient of the abscissa b a, divided by the transverse. To each of these tv^^o prod- ucts add 15 times the parameter, and divide the former by the lat- ter ; multiply this quotient by the ordinate, and the product is the length of half the curve nearly. OF CYLINDRICAL RINGS. To find the Convex Surface of a Cylindrical Ring— fig. 54. I^uiE.— To the thickness of the ring ab add the inner diameter c ; multiply this sum by the thickness, and the product by 9.8696, nd it will give the surface required. To find the Area of a Circular Ring— fig. 54. Rule —The difference of the areas of the two circles will be the area of the ring. OF LUNES. To find the Area of a Lune—fig. 21. Rule.— Find the areas of the two segments a deb, abce from which the lune is formed, and their difference will be the area re- quired.* OF CYCLOIDS. s n To find the Area of a Cycloid— fig. 22. Rule.— Multiply area of generating circle a 6 c by 3, and the prod- uct is the area. * If semicircles be described on the three sides of a right-angled triangle as diame- ters, two iunes will be formed, their united areas being equal to the area ol tne wi angle. MENSURATION OF SURFACES. 6*7 OF CYLINDERS. To find the Convex Surface of a Cylinder— fig. 25. Rule. — Multiply the circumference by the length, and the prod- uct will be the surface. OF CONES OR PYRAMIDS. To find the Convex Surface of a Right Cone or Pyramid — figs. 31 and 33. Rule. — Multiply the perimeter or circumference of the base by the slant height, and half the product will be the surface. To find the Convex Surface of a Frustrum of a Right Cone or Pyramid — figs. 32 and 34. Rule. — Multiply the sum of the perimeters of the two ends by the slant height or side, and half the product will be the surface. OF SPHERES. To find the Convex Surface of a Sphere or Globe— fig. 37. Rule — Multiply the diameter of the sphere by its circumference, and the product is the surface. OF CIRCULAR SPINDLES. To find the Convex Surface of a Circular Spindle — fig. 45. Rule. — Multiply the length /c of the spindle by the radius oc of the revolving arc; multiply the said arc fac by the central distance oe, or distance between the centre of the spindle and centre of the revolving arc. Subtract this last product from the former, double the remainder, multiply it by 3.1416, and the product is the surface. Note. — The same rule will serve for any zone or seg-ment, cut off perpendicularly to the chord of the re-volving arc ; in this case, then, the particular length of the part, and the part- of the arc which describes it, must be taken, in lieu of the whole length and whole arc. BY MATHEMATICAL FORMUL.'E. LINES. CIRCLE. Ratio of circumference to diameter, ^ = 3.1416. T -L r ^^' ^ Length of an arc = — - — nearly ; c the chord of the arc, and c' o the chord of half the arc. Length of 1 degree, radius being 1, = .0174533 " 1 minute, = .0002909 " 1 second, = .0000048 ELLIPSE. Circum/drcTicfi =l||p^J(a2+^>2) nearly, a and h being the axes.' PARABOLA. Length of an arc^ commencing at the vertex, =s \/(-q"+J') near- ly, a being the abscissa, and h the ordinate. 68 MENSURATION OF SURFACES. QUADRILATERALS. Half the product of the diagonals X the sine of their angle. CIRCLE, I>r2 ; or diam. 2 x. 78539816 ; or circum. ^ X. 0795774. CYLINDER. Curved-surface =: height X perimeter of base. SPHERICAL ZONE OR SEGMENT. 2prh ; or, the height of the zone or segment X the circumference of the sphere. CIRCULAR SPINDLE. 2p{rc—a^/r-—ic^) ; a being the length of the arc, and c its chord, or the length of the spindle. SPHERICAL TRIANGLE. pr^ ^'" ^ ; s being the sum of the three angles. 180 ANY SURFACE OF REVOLUTION. 2prXl; or, the length of the generating element x the circum ference described by its centre of gravity. Illustrations.— Let abcbe the side of a cylinder, br the radius ; then abcis the generating element, b the centre of gravity (of the line), and b r the radius of the circle described by abc. Then, ifa6c = 10, 5r=:5; 10x(5+5X 3.1416) = 314.16. Parabola. acX{^brXp), p being in this and all other instances = 3.1416, b the centre of gravity, and b r the radius of its circumference. MENSURATION OF SURFACES. 69 Or, take a umform piece of board or thick pasteboard, and cut out ^nH £'M fi^'* ^'^ ^'■'^'' '"'^•J"''"^'' ' ^^'eh both pieces together and then the figure separately, and say, as the gross weight is to the entire surface, so is the weight of the figure to its surface. CAPILLARY TUBE. Let the tube be weighed when empty, and again when filled with mercury ; let «, be the difference of thos^ weights in troy grains Tnd /the length of the tube in inches. " "oy grains, ana Diameter = .019353^/—. In which Thenp USEFUL FACTORS, p represents the Circwitiference of a Circle whose Diameter is 1. 4? iP iP iP iP ip 3F0-P 1 = 3.1415926535897932384626+ j = 6.283185307179+ = 12.566370614359+ = 1.570796326794+ = 0.785398163397+ = 4.188790 = .523598 = .392699 = .261799 = .008726 = .318309 = .636619 1.273239 .079577 4p Vp = Wp = 4' 360 - - =114.591559 |i» = 2.094395 - = 1.909859 P 36p =113.097335 1.772453 .886226 3.544907 .797884 .564189 70 MENSURATION OF SURFACES. Examples in Illustration of the foregoing Rules. Required the area, 1. Of the rhombus, fig. 3, a c 12 feet 6 inches, and its he^ aZ>, 9 feet 3 inches. Ans. 115.625 feet. 2. Of the ti'iangle a J c, fig. 5, a 5 being 10 feet, and cd o feet. Ans. 25 feet. 3. Of the triangle abc, fig. 7, its three sides measuring respectively 24, 36, and 48 feet. ^715. 418.282. 4. In the right-angled triangle a 6 c, fig. 7, the base is 56, and the height 33 ; what is the hypolhenuse 1 Ans. 65. 5. If the hypothenuse of a triangle be 53, and the base 45, what is the perpendic- ular ? Ans. 28. 6. Required the area of the trapezium, fig. 8, the diagonal ac 84, the perpendicu- lars 21 and 28. Ans. 2058. 7. Of the trapezoid, fig. 9, a & 10, <« °^ ^ = '"-^ "'"■ 22. The abscissa c b, fig. 18, is 12, and its ordinate a 6 6 ; what is the length of acd? ^715.30.198. 23. The transverse and conjugate diameters of a hyperbola, fig. 19, are 100 and 60, and the abscissa a 6 60 ; what is the area ? Ans. 4320. 24. What is the curve a c d of the hyperbola, fig. 20, the abscissa a 6 40 "? Ans. 59.85. MENSURATION OF SURFACES. 71 25. The chord ac, fig. 21, is 19, the heights ed 6.9, and c6 2.4 ; what is the area ot tne lune . ^^^ g^ 3 tbf; J]^^/A^''5'f '"" ''''''^^ ''*'' ^S- ^' ^^ ^ ^"^^^' diameter ; what is the area of me cycloid bcdl ^^^ 37.6992. conve Js^urfaTeS""^ ^ '''''^' ^°' ^^' '' ^ ^'^^' ^"""^ *^^ '^^''* ^^'^^^ ^^ ^^^^ ' ^^^* ^^ t^« .^W5u 70.686. 15?; Ji"^ thickness of a cylifidric ring, fig. 54, is 3 inches, and the inner diameter 12 mches ; what is the convex surface ? ^„5. 444.132. 29. What is the convex surface of a globe, fig. 37, 17 inches in diameter 1 ^715. 6.305 square feet. rnH?.;.^^'''?n^ ^^i""""^^^® ^f i^^ circular spindle, fig. 45, the length fc 14.142, the radius o c 10, and the central distance o e 7.071 inches ? Anl 190.82 inches 31. What is the surface of an octahedron, the linear side being 2 inches ? 22x3.46410 (tabular surface) = 13.85640 \/fnff. 72 AREAS OF THE SEGMENTS OF A CIRCLE. Table of the Areas of the Segments of a Circle., the diameter of which is Unity, and supposed to he divided into 1000 equal Parts. ^sIlS" ^^=-'^-- 'is' ^^^•^^-• .00004 .00011 .00021 .00033 .00047 .00061 .00077 .00095 .00113 .00132 .00153 .00174 .00196 .00219 .00243 .00268 .00294 .00320 .00347 .00374 .00403 .00432 .00461 .00492 .00523 .00554 .00586 .00619 .00652 .00686 .00720 .00755 .00791 .00827 .00863 .00900 .00938 .00976 .01014 .01053 .01093 .01133 .01173 .01214 .01255 .01297 .01339 .01381 .01424 .01468 .01511 .01556 .01600 .01645 .055 .056 .057 .058 .059 .060 .061 .062 .083 .064 .065 .066 .067 .068 .069 .070 .071 .072 .073 .074 .075 .076 .077 .078 .079 .080 .081 .082 .083 .084 .085 .086 .087 .088 .089 .090 .091 .092 .093 .094 .095 .096 .097 .098 .099 .100 .101 .102 .103 .104 .105 .106 .107 .108 .01691 .01736 .01783 .01829 .01876 .01923 .01971 .02019 .02068 .02116 .02165 .02215 .02265 .02315 .02365 .02416 .02468 .02519 .02571 .02623 .02676 .02728 .02782 .02835 .02889 .02943 .02997 .03052 .03107 .03162 .03218 .03274 .03330 .03.387 .03444 .03501 .03558 .03616 .03674 .03732 .03790 .03849 .03908 .03968 .04027 .04087 .04147 .04208 .04268 .04329 .04390 .04452 .04513 .04575 Versed Sine. Scg. Area. Versed Sine. .109 .04638 .163 .110 •.04700 .164 .111 .04763 .165 .112 .04826 .166 .113 .04889 .167 .114 .04952 .168 .115 .05016 .169 .116 .05080 .170 .117 .05144 .171 .118 .05209 .172 .119 .05273 .173 .120 .05338 .174 .121 .05403 .175 .122 .05468 .176 .123 .05534 .177 .124 .05600 .178 .125 .05666 .179 .126 .05732 .180 .127 .05799 .181 .128 .05865 .182 .129 .05932 .183 .130 .05999 .184 .131 .06067 .185 .132 .06134 .186 .133 .06202 .187 .134 .06270 .188 .135 .06338 .189 .136 .06407 .190 .137 .06476 .191 .138 .06544 .192 .139 .06614 .193 .140 .06683 .194 .141 .06752 .195 .142 .06822 .196 .143 .06892 .197 .144 .06962 .198 .145 .07032 .199 .146 .07103 .200 .147 .07174 .201 .148 .07245 .202 .149 .07316 .203 .150 .07387 .204 .151 .07458 .205 .152 .075.30 .206 .153 .07602 .207 .154 .07674 .208 .155 .07746 .209 .156 .07819 .210 .157 .07892 .211 .158 .07964 .212 .159 .08038 .213 .160 .08111 .214 .161 .08184 .215 .162 .08258 .216 AREAS OF THE SEGMENTS OF A CIRCLE. Table — (Continued). Versed Sine, Seg. Area. Versed Sine. Ses- Area. ^'ersed Siue. Seg. Area. Versed Sine. Seg. .217 .12563 .272 .17286 .327 .22321 .382 ,2" .218 .12645 .273 .17375 .328 .22415 .383 .2"; .219 .12728 .274 .17464 .329 .22509 .384 .2; .220 12811 .275 .17554 .330 .22603 .385 .27 .221 .12894 .276 .17643 .331 .22697 .386 .27 .222 .12977 .277 .17733 .332 .22791 .387 .2S .223 .13060 .278 .17822 .333 .22885 .388 .2S .224 .13143 .279 .17912 ..334 .22980 .389 .2S .225 .13227 .280 .18001 .335 .23074 .390 .28 226 .13310 .281 .18091 .336 .23168 .391 .28 227 .13394 .282 .18181 .337 .23263 .392 .28 228 .13478 .283 .18271 .338 .23358 .393 .28 229 .13562 .284 .18361 .339 .23452 .394 .28 230 .13646 .285 . 18452 .340 .23547 .395 .28 231 .13730 .286 . 18542 .341 .23642 .396 .28 232 .13815 .287 .18632 .342 .23736 .397 .29 233 .13899 .288 .18723 .343 .23831 .398 .29 234 .13984 .289 .18814 .344 .23926 .399 .29 235 .14068 .290 .18904 .345 .24021 .400 .29 236 .14153 .291 .18995 .346 .24116 .401 .29 237 .14238 .292 .19086 .347 .24212 .402 .29 238 .14323 .293 .19177 .348 .24307 .403 .29 239 .14409 .294 .19268 .349 .24402 .404 .29 240 . 14494 .295 .19359 .350 .24498 .405 .29 241 .14579 .296 .19450 .351 .24593 .406 .29 242 .14665 .297 .19542 .352 .24688 .407 .30 243 .14751 .298 .19633 .353 .24784 .408 .30 244 .14837 .299 .19725 .354 .24880 .409 .30 245 . 14923 .300 .19816 .355 .24975 .410 .30, 246 .15009 .301 .19908 .356 .25071 .411 .30^ 247 .15095 .302 .20000 .357 .25167 .412 .30. 248 .15181 .303 .20092 .358 .25263 .413 .30( 249 .15268 .304 .20184 .359 .25359 .414 .30' 250 .15354 .305 .20276 .360 .25455 .415 .30^ 251 .15441 .306 .20368 .361 .2555] .416 .30j 252 .15528 .307 .20460 .362 .25647 .417 .311 253 .'5614 .308 .20552 .363 .25743 .418 .31 254 .15701 .309 .20645 .364 .25839 .419 .311 255 .15789 .310 .20737 .365 .25935 .420 .3n 256 .15876 .311 .20830 .366 .26032 .421 .31^ 257 .15963 .312 .20922 .367 .26128 .422 .31." 258 .16051 .313 .21015 .368 .26224 .423 .3U 259 .16138 .314 .21108 .369 .26321 .424 .31( 260 .16226 .315 .21201 .370 .26417 .425 .317 261 .16314 .316 .21294 .371 .26514 .426 .316 262 .16401 .317 .21387 .372 .26611 .427 .3U 263 .16489 .318 .21480 .373 .26707 .428 .32C 264 .16578 .319 .21573 .374 .26804 .429 .32: 265 .16666 .320 .21666 .375 .26901 .430 .325 266 .16754 .321 .21759 .376 .26998 .431 .322 267 .16843 .322 .21853 .377 .27095 .432 .32^ 268 .16931 .323 .21946 .378 .27192 .433 .32£ 269 .17020 .324 .22040 .379 .27289 .434 .326 270 .17108 .325 .22134 .380 .27386 .435 .327 271 .17197 .326 .22227 G .381 .27483 .436 .328 74 AREAS OF THE SEGMENTS OF A CIRCLE. Table — (Continued). Versed Sine. Seg. Ai-ea. Versed Sine, Seg. Area. Versed Sine. S&g. Area. Versed Sine. Seg. Area, ,437 .32986 .453 .34576 .469 .36171 .485 .37770 ,438 .33085 .454 .34676 .470 .36271 .486 .37870 .439 .33185 .455 .34775 .471 .36371 .487 .37970 .440 .33284 .456 .34875 .472 .36471 .488 .38070 .441 .33383 .457 .34975 .473 .36571 .489 .38169 .442 .33482 .458 .35074 .474 •36671 .490 .38269 .443 .33582 .459 .35174 .475 .36770 .491 .38369 .444 .33681 .460 .35274 .476 .36870 .492 .38469 . 445 .33781 .461 .35373 .477 .36970 .493 .38569 .446 ,447 .33880 .33979 ,462 .463 .35473 .35573 .478 .479 .37070 .37170 .494 .495 .38669 .38769 .448 .84079 .464 .35673 .480 .37276 .496 .38869 ,449 .34178 .465 .35772 .481 .37370 .497 .38969 .450 .34278 .466 .35872 .482 .37470 .498 .39069 .451 .34377 .467 .35972 .483 .37570 .499 .39169 ,452 .34477 .468 .36072 .484 .37670 .500 .39269 USE OF THE ABOVE TABLE. To find the Area of a Segment of a Circle, Rule —Divide the height or versed sine by the diameter of tlie cu-cle, and find the quotient in the column of versed sines. Take the area noted in the next col- umn, and multiply it by the square of the diameter, and it will give the area re- quired. Example.— Required the area of a segment; its height being 10, and the diame- ter of the circle 50 feet. ^ r.<^r. -^ n 10-r-50=.2, and .2, per table, = .11182 ; then .11182X502 = 2^9.oo Ans. LENGTHS OF CIRCULAR ARCS. 75 Height. Table of the Lengths of Circular Arcs. Length. Height. I Length. Height. Length. Height. .100 1.0265 .156 1.0637 .212 1.1158 .268 1.1816 .101 1.0270 .157 1.0645 .213 1.1169 .269 1.1829 .102 1.0275 .158 1.0653 .214 1.1180 .270 1.1843 .103 1.0281 .159 1.0661 .215 1.1190 .271 1.1856 .104 1.0286 .160 1.0669 .216 1.1201 .272 1.1869 .105 1.0291 .161 1.0678 .217 1.1212 .273 1.1882 .106 1.0297 .162 1.0686 .218 1.1223 .274 1.1897 .107 1.0303 .163 1.0694 .219 1.1233 .275 1.1908 .108 1.0308 .164 1.0703 .220 1.1245 .276 1.1921 .109 1.0314 .165 1.0711 .221 1.1256 .277 1.1934 .110 1.0320 .]66 1.0719 .222 J. 1266 .278 1.1948 .111 1.0325 .167 1.0728 .223 1.1277 .279 1.1961 .112 1.0331 .168 1.0737 .224 1.1289 .280 1.1974 .113 1.0337 .169 1.0745 .225 1.1300 .281 1.1989 .114 1.0343 .170 1.0754 .226 1.1311 .282 1.2001 .115 1.0349 .171 1.0762 .227 1.1322 .283 1.2015 .116 1.0355 .172 1.0771 .228 1.1333 .284 1.2028 .117 1.0361 .173 1.0780 .229 1.1344 .285 1.2042 .118 1.0367 .174 1.0789 .230 1.1356 .286 1.2056 .119 1.0373 .175 1.0798 .231 1.1367 .287 1.2070 .120 1.0380 .176 1.0807 .232 1.1379 .288 1.2083 .121 1.0386 .177 1.0816 .233 1.1390 .289 1.2097 .132 1.0392 .178 1.0825 .234 1.1402 .290 1.2120 .123 1.0399 .179 1.0834 .235 1.1414 .291 1.2124 ,124 1.0405 .180 1.0843 .236 1.1425 .292 1.2138 .125 1.0412 .181 1.0852 .237 1 . 1436 .293 1.2152 .126 1.0418 .182 1.0861 .238 1.1448 .294 1.2166 .127 1.0425 .183 1.0870 .239 1.1460 .295 1.2179 .128 1.0431 .184 1.0880 .240 1.1471 .296 1.2193 .129 1.0438 .185 1.0889 .241 1.1483 .297 1.2206 .130 1.0445 .1-86 1.0898 .242 1 . 1495 .298 1.2220 .131 1.0452 .187 1.0908 .243 1.1.507 .299 1.2235 .132 1.0458 .188 1.0917 .244 1.1519 .300 1.2250 .133 1.0465 .189 1.0927 .245 1.1531 .301 1.2264 .134 1.0472 .190 1.0936 .246 1.1543 .302 1.2278 .135 1.0479 .191 1.0946 .247 1.1555 .303 1.2292 .136 1.0486 .192 1.0956 .248 1.1567 .304 1.2306 .137 1.0493 .193 1.0965 .249 1.1579 ..305 1.2321 .138 1.0500 .194 1.0975 .250 1.1591 .306 1.2335 .139 1.0508 .195 1.0985 .251 1.1603 .307 1.2349 .140 1.0515 .196 1.0995 .252 1.1616 .308 1.2364 .141 1.0522 .197 1.1005 .253 1.1628 .309 1.2378 .142 1.0529 .198 1.1015 .254 1.1640 .310 1.2393 .143 1.0537 .199 1.1025 .2.55 1.1653 .311 1.2407 .144 1.0544 .200 1.1035 .256 1.1665 .312 1.2422 145 1.0552 .201 1.1045 .257 1.1677 .313 1.2436 .146 1.0559 .202 1.1055 .258 1.1690 .314 1.2451 147 1.0567 .203 1.1065 .259 1.1702# .315 1.2465 148 1.0574 .204 1.1075 .260 1.1715 .316 1.2480 149 1.0582 .205 1.1085 .261 1.1728 .317 1.2495 150 1.0590 .206 1.1096 .262 1.1740 .318 1.2510 151 1.0597 .207 1.1006 .263 1.1753 .319 1.2524 152 1.0605 .208 1.1117 .264 1.1766 .320 1.2539 153 1.0613 .209 1.1127 .265 1.1778 .321 1.25.54 154 1.0621 .210 1.1137 .266 1.1791 .322 1.2569 .155 1.0629 1 .211 1 1.1148 .267 1.1804 1 .323 1.2584 Length. 76 LENGTHS OF CIRCULAR ARCS. Table— (Continued). Length. Height. Length. .2599 .369 .2614 .370 .2629 .371 .2644 .372 .2659 .373 1 .2674 .374 1 .2689 .375 ! .2704 .376 i .2720 .377 ! .2735 .378 ! .2750 .379 .2766 .380 ' .2781 .381 .2786 .382 .2812 .383 .2827 .384 .2843 .385 .2858 .386 .2874 .387 .2890 .388 .2905 .389 .2921 .390 .2937 .391 .2952 .392 .2968 .393 .2984 .394 .3000 .395 .3016 .396 .3032 .397 .3047 .398 .3063 .399 .3079 .400 .3095 .401 .3112 .402 .3128 .403 .3144 .404 .3160 .405 .3176 .406 .3192 .407 .3209 .408 .3225 .409 .,3241 .410 .3258 .411 L.3274 .412 1.3291 1.3307 1.3323 1.3340 1.3356 1.3373 1.3390 1.3406 1.3423 1.3440 1.3456 1.3473 1.3490 1.3507 1.3524 1.3541 1.3558 1.3574 1.3591 1.3608 1.3625 1.3643 1.3660 1.3677 1.3694 1.3711 1.3728 1.3746 1.3763 1.3780 1.3797 1.3815 1.3832 1.3850 1.3867 1.3885 1.3902 1.3920 1.3937 1.3955 1.3972 1.3990 1.4008 1.4025 1.4043 Height. Length. Height. .413 1.4061 .457 .414 1.4079 .458 .415 1.4097 .459 .416 1.4115 .460 .417 1.4132 .461 .418 1.4150 .462 .419 1.4168 .463 .420 1.4186 .464 .421 1.4204 .465 .422 1.4222 .466 .423 1.4240 .467 .424 1.4258 .468 .425 1.4276 .469 .426 1.4205 .470 .427 1.4313 .471 .428 1.4331 .472 .429 1.4349 .473 .430 1.4367 .474 .431 1.4386 .475 .432 1.4404 .476 .433 1.4422 .477 .434 1.4441 .478 .435 1.4459 .479 .436 1.4477 .480 .437 1.4496 .481 438 1.4514 .482 .439 1.4533 .483 .440 1.4551 .484 .441 1.4570 .485 .442 1.4588 .486 .443 1.4607 .487 .444 1.4626 .488 .445 1.4644 .489 .446 1.4663 .490 .447 1.4682 .491 .448 1.4700 .492 .449 1.4719 .493 .450 1.4738 .494 .451 1.4757 .495 .452 1.4775 .496 .453 1.4794 .497 .454 1.4813 .498 .455 1.4832 .499 .456 1.4851 .500 To find tlwLengVi of an Arc of a Circle by the foregoing Table. Bt-i f -Divide the heialit by the base, find the quotient in the column of heights, the arc required. u ■ „ mn ExAMPLE.-What is the length of an arc of a circle, the span or base being 100 feet, and the height 25 feet "? ,.• v«^ w inn 25-M00= .25, and .25, per table, gives 1.1591 ; which, being mulUphed by 100, = 115.9100, the length. LENGTH OF AN ELLIPTIC ARC. 77 Note. — When great accuracy is required, if, in the division of a height by the base, there should de a remainder. Find the lengths of the cuives from the two nearest tabular heights, and sub- tract the one length from the other. Then, as the base of the arc of which the length is required is to the remainder in the operation of division, so is the differ- ence of the lengths of the curves to the complement required, to be added to the length. Example.— What is the length of an arc of a circle, the base of which is 35 feet and the height or versed sine 8 feet 1 * 8-^35 =.228|5, .228 = 1.1333, .229 = 1.1344, 1.1333X35 = 39.6655, 1.1344X35 = 39.7040, 39.7040—39.6655 = .0385, difference of lengths. Hence, as 35 : 20 : : .0385 : .0220, the length for the remainder, and .0220-f- 39.6655 = 39.6875, and .6875X12, for inches = 8^ making the length of the arc 39 feet 84: inches. To find the length of an Elliptic Curve which is less than half of the entire Figure, Geometrically. — Let the curve of which the length is required he aba. Extend the versed sine hd to meet the centre of the curve in e. Draw the line c e, and from e, with the distance e b, describe bh; bisect c A in e, and from e, with the radius e i, describe k i, and it is equal half the arc a be. To find the length when the Curve is greater than half the entire Figure* Rule. — Find by the above problem the curve of the less portion of the figure, and subtract it from the circumference of the ellipse, and the remainder will be the length of the curve required. G2 78 LENGTHS OF SEMI-ELLIPTIC ARCS. Height. Table of the Lengths of Semi-elliptic Arcs. Lenstli. Height. ) Length. Height. Length. Height. Length. .100 .101 .102 .103 .104 .105 .110 .115 .120 .125 .130 .135 .140 .145 .150 .155 .160 .165 .170 .175 .180 .185 .190 .195 .200 .205 .210 .215 .220 .225 .230 .235 .240 .245 .250 .255 .260 .265 .270 .275 .280 .285 .290 .295 .300 .305 .310 1.0416 1.0426 1.0436 1.0446 1.0456 1.0466 1.0516 1.0567 1.0618 1.0669 1.0720 1.0773 1.0825 1.0879 1.0933 _ 0989 1.1045 1.1106 1.1157 1.1213 1270 1327 1384 1442 1501 1560 1620 1680 1.1741 1.1802 1.1864 1.1926 1.1989 1.2051 1.2114 1.2177 1.2241 1.2306 1.2371 1.2436 1.2501 1.2567 1.2634 1.2700 1.2767 1.2834 1.2901 315 320 325 330 ,335 ,340 ,345 .350 .355 .360 .365 .370 .375 .380 .385 .390 .395 .400 .405 .410 .415 .420 .425 .430 .435 .440 .445 .450 .455 .460 .465 .470 .475 .480 .485 .490 .495 .500 .505 .510 .515 .520 .525 .530 .535 .540 1.2960 1.3038 1.3106 1.3175 1.3244 1.3313 1.3383 1.3454 1.3525 1.3597 1.3669 1.3741 1.3815 1.3888 1.3961 1.4034 1.4107 1.4180 1.4253 1.4327 1.4402 1.4476 1.4552 1.4627 1.4702 1.4778 1.4854 1.4931 1.5008 1.5084 1.5161 1.5238 1.5316 1.5394 1.5472 1.5550 1.5629 1.5709 1.5785 1.5863 1.5941 1.6019 1.6097 1.6175 1.6253 1.6331 545 550 555 560 ,565 ,570 ,575 .580 .585 .590 .595 .600 .605 .610 .615 .620 .625 .630 .635 .640 .645 .650 .655 .660 .665 .670 .675 .680 .685 .690 .695 .700 .705 .710 .715 .720 .725 .730 .735 .740 .745 .750 .755 .760 .765 .770 1.6409 1.6488 1.6567 1.6646 1.6725 1.6804 1.6S83 1.6963 1.7042 1.7123 1.7203 1.7283 1.7364 1 . 7444 1.7525 1.7606 1.7687 1.7768 1.7850 1.7931 1.8013 1.8094 1.8176 1.8258 1.8340 1.8423 1.8505 1.8587 1.8670 1.8753 1.8836 1.8919 1.9002 1.9085 1.9169 1.9253 1.9337 1.9422 1.9506 1.9599 1.9675 1.9760 1.9845 1.9931 2.0016 2.0102 775 780 785 790 795 ,800 ,805 ,810 .815 .820 .825 .8.30 .835 .840 .845 .850 .855 .860 .865 .870 .875 .880 .885 .890 .895 .900 .905 .910 .915 .920 .925 .930 .935 .940 .945 .950 .955 .960 .965 .970 .975 .980 .985 .990 .995 .1000 2.0187 2.0273 2.0360 2.0446 2.0533 2.0620 2.0708 2.0795 2.0883 2.0971 2.1060 2.1148 2.123T 2.1326 2.1416 2.1505 1595 1685 1775 1866 1956 2.2047 2.2139 2.2230 2.2322 2.2414 2.2506 2.2597 2.2689 2.2780 2.2872 2.2964 2.3056 2.3148 2.3241 2.3335 2.3429 2.3524 2.3619 2.3714 2.3810 2.3906 2.4002 2.4098 2.4194 2.4291 I _^ To find the Length of the Curve of a Right Semi-Ellipse. Proceed with the foregoing table by the rules for ascertaining the lengths of cir- cular arcs, page 76. Example.— What is the length of the curve of the arch of a bridge, the spam being 70 feet, and the height 30.10 feet 1 30.10^70 =.430 = per table, 1.4627, and 1.4627X70 = 102.3890, the length re quired. SEMI-ELLIPTIC ARCS. 79 When the Curve is not that of a Right Semi-Ellipse^ the height being half of the Transverse Diameter. Rule. — Divide half the base by twice the height ; then proceed as in the forego- ing example, and multiply the tabular length by twice the height, and the product will be the length required. Example.— What is the length of the profile of arch (it being that of a semi-el- lipse), the height measuring 35 feet and the base 60 feet 1 60-^2=1:30-7 -35x2 = .428, the tabular length of which is 1.4597. Then, 1.4597x35X2=102.1790, the length required. Note. — When the quotient is not given in the column of heights, divide the dif- ference between the two nearest heights by .5 ; multiply the quotient by the excess of the height given and the height in the table first above it, and add this sum to the tabular area of the least height. Thus, if the height is 118, .115, per table, = 1.0567 .120, " = 1.0618 .0051-S-.5 = .00102 X (118 — 115) = .00306, which, added to 1.0567 = 1.05976, the length for 118. 80 AREAS OF THE ZONES OF A CIRCLE. Table of the Areas of the Zo7ies of a Circle. Height. Area. Height. Area. i Height. Area. Height. Area. .001 .00100 .115 .11397 .245 .23480 .375 .33604 .002 .00300 .120 .11883 .250 .23915 .380 .33931 .003 .00300 .125 .12368 .255 .24346 .385 .34253 .004 .00400 .130 .12852 .260 .24775 .390 .34569 .005 .00500 .135 .13334 .265 .25201 .395 .34879 .010 .01000 .140 .13814 .270 .25624 .400 .35182 .015 .01499 .145 .14294 .275 .26043 .405 .35479 .020 .01999 .150 .14772 .280 .26459 .410 .35769 .025 .02499 .155 .15248 .285 .26871 .415 .36051 .030 .02998 .160 .15722 .290 .27280 .420 .36326 .035 .03497 .165 .16195 .295 .27686 .425 .36594 .040 .03995 .170 .16667 .300 .28088 .430 .36853 .045 .04494 .175 .17136 .305 .28486 .435 .37104 .050 .04992 .180 .17603 .310 .28880 .440 •37346 .055 .05489 .185 .18069 .315 .29270 .445 .37579 .060 .05985 .190 .18532 .320 .29657 .450 .37805 .065 .06482 .195 .18994 .325 .30039 .455 .38015 .070 .06977 .200 .19453 .330 .30416 .460 .38216 .075 .07472 .205 .19910 .335 .30790 .465 .38466 .080 .07965 .210 .20365 .340 .31159 .470 .38853 .085 .08458 .215 .20818 .345 .31523 .475 ..38747 .090 .08951 .220 .21268 .350 .31883 .480 .38895 .095 .09442 .225 .21715 .355 .32237 .485 .39026 .100 .09933 .230 .22161 .360 .32587 .490 .39137 .105 .10422 .235 .22603 .365 .32931 .495 .39223 .110 .10910 .240 .23t)43 .370 .33270 .500' .39270 To find the Area of a Zone hy the above Tahle. Rule 1. — When the zone is greater than a part of a semicircle, take the height on each side of the diameter of the circle, of which it is a part; divide the heights by the diameter ; find the respective quotients in the column of heights, and take out the areas oppot^ite to them, multiplying the areas thus found by the square of the diameter or chord, and the products, added together, will be the area required. Note. — When the quotient is not given in the column of heights, divide the differ- ence between the two nearest heights by 5, and multiply the quotient by the excess be- tween the height giver^and the height in the table first above it, and add this sum to the tabular area of the least height. Thus, if the height is .333, .30416— .30790= .00374-^-5= .000748x3 (excess of 333 over 330) = .002244+.30416 = .306404, the area for 333. Example. — What is the area of zone, the diameter of the circle being 100, and the heights respectively 20 and 10, upon each side of it ? 20-1-100 = .200, and 200, per table, = .19453x1002 = 1945.3. 10-J-lOO = .100, and 100, per table, = .09933X 100^ = 993.3. Hence, 1945.3+993.3 = 2938.6 ^ns. RULE.- height. 'When the zone is less than a semicircle, proceed as in rule 1 for one Example. — What is the area of a zone, the longest chord being 10, and the height 4 ? 4-r-lO = .400 = .35182X 10^ = 35.182 .^ns. MENSURATION OF SOLIDS. 81 MENSURATION OF SOLIDS. 23. OF CUBES AND PARALLELOPIPEDONS. 24. To find the Solidity of a Cube— fig. 23, Rule. — Multiply the side of the cube by itself, and that product again by the side, and this last product will be the solidity. To find the Solidity of a ParaUelopipedon—fig. 24. Rule. — Multiply the length by the breadth, and that product by the depth, and this product is the solidity. OF REGULAR BODIES. To find the Solidity of any Regular Body. Rule. — Multiply the tabular solidity in the following table by the cube of the linear edge, and the product is the solidity. Table of the Solidities of the Regular Bodies when the Lirwar Edge is Number of Sides, Names. Solidities. 4 Tetrahedron. 0.11785 6 Hexahedron. 1. 00000 8 Octahedron. 0.47140 12 Dodecahedron. 7.66312 20 Icosahedron. 2.18169 OF CYLINDERS, PRISMS, AND UNGULAS. 25- 28. ^ 27 To find the Solidity of Cylinders^ Prisms, and Ungulas—figs. 25, 26, aTid 27. Rule. — Multiply the area of the base by the height, and the prod- uct is the solidity. 82 MENSURATION OF SOLIDS. To find the Solidity of an Ungula^fig. 28, when the section passes obliqvAy through the cylinder^ abed. Rule.— Multiply the area of the base of the cylinder by half the sum of the greater and less heights a e, of of the ungula, and the product is the solidity. When the Section passes through the base of the Cylinder and one of its sides— fig. 29, a be. Rule.— Frorii | of the cube of the right sine a d, of half the arc ag of the base, subtract the product of the area of the base, and the cosine df of said half arc. Multiply the difference thus found by the quotient of the height, divided by the versed sine, and the prod- uct is the solidity. Whe7i the S2ction passes ooliquely through both ends of the Cylinder^ adc e—fig. 30. Rule.— Find the solidities of theungulas adce and dbc, and the difference is the solidity required {conceiving the section to be con- tinued till it meets the side of the cylinder). Note.— For rules to ascertain the solidity of conical ungulas, see RyarCs Bonny- castle's Mensuration^ page 136 (1839). OF CONES AND PYRAMIDS. 33. c a To find the Solidity of a Cone or Pyramid—figs. 31 and 33. Rule.— Multiply the area of the base by the height c d, and I the product vi^ill be the content. To find the Solidity of the Frustrum of a Cone— fig. 32. Rule. — Divide the difference of the cubes of the diameters ah.cd of the two ends by the difference of the diameters ; this quotient, multiplied.by .7854, and again by J of the height, will give the so- lidity. To find tlie Solidity of the Frustrum of a Pyramid— fig. 34. Rule.— Add to the areas of the two ends of the frustrum the square root of their product, and this sum, multiplied by J of the height a b, will give the solidity. MENSURATION OF SOLIDS. 83 OF WEDGES AND PRISMOIDS. d To find the Solidity of a Wedge— fig. 35. Rule.— To the length of the edge of the wedge de add twice the length of the back a h ; multiply this sum by the height of the wedge df and then by the breadth of the back c a, and i of the product will be the solid content. To find the Solidity of a Prismoid—fig. 36. Rule.— Add the areas of the two ends ahc, def and four times the middle section g h, parallel to them, together ; multiply this sum by ^ of the height, and it will give the solidity. OF SPHERES. c To find the Solidity of a Sphere— fig. 37. Rule —Multiply the cube of the diameter by .5236, and the prod- uct IS the solidity. ^ To find th£ Solidity of a Spherical Segment— fig. 38. Rule.— To three times the square of the radius of its base a b, add the square of its height cb; then multiply this sum by the height and the product by .5236. ^ To find the Solidity of a Spherical Zone or Frustrum—fig. 39. Rule.—To the sum of the squares of the radius of each end ab.cd, add I of the square of the height b d of the zone ; and this th"^' rd^^^ ^^ ^^^ ^^^^^^' ^"^ ^^^ product by 1.5708, will give 84. MENSTJRATION OF SOLIDS. OF SPHEROIDS To find the Solidity of a Spheroid— fig. 40. Rule.— Multiply the square of the revolving axis cdhy the fixed axis ah; the product, multiplied by .5236, will give the solidity. To find the Solidity of the Segment of a Spheroid— figs, 41 and 42, I^uLE. When the base efts circular, or parallel to the revolving axis cd, fig. 41. Multiply the fixed atis a^> by 3, the height of the seg- ment^^a g by 2, and subtract the one product from the other ; then multiply the rem.ainder by the square of the height of the segment, and the product by .5236. Then, as the square of the fixed axis is to the square of the revolving axis, so is the last product to the content of the segment. Rvh^.—When the base ef is perpendicular to the revolving axis cdy fig. 42. Multiply the revolving axis by 3, and the height of the seg- ment c^ by 2, and subtract the one from the other; then multiply the rentainder by the square of the height of the segment, and the product by .5236. Then, as the revolving axis is to the fixed axis, so is the last product to the content. To find the Solidity of the Middle Frustrum of a Spheroid— figs, 43 and 44. Rule.— WAcTi the ends ef and gh are circular, or parallel to the re- volving axis c d, fig. 43. To twice the square of the revolving axis c d, add the square of the diameter of either end, ef or g h ; then multi- ply this sum by the length a ^ of the frustrum, and the product again i by .2618, and this will give the solidity. Rule.— T7/ien the ends ef and gh are elliptical or perpendicular to the revolving axis c d, fig. 44. To twice the product of the transverse and conjugate diameters of the middle section ab, add the product of the transverse and conjugate of either end ; multiply this sum by the length Ik of the frustrum, and the product by .2618, and this will give the solidity. * Spheroids are either Prolate or Oblate. They are prolate when produced by th revolution of a semi-ellipse about its transverse diameter, and oblate when produc( by an ellipse revolving about its conjugate diameter. MENSURATION OF SOLIDS. 85 OF CIRCULAR SPINDLES. 46. \ ! To find the Solidity of a Circular S'pindle—fig. 45. Rule.— Multiply the central distance oe by half the area of the revolving segment a c ef. Subtract the product from J of the cube fe of half the length ; then multiply the remainder by 12.5664 (or four times 3.1416), and the product is the solidity. To find the Solidity of the Frustrum, or Zone of a Circular Spindle — fig' 46. Rule.— From the square of half the length h i of the whole spin- dle, take I of the square of half the length n i of the frustrum, and multiply the remainder by the said half length of the frustrum ; mul- tiply the central distance o i by the revolving area* which generates the frustrum ; subtract the last product from the former, and the remainder, multiplied by 6.2832 (or twice 3.1416), will give the so- lidity. ELLIPTIC SPINDLES. 48, To find the Solidity of an Elliptic Spindle— fig. 47. Rule. — To the square of the greatest diameter a b, add the square of twice the diameter ef at i of its length ; multiply the sum by the length, and the product by .1309, and it will give the solidity nearly. To find the Solidity of a Frustrum or Segment of an Elliptic Spindle — fig. 48. Rule. — Proceed as in the last rule for this or any other solid formed by the revolution of a conic section about an axis, viz. : Add together the squares of the greatest and least diameters, ab.cd, and the square of double the diameter in the middle, between the two ; multiply the sum by the length ef and the product by .1309, and it will give the solidity. Note. — For all such solids, this rule is exact when the body is formed by the conic •action, or a part of it, revolving about the axis of the section, and will always be very near when the figure revolves about another line. * The area of the frustrum can be obtained by dividing its central plane into seg- ments of a circle, and triangles or parallelograms. H 86 MENSURATION OF SOLIDS. OF PARABOLIC CONOIDS AND SPINDLES. J? 50. a!^-—^i. To find tlie Solidity of a Paraholic Co7wid*—fig. 49. Rule.— Multiply the area of the base dchy half the altitude fg, and the product will be the solidity. Note.— This rule will hold for any seg-ment of the paraboloid, whether the base be perpendicular or oblique to the axis of the solid. To find the Solidity of a Frustrum of a Paraboloid— fig. 49. Rule. — Multiply the sum of the squares of the diameters ah and dchy the height ef, and the product by .3927. To find the Solidity of a Parabolic Spindle— fig. 50. Rule. — Multiply the square of the diameter ah hy the length rfc, and the product by .4188, and it will give the solidity. To find the Solidity of the Middle Frusf.rum of a Parabolic Spindle— fig- 51. Rule.— Add together 8 times the square of the greatest diameter- c8 7370.7 CIRCrMFEEENCES OF CIRCLES. 95 Circumferences of Circles, from 1 to 100. Dumeter Circumference Diameter. Circumference Diameter Circumference .1 Diameter. Circumfer'ce I .0490 5. 15.70 12. 37.69 19. 59.69 .0981 .1 16.10 4 38.09 4 60.08 ■h .i 16.49 4 38.48 ,i 60.47 fV .1963 •8 16.88 .1 38.87 60.86 1 .3926 •t 17.27 • i 39.27 ,i- 61.26 . 3 17.67 •1 39.66 ^5 61.65 ■h .5890 • i 18.06 1 40.05 '1 62.04 I .7854 7 • 3 18.45 •1 40.44 *| 62.43 5 .9817 6. 18.84 13. 40.84 20.' 62.83 18" 'i 19.24 1 • 8 41.23 4 63.22 f 1.178 .i 19.63 41.62 'i 63.61 tV 1.374 • 8 20.02 •f 42.01 4 64.01 • i 20.42 •i 42.41 A 64.40 i 1.570 .1 20.81 •I 42.80 5 • 8 64.79 9 1.767 •f 21.20 .4 43.19 .! 65.18 5 • i 21.57 7 • 8 43.58 4 65.58 ■ff 1.963 7. 21.99. 14. 43-. 98 21.' 65.97 1 I TS" 2.159 1 •3 22.38 .i 44.37 'i 66.36 3 2.356 •i 22.77 .i 44.76 •i 6^.75 T '-S 23.16 .1 45.16 67.15 tI 2.552 'i 23.56 A 45.55 • 1 67.54 7 2.748 •f 23.95 .1 45.94 • 8 67.93 1 c 2.945 24.34 .i 46.33 'i 68.32 tI 7 • S 24.74 .1 46.73 •1 68.72 1. 3.141 8. 25.13 15. 47.12 22.] 69.11 3.534 .1 25.52 .1 47.51 69.50 3.927 •t 25.91 .i 47.90 ^_ 69.90 4.319 26.31 .i 48.30 .- 70.29 4.712 • i 26.70 A 48.69 • r 70.68 5.105 .1 27.09 .8 49.08 •8 71.07 5.497 .1 27.48 .1 49.48 .4 71.47 5.890 7 •8 27.88 7 .8 49.87 • i 71.86 2 6.283 9. 28.27 16. 50.26 23.' 72.25 6.675 •8 28.66 .i 50.65 .1 72.64 7.068 •5 29.05 .i 51.05 .i 73.04 7.461 .- 29.45 .f 51.44 .t 73.43 7.854 .1 29.84 51.83 .i 73.82 8.246 •1 30.23 .1 52.22 .1 74.21 8.639 •1 30.63 .| 52.62 .i 74.61 9.032 • I 31.02 .1 53.01 'I 75. 3 9.424 10. 31.41 17. 53.40 24.] 75.39 9.817 •1 31.80 .1 53.79 75.79 10.21 • 1 32.20 .i 54.19 *^ 76.18 10.60 •^ 32.59 .1 54.58 •f 76.57 10.99 32.98 .•^ 54.97 76.96 11.38 •1 33.37 .1 55.37 • 1 77.36 11.78 • ^ 33.77 .1 55.76 .? 77.75 12.17 •1 34.16 7 .8 56.16 • ? 78.14 Jc 12.56 11. 34.55 18. 56.54 25. 78.54 12.95 'i 34.95 .1 56.94 4 78.93 13.35 •i 35.34 57.33 'i 79.32 13.74 •5 35.73 .i 57.72 .f 79.71 14.13 •^ 36.12 .1 58.11 80.10 14.52 •1 36.52 .1 .58.51 •1 80.50 14.92 4 36.91 37.30 .i 58.90 ^ 80.89 i 15.31 7 7 •5 59.29 4 81.28 96 CIRCUMFERENCES OF CIRCLES. Table— (Continued). Diameter Circumference. Diameter. ! Circumference 81.68 82.07 82.46 82.85 83.25 83.64 84.03 84.43 84.82 85.21 85.60 86. 86.39 86.78 87.17 87.57 87.96 88.35 88.75 89.14 89.53 89.92 90.32 90.71 91.10 91.49 91.89 92.28 92.67 93.06 93.46 93.85 94.24 94.64 95.03 95.42 95.81 96.21 96.60 96.99 97.38 97.78 98.17 98.56 98.96 99.35 99.74 100.1 100.5 100.9 101.3 101.7 102.1 102.4 102.8 103.2 33 34 35 36 37 38 39 103.6 104. 104.4 104.8 105.2 105.6 106. 106.4 106.8 107.2 107.5 107.9 108.3 108.7 109.1 109.5 109.9 110.3 110.7 111.1 111.5 111.9 112.3 112.7 113. 113.4 113.8 114.2 114.6 115. 115.4 115.8 116.2 116.6 117. 117.4 117.8 118.2 118.6 118.9 119.3 119.7 120.1 120.5 120.9 121.3 121.7 122. 1 122.5 122.9 123.3 123.7 124. 124.4 124.8 125.2 Diameter. Circumference. Diame ter. Circumfer'ce 40. 125.6 47. 147.6 -I 126. - 148. i 126.4 5 148.4 126.8 j; 148.8 1 127.2 ^ 149.2 1 127.6 1 149.6 I 128. f 150. i 128.4 1 150.4 41 128.8 48 150.7 1 129.1 - 151.1 1 129.5 _ 151.5 I 129.9 i 151.9 8 130.3 I 152.3 Jl 130.7 1 152.7 S .1 131.1 1 153.1 2 131.5 i 153.5 42 / 131.9 49 153.9 'i 132.3 ... 154.3 132.7 ,]. 154.7 f 133.1 •8 155.1 I 133.5 ,| 155.5 1 133.9 5 1.55.9 i 134.3 .1 166.2 ■^ 134.6 7 • 8 156.6 43 135. 50 157. 'i 135.4 •r 157.4 135.8 . :: 157.8 2. 136.2 /A 158.2 I 136.6 .- ■ 158.6 b 137. ,'.. 1.59. I 137.4 Jy 159.4 I 137.8 • 1 ■ 159.8 44 8 138.2 51 160.2 I 138.6 *i 160.6 139. • i 161. 3 139.4 161.3 ^- 139.8 .^ 161.7 1 140.1 1 162.1 140.5 162.5 7 140.9 i 162.9 45 g 141.3 52 163.3 1 141.7 'i 163.7 i 142.1 i 164.1 142.5 8 164.5 1 142.9 i 164.9 1 143.3 5 165.3 1 143.7 1 165.7 144.1 7 5 166.1 46 144.5 53 166.5 i 144.9 'l 166.8 i 145.2 4 167.2 3 8 145.6 8 167.6 .V 146. 1 168. 146.4 168.4 I 146.8 3 168.8 i 147.2 i 169.2 CIRCUMFERENCES OF CIRCLES. 97 Table — (Continued). Diamete r Circumference Diameter 1 Circumference Diameter. Circumference . Diameter [Circu 54. 169.6 61. 191.6 68. 213.6 75. 23 .| 170. .1 192. •1 214. •8 23 •i 170.4 •i 192.4 'i 214.4 • i 23 3 •8 170.8 •f 192.8 •8 214.8 .f 23 171.2 193.2 •t 215.1 •t 23 •1 171.6 •1 193.6 215.5 *t 23 .1 172. •1 193.9 •1 215.9 .i 23 .J 172.. 3 •i 194.3 .| 216.3 7 23 55. 172.7 62. 194.7 69. 216.7 76? 23 'i 173.1 'i 195.1 •1 217.1 •f 23 .1 173.5 .1 195.5 •1 217.5 23 3 •^ 173.9 195.9 •i 217.9 • i 23 • i 174.3 •1 196.3 .1. 218.3 24 5 •8 174.7 5 • 8 196.7 4 218.7 •f 24 175.1 •J 197.1 •f 219.1 A 24 1 /175.5 7 • 3 197.5 • i 219.5 .1 24 56. 175.9 63. 197.9 70. 219.9 77. 24 'i 176.3 1 •8 198.3 1 •8 220.3 .1 24 a 176.7 198.7 1 • 4 220.6 1 24* .f 177.1 •f 199. • 8 221. 3 •8 24: •i 177.5 •2" 199.4 •f 221.4 24: ^i 177.8 •f 199.8 221.8 • ¥ 24: •1 178.2 .| 200.2 • 1 222.2 •f 24^ ,1 178.6 7 •8 200.6 ,1 222.6 •3 24^ 57. 179. 64. 201. 71.' 223. 78. 24^ ,- 179.4 •i 201.4 •1 223.4 .1 24f •4 179.8 •5 201.8 .i 223.8 .1 24f .- 180.2 •t 202.2 • w 224.2 • I 24( •i 180.6 4 202.6 •t 224.6 A 24( .- 181. • s 203. 225. .1 24i .1 181.4 .1 203.4 • 1 225.4 .1 24'3 •i 181.8 7 •8 203.8 7 •8 225.8 7 .¥ 247 58. 182.2 65. 204.2 72. 226.1 79. 248 .1 182.6 •1 204.5 .1 226.5 • 1 248 •i 182.9 •t 204.9 .i 226.9 • ? 248 ;i 183.3 205.3 227.3 .i 24S . \ 183.7 'i 205.7 ."1 227.7 .* 24S 5 •8 184.1 5 206.1 5 • 3 228.1 1 25C .f 184.5 •5 206.5 .1 228.5 *| 25G 7 184.9 7 •8 206.9 228.9 * 7 .5 25G 39. 185.3 66. 207.3 73!" 229.3 80. 251 •i 185.7 •1 207.7 ,1 229.7 .i 251 •i 186.1 •i 208.1 •| 230.1 .i 252 .•2 186.5 .i 208.5 ••^ 230.5 .8 252 • i 186.9 •t 208.9 4 230.9 .i 252 • 1 187.3 209.3 • 8 231.3 .3 253 .f 187.7 .^ 209.7 231.6 . z 253 4 188.1 .1 210. 7 •8 2.32. .1 254 50. 188.4 67. 210.4 74. 232.4 81. 254 ■ i 188.8 .1 210.8 • i 232.8 .8 254 1 189.2 .i 211.2 .| 233.2 ^; ■ 255 [a 189.6 • 1 211.6 233.6 • i 255 •i 190. .2- 212. ,^ 234. •I 256 •8 190.4 .8 212.4 5 234.4 ft 256 n •^ 190.8 .1 212.8 *g 234.8 .1 256 7 191.2 7 •5 213.2 '■1 235.2 4 257 98 CIRCUMFERENCES OF CIRCLES. Table— (Continued). Diameter Circumference. Diameter. Circumference. Diameter. Circumference. Diameter." Circumfer'ca 82. 257.6 87. 273.3 92. 289. 97. 304.7 258. 'i 273.7 .1 289.4 ■ i 305.1 258.3 ,1 274.1 4 289.8 I 305.5 3 258.7 2 274.4 • i 290.2 *.| 305.9 • i 259.1 .1 274.8 4 290.5 'k 306.3 i 259.5 •1 275.2 ft • 8 290.9 •1 306.6 259.9 .1 275.6 .1 291.3 .1 307. 7 260.3 7 • S 276. 7 •8 291.7 7 •8 307.4 83. 260.7 88. 276.4 93. 292.1 98. 307.8 4 261.1 .1 276.8 .1 292.5 'i 308.2 261.5 4 277.2 .i 292.9 • ? 308.6 • 1 261.9 • s 277.6 293.3 • i 309.0 i| 262.3 278. • i 293.7 •t 309.4 .1 262.7 •1 278.4 • 1 294.1 .1 309.8 A 263.1 •f 278.8 .? 294.5 .^ 310.2 7 263.5 • t 279.2 7 •8 294.9 7 •8 310.6 84. 263.8 89. 279.6 94. 295.3 99. 311.0 264.2 •8 279.9 .1 295.7 • i 311.4 _ 264.6 280.3 1 296. ^1 311.8 •f 265. •¥ ^80.7 ,f 296.4 • i 312.1 265.4 ,i. 281.1 • i 296.8 .^ 312.5 • 1 265.8 .1 281.5 •1 297.2 •■J 312.9 • 1 266.2 .1 281.9 .? 297.6 ^ J 313.3 •1 266.6 7 •8 282.3 •i- 298. • i 313.7 8.5*.' 267. 90. 282.7 95. 298.4 100. 314.1 267.4 'i 283.1 4 298.8 • ? 267.8 • i 283.5 • i 299.2 • i 268.2 •t 283-9 3 •8 299.6 •t 269.6 •i 284.3 •i 300. 268.9 .1 284.7 .1 300.4 • 1 269.3 4 285.1 •t «00.8 7 . 8 269.7 7 •8 285.4 • i 301.2 86. 270.1 91. 285.8 96. 301.5 .1 270.5 •1 286.2 •8 301.9 270.9 • i 286.6 •? 302.3 '! 271.3 3 287. •8 302.7 • r 271.7 .^ 287.4 .-- 303.1 • 1 272.1 •t 287.8 303.5 .i 272.5 •I 288.2 . J: 303.9 4 272.9 ■l 288.6 .| 304.3 SQUARES, CUBES, AND ROOTS. 99 Table of Squares, Cubes, and Square and Cube Roots, of all lumbers from 1 to 1000. Number. Square. 1 1 4 8 9 27 16 64 25 125 36 216 49 343 64 512 81 729 100 1000 121 1331 144 1728 169 2197 196 2744 225 3375 256 4096 289 4913 324 . 5832 361 6859 400 8000 441 9261 484 10648 529 12167 576 13824 625 15625 676 17576 729 19683 784 21952 841 24389 900 27000 961 29791 1024 32768 1089 35937 1156 39304 1225 42875 1296 46656 1369 50653 1444 54872 1521 59319 1600 64000 1681 68921 1764 74088 1849 79507 1936 85184 2025 91J25 2116 97336 2209 103823 2304 110592 2401 117649 2500 125000 2601 132651 2704 140608 2809 148877 2916 157464 3025 166375 Square Root, 1. 1.414213 1.732050 2. 2.236068 2.449489 2.645751 2.828427 3. 3.162277 3.316624 3.464101 3.605551 3.741657 3.872983 4. 4.123105 4.242640 4.358898 4.472136 4.582575 4.690415 4.795831 4.898979 5. 5.099019 5.196152 5.291502 5.385164 5.477225 5.567764 5.656854 5 . 744562 5.830951 5.916079 6. 6.082762 6.164414 6.244998 6.324555 6.403124 6.480740 6.557438 6.633249 6.708203 6.782330 6.855654 6.928203 7. 7.071067 7.141428 7.211102 7.280109 7.348469 7.416198 Cube Root. 1. 1.259921 1.442250 1.587401 1.709976 1.817121 1.912933 2. 2.080084 2.154435 2.223980 2.289428 2.351335 2.410142 2.466212 2.519842 2.571282 2.620741 2.668402 2.714418 2.758923 2.802039 2.843867 2.884499 2.924018 2.982496 3. 3.036589 3.072317 3.107232 3.141381 3.174802 3.207534 3.239612 3.271066 3.301927 3.332222 3.361975 3.391211 3.419952 3.448217 3.476027 3.503398 3.530348 3.556893 3.583048 3.608826 3.634241 3.659306 3.684031 3.708430 3.732511 3.756286 3.779763 3.802953 100 SQUARES, CUBES, AND ROOTS. Table — (Continued). Number. Square. Cube. i Square Root. 56 3136 175616 7.483314 57 3249 185193 7.549834 58 3364 195112 7.615773 59 3481 205379 7.681145 60 3600 216000 7.745966 61 3721 226981 7.810249 62 3844 238328 7.874007 63 3989 250047 7.937253 64 4096 262144 8. 65 4225 274626 8.062257 66 4356 287496 8.124038 67 4489 300763 8.185352 68 4624 314432 8.246211 69 4761 328509 8.306623 70 4900 343000 8.366600 71 5041 357911 8.426149 72 5184 373248 8.485281 73 5329 389017 8.544003 74 5476 405224 8.602325 75 5625 421875 8.660254 76 5776 43S976 8.717797 77 5929 456533 8.774964 78 6084 474552 8.831760 79 6241 493039 8.888194 80 6400 512000 8.944271 81 6561 531441 9. 82 6724 551368 9.055385 83 6889 571787 9.110433 84 7056 592704 9.165151 85 7225 614125 9.219544 86 7396 636056 9.273618 87 7569 658503 9.327379 88 7744 6S1472 9.380831 89 7921 704969 9.433981 90 8100 729000 9.486833 91 8281 753571 9.539392 92 8464 778688 9.591663 93 8649 804357 9.643650 94 8836 830584 9.695359 95 9025 857375 9.746794 96 9216 .884736 9.797959 97 9409 912673 9.848857 98 9604 941192 9.899494 99 9801 970299 9.949874 100 lOOOO 1000000 10. 101 10201 1030301 10.049875 102 10404 1061208 10.099504 103 10609 1092727 10.148891 104 10816 1124864 10.198039 105 11025 1157625 10.246950 106 11236 1191016 10-295630 107 11449 1225043 10.344080 108 11664 1259712 10.392304 109 11881 1295029 10.440306 110 12100 1331000 10-488088 111 12321 1367631 10.535653 SQUARES, CUBES, AND ROOTS. 101 Table — (Continued). p Namber. Sqnare. 1 Cube. Square Root. Cube Root. 112 12544 1404928 10.583005 4.820284 113 12769 1442897 10.630145 4.834588 114 12996 1481544 10.677078 4.848808 '115 13225 1520375 10.723805 4.862944 116 13456 1560896 10.770329 4.876999 117 13689 1601613 10.816653 4.890973 118 13924 1643032 10.862780 4.904868 119 14161 1685159 10.908712 4.918685 120 14400 1728000 10.954451 4.932424 121 14641 1771561 11. 4.946088 122 14884 1815848 11.045361 4.959675 123 15129 1860867 11.090536 4.973190 124 15376 1906624 11.135528 4.986631 125 15625 1953125 11.180339 5. 126 15876 2000376 11.224972 5.013298 127 16129 2048383 11.269427 5.026526 128 16384 2097152 11.313708 5.039684 129 16641 2146689 11.357816 5.052774 130 16900 2197000 11.401754 5.065797 131 17161 2248091 11.445523 5.078753 132 17424 2299968 11.489125 5.091643 133 17689 2352637 11.532562 5.104469 134 17956 2406104 11.575836 5.117230 135 18225 2460373 11.618950 5.129928 136 18496 2515456 11.661903 5.142563 137 18769 2571353 11.704699 5.155137 138 19044 2628072 11.747344 5.167649 139 19321 2685619 11.789826 5.180101 140 19600 2744000 11^832159 5.192494 141 19881 2803221 11.874342 5.204828 142 20164 2863288 11.916375 5.217103 143 20449 2924207 11.958260 5.229321 144 20736 2985984 12. 5.241482 145 21025 3048625 12.041594 5.253588 146 21316 3112136 12.083046 5.265637 147 21609 3176523 12.124355 5.277632 148 21904 3241792 12.165525 5.289572 149 22201 3307949 12.206555 5.301459 150 22500 3375000 12.247448 5.313293 151 22801 3442951 12.288205 5.325074 152 23104 3511808 12.328828 5.336803 153 23409 3581577 12.369316 5.348481 154 23716 3652264 12.409673 5.360108 155 24025 3723875 12.449899 5.371685 156 24336 3796416 12.489996 5.3S3231 157 24649 3869893 12.529964 5.394690 158 24964 3944312 12-569805 5.406120 159 25281 4019679 12.609520 5.417501 160 25600 4090000 12.649110 5.428835 161 25921 4173281 12.688577 5.440122 162 26244 4251528 12-727922 5.451362 163 26569 4330747 12.767145 5.462556 164 26896 . 4410944 12.806248 5.473703 165 27225 4492125 12-845232 5.484806 166 27556 4574296 12.884098 5.495865 167 27889 4657463 12 12-922848 5.506879 102 SQUARES, CUBES, AND ROOTS. Table— (Continued). Square. 28224 28561 28900 29241 29584 29929 30276 30625 30976 31329 31684 32041 32400 32761 33124 33489 33856 34225 34596 34969 35344 35721 36100 36481 36864 37249 37636 38025 38416 38809 39204 39601 40000 40401 40804 41209 41616 42025 42436 42849 43264 43681 44100 44521 44944 45369 45796 46225 46656 47089 47524 47961 48400 48841 49284 49729 Cube. Square Root. Cube Root. 4741632 12.961481 5.517848 4826809 4913000 13. 13.038404 5.528775 5.539658 5000211 13.076696 5.550499 5088448 13.114877 5.561298 5177717 13.152946 5.572054 5268024 13.190906 5.582770 5359375 13.228756 5.593445 5451776 13.266499 5.604079 5545233 13.304134 5.614673 5639752 13.341664 5.625226 5735339 13.379088 5.635741 5832000 13.416407 5.646216 5929741 13.453624 5.656652 6028568 13.490737 5.667051 6128487 13.527749 5.677411 6229504 13.564660 5.687734 6331625 13.601470 5.698019 6434856 13.638181 5.708267 6539203 13.674794 5.718479 6644672 13.711309 5.728654 6751269 13.747727 5.738794 6859000 13.784048 5.748897 6967871 13.820275 5.758965 7077888 13.856406 5.768998 7189057 13.892444 5.778996 7301384 13.928388 5.788960 7414875 13.964240 5.798890 7529536 14. 5.808786 7645373 14.035668 5.818648 7762392 14.071247 5.828476 7880599 14.106736 5.838272 8000000 14.142135 5.848035 8120601 14.177446 5.857765 8242408 14.212670 5.867464 8365427 14.247806 5.877130 8489664 14.282856 5.886765 8615125 14.317821 5.896368 8741816 14.352700 5.905941 8869743 14.387494 5.915481 8998912 14.422205 5.924991 9123329 14.456832 5.934473 9261000 14.491376 5.943911 9393931 14.525839 5.953341 9528128 14.560219 5.962731 9663597 14.594519 5.972091 9800344 14.628738 5.981426 9938375 14.662878 5.990727 10077696 14.696938 6. 10218313 14.730919 6.009244 10360232 14.764823 6.018463 10503459 14.798648 6.027650 10648000 14.832397 6.036811 10793861 14.866068 6,045943 10941048 14.899664 6.055048 11089567 14.933184 6.064126 SQUARES, CUBES, AND ROOTS. 103 Table — (Continued). Number. Square. Cube. Square Root. Cube Root. 224 50176 11239424 14.966629 6.073177 225 50625 11390625 15. 6.082201 226 51076 11543176 15.033296 6.091199 227 51529 11697083 15.066519 6.100170 228 51984 11852352 15.099668 6.109115 229 52441 12008989 15.132746 6.118032 230 52900 12167000 15.165750 6.126925 231 53361 ^ 12326391 15.198684 6.135792 232 53824 12487168 15.231546 6.114634 2.33 54289 12649337 15.264337 6.153449 234 54756 12812904 15.297058 6.162239 235 55225 12977875 15.329709 6.171005 236 55696 13144256 15.362291 6.179747 237 56169 13312053 15.394804 6.188463 238 56644 13481272 15.427248 6.197154 239 57121 13651919 15.459624 6.205821 240 57600 13824000 15.491933 6.214464 241 58081 13997521 15.524174 6.223083 242 58564 14172488 15.556349 6.231678 243 59049 14348907 15.588457 6.240251 244 59536 14526784 15.620499 6.248800 245 60025 14706125 15.652475 6.257324 246 60516 14886936 15.684387 6.26.5826 247 61009 15069223 15.716233 6.274304 248 61504 15252992 15.748015 6.282760 249 62001 15438249 15.779733 6.291194 250 62500 15625000 15.811388 6.299604 251 63001 15813251 15.842979 6.307992 252 63504 16003008 15.874507 6.316359 253 64009 16194277 15.905973 6.324704 254 64516 16387064 15.937377 6.333025 255 65025 16581375 15.968719 6.341325 256 65536 16777216 16. 6.349602 257 66049 16974593 16.031219 6.357859 258 66564 17173512 16.062378 6.366095 259 67081 17373979 16.093476 6.374310 260 67600 17576000 16.124515 6.382504 261 68121 17779581 16.155494 6.390676 262 68644 17984728 16.186414 6.398827 263 69169 18191447 16.217274 6.406958 264 69696 18399744 16.248076 6.41.5068 265 70225 18609625 16.278820 6.423157 266 70756 18821096 16.309506 6.431226 267 71289 19034163 16.340134 6.439275 268 71824 19248832 16.370705 6.447305 269 72361 19465109 16.401219 6.455314 270 72900 19683000 16.431676 6.463304 271 73441 19902511 16.462077 6.471274 272 73984 20123648 16.492422 6.479224 273 74529 20346417 16.522711 6.487153 274 75076 20570824 16.552945 6.495064 275 75625 20796875 16.583124 6.502956 276 76176 21024576 16.613247 6.510829 277 76729 21253933 16.643317 6.518684 278 77284 21484952 16.673332 6.526519 279 77841 21717639 16.703293 6.634335 104 SQUARES, CUBES, AND ROOTS. Table — (Continued). Number. Square. Cube. Square Root. Cube Root. 280 78400 21952000 16.733200 6.542132 281 78961 22188041 16.763054 6.549911 282 79524 22425768 16.792855 6.557672 283 80089 22665187 16.822603 6.565415 284 80656 22906304 16.852299 6.573139 285 81225 23149125 16.881943 6.580844 286 81796 23393656 16.911534 6.588531 287 82369 23639903 16.941074 6.596202 288 82944 23887872 16.970562 6.603854 289 83521 24137569 17. 6.611488 290 84100 24389000 17.029386 6.61910S 291 84681 24642171 17.058722 6.626705 292 85264 24897088 17.088007 6.634287 293 85849 25153757 17.117242 6.641851 294 86436 25412184 17.146428 6.649399 295 87025 25672375 17.175564 6.656930 296 87616 25934336 17.204650 6.664443 297 88209 26198073 17.233687 6.671940 298 88804 26463592 17.262676 6.679419 299 89401 26730899 17.291616 6.686882 300 90000 27000000 17.320508 6.694328 301 90601 27270901 17.349351 6.701758 302 91204 27543608 17.378147 6.70917^ 303 91809 27818127 17.406895 6.716569 304 92416 28094464 17.435595 6.723950 305 93025 28372625 17.464249 6.73131(> 306 93636 28652616 17.492855 6.738665 307 94249 28934443 17.521415 6.745997 308 94864 29218112 17.549928 6.753313 309 95481 29503629 17.578395 6.760614 310 96100 29791000 17.608816 6.767899 311 96721 30080231 17.635192 6.775168 312 97344 30371328 17.663521 6.782422 313 97969 30664297 17.691806 6.7S9661 314 98596 30959144 17.720045 6.796884 315 99225 31255875 17.748239 6.804091 316 99856 31554496 17.776388 6.S112S4 317 100489 31855013 17.804493 6.818461 318 101124 32157432 17.832554 6.825624 319 101761 32461759 17.86057.1 6.832771 320 102400 32768000 17.888543 6.839903 321 103041 33076161 17.916472 6.847021 322 103684 33386248 17.944358 6.854124 323 104329 33698267 17.972200 6.861211 324 104976 34012224 18. 6.868284 325 105625 34328125 18.027756 6.87.5343 326 106276 34645976 18.055470 6.882388 327 106929 34965783 18.083141 6.889419 328 107584 35287552 18.110770 6.896435 329 108241 35611289 18.138357 6.903436 330 108900 35937000 18.165902 6.910423 331 109561 36264691 18.193405 6.917396 332 110224 36594368 18.220867 6.924355 333 110889 36026037 18.248287 6.931300 334 111556 37259704 18.275666 6.938232 335 U2225 87595375 18.303005 6.945149 SQUARES, CUBES, AND ROOTS. 105 Table — (Continued). Number. Square. Cube. Square Root. Cube Root. 336 112896 37933056 18.330302 6.952053 ' 337 113569 38272753 18.357559 6.958943 338 114244 38614472 18.384776 6.965819 339 114921 3S958219 18.411952 6.972682 340 115600 39304000 18.439088 6.979532 341 116281 39651821 18.466185 6.986369 342 116964 40001688 18.493242 6.993491 343 117649 40353607 18.520259 7. 344 118336 40707584 18.547237 7.006796 345 119025 41063625 18.574175 7.013579 346 119716 41421736 18.601075 7.020349 347 120409 41781923 18.627936 7.027106 348 ^ 121104 42144192 18.654758 7.033850 349 121801 42508549 18.681541 7.040581 350 122500 42875000 18.708288 7.047208 351 123201 43243551 18.7.34994 7.054003 352 123904 43614208 18.761663 7.060696 353 124609 43986977 18.788294 7.067376 354 125316 44361864 18.814887 7.074043 355 126025 44738875 18.841443 7.080698 356 126736 45118016 18.867962 7.087341 357 127449 45499293 18.894443 7.093970 358 128164 45882712 18.920887 7.100588 350 12SS81 46268279 18.947295 7.107193 360 129600 46656000 18.973666 7.113786 361 130321 47045881 19. 7.120367 362 131044 47437928 19.026297 7.126935 363 131769 47832147 19.052558 7.133492 364 132496 48228544 19.078784 7.140037 365 133225 48627125 19.104973 • 7.146569 366 133956 49027896 19.131126 7.153090 367 134689 49430863 19.157244 7.159599 368 135424 49836032 19.183326 7.166095 369 136161 50243409 19.209372 7.172580 370 136900 50653000 19.235384 7.179054 371 137641 51064811 19.261360 7.185516 372 138384 51478848 19.287301 7.191966 373 139129 51895117 19.313207 7.198405 374 139876 52313624 19.339079 7.204832 375 140625 • 52734375 19.364916 7.211247 376 141376 53157376 19.390719 7.217652 377 142129 535S2633 19.416487 7.224045 378 142884 54010152 19.442222 7.230427 379 143641 54439939 19.467922 7.236797 380 144400 54872000 19.493588 7.243156 381 145161 55306341 19.519221 7.249504 382 145924 55742968 19.544820 7.255841 383 146689 56181887 19.570385 7.262167 384 147456 56623104 19.595917 7.268482 385 148225 • 57066625 19.621416 7.274786 386 148996 57512456 19.646882 7.281079 387 149769 57960603 19.672315 7.287362 388 150544 58411072 19.697715 7.293633 389 151321 58863869 19.723082 7.299893 390 152100 59319000 19.748417 7.306143 391 152881 59776471 19.773719 7.312383 106 SQUARES, CUBES, AND ROOTS. Table — (Continued). Square, Cube. Square Root. Cube Root. 153664 60236288 19.798989 7.318611 154449 60698457 19.824227 7.324829 155236 61162984 19.849432 7.331037 156025 61629S75 19.874606 7.337234 156816 62099136 19.899748 7.343420 157609 62570773 19.924858 7.349596 158404 63044792 19.949937 7.355762 159201 63521199 19.974984 7.361917 160000 64000000 20. 7.368063 160801 64481201 20.024984 7.374198 161604 64964808 20.049937 7.380322 162409 65450827 20.074859 7.386437 163216 65939264 20.099751 7.392542 164025 66430125 20.124611 7.398636 164836 66923416 20.149441 7.404720 165649 67419143 20.174241 7.410794 166464 67911312 20.199009 7.416859 167281 68417929 20.223748 7.422914 168100 68921000 20.248456 7.428958 168921 69426531 20.273134 7.434993 ] 69744 69934528 20.297783 7.441018 170569 70444997 20.322401 7.447033 171396 70957944 20.346989 7.453039 172225 71473375 20.371548 7.459036 173056 71991296 20.396078 7.465022 173889 72511713 20.420577 7.470999 174724 73034632 20.445048 7.476966 175561 73560059 20.469489 7.482924 176400 74088000 20.493901 7.488872 177241 74618461 20.518284 7.494810 178084 75151448 20.542638 7.500740 178929 75686967 20.566963 7.506660 179776 76225024 20.591260 7.512571 180625 76765625 20.615528 7.5184-73 181476 77308776 20.639767 7.524365 182329 77854483 20.663978 7.530248 183184 78402752 20.688160 7.536121 184041 78953589 20.712315 7.541986 184900 79507000 20.736441 7.547841 185761 80062991 20.760539 7.553688 186624 80621568 20.784609 7.559525 187489 81182737 20.808652 7.565353 188356 81746504 20.832666 7.57117.a 189225 82312875 20.856653 7.576984 190096 82881856 20.880613 7.582786 190969 83453453 20.904545 7.588579 191844 84027672 20.928449 7.594363 192721 84604519 20.952326 7.600138 193600 85184000 20.976177 7.605905 194481 85766121 21. 7.611662 195364 86350388 21.023796 7.617411 196249 86938307 21.047565 7.623151 197136 87528384 21.071307 7.628883 198025 88121125 21.095023 7.634606 198916 88716536 21.118712 7.640321 199809 89314623 21.142374 7.646027 SQUARES, CUBES, A:^D ROOTS. 107 Table — (Continued). Square. 448 200704 89915392 449 201601 90518849 450 202500 91125000 451 203401 91733851 452 204304 92345408 453 205209 92959677 454 206106 93576664 455 207025 94196375 456 207936 94818816 457 208849 95443993 458 209764 96071912 459 210681 96702579 460 211600 97336000 461 212521 97972181 462 213144 98611128 463 214369 99252847 464 215296 99897344 465 216225 100544625 466 217156 101194696 467 218089 101847563 468 219024 102503232 469 219961 103161709 470 220900 103823000 471 221841 104487111 472 222784 105154048 473 223729 105823817 474 224676 106496424 475 225625 107171875 476 226576 107850176 477 227529 108531333 478 228484 109215352 479 229441 109902239 480 230400 13 0592000 481 231361 111284641 482 232324 111980168 483 233289 112678587 484 234256 113379904 485 235225 114084125 486 236196 114791256 487 237169 115501303 488 238144 116214272 489 239121 116930169 490 240100 117649000 491 241081 1 J 8370771 492 242064 119095488 493 243049 119823157 494 244038 120553784 495 245025 121287375 496 246016 122023936 497 247009 122763473 498 248004 123505992 499 249001 124251499 600 250000 125000000 501 251001 125751501 502 252004 126506008 ^03 253009 127263527 Square Root. 21.166010 21.189620 21.213203 21.236760 21.26029] 21.283796 21.307275 21.330729 21.354156 21.377558 21.400934 21.424285 21.447610 21.470910 21.494185 21.517434 21.540659 21.563858 21.587033 21.610182 21.633307 21,656407 21.679483 21.702534 21.725561 21.748563 21.771541 21.794494 21.817424 21.840329 21.863211 21.886068 21.908902 21.931712 21.954498 21.977261 22. 22.022715 22.045407 22.068076 22.090722 22.113344 22.135943 22.158519 22.181073 22.203603 22.226110 22.248595 22.271057 22.293496 22.315913 22.338307 22.360679* 22.383029 22.40.5356 22.427661 Cube Root. 7.651725 7.657414 7.663094 7.668766 7.674430 7.680085 7.685732 7.691371 7.097002 7.702624 7.708238 7.713844 7.719442 7.725032 7.730614 7.736187 7.741753 7.747310 7.752860 7.7.58402 7.763936 7.769462 7.774980 7.780490 7.785992 7.791487 7.796974 7.802453 7.807925 7.813389 7.818845 7.824294 7.829735 7.835168 7.840594 7.846013 7.851424 7.856828 7.862224 7.867613 7.872994 7.878368 7.883734 7.889094 7.894446 7.899791 7.905129 7.910460 7.915784 7.921100 7.926408 7.931710 7.937005 7.942293 7.947673 7.952847 108 SQUARES, CUBES, AND ROOTS. Table— (Continued). Square. 504 254016 505 255025 506 256036 607 257049 508 258064 509 259081 510 260100 511 261121 512 262144 513 263169 514 264196 515 2G5225 516 266256 • 617 267289 518 268324 619 269361 520 270400 521 271441 522 272484 523 273529 524 274576 525 275625 526 276676 527 277729 528 278784 629 279841 530 280900 631 281961 532 283024 533 284089 534 285156 535 286225 536 287296 637 288369 538 289444 639 290521 640 291600 541 292681 642 293764 643 294849 644 295936 645 297025 646 298116 547 299209 548 300304 549 301401 650 302500 651 303601 652 304704 653 305809 564 306916 655 308025 656 309136 657 310249 658 311364 559 312481 128024064 128787625 129554216 130323843 131096512 131872229 132651000 133432831 134217728 135005697 135796744 136590875 137388096 138188413 138991832 139798359 140608000 141420761 142236648 143055667 143877824 144703125 145531576 146363183 147197952 148035889 148877000 149721291 150568768 151419437 152273304 153130375 153990656 154854153 155720872 156590819 157464000 158340421 159220088 160103007 160989184 161878625 162771336 163667323 164566592 165469149 166375000 167284151 168196608 169112377 170031464 170953875 171879616 172808693 173741112 174676879 Square Root._ Cube Root. 22.449944 22.472205 22.494443 22.516660 22.538855 22.561028 22.583179 22.605309 22.627417 22.649503 22.671568 22.693611 22.715633 22.737634 22.759613 22.781571 22.803508 22.825424 22.847319 22.869193 22.891046 22.912878 22.934689 22.956480 22.978250 23. 23.021728 23.043437 23.065125 23.086792 23.108440 23.130067 23.151673 23.173260 23.194827 23.216373 23.237900 23.259406 23.280893 23.302360 23.323807 23.345235 23.366642 23.388031 23.409399 23.430749 23.452078 22.473389 23.494680 23.515952 23.537204 23.558438 23.579652 23.600847 23.622023 23.643180 7.958114 7.963374 7.968627 7.973873 7.979112 7.984344 7.989569 7.994788 8. 8.005205 8.010403 8.015595 8.020779 8.025957 8.031129 8.036293 8.041451 8.046603 8.051748 8.056886 8.062018 8.067143 8.072262 8.077374 8.082480 8.087579 8.092672 8.097758 8.102838 8.107912 8.112980 8.118041 8.12309G 8.128144 8.133186 8.138223 8.143253 8.148276 8.153293 8.158304 8.163309 8.168308 8.173302 8.178289 8.183269 8.188244 8.193212 8.198175 8.203131 8.208082 8.213027 8.217965 8.222898 8.227825 8.232746 8.237661 SQUARES, CUBES, AND ROOTS. 109 Table — (Continued). Number. Square. Cube. Square Root. Cube Root. 660 313600 175616000 23.664319 8.242570 561 314721 17655S48I 23.685438 8 . 247474 562 315844 177504328 23.706539 8.252371 563 316969 178453547 23.727621 8 . 257263 564 318096 179406144 23.748684 8.262149 565 319225 180362125 23.769728 8.267029 566 320356 181321496 23.790754 8.271903 567 321489 182284263 23.811761 8.276772 568 322624 183250432 23.832750 8.281635 569 323761 184220009 23.853720 8.286493 570 324900 185193000 23.874672 8.291344 571 326041 186169411 23.895606 8.296190 572 327184 187149248 23.916521 8.301030 573 328329 188132517 23.937418 8.305865 574 329476 189119224 23.958297 8.310694 575 330625 190109375 23.979157 8.315517 576 331776 191102976 24. 8.320335 577 332929 192100033 24.020824 8.325147 578 334084 193100552 24.041630 8.329954 579 335241 194104539 24.062418 8.334755 580 336400 195112000 24,083189 8.339551 681 337561 196122941 24.103941 8-344341 582 338724 197137368 24.124676 8.349125 583 339889 198155287 24.145392 8.353904 584 341056 199176704 24.166091 8.358678 585 342225 200201625 24.186773 8.363446 586 343396 201230056 24.207436 8.368209 587 344569 202262003 24.228082 8.372966 588 345744 203297472 24.248711 8.377718 689 346921 204336469 24.269322 8 . 382465 690 348100 205379000 24.289915 8.387206 591 349281 206425071 24.310491 8.391942 692 350464 207474688 24.331050 8.396673 593 351649 208527857 24.351591 8.401398 594 352836 209584584 24.372115 8.406118 595 354025 210644875 24.392621 8.410832 696 355216 211708736 24.413111 8.415541 697 356409 212776173 24.433583 8.420245 598 357604 213847192 24.454038 8.424944 699 358801 214921799 24.474476 8.429638 600 360000 216000000 24.494897 8.434327 601 361201 217081801 24.515301 8.439009 602 362404 218167208 24.535688 8.443687 603 363609 219256227 24.556058 8.448360 604 364816 220348864 24.576411 8.453027 605 366025 221445125 24.596747 8.457689 606 367236 222545016 24.617067 8.462347 607 368449 223648543 24.637370 8.466999 608 369664 224755712 24.657656 8.471647 609 370881 225866529 24.677925 8.476289 610 372100 226981000 24.698178 8.480926 611 373321 228099131 24.718414 8.485557 612 374544 229220928 24.7386.33 8.490184 613 375769 230346397 24.758836 8.494806 614 376996 231475544 24.779023 8.499423 615 378225 232608375 K 24.799193 8.5Q4034 no SQUARES, CUBES, AND ROOTS. Table — (Continued ). Number. Square. Cube. Square Root. Cube Root, 616 379456 233744896 24.819347 8.508641 617 380689 234885113 24.839484 8.513243 618 381924 236029032 24.859605 8.517840 619 383161 237176659 24.879710 8.522432 620 384700 238328000 24.899799 8.527018 621 385641 239483061 24.919871 8.531600 622 386884 240641848 24.939927 8.536177 623 388129 241804367 24.959967 8.540749 624 389376 242970624 24.979992 8.545317 625 390625 244140625 25. 8.549879 626 391876 245314376 i 25.019992 8.554437 627 393129 246491883 25.0.39968 8.558990 628 394384 247673152 25.059928 8.563537 629 395641 248858189 25.079872 8.568080 630- 396900 250047000 25.099800 8.572618 631 398161 251239591 25.119713 8.577152 632 399424 252435968 25.139610 8.581680 633 400689 253636137 25.159491 8.586204 634 401956 254S40104 25.179356 8.590723 635 403225 256047875 25.199206 8.595238 636 404496 257259456 25.219040 8.599747 637 405769 258474853 25.238858 8.604252 638 407044 259694072 25.258661 8.608752 639 408321 260917119 25.278449 8.613248 640 409600 262144000 25.298221 8.617738 641 410881 263374721 25.317977 8.622224 642 412164 264609288 25.337718 8.626705 643 413449 265847707 25.357444 8.631183 644 414736 267089984 25.377155 8.635655 645 416025 268336125 25.396850 8.640122 646 417316 269586136 25.416530 8.644585 647 418609 270840023 25.436194 8.649043 648 419904 272097792 25.455844 8.653497 649 421201 273359449 25.475478 8.657946 650 422500 274625000 25.495007 8.662301 651 423801 275894451 25.514701 8.666831 652 425104 277167808 25.534290 8.671266 653 426409 278445077 25.553864 8.675697 654 427716 279726264 25.573423 8.680123 655 429025 281011375 25.592967 8 . 684545 656 430336 2S2300416 25.612496 8.688963 657 431649 283593393 25.632011 8.693376 658 432964 284890312 25.651510 8.6977841 659 434281 286191179 25.670995 8.702188^ 660 435600 287496000 25.690465 8.706587. 661 436921 288804781 25.709920 8.710982. 662 438244 290117528 25.720360 8.715373 663 439569 291434247 25.748786 8.719759 664 440896 292754944 25.768197 8.724141 665 442225 294079625 25.787593 8.728518- 666 443556 295408296 25.806975 8.732891 667 444889 296740963 25.826343 8.737200 668 446224 298077632 25.845696 8.741624i 669 447561 299418309 25.865034 8.7459841 670 448900 300763000 25.884358 8.750340 671 450241 302111711 25.903667 8.754691 SQUARES, CUBES, AJMD ROOTS. Ill Table— (Continued). Number. Square. Cube. Square Root. Cube Root. 672 451584 303464448 25.922962 8.759038 673 452929 304821217 25.942243 8.763380 674 454276 306182024 25.961510 8.767719 675 455625 307546875 25.980762 8.772053 676 456976 308915776 26. 8.776382 677 458329 310288733 26.019223 8.780708 678 459684 311665752 26.038433 8.785029 679 461041 313046839 26.057628 8.789346 680 462400 314432000 2S. 076809 8.793659 681 463761 315821241 26.095976 8.797967 682 465124 317214568 26.115129 8.802272 683 466489 318611987 26.134268 8.806572 684 467856 320013504 26.153393 8.810868 685 469225 321419125 20.172504 8.815159 686 470596 322828856 26.191601 8.819447 687 471969 324242703 26.210684 8.823730 688 473344 325660672 26.229754 8.828009 689 474721 327082769 26.248809 8.832285 690 476100 328509000 26.267851 8.836556 691 477481 329939371 26.286878 8.840822 692 478864 331373888 26.305892 8.845085 693 480249 332812557 26.324893 8.849344 694 481636 334255384 26.343879 8.853598 695 483025 335702375 26.362852 8.857849 696 484416 337153536 26.381811 8.862095 697 485809 338608873 26.400757 8.866337 698 487204 340068392 26.419689 8.870575 699 488601 341532099 26.433608 8.874809 700 490000 343000000 26.457513 8.879040 701 491401 344472101 26.476404 8.883266 702 492804 345948088 26.495282 8.887488 703 494209 347428927 26.514147 8.891706 704 495616 348913664 26.532998 8.895920 705 497025 350402625 26.551836 8.900130 706 498436 351895816 28.570660 8.904336 707 499849 353393243 26.589471 8.908538 708 501264 354894912 26.608269 8.912736 709 502681 356400829 26.627053 8.916931 710 504100 357911000 26.645825 8.921121 711 505521 359425431 26.664583 8.925307 712 506944 360944128 26.683328 8.929490 .713 508369 362467097 26.702059 8.933668 714 509796 363994344 26.720778 8.937843 715 511225 365525875 26.739483 8.942014 716 512656 367061696 26.758176 8.946180 717 514089 368601813 26.776855 8.950343 718 515524 370146232 26.795522 8.954502 719 516961 371694959 26.814175 8.958658 720 518400 373248000 26.832815 8.962809 721 519841 374805361 26.851443 8.966957 722 521284 376367048 26.870057 8.971100 723 522729 377933067 26.888659 8.975240 724 524176 379503424 26.907248 8.979376 725 525625 381078125 26.925824 8.983508 726 527076 382657176 26.944387 8.987637 727 528529 384240583 26.962937 8.991762 112 SQUARES, CUBES, A2iD ROOTS. Table — (Continued). Number. Square. Cube. Square Root. Cube Roof. 728 529984 385828352 26.981475 8.995883 729 531441 387420489 27. 9. 730 532900 389017000 27.018512 9.004113 731 534361 390617891 27.037011 9.008222 732 535824 392223168 27.055498 9.012328 733 537289 393832837 27.073972 9.0164S0 734 538756 395446904 27.092434 9.020529 735 540225 397065375 27.110883 9.024623 736 541696 3986SS256 27.129319 9.028714 737 543169 400315553 27.147743 9.032802 738 544644 401947272 27.166155 9.036885 739 546121 403583419 27.184554 9.040965 740 547600 405224000 27.202941 9.045041 741 549081 406869021 27.221315 9.049114 742 550564 408518488 27.239676 9.053183 743 552049 410172407 27.258026 9.057248 744 553536 411830784 27.276363 9.061309 745 555025 413493625 27.294688 9.065367 746 556516 415160936 27.313000 9.069422 747 558009 416832723 27.331300 9.073472 748 559504 41S50S992 27.349588 9.077519 749 561001 420189749 27.367864 9.081563 750 562500 421875000 27.386127 9.085603 751 564001 423564751 27.404379 9.089639 752 565504 425259008 27.422618 9.093672 753 567009 42-6957777 27.440845 9.097701 754 568516 428661064 27.459060 9.101726 755 570025 430368375 27.477263 9.105748 756 571536 432081216 27.495454 9.109766 757 573049 433798093 27.513633 9.113781 758 574564 435519512 27.531799 9.117793 759 576081 437245479 27.549954 9.121801 760 577600 438976000 27.568097 9.125805 761 579121 440711081 27.586228 9.129806 762 580644 442450728 27.604347 9.133803 763 582169 444194947 27.622454 9.137797 764 583696 445943744 27.640549 9.141788 765 585225 447697125 27.658633 9.145774 766 586756 449455096 27.676705 9.149757 767 588289 451217663 27.694764 9.153737 768 589824 452984832 27.712812 9.157713 769 591361 454756609 27.730849 9.161686 770 592900 456533000 27.748873 9.165656 771 594441 458314011 27.766886 9.169622 772 595984 460099648 27.784888 9.173585 773 597529 461889917 27.802877 9.177544 774 599076 463684824 27.820855 9.181500 775 600625 465484375 27.838821 9.185452 776 602 176 467288576 27.856776 9.189401 77.7 603729 469097433 27.874719 9.193347 778 605284 470910952 27.892651 9.197289 779 606841 472729139 27.910571 9.201228 780 608400 474552000 27.928480 9.205164 781 609961 476379541 27.946377 9.209096 782 611524 478211768 27.964262 9.213025 783 613089 480048687 27.982137 9.216950 SQUARES, CUBES, AND ROOTS. 113 Table — (Continued). Number. Square. Cube. Square Root. Cube Root 784 614656 4^1890304 28. 9.220872 785 616225 483736025 28.017851 9.224791 786 617796 485587656 28.035691 9.228706 787 619369 487443403 28.053520 9.232618 788 620944 489303872 28.071337 9.237527 789 622521 491169069 28.089143 9.240433 790 624100 493039000 28.106938 9.244335 791 625681 494913671 28.124722 9.248234 792 627264 496793088 28.142494 9.252130 793 628849 498677257 28.160255 9.256022 794 630436 500566184 28.178005 9.2.^9911 795 632025 502459875 28.195744 9.263797 796 633616 504358336 28.213472 9.267679 797 635209 506261573 28.231188 9.271559 798 636804 508169592 28.248893 9.275435 799 638401 510082399 28.266588 9.279308 800 640000 512000000 28.284271 9.283177 801 641601 513922401 28.301943 9.287044 802 643204 515849608 28.319604 9.290907 803 644809 517781627 28.337254 9.294767 804 646416 519718464 28.354893 9.298623 805 648025 521660125 28.372521 9.302477 806 649636 523606616 28.390139 9.308327 807 651249 525557943 28.407745 9.310175 808 652864 527514112 28.425340 9.314019 809 654481 529475129 28.442925 9.317859 810 656100 531441000 28.460498 9.321697 811 657721 533411731 28.478061 9.325532 812 659344 535387328 28.495613 9.329363 813 660969 537366797 28.513154 9.333191 814 662596 539353144 28.530685 9.337016 815 664225 541343375 28.548204 9.340838 816 665856 543338496 28.565713 9.344657 817 667489 545338513 28.583211 9.348473 818 669124 547343432 28.600699 9.352285 819 670761 549353259 28.618176 9.356095 820 672400 551368000 28.635642 9.359901 821 674041 553387661 28.653097 9.363704 822 675684 555412248 28.670542 , 9.367505 823 677329 557441767 28.687976 9.371302 824 678976 559476224 28.705400 9.375096 825 680625 561515625 28.722813 9.378887 826 682276 563559976 28.740215 9.372675 827 683929 565609283 28.757607 9.386460 828 685584 567663552 28.774989 9.390241 829 687241 569722789 28.792360 9.394020 830 688900 571787000 28.809720 9.397796 831 690561 673856191 28.827070 9.401569 832 692224 575930368 28.844410 9.405338 833 693889 578009537 28.861739 9.409105 834 695556 580093704 28.879058 9.412869 835 697225 582182875 28.896366 9.416630 83G 698896 584277056 28.913664 9.420387 837 700569 586376253 28.930952 9.424141 838 702244 588480472 28.948229 9.427893 839 703921 590589719 K2 28.965496 9.431642 tu SQUARES, CUBES, AND ROOTS. Table— (Continued). Number. Square. Cube. Square Root. 840 705600 592704000 28.982753 841 707281 594823321 29. 842 708964 596947688 29.017236 843 710649 599077107 29.034462 844 712336 601211584 29.051678 845 714025 603351125 29.068883 846 715716 605495736 29.086079 847 717409 607645423 29.103264 848 719104 609800192 29.120439 849 720801 611960049 29.137604 850 722500 614125000 29.154759 851 724201 616295051 29.171904 852 725904 618470208 29.189039 853 727609 620650477 29.206163 854 729316 622835864 29.223278 855 731025 625026375 29.240383 856 732736 627222016 29.257477 857 734449 629422793 29.274562 858 736164 631628712 29.291637 859 737881 633839779 29.308701 860 739600 636056000 29.325756 861 741321 638277381 29.342801 862 743044 640503928 29.359836 863 744769 642735647 29.376861 864 746496 644972544 29.393876 865 748225 647214625 29.410882 866 749956 649461896 29.427877 867 751689 651714363 29.444863 868 753424 653972032 29.461839 869 755161 656234909 29.478805 870 756900 658503000 29.495762 871 758641 660776311 29.512709 872 760384 663054848 29.529646 873 762129 665338617 29.546573 874 763876 667627624 29.563491 875 765625 669921875 29.580398 876 767376 672221376 29.597297 877 769129 674526133 29.614185 878 770884 676836152 29.631064 879 772641 679151439 29.647932 880 774400 681472000 29.664793 881 776161 683797841 29.681644 882 777924 686128968 29.698484 883 779689 688465387 29.715315 884 781456 690807104 29.732137 885 783225 693154125 S9. 748949 886 784996 695506456 29.765752 887 786769 697864103 29.782545 888 788544 700227072 29.799328 889 790321 702595369 29.816103 890 792100 704969000 29.832867 891 793881 707347971 29.849623 892 795664 709732288 29.866369 893 797449 712121957 29.883105 894 799236 714516984 29.899832 895 801025 716917375 1 29.916550 ! SQUARES, CUBES, AND ROOTS. Table — (Continued). 115 Number. Square. Cube. Square Root. Cube Root. 896 802816 719323136 29.933259 9.640569 897 804609 721734273 29.949958 9.644154 898 806404 724150792 29.966648 9.647736 899 808201 726572699 29.983328 9.651316 900 810000 729000000 30. 9.654893 901 811804 731432701 30.016662 9.658468 902 813604 733870808 30.033314 9.662040 903 815409 736314327 30.049958 9.665609 904 817216 738763264 30.066592 9.669176 905 819025 741217625 30.083217 9.672740 906 820836 743677416 30.099833 9.676301 907 822649 746142643 30.116440 9.679860 908 824464 748613312 30.133038 9.683416 909 826281 751089429 30.149626 9.686970 910 828100 753571000 30.166206 9.690521 911 829921 756058031 30.182776 9.694069 912 831744 758550528 30.199337 9.697615 913 833569 761048497 30.215889 9.701158 914 835396 763551944 30.232432 9.704698 915 837225 766060875 30.24S966 9.708236 916 839056 768575296 30.265491 9.711772 917 840889 771095213 30.282007 9.715305 918 842724 773620632 30.298514 9.718835 919 844561 776151559 30.315012 9.722363 920 846400 778688000 30.331.501 9.725888 921 848241 781229961 30.347981 * 9.729410 922 850084 783777448 30.364452 9.732930 923 851929 786330467 30.380915 9.736448 924 853776 788889024 30.397368 9.739963 925 855625 791453125 30.413812 9.743475 926 857476 794022776 30.430248 9.746985 927 859329 796597983 30.446674 9.7.50493 928 861184 799178752 30.463092 9.753998 929 863041 801765089 30.479501 9.757500 930 864900 804357000 30.495901 9.761000 931 866761 806954491 30.512292 9.764497 932 868624 809557568 30.528675 9.767992 933 870489 8121662S7 30.545048 9.771484 934 872356 814780504 30.561413 9.774974 935 874225 817400375 30.577769 9.778461 936 876096 820025856 30.594117 9.782946 937 877969 822656953 30.610455 9.785428 938 879844 825293672 30.626785 9.788908 939 881721 827936019 30.643106 9.792386 940 883600 830584000 30.659419 9.795861 94] 885481 833237621 30.675723 9.799333 942 887364 835896888 30.692018 9.802803 943 889249 838561807 30.708305 9.806271 944 891136 841232384 30.724583 9.809736 945 893025 843908625 30.740852 9.813198 946 894916 846590536 30.757113 9.816659 947 896809 849278123 30.773365 9.820117 948 898704 851971392 30.789608 9.823572 949 900601 854670349 30.805843 9.827025 950 902500 857375000 30.822070 9.830475 951 904401 8600S5351 30.838287 9.833923 116 SqUAEES, CUBES, AND ROOTS. Table— (Continued). Number. Sqaure. Cube. Square Root. Cube Root. 952 906304 862801408 30.854497 9.837369 953 908209 865523177 30.870698 9.840812 954 910116 868250664 30.886890 9.844253 955 912025 870983875 30.903074 9.847692 956 913936 873722816 30.919249 9.851128 957 915849 876467493 30.935416 9.854561 958 917764 879217912 30.951575 9.857992 959 919681 881974079 30.967725 9.861421 960 921600 884736000 30.983866 9.864848 961 923521 887503681 31. 9.868272 962 925444 890277128 31.016124 9.871694 963 927369 893056347 31.032241 9.875113 964 929296 895841344 31.048349 9.878530 965 931225 898632125 31.064449 9.881945 966 933156 9014*^8696 31.080540 9.885357 967 935089 90423 J 063 31.096623 9.888767 968 937024 907039232 31.112698 9.892174 969 938961 909853209 31.128764 9.895580 970 940900 912673000 31.144823 9.898983 971 942841 915498611 31.160872 9.902383 972 944784 918330048 31.176914 9.905781 973 946729 921167317 31.192947 9.909177 974 948676 924010424 31.208973 9.9J2571 975 950625 926859375 31.224990 9.915962 976 95257J5 929714176 31.240998 9.919351 977 954529 932574833 31.256999 9.922738 978 956484 935441352 31.272991 9.926122 979 958441 938313739 31.288975 9.929504 980 960400 941192000 31.304951 9.932883 981 962361 944076141 31.320919 9.936261 982 964324 946966168 31.336879 9.939636 983 966289 949862087 31.352830 9.943009 984 968256 952763904 31.368774 9.946379 985 970225 955671625 31.384709 9.949747 986 972196 958585256 31.400636 9.953113 987 974169 961504803 31.416556 9.956477 988 976144 964430272 31.432467 9.959839 989 978121 967361669 31.448370 9.963198 990 980100 970299000 61.464265 9.966554 991 982081 973242271 31.480152 9.969909 992 984064 976191488 31.496031 9.973262 993 986049 979146657 31.511902 9.976612 994 988036 982107784 31.527765 9.979959 995 990025 985074875 31.543620 9.983304 996 992016 988047936 31.559467 9.986648 997 994009 991026973 31.575306 9.989990 998 996004 994011992 31.591138 9.993328 999 998001 997002999 31.606961 9.996665 1000 1000000 1000000000 31.622776 10. Additional use of this table can be made by the aid of the following Rules : To find the Square of a Number above 1000 =^when the Number is divisible by any Number without leaving a Remainder. Rule. — If the number exceed by 2, 3, or any other number of times, any SQUARES, CUBES, AND ROOTS. 117 number contained in the table, let the square affixed to that number in the table be multiplied by the square of 2, 3, 4, 5, or 6, &c., and the product will be the answer. Example. — Required the square of 1550. 1550 is 10 times 155, and the square of 155 in the table is 24025. Then 24025X10^ = 2402500 Ans. When the Number is an Odd Number. Rule.— Find the two numbers nearest to each other, which, added together, make that sum ; then the sum of the squares of these two numbers, as per table, multiplied by 2, will give the answer, exceeded by 1, which is to be subtracted, and the remainder is the answer required. Example.— What is the square of 1345 1 The nearest two numbers are j g 'g | = 1345. Then ner table \ 6732 = 452929 men, per table, j 6722- 452584 904513X2 =: 1809026—1 = 1809025 Ans. To find the Cube of a Number greater than is contained in the Table. Rule.— Proceed as in squares to find how many times the number exceeds one of the tabular numbers. Multiply the cube of that number by the cube of the number of times the number sought exceeds the number in the table, and the prod- uct will be the answer. Example.— What is the cube of 1200 1 1200 is 3 times 400, and the cube of 400 is 64000000. Then 64000000X3^ = 1728000000 Ans. To find the Squares of Numbers following each other m Arith- metical Progression. Rule.— Find the squares of the two first numbers in the usual way, and subtract the less from the greater. Add the difference to the greatest square, with the ad- dition of 2 as a constant quantity ; the sum will be the square of the next number. Example.— What are the squares of 1001, 1002, 1003, 1004, and 1005 '» 10002 = 1000000 9992= 998001 1999 Add . . 2 Add 10002 = 1000000 1002001 Square of 1001. Difference, 2001+2= 2003 1004004 Square of 1002. Difference, 2003+2= 2005 1006009 Square of 1003. Difference, 2005+2 = 2007 1008016 Square of 1004. Difference, 2007+2= 2009 1010025 Square of 1005. To find the Cubes of Numbers following each other in Arithmeti- cal Progression. Rule.— Find the cubes of the two first numbers, and subtract the less from the greater ; then multiply the least of the two numbers cubed, by 6; add the product, 118 SQUARES, CUBES, AND ROOTS. with the addition of 6, to the difference, and continue this the first series of diffe^ ences. For the second series of differences, add the cube of the highest of the above numbers to the difference, and the sum will be the cube of the next number. Example.— What are the cubes of 1001, 1002, and loos'? First Series. Cube of 1000 = 1000000000 Cube of 999 = 997002999 a 2997001 Difference. 999x6+6 = 6000 3003001 Difference of 1000. 6000 +6 = 6008 "3009007 Difference of 1001. 6006 +6 = 6012 3015019 Difference of 1002. Second Series. Cube of 1000 . =1000000000 Difference for 1000, 3003001 1003003001 = Cube of 1001. Difference for 1001, 3009007 1006012008 = Cube of 1002. Difference for 1002, 3015019 1009027027 = Cube of 1003. To find the Cube or Square Root of a higher Number than %s contained in the Table. Rule.— Find in the column of Squares or Cubes the number nearest to that num- ber whose root is required, and the number from which that square or cube is de- rived will be the answer when decimals are not of importance. Example.— What is the square root of 562500 ? In the table of Squares, this nmnber is the square of 750 ; therefore 7o0 is the square root required. Example.— What is the cube root of 2248090 1 In the table of Cubes, 2248091 is the cube of 131 ; therefore 131— is the cube root required, nearly. To find the Cube Root of any Number over 1000. Rule.— Find by the table the nearest cube to the number given, and call it the assumed cube. Multiply the assumed cube and the given number respectively by 2 ; to the product of the assumed cube add the given number, and to the product of the given number add the assumed cube. Then, as the sum of the assumed cube is to the sum of the given number, so is the root of the assumed cube to the root of the given number. Example.— What is the cube root of 224809 ? By table, the nearest cube is 216000, and its root is 60. 216000x2+224809 = 656809, And 224809x2+216000 = 665618. Then, as 656809 : 665618 : : 60 : 60.804+ To find the Sixth Root of a Number. Rule.— Take the cube root of its square root. Example.— What is the ^ of 441 ? ^441 = 21 , and .^21 = 2.7589 ^ns. SQUARES, CUBES, AND ROOTS. 119 TO FIND THE CUBE OR SQUARE ROOT OF A NUMBER CONSIST- ING OF INTEGERS AND DECIMALS. Rule.— Multiply the difference between the root of the integer part and the root of the next higher integer by the decimal, and add the product to the root of the integer given ; the sum will be the root of the number required. This is correct for the square root to three places of decimals, and in the cube root to seven. Example.— What is the square root of 53.75, V' 54 = 7.3484 y/ .53= 7.2801 .051225 53 = 7.2801 V'53.75 = 7.331325 ^/ , and the cube root of 843.75 1 ^844 = 9.4503 3/843 = 9.4466 .0037 ^ .002775 ^843 = 9.4466 4^843.75 = 9.449375 120 Table SIDES OF EQUAL SQUARES. of the Sides of Squares^equal in Area to Diameter, from 1 to 100. a Circle of any Side of equal Square. Diameter. Side of equal Square. 0.886 1.107 1.329 1.550 1.772 1.994 2.215 2.437 2.658 2.880 3.101 3.323 3.544 3.766 3.988 4.209 4.431 4.652 4.874 5.095 5.317 6.538 5.760 5.982 6.203 6.425 6.646 6.868 7.089 7.311 7.532 7.754 7.976 8.197 8.419 8.640 8.862 9.083 9.305 9.526 9.748 9.970 10.191 10.413 10.634 10.856 11.077 11.299 11.520 11.742 11.964 12.185 12.407 12.628 12. '850 13.071 15. .25 .5 .75 16. .25 .5 .75 17. .25 .5 .75 18. .25 .5 .75 19. .25 .5 .75 20. .25 .5 .75 21. .25 .5 .75 22. .25 .5 .75 23. .25 .5 .75 24. ' .25 .5 .75 25. .25 .5 .75 26. .25 .5 .75 27. .25 .5 .75 28. .25 .5 .75 13.293 13.514 13.736 13.958 14.179 14.401 14.622 14.844 15.065 15.287 15.508 15.730 15.952 16.173 16.395 16.616 16.838 17.059 17.281 17.502 17.724 17.946 18.167 18.389 18.610 18.832 19.053 19.275 19.496 19.718 19.940 20.161 20.383 20.604 20.826 21.047 21.269 21.491 21.712 21.934 22.155 22.377 22.598 22.820 23.041 23.263 23.485 23.706 23.928 24.149 24.371 24.592 24.814 25.035 25.257 25.479 Diameter. "297" Side of equal Squire. .25 .5 .75 30. .25 .5 .75 31. .25 .5 .75 32. .25 .5 .75 33. .25 .5 .75 34. .25 .5 .75 35. .25 o5 .75 36. .25 .5 .75 37. .25 .5 .75 38. .25 .5 .75 39. .25 .5 .75 40. .25 .5 .75 41. .25 Diameter. 42. .75 .25 .5 .75 25.700 25.922 26.143 26.365 26.586 26.808 27.029 27.251 27.473 27.694 27.916 28.137 28.359 28.580 28.802 29.023 29.245 29.467 29.688 29.910 30.131 30.353 30.574 30.796 31.017 31.239 31.461 31.682 31.904 32.125 32.347 32.568 32.790 33.011 33.233 33.455 33.676 33.898 34.119 34.341 34.562 34.784 35.005 35.227 35.449 35.670 35.892 36.113 36.335 36.556 36.778 36.999 37.221 37.443 37.664 37.886 43. .25 .5 .75 44. .25 .5 .75 45. .25 .5 .75 46. .25 .5 .75 47. .25 .5 .75 48. .25 .5 .75 49. .25 .5 .75 50, .25 .5 .75 51. .25 .5 .75 52. .25 .5 .75 53. .25 .5 .75 54. .25 .5 .75 55. .25 .5 .75 56. .25 .5 .75 {Side of equal j Square. 38.107 38.329 38.550 38.772 38.993 39.215 39.437 39.658 39.880 40.101 40.323 40.544 40.766 40.987 41.209 41.431 41.652 41.874 42.095 42.317 42.538 42.760 42.982 43.203 43.425 43.646 43.868 44.089 44.311 44.532 44.754 44.976 45.197 45.419- 45.640 45.862 46.083 46.505 46.526 46.748 46.970 47.191 47.413 47.634 47.856 48.077 48.299 48.520 48.742 48.964 49.185 49.407 49.628 49.850 .50.071 50 .293 SIDES OF EQUAL SQUARES. Table— (Continued). 121 Diameter. Side of equal Square. Diameter. Side of equal S^qiiare. Diameter. Side of equal Square. Diameter. Side of equal Square 67. 50.514 68. 60.263 79. 70.011 90. 79.760 .25 50.736 .25 60.484 .25 70.233 .25 79.981 .5 50.958 .5 60.706 .5 70.455 .5 80.203 .75 51.179 .75 60.928 .75 70.676 .75 80.425 68. 51.401 69. 61.149 80. 70.898 91. 80.646 .25 51.622 .25 61.371 .25 71.119 .25 80.868 .5 51.844 .5 61.592 .5 71.341 .5 81.089 .75 52.065 .75 61.814 .75 71.562 .75 81.311. 69. 52.287 70. 62.035 81. 71.784 92. 81.532 .25 52.508 .25 62.257 .25 72.005 .25 81.754 .5 52.730 .5 62.478 .5 72.227 .5 81.975 .75 52.952 .75 62.700 .75 72.449 .75 82.197 60. 53.173 71. 62.922 82. 72.670 93. 82.419 .25 53.395 .25 63.143 .25 72.892 .25 82.640 .5 53.616 .5 63.365 .5 73.113 .5 82.862 .75 53.838 .75 63.586 .75 73.335 .75 83.083 61 54.059 72. 63.808 83. 73.556 94. 83.305 .25 54.281 .25 64.029 .25 73.778 .25 83.526 .5 54.502 .5 64.251 .5 73.999 .5 83.748 .75 54.724 .75 64.473 .75 74.221 .75 83.970 62. 54.946 73. 64.694 84. 74.443 95. 84.191 .25 55.167 .25 64.916 .25 74.664 .25 84.413 .5 55.389 .5 65.137 .5 74.886 .5 84.634 .75 55.610 .75 65.359 .75 75.107 .75 84.856 63. 55.832 74. 65.580 85. 75.329 96. 85.077 .25 56.053 .25 65.802 .25 75.550 .25 85.299 .5 56.275 .5 66.023 .5 75.772 .5 85.520 .75 56.496 .75 66.245 .75 75.993 .75 85.742 64. 56.718 75. 66.467 86. 76.215 97. 85.964 .25 56.940 .25 66.688 .25 76.437 .25 86.185 .5 57.161 .5 66.910 .5 76.658 .5 86.407 .75 57.383 .75 67.191 .75 76.880 .75 86.628 65. 57.604 76. 67.353 87. 77.101 98. 86.850 .25 57.826 .25 67.574 .25 77.323 .25 87.071 .5 58.047 .5 67.796 .5 77.544 .5 87.293 .75 58.269 .75 68.017 .75 77.766 .75 87.514 66. 58.490 77. 68.239 88. 77.987 99. 87.736 .25 58.712 .25 68.461 .25 78.209 .25 87.958 .5 58.934 .5 68.682 .5 78.431 .5 88.179 .75 59.155 .75 68.904 .75 78.652 .75 88.401 67. 59.377 78. 69.125 89. 78.874 100. 88.622 .25 59.598 .25 69.347 .25 79.095 .25 88.844 .5 59.820 .5 69.568 .5 79.317 .5 89.065 .75 60.041 .75 69.790 .75 79.538 .75 89.287 USE OF THIS TABLE. To find a Square that shall have the same Area as a Given Circle. Example.— What is the side of a square that has the same area as a circle of 73i inches ? .^, . , j . .^ x. By table of Areas, page 93, opposite to 73.25 is its area, 4214.1 ; and in the above table, page 121, is 64.916, the side of a square that has the same area as a circle of 73^ inches in diameter. Example.— What should be the side of a square that would give the same area as a board that is 18 inches v^de and 10 feet long 1 • 122 SIDES OF EQUAL SQUARES. 18 inches is . 1.5 feet. ^10 15.0 feet. 14 4 square inches in a foot. 60F 600 150 2160.0 inches area. Bv table page 120, 2164.75 inches area have a diameter of 52.5 inches, which in the above table gives an equal side of 46.526, which is the answer very nearly. PLANE TRIGONOMETRY. ABC the three angles (A the right angle) ; a Z»c the three sides respectively o]^ posite to them ; R the tabular radius (1 or 1000000) ; S the area of the triangle, and y half its perimeter = ( — ^ — )' RIGHT-ANGLED TRIANGLES. QCasinth. E C -.^ , also = a. - Given. To find A C and B A. Hyp. B C, J R : B C : : sin. B : A C, and Angles, j R : B C : : sin. C : B A. 61VEN. To find B A and B C. AC, S R : AC : : tan. C : B A, and Angles. ^ R : A C : : sec. C : B C, J or sin. B . AC : : R : BC. Given. To find Angles and A C. Hyp. B C, < B C : R : : B A : sin. C, whose comp. is B. and leg B A. ^ R : B C : : sin. B : A C. Given. Both legs. I sin. B To find Angles and B C. A C : R : : B A : tan. C, whose comp. is B. Note. 1 fein. C : B A : : R : B C, I or R : AC : : sec. C : BC. -By sin. or tan. B or C is meant the sine or tangent of the angle B or C. Let A B C be a right-angled triangle, in which A B is as- sumed to be radius ; B C is the tangent of A, and A C its secant to that radius ; or, dividing each of these by the base, we shall have the tangent and secant of A respectively to radius 1. Tracing the consequences of assuming B C and A C each for radius, we obtain the following expressions : -Jb — — = tan. angle A. base -^ = sec. angle A. base P— ^* = sin. angle A. hyp. base perp. hyp. perp. base hyp. = tan. angle C. = sec. angle C = sin. angle C. 124 PLANE TRIGONOMETRY. OBLIQUE-ANGLED TRIANGLES. -A, G . C D A D / ^ O Given, the Angles and Side A B, to find B C and A C. Sin. C AB : : sin. A BC. Sin. C AB : : sin. B AC. Given, two Sides A B, B C, and the Angle C, to find Angle A and B, and Side A C . A B : sin. C : : B C : sin. A, which, added to C, and the sum subtracted from 180O, will give B. Sin. C : A B : : sin. E : A C. Given, A C, AB, and the included Angle A, to find Angles C and B, and Side B C. Subtract half the given angle A from 90° ; the remainder is half the %um of the other angles. Then, as the sum of the sides A C, A B is to their difference, so is the tangent of the half sum of the other angles to the tangent of half their differ- ence, which, added to and subtracted from the half sum, will give the two angles B and C, the greatest angle being opposite to the greatest side. Sin. B : A C : : sin. A : B C. Given, all three Sides, to find all the Angles. Let fall a perpendicular B D opposite to the required angle ; then, as A C : sum of A B, B C : : their difference : twice D G, the distance of the perpendicular from the middle of the base ; hence A D, C D are known, and the triangle A B C is divi- ded mto two right-angled tiiangles B C D, B A D ; then, by the rules in right-angled triangles, find the angle A or C. *«» & NATCKAL SINES, COSINES, AND TANGENTS. 125 Table of Natural Sines Cosines , and Tangents. D.M. Sine. Cosine. Tangent. D.M , Sine. Cosine. Tangent. .15 00436 99999 00436 14. 24192 97030 24933 .30 00872 99996 00873 .15 .30 24615 25038 96923 96815 25397 25862 .45 01309 99991 01309 .45 25460 96705 26328 1. 01745 99985 01745 15. 25882 96593 26794 .15 02181 99976 02182 .15 26303 96479 27263 .30 02618 99966 02619 .30 26724 96363 27732 .45 03054 99953 03055 .45 27144 96246 28203 2. 03490 99939 03492 16. 27564 96126 28675 .15 03926 99923 03929 .15 27983 96005 29147 .30 04362 99905 04366 .30 28402 95882 29621 .45 04798 99885 04Q03 .45 28820 95757 30097 3. 05234 99863 05241 17. 29237 95630 30573 .15 05669 99839 05678 .15 29654 95502 31051 .30 06105 99813 06116 .30 30071 95372 31530 .45 06540 99786 06554 .45 30486 95240 32010 4. 06976 99756 06993 18. 30902 95106 32492 .15 07411 99725 07431 .15 31316 94970 32975 .30 07846 99692 07870 .30 31730 94832 33460 .45 08281 99657 08309 .45 32144 94693 33945 5. 08716 99619 08749 19. 32557 94552 34433 .15 09150 99580 09189 .15 32970 94409 34922 .30 09585 99540 09629 .30 33381 94264 35412 .45 10019 99497 10069 .45 33792 94118 35904 6. 10453 99452 10510 20. 34202 93969 36397 .15 10887 99406 10952 .15 34612 93819 36892 .30 11320 99357 11394 .30 35021 93667 37388 .45 11754 99307 11836 .45 35429 93514 37887 7. 12187 99255 12278 21. 35837 93358 38386 .15 12620 99200 12722 .15 36244 93201 38888 .30 13053 99144 13165 .30 36650 93042 39391 .45 13485 99087 13609 .45 37056 92881 39896 8. 13917 99027 14054 22. 37461 92718 40403 .15 14349 98965 14499 .15 37865 92554 40911 .30 14781 98902 14945 .30 38268 92388 41421 .45 15212 98836 15391 .45 38671 92230 41933 9. 15643- 98769 16838 23. 39073 92050 42447 .15 16074 98700 16286 .15 39474 91879 42983 .30 16505 98629 16734 .30 39875 91706 43481 .45 16935 98556 17183 .45 40275 91531 44001 10. 17365 98481 17633 24. 40674 91355 44522 .15 17794 98404 18083 .15 41072 91176 45046 ,30 18224 98325 18534 .30 41469 90996 45572 .45 18652 98245 18986 .45 41866 90814 46100 11. 19081 98163 19438 25. 42262 90631 46630 .15 19509 98079 19891 .15 42657 90446 47163 .30 19937 97992 20345 .30 43051 90259 47697 .45 20364 97905 20800 .45 43445 90070 48234 12. 20791 97815 21256 26. 43837 89879 48773 .15 21218 97723 21712 .15 44229 89687 49314 .30 21644 97630 22169 .30 44620 89493 49858 .45 22070 97534 22628 .45 45010 89298 50404 13. 22495 97437 23087 27. 45399 89101 50952 ' .15 22920 97338 23547 .15 45787 88902 51503 .30 23345 97237 24008 .30 46175 88701 52056 .45 23769 97134 24470 .45 46561 88499 52612 126 NATtJRAL SINES, COSIxNES, AND TANGENTS. Table — (Continued ). D.M. Sine. Cosine. Tangent. DM. Sine. Cosine. Tangent. 28. 46047 88295 53170 37. 60182 79864 75355 .15 47332 88089 53731 .15 60529 79600 76041 .30 47716 87882 54295 .30 60876 79335 76732 .45 48099 87673 54861 .45 61222 79069 77428 29. 48481 87462 55430 38. 61566 78801 78128 .15 48862 87250 56002 .15 61909 78532 78833 .30 49242 87036 56577 .30 62251 78261 79543 .45 49622 86820 57154 .45 62592 77988 80258 30. 50000 86603 57735 39. 62932 77715 80978 .15 50377 86384 58318 .15 63271 77439 81703 .30 50754 86163 58904 .30 63608 77162 82433 .45 51129 85941 59493 .45 63944 76884 83169 31. 51504 85717 60086 40. 64279 76604 83910 .15 51877 85491 60681 .15 64612 76323 84656 .30 52250 85264 61280 .30 64945 76041 85408 .45 52621 85035 61881 .45 65276 75756 86165 32. 52992 84805 62486 41. 65606 75471 86928 .15 .53361 84573 6.3095 .15 65935 75184 87697 .30 63730 84339 63707 .30 66262 74896 88472 .45 54097 84104 64392 .45 66588 74606 89253 33. 54464 83867 64940 42. 66913 74314 90040 .15 54829 83629 65562 .15 67237 74022 90833 .30 55194 83389 66188 .30 67559 73728 91633 .45 55557 83147 66817 .45 67880 73432 92439 34. 55919 82904 67450 43. 68200 73135 93251 .15 56280 82659 68087 .15 68518 72837 94070 .30 56641 82413 68728 .30 68835 72537 94896 .45 57000 82165 69372 .45 69151 72236 95729 35. 57358 81915 70020 44. 69466 71934 96568 .15 57715 81664 70673 .15 69779 71630 97415 .30 58070 81412 71329 .30 70091 71325 98269 .45 58425 81157 71989 .45 70401 71019 99131 36. 58779 80902 72654 45. 70710 70710 1.00000 .15 59131 80644 73323 .15 71019 70401 1.00876 .30 59482 80386 73996 .30 71325 70091 1.01760 .45 59832 80125 74673 TANGENTS FROM 450 TO 9(P. D. Tangent. D. Tangent. D. Tangent. D. Tangent. D. Tangent 46 1.0355 55 1.4281 64 2.0503 73 3.2708 82 7.1153 47 1.0724 56 1.4826 65 2.1445 74 3.4874 83 8.1443 48 1.1106 57 1.5399 66 2.2460 75 3.7321 84 9.5144 49 1.1504 58 1.6003 67 2.3558 76 4.0107 85 11.4301 50 1.1918 59 1.6643 68 2.4751 77 4.3314 86 14.3007 51 1.2349 60 1.7321 69 2.6051 78 4.7046 87 19.0811 52 1.2799 61 1.8040 70 2.7475 79 5.1445 88 28.6363 53 1.3270 62 1.8807 71 2.9042 80 5.6712 89 57.2900 54 1.3764 63 1.9626 72 3.0776 81 6.3137 90 Infinite. SINES AND SECANTS OF ANGLES. 127 To find the Sine or Cosine of any Angle exceeding 45°, hy the foregoing Table. Subtract the angle given from 90, look in the table for the re- mainder, and opposite to it take out the sine for the cosine, and the cosine for the sine of the angle given. Example.— What is the sine and the cosine of 85°1 85°— 90° — 5°, and opposite to 5° in the table is 08716 and 99619, which are respectively the cosine and sine of 85° The sine of 90° is 100000, cosine 0. The sine of an arc, divided by the cosine, gives the natural tan- gent of that arc. To compute Tangents and Secants, Cos. : sin. : : rad. : tangent. Cos. : rad. : : rad. : secant. Sin. : COS. : : rad. : cotangent. Sin. : rad. ; : rad. : cosecant. To find the Secant of an Angle, Divide 1 by the cosine of that angle. Example.— The cosine of 21° 30' is .93041 ; = 1.07479. .93041 To find the Cosecant of an Angle. Divide 1 by the sine of the angle. Example. — The sine of 21° 30' is .36650; = 2.72951. .36650 To find the Versed Sine. Subtract the cosine from 1. Example. — The cosine of 21° 30' is .93042 ; 1— .93042 = .06958. To find the Cover sed Sine. Subtract the sine of the angle from 1. Example.— The sine of 21° 30' is .36650 ; 1— .36650 = .6335. To find the Chord of any Angle, Take the sine of half the angle and double it. Example.— The chord of 21° 30' is required. ^. ^21° 30' Sine of — ^ = .18653X2 = .37304. 128 MECHANICAL POWERS. MECHANICAL POWERS. Power is a compound of weight, or the expansion of a body, mul- tiplied by its velocity : it cannot be increased by mechanical means. The Science of Mechanics is based upon Weight and Power, or Force and Resistance. The weight is the resistance to be overcome, the power is the requisite force to overcome that resistance. When they are equal no motion can take place. The Powers are three in number, viz., Lever, Inclined Plane, and Pulley. Note.— The Wheel and Axle is a continual or revolving lever, the Wedge is a double inclined plane, and the Screw is a revolving inclined plane. LEVER. When the Fulcrum (or Support) of the Lever is between the Weight and the Power. Rule.— Divide the weight to be raised by the power, and the quotient is the difference of leverage, or the distance from the ful- crum at which the power supports the weight. Or, multiply the weight by its distance from the fulcrum, and the power by its distance from the same point, and the weight and power will be to each other as their products. Example.— A weight of 1600 lbs. is to be raised by a force of 80 lbs. ; required the length of the longest arm of the lever, the short- est being 1 foot. 15^>ll = 20feet,4n.. 80 Proof, by second rule. 1600 X 1=1600. 80X20 = 1600. Example.— A weight of 2460 lbs. is to be raised with a lever 7 feet long and 300 lbs. ; at what part of the lever must the fulcrum be placed 1 51— = 8.2 ; that is, the weight is to the power as 8.2 to 1 ; there- 300 ' 7 V 12 84 fore the whole length -— — - = — == 9.13 inches, the distance of o.-4~rl u.Z the fulcrum from the weight. Example.— A weight of 400 lbs. is placed 15 inches from the ful- crum of a lever ; what force will raise it, the length of the other arm being 10 feet 1 400X15 _,, . = 50 lbs., Ans. 120 NoTK.— Pressure upon fulcrum equal the sum of weight and power. MECHANICAL POWERS. 129 When tke fulcrum is at one Extremity of the Lever, and the Poioer or the Weight, at the other. . ' ^ Rule.— As the distance between the power or weight and ful- crum is to the distance between the w^eight or power and fulcrum 80 is the effect to the power. * Example.— What power will raise 1500 lbs., the weight beinff 5 feet from it, and 2 feet from the fulcrum 1 5+2 = 7 : 2 : : 1500 : 428.5714+^715. Example.— What is the weight on each support of a beam that is 30 feet long, supported at both ends, and bearing a weight of 6000 lbs. 10 feet from one end 1 30 : 20 : : 6000 : 4000 lbs. at the end nearest the weight ; and 30 : 10 : : 6000 : 2000 lbs. at the end farthest from th'e weight. Note.— Pressure upon fulcrum is the difference of the weight and the power. The General Rule, therefore, for ascertaining the relation of Power to Weight m a lever, whether it be straight or curved, is the power multiplied by its distance from the fulcrum, is equal to the weight multiplied by its distance from the fulcrum. Let P be called the power, W the weight, p the distance of P from the fulcrum, and w the distance of W from the fulcrum ; then P : W : :-w; : p, orPx;7 = Wxi^: and V w 'Wxw_ Vxv If several weights or powers act upon one or both ends of the lever, the condition of equilibrium is PXp+FX/+P^^X;?^ &c.,=z.WXl^;+W'x^^;^ &c. In a system of levers, either of similar, compound, or mixed kinds, the condition is * Px^x/x/;^^^ wXw'Xw" .Jt\^f"'^ lb-^,f^d./ each 10 feet,/' 1 foot ; and if tt- and z.,' be each 1 foot, and w'' 1 inch, then 1 X 120 X 120 X 12 __ 172800 12x12x1 — 144 — 1^^^ 5 ^hat is, 1 lb. will balance 1200 lbs. with levers of the lengths above given. rpJIt^Jnf";?'^'^^ ^-^^^ ""^ ^^^ l^""^'^ ^^ ^^® ^^^^^ formula are not considered, the centre of gravity being assumed to be over the fulcrums. If the arms of the lever be equally bent or curved, the distances from the fulcrum must be measured upon perpendiculars, drawn from the lines of direction of the weight and power, to a line run- nmg horizontally through the fulcrum ; and if unequally curved measure the distances from the fulcrum upon a line running hori- zontally through it till it meets perpendiculars falling from the ends of the lever. 130 MECHANICAL POWERS. WHEEL AND AXLE. The power multiplied by the radius of the wheel is equal to the weight multiplied by the radius of the axle. As the radms of the wheel is to the radius of the axle, so is the effect to the power. ^ ^^ . . , ^ When a series of wheels and axles act upon each other, either by belts or teeth, the weight or velocity will be to the power or unity as the product of the radii, or circumferences of the wheel-s, to the product of the radii, or circumferences of the axles. Example.— If the radii of a series of wheels are 9, 6, 9, 10, and 12, and their pinions have each a radius of 6 inches, and the weight applied be 10 lbs., what weight will it raise] 6X6X6X6X6 Or, if the 1st wheel make 10 revolutions, the last will make 75 in the same time. Note.— For a fuller treatise on wheels, see Grier's Mechanic's Calculator, pages 130 to 136. To find the Power of Cranes, <$fc. Rule.— Divide the product of the driven teeth by the product of the drivers, and the quotient is the relative velocity, which, multi- plied by the length of the winch and the force in lbs., and divided by the radius of°the barrel, will give the weight that can be raised. Example.— A force of 18 lbs. is applied to the winch of a crane, the length being 8 inches ; the pinion having 6, the wheel 72 teeth, and the barrel 6 inches diameter. -^ = 12X8X18 — 1728-^-3 = 576 lbs. weight. Let w represent length of winch, ^^^'P r ** radius of barrel, W. r Wr = vwV. vw p " force applied 7, " velocity, W " weight.raised. Example —A weight of 94 tons is to be raised 360 feet in 15 minutes, by a force the velocity of which is 220 feet per minute ; what is the power required ] OCA = 24 feet per minute. 15 ?i^= 10.2542 tons. 220 In a wheel and axle, where the axle has two diameters, the con- dition of equilibrium is W : P : : R : 4 [r—r') ; or, PxR==Wxi(r— r^j; .^^ that is, the weight is to the power as the lever by which the power works, is to half the difference of the radii of the axle ; MECHANICAL POWERS. 131 R representing radius of wheel, r " radius of large axle, r' " radius of less axle. INCLINED PLANE. Rule.— As the length of the plane is to its height, so is the weight to the power. Example.— Required the power necessary to raise 1000 lbs. up an inclined plane 6 feet long and 4 feet high. As 6 : 4 : : 1000 : 666.66 Ans. Wxh Let W represent weight, height of plane, length of plane, power, base of plane, pressure on plane. h Wxb -r-=P' To find the Length of the Base, Height, or Length of the Plane, when any two of them are given. Rule. — For the length of the base, subtract the square of the height from the square of the length of the plane, and the square root of the remainder will be the length of the base. For the length of the plane, add the squares of the two other di- mensions together, and the square root of their sum will be the length required. For the height, subtract the square of the base from the square of the length of the plane, and the square root of the remainder is the height required. Example. — The height of an inclined plane is 20 feet, and its length 100 ; what is its base, and the pressure of 1000 lbs. upon the plane "? -v/202— 100^ = 9600 = 97.98 the base. As 100 : 20 : : 1000 : 200 lbs. necessary power to raise the 1000 ^ 1000X97.98 lbs., and — = 979.8 the pressure upon the plane. If two bodies on two inclined planes sustain each other hy the aid of a cord over a pulley, their weights are directly as the lengths of the planes. Example.— If a body of 50 lbs. weight, upon an inchned plane, of 10 feet rise in 100, be sustained by another weight on an opposite plane, of 10 feet rise to 90 of an inclination, what is the weight of the latter 1 As 100 : 90 : : 50 : 45, the answer. When a body is supported by two planes, and if the weight be repre- sented by the sine of the angle between the two planes. The pressures upon them are reciprocally as the sines of the inclinations of those planes to the horizon, viz. : 132 MECHANICAL POWERS. ( Sine of the angle be- _, . , ^ ^1 tween the planes. The weight, ) , g^^g ^f t^g l3 ^^ The pressure upon one plane, > are as<^ ^^^ 1^^^ ^ The pressure upon the other plane, ) j g.^^ ^^ ^^^ ^^^^^ ^^ \ the other plane. Thus, if the angle between the planes was 90°, of one plane 60"^, and the other 30°— since the natural sines of 90°, 60°, and 30° are 1, .866, and .500— if the body weighed 100 lbs., the pressure upon the plane of 30° would be 86.6 lbs., and upon the plane of 60°, 50 lbs. = the centre of gravity being in the centre of the body. When the 'power does not act parallel to the plane, draw a line per- pendicular to the direction of the power's action from the end of the base line (at the back of the plane), and the intersection of this line on the length will determine the length and height of the plane. Note.— When the line of direction of the power is parallel to the plane, the power is least. The space which a body describes upon an inclined plane, when descending on the plane by the force of gravity, is to the space it would freely fall in the same time, as the height is to the length of the plane ; and the spaces being the same, the times will be in- versely in this proportion. Example. — If a body be placed upon an inclined plane 300 feet long and 25 feet high, what space will it roll down in one second by the force of gravity alone 1 As 300 : 25 : : *16.08 : 1.33 feet, Ans. If a body be projected down an inclined plane with a given velocity, then the distance which the body will be from the point of projec- tion in a given time will be tXv-\—Xl6.08t^ ; but if the body be * The distance a body will freely fall in one second by the force of gravity. The force of an inclined plane bears the same proportion to the force of gravity as the height of the plane bears to its length ; that is, the force which accelerates a body down an inclined plane is that fractional part of the force of gravity which is represented by the height of the plane divided by its length. Let h represent the height of the plane, I its length, t the time in seconds, s the space which a body will move through in a given time, v the velocity, and i the / h\ angle of inclination \^sm. ^ = jj* 16.08 hf^ tv Zr^ v^ ^ ^„ , - or ^' or -r— , or ^JT^^-;' ^r sin. zXl6.08«2. tj:=z — ,or — -, — ,or V — - — , or sin. 2X32.16t, or ^sm. zXo4.3s « = -' or ^5-^^, or V -— -, or ^oir «i„ ,' o^ V T 32.16 h' ^ 16.08 li 32.16 sin. t' ^ 16.08 sin. i l °' '*^- *■ = 32i6-t' "' 16:^7^' °' ^b^- The acceleratmg force on the plane is to the accelerating force of gravity as v^ is to 64.3X5. If sin. i = i, it shows that the length of the plane is twice its height, or ^ = 30©. If the proportion wliich the length of the plane bears to the height be given, sub- stitute these proportions for the length and height in the above rules, and the con- clusions will be equally true. MECHANICAL POWERS. 133 projected upward, then the distance of the body from the point of projection will he tXv—jXl^Mt^, WEDGE. WIi£n two Bodies are forced from mie another, in a direction Parallel to the back of the Wedge. Rule.— As the length of the wedge is to half its back, so is the resistance to the force. Example.— The length of the back of a double wedge is 6 inches and the length of it through the middle 10 inches ; what is the pow- er necessary to separate a substance having a resistance of 150 lbs. 1 As 10 : 3 : : 150 : 45 lbs., Ans. When only one of the Bodies is Movable. Rule.— As the length of the wedge is to its back, so is the resist- ance to the power. Example.— What power, applied to the back of a wedge, will raise a weight of 15,000 lbs., the wedge being 6 inches deep, and 100 Ion? on its base. ^ As 100 : 6 : : 15000 : 900 lbs., Ans. Note.— As the power of the wedge in practice depends upon the split or rift in the vyod to be cleft, or in the body to be raised, the above rules are only theo- reucal where a rift exists. SCEEW. As the screw is an inclined plane wound round a cylinder, the length of the plane is found by adding the square of the circumfer- ence of the screw to the square of the distance between the threads, and taking the square root of the sum, and the height is the distance between the consecutive threads. Rule.— As the length of the inclined plane is to the pitch or height of It, so IS the weight to the power. When a wheel or capstan is applied to turn the screw, the length of the lever IS the radius of the circle described by the handle of the wheel or capstan bar. Let P represent power. R W / P X length of lever, weight, length of the inclined plane, pitch of screw or height of plane, effect of power at circumference of screw, radius of screw. M 134 MECHANICAL POWERS. w :P, p :P, p • V^ p w, V : U p . X, r ■■ R, X : P. Then, by the above rules, As Z : j!7 I :W W; / p: I P :W r : R P: X R: r Example.— What is the power requisite to raise a weight of 8000 lbs. by a screw of 12 inches circumference and 1 inch pitch ^ 122+12 rr: 145, and ^145 =: 12.04159. Then, 12.0416 : : 1 : : 8000 : 664.36 lbs., Ans. And if a lever of 30 inches length was added to the screw, 12—3.1416 = 3.819+2+30 = 31.9095, length of lever. » Then, as 31.9095 : 1.9095 : : 664.363 : 39.756 lbs., Ans. Or, when the circumference described by the power is used (C), we have P : W: : ;? : C, C : ;> : : W : P, PXC=WX;?; thus, 39.756 : 8000 : : 1 : 201.227 = circumference described by lever, which is the hypothenuse of the triangle formed by the base and height of the inclined plane. When a hollow screw revolves upon one of less diameter and pitch (as designed by Mr. Hunter), the effect is the same as tb^t of a single screw, in which the distance between the threads is equal to the difference of the distances between the threads of the two screws. If one screw has 20 threads in an inch pitch, and the other 21, the power is to the weight as the difference between ^V ^^^ Tf* ^^ tIo — 1 to 420. In a complex machine, composed of the screw, and wheel, and axle, the relation between the weight and power is thus : Let X represent the effect of the power on the wheel, R " the radius of the wheel, p " the pitch of the screw, r " the radius of the axle, C " the circumference described by the power. Then, by the properties of the screw, PxC=a:X;?; and of the wheel and axle, a;XR==WXr. Hence we have PxCX2:XR = xX:pXWXr. Omitting the common multiplier, x, PXCXR — WX;?Xr; or P: W: -.pXr: CxR, andj^Xr :CXR : : P : W. • MECHANICAL POWERS. 135 Ex AMPLE. —■What weight can be raised with a power of 10 lbs. applied to a crank 32 inches long, turning an endless screw of 3^ inches diameter and one inch pitch, applied to a wheel and axle of 20 and 5 inches in diameter respectively ] Circumference of 64 = 201. 1 : 201 : : 10 : 2010. Radii of wheel and axle, 10 and 2.5. 2.5 : 10 : : 2010 : 8040 lbs., Arts., or 2.5X1 : 201X10:: 10: 8040. And when a series of wheels and axles act upon each other, the weight will be to the power as the continued product of the radii of the wheels to the continued product of the radii of the axles ; thus, W : P : : R3 : r^ ; or, r^ : R3 : : P : W, there being three wheels and axles of the same proportion to each other. Example. — If an endless screw, with a pitch of half an inch, and a handle of 20 inches radius, be turned with a power of 150 lbs., and geared to a toothed wheel, the pinion of which turns another wheel, and the pinion of the second wheel turns a third wheel, to the pinion or barrel of whi^ih is hung a weight, it is required to know what weight can be sustained in that position, the diameter of the wheels being 18, and the pinions 2 inches '! pXr' : CxR^ :P: W; or .5X1^ : 125.6X93 : : 150; which, when extended, gives .5 : 91562.4 : : 150 : 27468720 lbs., Ans. PULLEY. When only one Cord or Rope is used. Rule. — Divide the weight to be raised by the number of parts of the rope engaged in supporting the lever or movable block. * Example. — What power is required to raise 600 lbs. when the lower block contains six sheaves and the end of the rope is fasten- ed to the upper block, and what power when fastened to the lower block] _ = 50 lbs., 1st Ans. 6X2 ^^^ ■ = 46.15 lbs., 2d ^r?5.; 6x2+1 or W = nXP, n signifying the number of parts of the rope which sustain the lower block. 136 MECHANICAL POWERS. Wken TTwre than one Rope is used. In a Spanish burton, where there are two ropes, tw^o moveable pul- leys, and one fixed and one stationary pulley, with the ends of one rope fastened to the support and upper moveable pulley, and the ends of the other fastened to the lower block and the power^ the weight is to the power as 5 to 1. And in one where the ends of one rope are fastened to the sup- port and the power, and the ends of the other to the lower and up- per blocks, the weight is to the power as 4 to 1. I?i a system of pulleys, with any number of ropes, the ends being fas- tened to the support, W = 2^XP, n expressing the number of ropes. Example.— What weight will a power of 1 lb. sustain in a system of 4 movable pulleys and 4 ropes'? 1X2x2x2x3 — 16 \\)S,Ans, When fixed pulleys are used in the place of hooks, to attach the ends of the rope to the support, W = 3«XP. Example.— What weight will a powder of 5 lbs. sustain with 4 moveable and 4 fixed pulleys, and 4 ropes 1 5x3x3x3x3 = 405 lbs., Ans. When the ends of the rope, or the fixed pulleys, are fastened to the weight, W=r(2"— 1)XP, andW==(3^— 1)XP, which would give, in the above examples, 1X2X2X2X2= 16—1=: 15 lbs., 5X3X3X3X3 = 405— 1=404 lbs. CENTRES OF GRAVITY. 137 CENTRES OF GRAVITY. The Centre of Gravity of a body, or any system of bodies con- nected together, is the point about which, if suspended, all the parts are in equilibrio. If the centres of gravities of two bodies B C be connected by a line, the distances of B and C from the common centre of gravity A will be as the weights of the bodies ; thus, B : C : : CA : AB. SURFACES. 1. To find the Centre of Gravity of a Circular Arc. Radius of circle X chord of arc ,.■ ^ , • — — — — — = distance from the centre of the length of the arc circle. 2. Of a Parallelogram^ Rhombus, Rhomboid, Circle, Ellipse, Regular Polygon, or Lmne. The geometrical centre of these figures is their centre of gravity. 3. Of a Triangle. On a line drawn from any angle to the middle of the opposite side, at § of the distance from the angle. 4:. Of a Trapezium. Draw the two diagonals, and find the centres of gravity of each of the four triangles thus formed ; join each opposite pair of these centres, and the intersection of the two lines will be the centre of gravity of the figure. 5. Of a Trapezoid. On a line a, joining the middle points of the two parallel sides B h, the distance from B = ^x(~—r\, 3 ^ B+o / Q. Of a Sector of a Circle. 2 X chord of arc X radius of circle ... 5-— -j -r--^ = distance from the centre of 3 X length of arc the circle. 1. Of a Semicircle. 4 X radius of circle = distance from centre. oXo.i4lD 8. Of a Segment of a Circle. Chord of the segment 3 ,. . , -rr-- 7 — = distance from the centre. 12 X area of segment 9. Of a Parabola. Distance from the vertex, § of the abscissa. M2 138 CENTRES OF GRAVITY. 10. Of any Plane Figure. Divide it into triangles, and find the centre of gravity of each ; connect two centres together, and find their common centre ; then connect this and the centre of a third, and find the common centre of these, and so on, always connecting the last found common cen- tre to another centre till the whole are included, and the last com- mon centre will be that which is required. 11. Of a Cylinder^ Cone^ Fnistiim of a Cone^ Pyramid^ Frustum of a Pyramid, or Ungula. The centre of gravity is at the same distance from the base as that of the parallelogram, triangle, or trapezoid, which is a right section of either of the above figures. 12. Of a Sp/i£re, Spherical Segment, or ZoTie, At the middle of their height. SOLIDS, I. of a Sphere, Cylinder, Cube, Regular Polygon, Spheroid^ Ellipsoid^ Cylindrical Ring, or any Spindle. The geometrical centre of these figures is their centre of gravity. 2. Of a Right Ungula, Prism, or Wedge. At the middle of the line joining the centres of the two ends. 3. Of a Prismoid, or Ungula. At the same distance from the base as that of the trapezoid or triangle, which is a right section of them. 4. Of a Pyramid, or Cone. Distance from the base, i of the line joining the vertex and cen- tre of gravity of the base. b.Ofa Frustum of a Cone, or Pyramid. Distance from the centre of the smaller end, = i height x ^^^J^.^^^ ; R and r radii of the greater and less ends in a cone, and the sides of a pyramid. 6. Of a Parabolaid. Distance from the vertex, f of the abscissa. 1. Of a Friistum of a Paraboloid. Distance on the abscissa from the centre of the less end, i h -TjTT-T' ^ being the height. S.Ofa Spherical SegmeTit, Distance from the centre, ^ ^"T"/ , V being the versed sine, s the solid contents of the s segment, and r the radius of the sphere. GRAVITATION. 9. Of a Spherical Sector. 139 Distance from the centre, | (r — ). 10. Of any System of Bodies, Distance from a given plane, ♦ BD+B'D'+B-D-+, &c. ^ , . ^^ ,., ^ ^ = Yx — p/ , p// ^ > -^ bemg the solid contents or weights, and D the distances of their respective centres of gravity from the given plane. GRAVITATION. In bodies descending freely by their ovfn weight, the velocities are as the times, and the spaces as the square of the times. The times, then, will be 1, 2, 3, 4, &c. ; The velocities, then, will be 1, 2, 3, 4, &c. ; The spaces passed through as 1, 4, 9, 16, &c. ; And the spaces for each time as 1, 3, 5, 7, 9, &c. A body faUing freely will descend through 16.0833 feet m the first second of time, and will then have acquired a velocity which will carry it through 32.166 feet in the next second. Table exhibiting the Relation of Time^ Space, and Velocities. Seconds from the begin- nia^ of the Velocity acquired at the end of that time. Squares. Space fallen through in that tiiuc. Spaces. Space fallen through in the last Second oi the descent. fall. 1 32.166 1 16.08 1 16.08 2 64.333 4 64.33 3 48.25 3 96.5 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 15 241.25 9 289.5 81 1302.75 17 273.42 10 321.666 100 1608.. 33 19 305.58 11 353.832 121 1946.08 21 337.75 12 386. 144 2316. 23 369.92 and in the same manner the table might be continued to any extent. To find the Velocity a Falling Body will acquire in any Given Time, RuLE.—Multiply the time in seconds by 32.166, and it will give the velocity acquired in feet per second. Example^ — Required the velocity in 12 seconds. 12X32.166 = 386 feet, ^n*. 140 GRAVITATION. To find the Time which a Body will he in falling through a Given Space, Rule. — Divide the space in feet by 16.083, and the square root of the quotient will give the required time in seconds. Example. — How long will a body be in falling through 402.08 feet of space I ^402.08-M6.083 = 5, Ans. To find the Space through which a Body will fall in any Given Time, Rule. — Multiply the square of the time in seconds by 16.083, and it will give the space in feet. Example. — Required the space fallen through in 5 seconds. 52 =: 25X16.083 =402.08 feet, Ans. To find the Velocity a Body will acquire by falling from any Given Height. Rule. — Multiply the space in feet by 64.333, and the square root of the product will be the velocity acquired in feet per second. Example. — Required the velocity a ball has acquired in descend- ing through 579 feet. ^579x64.333== 193 feet, Ans. Or, when the time is given, multiply the time in seconds by 32.166, Thus, the time for 679 feet is 6 seconds; then, 6x32.166 = 192.996, Ans. To find the Space fallen through, the Velocity being given. Rule. — Divide the velocity by 8, and the square of the quotient will be the distance fallen through to acquire that velocity. Example. — If the velocity of a common ball is 579 feet per see-- ond, from what height must a body fall to acquire the same velocity T 579-^8 = 72.3752 = 5237 feet, Ans. To find the Time, the Velocity per Second being given. Rule. — Divide the given velocity by 8, and i of the quotient is the answer. Example. — How long must a bullet be falling to acquire a velo- city of 800 feet per second 1 800-^-8 = 100-^4 = 25 seconds, Ans. Let s represent the space described by any falling body, t the time, and V the velocity acquired in feet. tv v^ Then 5 = 16.08 i% or-, or -— . Z d4.o , 5 V 2s 2* V = 2-/16.08 5, or 32.16 t, or — . The distance fallen through iyi feet is very nearly equal to the square of the time in fourths of a second. GRAVITATION. 141 Example.— A bullet being dropped from the spire of a church, was 4 seconds in reaching the ground ; what was the height ^ 4x4xi6zrz256feet, Ans. Example.— What is the depth of a well, a buUet being 2 seconds m reaching the bottom 1 ^ 2X4X8 = 64 feet, ^n5. Or, more correctly, as in case 2, 4X4X16.0833 = 257.33 feet, and 2X2X16.0833= 64.33 feet. Bt/ Inversion. In what time will a bullet fall through 256 feet 1 ^256 = 16, and 16—4 = 4 seconds, Ans. Ascending bodies are retarded in the same ratio that descending bodies are accelerated. ^ To find the Space moved through by a Body projected upward or downward with a Given Velocity. If projected dovmward. Rule.— Multiply the square of the time in seconds by 16 083 the velocity of the projection in feet by the number of seconds the body IS m motion, and the sum of these products is the answer. If projected upward. Then the difference of the above products wiU give the distance of the body from the point of projection Or, ^Xu±16.083x/^ Example.— If a shot discharged from a gun return to the earth in 12 seconds, how high did it ascend 1 The shot is half the time in ascending. 12-^2 =. 6, and 6 ^ x 16.083 = 579 feet, Ans. Or, 62x16.083—192.96x6. Example.— If a body be projected upward with a velocity of 30 feet per second, through what space will it ascend before it beffins to return 1 ^ 302^64.3= 13.9 feet, ^715. Example.— If a body be projected upward with a velocity of 100 feet per second, it is required to find the place of the body at the end of 10 seconds. \^iio ^^f ^^^^' ^^^ ^P^^^ if gravity did not act, and 16.083xi0« = 1608.3, the loss arising from gravity. Hence 1000-1608.3 = 608.3 feet below the point of projection. To find the Velocity of a Falling Stream of Water per Second {the perpendicular distance being given) at the End of any Given Time. *^ ^ r.^T"'"^^ two bodies begin to descend from rest, and from the same point, the Wh? ^/"''^'r'^ P'"^"^' ^"?/?^ °^^^^ ^^"i"S freely, their velocities at all equal heights below the surface will be equal. ° •' » «" cquai The space through which a body will descend on an inclined plane is to the ^ane to uTfengTh.' '^ ^^^^ ^''^^^ ^ '^^ '^"'^ "^""^ ^ the height of thi If a body descend in a curve, it suffers no loss of velocity. 142 GRAVITIES OF BODIES. Example. — One end of a sluice is 30 inches lower than the other, what IS the velocity of the stream per second] By case 4, 30 inches = 2.5 feet, x 64.33== 160.82, and ^^160.82 = 12.65 feet, Ans. What is the distance a stream of water will descend on an incli- ned plane 10 feet high, and 100 feet long at the base, in 5 seconds'? 5- X 16.083 ==402.08, and 100 : 10 • : 402.08 : 40.20 feet, Ans. The momenlum with which a falling body strikes is equal to its weight multiplied by its velocity. If a weight of 4500 lbs. fall through 10 feet, with what force does it strike 1 yi0x64.33r=25.35x4500==114075Ibs.,^7i5. If a stream of salt water, running at the rate of 5 feet per second, strike a dam 15 by 4 feet, what is the pressure of the stream] Rule. — Multiply the height of the fall by the weight of the fluid, and the product by the area of the resisting body, and that product, again, by the velocity in feet per second. By case 5, 5-^8=^.625-==.390625, the height of the fall of the water, x64 lbs., the weight per cubic foot, =25 lbs. X(15x4)60 =1500x5=7500 pounds, Ans. Note. — Water being a yielding- substance, an allowance for loss of power should be made. PROMISCUOUS EXAMPLES. 1. Suppose a bullet to be 1 minute in falling, how far will it fall in the last second ] Space fallen through equal the square of the time ; then 1 minute =60 seconds. 60^ X 16.083=57898 distance for 60 seconds, 59^X16.083 =55984 " " 59 1914 *' " 1 " Ans. 2. Find the time of generating a velocity of 193 feet per second, and the whole space descended. 193-^32.166= 6 seconds, ) . 62 X 16.083=579 feet, S The velocity acquired at any period is equal to twice the mean velocity during that period. 3. Then, if a ball fall through 2316 feet in 12 secoiMs, with w^hat velocity will it strike 1 2316—12=193x2 = 386 feet, ^n5. SPECIFIC GRAVITIES. 143 GRAVITIES OF BODIES. The gravity of a body, or its weight above the earth's surface decreases as the square of its distance from the earth's centre in semi-diameters of the earth. Example.— If a body weigh 900 lbs. at the surface of the earth what will It weigh 2000 miles above the surface ? ' The earth's semi-diameter is 3993 miles (say 4000) Then 2000+4000=6000 or H semi-diameters and 900-^ 1.5^==: 400 lbs, Ans. Inversely, if a body weigh 400 lbs. at 2000 miles from the earth's surface, what will it weigh at the surface ? 400X1.5^=900 lbs, Ans. EXAMPLE.--.A body at the earth's surface weighs 360 lbs. : how high must i t be elevated to weigh 40 lbs. 1 v/360-^40=3, or 3 semi-diameters from the earth's surface, Ans ^/4000^3^=5656— 4000=1656 miles, Ans. fh.V.thfTT^ '^ ^r.^^'^'' ^' ^^"«^ ^^^ their^ densities different, the weight of a body on their surfaces will be as their densities. /a/ ^^'ir f'""^'^]''. ^^.^?^^^ ^nd their diameters different, the weight of them will be as their diameters. ujci^ul oj the{r%duc7s^''' "'''^ ^'''''^''' """' ^'^^ ^'-^'"'''^^ ^^' ^^i"^^ ^^'^^ ^e as wh^at wilM.T^^ a body weigh 10 lbs. at the surface of the earth, ^92 anT 00 Tlf^^' 'r^ '^'^^"^ ^^ '^^ '"^ • '^^'' ^^"^ities being ^^'i and 1 00, and their diam eters 8000 and 883000 miles. 883000X100X10-^8000^^392=281.5 lbs., Ans. SPECIFIC GRAVITIES. wJr. ^f ^^fi^^^^i^ of a body is the proportion it bears to the weight of another body of known density, and water is well adapt- ed for the standard ; and as a cubic foot of it weighs 1000 ounces avoirdupois, its weight is taken as the unit, viz., 1000. Tojind the Specific Gravity of a Body heavier than Water. en^J'^^thTn^f ^^1.'^ ^""^'^ ^"^. "^^^ ^^ ^^^^^' ^"d take the differ- f. ?ooo tn ?h. t^^./^^gh^lost in water is to the whole weight, sa. is 1000 to the specific gravity of the body. i^'^h«^''hnf""~^^^^ ''.^^^ 'P^^^^^ ^^^^^*y of a stone which weighs 15 lbs., but in water only 10 lbs. \ ^ 15—10=5."' 5 : 15 : : 1000 : 3000, Ans. When the Body is lighter than Water. ..^^A^l^^''^^ ^"^ ^^ ^ P^^^^ Of metal or stone, weigh the piece added and the compound mass separately, both in and out of watej^ 144 SPECIFIC GRAVITIES. find how much each loses in water by subtracting its weight in water from its weight in air, and subtract the less of these differ- ences from the greater ; then, As the last remainder is to the weight of the light body in air, so is 1000 to the specific gravity of the body. Example. — What is the 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. 1 20+24—8=36 24—21 z=3^ 33": 20 : : 1000 : 606, Ans. Of a Fluid. Rule. — Take a body of known specific gravity, weigh it in and out of the fluid ; then, as the weight of the body is to the loss of weight, so is the specific gravity of the body to that of the fluid. Example. — What is the specific gravity of a fluid in which a piece of copper (5. ^.=0000) weighs 70 lbs. in, and 80 lbs. out of if? 80 : 80—70 : : 9000 : 1125, Ans, To find the Quantities of two Ingredients in a Compound, or to discover Adulteration in Metals. Rule. — Take the differences of each specific gravity of the in- gredients and the specific gravity of the compound, then multiply the gravity of the one by the difference of the other ; and, as the sum of the products is to the respective products, so is the specific gravity of the body to the weights of the ingredients. Example. — A body compounded of gold {s. ^.=18.888) and silver {s. ^.=:10.635) has a specific gravity of 14 : what is the weight of each quantity of metal 1 18.888—14=4.888 X 10.535=51.595 silver, 14.— 10.535=3.465X18.888=65.447 gold, 65.447-f 51.495 : 65.447 : : 14 : 7.835 gold, > . 65.447+51.495 : 51.495 : : 14 : 6.165 silver, ] Proof of Spirittums Liquors. A cubic inch of proof spirits weighs 234 grains ; then, if an inch cube of any heavy body weigh 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 The magnitude of a body in cubic feet multiplied by its specific gravity, in the following table, gives its weight in avoirdupois SPECIFIC GRAVITIES. U5 SOLIDS. Divide the Specific Gravity by 16, and the quotient is the weight of a Cubic Foot in pounds. Metals. Antimony Arsenic , Bismuth Brass, common , Bronze, gun metal iJopper, cast wire-drawn Gold, pure, cast hammered 22 carats fine . 20 carats fine . Iron, cast bars Lead, cast Mercury, 32° 60O Platinum, rolled hammered Silver, pure, cast hammered .. Steel, soft tempered and hardened Tin, Cornish Zinc, cast Woods (Dry). Apple Alder Ash Beech Box, Dutch French Brazilian Campeachy Cherry Cocoa Cork Cypress Ebony, American .... Elder ... Elm Fir, yellow , white Hacmetac , Lignum vitae Live Oak , Logwood Mahogany Maple Mulberry , Oak, English heart, 60 years . . . Orange Pine, yellow white Poplar white Pear Plum Quince • SpecifxclY^'^^'^^ Graviiy "^ 6.712 5.763 9.823 7.820 8.700 8.788 8.878 19.258 19.361 17.488 15.709 7.20' 7.788 11.352 13.598 113.580 22.069 20.33 10.474 10.511 7.833 7.818 7.291 6.861 aCu bic In. Divide the Specific Gravity by 16, and the quotient is the weight of a Cubic Foot in pounds. .793 .800 .845 .852 .912 1.328 1.031 .913 .715 1.040 .240 .644 1.331 .695 .671 .65- .569 .592 1.333 1.120 .913 1.063 .750 .89' .932 1.170 .705 .660 .554 .383 .529 .661 .785 .705 .482 Lbs. .244 .208 .355 .282 .3 J 5 .317 .320 .697 .700 .633 .563 .280 .281 .410 .492 .491 .798 .736 .379 .381 .283 .283 .263 .248 Walnut Willow Yew, Dutch... Spanish - .029 .029 .031 .031 .033 .048 .037 .033 .026 .037 .009 .023 .048 .025 .024 .023 .021 .021 .048 .040 .033 .038 .027 .032 .033 .043 .025 .024 .020 .014 .019 .024 .029 .025 .017 * Well-seasoned J3m., 1839. Ash Beech Cherry Cypress Hickory, red Mahogany, St. Domingo . White Oak, upland James River. Pine, yellow white Poplar Stones and Earths. Alabaster, white yellow Amber Asbestos, starry Borax Brick Chalk Charcoal triturated Clay common soil Coral, red Coal, bituminous Newcastle Scotch Maryland Anthracite. Diamond Earth, loose Emery Flint, black white , Gla-ss, flint , white bottle green Granite, Scotch Susquehanna.. . Q-uincy Patapsco Grindstone — Gypsum, opaque Hone, white, razor Ivory Limestone, green white Lime, quick Manganese Marble, African Egyptian Parian Specific Gravity ,671 .585 .788 .722 .624 .606 .441 .838 .720 .687 .759 .541 .473 .587 2.730 2.699 1.078 3.073 L714 1.900 2.784 .441 1.380 1.930 1.984 2.700 1.270 1.270 1.300 1.355 1.436 1.640 3.521 1.500 4.000 2.582 2.594 2.933 2.892 2.732 2.642 2.625 2.704 2.652 2.640 2.143 2.168 2.876 1.822 3.180 3.156 .804 7.000 2.708 2.668 2.838 Weight of a Cu- bic In. Lbs. .024 .021 .028 .029 .026 .023 .022 .016 .030 .026 .025 .027 .020 .017 .021 .099 .098 .039 .111 .062 .069 .100 .016 .050 .070 .071 .098 .046 .046 .047 .049 .052 .059 .127 .054 .144 .094 .094 .099 .098 .099 .096 .095 .098 .097 .096 .077 .077 .104 .066 .115 .114 .029 .252 .098 .097 .103 * Ordnance Manual, 1841. N 146 SPECIFIC GRAVITIES. Table — (Continued) . Specific Gravity Divide the Specific Gravity by 16, and the quotient is the weight of a Cubic Foot in pounds. Stones and Earths. Marble, common French white Italian Mica Millstone Nitre Porcelain, China Pearl, Oriental Phosphorus Pumice Stone Paving Stone Porphyry, red Rotten Stone Salt, common Saltpetre Sand Shale Slate. Stone, Bristol common Sulphur, native Tale, black Acid, Acetic Nitric Sulphuric Muriatic Alcohol, piu-e of commerce. Ether, sulphuric Honey Human blood Milk Oil, Linseed. Divide the Specific Gravity by 16, and the quolient is the weight of a Cubic Foot in pounds. G^"''> bicln. Miscellaneous. Asphaltum Beeswax ■ Butter Camphor India rubber Fat of Beef Hogs.- ]Mutton Gamboge Gmipowder, loose - . • • shaken.. solid Gum Arabic Indigo Lard Mastic Spermaceti Sugar Tallow Atmospheric air Liquids. ELASTIC FLUIDS. 1 cubic foot of atmospheric air weighs 527.04 troy grains. Its assumed gravity of 1 is the unit for elastic fluids. Ammoniacal gas Azote Carbonic acid Carburetted hydrogen Chlorine • Chloro-carbonic Specific Gravity 1. 000 .597 .976 1.524 .555 2.470 3.389 .905 1.650 .965 .942 Lhs. .033 .058 .035 .034 .988 { .362 .933; .033 .923! .033 .936 1 .034 .923! .033 1.222 .044 .9001 .032 1.000 i .036 1.550 1.800 1.452 1.009 .947 1.074 .943 1.606 .941 .0012 .056 .065 .051 .037 .0.34 .038 .034 .058 .034 Oil, Olive Essential, turpentine Whale Proof Spirit Vinegar Water, distilled sea. Dead Sea Wine Port Champagne Hydrogen Oxygen Sulphuretted hydrogen *Steam,2120 Nitrogen Vapour of Alcohol " of Turpentine spirits • • . " of Water Smoke of bituminous Coal " of Wood * Weight of a cubic foot, 25S.3 grains. APPLICATION OF THE ABOVE. When the Weight of a Body is required. Rule.— Find the contents of the body in cubic feet or inches, ana multiply it by the factor in the table. SPECIFIC GRAVITIES. M7 Example. — What is the weight of a cube of Italian marble, the sides being 3 feet 1 33 X 2708 = 73116 oz. -^16 r= 4569.7 lbs, ^725. Or, of a 2 inch sphere of cast iron, 22x.5236x.260 weight of a cubic inch =1.089 lbs., Ans. Comparative Weight of Timber in a Green and Seasoned Stale. Timber Weight of a Cubic Foot. Green. Seasoned. lbs. OZ. lbs. OZ. English Oak .... 71.10 43. 8 Cedar 32. 28. 4 Riga Fir . 48.12 35. 8 American Fir 44.12 30.11 Elm . 66. 8 37. 5 Beech 60. 53. 6 Ash . 58. 3 50. Note. — The average weight of the timber materials in a vessel of war (English) is about 50 lbs. to the cubic foot, and for masts and yards about 40 lbs. — Edye's Js". C. Given the Diameter of a Balloon to find what Weight it luill raise. Rule. — As 1 cubic foot is to the specific difference between at- mospheric air, and the gas used to inflate the balloon, so is the ca- pacity of the balloon to the weight it will raise. Example. — The diameter of a balloon is 26.6 feet, and the gas used to inflate it is hydrogen ; what weight will it raise 1 Sp.gr. of air. Grains. Sp. gr. of hydr. Grains. 1.000 : 527.04 : : .070 : 36.89 wt. of 1 cubic foot of hydrogen. Then 1 : 527.04-^36.89 : : 26.63x.5236 : 4830293. grains, —7000 (grains in a lb.), =690.04 lbs., Ans. Given the Weight to be raised to find the Diameter of a Balloon. By inversion of the preceding rule. The weight to be raised is 690.04 lbs. ; what is the diameter'? 490.15 ==(527.04 — 36.89) : 1 : : 690.04x7000 : 9854.725 cubic feet, 4-. 5236, the cube root of the quotient, is 26.6 feet, Ans. 148 STREIS^GTH OF MATERIALS. STEENGTH OF MATERIALS, COHESION. The power of cohesion is that force by which the fibres or parti- cles of a body resist separation, and it is therefore proportional ta the number of fibres in the body, or to the area of its section. Table of the Cohesive Force of Metals, &c. Weight or Force necessary to tear asunder 1 Square Inch^ in Avoirdupois pounds. Metals. Copper, cast wire Gold, cast . wire . Iron, cast wire best bar medium bar inferior bar , Gold 5, Copper 1 . Brass . Copper 10, Tin 1 . 8, " 1 . 4, '^ 1 . 22500 61200 20000 30800 18000 50000 103000 75000 60000 30000 Compositions. 50000 45000 32000 Lead, cast . milled Platinum, wire Silver, cast . Steel, soft . razor . Tin, cast block Zinc, cast sheet . 36000 35000 Silver 5, Copper 1 " 4, Tin 1 . Tin 10, Antimony 1 " 10, Zinc 1 . " 10, Lead 1 . Ash . Beech . Box Cedar . Chestnut, sweet Cj'press Deal, Christiana Ehu . Fir, strongest American Lance wood Lignum vitee Locust . Mahogany . Woods. 16000 11500 20000 11400 10500 6000 12400 13400 12000 8800 23000 11800 20500 21000 Miscellaneous Substances. Brick Glass plate . . . . Hemp fibres glued together Ivory Marble 290 9400 92000 16000 9000 Mortar, 20 years , Slate . Stone, fine grain Whalebone . 880 3320 53000 40000 120 ooa 150000 5000 2600 16 GOO 4800!) 41000 11000 12914 6830 Mahogany, Spanish 12000 Maple . 10500 Oak, American white 11500 English 10000 seasoned 13600 Pine, pitch . 12000 Norway 13 COO Poplar . 7000 Quince 6000 Sycamore . 13000 Teak, Java . 1400O Walnut 7800 Willow 13000 52 12000 200 7600 To find the Strength of Direct Cohesion, Rule. — Multiply area of transverse section in inches by the weight given in the preceding tables, and the product is the strength in lbs. STRENGTH OF MATERIALS. 149 Example. — In a square bar of ordinary wrought iron, of 2 inches, what is the resistance 1 2X2X60000 r= 240000 lbs., Ans. Also, in a rod of cast steel i inch diameter, area of i=.1963x 120000 = 23556 lbs., Ans. The absolute strength of materials, pulled lengthwise, is in propor- tion to the squares of their diameters. The Lateral or Transverse Strens^th Of any beam, or bar of wood, &c., is in proportion to its breadth, multiplied by its depth squared, and in like-sided beams as the cube of the side of a section. Or, one end being fixed, and the other pro- jecting, is inversely as the distance of the weight from the section acted upon, and the strain upon any section is directly as the dis- tance of the weight from that section. The strength of a projecting beam is only one fourth of what it would be if supported at both ends, and the weight applied in the middle. The strength of a projecting beam is only one sixth of one of the same length, fixed at both ends, and the weight applied in the mid- dle. The strength of a beam to support a weight in the centre of it, when the ends rest merely upon two supports, compared to one, the ends being fixed, is as 2 to 3. Tables of the Transverse Strength of Timber. AMERICAN. One Foot in Lengthy and 1 iTich Square^ Weight suspended from oTie end. Breaking Greatest Weight Value for Materials. weight deflexion borne with general in lbs. in inches. safety. use. /White Oak 240 9. 196 72 ?> Sweet Chestnut 170 1.8 115 35 i Yellow Pine 150 1.7 100 30 White Pine 135 1.4 95 32 Ash 175 2.4 105 25 V Hickory .... 270 8. 200 75 One Foot in Lengthy and 2 Inches Square. White Pine 1087 1.5 800 32 Cylinder. On^ Foot in Length. White Pine, 2 inches diameter White Pine, 1 inch diameter . Breaking weight in lbs. 610 75 N2 Weight borne with safety. 460 56 Value for general 20 150 STRE^'GTH OF MATERIALS. Tables of the Transverse Strength of Cast and Wrought Iron. AMERICA ]N\ Weight susvcnded from one end,. Cylinder. One Foot in Length, and 3 Incfies Dimneter. Average of 18 ezperimeats with Gun Metal. Breaking weight in lbs. Weight borne with safety. Value for general use. ^Cast Iron, cold blast .... 12000 10000 350 SauARE Bar. One Foot in Length by 2 Inches. Gun Metal. Breaking weight in lbs. Weight borne with safety. Value for general Cast Iron, cold blast .... 5781 5000 500 The values above given are for iron of more than ordinary strength; if an inferior article is to be used, a corresponding de- duction should be made. SauARE Bar. 07ie Foot in Length by 1 Inch. Wrought Iron. Weight borne with perfect safety Deflexion from a hori- zontal plane without rup- ture. Weight that gave a per- manent bend. Deflexion in inches with last weight. Value for general use. t Wrought Iron . . ! 1520 53<^ 600 1 300 MISCELLANEOUS. Cast /row.— Square bar, side 2 inches, length 12 inches, supported at both ends, broke with 22728 lbs. applied in the middle. Cylinder 3 inches diameter, length 8i inches, broke with 17110 lbs. applied at one end. White Pme.— Cylinder | ins. diameter, length' 12 inches, broke with 68 lbs. applied at the end. Yellow Fine.—\ inch square, and 15 inches in length, broke with 125 lbs. applied at the end. Hickory and White Oak.— I inch square, and 12 inches in length, re- quired 82 lbs. to deflect them i an inch, the weight suspended from the end. The above and preceding experiments were made by the author in De- cember, 1840. * From the West Point Foundry Association at Cold Spring, Putnam county, N. Y. Specific gravity, 7210. t From the Ulster Iron Company, Saugerties, N. Y A fine specimen of ma- chinery iron. ^, J 1. I, u This specimen hroke with the greatest weight here given, when filed through the top to the depth of a i of an inch, and the fracture showed but very httle fibre. STRENGTH OF MATERIALS. 151 Mean Result of several Experiments by English Authors on Cast Iron. Square bar, 1 inch by 32, resting upon two supports, broke with 840 lbs. suspended in the middle. Square bar, of 1 inch, projecting 32 inches from a wall, broke with 278 lbs, applied ; and one, 2 inches deep by i an inch, required 643 lbs. to break it. Square bar, 1 inch by 32, the ends fixed in walls, required U70 lbs. suspended from the middle to break it. TO FIND THE TRANSVERSE STRENGTH. When a Rectangular Bar or Beam is Fixed at on^ End, and Loaded at the other. Rule.— Multiply the Value in the preceding table by the breadth, and square of the depth, in inches, and divide the product by the length in feet ; the quotient is the weight in pounds. Note.— When the beam is loaded uniformly throughout its length, the result must be doubled. Example. — What are the weights a cast and a wrought iron bar, projecting 30 inches in length, by 2 inches square, will bear'? 2X22x500-^2.5 = 1600 lbs., Ans. 2 X 2- X 300H-2.5 = 960 lbs., Ans. Or, if the Dimensions of a Beam be required, to support a Given Weight at its End. ^^^^ _WeighO<_len^^ p^^^^^^ of breadth, and square of the value in table depth. Example.— What is the depth of a wrought iron beam, 2 inches square, necessary to support 960 lbs. suspended at 30 inches from the fixed end '? 960 X2^ ^ 8, and 8-^2 == 4, and ^4 = 2, Ans. When the Bar or Beam is Fixed at both Ends, and Loaded in the Middle. Rule.— Multiply the Value in the preceding table by six times the breadth, and the square of the depth, in inches, and divide by the length in feet. Note.— When the weight is laid uniformly along its length, the result must be tripled. Example. — What weight will a bar of cast iron, 2 inches square and 5 feet in length, support in the middle, when fixed at the ends 1 500x6x2x22-^5 = 4800 lbs., Ans. When the Bar or Beam is Supported at both Ends, and Loaded in the Middle. Rule.— Multiply the Value in the preceding table by the square of 152 STRENGTH OF MATERIALS. the depth, and four times the breadth, in inches, and divide the product by the length in feet. Note.— When the weight is laid uniformly along its length, the result must be doubled. Example.— What are the weights a cast and a wrought iron bar, 60 inches between the supports, and 2 inches square, will bear 1 500 X 2^ X 2 X 4-r-5 — 3200 lbs., Ans. 300x22x2x^-^5 = 1920 lbs., Ans. Or, if the Dimensions be required to Support a Given Weight. nuLB.-^^'^^^ ^ ^^"^^^rz: product of four times the breadth, value in iable and square of the depth. Example.— What is the side of a square cast iron beam 2 feet in length, between supports, that will support 8000 lbs. in the centre 1 ?2£?^.^4^ = 8, and ^8 = 2.828, Ans. 500 ^ When the Breadth or Depth is required. Divide the product obtained by the preceding rules by the square of the depth, and you have the breadth; or by the breadth, and the square root of the quotient is the depth. Example.— If 128 is the product, and the depths, 128-7-82=2, the breadth ; And -^(128-^2) = 8, the depth. When the Weight is n^t in the Middle between the Supports Distance from nea rest end x weight ^ ^^^^^^^^ ^p^^ ^^pp^^ whole length farthest from the weight. Distance from farthest end X weight ^^^^„„^^ „^^„ o^r^r^^vf . — - — = pressure upon support whole length nearest the weight. When a Beam, supported at both Ends, bears two Weights at unequal Distances from the Ends. Let D = distance of greatest weight from nearest end, d = distance of least weight from nearest end, W =: greatest weight, w = least weight, L — whole length, / = length from least weight to farthest end, Z' = distance of greatest weight from farthest end. STRENGTH OF MATERIALS. 153 _, DXW IXw Then — | — +- ^ — = pressure at vj end ; and -r— -\ — = — = pressure at W end. In cylindrical beams or bars, the lateral strength is as the cube ol the diameter. The strength of a hollow cylinder is to that of a solid cylinder, of the same length and quantity of matter, as the greater diameter of .the former is to the diameter of the latter ; and the strength of hol- low cylinders of the same length, weight, and material, is as their greatest diameters. To find the Diameter of a Solid Cylinder^ Fixed at both Ends, to support a Given Weight in the Middle. Rule.— Multiply the length between the supports in feet by the weight in pounds ; divide by the value, and the cube root of one sixth of the quotient is the diameter in inches. Example. — What should be the diameter for a cylinder 2 feet in length between the supports, to bear 20000 lbs. 1 3,20000x2-^350 v^ ~ = 2.67+, Ans. o To find the Diameter of a Solid Cylinder, to support a Given Weight in the Middle, between the Supports. Rule.— Multiply the weight in pounds by the length in feet ; di- vide by the Value^ and the cube root of J the quotient is the diameter in inches. Example. — What is the diameter of a cast iron cylinder, 8 inches long between the supports, that will support 60000 lbs. suspended in the middle 1 3 60000 X. 66-^350 ^ ^^ ^ ^ ^=: 3 03, Ans. To find the Diameter of a Solid Cylinder when Fixed at one End, the Load applied at the other. Rule. — Multiply the length of the projection in feet by the weight to be supported in pounds ; divide by the given Value, and the cube root of the quotient is the diameter. Example. — What should be the diameter of a cast iron cylinder 8 inches long, to support 15000 lbs. 1 8 inches is .66 feet, y(15000x.66-f-350) = 3+ inches, Ans. Example.— What should be the diameter for 270000 lbs., at 12 inches from the end 1 -^(270000 X 1—350) =r: 9.17, Ans. 254!i STRENGTH OF MATERIALS. To find the Diameter of a Beam or Solid Cylinder when the Load is uniformly distributed over its Length. Rule.— Proceed as if the load was suspended at the end or in the middle until the quotient is obtained ; then, , ^ u ir .u:^ If for a cylinder with one end fixed, the cube root of half this quotient is the diameter ; ^r v^^if ,w^ nnn If the ends rest upon two supports, the cube root of half this quo- tient is the diameter ; ^ n • •, r ^x,- And if the ends are fixed, the cube root of one third of this quo- tient is the diameter. The Constant Divisor of 350 is for iron of great strength ; where an inferior article is to be used, it may be decreased to 250. Thus, 350 represents a weight of 9450 lbs. upon the end of a cyl- inder 3 inches in diameter and 1 foot in length ad ^50 ^nder the same circumstances is equal to a weight of 6750 lbs. 500 represents a weight of 4000 lbs. upon the end of a bar 2 mches square and 1 foot in length, and 400 upon the same bar is equal to a weight of 3200 lbs. The strength of an equilateral triangle, an edge up, compared to a square of the same area, is as 45 to 28. To ascertain the Relative Value of Materials to resist a Trans- verse Strain. T et V represent this value in a beam, bar, or cylinder one foot in length and one inch square, side or in diameter. 1 Fixed at one end. Weight suspended from the other. 2 Fixed at loth ends: Weight suspended from the middle. 3 Suvvorted at loth ends. Weight suspended from the middle. "~ Ud^ 4. Su^vorted at hath ends. Weight suspended at any other point than the middle. mnW IbdT' 5. Fixed at loth ends. Weight suspended at any other point than the middle. _ 2mnW ^""m'dF'' W representing the weight, I the length & the breadth, ^ the depth, m the distance from one end, and n the distance from the other. STRENGTH OF MATERIALS. 155 From which the value of any of the dimensions may be found, by the following formulae : \hi^ = W Ybd' W 7 ^w . .m In square beams, &c., b and d = \^"v~ ^^!I. = w ebd^Y w 7 ^^ -h IW In square beams, &c., b and d = \^^' Ud^Y = W. Ud^Y W ' IW U^Y' If/. In square beams, &c., Z> and iZ = -5/ /i^^V mn :W mTzW - m/zW bd'Y ' Id-'Y =bW 4V TFT" r^. In square beams, &c., 5 and d = ^-— ^— . -= W . 2mnW = 5. V In square beams, &c., b and d = ^ ,2mnYV _ 2mnW 3/V When the weight is uniformly distributed, the same formulae will apply, W representing only half the required or given weight. Mean Results of various Experiments hy English Authors. WOODS. Fixed at one end. Length in incli3s. Breadth in inches. Depth in inches. Breaking weight in lbs Riga Fir (dry) . Riga Fir (wet) . Yellow Pine (American) . White Pine (Canadian) . 60 60 60 60 2 2 2 2 2 2 2 2 153 162 176 112 SOLID AND HOLLOW CYLINDERS. Supported at each end. Length in inches. Diameter ex- ternal in ins Diameter inter- nal in inches. Deflexion in inches. Break ins: weight in lbs. Fir Ash . Ash 48 46 46 2 2 2 .5 1. 2. 3. 3.6 740 664 630 156 STRENGTH OF MATERIALS. Cast Iron of various Figures^ having equal Sectional Areas. Description of bar Distance between sup- 1 Breaking weight jjescripiion oi oar. p^^.^^ j^ mcbes. | in Jbs. Area of 1 square iucb. Square . . . . . *' through the diagonal 2 inches deep by i inch 3 inches deep by h inch 4 inches deep by \ inch Equilateral Triangles. Angle up Angle down .... 36 32 32 32 32 32 32 897 851 2185 3588 3979 1437 840 Oak, in seasoning, loses at least ^ of its original weight, and this process is facilitated by steaming or boiling. It loses more by the former process than the latter. By steaming, the specific gravity of a piece of oak was reduced from .... 1050 to 744 By boiling, from 1084 to 788 By exposure to the air, from .... 1080 to 928 Weight in air of a cubic foot of White Pine, before seasoning . Butt. Ounces. 658 Ounces. 432 " " when seasoned . 549 416 Pitch Pine, before seasoning . " *' when seasoned . 628 540 597 529 Spruce Spar, before seasoning . " "■ when seasoned . 587 541 580 554 Stiffness of Oak to Cast Iron is as . 1 to 13 Strength of Oak to Cast Iron is as . 1 to 4.5 Mean Specific Gravity of Yellow Pine of Pitch Pine . 558 777 DEFLEXION OF RECTANGULAR BEAMS. 1. The deflexions of the same beam, resting on props at each end, and loaded in the middle with weights, are as those weights. 2. The deflexion is inversely as the cube of the depth ; also, the depth being the same, the deflexion is inversely as the breadth. 3. The deflexion is directly as the cube of the length. Let I represent the length of a beam, b its breadth, d its depth, and W the weight with which it is loaded ; then the deflexion will vary as -rrr ; and if the deflexion is represented by e, then, When the Beam is Fixed at one End, and Loaded at the other ^ -_- = C, a constant quantity. bd^e 3/3 \Y When unifoniuy loaded . g, ,3 = C. STRENGTH OF MATERIALS. Wh£n SuppaHed at both Ends, aiid Loaded in the Middle, 157 32 bd' -=zC. When uniformly loaded 5 PW _ Hence it follows, that, to preserve the same stiffness in beams, the depth must be increased in the same proportion as the length, the breadth remaining constant. The deflexion of different beams arising from their own weight, having their several dimensions proportional, will be as the square of either of their like linear dimensions. Of three equal and similar beams, one inclined upward, one in- clined downward at the same angle, and the other horizontal, it has been determined that that which had its angle upward was the weakest, the one which declined was the strongest, and the one horizontal was a mean between the two. Barlow furnishes the following as some of the results obtained by him upon the deflexion of beams : Length. | Depth. | Breadth. r Lbs. i Deflexion in ins. Fir . Fir . Fir . 6 feet 3 " 6 " 2 inches 2 " H inches 2 180 , 120 r 180 1. .10 2. WROUGHT IRON. Supported at each end. The average of a number of experiments gave, for bars 33 inches in length, 1.9 inches broad, and 2 inches deep, a deflexion for every half ton of .024 inches. CAST IRON. Supported at both ends. Bars 33 inches in length, 1.3 inches in breadth, and 0.65 inches deep, deflected 0.27 inches with 162 lbs. applied. Fir battens. Supported at each end, 15 inches in Jength, and 1 inch square, broke with a weight of 440 lbs. ; 30 inches in length, and 1 inch square, broke with 240 lbs. Oak battens. Supported at each end, 2 feet long, H inches deep, and I inch m breadth, deflected 1.1 ins., and supported 408 lbs. Ash battens. Fixed at one end, 2 feet long, 2 inches deep, and 1 inch in breadth, deflected 6 inches with a weight of 434 lbs. Fir battens. Fixed at one end, same dimensions as last piece, de- flected 3.9 ins. with 276 lbs. Note 1. — When a weight is uniformly distributed over the length of a beam, the deflexion will be three eighths of the deflexion from the same weight applied at the extremity. 2. If the beam be a cylinder, the deflexion is 1.7 times that of a square beam, other tilings being equal. 3. If the load is unifonnly distributed over the length, the deflexion will be five eighths of the deflexion from the same load collected in the middle. COHESION. In page 148, the results given in the table are those of ultimate* resistance ; in practice, i of the weight there given will be suflicient O 158 STRENGTH OF THE JOURNALS OF SHAFTS. STRENGTH OF THE JOURNALS OF SHAFTS. Wlien tJie Weight is in the Middle of the Shaft. Apply the rule under the head of Strength of Materials, and the result is the diameter of the journals in inches. Example. — What should be the diameter of the journals of a shaft lOi feet long to support a wheel of 10,000 lbs. in the centre 1 Ans. 4.21 ins. TO RESIST TORSION. Water Wheels^ (^c. Rule. — Multiply pressure on the crank pin, or at the pitch line of the pinion, by the length of the crank or radius of wheel in feet ; divide their product by 125, and the cube root of the quotient is the diameter of the journal in inches if of wrought iron. If cast iron is to be used, add yq. Example. — What should be the diameter for the journal of a wa- ter-wheel shaft, the pressure on the crank pin being 594,000 lbs., and the crank 5 feet in length ] ^ == 13.5 inches, Ans. Example. — The pressure on a crank pin is 123.680 lbs., and the length of crank 5 feet. .123680x5 ^ r— =17+, Ans. 12o When tivo Shafts are used, as in Steam Vessels with one Engine, ^ „, /diameter for one shaft ^x3n ,. ^ . . ,^ Rule. — ^{ ) r= diameter m mches. Example. — The area of the journal of a single shaft is 113 inches ; what should be the diameter if two shafts are used \ ' Diameter for area of 113 = 12 inches. ' A^}^12<1= 10.9, Ans. 4 The examples above given are instances in successful practice ; where the diameter has been less, fracture has almost universally taken place, the strain being increased beyond the ordinary limit. Results of Experiments on Torsional Strain. Square bars, with a Journal 1 inch in diameter and i inch 271 length. Wrought Iron (Ulster Iron Co.), twisted with 326 lbs., and broke with 570 lbs. applied at the end of a lever 30 inches in length. Wrought Iron (Swede^), same length of lever, twisted with 367 ' lbs., and broke with 615 lbs. STRENGTH OF THE JOURNALS OF SHAFTS. 159 Cast Iron (Foundry), journal 1 inch long, same length of lever, broke with 436 lbs. The diameters for second and third movers are found by multi- plying the diameters ascertained by the above rules by .8 and .793 respectively. Grier, in his Mechanics' Calculator, gives the following rule for cast iron shafts : 240 X number horses' power >. -number revolutions per minute>' For wrought iron, multiply result by .963, for oak by 2.238, and for pine by 2.06. „,/ 240 X number horses' powder ^ , . . . , ^i ■ ; — : ^—. ) = diameter m mches. ^ ^numbfir revo utions ner mmute/ 160 GUDGEONS AND SHAFTS. GUDGEONS AND SHAFTS. To find the Dimensions of a Gudgeon, 0.30^(wl) = d, w representing the stress in 100 lbs., I the length in inches, and d the diameter in inches. If a Cylindrical Shaft has no other lateral stress to sustain than its own weight, and is Fixed at one E7id, dz=z. 00002U\ Let the stress supposed to be in the middle be n times the weight of the shaft ; then. When supported at both Ends, If the weight of the shaft be not taken i nto account, d = ^.0mi2 7d\ If the weight of the shaft is taken into account, d = ^. 00012 {ni.l)l\ When a Hollow Shaft is supported at each End, , ^,moisld~~ d=V 1 +DS '^ representmg the stress mlbs., / the length in inches, D the interior diameter, and d the diameter in inches. When a Hollow Shaft is Fixed at each End, and Loaded in the Middle, , 3 :00048w,/ «= V — ^ — +D^- For hollow Cylindrical Shafts, supported at one End, ^ — ^.00048 'i/;/+D3. If the hollow shaft support the weights at distances m and n from each end, and is supported at each end, ^=^.00048^i^+D3. The last four formulas do not take into account the weight of the shaft. The above is for Cast Iron. For Cyliiidrical Shafts of Cast Iron to resist Torsion, {Buchanan.) Let P be the number of horses' power, and II the revolutions of the shaft in a minute ; then V _j^ -d. For Wrought Iron, multiply this result by .963 ; for Oak, by 2.238 ; for Pine, by ^2.06. If a shaft has to sustain both lateral stress and torsion, then, For cast iron, ,,/240P wP^ TEETH OF WHEELS. 161 TEETH OF WHEELS. To Construct a Tooth. Divide the pitch into 10 parts. Let 3.5 of these parts be below the pitch line, and 3.0 of them above. The thickness should be 4.7 of the pitch The length should be 6.5 of the pitch. The Diameter of a wheel is measured from the pitch line. The wood used for teeth is about i the strength of cast iron, therefore they should be twice the depth to be of equal strength. To find the Diameter of a Wheel, the Pitch and Number of Teeth being given. Pitch X number of teeth 3:[4i6 = '^'^"'^*^''- Note. — The pitch, as found by this rule, is the arc of a circle ; the true pitch required is a straight line, and must be measured from the centres of two contigu- ous teeth. To find the Pitch, the Diameter and Number of Teeth being sriven. Diameter x 3.1416 number of teeth ; pitch. To find the Radius. Pitch X number of teeth 3.1416 2 = radius. To find the Number of Teeth. 2 X radius x 3.1416 pitch : number of teeth. Dimensions of Wheels in operation. Diameter. Breadth. Pitch. Length of teeth. Thickness of teeth. Velocity per second. Pressure. Feet. Ins. Inches. Inches. Inches. Inches. Feet Lbs. 10 7. 2.8 1.625 1.3 3. 11000 6 12. 4.2 2.25 1.9 6.6 20000 7 10 4.5 1.9 1.125 .875 1.1 3300 14 4 8. 3. 1.75 1.4 1.87 9000 162 VELOCITY OF WHEELS. VELOCITY OF WHEELS. The relative velocity of wheels is as the number of their teeth. To find the Velocity or Number of Turns of the last Wheel to one of the first. Rule. — Divide the product of the teeth of the wheels that act as drivers by the product of the driven, and the quotient is the number. Example. — If a wheel of 32 teeth drive a pinion of 10, on the axis of which there is one of 30 teeth, acting on a pinion of 8, what is the number of turns of the last 1 32 30 960 lo^-s-^W^'"'^"" To find the Proportion that the Velocities of the Wheels m a train should hear to one another. Rule. — Subtract the less velocity from the greater, and divide the remainder by one less than the number of wheels in the train ; the quotient is the number, rising in arithmetical progression from the less to the greater velocity. Example. — What are the velocities of three wheels to produce 18 revolutions per minute, the driver making 3 revolutions per minute 1 18— 3 = 15 _ then 34-7.5 = 10.5, 3—1 = 2 ' and 10.5+7.5 = 18 ; thus, 3, 10.5, and 18 are the velocities of the three wheels. To find the Number of Teeth required in a Train of Wheels to produce a certain Velocity. Rule. — As the velocity required is to the number of teeth in the driver, so is the velocity of the driver to the number of teeth in the driven. Example. — If the driver has 90 teeth, makes 2 revolutions, and the velocities required are 2, 10, and 18, what are the number of teeth in each of the other two ] 2d w^heel, 10 : 90 : : 2 : 18 teeth. 3d wheel, 18 : 90 : : 2 : 10 teeth. STRENGTH OF WHEELS. 163 STRENGTH OF WHEELS. The strength of the teeth of wheels is directly as their breadth and as the square of their thickness, and inversely as their length. The stress is as the pressure. To find the Thickness of a Tooth, the Strain at the Pitch Line being given. Rule.— Divide the pressure in pounds at the pitch line by 3000, and the square root of the quotient is the thickness of the tooth in inches. Example.— The pressure is 9000 lbs., what fs the thickness of the tooth required 1 9000 "^3000 — ^•'^^^ inches, Ans. The Breadth should be 2.5 times the pitch. Therefore, as the thickness should be 0.47 of the pitch, the pitch for the above example will be 3.685 inches, and the breadth 3.685 X2.5 = 9.2125 inches. To find the Horses'' Power of a Tooth, the Dimensions and Velocity being given. Thickness ^ x3000 — pressure. Pressure x velocity in feet per minute 33000 = ^''^^^^' P^^^^- Thickness x 2.1277+ = the pitch. Thickness x 1.5384+ = the length. To find the Dimensions of the Arms of a Wheel. Rule.— Multiply the power at the pitch line by the cube of the length of the arms, and divide this product by the product of the number of arms and 280 ; the quotient will be the breadth and cube of the depth. Example.— If the power be 1600, the diameter of the wheel 10 feet, and the number of arms 6, what will be the dimensions of each arml 1600X10-^23 200000 6x280 — ^ ^QQQ = 119 ; if the breadth be 5 inches, then 119 — = 23.8, and ^ of 23.8 = 2.87, the depth. 02 164 GENERAL EXPLANATIONS CONCERNING WHEELS. GENERAL EXPLANATIONS CONCERNING WHEELS. Pitch Lines. — The touching circumferences of two or more wheels, which act upon each other. Pitch of a Wheel. — The distance of two contiguous teeth, measured upon their pitch line. Length of a Tooth. — ^The distance from its base to its extremity. Breadth of a. Tooth. — The length of the face of the wheel. Spur Wheels.— Wheels that have their teeth perpendicular to their axis. Bevel Wheels. — Wheels having their teeth at an angle with their axis. Crown Wheels. — Wheels which have their teeth at a right angle with theL axis. Mitre Wheels. — Wheels having their teeth at an angle of 45° with their axis. Spur Gear. — Wheels acting upon each other in the same plane. Bevel Gear. — Wheels acting upon each other at an angle. When two wheels act up'on one another, the greater is called the spur or driver ^ and the lesser the pinion or driven. When the teeth of a wheel are made of a different material from the wheel, they are called cogs. Table of the Strength of Teeth and Arms. Teeth. With 6 Arms, Pressure in lbs. Horses' power at 3 feet per second. Pitch in inches. Thickness in inches. Breadth in inches. Depth for 1 foot radius in inches. Breadth of rib in inches. 22 .25 .25 .119 .75 0.87 .25 85 .5 .50 .238 1.25 1.24 .42 191 1. .75 .357 1.75 1.67 .60 337 2. 1. .475 2.50 1.76 .80 520 3. 1.25 .590 3. 2. 1. 800 4. 1.50 .730 4. 2.20 1.30 1040 5. 1.75 .835 4.25 2.40 1.40 1370 7. 2. .955 5. 2.50 1.70 1720 9. 2.25 1.070 5.50 2.70 1.80 2100 10.5 2.50 1.190 6. 2.85 2. 2560 13. 2.75 1.310 6.75 3. 2.20 3000 15. 3. 1.430 7.25 3.20 2.40 3600 18. 3.25 1.550 8. 3.30 2.60 4150 21. 3.50 1.670 8.50 3.40 2.80 4800 24. 3.75 1.790 9.25 3.50 2.90 5700 27.5 4. 1.910 10.25 3.60 3.40 6300 31.5 4.25 2.025 10.50 3.70 3. .50 6900 34.5 4.50 2.150 11. 3.80 3.70 7700 38.5 4.75 2.270 11.75 3.90 3.90 8500 42.5 5. 2.390 12.25 4. 4. Tredgold. HORSE POWER — ANIMAL STRENGTH. 165 HOESE POWER. As this is the universal term used to express the capabihty of first movers of magnitude, it is very essential that the estimate of this power should be uniform ; and as it is customary, in Europe, to es- timate the power of a horse equivalent to the raising of 33000 lbs. one foot high in a minute^ there can be no objection to such an esti- mate here. The estimate, then, of a horse's power in the calculations in this work, is 33000 pounds avoirdupois, raised through the space of one foot in height in one minute, and in this I am supported by the practice of a majority of the manufacturers of steam-engines in this country. ANIMAL STRENGTH. MEN. The mean effect of the power of a man, unaided by a machine, working to the best possible advantage, and at a moderate estima- tion, is the raising of 70 lbs. 1 foot high in a second, for 10 hours in a day. Two men, working at a windlass at right angles to each other, can raise 70 lbs. more easily than one man can 30 lbs. Mr. Bevan's results with experiments upon human strength are, for a short pe- nod, With a drawing-knife an auger, both hands a screw-driver, one hand a bench vice, handle a chisel, vertical pressure a windlass pincers, compression a hand-plane . a hand-saw a thumb-vice . a brace-bit, revolving a force of 100 lbs. 100 " 84 " 72 " 72 " 60 " 60 " 50 " 36 " 45 " 16 " Twisting by the thumb and fingers only, ) u and with small screw-drivers . . j ^^ By Mr. Field's experiments in 1838, the maximum power of a strong man, exerted for 2^ minutes, is = 18000 lbs. raised one foot in a minute. A man of ordinary strength exerts a force of 30 lbs. for 10 hours in a day, with a velocity of 2| feet in a second, = 4500 lbs. raised one foot in a minute, = i of the work of a horse. A foot-soldier travels in 1 minute, in common time, 90 steps, = 70 yards. in quick time, 110 " =: 86 " in double quick-time, 140 *' = 109 " He occupies in the ranks, a front of 20 inches, and a depth of 13, without a knap- sack ; the interval between the ranks is 13 inches. Average weight of men, 150 lbs. each. 5 men can stand in a space of 1 square yard. A man travels, without a load, on level ground, during 8| hours a day, at the rate of 3.7 miles an hour, or 314: miles a day. He can carry 111 lbs. 11 miles in a day. 166 ANIMAL STRENGTH. A porter going short distances, and returning unloaded, carries 135 lbs. 7 miles a day. He can carry, in a wheelbarrow, 150 lbs. 10 miles a day. The muscles of the human jaw exert a force of 534 lbs. HORSES. A horse travels 400 yards, at a walk, in 4-^ minutes ; at a trot, in 2 minutes ; at a gallop, in 1 minute. He occupies in the ranks a front of 40 inches, and a Septh of 10 feet ; in a stall, from 3^ to 4^ feet front ; and at picket, 3 feet by 9. Average weight = 1000 lbs. each. K horse, carrying a soldier and his equipments (say 225 lbs.), trav- els 25 miles in a day (8 hours). A draught horse can draw 1600 lbs. 23 miles a day, weight of car- riage included. The ordinary work of a horse may be stated at 22.500 lbs., raised 1 foot in a minute, for 8 hours a day. In a horse mill, a horse moves at the rate of 3 feet in a second. The diameter of the track should not be less than 25 feet. A horse power in machinery is estimated at 33.000 lbs., raised 1 foot in a minute ; but as a horse can exert that force but 6 hours a day, one machinery horse power is equivalent to that of 4.4 horses. The expense of conveying goods at 3 miles per hour per horse teams being 1, the expense at 4| miles will be 1.33, and so on, the expense being doubled when the speed is 5^ miles per hour. The strength of a horse is equivalent to that of 5 men. Table ^/ the Amount of Labour a Horse of average Strength is capable of performing^ at different Velocities^ on Canals^ Railroads^ and Turrh- pikes. Force of traction estimated at 83.3 lbs. Velocity in miles Duration of the day's work. Useful effect for one day in tons, drawn one mile. per hour. On a Canal. On a Railroad. On a Turnpike. Miles. Hours. Tons. Tons. Tons. 2i Hi 520 115 14 3 8 243 92 12 3^ H^ 153 82 10 4 ^\ 102 72 9. 5 2A 52 57 7.2 6 2 30 48 6. 7 1^ 19 41 5.1 8 \\ 12.8 36 4.5 9 tV 9.0 32 4.0 10 i 6.6 28.8 3.6 The actual labour performed by horses is greater, but they are injured by it. HYDROSTATICS. 167 HYDROSTATICS. Hydrostatics treat of the pressure, weight, and equilibrium of non-elastic fluids. The pressure of a fluid at any depth is as the depth of the fluid. The pressure of a fluid upon the bottom of the containing vessel is as the base and perpendicular height, whatever may be the figure of the containing vessel. Fluids press equally in all directions. The Centre of Pressure is that point of a surface against which any fluid presses, to which, if a force equal to the whole pressure were applied, it would keep the surface at rest. The centre of pressure of a parallelogram is at | of the line (meas- uring downward) that joins the middles of the two horizontal sides. In a triangular plane, when the base is uppermost, the centre of pressure is at the middle of the hne, raised perpendicularly from the vertex ; and when the vertex is uppermost, the centre of pressure is at I of a line let fall perpendicularly from the vertex. OF PRESSURE. The pressure of a fluid on any surface, whether vertical, ohhque, or horizontal, is equal to the weight of a column of the fluid, whose base is equal to the surface pressed, and height equal to the distance of the centre of gravity of the surface pressed, below the surface of the fluid. To find the Pressure of a Fluid upon the Bottom of the Contain- ing Vessel. Rule.— Multiply area of base in feet by height of fluid in feet, and their sum by the weight of a cubic foot of the fluid. Example. — What is the pressure upon a surface 10 feet square, the water (fresh) being 20 feet deep '? 102 X 20 X 62.5 = 125000 lbs., Ans. The side of any vessel sustains a pressure equal to the area of the side, multiplied by half the depth. The pressure upon an inclined, curved, or any surface, is as the area of the surface, and the depth of its centre of gravity below the fluid. Example. — What is the pressure upon the sloping side of a pond 100 feet square, the depth of the pond being 8 feet '? 1002x^X62.5 = 625000 lbs., Ans. Or, on a hemisphere just covered with water, and 36 inches in diameter, 3X3.1416X—X—X62.5=: 662.5, Ans. M At 168 HYDROSTATICS. The pressure upon a number of surfaces is found by multiplying the sum of the surfaces into the depth of their common centre of gravity, below the surface of the fluid. CONSTRUCTION OF BANKS. A bank, constructed of a given quantity of materials, will just resist the pressure of tiie water when the square of its thickness at the base is to the square of its perpendicular height, as the weight of a given bulk of water is to the weight of the same bulk of the material the bank is made of, increased by twice the aforesaid weight of the given bulk of water. Thus, if the bank is made of a stone 2 times heavier than water, the thickness of the base should be to the height, as 3 to 6. If the height, compared to the thickness of the base, be as 10 to 7, stability is al- ways ensured, whatever the specific gravity of the material may be. The bottom of a conical, pyramidal, or cylindrical vessel, or of one the section of which is that of an inverted frustrum of a cone or pyramid, sustains a pressure equal to the area of the bottom and the depth of the fluid. FLOOD GATES. To find the Strain which a Fluid will exert to make it turn upon its Hinges, or open. Rule. — Multiply \ of the square of the height by the square of the breadth, and take a bulk of water equal to the product. Example. — If the gate is 6 feet square, _X62 =324 cubic feet, or 20250 lbs. To find the Strain the Water exerts upon its Hinges. Rule. — Multiply ^ of the breadth by the cube of the height, and take a bulk of water equal to the product. Example. — With the same gate, ^ X 63 := 216 cubic feet, or 13500 lbs. PIPES. To find the Thickness of a Pipe. RuLE.T— Multiply the height of the head of the fluid in feet by the diameter of the pipe in inches, and divide their product by the co- hesion of one square inch of the material of which the pipe is com- posed. By experiment it has been found that a cast iron pipe, 15 inches in diameter, and % of an inch thick, will support a head of water of 600 feet ; and that one of oak, of the same diameter, and 2 inches thick, will support a head of 180 feet. The cohesive power of cast iron, then, would be 12,000 lbs. ; of oak, 1350 lbs. That of lead is 750 lbs. ; and wrought iron boiler plates, riveted together ^ is from as to 30,000 lbs. HYDROSTATICS. 169 In conduit pipes, lying horizontal, and made of lead, their thickness, compared to their diameter, should be. As 2|, 3, 4, 5, 6, 7, 8 lines, To 1, Ih 2, 3, 4^, 6, 7 inches. And when made of iron, As 1, 2, 3, 4, 5, &c., lines, ' To 1, 2, 4, 6, 8, &c., inches. The tenacity of lead is increased to 3000 by the addition of 1 part of zinc in 8. HYDROSTATIC PRESS. To find the Thickness of the Metal to resist a Given Pressure. Let ;? = pressure per square inch in pounds, r = radius of cylin- der, and c = cohesion of the metal per square inch. Then -^ = thickness of metal. The cohesive force of a square inch of cast iron is frequently estimated nt 18000 lbs. P 170 HYDRAULICS AND HYDRODYNAMICS. HYDRAULICS AND HYDRODYNAMICS. Hydraulics treats of the motion of non-elastic fluids, and Hy- drodynamics of the force with which they act. » Descending water is actuated by the same laws di^ falling bodies. Water will fall through 1 foot in i of a second, 4 feet in i of a second, and through 9 feet in | of a second, and so on. The velocity of a fluid, spouting through an opening in the side of a vessel, reservoir, or bulkhead, is the same that a body would acquire by falling through a perpendicular space equal to that he tween the top of the water and the middle of the aperture. Then, by rule 4 in Gravitation, ^ height X 64.33 = velocity. Example.— What is the velocity of a stream issuing from a head of 10 feet 1 • V^ 10 X 64.33 = 25.36 feet, ^ . Or, V10X8 == 25.30 feet, ) ^'^''' If the velocity be 50.72 feet per second, what is the head] 50.722-^64.33=140 feet, ) Or, 50.72 -^8^ = 40.2 feet, S This would be true were it not for the effect of friction, which m pipes and canals increases as the square of the velocity. The mean velocity of a number of experiments gives 5.4 feet for a height of one foot. The theoretical velocity is {^/&^) 8. OF SLUICES. To find the Quantity of Water which will flow out of an Opening. Rule.— Multiply the square root of the depth of the water by 5.4; the product is the velocity in feet per second. This, multiplied by the area of the orifice in feet, will give the number of cubic feet per second. Example.— If the centre of a sluice is 10 feet below the surface of a pond, and its area 2 feet, what quantity of water will run out in one second 1 -^lOx 5.4X2 = 34. 1496 feet, Ans. Note.— If the area of the opening is large compared with the head of the water take § of this velocity for the actual velocity. OF VERTICAL APERTURES OR SLITS. The quantity of water that will flow out of one that reaches as high as the surface is § of that which would flow out of the same aperture if it were horizontal at the depth of the base. Q^^ velocity at bottom X depth x 2 ^ ^^^^^^^ ^^ ^^^ ^ ^^^^^^ O of cubic feet per second. A- a ,'-<" y^ o / C i5 HYDRAULICS AND HYDRODYNAMICS. 171 OF STREAMS OR JETS. To find the Distance a Jet will he projected from a Vessel through an opening in the Side. Rule. — B C will always be equal to twice the square root of A O X OB. If is 4 times as deep below A, as fl is, will discharge twice the quan- tity of water that will flow from a in the same time, because 2 is the square root of A o, and 1 is the square root of A a. Note.— The water will spout the farthest when o is equidistant from A and B ; and if the vessel is raised above a plane, B must be taken upon the plane. The quantities of water passing through equal holes in the same time are as the square roots of their depths. Example.— A vessel 20 feet deep is raised 5 feet above a plane ; how far will a jet reach that is 5 feet from the bottom] ^15x10 X 2 = 24.48 feet, Ans. When a prismatic vessel empties itself by a small orifice, in the time of emptying itself, twice the quantity would he discharged if it were kept full by a new supply. To find the Vertical Height of a Stream projected from a Pipe. Rule.— Ascertain the velocity of the stream by computing the quantity of water running or forced through the opening ; then, by rule 5 in Gravitation, page 140, find the required height. Example.— If a fire-engine discharges 16.8 cubic feet of water through a | inch pipe in one minute, how high will the water be projected, the pipe being directed vertically] 1 6.8 X 1728 -^ area of | -finches in a foot -^ seconds in a minute = 91.6, or velocity of stream in feet per second ; then, by rule, page 140, 91.6-7-8 = 11.45, and 11.452 = 131.10 feet, Ans. Note.— This rule gives a theoretical result; the result in practice is somewhat VELOCITY OF STREAMS. In a stream, the velocity is greatest at the surface and in the middle of the current. To find the Velocity of a River or Brook. Rule. — Take the number of inches that a floating body passes over in one second in the middle of the current, and extract its square root ; double this root, subtract it from the velocity at top, and add 1 ; the result will be the velocity of the stream at the bot- tom ; and the mean velocity of the stream is equal the velocity at the surface — v^ velocity at the si^rface +.5. 172 HYDRAULICS AND HYDRODYNAMICS. Example. — If the velocity at the surface and in the middle of a stream be 36 inches per second, what is the mean velocity 1 ^36x2— 36+1 =25, the velocity at bottom. 36_^36+.5rrr30.5, Ans. To find the Velocity of Water running through Pipes. Rule. — Divide height of head in inches by length of pipe in inch- es, and the square root of the quotient, multiplied by 23.3, will give the velocity in inches at the orifice. Example. — What is the velocity when the head is 9 feet, the pipe 24 inches long and 2^ inches bore 1 ,/ 108+-24 X 23.3 = 49.49 inches per second, Ans. Quantities of Water discharged from Orifices of various forms^ the Altitude being constant^ at 34.642 Inches. Cubic inches Nature and dimensions of the tubes and orifices. discharged in a minute. 1. A circular orifice in a thin plate, the diameter being 1.7 inches 10783 2. A cylindrical tube 1.7 inches in diameter, and 5.117 inches long 14261 3. A short conical adjutage, 1.7 inches in diameter . 10526 4. The same, with a cylinder 3.41 inches long added to it 10409 5. The same, the length of the cylinder being 13.65 inch- es long 9830 6. The same, the length of the cylinder being 27.30 inch- es long 9216 Results prove that the discharge of water through a straight cylindrical pipe of an unlimited length may be increased only by al- tering the form of the terminations of the pipe, by making the inner end of the pipe of the same form as the veyia contracta, and the ex- tremity a truncated cone, having its length about 9 times the diam- eter of the cylinder or pipe attached, and the aperture at the outlet to the diameter of the cylinder as 18 is to 10. By giving this form, the discharge is over what it would be by the cylinder alone as 24 is to 10. "WAVES. The undulations of waves are performed in the same time as the oscillations of a pendulum, the length of which is equal to the breadth of a wave, or to the distance between two neighbouring cavities or eminences. HYDRAULICS AND HYDRODYNAMICS. 173 05 < P CD H- I-. H- h- H- tS JO W W 00 rf^ hf^ Ol 05 00 ^ 4^ en oj^;dm CJ«COC;i;D>f^OOOOO f^Oi05«^00«OOOt50KJ«00>^Oa5CO — Oi «x> CO 00 ►fi>. t— H- OS oi<{ ts CO f-" hf^ c;i 05 CO <;d to f».oooo)-' OCnO'^OSOSt-ii-'rf^H-'COMMCOtOOSCOffi^tnOiO -t5MC0C0C0>f*.CnO5COC0 05'>^oo«oOH-'h->»^0500i— 'Cooooc:noa5C;iOi— '00 00 c;i w H- t5 fs» copies coo -WWMCOC04SkC;iCn. 00 00 »f!k 00 ^i 05 <^ 00 05 c;t H- CO 05 as' ►_Oh-CO>f^050-f!». to ^ 1-1 ^ Oi I— to M»-'0«ooo^aic;n>f!».>fs»'Cocotowt5i-'i-'H-'i--» P2 4 "" 2 TJ S3 174 HYDRAULICS AND HYDEODYNAIMICS. GENERAL RULES. Discharge by Horizontal Pipes. 1. The less the diameter of the pipe, the less is the proportional discharge of the fluid. 2. The greater the length of the discharging pipe, the greater the diminution of the discharge. Hence, the discharges made in equal times by pipes of different lengths, of the same diameter, and under the same altitude of water, are to one another in the inverse ratio of the square roots of their lengths. 3. The friction of a fluid is proportionally greater in small than in large pipes. 4. The velocity of water flowing out of an aperture is as the square root of the height of the head of the water. Theoretically the velocity would be -y/ height X8. In practice it is ^ height X 5.4 = velocity in feet per second. Discharge by Vertical Pipes. The discharge of fluids by vertical pipes is augmented, on the principle of the gravitation of falling bodies ; consequently, the greater the length of the pipe, the greater the discharge of the fluid. Discharge by Inclined Pipes. A pipe which is inclined will discharge in a given time a greater quantity of water than a horizontal pipe of the same dimensions. Deductions from various Experiments. 1. The areas of orifices being equal, that which has the smallest perimeter will discharge the most water under equal heads ; hence circular apertures are the most advantageous. . 2. That in consequence of the additional contraction of the fluid vem, as the head of the fluid increases the discharge is a little diminished. 3. That the discharge of a fluid through a cylindrical horizontal tube, the diam- eter and length of which are equal to one another, is the same as through a simple orifice. -, •,. r- v, - 4. That the above tube may be increased to four times the diameter of the ori- fice with advantage. 5. The velocity of motion that would result from the direct, unretarded ac- tion of the column of a fluid which produces it, being a constant, or .8. The velochy through an aperture in a thin plate, with the same pressure, is 5. Through a tube from two to three diameters in length, projecting outward, 6.5 Through a tube of the same length, projecting inward 5.45 Through a conical tube of the form of the contracted vein . . . . 7.9 Curvilineal and rectangular pipes discharge less of a fluid than rectilineal pipes. Discharge from Reservoirs receiving no Supply of Water. For prismatic vessels the general law applies, that twice as much would be dis- charged from the same orifice if the vessel were kept full during the time which is required for emptying itself. Discharges from Compound or Divided Reservoirs. The velocity in each may be considered as generated by the difference of the heights in the two contiguous reservoirs ; consequently, the square root of the dif- ference will represent the velocity, which, if there are several orifices, must be inversely as their respective areas. Discliarge by Weirs and Rectangular Notches. The quantity of water discharged is found by taking § of the velocity due to the mean height, using 5.1 for the coefficient of tlie velocity. Example.— What quantity of water will flow from a pond, over a weir 102 inch- es in length by 12 inches deep ? ly/^ foot X 5.1 X 8.5 area of weir = 28.9 cubic feet in one second. HYDRAULICS AND HYDRODYNAMICS. 175 Table of the Rise of Water in Rivers^ occasioned by the erection of PierSj 4*c. l-J Amount of obstruction compared with area of section of the river. 9 '^£^ 1 3 3 4 5 6 7 8 TTT >S!di To To To To TTT 10 To- To- Feet. Faet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. 1 .0157 .0377 .0698 .1192 .2012 .3521 .6780 1.609 6.639 2 .0277 .0665 .123J .2102 .3548 .6208 1.196 2.838 11.71 3 .0477 .1144 .2118 .3618 .6107 1.069 2.0.58 4.885 20.15 4 .0760 .1822 .3372 .5759 .9719 1.701 3.276 7.775 32.07 5 .1165 .2793 .5168 .8782 1.490 2.607 5.020 11.92 49.15 6 .1558 .3736 .6912 1.181 1.993 3.487 6.715 15.94 65.75 7 .2078 .4983 .9221 1.575 2.658 4.651 8.958 21.26 87.71 8 .2678 .6^123 1.188 2.030 3.426 5.995 11.54 27.40 113.0 9 .3359 .80.54 1.490 2.557 4.296 7.517 14.48 34.36 141.7 10 .4119 .9877 1.827 3.122 5.268 9.219 17.75 42.14 173.8 Velocity of Water in Pipes or Seioers. The time occupied in an equal quantity of water through a pipe or sewer of equal lengths, and with equal falls, is proportionally as follows: In a right line as 90, in a true curve as 100, and in passing a right angle as 140. The resistance that a body sustains in moving through a fluid is in proportion to the square of the velocity. The resistance that any plane surface encounters in moving through a fluid with any velocity is equal to the weight of a column whose height is the space a body would have to fall through in free space to acquire that velocity, and whose base is the surface of the plane. Rx AMPLE. — If a plane, 10 inches square, move through water at the rate of 8 feet per second, then 82-^-64=1.=: the space a body would require to fall to ac- quire a velocity of 8 feet per second ; and as 1 foot= 12 inches, then 10X12 =:; 120 cubic inches, = the column of water whose height and base are required. Cub. Inches. Ounces. As 1728 : 120 : : 1000 : 69.4, or 4.3 lbs., which is the amount of resistance met ^vith by the plane at the above velocity. And it is the same, whether the plane moves against the fluid or the fluid against the plane. The following Table shows the results of experiments with a plane one foot square, at an immersion of 3 feet below the surface, and at different velocities per second. Velocity. Resistance. Velocity. Resistance. Velocity. Re^tance. 5 feet 6 " 7 " 29.5 lbs. 40. - 54.6 " 8 feet 9 " 10 " 71.7 lbs. 90.6 " 112. " 11 feet 12 " 13i " 136.3 lbs. 162.1 " 213. " 176 WATER WHEELS. WATER WHEELS, This subject belongs properly to Hydrodynamics, but a separate classification is here deemed preferable. Water Wheels are of three kinds, viz., the Overshot^ Undershot^ and Breast. The Overshot Wheel is the most advantageous, as it gives the greatest power with the least quantity of water. The next in or- der, m point of efficiency, is the Breast Wheel, w^hich may be con- sidered a mean betw^een the overshot and the Undershot. For a small supply of water w^ith a high fall, th^ first should be employed ; where the quantity of w^ater and height of fall are both moderate, the second form should be used. For a large supply of water with a low fall, the third form must be resorted to. Before proceeding to erect a water wheel, the area of the stream and the head that can be used must be measured. Find the velocity acquired by the water in falling through that height by the rule, viz. : Extract the square root of the height of the head of the w^ater (from the surface to the middle of the gate), and multiply it by 8. Note. — Where the opening is small, and the head of water is great, or proper tionally so, use from 5.5 to 8 for the multiplier. Example. — The dimensions of a stream are 2 by 80 inches, from a head of 2 feet to the upper surface of the stream ; what is the ve- locity of the w^ater per minute, and what is its w^eight 1 2 feet and i of 2 inches = 25 inches r= 2.08 feet, v'2.08x*6.5x 60 z=i 561.60 feet velocity per minute. And 80X2X561.6 feet X12 inches, -^1728= 624 cutjic feet, X 62^ lbs. = 39000 lbs. of w^ater discharged in one minute. To find the Power of an Overshot Wheel. Rule. — Multiply the weight of water in lbs. discharged upon the wheel in one minute by the height or distance in feet from the lower edge of the wheel to the centre of the opening in the gate ; divide the product by 50000, and the quotient is the number of horses' power. Example. — In the preceding example, the weight of the water discharged per minute is 39000 lbs. If the height of the fall is 23 feet, the diameter of the wheel being 22, what is the power of the wheel ] 23 feet — 8 inches clearance below = 22.4 = 22.33. 39000x22.33-^50000=17.41 horses' power, Ans. To find the Power of a Stream. Rule. — Multiply the weight of the water in lbs. discharged in one minute by the height of the fall in feet ; divide by 33000, and the quotient is the answer. * Estimate of velocity. WATER WHEELS. 177 Example.— What power is a stream of water equal to of the fol- lowing dimensions, viz. : 1 foot deep by 22 inches broad, velocity 350 feet per mmute, and fall 60 feet ; and what should be the size of the wheel applied to it ] 12;< 22X350X12—1728X621X60 feet -f-33000 = 72.9, Ans. Height of fall 60 feet, from which deduct for admission of water, and clearance below, 15 inches, which gives 58.9 feet for the diam- eter of the wheel. Clearance above 3 ) , ^ . , below 12 I 1^ inches. The power of a stream, applied to an overshot wheel, produces effect as 10 to 6.6. Then, as 10 .- 6.6 : : 72.9 : 48 horses' power equal that of an over- shot wheel of 60 feet applied to this stream. When the fall exceeds 10 feet, the overshot wheel should be applied, the ff ^^^^^^ ^^^ ^^'^^^^ ^^ ^^ proportion to the whole descent, the greater will be to^tiie^^^^^ ^s ^s the quantity of water and its perpendicular height multiplied The weight of the arch of loaded buckets in pounds, is found by multiplyinff 4 of their number, X the number of cubic feet in each, and that product by 40. ® To find the Power of an Undershot Wheel ivhen the Stream is confined to the Wheel. Rule.— Ascertain the weight of the water discharged against the floats of the wheel in one minute by the preceding rules, and divide it by 100000 ; the quotient is the number of horses' power;; NoTE.-The 100000 is obtained thus : The power of a stream, applied to an un- dershot wheel, produces effect as 10 to 3.3 ; then 3.3 : 10 : : 33000 • 100000 ^ .rrn; T ^^^ Opening is nbove the centre of the floats, multiply the weight of the water by the height, as m the rule for an overshot wheel. Example.— What is the power of an undershot wheel, applied to a stream 2 by 80 inches, from a head of 25 feet ? \/25x6. 5x60 — 1950 feet velocity of water per minute, and 2X80 = 160 mches X 1950 X12-M 728 =2166.6 cubic feet X62.5 = *135412 lbs. of water discharged in one minute : then 135412— 100000 = 1.35 horses' power. "^ Note.— The maximum work is always obtained when the velocity of the wheel IS half that of the stream. Let V represent velocity of float boards, and v velocity of water ; then -^^^- X force of the water, will be the force of the efiective stroke. V Till l^^i^^ °^ ^^ undershot wheel to the power expended is, at a medium, one half that of an overshot wheel. The virtual or effective head being the same, the effect will be very nearly as the quantity of water expended. When the fall is below 4 feet, an undershot wheel should be applied. To find the Power of a Breast Wheel. Rule.— Find the effect of an undershot wheel, the head of water of which is the difference of level between the surface and where it strikes the wheel (breast), and add to it the effect of that of an over- s hot wheel, the height of the head of which is equal to the diflfer- * Equal 160xi2-i-1728x62.5xi950 = momentum of water and its velocity. 178 WATER WHEELS. ence between where the water strikes the wheel, and the tail w^ater ; the sum is the effective power. Example. — What would-be the power of a breast w^heel applied to a stream 2x80 inches, 14 feet from the surface, the rest of the fall being 11 feet? ^14x6,5x60 — 1458.6 feet velocity of water per minute. And 2x80x1458x12-^1728 = 1620 cubic feet X 62.5 =:= 101250 lbs. of water discharged in one minute. Then 101250-MOOOOO = 1.012 horses' power as an undershot. v^llX6.5x60 = 1290 feet velocity of water per minute. And 2x80x1290x12-^1728 rr: 1433 cubic feet X 62.5 = 89562 lbs. of water discharged in one minute. Xll height of fall -^50000= 19.703 horses, which, added to the above, =20.715, Ans. When the fall exceeds 10 feet, it may be divided into two, and two breast wheels applied to it. When the fall is between 4 and 10 feet, a breast wheel should be applied. The power of a water wheel ought to be taken off opposite to the point where the water is producing its greatest action upon the wheel. BARKER S MILL. The effect of this mill is considerably greater than that which the 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. Fo7' a description of it, see Griefs Mechanics^ Calculator, page 234. Make each arm of the horizontal tube, from the centre of motion to the centre of the aperture of any convenient length, not less than ^ of the perpendicular height of the w^ater's surface above these centres. Multiply the length of the arm in feet by .61365, and the square root of the product will be the proper time for a revolution in sec- onds ; then adapt the geering to this velocity. Or, if the time of a revolution be given, multiply the square of it by 1.6296 for the pro- portional length of the arm in feet. Divide the continued product of the breadth, depth, and velocity of the stream in feet by 14.27 ; multiply the quotient by the square root of the height, and the result is the area of either aperture. Multiply the area of either aperture by the height of the head ot water, and this product by 56 ; the result is the moving force in lbs. at the centre of the apertures. Example. — If the fall bQ 18 feet from the head to the centre of the apertures, then the arm must not be less than 2 feet (as i of 18 = 2), v/2x.61365 = 1.107, the time of a revolution in seconds; the breadth of the race 17 inches, the depth 9, and the velocity 6 feet per second ; what is the moving force 1 17 inches = 1.41 feet, 9 inches = .75 feet; then 1.41 X. 75x6— 14.27Xx/18xl8x56 = 1895 1bs., .4/i5. WATER WHEELS. 179 To find the Centre of Gyration of a Water Wheel. Rule.— Take the radius of the wheel, the weight of its arms, and the weight of its rim, as composed of floats, shrouding, &c. Let R represent the weight of rim, " r '' the radius of the wheel, " A " the weight of arms, " W " the weight of the water in action when the buck- ets are filled, as in operation. Then v/(RXr2 X2+A xr- x2+Wxr2-f.R+ATWx2)rr centre ot gyration. Example.— In a wheel 20 feet diameter, the weight of the rim is 3 tons, the weight of the arms 2 tons, and the weight of the water I ton ; what is the distance of the centre of gyration from the cen- tre of the wheel 1 R =3 tons X10=X2 = 600 A =2 " Xl02x2=:400 W=l " X102 . . =100 3+2+1= 6 X2=:-Y^i= 91.6, the square root of which is S.5, or 9i feet, Ans. Notes.— At the mill of Mr. Samuel Newlin, at Fishkill Creek, N. Y., 5 barrels of flour can be ground, and 400 bushels of grain elevated 36 feet per hour wilh a stream and overshot wheel of the following dimensions, viz. : Height of head to centre of opening, 24^ inches ; opening, 1% by 80 inches ; wheel, 22 feet diameter by 8 feet face ; 52 buckets, each 1 foot in depth. The wheel making 3^ revolutions, driving 3 run of 5i feet stones 130 turns in a mmute, with all the attendant machinery. This is a case of maximum effect, in consequence of the gearing being well set up, and kept in good order. At the furnace of Mr. Peter Townsend, Monroe Works, N. J., 30 to 34 tons of No. 1 Iron are made per week, with the blast from two 5 feet by 5 feet 1 inch blowing cylinders. The wheel (overshot) being 24 feet diameter, by 6 feet in width, having /O buckets of 14 inches in depth. The stream is % by 51 inches liaving a head 6^ feet ; the wheel and cylinders each making 4^ revolutions per Rocky Glen Factory, Fishkill, N. Y., containing 6144 self-acting mule spindles, 160 looms, weaving printing cloths 27 inches wide of No. 33 yarn (33 hanks to a pound), and producing 24,000 hanks in a day of 11 hours, is driven by a breast wheel and stream of the following dimensions, viz. : Stream 18 feet by 3 inches, head 20 feet, height of water upon wheel 16 feet, di- ameter of wheel 26 feet 4 inches, face of wheel 20 feet 9 inches, depth of buckets 15f inches, number of buckets 70, Revolutions, 4^ per minute. 180 PNEUMATICS. PNEUMATICS. WEIGHT, ELASTICITY, AND RARITY OF AIR. The pressure of the air at the surface of the earth is, at a mean rate, equal to the support of 29.5 inches of mercury, or 33.18 feet of fresh water. It is usually estimated in round numbers at 30 inches of mercury and 34 feet of water, or 15 lbs. pressure upon the square inch. . . , ,. The Elasticity of air is inversely as the space it occupies, and di- rectly as its density. When the altitude of the air is taken in arithmetic proportion, its Rarity will be in geometric proportion. Thus, at 7 miles above the surface of the earth, the air is 4 times rarer or lighter than at the earth's surface ; at 14 miles, 16 times ; at 21 miles, 64 times, and so on. The weight of a cubic foot of air is 527.04 grams, or 1.205 ounces avoirdupois. j • -nnA At the temperature of 33°, the mean velocity of sound is 1100 feet per second. It is increased or diminished half a foot for each degree of temperature above or below 33°. To compute Distances hy Sound. Rule.— Multiply the time in seconds by 1100, and the product is the distance in feet. Example.— After observing a flash of lightning, air at 60°, it was 5 seconds before I heard the thunder ; what was the distance of the cloud 1 1100+- 50—33 X 6-^5280 = 1.049 miles, Ans, To compute ivhat Degree of Rarefaction may he effected in a Vessel, Let the quantity of air in the vessel, tube, and pump be represented by 1, and . the proportion of 'the capacity of the pump to the vessel and tube by .33 ; conse- quently, it contains ^ of the air in the united apparatus. a s c f\.^ r..i Upon the first stroke of the piston this fourth will be expelled, and | of the ori- ginal quantity will remain : ^ of this will be expelled upon the second stroke, which . is equal to ^V of the original quantity ; and, consequently, there remains in the ap- paratus ^ of the original quantity. Calculating in this way, the following table ^ is easily made : No. of stroke*.. 1 Air expelled at each stroke^ I Air remainiDg in the vessel. 3_ 16 • 9^ 64 27 256 81 1024 _ 3 "~4X4 3X3 '4X4X4 3X3X3 '4X4X4X4 _ 3X3X3X3 '4X4X4X4X4 16 27 64 ?1 256 243 1024 _3X3 "4X4 3X3X3 "~ 4X4X4 3X3X3X3 ""4X4X4X4 3X3X3X3X3 '4X4X4X4X4 PNEUMATICS. 181 And so on, continually multiplying the air expelled at the preceding stroke by 3 and dividing it by 4 ; and the air remaining after each stroke is found by multiDlv- ing the air remaining after the preceding stroke by 3, and dividing it by 4, Measurement of Heights by Means of the Barometer. Jlpproximate Rule. For a mean temperature of 550, X = required difference in height in feet, h^ = the height of the mercury at the lower station, h' = the height of the mercury at the upper station, X = 55.000 X h-\-k' ■^^^ ^ko °^ ^^^^ result for each degree which the mean temperature of the air at the two stations exceeds 550, and deduct as much lor each degree below 5oO. *""vi* Velocity and Force of Wind. Miles in an hour. Feet in a minute. 1 • 88 2 176 3 264 4 352 5 440 6 528 8 704 10 880 15 1320 20 1760 25 2200 30 2640 35 3080 40 3520 45 3960 50 4400 60 5280 80 7040 100 8800 Pressure on a square foot in pounds avoir- dupois. .005 .020 ) .045 ] .080 .125 J .180 C .320 S .500 ) 1.125 ] 2.000 ) 3.125 S 4.500 ) 6.125 S 8.000 ) 10.125 ] 12.500 18.000 32.000 50.000 Description. Barely observable. Just perceptible. Light breeze. Gentle, pleasant wind. Brisk blow. Very brisk. High wind. Very high. Storm. Great storm. Hurricane. Tornado, tearing up trees, &c. To find the Force of Wind acting perpendicularly upon a Surface. RuLE.--Multiply the surface in feet by the square of the velocity in feet, and the product by .002288 ; the result is the force in avoir- idupois pounds. Q 182 STATICS. STATICS. PRESSTJRE OF EARTH AGAINST WALLS. A B 1j I^ ^ . Let AB C D be the vertical section of a wall, behind which is a bank of earth, AD/.; let DG be the line of rupture, or natural slope which the earth would assume but for the resistance of the wall. . - In sandy or loose earth, the angle G D H is generally 30« ; in firmer earth it is 36°, and in some instances it is 45°. The angle formed with the vertical by the earth, AD G that ex- erts the greatest horizontal stress against a wall, is half the angle which the natural slope makes with the vertical. If the upper surface of the earth and the wall which supports it are both in one horizontal plane. Then the resultant In of the pressure of the bank, behind a verti- eal wall, is at a distance D w of J A D. In ve^retable earths, the friction is J the pressure ; in sands, ^. The tine of rupture A G in a bank of vegetable earth is = .618 of A D. When the bank is of sand, it is .677 of A D. If of rubble, it is .414 of A D. Thichiess of Walls, both Faces Vertical. Brick. Weight of a cubic foot, 109 lbs. avoirdupois, bank of vegetable earth be- ^"^Unhewn stones. 135 lbs. per cubic foot, bank as before, A B = .15 A D. Brick. Bank clay, well rammed, A B = .17 A D. a « — i-l A n • Hewn freestone. 170 lbs. per cubic foot, bank of vegetable earth, A B = .13 A D , if the bank is of clay, A B = .14 AD. Bricks. Bank of sand, A B = .33 A D. Unhewn stone. Bank of sand, A B = .30 A D. Hewn freestone. Bank of sand, A B = .26 A D. When the bank is liable to be saturated with water, the thickness of the walll must be doubled. For farther notes, and for the EquUibrium of Piers, see Gregory's Mathematics^ pages 220 to 224. DYNAMICS. 183 DYNAMICS. Dynamics is the investigation of body^ force, velocity, space, and time. Let them be represented by their initial letters hfv s t, gravity by g, and momentum or quantity of motion by m; this is the effect pro- duced by a body in motion. Force is motive, and accelerative or retardative. Motive force, or momentum, is the absolute force of a body in motion, and is the product of the weight or mass of matter in the body, multiplied by its velocity. Accelerative or retardative force is that which respects the velo- city of the motion only, accelerating or retarding it, and is found by the force being divided by the mass or weight of the body. Thus, if a body of 4 lbs. be acted upon by a force of 40 lbs., the*^ accelera- ting force is 10 lbs. ; but if the same force of 40 act upon another body of 8 lbs., the accelerating force then is 5 lbs., only half the former, and will produce only half the velocity. Uniform Motion, The space described by a body moving uniformly is represented by the product of the velocity into the time. With momenta, m varies as b v. Example.— Two bodies, one of 20, the other of 10 lbs., are impelled by the same momentum, say 69. They move uniformly, the first for 8 seconds, the second for 6 ; what are the spaces described by both 1 * 60 o . 60 ^ - = .,or- = 3,and- = 6. Then «v = 3X8 = 24 = 5, and 6x6 = 36 = ^. Thus the spaces are 24 and 36 respectively- Motion Uniformly Accelerated. In this motion, the velocity acquired at the end of any time whatever, is equal to the product of the accelerating force into the time, and the space described is equal to the product of half the accelerating force into the square of the time. The spaces described in successive seconds of time are as the odd numbers, 1, 3, 5, 7, 9, &c. Grav-ity is a constant force, and its efiect upon a body falling freely is represented by^. The following theorems are applicable to all cases of motion uniformly accelera- ted by any constant force : v = Y =gFt =:y/2gfs. ts V s ~~^ -gF -^IgF' F— — — -?1 —J^ ^ gt ~~g^ "^gs When gravity acts alone, as when a body falls in a vertical line, F is omitted and we have, s = igt^ = — = ^tv, v = gt =—=^2gs. 184* DYNAMICS. ^~" £ t2 —05- Note.— g is obviously 32.166 from what has been given in rules for Gravitation, cuiid is the force of gravity. If, instead of a heavy body falling freely, it be propelled vertically upward or downward with a given velocity, v, then sz^tv::^hgf'\ an expression in which — must be taken when the projection is upward, and -j- when it is downward. Motion over a Fixed Pulleij. Let the two weights which are connected by the cord that goes over the pulley be represented by W and w ; then W — w W+w' = F in the formulce where F is used ; so that W- -hgt''' Or, if the resistance of the friction and inertia of the pulley be represented by r, then V7—W „ Example. — If by experiment it is ascertained that two weights of 5 and 3 lbs. over a pulley, the heavier weight descended only 50 feet in 4 seconds, what is the measure of r ? If r is not considered, the heavier weight would fall 64^ feet. Then „^^~^ ^gt~ — 50 feet. And, as 5+3+ r : 5+3 : That is . r : 5+3 : Whence W+MJ+r : 64i : 50 ; : 14^ : 50. _8X14J 50 =: 2.293 lbs., Ans. Table of the Effects of a Force of Traction of 100 lbs. at different Velo- cities J on Canals ^ Railroads^ and Turnpikes. Velocity. On a Canal. On a Railroad. On a Turnpike. Miles Feet per JIass Useful Mass Useful Mass Useful perhr. second. moved. etfect. moved. effect. moved. effect. lbs. lbs. lbs. lbs. lbs. lbs. 2^ 3.66 55.500 39.400 14.400 10.800 1.800 1.350 3 4.40 38.542 27.361 14.400 10.800 1.800 1.350 8^ 5.13 28.316 20.100 14.400 10.800 1.800 1.350 4 5.86 21.680 15.390 14.400 10.800 1.800 1.350 5 7.33 13.875 9.850 14.400 10.800 1.800 1.350 6 8.80 9.635 6.840 14.400 10.800 1.800 1.350 7 10.26 7.080 5.026 14.400 10.800 1.800 1.350 R 11.73 5.420 3.848 14.400 10.800 1.800 1.350 9 13.20 4.282 3.040 14.400 10.800 1.800 1.350 10 14.66 3.468 2.462 14.400 10.800 1.800 1.350 13.5 19.9 U900 1.350 14.400 10.800 1.800 1.350 The load carried, added to the weight of the vessel or carriage which contains it, forms the total mass moved, and the useful effect is the load. The force of traction on a canal varies as the square of the velocity ; on a rail- road or turnpike the force of traction is constant, but the mechanical power neces- sary to move the carriage increases as the velocity. PENDULUMS. 185 PENDULUMS. The Vibrations of Pendulums are as the square roots of their lengths. The length of one vibrating seconds in New- York at the level of the sea is 39.1013 inches. To find the Length of a Pendulum for any Given Number of Vibrations in a Minute. Rule. — As the number of vibrations given is to 60, so is the square root of 39.1013 (the length of the pendulum that vibrates seconds) to the square root of the length of the pendulum required. Example. — What is the length of a pendulum that will make 80 vibrations in a minute? As v'39. 1013x60 = 375, a constant number, 375 Then _ — 4.6875, and 4.68752 = 21.97 inches, Ans. The lengths of pendulums for less ' or greater times is as the square of the times ; thus, for i a second it would be the square of on lAlO h, or — '-- — = 9.7753 inches, the length of a i second pendulum at New- York. To find the Number of Vibrations in a Minute, the Length of the Pendulum being given. Rule. — As the square root of the length of the pendulum is to the square root of 39.1013, so is 60 to the number of vibrations required. Example. — How many vibrations will a pendulum of 49 inches long make in a minute \ ->/49 : v^39.1013 : : 60 : number of vibrations. 375 Or, -— = 53.57 vibrations, Ans. To find the Length of a Pendulum, the Vibrations of which will be the same Number as the Inches in its Length. Rule. — Square the cube root of *375, and the product is the an- swer. Example.— ^375 = 7.211247, and 7.2112472 r= 52.002, Ans. The Length of a Pendulum being given, to find the Space through which a Body will fall in the Time that the Pendulum makes one Vibration. Rule.— Multiply the length of the pendulum by 4.93482528, and it will give the answer. * 375 is the constant for tlie latitude of New-York ; in any other place, multiply the square root of the length of the pendulum at that place by 60. Q2 186 CENTRE OF GYRATION. Example. — The length of the pendulum is 39.1013 inches ; what is the distance a body will fall in one vibration of it'? 39.1013x4.9348 = 192-9578 inches, or 16.8298 feet, Ans. All vibrations of the same pendulum, whether great or small, are performed very nearly in the same time. In a Simple Pendulum, which is, as a ball, suspended by a rod or line, supposed to be inflexible, and without weight, the length of the pendulum is the distance from its centre of gravity to its point of suspension. Otherwise, the length of the pendulum is the distance from the point of suspension to the Centre of Oscillation.,* which does not coincide with the centre of gravity of the ball or bob. CENTRE OF GYRATION. The Centre of Gyration is the point in any revolving body, or system of bodies, that, if the whole quantity of matter were collect- ed in it, the angular velocity w^ould be the same ; that is, the mo- mentum of the body or system of bodies is centred at this point. If a straight bar, equally thick, was struck at this point, the stroke would communicate the same angular velocity to the bar as if the whole bar w^as collected at that point. To find the Centre of Gyration. Rule 1. — Multiply the weight of the several particles by the squares of their distances in feet from the centre of motion, and divide the sum of the products by the weight of the entire mass ; the square root of the quotient will be the distance of the centre of gyration from the centre of motion. 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 the distance of the centre of gyration from the centre of motion (the fulcrum) 1 3X1'=:3. 4x22zrri6. ^t^ =-^ = 2.71, and v^2.71 = 1.64 feet, Ans. That is, a single weight of 7 lbs., placed at 1.64 feet from the ful- crum, and revolving in the same time, would have the same impetus as the two weights in their respective places. * See Centre of Oscillation. CENTRE OF GYRATION. 187 Rule 2. — Multiply the distance of the centre of oscillation, from the centre or point of suspension, by the distance of the centre of gravity from the same point, and the square root of the product will be the answer. Example. — The centre of oscillation is 9 feet, and that of gravity is 4 feet from the centre of the system, or point of suspension ; at what distance from this point is the centre of gyration 1 9x4 = 36, and -/36 — 6 feet, Ans. The following are the distances of the centres of gyration from the centre of motion in various revolving bodies, as given by Mr. Farey : In a straight, uniform Rodj revolving about one end ; length of rod X-5773. In a circular Plate, revolving on its centre ; the radius of the circle X.VOTl. In a circular Plate, revolving about one of its diameters as an axis : the radius X.5. In a Wheel of uniform thickness, or in a Cylinder revolving about the axis ; the radius X.7071. In a solid Sphere, revolving about one of its diameters as an axis ; the radius X.6325. In a thin, hollow Sphere, revolving about one of its diameters as an axis ; the radius X. 8164. In a Cone, revolving about its axis; the radius of the circular base X.5477. In a right-angled Cone, revolving about its vertex ; the height of the cone X.866. In a Paraboloid, revolving about its axis ; the radius of the circular base X.5773. In a straight Lever, the arms being R and r, the distance of the centre of ervra- tion from the centre of motion = y/Trr—^- — . 3(R— r) Note. — The weight of the revolving body, multiplied into the height due to the ve- locity with which the centre of gyration moves in its circle, is the energy of the body^ or the mechanical power which must be communicated to it to give it that motion. Example. — In a solid sphere revolving about its diameter, the diameter being 2 feet, the distance of the centre of gyration is 12x.632o = 7.59 inches. 188 CENTRES OF PERCUSSION AND OSCILLATION. CENTKES OF PERCUSSION AND OSCILLATION. The Centres of Percussion and Oscillation being in the same point, their properties are the same, and their point is, that in a body revolving around a fixed axis, which, when stopped by any force, the whole motion, and tendency to motion, of the revolving body is stopped at the same time. It is also that point of a revolving body v^hich would strike any obstacle with the greatest effect, and from this property it has re- ceived the name of percussion. As in bodies at rest, the whole weight may be considered as col- lected in the centre of gravity ; so in bodies in motion, the whole force may be considered as concentrated in the centre of percus- sion : therefore, the weight of a bar or rod, multiplied by the dis- tance of the centre of gravity from the point of suspension, will be equal to the force of the rod, divided by the distance of the centre of percussion from the same point. Example. — The length of a rod being 20 feet, and the weight of a foot in length equal 100 oz., having a ball atmched at the under end weighing 1000 oz., at what point of the rod from the point of suspension will be the centre of percussion 1* The weight of the rod is 20X100 = 2000 oz., which, multiplied bv half its length, 2000X10 = 20000, gives the momentum of the rod. The weight of the ball = 1000 oz., multiplied by the length of rod, = 1000X20, gives the momentum of the ball. Now the weight of the rod multiplied by the square of the length, and divided by 2000 V 202 3, = — -^— — = 268666, the force of the rod, and the weight of the ball multi- plied by the square of the lensth of the rod, 1000x20^ = 400000, is the force of the V n *-u r .u . / ■ 266666+400000 ^ , ^^ . ^ ball : therefore, the centre of percussion = — - — ^^ ^^^^.^ = 16.66 feet. ' ^ 20000-1-20000 Example. — Suppose a rod 12 feet long, and 2 lbs. each foot in length, with 2 balls of 3 lbs. each, one fixed 6 feet from the point of suspension, and the other at the end of the rod ; what is the distance between the points of suspension and percus- sion? 12X 2X6 = 144, momentum of rod, 3X 6 =18 " of 1st ball, 3X12 = 36 " of 2d " 198 ^^^^ = 1152, force of rod, 3X 36= 108 " of 1st ball, 3X144 = _432 " of 2d ball, 1692 1692 therefore the centre of percussion = — — = 8.545 feet from the point of suspension* luo As the centre of percussion is the same with the centre of cscillation in the non-ap- plication to practical purposes, the following is the easiest and simplest mode of finding it in any beam, bar, &c. : Suspend the body very freely by a fixed point, and make it vibrate in small arcs, counting the number of vibrations it makes in any time, as a minute, and let the number of vibrations made in a minute be called n ; then shall the distance of the centre of oscillation from the point of suspension be = — ^ — inches. For the length of the pendulum vibrating seconds, or 60 times in a minute, being 39|^ inch- '^ ^adXa-j-a : 20 feet long, -- 100 oz. weight of a foot in lengtli, \ ^""^" ^"^ _ centre of percussion. :1000 " fixed at end, ' i-.w.j... CENTRES OF PERCUSSION AND OSCILLATION. 189 es, and the lengths of the pendulums being reciprocally as the square of the num ber of vibrations made in the same time, therefore n^ : 60^ : : 39|^ : — - = — being the length of the pendulum which vibrates n times in a minute, or the distance of the centre of oscillation below the axis of motion. There are many situations in which bodies are placed that prevent the applica- tion of the above rule, and for this reason the following data are given, which will be found useful when the bodies and the forms here given correspond : 1. If the body is a heavy, straight line of uniform density, and is suspended by one extremity, the distance of its centre of percussion is § of its length. 2. In a slender rod of a cylindrical or prismatic shape, the breadth of which is very small compared with its length, the distance of its centre of percussion is nearly § of its length from the axis of suspension. If these rods were formed so that all the points of their transverse sections were equidistant from the axis of suspension, the distance of the centre of percussion would be exactly § of their length. 3. In an Isosceles triangle, suspended by its apex, and vibrating in a plane per- pendicular to itself, the distance of the centre of percussion is J of its altitude. A line or rod, whose density varies as the distance from its extremity, or the point of suspension ; also Fly-wheels, or wheels in general, have the same relation as the isosceles triangle, the centre of percussion being distant from the centre of suspen sion I of its length. 4. In a ver>^ slender cone or pyramid, vibrating about its apex, the distance of its centre of percussion is nearly f of its length. The distance of either of these centres from the axis of motion is found thus : If the Axis of Motion be in the vertex of the figure, and the motion be flatwise ; then, in a right line, it is § of its length. In an Isosceles Triangle = ^ of its height. In a Circle = J of its radius. In a Parabola t= f of its height. But if the bodies move sidewise, it is. In a Circle = 5: of its diameter. In a Rectangle, suspended by one angle, = § of the diagonal. In a Parabola, suspended by its vertex, = -| axis -f- J parameter ; but if suspend* ed by the middle of its base, = A axis -}- ^ parameter. ^ , ^ ^ ^. , 3 X arc X radius In the Sector of a Circle = -— — r — 5 •^ 4 X chord ^ A . . radius of base ^ In a Cone = 4 axis H -— : . 5 ' 5X axis In a Sphere = — r radius + 1, t representing the length of the thread 5[t X radius) by which it is suspended. Example.— What must be the length of a rod without a weight, so that when hung by one end it shall vibrate seconds 1 To vibrate seconds, the centre of oscillation must be 39.1013 inches from that of suspension ; and as this must be § of the rod, Then 2:3:: 39.1013 : 58.6519, Ans. Example.— What is the centre of percussion of a rod 23 inches long ? § of 23=15.3 inches from the point of suspension or motion. Example. — In a sphere of 10 inches diameter, the thread by which it is suspend- ed being 20 inches, where is the centre of percussion or oscillation 1 These centres are in the same point only when the body is symmetrical with regard to the plane of motion, or when it is a solid of revolution, which is com- monly the case. 190 CENTRAL FORCES. CENTRAL FORCES. All bodies moving around a centre or fixed point have a tendency to fly off in a straight line : this is called the Centrifugal Force ; it is opposed to the Centripetal Force, or that power which maintains the body in its curvilineal path. The centrifugal force of a body, moving with different velocities in the same circle, is proportional to the square of the velocity. Thus, the centrifugal force of a body making 10 revolutions in a minute is four times as great as the centrifugal force of the same body making 5 revolutions in a minute. To find the Centrifugal Force of any Body. Rule 1. — Divide the velocity in feet per second by 4.01, also the square of the quotient by the diameter of the circle ; the quotient is the centrifugal force, assuming the weight of the body as 1. Then this, multiplied by the weight of the body, is the centrifugal force. Example. — What is the centrifugal force of the rim of a fly- wheel 10 feet in diameter, running with a velocity of 30 feet in a second'? 30—4.01 X7.48-M0=: 5.59 times the weight of the rim, Ans. Note. — When great accaracy is required, find the centre of gyration of the body and take twice the distance of it from the axis for the diameter. Rule 2. — Multiply the square of the number of revolutions in a minute by the diameter of the circle in feet, and divide the product by the constant number 5870 ; the quotient is the centrifugal force when the weight of the body is 1. Then, as in the previous rule, this quotient, multiplied by the weight of the body, is the centrifu- gal force. Example. — What is the centrifugal force of a grindstone, weigh- ing 1200 lbs., 42 inches in diameter, and turning with a velocity of 400 revolutions in a minute 1 400 2 v*^ 5 ——^-^X 1200= 114480 lbs., Ans. The central forces are as the radii of the circles directly, and the squares of the times inversely ; also, the squares of the times are as the cubes of the distances. Hence, let v represent velocity of body in feet per second, w " weight of body, r " radius of circle of revolution, c " centrifugal force. Then -^ =c, and -^ =r; , cX32Xr , yrX32Xc\ and = 2C, and Vv ) = r. CENTRAL FORCES. 191 Dr. Brewster has famished the following : 1 The centrifugal forces of two unequal bodies, having the same velocity, and at the same distance from the central body, are to one another as the respective quantities of matter in the two bodies. 2 The centrifugal forces of two equal bodies, which perform their revolutions in the same time, but are different distances from their axis, are to one another as their respective distances from their axis. 3. The centrifugal forces of two bodies, which perform their revolutions in the same time, and whose quantities of matter are inversely as their distances from the centre, are equal to one another. 4. The centrifugal forces of two equal bodies, moving at equal distances from the central body, but with different velocities, are t|pone another as the squares of their velocities. 5. The centrifugal forces of two equal bodies, moving with equal velocities at different distances from the centre, are inversely as their distances from the centre. 6. The centrifugal forces of two unequal bodies, moving with equal velocities at different distances from the centre, are to one another as their quantities of matter multiplied by their respective distances from the centre. 7. The centrifugal forces of two unequal bodies, having unequal velocities, and at different distances from their axis, are, in the compound ratio of their quantities of matter, the squares of their velocities, and their distances from the centre. The weight of the rim of a fly-wheel for a 20 horse engine is 6000 lbs., the diam- eter 16 feet, and the revolutions 45 ; what is its centrifugal force 1 33120 lbs., Ans. Summary.— Let b represent any particle of a body B, and d its distance from the axis of motion, S. G, O, R, the centres of Gravity, Oscillation, and Gyration, Then^ = SG. • s*fB = -- 192 FLY-WHEELS GOVERNORS. FLY-WHEELS. To find the Weight of Fly-wheels. Rule. — Multiply the horses' power of the engine by 2240, and divide the product by the square of the velocity of the circumfer- ence of the wheel in feet per second ; the quotient will be the weight in 100 lbs. Example. — The powef'of an engine is 35 horses, the diameter of the wheel 14 feet, and the revolutions 40 ; what should be the weight of the wheel \ 35X2240-^40X14X3.1416^602 :=I|^Xl00=r::9130 lbs. 858.5 The weight of the wheel in engines that are subjected to irregu- lar motion, as in the cotton-press, rolling-mill, &c., must be greater than in others where so sudden a check is not experienced, and 3000 would be a better multiplier in such cases. GOVERNORS.' The Governor acts upon the principle of central forces. When the balls diverge, the ring or the vertical shaft raises, and that in proportion to the increase of the velocity squared ; or, the square roots of the distances of the ring from the top, corresponding to two velocities, will be as these velocities. Example. — If a governor make 6 revolutions in a second w^hen the ring is 16 inches from the top, w^hat will be the distance of the ring when the speed is increased to 10 revolutions in the same timel As 10'' : 6' : : ^16 inches : 2.4 inches, which, squared, is 5.76 inches, the distance of the ring from the top. A governor performs in one minute half as many revolutions as a pendulum vibrates, the length of which is the perpendicular dis- • tance between the plane in which the balls move and the centre of suspension. GUNNERS. 193 GUNNERY. It has been ascertained by experiment that the velocity of the ball projected from a gun varies as the square root of the charge direct- ly, and as the square root of the weight of the ball reciprocally. — Hut ton. The same author furnishes the following practical rules : To find the Velocity of any Shot or Shell. Rule. — As the square root of the weight of the shot is to the square root of the weight of treble the weight of the powder, both taken in pounds, so is 1600 to the velocity in feet per second. Example. — What is the velocity of a shot of 196 lbs., projected with a charge of 9 lbs. of powder 1 14 : 5.2 : : 1600 : 594, Ans. When the Range for one Charge is given, to find the Range for another Charge, or the Charge for another Range. Rule. — The ranges have the same proportion as the charges ; that is, as one range is to its charge, so is any other range to its charge, the elevation of the piece being the same in both cases. Example.— If, with a charge of 9 lbs. of powder, a shot range 4000 feet, how far will a charge of 6} lbs. project the same shot at the same elevation 1 9 : 6.75 : : 4000 : 3000, Ans. Given the Range for one Elevation, to find the Range at another Elevation. Rule.— As the sine of double the first elevation is to its range, so is the sine of double another elevation to its range. Example.— If a shot range 1000 yards when projected at an ele- vation of 45°, h'ow far will it range when the elevation is 30^ 16', the charge of powder being the same 1 Sine of 45° X 2 =100000, Sine of 30° 16^x2=: 87064. Then, as 100000 : 1000 : : 87064 : 870.64, Ans. Example. — The range of a shell at 45° elevation being 3750 feet, at what elevation must a gun be set for a shell to strike an object at the distance of 2810 feet with the same charge of powder? As 3750 : 100000 : : 2810 : 74934, the sine for double the eleva- tion of 240 16', or of 65° 44', Ans. R 194 FRICTION. FRICTION. Experiments upon the effect of this branch of mechanical science are as yet not of such a nature as to furnish deductions for very satisfactory rules. ..-u ^ i u • The friction of planed woods and polished metals, without luDri- cation, upon one another, is about i of the pressure. Friction does not increase with the increase of the rubbing sur- The friction of metals is nearly constant ; that of woods seems to increase with action. . The friction of a cylinder rolling upon a plane is as the pressure, and inversely as its diameter. .• ^i The friction of wheels is as the diameter of their axes directly, and as the diameter of the wheel inversely. Friction is at a maximum after a state of rest ; the addition is as the fifth root of the time. The following are the results of some experiments, -without lubrication, as given by Adcock : FRICTION AFTEH A STATE OF REST. At a maximum, oak on oak, ^ to ^^^ of the weight, according to the magnitude of the surface ; for oak on pine, 73 ; for pine on pine, — ? for elm on elm, 2Ts of the weight, the fibres moving longitudinally. ^ When they cross at right angles, the friction of oak is — ; for iron on oak, — ; for iron on iron, ^; for iron on brass, \, the surfaces well polished; but when . larger, and not so smooth, — . For iron on copper, with tallow, the friction is ^ of the weight ; when olive : oil is used, the friction is increased to ^. The Friction on a level Railroad of a Locomotive is about \ ; that is, an en^ne weighing 10 tons has a tractive power of 2 tons by the friction of the surfaces of its ♦vheels upon the rails. FRICTION OF BODIES IN MOTION, Without Lubrication. When the surfaces are large, the friction increases with velocity. , ^, . , For a pressure of from 100 to 4000 lbs. on a square foot, for oak on oak, the fric- tion is about —., besides a resistance of about 1§ lbs. for each square foot, independ- ent of the pressure. When the surface is very small, the friction is somewhat i diminished. For oak on pine, the friction is ~ ; for pine on pine, g ; for iron or i copper on wood, ^, which is much increased by an increase of the velocity ; for : iron on iron, g^ ; for iron on copper, -^^ ; after much use, ^ at all velocities. Where the unctuous matter is interposed between the surfaces, the hardest were found to diminish the friction most where the weight was great. Tallow, applied between oak, reduced the friction to ^\ of the pressure. When the surfaces arc very small, tallow loses its effect, and" the friction is increased to^^^; the adhesion j was about 7 lbs. per square foot. FRICTION. 195 With tallow between iron on oak, the friction is 3*^ ; with brass on oak, JL • for iron on iron, the friction is y^^, adhesion 1 lb. for 15 square inches ; on copper, ■y^, adhesion 1 lb. for 13 square inches ; with soft grease or oil, the friction of iron on copper and brass was ^ and ^. On the whole, in most machines, I of the pressure is a fair estimate of the fric- tion. FRICTION ON AXES. For axes of iron on copper, -^ where the velocity was small, the friction being always a little less than for plane surfaces. An axis of iron, with a pulley of giia iacum, gave, with tallow, -^^, FRICTION AND RIGIDITY OF CORDAGE. Wet ropes, if small, are a little more flexible than dry; if large, a little less flexi- ble. Tarred ropes are stifler by about ^, and in cold weather somewhat more. FRICTION OF PIVOTS. When the angle of the summit of the pivot is about IS^ or 20© the friction for garnet is y^V"? ^o -^\-^; agate, -g^^; rock crystal, ^|^; glass, ^^ ; and steel (tempered), 3^. At an angle of 45° the friction is much reduced, and the friction of agate and steel are then nearly equal. Notes. — In general, friction is increased in the ratio of the weight. Between woods, the friction is less when the grains cross each other than when they are placed in the same direction. Friction is greater between surfaces of the same kind than between surfaces of different kinds. '' The best Lubricators are, and in the following order : Tallow, Soft Soao. Lard, Oil, and Black-lead. ^' ' 196 HEAT. HEAT. Heat, in the ordinary application of the word, signifies, or, rather, implies the sensation experienced upon touching a body hotter, or of a higher temperature than the part or parts which we bring into contact with it ; in another sense, it is used to express the cause of that sensation. To avoid any ambiguity that may arise from the use of the same expression, it is usual and proper to employ the word Caloric to sig- nify the principle or cause of the sensation of heat. Caloric is usually treated of as a material substance, though its claims to this distinction are not decided ; the strongest argument in favour of this position is that of its power of radiation. On touching a hot body, caloric passes from it, and excites the feeling of warmth; when we touch a body having a lower temperature than our hand, caloric passes from the hand to it, and thus arises the sensation of cold. COMMUNICATION OF CALORIC. Caloric passes 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 includes such as metals, which allow caloric to pass freely through their substance, and the latter comprises those that do not give an easy passage to it, such as stones, glass, wood, charcoal, &c. Table of the relative Conducting Power of different Bodies. Platinum . Copper Zinc . Lead Porcelain . Fire-clay . With Water as the Standard. Gold . . 1000 Silver . 973 Iron . . 374 Tin . . 304 Marble . 24 Fire-brick . 11 Water Pine . Lime Oak . 10 39 39 33 Elm . Ash . Apple Ebony 981 898 363 180 12.2 11.4 32 31 28 22 Relative Conducting Power of different Substances compared with each other. Hare's fur . . 1.315 Cotton . . . 1.046 Eider-down . . 1.305 Lint .... 1.032 Beaver's fur . . 1.296 Charcoal . . . .937 Raw silk . . . 1.284 Ashes (wood) . . .927 Wool . . . 1.118 Sewing-silk . . .917 Lamp-black . . 1.117 Air . . .' . .576 Relative Conducting Power of Fluids. Mercury . . . 1.000 I Proof Spirit . . .312 Water . , . .357 | Alcohol (pure) . . .232 Blackened tin . 100 Clean "... 12 Scraped "... 16 Ice 85 Mercury .... 20 Polished iron . 15 Copper .... 12 HEAT. 197 RADIATION OF CALORIC. When heated bodies are exposed to the air, they lose portions of their heat, by projection in right lines into space, from all parts of their surface. Bodies which radiate heat best absorb it best. Radiation is affected by the nature of the surface of the body; thus, black and rough surfaces radiate and absorb more heat than light and polished surfaces. Table of the Radiating Power of different Bodies. Water 100 Lamp-black .... 100 Writing paper .... 100 Glass 90 India ink 88 Bright lead .... 19 Silver 12 Reflection of Caloric is the reverse of Radiation^ and the one increases as the other diminishes. SPECIFIC CALORIC. Specific Caloric is that which is absorbed by different bodies of equal weights or volumes when their temperature is equal, based upon the law, acknowledged as universal, that similar quantities of different bodies require unequal quantities of caloric at any given temperature. Dr. Black termed this, capacity for caloric; but as this term was supposed to be suggested by the idea that the caloric present in any substance is contained in its pores, and, consequently, the capacities of bodies for caloric would be inversely as their densities; and such not being the case, this w^ord is apt to give an incorrect notion, unless it is remembered that it is but an ex- pression of fact, and not of cause ; and to avoid error, the word specific was propo- sed, and is now very generally adopted. It is important to know the relative specific caloric of bodies. The most conve- nient method of discovering it is by mixing different substances together at differ- ent temperatures, and noting the temperature of the mixture ; and by experiments it appears that the same quantity of caloric imparts twice as high a temperature to mercury as to an equal quantity of water; thus, when water at 100° and mer- cury at 40O are mixed together, the mixture will be at 80°, the 20° lost by the water causing a rise of 40° in the mercury ; and when weights are substituted for meas- ures, the fact is strikingly illustrated ; for instance, on mixing a pound of mercury at 40O with a pound of water at 160O a thermometer placed in it will stand at 1550. Thus it appears that the same quantity of caloric imparts twice as high a temperature to mercur>- as to an equal volume of water, and that the heat which gives 50 to w^ater will raise an equal weight of mercury 1150, being the ratio of 1 to 23. Hence, if equal quantities of caloric be added to equal weights of water and mercury, their temperatures will be expressed in relation to each other by the 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 cal- oric as the mercury. The rule for Ending by calculation, combined with experiment, the relative capacities of different bodies, is this : Multiply the weight of each body by the number of degrees lost or gained by the mixture, and the capacities of the bodies will be inversely as the products. Or, if the bodies be mingled in unequal quantities, the capacities of the bodies will be reciprocally as the quantities of matter, multi- plied into their respective changes of temperature. The general facts respecting specific caloric are as follows : 1. Every substance has a specific heat peculiar to itself, whence a change of composition will be attended by a change of capacity for caloric. R3 198 HEAT. ^ The specific heat of a body varies with its form. A solid has a less capacity for caloric than the same substance when in the state of a liquid ; the specific heat of water, for instance, being 9 in the solid state, and 10 m the liquid. ^ 3 The specific heat of equal weights of the same gas increases as-tne density decreases ; the exact rate of increase is not known, but the ratio is less rapid than the diminution in density. . 4 Change of capacity for caloric always occasions a €^hange of temperature. In- crease in the former is attended by diminution of the latter, and vice versa. Tables of the Specific Heat of various Substances. Air Hydrogen Carbonic acid Oxygen defiant gas . 1. Air taken as unity. Equal volumes. Equal weights. 1.000 1.000 .903 12.340 1.258 .828 .976 .884 1.553 1.576 The specific heat of the foregoing compared with that of an equal quantity of water. Water Air . Oxygen . 1.000 Hydrogen. . . 3.293 . 2.669 Carbonic acid . . .221 . 2.361 defiant gas . . .420 2. Water taken as unity. . .0288 Tellurium . . .0912 . .0293 Copper . . . .0949 . .0298 Nickel . . . .1035 . .0314 Iron 1100 ■ . .0514 Cobalt . •. . .1498 . .0557 Sulphur . . . .1880 . .0927 Mercury . . . .0330 N.— If 1 lb. of coal will heat 1 lb. of water to 100©, -j^ Q) of a lb. will heat 1 lb. of mercury to lOOO. The term Capacity for heat means the relative powers of bodies, in receiving and retaining heat, in being raised to any given temperature ; while Specific applies to the actual quantity of heat so received and retained. When a body has its density increased, its capacity for heat is diminished. The rapid reduction of air to i of its volume evolves heat suflicient to inflame tmder. Table showing the relative Capacity for Heat of various Bodies. Bismuth Lead Gold Platinum Tin . Silver Zinc Illustration.- Equal weights. Equal vol. Equal weights. Equal vol. Glass .187 .448 Silver .082 .833 Iron . .126 .993 Tin . .060 — Brass .116 .971 Gold . .050 .966 Copper .114 1.027 Lead . .043 .487 Zinc . .102 — Latent Caloric is that which is insensible to the touch, or incapable of being detected by the thermometer. The quantity of heat necessary to enable ice to assume the fluid state is equal to that which would raise the temperature of the same weight of water 140° ; and an equal quantity of heat is set free from water when it assumes the solid form. Ung If 5i lbs. of water, at the temperature of 32°, be placed in a vessel, communica- ng with another one (in which water is kept constantly boihng at the tempera- HEAT. 199 ture of 2120), until the former reaches this temperature of the latter quantity, then let it be weighed, and it will be found to weigh 6^ lbs., showing that 1 lb. of water has been received in the form of steam through the communication, and reconvert- ed into water by the lower temperature in the vessel. Now this pound of water, received in the form of steaip, had, when in that form, a temperature of 2120 It is now converted into the liquid form, and still retains the same temperature of 212° ; but it has caused 5^ lbs. of water to rise from the temperature of 32° to 212^, and this without losing any temperature of itself. It follows, then, that in returning to the liquid state, it has parted with 5| times the number of degrees of temperature between 32^ and 212^ which are equal 180O and 1800x5^ = 9900. Now this lieat was combined with the steam; but as it is not sensible to a thermometer, it is called Latent. It is shown, then, that a pound of water, in passing from a liquid at 2120 to steam at 2120, receives as much heat as would be sufficient to raise it through 990 thermometric degrees, if that heat, instead of becoming latent, had been Sensible. The sum of the Sensible and Latent heat of Steam is always the same at any one temperature; thus, 990o+212° == 1202°. If to a pound of newly-fallen snow were added a pound of water at 172°, the snow would be melted, and 32^ will be the resulting temperature. Latent Heat of Steam, and several Vapours. Steam . Alcohol . Ether . Sensible Caloric is free and uncombined, passing from one sub- stance to another, affecting the senses in its passage, determining the height of the thermometer, and giving rise to all the results which are attributed to this active principle. To reduce the Degrees of a Fahrenheit Thermometer to those of Reaumur and the Centigrade. FAHRENHEIT TO REAtJMUR. Rule. — Multiply the number of degrees above or below the free2> ing point by 4, and divide by 9. Thus, 212°— 32 = 180x4 = 720-^9=80, Ans. —24°— 32= 8X4= 32-4-9=3.5,^715. FAHRENHEIT TO CENTIGRADE. Rule. — Multiply the number of degrees above or below the freez- ing point by 5, and divide by 9. Thus, 212°— 32 = 180x5 = 900-^9 = 100, Ans. Medium Heat of the globe is placed at 50° ; at the torrid zone, 75° ; at moderate climates, 50° ; near the polar regions, 36°. The extremes oi natural heat are from —70° to 120° ; of artificial heat, from —91° to 36000°. 990° Nitric acid 632° 442° Vinegar . 875° 302° Lead 610° 200 HEAT. EVAPORATION. Evaporation produces cold, because heat must be absorbed to form vapour. Evaporation proceeds only from the surface of the fluids, and therefore ochet things equal must depend lipon the extent of surface exposed. When a liquid is covered by a stratum of dry air, evaporation is rapid, even when the temperature is low. As a large quantity of caloric passes from a sensible to a latent state during the formation of vapour, it follows that cold is generated by evaporation. CONGELATION AND LIQUEFACTION. Freezing water gives out 140^ of heat. Water may be cooled to 20°. All soUds absorb heat when becoming fluid. The particular quantity of heat which renders a substance fluid is called its cal- oric of fluidity, or latent heat. The heat absorbed in liquefaction is given out again in freezing. Fluids boil in vacuo with 124° less of heat than when under the pressure of the at- .ere. On Mont Blanc water boils at 187^. DISTILLATION. Distillation is the depriving vapour of its latent heat, and, though it may be ef- fected in vacuo with verv litlle heat, no advantage in regard to a saving of fuel is gained, as the latent heat of vapour is increased in proportion to the diminution of sensible heat. Table of Effects upon Bodies by Heat. Chinese porcelain, softened . Cast iron, thoroughly smelted " " begins to melt Smith's forge, greatest heat Stone-ware, bakes Welding heat of iron (greatest) " (least) Plate glass, working heat Fine gold, melts . Fine silver, melts . Copper, melts Brass, melts . Red heat, visible by day Iron, red hot in twilight Common fire Iron, bright red in the dark Zinc, melts . Quicksilver, boils Linseed oil, boils . Lead, melts . Bismuth, melts Tin, melts Tin and bismuth, equal parts, melt . Tin 3 parts, bismuth 5, and lead 2, melt Alcohol, boils Ether, boils . Human blood (heat of) Strong^ines, freeze . Brandy, freezes . Mercury, melts Wedgewood's zero is 1077° of Fahrenheit, 130O of Fahrenheit. Wedzewood. Fahrenheit. . 1560 213570 150O 205//^ 130O 179770 125<^ 173270 102O 143370 950 134270 90O 127770 570 84870 320 52370 280 47170 270 45870 210 3807O 10770 8840 790O 7520 700O 66OO 6OOO \ * 5940 4760 ] 4420 2830 2120 1740 980 * 980 200 70 * _ —390 and each of his deg rees is equal to HEAT. 201 MISCELLANEOUS. FRIGOEIFIC MIXTURES. parts 1 Nitrate of Ammonia 1 part ) Water . . . 1 " i Phosphate of Soda 9 parts ^ Nitrate of Ammonia 6 Dilute Nitric Acid 4 Sulphate of Soda 8 parts ; Muriatic Acid . 5 " j Snow ... 2 parts > Muriate of Lime . 3 " i Snow ... 8 parts ) Dilute Sulphuric Acid 10 " ) Snow ... 3 parts ) Potash . . . 4 " i Thermometer falls, or degrees of cold produced. 460 710 5(P 530 220 830 Degrees of Fahrenheit. From + 50O to -f- 4P From +500 to— 210 From +500 to qo From— 150 to— 68O From— 680 to— 90O From 4- 320 to— 510 EFFECTS OF HEAT. Fahrenheit. Wedgewood, —900 — Greatest cold ever produced. —500 — Natural cold at Hudson's Bay. — Snow and salt, equal parts. +430 — Phosphorus burns. COO to 770 — Vinous fermentation. 780 — Acetous fermentation begins. 88O — Acetification ends. 6380 — Lowest heat of ignition of iron in the dark. 8OOO — Charcoal burns. 8490O 57 Working heat of plate glass. 143370 102 Stone ware, fired. I68O70 124 Greatest heat of plate glass. 251270 185 Greatest heat observed. EXPANSION OF SOLIDS.. At 2120, the length of the bar at 320 considered as 1.0000000. Glass Platina . Cast Iron Steel " annealed Forged Iron . Iron wire .0008545 .0009542 .0011112 .0011899 .0012200 .0012575 .0014410 Gold 0014950 Copper 0017450 Brass 0019062 Silver 0020100 Tin 0026785 Lead 0028436 Zinc 0029420 To find the expansion in Surface, double the above ; in Volume, triple them. Table of the Expansion of Air by Heat. By Mr. Dalton. Fahrenheit. Fahrenheit. Fahrenheit. 320 , 330 . 340 , 350 . 1000 , 1002 1004 1107 40O , 1021 450 1032 500 . 550 , 6OO , 650 . 70O , 750 . 1043 8OO 1055 850 1066 900 1077 1000 1089 2000 1099 2120 1110 1121 1132 1152 1354 1376 202 HEAT. MELTING POINT OF ALLOYS. Lead 2 parts, Tin 3 parts, Bismuth 5 parts, melts at . . . " 5 " melts at . . . 2120 2460 " 1 " melts at . 2860 a 2 u " 1 " melts at . 3360 Lead 2 parts, "3 || melts at . " 1 " melts at . 3340 3920 « 2 " "1 " common solder melts at . . . 4750 « 1 " " 2 " soft solder melts at . . . 36(P GUNPOWDER. 203 GUNPOWDER. PROPORTIONS OF INGREDIENTS. In the United States. Saltpetre. Charcoal. Military service .... J ^g] j4[ { 78*. 12*. Sporting i 77. 13. In England. Military service ..... 75. 15. o .• S 78. 14. Sporting J 75. 17. In France. Military service 75. 12.5 o -*• S 78. 12. Sporting J 76. 14. Blasting 62. 18. Sulphur. 10. 10. 10. 10. 10. 8. 8. 12.5 10. 10. 20. GRANULATION. Diameter of sieve holes for Cannon powder . . .070 to .100 inches Musket " . . .050 " .070 " " " Rifle " . . .025 " .035 " DENSITY OF POWDER. Size of Grain. Specific Gravity. Number of grains in 10 troy grains. Weight of 1 cubic foot Cubic Loose. Shaken. lb. loose. "Cannon Musket Rifle . . Sporting 1.630 1.538 1.535 1.800 350 700 16.000 35.000 oz. 922 900 860 885 oz. 1.000 990 960 1.035 30 31 32 31 To find how much Powder will fill a Shell. Multiply the cube of the interior diameter in inches by .01744. Example. — How much powder will fill a shell, the internal diam- eter being 9 inches '? 93 x.01744= 12.71 lbs., ^715. DIMENSIONS OF POWDER BARRELS. Whole length " . . 20.5 inches. Length, interior in the clear 18. " Interior diameter at the head 14. *' " " at the bilge 16. " Thickness of staves and heads Weight of barrels about Proof of Powder.— One oz. with a 24 lb. ball, at any one time, must not be less than 250 yards ; but none ranging below 225 yards is received. Powder in magazines that does not range over 180 yards is considered unservice- able. Good powder averages from 280 to 300 yards ; small grain from 300 to 320 yards. The greatest initial velocity is obtained by powder of great specific gravity and of very coarse grain, giving 130 grains to 10 grains troy. 25 lbs. The mean range of new, proved 204) LIGHT TONNAGE. LIGHT. Light is similar to caloric in many of its qualities, bei^g emitted in the form of rays, and subject to the same laws of reflection. It is of two kinds, J^atural and Artificial; the one proceeding from the sun and stars, the other from heated bodies. Solids shine in the dark only when heated from 600^ to 700^ and in daylight when the temperature reaches iOOOO. Relative intensity of light from the burning of various bodies is, for wax, 101 parts ; tallow, 100 ; oil in an Argand lamp, 110 ; in a common lamp, 129 ; and an ill-snuffed candle, 229. By experiments on coal gas, it appears that above 20 cubic feet are required to produce light equal in duration and in illuminating powers to a pound of tallow candles, six to a pound, set up and burned out one after the other. In distilling 56 lbs. coal, the quantity of gas produced in cubic feet when the dis- tillation was effected in 3 hours was 41.3, in 7 hours 37.5. in 20 hours 33.5, and in 25 hours 31.7. TONNAGE. By a law of Congress, the tonnage of vessels is found as follows : FOR A DOUBLE-DECKED. Take the length from the fore part of the stem to the after side of the sternpost above the upper deck ; the breadth at the broadest part above the main wales ; half of this breadth must be taken Is the depth of the vessel ; then deduct from the length § of the breadth, multiply the remainder by the breadth, and the product by the depth ; divide this last product by 95, and the quotient is the tonnage. Example. — What is the tonnage of a ship of the line, measuring, as above, 210 feet on deck, and 59 feet in breath ^ 59-1-2 = 29.5, depth. 210 — fof 69 = 174.6X59X29. 5-^95 = 3198.8 tons. FOR A SINGLE-DECKED. Take the length and breadth as above directed for a double-deck- ed, and deduct from the length § of the breadth ; take the depth from the under side of the deck-plank to the ceiling of the hold ; then proceed as before. Example.— -The length of a vessel is (as above) 223 feet, the breadth 39i feet, and the depth of hold 23^ feet ; what is the ton- nage '? 223— f of 39.5 =- 199.3 X 39.5 x23.5-^95 ==1947.3 tons. A ton will stow 3^ bales cotton. Note.— The burden of similar ships are to each other as the cubes of their like dimensions. TONNAGE. 205 CARPENTERS' MEASUREMENT. FOR A SINGLE-DECKED. Multiply the length of keel, the breadth of beam, and the depth of the hold together, and divide by 95. FOR A DOUBLE-DECKED. Multiply as above, taking half the breadth of Ij^am for the depth of the hold, and divide by 95. To find the Tonnage of English Vessels. Rule. — Divide the length of the upper deck between the afterpart of the stem and the forepart of the sternpost into 6 equal parts, and note the foremost, middle, and aftermost points of division. Measure the depths at these three points in feet, and tenths of a foot, also the depths from the under side of the upper deck to the ceiling at the limber strake : or, in case of a break in the upper deck, from a line stretched in continuation of the deck. For the breadths, divide each depth into 5 equal parts, and measure the inside breadths at the following points, viz. : at \ and at I from the upper deck of the foremost and aftermost depths, and at | and f from the upper deck of the midship depth. Take the length, at half the midship depth, from the afterpart of the stem to the forepart of the ^sternpost. Then, to twice the midship depth, add the foremost and aftermost depths for the 5^771 of the depths ; and add together the foremost upper and lower breadths, 3 times the upper breadth with the lower breadth at tlie midship, and the upper and twice the lower breadth at the after division for the sum of the breadths. Multiply together the sum of the depths, the sum of the breadths, and the length, and divide the product by 3500, which will give the number of tons, or register. If the vessel have a poop or half-deck, or a break in the upper deck, measure the in^e mean length, breadth, and height of such part thereof as may be included within the bulkhead ; multiply these three measurements together, and divide the product by 92r4. The quotient will be the number of tons to be added to the result as above found. For Open 'Vessels. The depths are to be taken from the upper edge of the upper strake. ^ For Steam Vessels. The tonnage due to the engine-room is deducted from the total tonnage calculated by the above rule. To determine this, measure the inside length of the engine-room from the fore- most to the aftermost bulkhead ; then multiply this length by the midship depth of the vessel, and the product by the inside midship breadth at 0.40 of the depth from the deck, and divide the final product by 92.4. S 206 PILING OF BALLS AND SHELLS. PILING OF BALLS AND SHELLS. To find the Number of Balls in a Triangular Pile. Rule.— Multiply continually together the number of balls in one side of the bottom row, and that number increased by 1 ; also, the same number increased by 2 ; \- of the product will be the answer. Example.— Wh^t is the number of balls in a pile, each side of the base containing 30 balls ] • 30x31x32-^-6 = 4960, Arts. To find the Number of Balls in a Square Pile. Rule.— Multiply continually together the number in one side of the bottom course, that number increased by 1, and double the same number increased by 1 ; ^ of the product will be the answer. Example.— How many balls are there in a pile of 30 rows 1 30 X 31 X 61-^6 = 9455, Arts. To find the Number of Balls in an Oblong Pile. Rule.— From 3 times the number in the length of the base row subtract One less than the breadth of the same ; multiply the re- mainder by the same breadth, and the product by one more than the same, and divide by 6. Example.— Required the number of balls in an oblong pile, the numbers in the base r ow b eing 16 an d 7 ] 16x3— 7^X7x7-t-l~6=:392, Ans. To find the Number of Balls in an Incomplete Pile. Rule.— From the number in the pile, considered as complete, subtract the number conceived to be in the upper pile which is want- ing. WEIGHT AND DIMENSIONS OF BALLS AND SHELLS. 207 WEIGHT AND DIMENSIONS OF BALLS AND SHELLS. The weights of these may be found by the rules in Mensuration ; also, in the tables, pages 233, 236, and 255. To find the Weight of an Iron Ball from its Diameter. An iron ball of 4 inches diameter weighs 8.736 lbs. Therefore, ^ of the cube of the diameter is the weight, for the weight ol spheres is as the cubes of the diameters. Example. — What is the weight of a ball 10 inches in diameter 1 !^ of 102 = 136.5 lbs., ^/25. To find the Diameter from the Weight. Example. — What is the diameter of an iron ball, its weight being 99.5 lbs. ] v^sli ^ ^^-^ — ^ inches, Ans. Or, multiply the cube of the diameter in inches by .1365, and the sum is the weight. And divide the weight in pounds by .1365, and the cube root of the product is the diameter. To find the Weight of a Leaden Ball. A leaden ball of 4 inches diameter weighs 13.744 lbs. Therefore, ^^^ of the diameter is the weight. Example. — What is the weight of a leaden ball 10 inches in di- ameter 1 ^^ of 103 ^ 214.7 lbs., Ans. Inversely, v^ j^ X weight = diameter. Or, multiply the cube of the diameter in inches by .2147, and the sum is the weight. And divide the weight in pounds by .2147, and. the cube root of the product is the diameter. To find the Weight of a Cast Iron Shell. Multiply the difference of the cubes of the exterior and interior diameter in incheS by .1365. Example. — What is the weight of a shell having 10 and 8.50 inch- es for its diameters "? 103— 8.53 X. 1365 = 52.6 lbs., ^7w. 208 WINDING ENGINES. WINDING ENGINES. In winding engines, for drawing coals, water, &c., out of a pit : where it is wanted to give a certain number of revolutions, it is ne- cessary to know the diameter of the drum and the thickness of the rope. Where flat ropes are used, and are wound one part over the other, To find the Diameter of the Drum. Rule.— Multiply the depth of the pit in inches by the thickness of the rope in inches for a dividend. Multiply the number of revolutions by 3.1416, and the product by the thickness of the rope in inches for a divisor. Divide the one by the other, and from the quotient subtract the product of the thickness of the rope and the num.ber of revolutions ; the remainder is the diameter in inches. Example.— If an engine make 20 revolutions, the depth of the pit being 600 feet, and the thickness of the rope 1 inch, what is the di- ameter of the drumj 600X12X1-^20X3.1416X1— IX 20 = 94.5 inches, Ans. To find the Diameter of the Roll. Rule.— To the area of the drum add the area or edge surface of the rope, and the diameter of the circle having that area is the di- ameter of the roll. Example.— What is the diameter of the roll in the preceding ex- ample % Area of 94.5 = 7013.8+ area of 7200 X 1 = 14213.8, and y/ of this sum H-.7854 — 134.5, Ans. Or, the radius of the drum is increased the number of the revo- lutions, multiplied by the thickness of the rope ; as, -^-f-20xl = 67.25. To find the Number of Revolutions. , Rule. — To the area of the drum add the area of the edge surface of the rope ; then find by inspection in the table of areas, or by cal- culation, if necessary, the diameter that gives the exact area; sub- tract the diameter of the drum from this, and divide the remainder by twice the thickness of the rope ; the quotient is the number of revolutions. Example.— The length of a rope is 2600 inches, its thickness 1 inch, and the diameter of the drum 20 inches. Required the num- ber of revolutions. Area of 20 + area of rope =314.15+2600 = 2914.15, the diame- ter of which is 60.91, and 60.91— 20-MX2 = 20.45 revolutions. FRAUDULENT BALANCES. 209 To find the Place of Meeting of the Ascending and Descending Buckets when two or more are used. Meetings will always be below half the depth of the pit, and To find this Depth, Take the circumference of the druni for the length of the first turn ; then, to the diameter of the drum add twice the thickness of the rope, multiplied by the number of revolutions, less 1, for a diameter, and the circumference of this diameter is the length of the last turn ; add these two lengths together, multiply their sum by half the num- ber of revolutions, and the product is the depth of the pit. Example. — The diameter of a drum is 9 feet, the thickness of the rope 1 inch, and the revolutions 20 ; what is the depth of the pit, and at what distance from the top will buckets meet % 9x3.1416 =28.27, length of first turn; 2 V 1 v2n 1 9+ = 12.166X3.1416 = 38.23, length of last turn ; 20 28.27+38.23 X— = 665 feet, or depth of pit. 2 At 10 revolutions the buckets will meet. Therefore, add 9 times twice the thickness of the rope to the diameter of the drum ; to the circumference of this diameter add the length of the first turn, multiply their sum by half the number of turns to meetings, and the product is the distance from the bottom of the pit at which the buckets will meet. Q V 1 v2 10 —iiir — 1.5+9x3.1416+28.27x- =306.25 feet, ^715. FRAUDULENT BALANCES. In order to detect them, after an equilibrium has been established between the weight and the article weighed, transpose them, and the weight will preponderate if the article weighed is lighter than the weight, and contrariwise. Then, To ascertain the True Weighty Let the weight which will produce equilibrium after transposition be found, and with the former weight be reduced to the same de- nomination of weight ; and let the two weights thus expressed be multiplied together, and the square root of the product will be the true weight. Example. — If one weight be 7 lbs., and the other 91, 7x91 = 64, and the square root of 64 is 8 ; hence 8 lbs. is the true weight. Or, let a = length of longest arm, 1 A = greatest weight, h = length of shortest arm, I B = least weight. Then Wa = Ab, and W6 = Ba ; multiplying these two equations, we have W^aJ =r ABa^ or W^ = AB, and W = ^AB. S2 210 MEASURING OF TIMBER. MEASURING OF TIMBER. Sawed or hewn timber is measured by the cubic foot. The unit of board measure is a superficial foot 1 inch thick. To measure Round Timber. Multiply the length in inches by the square of \ the mean girth in inches, and the product, divided by 1728, will give the contents in cubic feet. When the length is given in feet, and the girth in inches, divide by 144. When all the dimensions are in feet, the product is the content without a division. Or, ^^^^ -^144, L the length in feet, and C half the sum of the 16 circumferences of the two ends in inches. Or, ascertain the contents by the rules in Mensuration of Solids, page 82, and multiply by .75734. Example.— The girths of a piece of timber are 31.416 and 62.832 inches, and its length 50 feet ; required its contents. 31.416+62.832_^^^^^^g ^^^ 11.7812x50-^-144 = 48.1916 cu- 2 bic feet, Ans. Or, ^^^^^-^^^^144=: 48.1916 cubic feet. Or, 103— 203-20— 10X.7854X^= 63.632X.75734 = 48.1916 o cubic feet, Ans. To measure Square Timber. Multiply the length in inches by the breadth in inches, and the product by the depth in feet ; divide by 144, and the quotient is the content. Note.— When all the dimensions are in feet, omit the divisor of 144. BOARD MEASURE. Multiply the length by the breadth, and the product is the content. Note.— This rule only applies when all the dimensions are in feet. When either the length or breadth are given in inches, divide their product by 12; and when all the dimensions are in inches, divide it by 144. Pine spars, from 10 to 4i inches in diameter inclusive, and spruce spars, are to be measured^by taking the diameter, clear of bark, at J of their length from the large end. Spars are usually purchased by the inch diameter ; all under 4 inches are considered poles. Spruce spars of 7 inches and less should have 5 feet in length for every inch diameter. Those above 7 inches should have 4 feet in length for every inch diameter. STEAM. 211 STEAM. • Steam, aris/.ng from water at the boiling point, is equal to the pressure of the atmosphere, which is in round numbers 15 lbs. on the square inch. Table of the Expansive Force of Steam, from 212° to 352i°. (From experiments of Committee of Franklin Institute.) The unit is the atmospheric pressure, 30 inches of mercury. Degrees of heat. Pressure. Decrees of heat. Pressure. Degrees of heat. Pressure. 212.0 1. 298.50 4.5 331. o 7.5 235.0 1.5 304.50 5. 336.0 8. 250.O 2. 310.O 5.5 340.50 8.5 264.0 2.5 315.50 6. 345.0 9. 275.0 3. 321.0 6.5 349.0 9.5 284.0 3.5 326.0 7. 352.0 10. 291.50 4. Under the pressure of the atmosphere alone, water cannot be heated above the boiling point. It has already been stated (see Heat) that the sum of sensible and latent heats is 1202O, and that 140O of sensible heat becomes latent upon the liquefaction of ice ; also, that 1 lb. of water converted into steam at 2120 will heat 5^ lbs. of water at 320 to 2120, and that the sum is 6^ lbs. of water. Table of the Volume of Air and Force of Vapour. Temperature. Volume of air or vapour. Force of vapour in inches of mer- cury. Temperature. Volume of air or vapour. Force of vapour in inches of mer- cury. OO 320 520 720 920 1120 1000 1071 1123 1183 1255 1354 .032 .172 .401 .842 1.629 2.950 1320 1520 1720 1920 2120 1491 1689 1930 2287 2672 5.070 8.330 13.170 20.160 30. To ascertain the Number of Cubic Inches of Water, at any Given Temperature, that must be mixed ivith a Cubic Inch of Steam to reduce the Mixture to any Required Temperature. Rule. — From the required temperature subtract the temperature of the water ; then find how often the remainder is contained in the given temperature, subtracted from 1202^, and the quotient is the answer. Example.— The temperature of the condensing water of an engine is 80°, and the required temperature lOOO ; what is the proportion of condensing water to that evaporated ? 100— 80-8-1202— 100 = -gQ- = 55.5, Ans, 212 STEAM. Again, the temperature is 60^, and the required temperature KXP. 1202— 100-f-(100— 60) = ^ = 27.5, ^ns. Or, let w represent temperature of condensing water, t the required teraperaturey and h the sum of sensible and latent heats. h — t Then = water required. t — w To ascertain the Quantity of Steam required to raise a Given Quantity of Water to any Given Temperature. Rule. — Multiply the water to be warmed by the difference of temperature be- tween the cold water and that to which it is to be raised, for a dividend ; then to the temperature of the steam add 990^, and from that sum take the required tem- perature of the water for a divisor ; the quotient is the quantity of steam in the same terms as the water. Example. — What quantity of steam at 212^ will raise 100 cubic feet of water at 80O to 2120 T: 100x2120 — 80 o 9o_j_QQno— o 1 oo ~ •'^^•^ c\i^i\c feet of water formed into steam, occupying (13.3X 1694) 22586.6 cubic feet of space. Table of the Boiling Points corresponding to the Altitudes of the Barometer between 26 and 31 Inches. Barometer. Boiling point. Barometer. Boiling point. Barometer. Boiling point. 26. 26.5 27. 27.5 204.91O 205.790 206.67O 207.550 28. 28.5 29. 29.5 2O8.43O 209.31O 210.190 211.070 30. 30.5 31. 212.0 212.880 213.760 A cubic inch of water, evaporated under the ordinary atmospheric pressure, is converted into 1694 cubic inches of steam, or, in round numbers, 1 cubic foot, and gives a mechanical force equal to the raising of 2200 lbs. 1 foot high. The Pressure of Steam being given, to find its Temperature. Rule. — Multiply the 6th* root of the pressure in inches by 177, and subtract 100 from the product. Example. — If the pressure is 240 inches of mercury, what is the temperature 1 6/240 = 2.493X177—100=341.61, .dns. For sea water, multiply by 177.6 ; when -^^ saturated, by 178.3 ; and when ^^ saturated, by 179. Table of the Density of Steam under different Pressures. Atmospheres. Density. Volume. Atmospheres. Density. Volume. 1 .00059 1694 10 .00492 203 2 .00110 909 12 .00581 172 3 .00160 625 14 .00670 149 4 .00210 476 16 .00760 131 5 .00258 387 18 .00849 117 6 .00306 326 20 .00937 106 8 .00399 250 The volumes are not direct, in consequence of the increase of heat. See obser- vations, page 198. * See page 118 for rule to find this root. STEAM. 213 Table of the Expansive Force of Steam in Atmospheres. Temperature. Pressure in atmospheres. Temperature. Pressure in atmospheres. Temperature. Pressure in atmospheres. 212.0 1 331.20 7 413.80 19 242P H 341. 80 8 4I8.50 20 250.60 2 350.80 9 423.0 21 264.0 2i 359.0 10 427.30 22 2T7.20 3 366.80 n 431.40 23 285.20 3^ 374.0 12 435.60 24 293.80 4 380.60 13 438.70 25 301. o 4^ 387.0 14 457.20 30 308.O 5 392.60 15 472.80 35 314.40 5.V 398.50 16 486.60 40 320.40 6 403.80 17 499.10 45 326.30 6i 409.O 18 510.60 50 Note. — This table gives results slightly differing from that furnished by the Franklin Institute, being about 3.5^ for every 5 atmospheres. Table of the Pressure, Specific Gravity, and Weight of a Cubic Foot of Steam at different Temperatures. Pressure in ins. "Weight of a cub. Spec, gravity, Pressure in ins. Weight of a cub. Spec, gravity, of mercury. foot in grains. air being 1. of mercury. foot in grains. air being 1. .55 1. 2. 3. 4. 7.5 15. 22.5 30. 35. 45. 52.5 60. 6.10 10.70 20.50 30. 39. 71. 135. 196. 254.70 292. 363. 427. 483. .0115 .0202 .0388 .0568 .0744 .134 .255 .371 .484 .553 .687 .810 .915 75. 90. 105. 120. 150. 180. 210. 240. 270. 300. 600. 900. 1200. 593.50 1.123 700. 1.33 810. 1.53 910. 1.728 1110. 2.12 1317. 2.5 1520. 2.88 1660. 3.25 1910. 3.61 2100. 3.97 3940. 7.44 5670. 10.75 7350. 13.88 A pressure of 1, 5, 10, 20, 40, and 50 lbs. on a square inch, will raise a mercurial gauge respectively 1.01, 5.08, 10.16, 20.32, 40.65, and 50.80 inches. The mean is 1.0159 inches. A column of mercury 2 inches in height will counterbalance a pressure of .98 lbs. on a square inch. The practical estimate of the velocity of steam, when flowing into a vacuum, is about 1400 feet in a second when at an expansive power equal to the atmosphere ; and when at 20 atmospheres, the velocity is increased but to 1600 feet. And when flowing into the air under a similar power, about 650 feet per second, increasing to 1600 feet for a pressure of 20 atmospheres. The elasticity of the vapour of spirit of wine, at all temperatures, is equal to 2.125 times that of steam. 214. STEAM. STEAM ACTING EXPANSIVELY. To find the Mean Pressure of the Steam on the Piston. Rule.— Divide the length of the stroke, added to the clearance in- the cylinder at one end, by the length of the stroke at which the steam is cut off, added to the clearance, and the quotient will ex- press the relative expansion it undergoes. Find in the following table, in the column of expansion, a number corresponding to this ; take out the multiplier opposite to it, and multiply it into the full pressure of the steam per square inch as it enters the cylinder. Expansion. Table showing the Mean Pressure of Steam Multiplier. Expansion, i Multiplier. Expansion. I.O 1.000 3.4 .654 5.8 .479 1.1 .995 3.5 .644 5.9 .474 1.2 .985 3.6 .634 6. .470 . 1.3 .971 3.7 .624 6.1 .466 1.4 .955 3.8 .615 6.2 .462 1.5 .937 3.9 .605 6.3 .458 1.6 .919 4. .597 6.4 .454 1.7 .900 4.1 .588 6.5 .450 1.8 .882 4.2 .580 6.6 .446 1.9 .864 4.3 .572 6.7 .442 2. .847 4.4 .564 6.8 .438 2.1 .830 4.5 .556 6.9 .434 2.2 .813 4.6 .549 7. .430 2.3 .797 4.7 .542 7.1 .427 2.4 .781 4.8 .535 7.2 .423 2.5 .766 4.9 .528 7.3 .420 2.6 .752 5. .522 7.4 .417 2.7 .738 5.1 !516 7.5 .414 2.8 .725 5.2 .510 7.6 .411 2.9 .712 5.3 .504 7.7 .408 3. .700 5.4 .499 7.8 .405 3.1 .688 5.5 .494 7.9 .402 3.2 .676 5.6 .489 8. .399 3.3 .665 5.7 .484 Multiplier* Example.— Suppose the steam to enter the cylinder at a pressure of 20 lbs. per square inch, and to be cut off at :i the length of the stroke of the piston. The stroke being 8 feet, 8 feet = 96 inches + 1 for clearance := 97, i = 24 inches -|- 1 *' ■= 25. Then 97-^25 = 3.88, the relative expansion which falls between 3.8 and 3.9. Referring to the table, the multiplier for 3.8 is .615, and the difference between that and the multiplier for 3.9 is .010. Hence, multiplying .010 by .8, and subtracting the product .008 from .615, the remainder, .607, is the multiplier for 3.88. Therefore, .607 .X20 lbs. = 12 140 lbs. ix?r square inch, the mean effective pressure of the piston required. Specific gravity of steam at the pressure of the atmosphere .490, air being 1. STEAM. 215 FOR WARMING APARTMENTS. Every cubic foot of water evaporated in a boiler at the pressure of the atmo- sphere will heat 2000 feet of enclosed air to an average temperature of 750,Tnd each square foot of surface of steam-pipe will warm 200 cubic feet of space. The force of steam is the same at the boiling point for every fluid. LOSS BY RADIATION. To ascertain the Loss of Heat per Square Foot in a Second. """' T = feTerS oltK^'^^ ^^' ''' ^ '^'^ ^^^ ^^^^ '' '^^ «^--' I = length of the pipe in feet, d = diameter in inches, V = velocity in feet per second, R = radiation in degrees of heat. l.7l(T-t)__ ^ ^» * Tredgold, 216 STEAM-ENGINE. STEAM-ENGINE. It is not consistent with the plan of this work to enter fully into details of the steam-engine, and this article will be confined exclu- sively to some practical rules, the utility of which have been tested and their use adopted. CONDENSING ENGINES. Cylinder. The thickness of the metal is found by the following formula : — — ^x -i-T = thickness in inches, P representing pres- 10000 d—2.5^'' ' ^ sure of steam in lbs., and d diameter of cylinder. For cylinders over 30 inches diameter, divide by 9000 ; over 40 inches, by 8000 ; over 50 inches, by 6000 ; and over 60 inches, by 5000. Condenser, The capacity of it should be i that of the cylinder. Air-pump. The capacity of it should be -J that of the cylinder. Steam and Exhaust Valves. Their diameter should give an area of 10 square inches for every 10000 cubic inches contained in the cylinder, and should lift J their diameter. Foot and Delivery Valves. Their dimensions should give an area of-^Q that of the airpump. Force Pumps. Their capacity 'should be yl^ to yi^ that of the cylinder. Injection Cocks. Their area should be sufficient to supply 70 times the quantity of water evaporated when the engine is working at its maximum, and in marine engines there should be three of them to > each condenser, viz., a Side, Bottom, and Bilge. The Side injection should have yV of an inch diameter of pipe forr every inch diameter of cylinder, the Bottom injection should have -j^* of an inch diameter of pipe for every inch diameter of cyhnder ; ; the Bilge injection is usually a branch of the Bottom injection pipe, , and may be of less capacity. Piston Rod. Its diameter should be ^ that of the cylinder. Beam. Its length from centres should be twice the stroke of the piston, and its depth -^ of its length. The strap at its smallest di- mensions should have at least y^^ the area of the piston rod, and its depth equal half of its breadth. * The proportion here given will admit of a sufficient quantity of water when the- engine is in operation in the Gulf Stream, where the water is at times at the tem- perature of 84°, and the quantity of water (wlien the steam is at 10 lbs. pressure) required to give it and the water of condensation a temperature of 100^, is 70 times ^hat of the quantity evaporatetl STEAM-ENGINE. 217 Beam Centres, The end centres should have each one, and the mam centre two and a half times the area of the piston rod The proportion for the strap, is when the depth of the beam is J that of the stroke ; consequently, when the depth is less, the area must be increased. Connecting Rods. Their diameter in the neck should be the same as that of the piston rod. The diameter of the centre of the bodv IS found in the following manner ; As .75 the stroke of the piston is to the length of the body of the rod so IS the area of the neck to the area of the centre of the body \\ hen two rods are used, each diameter should be JL that of the piston rod. ^ ^ u. ^^ti ■ ^°^'\ = '^^^" "''''* *''°"'' to afford 1 lb. of oxygen ; and making a due allowance for loss, nearly 90 cubic feet of air will be required in the furnace of a boiler for each lb. of oxygen. The quantity of air and smoke for one cubic foot of water converted into steam at 220O is, for coal about 2000, and for hard wood about 4000 cubic feet. Table showing the Results of Mr. BulVs Experiments upon Wood. Woods. Weight of a cord. Compara- tive value per rord. Woods. Weight of a cord. Compara- tive value per cord. Shell -bark Hickory Pig-nut Hickory . Red-heart Hickory White Oak . . . Red Oak . . . . Lbs. . 4469 4241 3705 3821 3254 100 95 81 81 69 Hard Maple . . . Jersey Pine . . . Yellow Pine . . . White Pine . . . Lbs. 2878 2137 1904 1868 60 54 43 42 Pounds of Ice melted by the following Fuels : Good coal Coke . Charcoal 90 94 95 Wood (hard) Peat . Hydrogen gas 92 19 370 When bituminous coal is subjected to destructive distillation, about § of its weight is left, in the form of coke. Kelative Value oj Seasoned oak f the pit 125 owing riuls by Weight: Charcoal . . . . 285 " " artificially 140 Peat 115 Hickory 137 Welsh coal . 312 White pine . 137 Newcastle *' . . . 309 Yellow pine 145 Belgium " . . . 316 Good coke . 285 Anthracite, French 290 Inferior *' . . . 222 " Pennsylvamia . 250 ANALYSIS OF FUELS. Carbon . Hydrogen Nitrogen . Oxygen . Volatile matter Charcoal Ashes fewcastle Coal, cakin? kind. 75.28 4.18 15.96 4.58 Cannel Coal. 64.72 21.56 13.72 0.00 100. Ash. Maple. 81.3 79.3 17.9 20. .7 .7 Cumberland Coal, American. 80. Bitumen, 18.40 Ash, 1.60 100. Oak. 76.9 22.7 .4 100. Chestnut. Norway Pine. 76.3 80.4 23.3 19.2 .4 .4 An increase in the rapidity of combustion is accompanied by a diminution in the evaporative efficiency of the combustible. 226 COMBUSTION. ANTHRACITE COAL. The results of late and accurate observations upon the burning of anthracite coal, with the aid of a blast, gives an expenditure of 5 lbs. per horse power per hour. The best anthracites contain about 95 per cent, of inflammable matter, principally carbon. 1.84 tons coal are required for the smelting and heating of the blast to make 1 ton pig iron. 578304 cubic feet of air are required for the blast to make 1 ton of iron. CHARCOAL. The best quality is made from oak, maple, beech, and chestnut. Wood will furnish, when properly burned, about 16 per cent, of coal. A bushel of coal from hard wood weighs about 30 Ifts., and from pine 29 lbs. COKE. Sixty* bushels Newcastle coals (lumps) will make 92 bushels good coke, and 60 bushel's (fine) will make 85 bushels of a similar quality. 60 bushels Newcastle and Picton coal (one half of each) makes 84 bushels infe- rior ; 60 bushels Picton, or Virginia coal, makes 75 bushels of bad. A bushel of the best coke weighs 32 lbs. Coal furnishes 60 to 70 per cent, of coke by weight. 1 lb. in a common locomotive boiler will evaporate 7^ lbs. water at 212° into MISCELLANEOUS. One pound of anthracite coal in a cupola furnace will melt 5 lbs. of cast iron ; 80 bushels bituminous coal in an air furnace will melt 10 tons cast iron. When one bushel bituminous coal per hour will produce steam of the expansive force of 15 lbs. per square inch, 1* bushels will give 50 lbs., and 2 bushels 120 lbs. One lb. of Newcastle coal converts 7 lbs. of boiling water into steam, and the time, 6 times that necessary to raise it from the freezing to the boiling point. And a bushel will convert 10 cubic feet of water into steam. * Winchester bushel = 2150.42 cubic inches. WATER. 227 WATER. Fresh Water. The constitution of it by weight and measure is, By weight. By measure. Oxygen 88.9 1 Hydrogen 11.1 2 One cubic inch at 62°, the barometer at 30 inches, weighs 252.458 grains, and it is 830.1 times heavier than atmospheric air. A cubic foot weighs 1000 ounces, or 62 i lbs. avoirdupois ; a col- umn 1 inch square and 1 foot high weighs .434028 lbs. It expands i of its bulk in freezing, and averages .0002517, or ^-X^ for every degree of heat from 40° to 212°. Maximum density, 39.38°. Table of Expansion at different Temperatures. Temperature. Expansion. Temperature. Expansion. 12° 22° 32° 40° .00236 .00090 .00022 .00000 64^ 102° 212° .00159 .00791 .04330 Showing an increase in bulk from 40° t-o 212° of 5^^, equal to 1 cubic foot in every 23.09 feet. The height of a column of ( ] ^\- P^^ ^9^^^^ ^n^^' ^^ 2.31 feet, water at 60°, equivalent to the ( ] ' circular " - 2.94 - pressure of ) 1 mch of mercury, "1.133*^ \ the atmosphere . *' 34. River or canal water contains ^ ^ of its volume of gaseous mat- Spring or well water " __i_ i ter. A cubic inch weighs .03611 of a lb., and at 212° has a force of 29.56 inches mercury. Sea Water, according to the analysis of Dr. Murray, at the spe- cific gravity of 1.029, contains. Muriate of soda Sulphate of soda Muriate of magnesia Muriate of lime 220.01 = -1^ 33.16=^ 1 23? "2T^ 303.09=: 3L 42.08 : 7.84 = - Table showing the Deposites that take place at different Degrees of Saturation and Temperature. When 1000 parts were reduced by evaporation. Quantity of sea water. 1000 299 102 Boiling point. 214° 217° 228° Salt in 100 parts. 3. 10. 29.5 Nature of deposi te. None. Sulphate of lime. Common salt. 228 WATER* Boiling Point at different Degrees of Saturation. Proportion of salt in 100 parts by weight. Saturated ) solution 5 36.37 33.34 30.30 27.28 24.25 21.22 Boiling point. 226.° 224.9° 223.7^ 222-5° 221.4° 220.2° Proportion of salt in 100 pans by weight. 18.18 15.15 12.12 9.09 6.06 3.03 Sea water \ Boiling point. 219.0 217.9° 216.7° 215.5° 214.4° 213.2° Salt Water. A cubic foot of it weighs 64.3 lbs. ; a cubic inch, .03721 lbs. The height of a column of j water at 60°, equivalent to the pressure of . (Specific gravity, 1029). I 1 lb. per square inch, is 2.37 feet, 1 " " circular " " 3.02 " 1 inch of mercury, " 1.165" the atmosphere . ''34.98 " MOTION OF BODIES IN FLUIDS. 229 MOTION OF BODIES IN FLUIDS. Table of the Weights required to give different Velocities to several different Figures. The diameter of all the figures but the small hemisphere is 6.375 inches, and the altitude of the cone 6.625 inches. The small hemisphere is 4.75 inches. The angle of the side of the coae and its axis is, consequently, 25° 42' nearly. Velocity Cone. Whole globe. Cylinder. Hennisphere. Small hem- per second. Vertex. Base. Flat. Round. isphere. feet. oz. oz. oz. oz. oz. oz. oz. 3 .028 .064 .027 .050 .051 .020 .028 4 .048 .109 .047 .090 .096 .039 .048 5 .071 .162 .068 .143 .148 .063 .072 6 .098 *.225 .094 .205 .211 .092 .103 7 .129 .298 .125 .278 .284 .123 .141 8 .168 .382 .162 .360 .368 .160 .184 9 .211 .478 .205 .456 .464 .199 .233 10 .260 .587 .255 .565 .573 .242 .287 12 .376 .850 .370 .826 .836 .347 .418 15 .589 1.346 .581 1.327 1.336 .552 .661 16 .675 1.546 .663 1.526 1.538 .634 .754 20 1.069 2.540 1.057 2.528 2.542 1.033 1.196 Propor. number 126 291 124 285 288 119 140 From this table several practical inferences may be drawn. 1. That the resistance is nearly as the surface, the resistance in- creasing but a very little above that proportion in the greater sur- faces. 2. The resistance to the same surface is nearly as the square of the velocity, but gradually increasing more and more above that proportion as the velocity increases. 3. When the hinder parts of bodies are of different forms, the re- sistances are different, though the fore parts be alike. 4. The resistance on the base of the hemisphere is to that on the convex side nearly as 2.4 to 1, instead of 2 to 1, as the theory as- signs the proportion. 5. The resistance on the base* of the cone is to that on the vertex nearly as 2.3 to 1. And in the same ratio is radius to the sine of the angle of the inclination of the side of the cone to its path or axis. So that, in this instance, the resistance is directly as the sine of the * This is a complete refutation of the popular assertion, that a taper spar will tow in water easiest when the base is foremost. u 230 MOTION OF BODIES IN FLUIDS. angle of incidence, the transverse section being the same, instead of the square of the sine. 6. Hence we can find the altitude of a column of air, the pressure of which shall be equal to the resistance of a body moving through it with any velocity. Thus, let a = the area of the section of the body, similar to any of tliose in the table, perpendicular to the direction of motion, R = the resistance to the velocity, in the table, and X = the altitude sought, of a column of air whose base is a and its pressure R. Then ax = the contents of the columns in feet, and 1.2 ax, or ^ ax its weight in ounces. R Therefore, 6 ^ x = R, and a; = #X~ is the altitude sought in feet, namely, 5 of the ^ ° a 6 quotient of the resistance of any body divided by its transverse section, which is a constant quantity for all similar bodies, however different in magnitude, since the resistance R is as the section a, as by article 1. When a = |- of a foot, as in all the figures in the foregoing table except the small R It; hemisphere, then x = | X — , becomes x = — R, where R is the resistance in the Q a ^ table, to the similar body. If, for example, we take the convex side of the large hemisphere, whose resist- ance is .634, or at a velocity of 16 feet per second, then R = .634, and x = — R = 2.3775 feet, is the altitude of the column of air whose pressure is equal to the re- sistance on a spherical surface, with a velocity of 16 feet. And to compare the above altitude with that which is due to the given velocity, it will be 322 ^iqz. . iq . 4^ t^e altitude due to the velocity 16, which is near double the altitude that is equal the pressure. And as the altitude is proportional to the square of the velocity^ therefore, in small velocities the resistance to any spheri- cal surface is equal to the pressure of a column of air on its great circle, whose al- titude is ^, or .594 of the altitude due to its velocity. But if the cj^linder be taken, where resistance R = 1.526, then x=: -^Rr=5.72, which exceeds the height 4, due to the velocity, in the ratio of 23 to 16 nearly. And the difference would be still greater if the body were larger, and also if the velocity were more. If any body move through a fluid at rest, or the fluid move against the body at rest, the force or resistance of the fluid against the body will be as the square of the velocity and the density of the fluid ; that is, R:=zdv^. For the force or resistance is as the quantity of matter or particles struck, and the velocity with which they are struck. But the quantity or number of particles struck in any time are as the velocity and the density of the fluid. Therefore, the resistance or force of the fluid is as the density and square of the velocity. The resistance to any plane is also more or less, as the plane is greater or less, and therefore the resistance on any plane is as the area of the plane a, the density of the medium, and the square of the velocity ; that is, R = adv^. If the motion be not perpendicular, but oblique to the plane, or to the face of the body, then the resistance in tlie direction of the motion will be diminished in the MOTION OF BODIES IN FLUIDS. 231 triplicate ratio of radius to the sine of the angle of inclination of the plane to the direction of the motion, or as the cube of radius to the cube of the sine of that an- gle. So that R = adv^s^, 1 = radius, and s = sine of the angle of inclination. The real resistance to a plane, from a fluid acting in a direction perpendicular to its face, is equal to the weight of a column of the fluid, whose base is the plane and altitude equal to that which is due to the velocity of the motion, or through which a heavy body must fall to acquire that velocity. The resistance to a plane rimning through a fluid is the same as the force of the fluid in motion with the same velocity on the plane at rest. But the force of the fluid in motion is equal to the weight or pressure which generates that motion, and this is equal to the weight or pressiu-e of a column of the fluid, the base of which is the area of the plane, and its altitude that which is due to the velocity. 1. If a be the area of a plane, v its velocity, n the density or specific gravity of the fluid, and i ^=: 16.0833 feet ; then, the altitude due to the velocity v being — , therefore aXnX^ = -g— , will be the whole resistance or force R. 2. If the direction of motion be not perpendicular to the face of the plane, but ODhque to it, in an angle ; then R = . . 3. If W represent the weight of the body, a being resisted by the absolute force R ; then the retarding force /, or 5 will be ^^!!^ The resistance to a sphere moving through a fluid, is but half the resistance to its great circle, or to the end of a cylinder of the same diameter, moving with an equal velocity. ^ = ^^ > being the half of that of a cylinder of the same diameter, R repre- senting radius. Illustration.— A 9 lb. iron ball, the diameter being 4 inches, when projected at a velocity of 1600 feet per second, will meet a resistance which is equal to a weight of 132.66 lbs. over the pressure of the atmosphere. 232 AIE. AIR. 100 Cubic Inches of atmospheric air, at the surface of the earth, when the barometer is at 30 inches, and at a temperature of 60^' weighs 30.5 grains, being 830.1 times lighter than water. ' Specific gravity compared with loaier, .001246. The atmosphere does not extend beyond 50 miles from the earth's surface. The mean weight of a column of air a foot square, and of an al- titude equal to the height of the atmosphere, is equal to 2116 8 lbs avoirdupois. It consists of oxygen 20, and nitrogen 80 parts ; and in 10 000 parts there are 4.9 parts of carbonic acid gas. The mean pressure of the atmosphere is usually estimated at 14 7 lbs. per square inch. 13.29 cubic feet of air weigh a lb. avoirdupois, hence 1 ton of air wall occupy 29769.6 cubic feet. The rate of expansion of air, and all other Elastic Fluids, for all temperatures, is uniform. From 320 to 212^ they expand from 1000 to 1376, equal to -1-* of their bulk for every degree of heat. "* "^^ See Heat, page 201. '- 5-|g equal .002087 for each degree. DIMENSIONS AND WEIGHTS OF GUNS, SHOT, AND SHELLS. 233 to < O S^« 3 S^ S ■= i" M s-§ i-5-§ C5 Co 00 c§ Cos" to CO CO »^ ^ I— • rf»- t3 t3 to to 00 O ^ 3 CD r^ to O 00 CO cn CO to OJOO CTK? cn cn OOt— ' Ci * C5tO OO COO cn ►— ocn to tototototooto" CO Cn cn -c;i OS CO cn cn 05 05 P>- 05 >f^ 05 05 »— ' ►^cn Cn^ to05i^ = 3 >f^ 4^ 05 to ►^ffi-oc;! H- 4^ to CO " * bo" to to 00 cn oto t;^ — toe 5*^^tf»' >f^ tf^ »— j^ to 4i>- CO t^ rfii ©o o o ;o©coo^oo U2 234 WEIGHT AND DIMENSIONS OF LEADEN BALLS. WEIGHT AND DIMENSIONS OF LEADEN BALLS. Table showing the Number of Balls in a Pound, from l.-^ihs to TWF ^f ^^ ^^^^ Bore. Diam. Diam. Number Diam. Diam. Number Diam. Diam. in parts of in decimals of balls in a in parts of in decimals of balls in a in parts of in decimals of balls in a an inch. of an inch pound. an inch. of an inch. pound. an inch. of an inch. pound. 1.670 1 .570 25 .301 170 *i-nr 1.326 2 .537 30 .295 180 1.157 3 .510 35 .290 190 1.051 4 * 1 2 .505 36 .285 200 * H .977 5 .488 40 .281 210 .919 6 .469 45 .276 220 * i .873 7 .453 50 .272 230 .835 8 *Tt .426 60 .268 240 * T .802 9 .405 70 .265 250 .775 10 .395 • 75 .262 260 .750 11 .388 80 .259 270 .730 12 * i .375 88 .256 280 .710 13 .372 90 * i .252 290 *u .693 14 .359 100 .249 300 .677 15 .348 110 M7 310 .662 16 .338 120 .244 320 .650 17 .329 130 .242 330 * 5 .637 18 .321 140 .239 340 .625 19 .314 150 .237 350 .615 20 .307 160 * The exact decimals would be as follows : 1 5 7 H 1.3125 .9375 .8750 13 1 .8125 .5000 7 TF 5 1 4 .4375 .3125 .2500 Expansion of Shot heated to a White Heat. Expansion Inches. 0.11 Inches. 0.10 Inches. 0.08 Inches. 0.06 Inches. 0.04 Experiment at Fort Monroe, 1839. WEIGHT AND DIMENSIONS OF SHOT. GRAPE. CALIBRE OF 8 Inch. 42 32 24 18 12 Diameter of high gauge . " low gauge . Inches. 3.60 3.54 Inches. 3.17 3.13 Inches. 2.90 2.86 Inches. 2.64 2.60 Inches, 2.40 2.36 Inches. 2.06 2.02 Mean weight in lbs. . . 6.24 4.25 3.25 2.45 1.83 1.19 WEIGHT AND DIMENSIONS OF SHOT. 235 CANISTER. 42 32 24 and 8 inch how- iizer. 18 12 9 and 24 1b. how. ilzer. 6 12 lb. howitzer. Calibre of Field. Mountain Diam. of high gauge, " low gauge, Ins. 2.26 2.22 Ins. 2.06 2.02 Ins. 1.87 1.84 Ins 1.70 1.67 Ins. 1.49 1.46 Ins. 1.35 1.32 Ins. 1.17 1.14 Ins. 1.08 1.05 Ins. Musket ball. Mean weight in lbs., 1.57 1.19 .90 .67 .45 .33 .235 .17 .056 CARCASSES, lis inch. 1 10 inch.) 8 inch. I 42 i 32 t 24 i 18 I 12 Mean weight in lbs. . . | 194 | 87.63 | 43.62 | 29.45 | 21.60 | 15.84 | 12.15 | 8 BRONZE FOR CANNON. Copper 90, Tin 10. Specific gravity is greater than the mean of copper and tin, vix.* 8.766. 236 DIMENSIONS AND WEIGHTS OF GUNS, SHOT, AND SHELLS. ffi m o m m iz; P O o Eh o 1^ o I— I iz; o s «! • ■A w ^^/^ c4 zT it 1-3 1-5 i^ g is l?s 21 r 3 II iJ «s £ c^ sss £:a £ s S £ c ) - ceo c . c (S >.= U >^^ 000 > ^ X > > 00 >.^ > >x > > > > > ct ce "-1 K r; rf rt ^^ ce rt c; rt ^ ^^ ^<^^ ^0 ^^ ^^ « ^ -iiaoq ^ coo c ^^ £ g aqj IV ^ ^1 ^?^^ o\^ §»§ = ^1 ^. ^ ^s^ ^^ ^^ ^^ •paDnpan ^ 1 •itiwiipjO a o •umrairei^ 5 d oc S SSod CO oco XX i 'uinuiiaiiv =' i 00 ^11 II c c II c Q ^ X coo c c ^ •ranuiixEj^ a ^ ^ o> o\%% c^^ g^ i^ '^N -^ -«q ,—1 rt a '<}< Ci 0< c^o^c^ Cil- T-X 1-1 >/5 C5 COC5 xo Clt^- VOL'S H U4 ^ •2a u J ■rj^ ^ C^Tf CD f .t^ r-e< asTBqox bM 00 Ci CiOOO xo dco rf rr 5 g^£° ^^ ood d t^o o»^ CV »0 t^ oc 1-1 10 00 — ^ 00 x-t r^ C? \ri C5M ^SSSd 1>C5 o*G ^ &5c to coo COCO Irtlrf r«-«o ' >^^^ • • . >'d ^ . . i ^ ^ c Ills- ts ^ * * rt CJ rt eS &0 %-r ««-. '^3 0^ £ V4 S) 03 a> *: ^ 1 ^ XS 'U ^'S T3 ^ .2 s « ^ ???5S?J ?5 ss §8 PENETRATION OF SHOT AND SHELLS. 237 PENETRATION OF SHOT AND SHELLS. ^ PENETRATION IN MASONRY. Experiments at Fort Monroe Arsenal in 1839. Calibre. Charge. Elevation. Distance. Mean penetrat on. Dressed granite. Poioniac freestone. Hard briCK. Shot. 32 Pounder (Gun) . Shell 8 Inch Howitzer ) Seacoast ) Lbs. 8 6 1° 1° 35' Yards. 880 880 Inches. 3.5 1. Inches. 12. 4.5 Inches. 15.25 8.5 The solid shot broke against the granite. The shells broke into small fragments against each of the three materials. PENETRATION IN WHITE OAK. Experiments at New- York Harbour in 1814. Calibre. Charge. Distance. Penetration. Remarks. 32 Pounder . j Lbs. 11 11 Yard?. 100 150 Inches. 60 54 Shot wrapped so as to destroy the wind- age. PENETRATION IN COMPACT EARTH (Half sand, half clay). Calibre. Charge. Distances in yards. 27 109 32S 1094 Shot. Inches. Inches. Inches. Inches. 6.885 * 109.1 102.4 93.4 69.7 Shells. 8.782 4.4 lbs. *48.4 *45.3 38.6 23.2 Musket 154 grains 9.85 8.6 4.3 The penetrations in other kinds of earth are found by multiplying the above by 63 for sand mixed with gravel; by 0.87 for earth mixed vi^ith sand and gravel, weighing 125 lbs. to a cubic foot; by 1.09 for compact mould and fresh earth mixed with sand, of half clay; by 1.44 for wet potter's clay; by 1.5 for light earth, settled- and by 1.9 for light earth, fresh. * With this charge, and at these distances, the shells were often broken. 238 PENETRATION OF SHELLS. PENETRATION OF SHELLS. Eleva- Distance In Compact Earth, In Oak Wood, In Masonry, tion. Sins. 10 ins. 12 ins. 8 ins. 10 ins. 12 ins. S ins. 10 ins. 12 ins. Yards. Inches. Inches. Inches. Inches. Inches. laches. Inches. Inches. Inches. 30<^| 656 7.8 17.7 19.6 3.9 7.8 8.6 1.9 3.5 3.9 1312 9.8 25.6 27.5 4.7 11.8 13.7 2.3 4.7 5.1 45° "I 656 11.8 19.6 21.6 5.9 9.8 10.6 3.1 3.9 4.3 1312 15.7 27.5 29.5 7.8 13.7 15.7 3.9 5.5 5.9 60oj 656 19.6 29.5 31.5 8.6 13. 14.5 4.3 5.9 6.3 1312 21.6 31.5 33.4 9.8 13.7 15.7 4.7 6.3 6.6 Falling with ( maximum < 23.6 33.4 35.4 9.8 13.7 15.7 4.7 6.6 7. velocity. ( The penetration in other kinds of earth and stone may be obtain- ed by using the coefficients given for the other tables. For woods, use for beech and ash 1, for elm 1.3, for white pine and birch 1.8, and for poplar 2. 144 grains of powder in a musket, at 5 yards' distance, will pro- ject a ball 3 inches into seasoned wh*ite oak, and 100 grains in a rifle, at the same distance, 2.05 inches. MISCELLANEOUS. 239 MISCELLANEOUS. RECAPITULATION OF WEIGHTS OF VARIOUS SUBSTANCES. Cubic foot in pounds. Cubic inch in pounds. *Cast Iron .... 450.55 .2607 t Wrought Iron . 486.65 .2816 Steel 489.8 .2834 tCopper . 555. .32118 Lead 708.75 .41015 Brass 537.75 .3112 Tin 456. .263 is White Pine . 29.56 .0171 Salt Water (sea) 64.3 .03721 Fresh Water . 62.5 .03611 Air ... • .07529 Steam . .0350 — Weights of a Cubic Foot of various Substances in ordinary use. Loose earth or sand Lbs. . 95 Clay and stones Lbt. . 160 Common soil . . 124 Cork 15 Strong soil . 127 Tallow 59 Clay . 135 Brick . 125 SLATING. Sizes of Slates. Doubles . 14 by 6 inches, Ladies' 15 '' 8 Countess . 22 *'ll Duchess . 26 *-15 Imperial and Patent 32 "26 Rags and Queens 39 "27 * From the West. Point Foundry Association at Cold Spring, N. Y. Other ex- periments have given .2613 as the weight of a cubic inch, t Ulster Iron Company, Saugerties. N. Y. t From Phelps, Dodge, & Co.'s Works, in Derby, Conn. 240 MISCELLANEOUS. CAPACITY OF CISTERNS IN U. S. GALLONS. For each 10 Inches m Depth. 2 feet diameter 2i 3 3^ 4 4i 5 5i 6 6i 7 7i 19.5 30.6 44.06 59.97 78.33 99.14 122.40 148.10 176.25 206.85 239.88 275.40 8 feet diameter . 313.33 8i . 353.72 9 . 396.56 9i . 461.40 10 . 489.20 11 . 592.40 12 . 705. 13 . 827.4 14 . 959.6 15 . 1101.6 20 . 1958.4 25 . 3059.9 TABLE OF COMPOSITIONS BRASS, ETC. Copper. Tin. Zinc. 2 1 3 1 4 i 6 .5 i 8 9 3 1 10 1 78 22 5.6, and leaa 4.3 J 80 10 For Yellow Brass. Spelter. Lathe brushes. Shaft bearings. (hard). Wheels, boxes, cocks, &c. Gun metal. Brass. Valves. Bells and Gongs. SIZES OF NUTS, EQUAL IN STRENGTH TO THEIR BOLTS. Diameter of bolt Short diameter Diameter of bolt Short diameter Diameter of bolt Short diameter in inches. of nut in inches. in inches. of nut in inche?. in inches. of nut in inches- 1 '3 If 2tV 21 4rt 3 5 n 2H 2| 4| ^ 7 If 2i 2|- 4il 1 ItV 1^^ • 3^ n H 3 4^ 1^ li 31 3 5| i lA 2 3^ 3| 5i 1 i| 2| 3| 3i 6/* u 2 21 4 sr 6| u n 21 4| 4 n Note. — The depth of the head should equal the diameter of the bolt ; the den- of the nut should exceed it in the proportion of 9 or 10 to 8. MISCELLANEOUS. 241 SCREWS. Table showing the Number of Threads to an Inch in V thread Screws. Diam. in inches, No. of threads, Diam. in inches, No. of threads. 20 H 6 5 7 1 1 18 16 14 12 11 10 9 7 Ij Ig- A& 2^ ^2 -^4 O o^- 3-2 5 4J 4i 4 4 3i 3| 3^ 3^ ^4 2^ 6 2* Diam. in inches, 3| 4 4]^ ^ 4J 5 5j No. of threads, 3 3 2 J 2 J 2| 2| 2f The depth of the threads should be half their pitch. The diameter of a sc^ew, to work in the teeth of a wheel, should be such that the angle of the threads does not exceed 10°. Table of the Strength of Copper at different Temperatures, Temperature. strength in lbs. Temperature. strength in lbs. Temperature. strength in lbs. 122° 33079 482° 26981 801° 18854 212^ 32187 545° 25420 912° 14789 302° 30872 602° 22302 1016° 11054 392° 27154 Franklin Institute. DIGGING. 23 cubic feet of sand, or 18 cubic feet of earth, or 17 cubic feet of clay, make a ton. 18 cubic feet of gravel or earth before digging, make 27 cubic feet when dug. COAL GAS. A chaldron of bituminous coal yields about 10.000 cubic feet of gas. Gas pipes i inch in diameter supply a light equal to 20 candles. 1.43 cubic feet of gas per hour give a light equal to one good candle. 1.96 cubic feet equal four candles. 3. '* " '« ten X 242 MISCELLANEOUS. ALCOHOL Is obtained by distillation from fermented liquors. Proportion of Alcohol in 100 parts of the following Liquors : Scotch Whiskey 54.32 Sherry 19.17 Irish " . 53.9 Claret 15.1 Rum 53.6S Champagne 13.8 Brandy 53.39 Gooseberry 11.84 Gin . 51.6 Elder 8.79 Port . 22.9 Ale . 6.87 Madeira . 22.27 Porter 4.2 Currant . 20.55 Cider 9.8 to 5.2 Teneriffe . 19.79 Prof Brande, WEIGHT OF COMPOSITION SHEATHING NAILS. Number- Length in Number in Length in Number in Length in Number in inches. a pound. inches. a pound. inches. a pound. 1 1 290 6 1 190 10 11 101 2 7 260 7 H 184 11 11 74 3 1 212 8 U 168 12 2 64 4 1| 201 9 H 110 13 2i 59 5 1 u 199 CEMENT. Ashes 2 T5JiTt^ ^ Clav ' 3 " ( Mixed with oil, will resist the weather equal to Sand, 1 - S '^^'^^'■ HYDRAULIC CEMENT. A barrel contains 300 lbs., equal to 4 struck bushels. BROWN MORTAR. One third Thomaston lime. Two thirds sand, and a small quantity of hair. MISCELLANEOUS. 243 BRICKS, LATHS, ETC. Dimensions. Common brick Front brick . 8 to 7|x4iX2i inches. 8i X4^X2^ ** 20 common bricks to a cubic foot, when laid ; 15 " " *' a foot of 8 inch wall, when laid. Laths are l\ to H inches by four feet in length, are usually set i of an inch apart, and a bundle contains 100. Stourbridge fire-brick, 9ix4|x2i inches. HAY. 10 cubic yards of meadow hay weigh a ton. When the hay is ta- ken out of large or old stacks, 8 and 9 yards will make a ton. 11 to 12 cubic yards of clover, when dry, weigh a ton. HILLS IN AN ACRE OF GROUND. 40 feet apart 27 hills, 8 feet apart 680 hills, 35 '' 35 6 " 1210 *' 30 " 48 5 " 1742 " 25 " 69 3i " '* . 3556 " 20 *' 108 3 " " . 4840 " 15 " 193 2^ " " . 6969 *' 12 '' 302 2 " *' . 10890 ^* 10 " 435 1 " " . 43560 " DISPLACEMENT OF ENGLISH VESSELS OF WAR, WHEN LAUNCH- ED AND WHEN READY FOR SEA. Weight of hull, launched . Weight received on board . Weight complete . . . . 120 80 74 Razee. 50 52^ 46 Tons. 795 670 28 Corv. 18 Tons. 281 326 607 Tons. 2467 2142 4609 Tons. 1882 1723 Tons. 1617 1359 Tons. 1448 1044 2492 Tons. 1042 1067 2109 Tons. 413 370 783 3605 2976 1465 Brig. IS Ton€. 213 242 455 Edye's JV*. C. 244* MISCELLANEOUS. WEIGHT OF LEAD PIPE PER YARD, From i to 4-| Inches Diameter. Weight ii lbs. and oz Weight in lbs. and oz. i inch medium . 3 H inch extra light 9 " strong . 4 " light 13 l' inch light . 3 " medium . 15 8 " medium . 4 " strong 19 — '' strong . 5 1} inch medium . 16 — *' extra strong 6 6 " strong 20 — 1 inch light . 5 — 2 inch ligtit 16 12 " medium . 6 8 " medium . 20 — " strong . 7 8 " strong 23 — " extra strong 8 4 2^ inch light 25 — i inch extra light 5 — " medium . 30 — " light . 6 4 ''' strong 35 — " medium . 8 — 3 inch light 30 — " strong . 9 12 " medium . 35 — " extra strong 10 8 " strong 44 — 1 inch extra light 6 14 3§ inch medium . 45 — " hght . 8 5 " strong 54 — *' medium . 10 5 " extra strong ^70 — *' strong . 12 4 4 inch waste, light li inch extra light 8 5 " " medium '21 — " light . 9 12 '' '' strong, 26 — " medium . 11 — 4i inch '' light, — — '' strong . 12 8 " '* medium 24 — " extra strong 14 10 '' " strong, 29 — Very light Pipe. Diametar. Weight in ibs. and oz. Diameter. Weight in lbs. and oz. i inch 1 J inch 3 6 § " . H 1 " 5 10 i " 2 li - 6 14 1 " . 2,^ TIN. Size of sheet. Mean thickness. Mean weight of one sheet. Description. 2^0. oil wire gau^^e. Thickness of sheet. Single . Double X Inches. 10x14 10X14 31 27 Inches. .0125, (or 80 to 1 inch) .0181, (or 55 to 1 inch) Lbs. 0.5 0.75 There are usually 225 sheets in a box. MISCELLANEOUS. 24.& RELATIVE PRICES OF AMERICAN WROUGHT IRON. Round. Square 4 inches 27 4 inches 27 3^ to 24 inches . 26 to 21 3i to 2i inches 26 to 20 2i - i " 19 21 - 1 - 21 20 to 29 li" i " 19 to 26 i^/a^ Hoops. i and \ inch to | 26 to 28 H to 4 inch 24 to 33 i "' 1 " '^ 1X4 19 i " i " " iXT%- 19 to 23 J5a7Z(i . 20 Illustration. — If 4 inch round iron is worth $135 per ton, then band iron is worth $100 per ton, for 27 is to 20 as 135 to 100. POWER REQUIRED TO PUNCH IRON AND COPPER PLATES-. Through an Iron Plate, with a Punch ^ Inch in Diameter, .08 inches thick .17 " .24 " " 6025 lbs. 11950 " 17100 " Through a Copper Plate, with a Punch \ Inch in Diameter, .08 inches thick .17 *' 3983 lbs. 7833 " The force necessary to punch holes of different diameters through metals of various thicknesses, is directly as the diameter of the hole and the thickness of the metal. To ascertain the Force necessary to Punch Iron or Copper Plates, Rule.— Multiply, if for iron, 150000, and if for copper, 96000, l^ the diameter of the punch and the thickness of the plate, each la inches ; the product is the pressure in pounds. The use of oil reduces the above 8 per cent. X2 24f6 WEIGHT OF SQUARE EOLLED IRON. IRON. Cast Iron expands ywtowo ^^ ^^^ length for one degree of heat ; greatest change in the s^hade in this dimate, ytto ^^ i^^ length ; exposed to the sun's rays, yoVo 5 shrinks in cooHng from -^^ to ^ of its length ; is crushed by a force of 93,000 lbs. upon a square inch ; will bear, without permanent alteration, 15,300 lbs. upon a square inch, and an extension of y—- ^^ ^^^ length. Weight of modulus of elasticity for a base of an inch square, 18,400,000 lbs. ; height of modulus of elasticity, 5,750,000 feet. Wrought Iron expands ytto^ ^^ ^^^' ^^"^^^ ^^^ ^"^ degree of heat ; will bear on a square inch, without permanent alteration, 17,800 lbs., and an extension in length of y^Vo ? cohesive force is diminished 30V0 ^Y ^^ increase of 1 degree of heat. Weight of modulus of elasticity for a base of an inch square, 24,920,000 lbs. ; height of modulus of elasticity, 7,550,000 feet. Compared with cast iron, its strength is 1.12 times, its extensibility 0.86 times, and its stiffness 1.3 times. WEIGHT OF SQUARE ROLLED IRON, From i Inch to 12 Inches, AND ONE FOOT IN LENGTH. Size in Weight in Size in Weight in Size in Weight in Size in Weight in iucbbs. pounds. inches. pounds. inches. pounds. inches. pounds. _!_ .013 2. 13.520 4.1 64.700 T.i 190.136 16^ 2.1 15.263 4.i 68.448 1,1 203.024 ¥ .053 2.i 17.112 4.| 72.305 8. 216.336 1 .118 2.t 19.066 4.f 76.264 8.i 230.068 .211 2.i 21.120 4.| 80.333 8.i 244.220 5 i .475 2.| 23.292 5. 84.480 %.% 258.800 2.1 25.560 5.1 88 . 784 9. 273.792 i .845 2.1 27.939 5.i 93.168 9.i 289.220 1 1.320 3. 30.416 5.f 97.657 9.i 305.056 1 1.901 3.8 33.010 5.i 102.240 9.f 321.332 7 2.588 3.5 35.704 5.1 106.953 10. 337.920 3.380 3.f 38.503 5.t 111.756 10. i 355.136 t 4.278 3.i 41.408 5.1 116.671 10.^ 372.672 5.280 3.1 44.418 6. 121.664 10.^ 390.628 3 6.390 3.1 47.534 6.i 132.040 11. 408.960 1 7.604 3.1 50.756 6.^ 142.816 11. i 427.812 5 8.926 4. .54.084 6.i 154.012 11. i 447.024 'i 10.352 4.1 57.517 7. 165.632 11. i 466.684 7 •8 11.883 4.i 61.055 7.i 177.672 12. 486.656 Example.— What is the weight of a bar of rolled iron 1^ inches square and 12 Inches long 1 * In column 1st find H, and opposite to it is 7.604 pounds, which is 7 lbs. and y^^ WEIGHT OF ROUND ROLLED IRON. 247 of a lb. If the lesser denomination of ounces is required, the result is obtained as follows : Multiply the remainder by 16, pointing off the decimals as in multiplica- tion of decimals, and the figures remaining on the left of the point indicate the number of ounces. Thus, -M^ of a lb. = .604 10 jg 9.664 ounces. The weight, then, is 7 lbs. 9.^^^^ ounces. If the weight for less than a foot in length was required, the readiest operation is this: Example. — What is the weight of a bar 6^: inches square and 9^ inches long ? In column 5th, opposite to 6^, is 132.040, which is the weight for a foot in length. 6^X12 inches = 132.040 6. " isi = 66.020 3. " is ^ of 6= 33.010 •i " is ^ of 3= 5.5016 4 " is i of i = _2.7508 WEIGHT OF ROUND ROLLED IRON, From 4 Inch to l^ Inches Diameter, AND ONE FOOT IN LENGTH. Diameter Weight in Diameter Weight in Diameter Weight in Diameter Weight in in inches. pounds. in inches. pounds. in inches. pounds. in inches. pounds. 3 .010 2.i 11.988 4.i 53.760 7.1 159.4.56 .041 2.1 2.1 13.440 14.975 4.1 4.1 56.788 59.900 8. 84 169.8.56 180.696 •TO .119 2.i 16.688 4.| 63.094 8.i 191.808 .165 2.| 18.293 5. 66.752 8.i 203.260 •8 .373 2.f 20.076 5.1 69.731 9. 215.040 •^ .663 2.| 21.944 5.i 73.172 9.i 227.152 ,- 1.043 3. 23.888 5.f 76.700 9.^ 239.600 .f 1.493 SA 25.926 5.i 80.304 9.1 252.376 .1 2.032 3.1 28.040 5.1 84.001 10. 266.288 2.654 3.f 30.240 5.1 87.776 10. i 278.924 1«8 3.360 3.i 32.512 5.1 91.634 10. i 292.688 4.172 3.1 34.886 6. 95.552 10. i 306.800 1.5 5.019 3.1 37.332 6.1 103.704 11. 321.216 l.i 5.972 3.1 39.864 6.i 112.160 11.1 336.004 7.010 4. 42.464 6.1 120.960 11. i 351.104 1 .5 8.128 4.1 45.174 7. 130.048 11. i 366.536 1.^ 9.333 4.i 47.952 7.i 139.544 12. 382.208 2. 10.616 4.f 50.815 7.i 149.328 The application of this table is precisely similar to that of the preceding one. 248 WEIGHT OF FLAT ROLLED IRON. WEIGHT OF FLAT ROLLED IRON, From ixk Inch /o 5f X6 Inches y AND ONE FOOT IN LENGTH. Breadth Thic fness Weight in Breadth Thickness Weight in Breadth Thickness in inchea. iu in ches. pounds. in inches. in inches. pounds. ia inches. in inches. •i . i 0.211 I.i 14 5.808 2. .1 0.422 1.^ 4 0.633 .1 8 1 0.634 4 1.266 •8 .1 ; 8 0.264 1.900 • i i 0.528 ^ 2.535 • 1 t 0.792 .1 3.168 .t i 1.056 A 3.802 7 •5 .1 0.316 7 •8 4.435 I 0.633 1. 5.069 1 .1 .1 0.950 1.1 5.703 I.i ■ ■ 1.265 I.i 6.337 l»t 'y 1.584 l.t 6.970 I.i •i : 1 ' 0.369 i.| .¥ 0.686 l.| 0.738 •t 1.372 l.| 1.108 2.059 l.f 1.477 A 2.746 2.| •8 1.846 ' 5 3.432 •i 2.217 I 4.119 1. ! 8 0.422 '.i 4.805 • i 0.845 5.492 • 1 1.267 1 .8 6.178 .3 ^ 1.690 6.864 .1 1 2.112 1 ,^ 7.551 I 2.534 I.i 8.237 l'.| 1 2.956 14 .8 0.739 1 i ■g 0.475 ,5 1.479 l.§ z 0.950 2 2.218 1 .i 1.425 A 2.957 l.| ^ 1.901 3.696 I.i ^ 2.375 4.435 ^ 1 7 I»8 I 2.850 • 8 5.178 2. i 3.326 5.914 2.i 1 3.802 1 .8 6.653 i.i i 0.528 1 .4 7.393 ;f f 1.056 ■, 8.132 1.584 I.i 8.871 •1 ^ 2.112 1.^ 9.610 •i 1 2.640 I.i 0.792 • i 5 3.168 1.584 7 3.696 .1 2.376 1 .^ 1 4.224 .i 3.168 1«^ 1 4.7.52 .1 3.960 I.i i.i 0.580 1 4.7.52 1.161 .? 5.544 l.| '; 1.742 6.336 l.f 2.325 1 1 1 . 8 7.129 1.1 ,. : 2.904 I.i 7.921 2. .i- 3.484 8.713 2.- 7 'IS 4.065 I.i 9.505 3.i 1 4.646 l.| 10.297 .? 1 '.i 6.327 l.| 11.089 .1 pounds. WEIGHT OF FLAT ROLLED IRON. 24.9 Breadth Thickness Weight in in inches in inches pounds. 2.f •i 4.013 •1 5.016 •4 6.019 7 •8 7.022 1. 8.025 l.f 9.02S l.i 10.032 14 11.035 l.i 12.038 l.| 13.042 l.t 14.045 1.1 15.048 2. 16.051 24 17.054 2.i 18.057 2.i .^ 1.056 .1 2.112 • i 3.168 •i 4.224 .1 5.280 .1 6.336 7 •8 7.392 1. 8.448 l.i 9.504 l.| 10.560 i.i 11.616 l.i 12.672 1.1 13.728 1.^ 14.784 1.1 15.840 2. 16.896 2.1 17.952 2.i 19.008 2.^ 20.064 2-1 .¥ 1.109 .? 2.218 .- 3.327 .i 4.436 i 5.545 • ? 6.654 • 8 7.763 1. 8.872 1.1 9.981 l.i 11.090 1.^ 12 199 l.i 13.308 l.| 14.417 l.i 15.526 1-J 16.635 2. 17.744 2.1 18.853 2.i 19.962 2.| 21.071 2.i 22.180 4 1.162 Table — (Continued). Breadth Thickness Weight in in inches, in inches pounds. 2.2 •i 3 •8 •i •4 7 .? 1. 1.1 l.i l.i l.i 1.1 l.£ l.| 2. 2.i 2.i 2.^ 2.i 2.1 1. l.i l.i l.f l.i 1.1 lA 1.1 2. 2.i 2.i 2.i 2.i 2.1 2.^ 1. l.i l.i l.f 1.1 2.323 3.485 4.647 5.808 6.970 8.132 9.294 10.455 11.617 12.779 13.940 15.102 16.264 17.425 18.587 19.749 20.910 22.072 23.234 24.395 1.215 2.429 3.644 4.858 6.072 7.287 8.502 9.716 10.931 12.145 13.360 14.574 15.789 17.003 18.218 19.432 20.647 21.861 23.076 24.290 25.505 26.719 1.267 2.535 3.802 5.069 6.337 7.604 8.871 10.138 11.406 12.673 13.940 15.208 1^.475 Breadth Thickness Weight ia in inches. in inches pounds. 3. l.£ 17.742 1.1 19.010 2. 20.277 2.i 22.811 2.i 25.346 2.;: 27.881 3.i • V 1.373 .:: 2.746 4.119 .1- 5.492 • 1 6.865 .4 8.237 .1 9.610 1. 10.983 l.i 12.356 l.i 13.730 l.i 15.102 l.i 16.475 l.f 17.848 1.5 19.221 l.| 20.594 2. 21.967 2.i 24.712 2.i 27.458 2.^ 30.204 3.^ 32.950 3.i • i? 1.479 .:: 2.957 I 4.436 5.914 • z 7.393 8.871 .8 10.350 1. 11.828 l.i 13.307 l.i 14.785 l.f 16.264 l.i 17.742 1.1 19.221 l.i 20.699 l.f 22.178 2. 23.656 2.i 26.613 2.i 29.570 2.1 32.527 3. 35.485 3.i 38.441 3.i .i 1.584 .i 3.168 *" *! 4.752 6.. 336 •1 7.921 A 9.505 7 .8 11.089 I. 12.673 250 WEIGHT OF FLAT ROLLED IRON. Table — (Continued). Breadth rhickness Weight in Breadth Thickness Weight in Breadth Thickness Weight in in inches. n inches. pounds. n inches. n inches. pounds. n inches. in inches pounds. 3. J l.| 14.257 4.i 2.i 34.217 5.i 2.i 44.355 1.- 15.841 2.i 38.019 2.5 48.791 17.425 2.^ 41.820 3. 53.226 1,- 19.009 3. 45.623 3.i 57 . 662 20.594 3.i 49.425 3.i 62.097 1 ^ 22.178 3.i 53.226 3.5 66.533 23.762 s.i 57.028 4. 70.968 J. • g 2. 25.346 4. 60.830 4.i 75.404 2.i 2.*- 28.514 4.^ 64.632 ^•t 79.839 31.682 44 .i 4.013 4.5 84.275 88.710 2.1 3. 34.851 .i 8.026 5. 38.019 .1 12.039 5.^ i 4.647 1:1 41.187 44.355 1. l.i 16.052 20.066 9.294 13.940 4. .2 1.690 3.380 l.i l.f 24.079 28.092 l.i 18.587 23.234 6.759 2. 32.105 l.i 27.881 10.138 2.i 36.118 1.5 32.527 • 4 1. 13.518 2.i 40.131 2. 37.174 1.1 16.897 2.i 44.144 2.i 41.821 l.i 14 20.277 3. 48.157 2.i 46.468 23.656 3.1 52.170 2.i 51.114 2. 27.036 3.i 56.184 3. 55.761 2.i 30.415 3.1 60.197 3.i 60.408 2.i 33.795 4. 64.210 3.i 65.055 2.1 37.174 4.i 68.223 3.1 69.701 <4( • 4- 3. 40.554 4.i 72.235 4. 74.348 3.i 3.1 43.933 5. .z 4.224 4.i 78.995 47.313 .i 8.449 4.i 83.642 3.5 50.692 .f 12.673 4.| 88.288 44 1 1.795 1. 16.897 5. 92.935 . 8 3.591 l.i 21.122 5.i 97.582 7.181 l.i 25.346 5.i •i 4.858 '3 10.772 l.i 29.570 .i 9.716 r. 14.364 2. 33.795 .i 14.574 l.i 17.953 2.i 38.019 1. 19.432 l.i 21.544 2.i 42.243 l.i 24.290 1.5 25.135 2.1 46.468 r.i 29.148 2. 28 . 725 3. 50.692 1.5 34.006 2.i 32.316 3.i 54.916 2. 38.864 2.i 35.907 3.i 59.140 2.i 43.722 2.5 39.497 3..I 63.365 2.i 48.580 3. 43.088 4. 67.589 2.1 53.437 3.5 46.679 4.i 71.813 3. 58.296 3.1 4.. 50.269 4.i 76.038 3.i 63.154 53.860 4.5 80.262 3.i 68.012 57.450 5.i • i 4.436 3.1 72.870 4.i ^.i 3.802 .i 8.871 4. 77.728 7.604 .5 13.307 4.i 82.585 11.406 1. 17.742 4.i 87.443 1 . 15.208 l.i 22.178 4.1 92.301 1.^ 19.010 l.i 26.613 5. 97.159 1.? 22.812 14 31.049 5.i 102.017 26.614 2. 35.484 5.i 106.876 2. . 30.415 2.i 39.920 6. 116.592 WEIGHT OF FLAT ROLLED IRON. 251 Examples.— What is the weight of a bar of iron 5^ inches in breadth by ^ inches thick 1 In column 4, page 250, find 54:, and below it, in column 5, ^ ; and opposite to that is 13.307, which is 13 lbs. and -^^ of a lb. For parts of a lb. and of a foot, operate precisely similar to the rule laid down for table, page 247. WEIGHTS OF A SQUARE FOOT OF IRON IN AVOIRDUPOIS POUNDS. THICKNESS BY WIRE GAUGE. No. on gauge . 1 |2|3|4|5|6|7|8|9|10|11 Pounds . . 12.5 I 12 I 11 1 10 1 9 I 8 I 7.5 I 7 I 6 I 5.68 1 5 No. on gauge .12 I 13 I 14 I 15 I 10 i 17 I 18 | 19 1 20 l 21 I 22 Pounds . . 4.62 1 4.31 I 4 I 3.95 I 3 I 2.5 I 2.18 I 1.93 I 1.62 I 1.5 I 1.37 Number 1 is y^^, number 4 is J, and number 11 is ^ of an inch. CAST IRON. To ascertain the weight of a cast iron Bar or Rod, find the weight of a UTOUght iron bar or rod of the same dimensions in the preceding tables, and from the weight deduct the ^^i^ th part ; or say. As 486.65 : 450.55 : : the weight in the table : to the weight required. Thu^ What is the weight of a piece of cast iron 4X3|X12 inches 1 In table, page 250, the weight of a piece of wrought iron of these dimensions is 50.692 lbs. 486.65 : 450.55 : : 50.692 : 46.93 lbs. Or, by an easier mode, though not so minutely correct, As 281 : 260 : : 50.692 : 46.90 lbs. To find the Weight of a piece of Cast or Wrought Iron of any size or shape. By the rules given in Mensuration of Solids (see page 81), ascertain the number of cubic inches in the piece, multiply by the weight of a cubic inch, and the product will be the weight in pounds. EXAMPLES. What is the weight of a block of wrought iron 10 inches square by 15 inches io length 1 10X10X15 = 1500 cubic inches. .2816 weight of a cubic inch. 422.4000 pounds. What is the weight of a cast iron bail 15 inches in diameter ? By table, page 255= 176.7149 cubic inches. .2607 Weight of a cubic inch. 460.6957 pounds. 252 WEIGHT OF CAST IRON PIPES. WEIGHT OF CAST IRON PIPES OF DIFFERENT THICKNESSES, From 1 Inch to 36 Inches Bore, AND ONE FOOT IN LENGTH. Bore. Thi ,kness ; Weight. Bore. ITbic kness Weight. Bore. Thi :kness Weight. Inches. Inc hes. Pounds. Inches. Inc hes. Pounds. Inches. In :hes. Pounds. 1. ;| 3.06 6. 4 49.60 11. i 4 58.82 5.05 , ; 58.96 4 74.28 14 4 3.67 6.i Y 34.32 4 90.06 4 6. - 43.68 4 106.14 1.^ 4 6.89 53.30 1 122.62 4 9.80 - 63.18 12. 'i 61.26 14 2 7.80 7. 1 36.66 1 77.36 * 11.04 1 1 46.80 f 93.70 2. 1 8.74 i 5 56.96 i 110.48 i 12.23 I 1 67.60 1 127.42 24 3 9.65 1 78.39 12. i i 63.70 i 13.48 74 i 39.22 1 80.40 24 1 10.57 1 49.92 i 97.40 14.66 1 60.48 7 8 114.72 1 19.05 1 71.76 1 132.35 24 1 11.54 l' 83.28 13. i 66.14 1 15.91 8. i i 41.64 1 83.46 1 20.59 5 52.68 i 101.08 3. f 12.28 1 64.27 7 8 118.97 i 17.15 ■g 76.12 1 137.28 1 22.15 1, 88.20 13. f i 68.64 i 27.56 8.i 'k 44.11 1 86.55 34 1 18.40 5 8 56.16 ^ 104.76 23.72 i 68. 7 8 123.30 1 20.64 7 8 80.50 1 142.16 34 1 19.66 1 93.28 14. i 71.07 25.27 9. 1 46.50 1 89.61 1 31.20 58.92 i 108.46 34 1 20.90 £ 71.70 1 127.60 26.83 7 84.70 1 147.03 2 33.07 1 97.98 14.^ . i 73.72 4. i 22.05 9.^ i 48.98 . f 92.66 1 28.28 1 62.02 j i 112.10 i 34.94 i i 75.32 1 I 131.86 44 i 23.35 ! 1 88.98 i 1 151.92 29.85 1 102.90 15. 1 . i 75.96 2 36.73 10. ^ 51.46 1 1 95.72 44 1 24.49 1 1 65.08 f 115.78 1 31.40 1 1 78.99 i 136.15 I 38.58 1 i 93.24 , 1 1 156.82 44 1 25.70 1 108.84 15. i i 1 78.40 32.91 104 1 53.88 98.78 1 40.43 68.14 % 119.48 5. ^ 26.94 4 82.68 1 140.40 1 34.34 4 97.44 1 161.82 1 42.28 1 112.68 16. 1 80.87 54 29.40 11. 4 56.34 101.82 1 37.44 71.19 I 123.14 4 45.94 4 86.40 1 144.76 6. 1 31.82 1 • t 101.83 1 166.60 4 40.56 ; 1 117.60 16.^ ■i 83.30 WEIGHT OF CAST IRON PIPES. 253 TABLE~(Continued ). Bore. Thic inesi Weight. Bore. Thic kness Weight. Bore. jThic kness Weight. Inches. Inc les. Pouads. Inches. Inc hes. Pounds. Inches. Inc hes. Pounds. 16. i 104.82 22. 5 8 138.60 30. 1. 303.86 7 126.79 5 167.24 1 343.20 149.02 7 3 196.46 31. '. f 233.40 i! 8 171.60 225.38 i 273.40 17. 1 85.73 23. 5 144.77 313.68 107.96 2 174.62 i 354.24 I 130.48 _ 204.78 32. 1 240.76 1 153.30 235.28 1 281.94 1 ^ 176.58 24. 1 150.85 323.49 17. i 1 88.23 111.06 g 181.92 213.28 33. 1 365.29 248.10 I 134.16 245.08 7 290.50 4- 7 157.59 25. 1 156.97 333.24 1 8 181.33 i 189.28 i 376.26 18. 1 114.10 1 221.94 34. i 255.45 8 137.84 254.86 i 298.88 1 161.90 26. i 196.62 342.88 1 8 186.24 1 230.56 I 387.13 19. 120.24 264.66 i 431.76 •■ 145.20 27. I 204.04 35. I 262.70 170.47 7 239.08 7 3 307.62 1 8 195.92 274.56 352.86 20. 5 126.33 28. 1 211.32 i 39S.10 7 1 52 . 53 8 247.62 ? 443.96 179.02 284.28 36. i 270.18 1 • 8 205.80 29. t 218.70 i 316.36 21. 4 132.50 7 8 256.20 362.86 159.84 294.02 '•I 409.34 • ? 7 • 8 187.60 30. 'i 226.20 • i 456.46 1 215.52 .1 264.79 Note.— These weights do not include any allowance for spigot and faucet ends. Y 254! WEIGHT OF A SQUARE FOOT OF CAST IRON, ETC. WEIGHT OF A SQUARE FOOT OF CAST AND WROUGHT IRON, COPPER, AND LEAD, From ^th to 2 Inches thick. Thickness. Cast Iron. Wrought Iron. Hard rolled. Copper. Hard roiled. Lead. Pounds. Pounds. Pounds. Pounds. •tV 2.346 2.517 2.890 3.691 ■i 4.693 5.035 6.781 7.382 .1% 7.039 7.552 8.672 11.074 • i 9.386 10.070 11.562 14.765 •A 11.733 12.588 14.453 18.456 .§ 14.079 15.106 17.344 22.148 -T^ 16.426 17.623 20.234 25.839 • i 18.773 20.141 23.125 29.530 9 •TS 21.119 22.659 26.016 33.222 •1 23.466 25.176 28.906 36.913 11 •Tff 25.812 27.694 31.797 40.604 4 28.159 30.211 34.688 44.296 •H 30.505 32.729 37.578 47.987 7 "5 32.852 35.247 40.469 51.678 •\i 35.199 37.764 43.359 55.370 1 inch 37.545 40.282 46.250 59.061 1.1 42.238 45.317 52.031 66.444 l.i 46.931 50.352 57.813 73.826 i.i 51.625 55.387 63.594 81.210 14 56.317 60.422 i 69.375 88.592 1.1 61.011 65.458 75.156 95.975 i.i 65.704 70.493 80.938 103.358 i.| 70.397 75.528 86.719 110.740 2. 75.090 80.563 92.500 118.128 Note.— The Specific Gravity of the Wrought Iron is that of Pennsylvania plates, and of the Copper, that of plates from the works of Messrs. Phelps, Dodge & Co., in Connecticut. The Lead, a mean from several places. WEIGHT AND CAPACITY OF CAST IKON AND LEAD BALLS. 25& WEIGHT AND CAPACITY OF CAST IRON AND LEAD BALLS, From 1 to 20 Inches in Diameter. Diameter in inches. Capacity in cubic inches. CAST IRON. Pounds. LEAD. Pounds. 1. .5235 .1365 .2147 l.i 1.7671 .4607 .7248 2. 4.1887 1.0920 1.7180 2.^ 8.1812 2.1328 3.3554 3. 14.1371 3.6855 5.7982 3.i 22.4492 5.8525 9.2073 4. 33.5103 8.7361 13.744 4.i 47.7129 12.4387 19.569 5. 65.4498 17.0628 26.843 5.i 87.1137 22.7206 35.729 6. 113.0973 29.4845 46.385 6.i 143.7932 37.4528 58.976 7. 179.5943 46.8203 73.659 7.^ 220.8932 57.5870 90.598 8. 268.0825 69.8892 109.952 8.i 321.5550 83.8396 131.883 9. 381.7034 99.5103 156.553 9.i 448.9204 117.0338 184.121 10. 523.5987 136.5025 214.749 11. 696.9098 181.7648 285,832 ^ 12. 904.7784 235.8763 371.096 13. 1150.346 299.6230 471.806 14. 1436.754 374.5629 589.273 15. 1767.145 460.6959 724.781 16, 2144.660 559.1142 879.616 17. 2572.440 670.7168 10.55.066 18. 3053.627 796.0825 1252.422 19. 3591.363 936.2708 1472.970 20. 4188.790 1U92.0200 1717.995 256 WEIGHT OF COPPER RODS AND PIPES. WEIGHT OF COPPER EODS OR BOLTS, From i to 4: Inches in Diameter, •TF 3 • ¥ •1 6 9 •T6- 5. • 8 1 1 •rg- 3 • 4 13 •TF 7 • ¥ AND ONE FOOT IN LENGTH. Pounds, .1892 .2956 .4256 .5794 .7567 .9578 1.1824 1.4307 1.7027 1.9982 2.3176 2.6605 3.0270 1 YE 3.4170 1 8 3.8312 3 TF 4.2688 1 4 4.7298 TF 5.2140 3 8 5.7228 7 TF 6.2547 1 2 6.8109 T% 7.3898 3 7.9931 3 4 9.2702 7 8- 10.6420 2. 12.1082 2.1 13.6677 2.i 15.3251 2.t 17.0750 2.i 18.9161 2.1 20.8562 2.1 22.8913 2.1 25.0188 3. 27.2435 3.1 29.5594 34 31.9722 3.f 34.4815 3.^ 37.0808 3.1 39 . 7774 3.i 42.5680 3.1 45.4550 4. 48.4330 WEIGHT OF RIVETED COPPER PIPES, From 5 to 30 Inches in Diameter, from 3 to j^ths thick, AND ONE FOOT IN LENGTH. Diam. Thickness Weight in Diam. Thickness Weight in Diam. Thickness Weight in in ins. in 16;hs, pounds. in ins. in 16ihs. pounds. 30.598 in ins. in I6ths. pounds. 5. 3 12.497 9.i 4 19. 4 60.142 5. 4 16.880 10. 4 32.208 19. 5 75.233 5.i 3 13.628 11. 4 35.200 20. 5 78.208 5.i 4 18.395 12. 4 38.456 21. 5 82.984 6. 3 14.765 13. 4 41.456 22. 5 86.771 6. 4 19.908 14. 4 44.640 23. 5 90.571 6.^ 3 15.897 15. 4 47.646 24. 5 94.308 6.i 4 21.415 15. 5 59.588 25. 5 98.122 7. 3 17.034 16. 4 .50.752 26. 5 101.897 7. 4 22.932 16. 5 63.470 27. 5 105.700 7.-^ 4 24.447 17. 4 53.856 28. 5 109.446 8. 4 25.961 17. 5 67.. 344 29. 5 113.221 8.-^. 4 27.471 18. 4 57.037 30. 5 116.997 9. 4 28.985 18. 5 71.258 The above weights include the laps on the sheets for riveting and caulking. The weights of the rivets are not depends upon the distance they are ter of the pipe. added; the number per lineal foot of pipe placed apart, and their size upon the diame- COPPER, LEAD, AND BRASS. 257 COPPER. To ascertain the Weight of Copper. Rule. — Find by calculation the number of cubic inches in the piece, multiply them by .32118, and the product will be the weight in pounds. Example. — What is the weight of a copper plate ^ an inch thick by 16 inches square 1 162 = 256 .5 for ^ an inch. 128.0X. 32118 = 41.111 pounds. LEAD. To ascertain the Weight of Lead, Rule. — ^Find by calculation the nimiber of cubic inches in the piece, and multi' ply the sum by .41015, and the product will be the weight in pounds. Example.— What is the weight of a leaden pipe 12 feet long, 3| inches in diam- eter, and 1 inch thick ? By rule in Mensuration of Surfaces^ to ascertain the area of cylindrical rings f Area of (3H-1+1) = 25.967 " " si = 11.044 Difference, 14.923, or area of ring. 144 = 12 feet. 2148.912X.41015 = 881.376 pounds. BRASS. To ascertain the Weight of ordinary Brass Castings, Rule.— Find the number of cubic inches in the piece, multiply by .3112, and the product will be the weight in pounds. Y2 258 CABLES AND ANCHORS. CABLES AND ANCHORS. Table showing le Size of Cables and Anchors proportioned to the Tonnage of Vessels. Tonnage of vessel. Cables. Circumference Chain Cables. Diameter in Proof in Weisht of Anchor in Weight of a fathom of Weight of a fathom of in inches. inches. pounds. Chain. Cable. 6 3. •1% •I 56 5.i 2.1 8 4. 3 • 8 14 84 8. 4. 10 4.i ■^ 2.i 112 11. 4.6 15 5.1- • i 4. 168 14. 6.5 25 6. •1 5. 224 17. 8.4 40 6.^ .f 6. 336 24. 9.8 60 7. •rk 7. 392 27. 11.4 75 7.i 3 • 4 9. 532 30. 13. 100 8. ■H 10. 616 36. 15. 130 9. 7 12. 700 42. 18.9 150 9.^ ■U 14. 840 50. 21. 180 10. i 1. 16. 953 56. 25.7 200 11. i-tV 18. 1176 60. 28.2 240 12. i-i 20. 1400 70. 33.6 270 12. i l-T^ 21. 1456 78. 36.4 320 13.^ l-i 22. i 1680 86. 42.5 360 14. 1-A 25. 1904 96. 45.7 400 14. i 1-1 27. 2072 104. 49. 440 15.^ 1-tV 30. 2240 115. 56. 480 16. l.i 33. 2408 125. 59.5 520 16.^ 1-T^ 36. 2800 136. 63.4 570 17. 1-1 39. 3360 144. 67.2 620 17. i 1-U 42. 3920 152. 71.1 680 18. 1-f 45. 4200 161. 75.6 740 19. l-H 49. 4480 172. 84.2 820 20. I-i 52. 5600 184. 93.3 900 22. 1-il 56. 6720 196. 112.9 1000 24. 2. 60. 7168 208. 134.6 The proof in the U. S. Naval Service is about 12^ per cent, less than the above. The utmost strength of a good hemp rope is 6400 lbs. to the square inch ; in prac- tice it should not be subjected to more than half this strain. It stretches from 4 to •^, and its diameter is diminished from 5^ to ^ before breaking. A difference in the quality of hemp may produce a difference of ^ in the strength of ropes of the same size. The strength of Manilla is about J that of hemp. White ropes are one third more durable. CABLES. 259 CABLES. Table showing what Weight a good Hemp Cable will bear with Safety. ' Circumference. Pounds. Circumference. Pounds. Circumference. Pounds. 6. 4320. 10.25 12607.5 14.50 25230. 6.25 4687.5 10.50 13230. 14.75 26107.5 6.50 5070. 10.75 13867.5 15. 27000. 6.75 5467.5 11. 14520. 15.25 27907.5 7. 5880. 11.25 15187.5 15.50 28830. 7.25 6307.5 11.50 15870. 15.75 29767.5 7.50 6750. 11.75 16567.5 16. 30720. 7.75 7207.5 12. 17280. 16.25 31687.5 8. 7680. 12.25 18007.5 16.50 32670. 8.25 8167.5 12.50 18750. 16.75 33667.5 8.50 8670. 12.75 19507.5 17. 34680. 8.75 9187.5 13. 20280. 17.25 35707.5 9. 9720. 13.25 21067.5 17.50 36750. 9.25 10267.5 13.50 21870. 17.75 37807.5 9.50 10830. 13.75 22687.5 18. 38880. 9.75 11407.5 14. 23520. 18.25 39967.5 10. 12000. 14.25 24367.5 To ascertain the Strength of Cables. Multiply the square of the circumference in inches by 120, and the product is the weight the cable will bear in pounds, with safety. To ascertain the Weight of Manilla Ropes and Hawsers. Multiply the square of the circumference in inches by .03, and the product is the weight in pounds of a foot in length. This is but an approximation, and yet it is sufficiently correct for many purposes. Table showing what Weight a Hemp Rope will bear with Safety. Circumference. Pounds, Circumference. Founds. Circumference. Poundi. 1. 200. 3.i 2450. 6. 7200. i.i 312.5 3.1 2812.5 6.i 7812.5 1.4 450. 4. 3200. 6.1 8450. l.J 612.5 44 3612.5 6.1 9112.5 2. 800. 4.i 4050. 7. 9800. 2.i 1012.5 • 4.i 4512.5 7.{ 10512.5 2.i 1250. 5. 5000. 7.i 11250. 2.1 1512.5 5.1 5512.5 7.1 12012.5 3. 1800. 5.i 6050. 8. 12800. 3.i 2112.5 5.1 6612.5 260 CABLES. To ascertain the Strength of Ropes, Multiply the square of the circumference in inches by 200, and it gives the weight the rope \n\\ bear in pounds, with safety. To ascertain the Weight of Cable-laid Ropes. Multiply the square of the circumference in inches by .036, and the product is the weight in pounds of a foot in lengtli. To ascertain the Weight of Tarred Ropes and Cables. Multiply the square of the circumference by 2.13, and divide by 9 ; the product is the weight of a fathom in pounds. Or, multiply the square of the circumference by .04, and the product is the weight of a foot. For the ultimate strength, divide the square of the circumference in inches by 5 ; the product is the weight in tons. A square inch of hemp fibres will support a weight of 92000 lbs. BLOWING ENGINES. 261 BLOWING ENGINES. The object of a blast is to supply oxygen to furnaces. The quantity of oxygen in the same bulk of air is different at dif- ferent temperatures. Thus, dry air at 85° contains 10 per cent, less oxygen than when at the temperature of 32^ ; when saturated with vapour, it contains 12 per cent. less. Hence, if an average supply of 1500 cubic feet per minute is re- quired in winter, 1650 feet will be required in summer. The pressure ordinarily required for smelting purposes is equal to a column of mercury from 3 to 7 inches. The capacity of the Reservoir should exceed that of the cylinder or cylinders, and the area of the pipes leading to it should be ^-^ of the area of the cylinder. The quantity of air at atmospheric density delivered into the reservoir, in conse- quence of escapes through the valves, and the partial vacuum necessary to produce a current, will be about j less than the capacity of the cylinder. To find the Poiver ivhen the Cylinder is Double Acting, Let P represent pressure in lbs. per square inch, V " the velocity of the piston in feet per minute, a " the area of the cylinder in inches, 1.25 " the friction necessary to work the machinery. Then Fva 1.25 = the power in lbs. raised 1 foot high per minute. ^ , „^ When Single Acting. Pi?al.25 ^ ^ — = the power in lbs. raised 1 foot high per minute. Air expands nearly 2| times its bulk while in the fire of an ordinary furnace. Dimensions of a Fui^nace, Engines, SfC, Furnace. At Lonakoning (Md.). Diameter at the boshes 14 feet, which fall in, 6.33 inches in every foot rise. Engine. Diameter of cylinder 18 inches, length of stroke 8 feet. Averaging 12 revolutions per minute, with a pressure of 50 lbs. per square inch. Boilers, Five : each 24 feet in length, and 36 inches in diameter. Blast Cylinders. 5 feet diameter, and 8 feet stroke. At a pressure of from 2 to 2^ lbs. per square inch, the quantity of blast is 3770 cubic feet per minute, requiring a power of about 50 horses to supply it. 180 tons air is required to make 10 tons pig iron, and burn the coke from 50 tons coal. The ore yielding about 33 per cent, of iron. Steam Boilers. Two cylinders, 12 inches in diameter and 12 inches stroke, aided by exhausting into a condenser, and with steam of 30 lbs. pressure per square inch, will make 50 revolutions per minute, and drive 4 blowers, each 54 inches in diame- ter, and 30 inches wide, 300 revolutions in a minute, furnishing the necessary blast for burning anthracite coal on a grate surface of 108 square feet ; supplying 4400 cubic feet steam per minute, at a pressure of 30 lbs. per square inch. 35 cubic feet of steam used in the cylinder of a blowing engine will drive blowers 4 feet in diameter by 26 inches face, and furnish the necessary blast to an anthracite fire, for generating 1150 cubic feet steam, the time 1 minute, and pressure per square inch 35 lbs. 262 MISCELLANEOUS NOTES. MISCELLANEOUS NOTES. ON MATERIALS, ETC. Wood is from 7 to 20 times stronger transversely than longitu- dinally. In Buffon's experiments, b, d, and I being the breadth, depth, and length of a piece of oak in inches, the weight that broke it in pounds wa3M^(51pio). The hardness of metals is as follows : Iron, Platina, Copper, Silver Gold, Tin, Lead. ■ tt , A piece spliced on to strengthen a beam should be on its convex side. Springs are weakened by use, but recover their strength if laid by. A pipe of cast iron 15 inches in diameter and .75 inches thick v/ill sustain a head of water of 600 feet. One of oa^, 2 inches thick, and of the same diameter, will sustain a head of 180 feet. When the cohesion is the same, the thickness varies as the height X the diameter. When one beam is let in, at an inclination to the depth of another, so as to bear in the direction of the fibres of the beam that is cut ; the depth of the cut at right angles to the fibres should not be more than i of the length of the piece, the fibres of which, by their cohe- sion, resist the pressure. Metals have five degrees of lustre— splendent, shining, glistening, glimmering, and dull. The Vernier Scale is |^, divided into 10 equal parts ; so that it divides a scale of lOths into lOOths w^hen the lines meet even in the two scales. A luminous point, to produce a visual circle, must have a velocity of 10 feet in a second, the diameter not exceeding 15 inches. Tides. The difference in time between high water averages about 49 minutes each day. In Sandy soil, the greatest force of a pile-driver will not drive a pile over 15 feet. A fall of yV of an inch in a mile will produce a current in rivers. Melted snow produces about | of its bulk of w^ater. All solid bodies become luminous at 800 degrees of heat. At the depth of 45 feet, the temperature of the earth is uniform throughout the year. A Spermaceti candle .85 of an inch in diameter consumes an inch Ik lent^th in 1 hour. " Silica vi the base of the mineral world, and Carbon of the organized. Sound passes m water at a velocity of 4708 feet per second. MISCELLANEOUS NOTES. 263 SOLDERS. For Lead, melt 1 part of Block tin ; and when in a state of fusion, add 2 parts of Lead. Resin should be used with this solder. For Tin, Pewter 4 parts, Tin ], and Bismuth 1 ; melt them to- gether. Resin is also used with this solder. For Iron, tough Brass, with a $mall quantity of Borax. CEMENTS. Glue. Powdered chalk added to common glue strengthens it. A glue w^hich will resist the action of water is made by boiling 1 pound of glue in 2 quarts of skimmed milk. Soft Cement. For steam-boilers, steam-pipes, &c. Red or white lead in oil, 4 parts ; Iron borings, 2 to 3 parts. Hard Cement. Iron borings and salt water, and a small quantity ot sal ammoniac with fresh water. Inside work. Outside work. 80 9 PAINTS. White Paint, TTTi •ill -. insiae v White-lead, ground m oil . . 80. Boiled oil 145 Raw oil Q Spirits turpentine .... 8. 4 New wood work requires about 1 lb. to the square yard for 3 coats. Lead Colour. White-lead, ground in oil, 75 Lampblack . . . i Boiled linseed oil . 23 Litharge ... .5 Japan varnish . . .5 Spirits turpentine . 2.5 The turpentine and varnish are added as the paint is required for use or transportation. White-lead, in oil . 78. Boiled oil-. . . 9.5 Raw oil . . . 9,5 Gray, or Stone Colour. Spirits turpentine . 3. Turkey umber . . .5 Lampblack . . .25 1 square yard of new brick work requires, for 2 coats, 1.1 lb. • for 3 coats, 1.5 lbs. * Cream Colour. White-lead, in oil French yellow . Japan varnish . Raw oil . Spirits turpentine Uc^^^^oixh^ ^^ ^^^ ^^'""^ ^'^^^ requires, for 1st coat, 0.757 for 1st coat. 2d coat. 66.6 70. 3.3 3.3 1.3 1.3 28. 24.5 225 2.25 264 MISCELLANEOUS NOTES. Lampblack . . 28 Litharge ... 1 Black Paint {for Iron). Linseed oil, boiled . 73 Spirits turpentine . 1 Japan varnish . . 1 The varnish and turpentine are added last. Liquid Olive Colour. Olive paste Boiled oil . . 61.5 . 29.5 Dryings Japan varnish Spirits turpentine 5.5 3.5 2. Paint for Tarpaulins {Olive), Liquid olive colour . 100 I Spirits turpentine . 6 Beeswax ... 61 1 square yard requires 2 lbs. for 3 coats. Dissolve the beeswax in the turpentine, and mix the paint warm. Lacker for Iron Ordnance. Black-lead, pulverized 12 Red-lead . 12 Litharge . 5 Lampblack 6 Linseed oil 66 Boil it gently for about 20 minutes, stirring it constantly during that time. Lacker for Small Arms, or for Water Proof Paper. Beeswax . . . 18. I Spirits turpentine . 80 Boiled linseed oil . 3.5 | Heat the ingredients in a copper or earthen vessel over a gentle fire, in a water bath, until they are well mixed. Lacker for Bright Iron Work. Linseed oil, boiled . 80.5 I Litharge ... 5.5 White-leadjgroundinoil, 11.25 I Pulverized rosin . 2.75 Add the litharge to the oil ; let it simmer over a slow fire for 3 hours ; strain it, and add the rosin and white-lead ; keep it gently warmed, and stir it until the rosici is dissolved Staining Wood and Ivor^^ /-^- j A V-A Yellow. Dilute nitric acid will produce it on wood. Red. An infusion of Brazil wood in stale urine, in the proportion of a lb. to a gallon for wood, to be laid on when boiling hot, and should be laid over with alum water before it dries. Or, a solution of dragon's blood, in spirits of wine, may be used. Black. Strong solution of nitric acid, for wood or ivory. Mahogany. Brazil, Madder, and Logwood, dissolved in water and put on hot. Blue. Ivory may be stained thus : 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 j[iitrous acid. I .'^^ <^ <<'i ^ '-^/v. i ,A ~ s '■^^<^' .-N -~ ^^A V^^ X^^^ aO* .^^ '% ." A*"-<>, 'u , . . * /. ■' ^^ v^' ■>* O " ^0 •>^ r; ,^^ ^^ ~ \ ^^. M' 'X % %■ < ' " '^ 1 ' H % y^^ •=* ■'' r^ vj_ 7 ., X ^^ °-<- \ /TJ^' - ,0( .^^P^^^x^