EDITOR’S PREFACE TO THE NINTH EDITION. CD cr> a»!h. *• This present edition of the following work is the result of a very careful and thorough examination of the former impression. I have not only gone over all the calculations, and corrected whatever numerical errors I could detect, but I have also been anxious to present the several steps of each arithmetical process in the most compact and improved form. Much numerical work is often thrown away by neglecting to take ad¬ vantage of cancelling operations. I have often had occasion to notice this in the revision of the last edition: in the present I have taken care to introduce every simplification into the arithmetical forms that such forms admitted; and I have reason to hope that those who have hitherto countenanced this work will find it even more acceptable in its present improved state. A 2 IV PREFACE. I would notice, in conclusion, that the Table of useful Numbers often required in calculation, together with their Logarithms, at pages 169 to 180 inclusive, which has been computed at considerable labour, will furnish particulars of interest and utility in many prac¬ tical inquiries. SAMUEL MAYNAKD. CONTENTS. PAGE An explanation of the Signs or Characters made use of in the work 1 Of Weights and Measures. 2 2 3 „ oysteme usuei . Foreign Weights, in Pounds Avoirdupois 3 Long Measure.. 4 French Long Measure, Decimal System.4 French Long Measure, .Decimal System.4 „ Systeme Usuei . . . . . .4 The Lineal Foot of various countries, given in English inches . 4 Length of a Mile in different countries, given in English yards . 5 Superficial Measure.5 French Superficial Measure.5 Solid Measure . . . . • • • . 5 Number of Cubic Feet in a Ton of various bodies ... 6 Imperial Wine Measure.6 French Measures of Capacity . . . • • .6 The British Imperial Gallon, compared with Foreign Measures of Capacity . . . v .7 Imperial Ale and Beer Measure.7 Imperial Dry Measure.8 Heaped Imperial Measure of Capacity, for Coals, Culm, &c. . . 8 Dimensions of Drawing Paper.8 Dimensions of Imperial Conical Liquid Measures .... 9 Dimensions of Imperial Cylindrical Dry Measures ... 9 Decimal Fractions. Numeration . Reduction n vi CONTENTS. PAGE A Table of Reciprocals.14 Addition.15 Subtraction.16 Multiplication.16 Division.18 A Table of the Fractional parts of an Inch.19 Of the Square Root.20 Of the Cube Root.22 To extract the Root of any Number that consists of more than One Figure.22 To find the Root of a Number consisting of Integers and Decimals 27 Practical Geometry. 1. To divide a given line into two equal parts .... 2. To divide a given angle into two equal parts .... 3. From any given point in a right line to erect a perpendicular . 4. Through a given point to draw a straight line parallel to a given straight line. 5. To divide a right line into any number of equal parts 6. To divide a triangle into two equal parts, and still retain its original form.. • . . 7. Through any three points out of a right line to describe the cir¬ cumference of a circle.. . 8. To find the centre of a given circle . . . . 9. From a given point to draw a tangent to a circle 10. To find a mean proportional between two given right lines, or the side of a square equal to a given rectangle . 11. To find the side of a square which shall be equal in area to a given triangle.. 12. Upon a given right line to construct a square .... 13. To make a rectangle equal to a given triangle . 14. To produce a rectangle equal to a given square 15. To make a triangle equal to a given quadrilateral figure . 16. To circumscribe a square about a given circle . 17. Upon a right line to describe an octagon. 18. In a given circle to inscribe any regular polygon . . . 19. To find the side of a square that/ shall be any number of times the area of a given square.. • 20. To find the diameter of a circle that shall be any number of times the area of a given circle . . . ... 21. To divide a given circle into any number of concentric parts equal to each other . . • • _ • • • 22. To find the side of a square nearly equal in area to a given 23. To find a right line that shall be nearly equal to any given arc of a circle . . • 24 To describe tbe largest possible circle in a triangle . _ . 25. To describe an ellipsis, the transverse and conjugate diameters being given. 29 29 29 30 31 31 31 32 32 32 36 37 37 37 38 . I eo eo eo ^ ^ co J55 CONTENTS. VII PAGE 26. To describe a parabola, any ordinate to the axis and its abscissa being given.38 27. To draw, at the circumference of a circle, lines tending towards the centre when the centre is inaccessible . . . .39 28. To describe an elliptical arc, the width and rise of span being given.40 Mensuration of Superficies. 1. To find the area of any parallelogram. 2. To find the area of a trapezoid. 3. To find the area of a triangle. 4. Any two sides of a right-angled triangle being given, to find the third side. 5. To find the area of any regular polygon, from 3 to 50 sides A table of multipliers for regular polygons, from 3 to 50 sides, with the names of the regular polygons .... 6. The side of a regular polygon being given, to find the radius of the circumscribing circle to contain that polygon 7. The radius of a circle being given, to find the length of side of a regular polygon inscribed in it. 8. The radius of a circle inscribed in a regular polygon being given, to find the radius of the circle circumscribing that polygon . 9. The diameter of a circle being given, to find the circumference; or the circumference being given, to find the diameter 10. To find the length of any arc of a circle. 11. To find the diameter of a circle, by having the chord and versed sine given. 12. To find the area of a parabola or its segment .... A table of versed sines ........ A table of the relative proportions of the circle, its equal and inscribed squares. Examples illustrative of the preceding table .... Some of the properties of a circle ...... 13. The area of a circle given, to find the diameter 14. To find the area of a sector of a circle. 15. To find the area of a segment of a circle ..... 16. To find the area of a circular ring or space included between two concentric circles. 17. To find the area of an ellipsis. 44 44 44 45 47 49 49 50 51 52 52 53 53 54 55 55 56 56 Mensuration of Solids. 1. To find the convex surface and solid content of a cylindrical body, or any figure of a cubical form.59 . 2. To determine the dimensions of any cylindrical vessel, whereby to contain the greatest cubical contents, bounded by the least superficial surface.60 3. To find the surface and solid content of a cone or pyramid . 61 CONTENTS. 4. To find the surface of the frustum of a cone or pyramid . . 61 5. To find the solid content of the frustum of a cone . . .62 6. To find the solid content of the frustum of a pyramid . . 63 7. To find the solidity of a wedge.63 8. To find the convex surface and solid content of a sphere or globe 64 9. To find the convex surface and solid content of the segment of a sphere.65 10. To find the solidity of a spheroid ...... 66 11. To find the solidity of the segment of a spheroid when the base is circular or parallel to the revolving axis . ._ . . 66 12. To find the convex surface and solid content of a cylindric ring 67 Of Timber Measure. To find the superficial content of timber . ... 69 Superficial measure by the engineer’s slide rule . . . • /0 To find the solidity of timber.71 By the slide rule . . •.*73 To find the transverse section of the strongest beam that can possibly be cut out of a round piece of timber. .73 To determine the greatest rectangle that can possibly be obtained in a given triangle.73 On the Strength of Materials. On the cohesive strength of bodies. A table of the ultimate cohesive strength of various bodies . 1. To find the ultimate cohesive strength of square, round, and rectangular bars.. 2. The weight of a body being given, to find the dimensions of a bar or rod to sustain that weight . . . . . .76 A table of the cohesive strength and weight of chains 75 77 On the Transverse Strength of Bodies. A table of the transverse strength of various bodies . . .78 1. To find the ultimate transverse strength of rectangular beams . 78 2. To find the breadth or depth of beams intended to support a permanent weight.80 On Torsion or Twisting. A table of the strength of various bodies to resist torsion . . 84 A table of the proportionate length of bearings, or journals for shafts of various diameters ....... 85 Of the Mechanical Powers. Wheel and axle.90 To calculate for the different parts of a crane, as respects mechanical advantage.91 CONTENTS. IX PAGE To find the thickness of cast iron for a crane post, when fixed at one end and loaded at the other. The pulley. The inclined plane. The wedge . The screw. A table of the estimated power of man or horse as applied to machinery ... . 97 97 Of Falling Bodies. To find the velocity a falling body will acquire in any given time . 99 To find the velocity a body will acquire by falling from any given height.99 To find the space through which a body will fall in any given time 100 A table of accelerated motion of falling bodies .... 100 On Pendulums .101 On the Velocity of Wheels, Drums, Pulleys, &c. . .102 A table for finding the radius of a wheel when the pitch is given, or the pitch of a wheel when the radius is given . . . 105 Regular approved proportions for wheels with flat arms in the middle of the ring, and ribs or feathers on each side .... 106 On the Maximum Velocity and Power of Water-wheels. 1. Of Undershot wheels.108 2. Of breast and overshot wheels.108 To find the circle of gyration in a water-wheel . . . .Ill A table of angles for windmill sails.Ill The velocity of threshing machines, millstones, boring irons. &c. . 112 Of Pumps and Pumping Engines. To find the quantity of water discharged by a pump of a given diameter in a given time, the velocity of the water being also given.. . . 114 To find the diameter of a pump and number of horses’ power that will discharge a given quantity of water in a given time . . 114 To find the time a cistern will take in filling, when a known quan¬ tity of water is going in, and a known portion of that Water going out, in a given time . . ..115 To find the time a vessel will take in emptying itself of water . 116 On the pressure of fluids.116 To find the number of imperial gallons contained in a yard of cir¬ cular pipe of any given diameter.117 To find the weight that a given power can raise by means of a hydrostatic press.117 To find the diameter of a cylinder to work a pump of a given dia¬ meter, for a given depth.118 CONTENTS. FAGE To find the diameter of a pump that a cylinder of a given diameter can work at a given depth.. .118 To find the depth from which a pump of a given diameter will work by means of a cylinder of a given diameter . . . .119 Approximate Rules for Calculating Liquids. To find the number of imperial gallons contained in any square or rectangular cistern.120 Any two dimensions of a square or rectangular cistern being given, to find the third, that shall contain any number of imperial gal¬ lons required. 120 To find the content of a cylinder in imperial gallons . . . 121 The length of a cylinder given, to find the diameter; or the dia¬ meter given, to find the length that shall contain any number of imperial gallons required.121 On Steam and tlue Steam Engine. To calculate the effect of a lever and weight upon the safety-valve of a steam boiler, &c.. To find the proper diameter for a safety-valve .... A table of the elastic force of steam on a square inch A table of the elastic force of steam on a circular inch . Proportions of fuel.. To find the height of a column of water to supply a steam boiler against any pressure of steam required. A table of velocities for marine engines. To find the surface of the floats or paddle boards .... Principles upon which the rule is founded for calculating the power of a steam engine. General Rules. Hyperbolic logarithms. The steam way. Of the slide valve. The eccentric. The cold water pump. To find the necessary quantity of water for a boiler . . The air pump. The beam. The parallel motion. The connecting rod. The fly wheel. The governor or regulator. Of high pressure steam engines. 124 126 127 128 128 129 130 147 148 Miscellanies. Approximate rules to find the weight of beams or bars of cast and wrought iron . . . ..150 CONTENTS. XI PAGE The dimensions of a cast iron ring being given, to find its weight nearly.150 To find the weight of any cast iron ball whose diameter is given . 151 To find the diameter of a cast iron ball when the weight is given . 151 A table containing the weight of a square foot of copper and lead in lbs. avoir..152 A table of the weight of a square foot of sheet iron in lbs. avoir. . 152 A table of the weight of a square foot of boiler-plate iron in lbs. avoir..152 A table of the weight of solid cylinders of cast iron in lbs. avoir. . 153 A table for finding the weight of a square foot of malleable iron, copper, and lead pipes.153 A table containing the weight of wrought iron bars in lbs. avoir. . 154 Table of proportional dimensions of 6-sided nuts for bolts . . 154 A table of the weight of flat bar iron in lbs. avoir. . . . 155 A table of the specific gravity of water at different temperatures . 155 A table of the weight of cast iron pipes in lbs. avoir. . . . 156 A table of the weight of cast iron balls in lbs. avoir. . . . 157 A table of the weight of a square foot of millboard in lbs. avoir. . 157 A table containing some of the properties of various bodies . .158 A table showing the expansion of water by heat .... 158 Proportions of cement for cast iron.159 A table of the boiling point of water holding various proportions of salt in solution.159 To reduce any number of degrees of temperature on Fahrenheit’s scale to the number of degrees of an equal temperature on Reau¬ mur’s scale ; and also to the number of degrees of an equal tem¬ perature on the centigrade scale or otherwise .... 159 Fahrenheit and Reaumur.160 Fahrenheit and Centigrade.160 Reaumur, Centigrade, and Fahrenheit.160 A table of brickwork.162 A table of specific gravities.163 Rules for making or correcting the gauge points on the engineer’s slide rule.. • . 165 Decimal approximations for facilitating calculations in mensura¬ tion ..166 Decimal equivalents to fractional parts of lineal measure . . 167 A table containing the price of metal, or other materials, by the Ion, cwt., qr., or lb .168 A table of useful numbers often required in calculations, together with their logarithms.169 CONTENTS. xii APPENDIX. Containing Mathematical Tables. TABLE PAGE I. Containing the circumferences and areas of circles, sides of equal squares, squares and cubes of the diameters, from \th to 100 inches, feet, yards, chains, miles, &c., advancing by \th, and also the side of equal square, advancing at an equal ratio.. • • [1 II. Containing the circumferences and areas of circles ; also, the side of a square of equal area, and the contents of cylinders in cubic yards, and imperial gallons at 1 foot in depth, from 1 to 50 feet diameter, advancing by an inch . . [21 III. Containing the square and cube roots of all numbers from 1 to 1000; and the difference existing between each root, by which the process for obtaining the roots of numbers, consisting of integers and decimals, is considerably facili¬ tated .[37 IV. Containing the surface and solidity of spheres, the edge or side of equal cubes, the lengths of equal cylinders,_ and the weight of equal volumes of water in lbs. avoirdupois . [49 V. Containing the weight of cylindrical columns of water, each 1 foot in length, and of various diameters, in lbs. avoirdupois [55 VI. Combining the specific gravities and other properties of bodies. Water the standard of comparison, or 1000 . . . [60 VII. Containing the circumferences, areas, squares, and cubes of fractional numbers.. • [61 A concise method of verifying dates, in accordance with the Julian and Gregorian Calendars; or for finding the day of the week, corresponding to any proposed date of the month after Christ, without limitation ... . [94 THE MILLWRIGHT AND ENGINEER’S POCKET COMPANION. EXPLANATION OS' THE SIGNS OR CHARACTERS MADE USE OF IN THE FOLLOWING WORK. = signifies Equality; as 4 added to 3 = 7, denotes 4 and 3 added together are equal to 7. + signifies Addition; as 5 + 3 = 8. — signifies Subtraction; viz. that the latter of the two numbers or quantities between which it is placed is to be subtracted from the former: thus, 5 — 3 = 2. X signifies Multiplication; as 5 x 3 = 15. — signifies Division; viz. that the former of two quantities or numbers between which it is placed is to be divided by the latter : thus, 15-7-5 = 3: the divided quantity or number is called the Dividend, that which divides is called the Divisor. Sometimes the divisor is written under the dividend, with a line drawn between them ; as ^ = 3. { denotes Proportionals, signifying that the numbers or quantities between which they are placed are propor¬ tionals : thus, 2 : 4 ;; 8 ; 16, denotes that the number 2 bears the same ratio to 4 as 8 does to 16, and is usually read 2 is to 4 as 8 is to 16. ( ) Parentheses are used to connect two or more quantities to¬ gether into one: thus, (3 + 5) X 3 = 24, denotes that the sum of 3 and 5 (which is equal to 8) multiplied by 3 is equal 3 2 denotes 3 X 3 = 0, or the square of 3. 3 3 denotes 3 X 3 X 3 = 27, or the cube of 3. V denotes square-root; as, ^/0 = 3. V denotes cube-root; as, %/27 = 3. B 2 01? 'WEIGHTS AND MEASURES. 0E WEIGHTS AND MEASURES. Avoirdupois Weight is the only weight made use of in mecha¬ nical calculations; and all metals, save gold and silver, are weighed by it: hence it is not requisite here to take any other into consideration. Fr. Grammes. 1 Dram... 1'772 16 Drams .= 1 Ounce... 28 347 16 Ounces.= 1 Pound.—. 453*544 28 Pounds. = 1 Quarter. =- 12*699 kilog. 4 Quarters.~ ] Hundred-wt. .. — .... 50*797 »» 20 Hundred-wt... = 1 Ton.= ....1015*939 „ Note.— 5760 Troy grains = 1 pound Troy; and 7000 Troy grains = 1 pound Avoirdupois: hence, 175 pounds Troy = 144 pounds Avoirdupois. Or, Avoirdupois. lbs. X 1*21528 = Troy lbs. Do.ounces X ‘9115 = Do. ounces. Troy. lbs. X *823 = Avoir, lbs. * Do.ounces X 1*1 = Do. ounces. Do.grains X *03657 = Do. drams. Also, Avoirdupois. lbs. x *00893 = Cwts. And, Do. lbs. X *000447 = Tons, nearly. TABLES Showing the relative Proportion betv>een Foreign Weights and the Avoirdupois Pound. 1. FRENCH WEIGHTS.-DECIMAL SYSTEM. 1 Milligramme .— *0154 Troy grains. 1 Centigramme.= *1543 „ 1 Decigramme.= 1*5434 „ 1 Gramme.= 15*4340 „ 1 Decagramme. = 154*3400 „ 1 Hectogramme. = 1543*4000 „ 1 Kilogramme.= 2*20486 lbs. Avoirdupois. 1 Myriagramme.= 22*0486 „ 1 Quintal.= 1 cwt. 3 qrs. 24§ lbs. nearly. 1 Millier or Bar = 100 Quintals = 9 tons 16 cwt. 3 qrs. 12£ lbs. OF WEIGHTS AND MEASURES. 3 The Kilogramme 2. systeme 1000 Grammes USUEL. = 2 lbs. 3 oz. 41 Drs. Avoir. The Livre Usuel 500 = 1 „ 1 „ 1so ,, peso sottile •7687 1-068 1075 •719 3-125 3 001 1-750 1125 1-19 1-781 1-964 •9026 •9217 1-01 2-205 2-82 •75 1-235 1-0517 6641 Note. —America, the British West Indies, Gibraltar, and Tas¬ mania use the pound Avoirdupois, as in England. Example 1. —Suppose I purchase an article in London which weighs 50 lbs. Avoirdupois, what will it weigh in Amsterdam according to their new weight ? 50 lbs. — 2-205 = 22-676 lbs. of Amsterdam. Ex. 2.—An article that weighs 60 lbs. in Leghorn, according to their weight, what will it weigh in lbs. Avoirdupois ? •749 X 60 = 44-94 lbs. Avoirdupois nearly. B 2 4 OF ‘WEIGHTS AND MEASURES. LONG MEASURE. Fr. Metres. 12 Inches.= 1 Foot. = ’3048 3 Feet.=1 Yard. = -3144 6 Feet.= 1 Fathom.= 1-8288 5| Yards. = 1 Pole or rod .... = 5-0292 40 Poles.= 1 Furlong. = 201-1070 8 Furlongs or 1760 yards.. = 1 Mile.= 1609 3436 3 Miles.= 1 League. = 4828-0307 Surveying Chain = 22 yards, consists of 100 links, and each link = 7'92 inches. FRENCH LONG MEASURE.—DECIMAL SYSTEM. French. English. 1 Millimetre.= -03937 inches. 1 Centimetre. = -39370 ,, 1 Decimetre.= 3*93701 „ 1 Metre. = 3937009 „ 1 Decametre.= 32-80841 feet. 1 Hectometre. = 328-08408 „ 1 Kilometre. = 1093*61358 yards. 1 Myriametre... = 10936-13583 „ Usuel. 1 Ligne .. 1 Pouce .. 1 Pied ...., 1 Aune ... 1 Toise ... SYSTEME USUEL. Metrical. 2-31 Millimetres .. 2- 77 Centimetres .. 3- 33 Decimetres .... 12 Decimetres.... 2 Metres. English. •091 inches. 1*090 „ 13110 „ 3 feet 11 -24 inches. 6 „ 6*74 „ THE LINEAL FOOT OF YARIOUS COUNTRIES, GIVEN IN ENGLISH INCHES. Amsterdam & Ant-"l werp./ Bahia, Lisbon, andl Rio de Janeiro.. J Bergen, Copenha- - ! gen, Cape Town, I Christiania, and [ Hamburgh = 11143 - 12-944 = 12-36 Inches. Canton.= 12-65 Dantzic and Memel =11-3 Port-au-Prince ) and \ = 12-8 Port Louis .j Riga.= 1079 Stockholm.= 11-684 Venice. = 1368 Note. —The English foot is used generally throughout America, the British West Indies, Russia, and Tasmania. OF "WEIGHTS AND MEASURES. 5 LENGTH OF 1 MILE IN DIFFERENT COUNTRIES, GIYEN IN ENGLISH YARDS. Dantzic. Yards. Poland. Yards. Denmark. . 8244 Portugal. . 6760 Flanders. . 6864 Prussia. Germany. Russia. . 1167 Hanover. Scotland. . 1984 Holland . .. 8101 Spain. . 4635 Hungary.. Sweden. Ireland. .. 2240 Switzerland .... . 9153 Netherlands. Tuscany. SUPERFICIAL MEASURE. 144 Square inches.= 1 Sq. foot . Fr. Sq. Metres. . = *0929 9 Square feet.... . = 1 Sq. yard . . = *8361 30| Square yards .... . = 1 Sq. pole . . = 25*2928 40 Square poles ... . — 1 Rood- . - 1011*7136 4 Roods, or 4840 Sq. yards . . = 1 Acre .... . - 4046*8543 A Scotch Acre contains 6084 square yards, And an Irish Acre contains 7840 square yards. FRENCH SUPERFICIAL MEASURES. 1 Centiare.= 1*1960 square yards. 1 Arc (a square decametre). = 119*5991 „ 1 Decare. = 1195*9907 „ 1 Hectare. = 11959*9067 or 2 acres 1 rood 35 perches. SOLID MEASURE. Fr. Cubic Metres. 1728 Cubic inches.= 1 Cubic foot .= *0283 27 Cubic feet.= 1 Cubic yard ...... = *7645 42 Cubic feet.= 1 Ton of Shipping .. = 1*1892 A Load of unhewn timber.= 40 cubic feet. „ squared do.= 50 „ „ 1 inch plank.= 600 square feet. „ 1J inch do.= 400 „ „ 2 inch do.= 300 „ B 3 6 03? WEIGHTS AND MEASURES. NUMBER 0E CUBIC EEET IN A TON OE VARIOUS BODIES. Names of Bodies. Cubic feet in a ton. Names of Bodies. Cubic feet in a ion. Marble. 1306 Beech. 42 (Tranite t T ... t , .. 13-5 Teak. 48 Common Stone .. 14-22 Mahogany. 34 Paving do.... 14-82 Lignum Vitse.... 27 Sand. 23-6 Maple & Riga Fir 47*8 Coal.. 28-6 Larch. 65-8 Tallow. 38 Pitch Pine. 54-2 English Oak .... 37 Oil. 39 American do. 41 Proof Spirits .... 38-6 Ash. 42 Distilled Water.. 35-8 Elm. 53 Sea do.... 34-8 A Gallon of Linseed Oil weighs. 9 32 lbs. Avoirdupois. „ Distilled Water.10 „ „ Sea Water. 10 32 „ Proof Spirits. 9-27 „ IMPERIAL WINE MEASURE. 1 Gill.±= 8-665 cubic inches. 4 Gills.= 1 Pint _= 34-659 „ 2 Pints.= 1 Quart.... = 69318 „ 4 Quarts.... = 1 Gallon ... = 277'273843570 „ 10 Gallons .. = 1 Anker ... = 1604 cubic feet. 18 Gallons .. = 1 Runlet... = 2-888 „ 42 Gallons .. = 1 Tierce ... = 6-739 „ 63 Gallons .. = 1 Hogshead =10-109 „ 84 Gallons .. = 1 Puncheon — 13-478 „ 126 Gallons .. = 1 Pipe .... = 20-218 „ 252 Gallons .. = 1 Tun .... = 40-435 „ FRENCH MEASURES OE CAPACITT. 1 Millitre. = -06102 cubic inches. 1 Centilitre.... = -61024 „ 1 Decilitre.= 6-10238 „ 1 Litre (a cubic decimetre).= 61-02379 „ 1 Decalitre.= 610-23792 „ 1 Hectolitre.. ..= 3-5315 cubic feet. .1 Kilolitre.. = 35-3147 „ 1 Myrialitre ...... = 353-14694 „ The Litron usuel.................. ■ = 62-45 cubic inches. OF WEIGHTS AND MEASHEES. 7 A TABLE Showing the relative Value between the British Imperial Gallon and Foreign Measures of Capacity. Places and Names of Imp. Places and Names of Imp. Measures. Gall. Measures. Gall. Amsterdam, .mingle •266 Havannah... .arroba 3-415 „ ....kan Antwerp... .stoopen •220 Leghorn..wine fiasco •499 •608 .oil fiasco •443 ..litre •220 Lisbon.almude 3-641 Barcelona.. cortane 2-270 Malta.caffiso 4-582 Bordeaux.velte 1-672 Mocha.cuda 1-666 Cadiz .. great arroba 3-540 Naples.. wine barilla 9164 „ .. small arroba 3 124 ,, . ...oilstaja 2-226 Constantinople.alma 1146 Oporto.almude 5-311 Dantzic.. beer anker 12925 Petersburgh.. wedro 2-707 „ .. wine anker 9-915 Rotterdam... .stoop •564 Genoa.. wine barilla 16-349 Stockholm... .kanne •575 „ ... .oil barilla 14162 Trieste.boccali •312 Gibraltar.... gallon •909 Venice .. wine sechii 2-377 Hamburgh, .stubjen •797 „ ....oilmiro 3-356 Note. —The new Imperial standards of capacity, length, and weight, are not only to be adopted throughout Great Britain and Ireland, but also in all its colonies and dependencies. 4 2 4 9 18 38 54 72 108 IMPEEIAL ALE AND BEEE MEASHEE. 1 Gill. — 8-665 cubic inches. Gills. — 1 Pint. — 34-659 „ Pints . — 1 Quart. .. — 69-318 Quarts .... 1 Gallon. 277-274 „ 1-444 cubic feet. Gallons... — 1 Firkin....... , , - Gallons... — 1 Kilderkin.... — 2-888 „ Gallons ... — 1 Barrel. .. — 5-776 Gallons.. . — 1 Hogshead.... — 8-665 „ Gallons ..., — 1 Puncheon ... — 11-553 Gallons .... 1 Butt. 17-330 Note.— The old Ale Gallon contained 282 cubic inches, and the old Wine Gallon contained 231; hence, Imperial Gallons .... Imperial Gallons .... Old Ale Gallons .... Old Wine Gallons.... Cubic feet. Cubic inches. X ’98324 = Old Ale Gallons. X 1-20032 = Old Wine Gallons. X 1-01704 = Imperial Gallons. X -83311 = „ X 6-23210 = „ X -00361 = „ B 4 8 03? WEIGHTS AND MEASIJBES. 4 Gills..., 2 Pints .. 4 Quarts. 2 Gallons. 4 Pecks .. 8 Bushels 40 Bushels 80 Bushels IMPERIAL DRY MEASURE. 1 Gill.= 8-665 cubic inches. — 1 Pint.= 34-659 ,, = 1 Quart.= 69-318 „ = 1 Gallon.= 277-274 = 1 Peck.= 554-548 „ = 1 Bushel.= 1-2837 cubic feet. = 1 Quarter.= 10*2700 „ = 1 Wey.= 51-3500 „ = 1 Last. = 102-6999 „ Note.— The Winchester bushel contained 2150-42 cubic in¬ ches, and the Imperial bushel contains 2218*192 cubic inches; hence, Imperial bushels. X 1-0315157 = Winchester bushels, and Winchester bushels.. X -9694472 = Imperial bushels. A bushel of wheat is reckoned = 60 lbs. Avoirdupois. „ barley „ = 47 „ oats „ = 38 ,, pease „ — 64 „ beans „ = 63 HEAPED IMPERIAL MEASURE OE CAPACITY FOR COALS, CULM, LIME, FISH, POTATOES, FRUIT, AND OTHER GOODS. The Gallon. 2 Gallons .... = 1 Peck. 4 Pecks. = 1 Bushel .... 3 Bushels.... = 1 Sack. 12 Sacks.= 1 Chaldron .. 351-936 cubic inches. 703871 „ 2815-487 4-888 cubic feet, nearly 58-656 „ DIMENSIONS OF DRAWING PAPER IN FEET AND INCHES. Wove Antiquarian .. -iDfluble Elephant.... Atlas. Columbier. Elephant. ^Imperial. Super royal. Royal. Medium. Demy. 4 feet 4 inches 3 „ 4 „ by 2 feet 7 inches, by 2 „ 2 by 2 by 1 by 1 by 1 by 1 by 1 by 1 by 1 11 10 | H 7 7 6 3 * NUMERATION OP DECIMALS. 9 DIMENSIONS OP IMPERIAL CONICAL LIQUID MEASURES. Diameters. Two Gallon ..Top2§ in. ..Bottom ll§in. ..Depth 12-66498 in. Gallon. „ 2 „ 9 „ 10-28258 Half Gallon.. „ 1| „ 7 „ 8-23406 Quart. „ 1| „ 5£ „ 6 33249 Pint. 1 „ 4£ „ 5-14129 Half Pint.. | „ 4-11703 GUI. | „ 2 1 „ 3-16625 DIMENSIONS OP IMPERIAL CYLINDRICAL DRY MEASURES. Diameters and Depths. Eighth of a Peck.a cylinder of 4-45232 inches. Forpit or Half Gallon.. „ 5*60957 Gallon or Half Peck .. „ 7’06762 Peck. „ 8-90464 Half Bushel. „ 11-21914 Bushel. „ 14 13524 Quarter. „ 28 27048 Note.—M ultiply the decimal part by 8, the product will equal eighths of an inch, and decimal parts of an eighth of an inch. DECIMAL FEACTIONS. A Decimal Fraction derives its name from the Latin decern , “ ten,” which denotes the nature of its numbers, representing the parts of an integral quantity, divided into a tenfold proportion. NUMERATION Teacheth to read or write any number proposed, either by words or characters. In Decimal Fractions, the integer, or whole thing, #s a gallon, a pound, a yard, &c., is supposed to be divided into ten equal parts, called tenths; those tenths into ten equal parts, called hundredths; and those hun- 10 NUMEBATION OF DECIMALS. dredths into ten equal parts, called thousandths ; and so on, without end. So that a denominator of a de¬ cimal being always known to consist of a unit, with as many ciphers as the numerator has places, is, therefore, never expressed, being understood to be 10,100,1000, 10000, &c., according as the numerator consists of 1, 2, 3, 4, or more figures; thus, instead of T 2 ^, iVoV> the numerators only are written with a dot or comma before them, thus, *2, ’24, '211. If a unit or whole quantity of any description, as a gallon, a pound, a foot, &c., be divided into ten equal parts, the decimal represents as many of those parts as the decimal figure expresses,—thus, *7 means seven of those parts, or seven-tenths; but if the decimal consisted of two figures, unity would be understood to be divided into a hundred equal parts, of which the decimal represents as many as the figure expresses,— thus, '65 means sixty-five of those parts, or sixty-five hundredths ; and if the decimal consisted of three figures, unity would be supposed to be divided into a thousand equal parts, of which the decimal represents as many as the number expresses,—thus, *625 is six hundred and twenty-five of those parts; or, suppose the decimal '0625, unity would be understood to be divided into 10000 equal parts; but the value of decimal figures is made more plain by means of the following TABLE. Tenths. ‘5 Hundredths... '56 Thousandths. '567 Ten thousandths. '5678 Hundred thousandths. '56789 Thus, '5 is read five-tenths ; '56 is read five-tenths and six hundredths, or fifty-six hundredths ; '567 is read five-tenths, six hundredths, and seven thousandths, or five hundred and sixty-seven thousandths; and so on, as in the table. SEDUCTION OE DECIMALS. 11 Ciphers to the right hand of decimals cause no dif¬ ference in their value ; for -5, -50, '500 are decimals of the same value, beiug each equal to \; that is, -5 = y 5 ^, •50 = •500 = But if ciphers are placed on the left hand of decimals, they diminish their value in a tenfold proportion ; thus, -3, -03, *003, are three- tenths, three-hundredths, and three-thousandths, and answer to the vulgar fractions T 3 ^, re¬ spectively. A whole number and decimal are thus expressed, 85-75, 85-04, Ac. REDUCTION OE DECIMALS. By reduction we change vulgar fractions, and the lesser parts of coin, weight, measure, &c., into decimals, and find the value of any decimal given. Because decimals increase their value towards the left hand, and decrease their value towards the right hand, in the same tenfold proportion with integers, or whole numbers, they may be annexed to whole numbers, and worked in all respects as whole numbers ; hence, if simple arithmetic be well understood, there is little more to be learned than the placing of the separating point— the rule for which ought to be well attended to. 1. To reduce a vulgar fraction to a decimal of an equal value. Rule. —Add a cipher, or ciphers, to the numerator, and divide by the denominator, the quotient will be the decimal required. b 6 12 SEDUCTION OE DECIMALS. Example. —Reduce to a decimal. 32)14 0000 (-4375 Thus you may take any number of 128 ciphers at pleasure, but their number 120 will be best ascertained when the work 96 is finished; then you must have as - many decimal figures as you have taken 2 ^J annexed ciphers in dividing; and if " there are not so many in the quotient, 160 you must prefix ciphers to the left hand 160 of it—thus, 10 ”°°° - = -03125. Sometimes the quotient figures will repeat continu¬ ally, as thus, ^ = -6', then it is called a repetend, and the first figure which repeats may be dashed, to distinguish it from a terminate decimal. Sometimes two, three, or more figures will repeat, as ■i|. } thus, l2 3 3 ~ = ipp = -3'6'; such are called com¬ pound repetends or circulates, and the first and last of the repeating figures may be dashed. 2. To reduce the lesser parts of coin, weights, mea¬ sures, Sfc., to decimals. Rule. —Divide the least name by such number as will reduce it to the next greater ; to the decimal so obtained prefix the given number of the same name, then divide by such number as will reduce it to the next greater, always annexing ciphers to the dividend, as occasion may require: thus proceed till it be reduced to the decimal of the required integer. Or, reduce the given parts to a simple quantity, by reducing them to the lowest name mentioned; annex ciphers thereto, and divide by such numbers as will reduce them to the name required. Or, reduce the given parts to a vulgar frac¬ tion, and that fraction to a decimal. 6 REDUCTION OE DECIMALS. 13 Example 1. —Eeduce 17 s. 10 \d. to the decimal of a pound sterling. ^ 1-0. ,10+ ‘5 10-500 Here -y d . = 5 d. ; and — s. = — s. = *875s. ; ,, 17 + -875. 17-8750, , then- 20 -^ = —20— = '^9375Z. is the decimal required. Example 2.—Eeduce 2 feet 9 inches to the decimal of a yard. Here 2 ft. Qin. = 33 inches; and 3 Gin. = 1 yard; hence, by vulgar fractions, = 11222?= -91 6'6yd. as required. To find the value of any given decimal. Eule. —Multiply the decimal given by the number of parts of the next inferior denomination, cutting off the decimals from the product; then multiply the re¬ mainder by the next inferior denomination ; thus pro¬ ceeding till you have brought the least known parts of the integer. Example 1 . —Eequired the value of -89375 of a pound sterling. •89375/. 20 17-87500 shillings. 12 10-500 pence. 4 2-0 farthings. - Ans. 17s. 10£tf. Example 2.—Eeduce -625 of a hundred weight to its proper terms. Here '625 x 4 qrs. = 2*500 qrs., and 5 x 28 lbs. = 14*0 lbs .; whence 2 qrs. 14 lbs. = Ans. 14 A TABLE OF RECIPROCALS, FOR OBTAINING DECIMAL EQUIVALENTS. No. Recip. No. Recip. No. Recip. ' No. Recip. No. Recip. 1 I’000000 51 •019608 101 •009901 151 •006623 201 •004975 2 •500000 52 •019231 102 •009804 152 •006579 202 •004950 3 •333333 53 •018868 103 •009709 153 •006536 203 •004926 4 •250000 54 •018519 104 •009615 154 •006494 204 •004902 5 •200000 55 •018182 105 •009524 155 •006452 205 •004878 6 •166667 56 •017857 106 •009434 156 •006410 206 •004854 7 •142857 57 •017544 107 •009346 157 006369 207 •004831 8 •125000 58 •017241 108 •009259 158 •000329 208 •004808 9 •111111 59 •016949 109 •009174 159 •006289 209 •004785 10 •100000 60 •016667 110 •009091 160 •006250 j 210 004762 11 •090910 61 •016393 111 •009009 161 •006211 211 •004739 12 •083333 62 •016129 112 •008929 ; 162 •006173 212 •004717 13 •076923 63 •015873 113 •008850 163 •006135 213 •004695 14 •071429 64 •015625 114 •008772 164 •006098 214 •004673 15 •066667 65 •015385 115 •008696 165 •006061 215 •004651 16 •062500 66 •015152 116 •008621 166 •006024 216 •004630 17 •058824 67 •014925 117 •008547 167 •005988 217 •004608 18 •055556 68 •014706 118 •008475 168 •005952 218 •004587 19 •052632 69 •014493 119 •008403 169 •005917 219 •004566 20 •050000 70 •014286 120 •008333 170 •005882 220 •004545 21 •047619 71 •014085 121 •008264 171 •005848 221 •004525 22 •045455 72 •013889 122 •008197 172 •005814 222 •0045U5 23 •043478 73 •013699 123 •008130 173 •005780 I 223 •004484 24 •041667 74 •013514 124 •008065 174 •005747 224 •004464 25 040000 75 •013333 125 •008000 175 •005714 225 -004444 26 •038462 76 •013158 126 •007937 | 176 •005682 226 •004425 27 •037037 77 •012987 127 •007874 ; 177 •005650 227 •004405 28 •035714 78 •012821 128 •007813 : 178 •005618 228 •004386 29 •034483 79 •012658 129 •007752 i 179 •005587 229 •004367 30 •033333 80 •012500 130 •007692 ; 180 •005556 230 •004348 31 •032258 81 •012346 131 •007634t 181 •005525 231 004329 32 •031250 82 •012195 132 •007576: 182 •005495 232 •004310 33 •030303 83 •012048 133 •007519 ; 183 •005464 233 •004292 34 •029412 84 •011905 134 •007463 ; 184 *005435 234 004274 35 •028571 85 •011765 135 •007407 185 •005405 235 •004255 36 •027778 •011628 136 •007353 186 •005376 236 •004237 37 •027027 87 •011494 137 •007299 i 187 •005348 237 •004219 38 •026316 88 •011364 138 007246- 188 •005319 238 •004202 39 •025641 •011236 139 •007194 189 •005291 $39 •004184 40 •025000 90 •011111 140 •007143 190 •005263 240 •004167 41 •024390 91 •010989 141 •007092 191 •005236 241 •004149 42 •023810 92 •010870 142 •007042 192 •005208 242 •004132 43 •023256 93 •010753 143 •006993 | 193 •005181 243 •004115 44 •022727 94 •010638 144 •006944 | 194 •005155 244 •004098 45 •022222 95 •010526 145 •006897 195 •005128 245 •0041.82 46 •021739 96 •010417 146 •006849 j 196 •005102 246 •004065 47 •021277 97 •010309 147 •006803 197 •005076 247 •004049 48 •020833 98 •010204 148 •006757 | 198 •005051 248 •004032 49 •020408 99 ■010101 149 •006711 199 •005025 249 •004016 50 •020000 100 •010000 150 •006667 200 •005000 250 •004000 The numbers in the table are the denominators of the fraction: hence, multiply the reciprocal of the denominator by the numerator of the fraction, »nd the product is the decimal equivalent. Thus, suppose the decimal equivalent of 7-16 ths be required Reciprocal of 16 = -0625 x 7 = ‘4375 its decimal equivalent. ADDITION OP DECIMALS. 15 ADDITION OF DECIMALS. Rule. —Arrange the numbers under each other, according to their several values ; find the sum, as in addition of whole numbers, and cut off for decimals as many figures to the right hand as there are decimals in any one of the given numbers. Example.— What is the sum of 23*45, 7*849, 543*2, 8*6234, 253*004 ? If any of the decimals be repetends, bring all to the form of repetends, and make them begin and end together; then, in adding, in¬ crease the sum of the first column by as many units as would arise to carry to it if they were continued farther; and you will have a circulate in the sum beginning and ending like the others. Thus, 36249 ' = 3 - 625 . 23 - 45 ’ 7 - 849 543-2 8 - 6234 253004 836 1264 • 7500 ' • 6666 ' • 8888 ' • 8750 ' • 4444 ' The repetend of *6', the circulate of 6'9' and *3'72', are continued till their periods begin and end together. It may easily be observed that there would be two to carry to the first column if it were carried any farther. Note. —It is not always necessary to attend to the rule for repetends and circulates; three or four decimal figures, accord¬ ing to the rule, being sufficiently near the truth for common cal¬ culations. 2 - 50 ' 00000 ' 3 - 66 ' 66666 ' 7 ’ 69 ' 696 .‘) 6 ' 14 - 3723723 ' 28 - 23 ' 00087 ' 16 SUBTRACTION OR DECIMALS. SUBTRACTION OF DECIMALS. Rule. —Place the numbers directly under each other according to their several values, subtract as in whole numbers, and cut off for decimals, as in addition. Example. —Subtract 35-87043 from 132-005. 132 005 Jf both be single repetends, make them 35 87043 en( j together; and if there be occasion to 96-13457 f> orrow at the first figure, borrow 9 only - instead of 10 ; Thus, -83' If both be circulates, or one a repetend * 66 ' and the other a circulate, continue both till . 16 / their periods begin and end together ; then - if there should be occasion to borrow at the following figure, where they continued that figure farther, carry one to the first figure ; and if the numbers be in different denominations, reduce them till they be alike. Subtract •§£-§ from 1-f ; thus, If = 1-6W and sis ~ -8 34' •831' MULTIPLICATION OF DECIMALS. Rule. —Place the factors under each other, and mul¬ tiply them together, as in whole numbers ; then point off as many figures from the product (counting from right to left) as there are decimal places in both factors; observing, if there be not enough, to annex as many ciphers to the left hand of the product as will supply the deficiency. Example. —Multiply -4375 by -125. •4375 Here the product of -4375 by ;125 is 125 -0546875 ; and as there are three places of 21875 decimals in the multiplier, and four in the 8750 multiplicand, a cipher must be added on the 4375 left hand of the product to reduce it to its MULTIPLICATION OP DECIMALS. 17 To multiply a repetend by a single figure, add 1 to tbe first product for every 9 therein, so will you have a repetend in the product; and if there be several figures in the multiplier, do so with each product, and make them begin and end together; then add them as so many repetends. If the multiplicand be a circulate, consider the increase that would arise to the first product if the multiplicand were continued farther: thus do with each product, make them begin and end together, and add them by the rule for adding circulates. To contract the operation so as to retain only as many decimals in the product as may he thought necessary. Rule.—P lace the unit figure of the multiplier under that figure of the multiplicand whose place is the last to be retained in the product, and dispose of the rest so that they may all stand in contrary order to that in which they are usually placed. Then, in multiplying, reject all the figures to the right hand of the multiplying digit, and set down the product so that the right-hand figures may fall in a straight line under each other; observing to increase the first figure of every line with what would arise by carrying 1 from 5 to 14,—2 from 15 to 24,—3 from 25 to 34, &c., from the product of the two preceding figures when you begin to multiply ; and the sum will be the product required. Example. —Multiply 27‘14986 by 92-41035, retain¬ ing four decimals in the product. Common way. Contracted way. 27-14986 2714986 92-41035 53014-29 13 574930 24434874 81 44958 542997 2714 986 108599 108599 44 2715 542997 2 81 24434874 14 2508-9280 650510 2508-9280 18 DIVISION OF DECIMALS. DIVISION OF DECIMALS. Rule. —Prepare your decimals as directed for multi¬ plication, divide as in whole numbers, cut off as many figures for decimals in the quotient as the number in the dividend exceeds the number in the divisor, namely, make the number of decimal figures in the divisor and quotient together equal to the number in the dividend. Example. —Divide 173 5425 by 3 75. 3-75) 173-5425(46278 1500 2354 2250 1042 750 2925 2025 Although you may take addi¬ tional ciphers at pleasure, care must be had in reckoning the number taken in dividing for decimals in the dividend ; and if you put the deci¬ mal point in the quotient at any part of the operation, continuing the operation afterwards will not cause the point to be removed. 3000 3000 If there should not be so many figures in the quo¬ tient as there should be decimals, prefix ciphers on the left hand to make up the number. Example. —Divide 1'4850 by 247‘5. Thus, -^- s 5 ° = -006. And if there be not as many decimal figures in the dividend as in the divisor, you may annex a sufficient number of ciphers; and if there be not a remainder, you must add ciphers to the right hand of the quotient till you have taken as many in the dividend as will make the decimal figures therein equal to those in the divisor : thus,— Vrrr = 6000 ‘ DECIMAL TABLE. 19 A TABLE Of the Fractional Parts of an Inch when divided into thirty-two parts; likewise a Foot of twelve inches reduced to Decimals. Parts. Decimals. Parts. Decimals. i + & = -96875 t + A = - 468 75 11 = -9166' 3 + A = -9375 1 + A = -4375 10 = -8333' | + 3 *5 = -90625 t + A = -40625 9 = -75 3 = -875 f = -375 8 = -6666' f + A = -84375 i + A = -34375 7 = -5833' f + A = -8125 1 + t's = -3125 6 = -5 I + A = -78125 i + A = *28125 5 = -4166' I =-75 i = -25 4 = -3333' 1 + A = *71875 3 + as = -21875 3 = -25 1 + A = -6875 3 + A = ’1875 2 = *1666' 1 + 3*5 = -65625 3 + A = *15825 1 = -0833' = -625 i = -125 S = -072916' i + & = '59375 A = -09375 f = -0625 i + A = -3625 A = *°625 | = -052083' i + A = -53125 J =-5 3*3 = -03125 } = -04166' | = -03125 i = -02083' } = -010416' The utility of this table will appear evident by means of the following example :— Suppose a board or plate to be 30^ inches long, 8f inches broad, and § + of an inch in thickness ; re¬ quired its contents in cubic inches. Here 30-25 x 8 625 = 260-90625, and 260-90625 X ’4375 = 114-14648, &c., the number of cubic inches. = Ans. 20 SQUAEE BOOT. OF THE SQUAEE EOOT. When a number is multiplied by itself, as 6 X 6, or 9x9, &c., it produces the square or second power of that number; and the number itself is called the root of that square. A root consisting of a single figure is found by in¬ spection of the following table :— Root 1 2 3 4 5 6 7 8 9 Squares 1 4 9 16 25 36 49 64 81 Cubes 1 8 27 64 125 216 343 512 729 To extract or find the square root of any number which consists of more figures than one. Eule. —Make a point or dot over every second figure, commencing at the right hand, by which the given square will be pointed into periods of two figures each, except the first or left-hand period, which will sometimes have but one. The unit figure must always be the latter figure in the period; for the decimal point must be between the periods, and not in the middle of a period. Find the greatest root in the first period, which write in the quotient or root, and the square thereof under the same period; subtract therefrom, and to the re¬ mainder annex the next period for a dividend. Double the quotient for a divisor; see how often the divisor is contained in the dividend, with this con¬ sideration, that the answer must be the unit’s figure of the divisor. Write the answer in the quotient, also in the unit place of the divisor ; then multiply the divisor so com¬ pleted by the last quotient figure; write the product under the dividend, and subtract therefrom; to the remainder annex the next period for a new dividend. SQUARE ROOT. 21 Thus proceed with every period ; and if there be still a remainder, annex pairs of ciphers for additional periods, till you have a competent number of decimals in the root. Vulgar fractions, &c., may be reduced to decimals. The periods which are whole numbers give whole numbers, and decimal periods give decimals in the root. Example 1.—"What is the square root of 76176 ? 76i76(276 4 47)361 32« The proof — 276 X 276 = 76176. 546)3276 3276 Example 2.—Required the square root of *75. •75(-866 64 166)1100 966 The proof = •866 2 + -000044 = -75. 1726)10400 10356 44 remainder. Example 3.—Required the square root of -000854. -000854(-029 and -029 4 -029 49)454 261 441 58 Remainder 13 -000841 -- +13 •000854 = Proof. 22 CUBE ROOT. OE THE OTJBE EOOT. When a square is multiplied again by its root, as 6 x 6 x 6, it produces the cube or third power of that root. Single cubes are found by inspection of the preceding table. To extract the root of any number that consists of more than one figure. Rule.— Point the given cube into periods of three figures, and so that the unit figure be the last in its period; then from the first period subtract the greatest cube it contains ; put the root as a quotient, and to the remainder bring down the next period for a dividend. Eind a divisor by multiplying the square of the root by 300 ; see how often it is contained in the dividend ; and the answer gives the next figure in the root. Multiply the divisor by the last figure in the root. Multiply all the figures in the root by 30, except the last; and that product by the square of the last. Cube the last figure in the root; add these three last-found numbers together, and subtract this sum from the divi¬ dend ; to the remainder bring down the next period for a new dividend, and proceed as before. Example. —Required the cube root of 444194947. 444194947(763 343 7 X 7 x 300 = 14700)101194 95976 X 76 X 300 = 1732800)5218947 5218947 76 1. Divisor 14700 _6 88200 7 X 30 X 36 = 7560 6 X 6 X 6 = 216 95976 2. Divisor 1732800 _3 5198400 76 X 30 X 9 = 20520 3 X 3X3 =_27 5218947 CUBE BOOT. 23 Besides the preceding, there is another, and, perhaps, a better way of extracting the cube root, and which we shall attempt to render as intelligible as possible, by an explanation of the following example:— Example. —Bequired the cube root of 926859375. 1 at Cipher. 2nd Cipher. , «, ,• . •_ , 0 9 9 9 18 9 270 7 277 7 284 7 2910 5 2915 the required root. Place it also under the first cipher, and by going through the process of addition, we ob¬ tain 9. Multiply this by 9, and place the product under the second cipher. By again going through the form of addition we get 81, which being multiplied by 9, becomes 729. This is placed under the first period of figures 926, and subtracted from it, and to the re¬ mainder the second period 859 is annexed. Then recommencing at the column of the first cipher, we again place a 9, and add up. To the sum we add another 9, and obtain 27. We now multiply the 18 by 9, and place the product 162 under the 81 oi the second column, and by addition we obtain 243. We now turn 81 162 24300 1939 262&0 1988 9268o9375(975 729 197859 183673 14186375 14186375 2822700 14575 Alter having di¬ vided the number into periods of three figures each as be¬ fore, place before it two ciphers, at mode¬ rate distances from each other, and from the number itself, as represented in the example; then find, by a reference to the table, page 20, the nearest cube contained in 926,the firstperiod, andplace its root, which is 9, in the quotient, as ■fir'ai: -fiofiTPA nf 24 CUBE BOOT. to the first column, and annex one cipher to the 27, making it 270; and in a similar manner annex two ciphers to the 243 of the second column. All this is done preparatory to finding the second figure of the quotient, or of the required root. To find that figure we divide the number 197859 of the third column by 24300 of the second colu'mn. The quotient would appear to be 8: this, however, is found on trial to be too large, and we therefore take 7, which answers. We add this 7 to the first column, and multiply the sum 277 by 7, placing the product 1939 in the second column. Then, by adding up, we obtain 26239, the product of which by 7 we place in the third column and subtract, and to the remainder we annex the last period of figures 375. Hecommencing at the first column, we add 7 to the 277, and to the sum 284 we add 7 again. We now multiply the sum 284 by 7, and place the product in the second column and add up. Then to the 291 in the first column we annex one cipher, and to the 28227 in the second column two ciphers. To find the third figure of the root we divide the number 14186375 in the third column by the 2822700 in the second column, and the quotient is found to be 5. This we add to the first column, and multiply the sum 2915 by 5, and place the product under the 2822700 of the second column. Upon adding we obtain 2837275, whose product by 5 is placed under the 14186375 of the third column, and, being exactly equal to it, there is no remainder upon subtracting ; consequently the work is finished, and 975 is the required cube root. N.B.—The above method is not only useful for extracting the cube root, but also for that of any other root, attention being paid to the following directions;— CUBE ROOT. 25 Instead of dividing the number whose root is to be extracted into periods of three figures each, we divide it into periods of as many figures each as correspond to the order of the root; as, for example, four figures for the fourth root, five for the fifth, &c.; but the number of ciphers which we employ must be one less than that which corresponds to the order of the root, as three ciphers for the fourth root, and four for the fifth root. It must be borne in mind, too, that every new figure which we place in the quotient must be added to the first column as many times as correspond to the order of the root, and that the number of pro¬ ducts added to the second column be one less than the above number, and that added to the third two less, and so on. Lastly, before finding a new figure for the quotient, we annex one cipher to the number in the first column, two to that in the second, three to that in the third, and so on, until we arrive at the last column, where, instead of annexing ciphers, we bring down the num¬ bers that make up the next period. If, after extracting the root, we have a remainder, we can continue the quotient to decimals, by annexing to the remainder as many ciphers as there are figures in a period, and then proceeding as before ; and if the num¬ ber whose cube root is to be extracted consist of a whole number and a decimal, we divide it into periods by commencing at the unit’s place, and proceeding to¬ wards the left hand to divide the whole number, and towards the right to divide the decimal; and if the decimal do not contain a sufficient number of figures to make up the last period, we supply the deficiency with ciphers. The following example will, perhaps, make the sub¬ ject a little more plain: c 26 CUBE BOOT. Example. —Required the fourth root of 285762-321. 1st Cipher. 2nd Cipher. 3rd Cipher. 12 12 2400 249 2907 267 317400 921 318321 922 285762-3210(23-12 32000 7947 39947 8721 102268890000 98739268336 ..3529621664 rem. 48986321 319243 49305564000 64070168 921 922 923~ 32016600 18484 Involution and Evolution of numbers are very con¬ veniently performed upon the Engineer's Slide Rule, for when the slide is set straight at both ends, C is a line of squares, and D a line of roots ; consequently, against any number upon D is its square upon C, and against any number upon C is its root upon D. CUBE ROOT. 27 Example 1.—What is the square of 16 ? Opposite 16 upon D is 256, the square number upon C. Example 2.—Eequired the square root of 625. Opposite 625 upon C is 25 upon D, the root required. The cube root is performed by inverting the slide, and setting the number to be cubed upon B to the same number upon D, and against 1 or 10 upon D is the cube number upon B. Also, set the cube number upon B to 1 or 10 upon D, and where two numbers of equal value meet upon the lines B and D is the root required. Example 1.—Eequired the cube of 9. Set 9 upon B to 9 upon D, and against 10 upon D is 729 upon B. Example 2.—Eequired the cube root of 343. Set 343 upon B to 10 upon D, and against 7 upon B is 7 upon D, the root required. These lines also serve to multiply the square of any number, any number of times: thus,— To find the product of 6 times 6, multiplied by 3. Set 3 upon B to 6 upon D, and against 10 upon D is 108 upon B. To find the root of a number consisting of integers and decimals. Eule. —Multiply the difference between the root of the integral part of the given number, and the root of the next higher integer number, by the decimal part of the given number, and add the product to the root of the integral part of the number given; the sum will be the root of the number required, correct in all cases of the square root to 3 places, and in the cube root to 7, if the roots of the integral parts are correct to 4 places of decimals. c 2 28 SQUA.BE AFTD CUBE BOOTS OF NUFBEBS. Example 1.—Eequired the square root of 602. >761 = 78102 /6‘0 = 7-7460 difference *0642 X 2 Product = -01284 The n + 77459 .*./60-2 = 7-75874 as required, correct to -3 places of decimals. Example 2.—Eequired the cube root of 843-75. V844 = 9 4503 V843 _ 9-4466 difference -0037 X -75 Product = . -002775 Then + 94466 .\V843-75 — 9 449375 as required. If the square root is required, correct to more places ■ of decimals, the following rule is correct to 7 places:— Multiply the root of the nearest integer number by twice the difference between that and the given num¬ ber, and divide the product by 3 times the integer number added to the given number; and the quotient added to the root of the integer number will be the root of the given number nearly. Then, the root of 60'2 will stand thus, a/60 = 7-7460 x -4 = -2 X 2 (60 X 3) + 60-2 = 240-2)3-09840(-01289 + 7-7460 = 7'75889 2 402 the root required. 6964 4804 21600 19216 23840 If the number consist wholly of decimals, the root will be decimals also. 29 PRACTICAL GEOMETRY. Geometry is the science which treats of that species of quantity called magnitude, as represented by lines, surfaces, and solids. Practical Geometry is that art by which we are ena¬ bled to turn the rules of the science to a practical account. PBOBLEM i. To divide a given line into two equal 'parts. From A and B as centres, with any distance greater than half the ; > length of the line, describe arcs cut¬ ting each other in m and n ; then /\_ 3 a line drawn through the points m and n will divide the line into two equal parts, as required. pboblem rr. To divide a given angle into two equal parts. From the point C as a centre with any distance at pleasure, de¬ scribe the arc A B ; and from A and B as centres, with the same or any other convenient distance, describe arcs cutting each other in n; then a line drawn from the point C, through n , will divide the angle as required. PBOBLEM in. From any given point in a right line to erect a perpendicular. 1 2 3 4 . a ^ - -- 1.—On each side of the point A, take equal distances, c 3 30 PEACTICAL GEOMETBY. as AS, Ac; from b and c, as centres, with any radius greater than b A or c A, describe arcs cutting each other in n; then will a line drawn from the point A through n he the perpendicular required. 2 . —Take any point o, and with o as centre, and o B as radius, describe the arc cutting the line in m and B ; draw a line from m through the centre o, and continue it until it cut the opposite side of the arc in n ; then the line which joins n and B is the perpendicular required. 3 . —With the point B as centre, and with any radius, describe the arc l m n, cutting the line in l; with l as centre, and the same radius, cut the arc in m ; and with m as centre, and the same radius, cut the arc again in n. Now, with m and n as centres, and with any radius, describe arcs cutting each other in r, then the line joining r and B is the perpendicular required. 4 . —From the point B, on the line A B, take three equal parts (as feet, inches, &c.) to m: and from m and B as centres, describe arcs cutting each other in n, making the distance from B to n four parts, and from m to n five parts, then will the line B n be the perpen¬ dicular required. PBOBLEM IT. Through a given point o, to d/raw a straight line C D, parallel to a given straight line A B. Take any point m in the line r 0 s n A B; with m as a centre and c - — | radius m o, describe the arc on; / j with n as a centre and the same A • _ X / b distance describe the arc s m. n m Take the arc o n in your com¬ passes, and apply it from m to s; through o s draw C D, and it will be the parallel as required. PRACTICAL GEOMETRY. 31 PROBLEM V. To divide a right line into any number of equal parts. Let A B be tbe line that is to be divided; then at the * s....”-. point A draw a line making A j w any angle with the line A B B and at B draw another line \ % parallel to it. Upon each of ** these lines, beginning at the points A and B, cut off as many equal parts as you require the line A B to be divided into, as A 1, 1 2, 2 3, &c., B 1, 1 2, 2 3, &c.; then draw lines joining the points A and 5, 1 and 4, 2 and 3, &c., and the lines so drawn will cut A B into the required number of equal parts. PROBLEM VI. To divide a triangle into two equal parts , and still retain its original form. Let A B 0 be the given triangle to be divided, bisect one of its sides as A B, and describe the semicircle A G- B; bisect the semicircle in G-, and at a distance from A, equal to A Gr or B G, draw the line xy, parallel to B C, which is the line of equal division as required. problem vn. Through any three points out of a right line to describe the circumference of a circle. From the middle point as a centre, with any convenient distance, describe the circle, or arcs of a circle, as A and B, and from the other points, with the same distance, describe arcs cutting the circle in C D and E E ; draw lines through C D and E E, and where 32 PRACTICAL GEOMETRY. 1 they intersect each other at o is the centre of the circle required. PROBLEM Till. To find the centre of a given circle. Draw any chord A B, bisect it in D, and through D draw E C perpendicular to A B ; then bisect E 0, and the point of section /will be the centre of the circle. PROBLEM IX. From a given point to draw a tangent to a circle. Let Abe the point from which it is B ^ A required to draw a tangent to the "x/ circle D E E. Join A and the centre / (\/\ C, and upon the line A C describe the f Vk } semicircle ADC, then through D, V J the point at which the circles intersect, E P draw the line A B, which will be the tangent required. PROBLEM x. To find a mean proportional between two given right lines , or the side of a square equal to a given rectangle. Upon a right line as a diameter equal to both given lines, describe the semicircle ABC, and where the two lines meet, or between their re¬ spective lengths, erect a perpendi- A cular to the semicircle at B, and the perpendicular will be the mean proportion or side of the required square, equal to the given rectangle. PRACTICAL GEOMETRY. 33 PROBLEM XI. To find the side of a square which shall he equal in area to a given triangle. Let A B C be the given tri¬ angle. From C let fall C D per¬ pendicular to A B, and produce the > line B A to E, making A E equal l to half the perpendicular 0 D; then upon E B describe the semicircle E F B, and from the point A erect a perpendicular cutting the circle in F; then A F will be the side required. PROBLEM XII. Upon a given right line to construct a square. Let A B he the line upon which f it is required to construct a square. ^ With A as centre, and A B as radius, describe the arc BCD; /_£ - with B as centre, and the same ^T 0 radius, cut the arc in C; and with C as centre, and the same radius, cut the arc again in D ; then with C and D as centres, and with equal radii, describe arcs cutting each other in F, and from F draw F A, cutting the circle in G-. Then with Q- and B as centres, and the distance A B as radius, describe arcs cutting each other in M; join G M and B M, and the figure G M B A is the required square. PROBLEM XIII. To make a rectangle equal to a given triangle. Let A B C be a triangle, to which it is required to make a rectangle equal. Bisect A B in D, and at D erect a perpendicular; from B draw a line parallel to D E, and from C * c 5, 34 PEACTICAL GEOHETEY. draw a line parallel to A B; then the figure DEFB is the rectangle required. PEOBLEM XIV. To produce a rectangle equal to a given square. 3 : Suppose AB CD he the given 1 square, also B E one end of the re- 1 quired rectangle, draw E E parallel to B C, join B N, continue the side of the square B C, and draw the line A Gi parallel with B N, until it intersects at G-, then B Gr is the side of the rectangle required. peoblem xv. . To make a triangle equal to a given quadrilateral Jigu/re. Let A B C D be the given quadrilateral figure. Join D and B, and from C draw C E parallel to B D; produce A B to E, and join D and E; then the triangle * A D E is the triangle required. PEOBLEM XVI. To circumscribe a square about a given circle. Draw two diameters at right angles as m n and 0 P; from m n D 0 P as centres, with the radius of t the circle, describe arcs cutting each m other in A B C and D ; join A B, s B C, C D, D A, and A B C D will ^ he the square required. And from A as a centre, with the distance A 0 , cut PEACTICAL GEOMETET. 35 the lines A B, A D, in 2 and 7 ; from B as a centre cut the lines B A, B C, in 1 and 4; from C as a centre cut the lines C B, C D, in 3 and 6; and from D as a centre cut the lines D C, D A, in 5 and 8 ; join 1, 8; 2, 3; 4, 5; and 6, 7; and 1, 2, 3, 4, 5, 6, 7, 8, will be a regular octagon. peobleSt XVII. Upon a right line to describe cm octagon . On the extremities of one side A B erect the perpendiculars A F and B E; continue the line A B to A m and B n, forming the angles m A r and nBs; bisect the angles with the lines A H and B C; make each of those lines equal to A B; make H G- and C D the same length, and parallel to A F and B E: from G and D as centres, with the radius A B, describe arcs cutting A F and B E; join G F, F E, and E D, then A B C D E F G H will be the octagon required. PEOBLEM XVIII. In a given circle to inscribe any regular polygon. Divide the diameter A B into as many equal parts as the polygon is required to have sides; from A and B as centres, with the distance A B, describe arcs cutting each other in C; draw a line through the second division, meeting the circumference at D; join A D, and it will be the Bide of the polygon required. c 6 36 PRACTICAL GEOMETRY. PROBLEM XIX. To find the side of a square that shall he any number of times the area of a given square. Let A B C D be the given square, then will the diagonal B D be the side of a square AEE G, double in g ^ area to the given square A B C D; i and if the diagonal be drawn from B to G-, it will be the side of a square AHKL, three times the area of the square ABCD, or the diagonal B L “ ~ will equal the side of a square four times the area of ithe square A B C D, && problem xx. To find the diameter of a circle that shall be any number of times the area of a given circle. Let ABCD be the given circle; .draw the two diameters A B and C D at right angles to each other, and the chqrd A D will be the radius of the .circle o P, twice the area of the given circle nearly; and half of this chord A D will be the radius of a circle that will contain half the area, &c. problem xxi. To divide a given circle into any number of co-centric parts equal to each other. Upon the radius A B describe the semicircle A e d B; divide A B into the proposed number of equal / parts, as 1,2, &c.; erect the perpen- I diculars 1 e , 2 d, &, c., meeting the semicircle in e and d; then from the centre A, and radii A e, A d, &c., describe circles ; so shall the circle be divided into the proposed number of equal parts as required. PRACTICAL GEOMETRY. 37 PROBLEM XXII. To find the side of a square nearly equal in area to a given circle. Draw the two diameters A B and C D at right angles to each other, c bisect the radius 0 C by a line from one end of the diameter at A, meet¬ ing the circumference in E, then will the line A E be the side of a square nearly equal in area to the given circle. ^ And if the line E E be drawn parallel to C D, it will be £ of the circumference nearly. Or three times the diameter A B or C D, and once the versed sine Q H, of the angle A 0 D, will be the circumference nearly. PROBLEM XXIII. To find a right line that shall he nearly equal to any given arc of a circle. Divide the chord A B into four equal parts, set one part on the circumference from B to D, draw a line from C, the first division on the chord; and twice the length of the line C D will be the length of the arc nearly. PROBLEM XXIY. To describe the largest possible circle in a triangle. Let A B C be a triangle; bisect the two angles A and B (by Problem II.), and from D, where the lines A D and B D meet, draw D E perpen- a dicular to A B; then, with D as centre, and D E as radius, describe the circle E E Q-, which is the one required. 38 PRACTICAL GEOMETRY. PROBLEM XXV. To describe an ellipsis, the transverse and conjugate diameters being given. Prom o as a centre, with the difference of the transverse and conjugate semi-diameters, set off o c and o d; draw the diagonal c d, and continue the line o c to k, by the addition of half the dia¬ gonal c d, then will the distance o Tc be the radius of the centres that will describe the ellipsis; draw the lines A B, CD, C E, and B H, cutting the semi¬ diameters of the ellipsis in the centres k B m n ; then with the radius m s, and with k and m as centres, de¬ scribe the arcs D H and A E; also, with the radius n r, and with n and B as centres, describe the arcs E D and A H, and the figure AEDH will be the ellipsis re¬ quired. PROBLEM XXVI. To describe a parabola, any ordinate to the axis and its abscissa being given. Let Y B and B o be the given c abscissa and ordinate; bisect B o in m, join Y m, and draw m n per¬ pendicular to it, meeting the axis in n; make Y C and V F each equal to B n, then will F be the v focus of the curve. Take any number of points, r, r, &c., in the axis, and draw o the double ordinates of an indefi¬ nite length. From F as a centre, with the radii C F, C r, &c., describe arcs cutting the corresponding ordinates in the points oooo, &c., and the curve oYo drawn through all the points of intersection will be the parabola required. PRACTICAL GEOMETRY. 39 PROBLEM XXYII. To draw at the circumference of a circle lines tending towards the centre when the centre is inaccessible. Mark off upon any por¬ tion of the circumference any number of equal ^, parts, and with any radius * greater than the length of one division, but less than that of two, and with the centres A, B, C, D, &c., describe arcs cutting each other in b, c, d, &c. ; then the required lines may be drawn by joining a A, b B, c C, and so on. To draw the end lines a A, g GL With the distance A b as radius, and with the centres B and F, describe the arcs a and g ; then, with the dis¬ tance B b as radius, and the centres A and Gr, describe arcs cutting the former ones, and at the points of inter¬ section a and g draw A a and g Gr, which will be the required straight lines. From any given point in the circumference (as A in the annexed figure) to draw a line tending towards the centre. With A as centre, and with any d>^ radius, cut the circle in B and 0; then with B and C as centres and a radius greater than the former, de¬ scribe arcs cutting each other in D; , join D and A, and the line D A will be the one required. To draw from a point , without the circumference , a line tending towards the centre. Let A be the given point. With A as centre, and with any convenient radius, /\ describe an arc cutting the circumference / \ in B and C; with B and C as centres, and the distance B A as radius, describe arcs ^ cutting each other in A and D ; then join \ / A and D, and the line so drawn will be the one required. 40 PRACTICAL GEOMETRY. PROBLEM 3XVTII. To describe an elliptical arc, the width and rise of span being given. Bisect the chord or width of q span A B, and at the point of n section C erect a perpendicular C D equal to the height of span; erect a perpendicular also at A, making A q equal to C D ; join q and D, and bisect A q in n, and A C in r; join n and D, and from q through r draw the line q l, meeting the line D C produced. Now, bisect the line s D in f and at the point of section erect at right angles the line f g, also meeting the line D C pro¬ duced, join g and q, and the line so drawn will cut the line A B in Tc; make C P equal to C Tc, and through P draw the line g i ; then, with g as centre and g B as radius, describe the arc sDi, and with Tc and P as centres and Tc A and P B as radii, describe the arcs A s and B i ; the construction of the arc will then be completed. MENSURATION OP SUPERFICIES. 41 MENSURATION. Mensuration is the method of calculating the com¬ parative magnitudes of figures ; and it is divided into two parts,—Mensuration of Superficies or Surfaces, and Mensuration of Solids. The magnitude of a surface is called its area, and is the space enclosed between its boundary lines. The magnitude of a body is called its solid contents, and is expressed in cubic feet, inches, &c. MENSURATION OF SUPERFICIES. Square. Rectangle. Rhombus. Rhomboid. Fig. 1. Fig. 2. Fig. 3. Fig. 4. A Square is a quadrilateral figure which has all its sides equal, and all its angles right angles. A Rectangle is a four-sided figure which has its angles right angles, and its opposite sides parallel. A Rhombus is a parallelogram whose sides are equal, but whose angles are not right angles. A Rhomboid is a parallelogram whose adjacent sides are unequal, and whose angles are not right angles. A Trapezoid is a four-sided figure which has but two of its sides parallel. A Circle is a figure bounded by one line called the circumference; and is such, that all lines drawn to the cir¬ cumference from a certain point within the figure called the centre are equal to each other. Any of these lines is called 42 MENSURATION a radius; and a line drawn through the centre, termi¬ nating both ways in the circumference, is called a diameter. The portion of circle cut off by a diameter is called a semicircle. An Arc of a circle is any portion of the circumference. A Segment of a circle is a figure contained by an arc and its chord. A Versed Sine is a line drawn from the middle of a chord perpendicular to the circumference. A Sector of a circle is a figure contained by two radii and an arc, as A C B E. Some useful Properties of Numbers employed in the Solutions of the following Problems. 1. Half the sum of any two numbers increased by half their difference, will give the greater number; and half their sum diminished by half their difference, will give the less number. 2. The quotient arising from the division of the sum of two or more numbers, is equal to the sum of the quotients arising from the division of the parts separately, by the same divisor. 3. Any three of the four following quantities of a division sum, viz. divisor, dividend, quotient, and remainder, being given, the fourth may be found by the following formula. Let d = the divisor; d = the dividend; q = the quotient; and r = the re¬ mainder. Then we shall have d = D =(dX{)+r;} =^T; and r = D-(g X d). q d 4. An even number cannot divide or measure an odd number, nor a greater a less. 5. A given number is divisible by 2 if the last digit is even ; it is divisible by 4 if the last two digits are divisible by 4; it is divi¬ sible by 8 if the last three digits are divisible by 8; and in general it is divisible by 2", if the last n digits are divisible by 2 n . 6. A number is divisible by 3 if the sum of the digits is divi¬ sible by 3; such number also may be divided by C if, besides this, the last digit is even; it is also divisible by 9 if the sum of the digits can be divided by 9. The method of proof, by casting out the nines, in Addition, Multiplication, and Division, depends upon this theorem. 7. Every number that has the last digit 5 or 0, such number is divisible by 5 in both cases, and by ten in the latter case. OE STTPEBEICIES. 48 8. A number is divisible by 11 when the sum of the digits in the odd places (counting from the right or unit’s place) be equal to the sum of the digits in the even places; or if the difference of these sums be divisible by 11, the number itself is divisible by 11 . 9. If any two numbers be separately divided by 9 or 3, and the two remainders multiplied together, and that product divided by 9 or 3, the last remainder will be the same as if you divided the product of the first two numbers by 9 or 3. 10. If any number ending with 1, 3, 7» or 9, be the numerator or denominator of a fraction, and will not divide by 3, 7, or 9, that fraction is generally in its lowest terms, for Every number must terminate in one) or other of the ten digits . . . / 0 1 2 3 4 5 6 7 8 9 But no even number can be a prime) number ; hence, take away We have remaining 7 . No number terminating in 0 or 5 can) „ be prime.J _ ' Hence it follows that every prime) ,5.7.9 7.9 number must terminate in one or > . 1 . other of these four digits . . . J But of such numbers none the sum of whose digits is a multiple of 3 can'be prime. Nor any terminating in 1, 3, 7> 9, that is, in any power or multiple of another number. 11. Every prime number is of one of the forms 8n + 1, 8ra + 3, 8a+ 5, 8« + 7; 12a + 1,12a+ 5, 12a + 7, 12a + 11; 4a ±1; 6a ± 1; 16a ± 1, 16a ± 3,16a ± 5, 16a ± 7; 60a ± 1, 60a ± 7, 60a ± 11, 60a ± 13, 60a ± 17, 60a ± 19, 60a ± 23, 60a ± 29. 12. When any prime number (above the number 3) is either increased or diminished by unity, the result is always divisible by 6; thus, take for example, (173 -f- 1) — 6 = 174 — 6 = 29, and (5749 - 1) 6 = 5748 -f- 6 = 958. ‘ 13. When any set of numbers are placed in the form of frac¬ tions, with the sign of addition or subtraction between them, and should the whole of these numbers that are in the numerator and denominator contain a common divisor, they may be abbreviated by dividing each of them by the common divisor. 14. When the numerator is equal to the denominator, the frac¬ tion is equal to the integer; thus, | = 1. 15. When the numerator is greater than the denominator, the fraction is greater than the integer; as, § = 1§. 16. If the numerator and denominator of a fraction be either multiplied or divided by the same number, the product or quo¬ tient will be a new fraction equal in value to the former; thus, | *r* | or | x | = §g, all of which have the same value, for 44 MENSURATION PROBLEM I. To find the area of any ‘parallelogram. Rule. —Multiply the length by the perpendicular height, and the product will be the area. Example. —Required the area of a rhomboid A B D 0 {Fig. ±,page 41) whose length A B = 20*5, and perpen¬ dicular height a A = 11*75. Here ABx«A = 205 X 1175 = 240*875, the area of the rhomboid A B D C. Note.—I n a square {Fig. 1), or rectangle (Fig. 2, page 41), the perpendicular height is the breadth; therefore, to find the areas of a square and rectangle, multiply the length by the breadth. PROBLEM U. To find the area of a trapezoid. Rule.—A dd together the two parallel sides, multiply their sum by the breadth or height, and half the pro¬ duct is the area. Example. —Required the area of a trapezoid whose sides A B and C D are 14*5 and 10*25, and breadth a A = 7*25. Here (AB + CD) x a A = (14*5 + 10*25) x 7*25 . 2 the area of the trapezoid ABD C. : 89*71875, PROBLEM III. To find the area of a triangle. Rule. —Multiply one of its sides as a base by a perpendicular let fall from the opposite angle, and take half the product for the area. Or, from half the sum of the three sides subtract each side separately, and multiply the three remainders so obtained and the half sum together, and the square root of the product will be the area. OS' SUPERFICIES. 45 Example 1.—Required the area of a triangle ABC, ■whose base A B = 165, and perpendicular D C = 10-25. TT ABxDC 16-5x10-25 Here-g- = -2- = 84-5625, the area of the triangle ABC. Example 2.—What is the area of the triangle ABC, whose sides are B C = 8, A C = 12, and A B = 16 respectively ? Here |(AB + AC + BC) = | (16 + 12 + 8) = 18 = half the sum of the three sides. 18 - 16 = 21 18 — 12 = 6 > the three remainders. 18 - 8 = 10 J Then V (18 x 2 X 6 x 10) = V 2160 = ^ (3 2 X 4 2 x 15) = ]2 V 15 = 12 x 3-873 = 46476 = the area of the triangle A B C = Ans. PROBLEM IV. If any two sides of a right-angled triangle he given , the third side may he found by the following rules. 1. To the square of the base add the square of the perpendicular; and the square root of the sum will be the hypotenuse or longest side. 2. Multiply the sum of the hypotenuse, and one side by their difference; and the square root of the product will be the other side. Example 1.—Given the base A B = 16, and per¬ pendicular B C = 12 ; required the length of the hypo¬ tenuse A C. Here V (AB ! + B C 1 ) = V (16 2 + 12 2 ) = s/ (256 + 144) = V 400 = -- A -g = (19 8 x 8) - 34-4 _ 158-4-34-4 A 3 3 = = 4T3 / 3, the length of the arc ABC. = Ans. d 2 52 MENSURATION PROBLEM XI. To find the diameter of a circle by having the chord and versed sine given. Rule. —Divide the square of half the chord by the versed sine, to the quotient of which add the versed sine, and the sum will be the diameter. Or, if the sum of the squares of the semichord and versed sine be divided by the versed sine, the quotient will be the diameter of the circle to which that segment corresponds. c Example. —Given the chord A B a = 24, and versed sine CD = 8; re¬ quired the diameter of the circle C E. Here { C d} + CD = (12 ! -=-8) + 8 = (H4 ~ 8) + 8 = 18 + 8 = 26, the diameter. = Ans. Or thus, | (A_B) 2 + CD ! ) -r 8 = (12 2 + 8 2 ) 8 = (144 + 64) ■— 8 = 208 -i- 8 = 26, the diameter, as before. PROBLEM XII. To find the area of a parabola, or its segment. Rule. —Multiply the base by the perpendicular height, and two-thirds of the product is the area. Example. —What is the area of a parabola, whose base is 20 feet, and perpendicular height 12 ? Here § x 20 X 12 = 2 x 20 X 4 = 160 feet, the OF SUPERFICIES. 53 Table of Versed Sines, in inches, quarters, and fractions of a quarter, by which to ascertain the diameters of circles corresponding to any segment or part of a circle having a chord of three feet. Table of the relative Proportions of the Circle, its equal and inscribed squares. 1. The Diameter of a circle .. X -88621 = the side of an equal 2. „ Circumference .... X -2821 / square. 3. „ Diameter. x -70711 = the side of an in¬ 4. „ Circumference. X -2251 / scribed square. 5. Area X -63661 = the area of an in¬ scribed square. 6. „ Side of inscribed square X 14142 = the diameter of a circumscribing circle 7- „ Side of inscribed square X 4-443 = the circumference of a circumscribing circle. = the diameter of an 8. „ Side of a square .... X 1'128 equal circle. 9. „ Side of a square .... X 3-545 = the circumference of an equal circle. D 3 54 MENSURATION Examples illustrative of the preceding table. Example 1.—The diameter of a circle is 12*5; re¬ quired the side of a square equal in area to the given circle. Here 12*5 x *8862 = 11*07750, side of equal square. = Ans. Ex. 2.—The circumference of a circle being 53*4; required the side of a square equal in area. Here 53*4 x *2821 = 15*06414, side of equal square. = Ans. Ex. 3.—The diameter of a circle being 18 ; required the side of the greatest square that can be inscribed therein. Here 18 X *7071 = 12*7278, side of inscribed square. = Ans. Ex. 4.—The circumference of a circle is 86; required the side of inscribed square. Here 86 X *2251 = 19*3586, side of inscribed square. = Ans. Ex. 5.—The area of a circle being 371*5 ; require* the area of the greatest square that can be inscribe* within the circle. Here 371*5 x *6366 = 236*49690, area of the square. = Ans. Ex. 6.—The side of a square being 19*375 ; required the diameter of its circumscribing circle. Here 19*375 X 1*4142 = 27*4001250, diameter. = Ans. Ex. 7.—Required the circumference of a circle to circumscribe a square, each side being 19*375. Here 19*375 X 4*443 = 86*083125, circumference of the circle. = Ans. Ex. 8.—The side of a square being 13*5 ; required the diameter of a circle equal in area to the given square. Here 13*5 X 1*128 = 15*228, diameter of the circle. = Ans. OP STJPEE.MO'IES. 55 Ex. 9.—The side of a square being 13 5 ; required the circumference of a circle equal in area to the given square. Here 13*5 X 3645 = 47 , 8575, circumference of the circle. = Ans. Some of the properties of a circle. 1. —It is the most capacious of all plain figures, or contains the greatest area within the same perimeter or outline. 2. —The areas of circles are to each, other as the squares of their diameters, or of their radii. 3. —Any circle whose diameter is double that of another, contains four times the area of the other. 4. —The area of a circle is equal to the area of a triangle whose base is equal to the circumference, and perpendicular equal to the radius. 5. —The area of a circle is equal to the rectangle of its radius, and a right line equal to half its circumference. 6. —The area of a circle is found by squaring the diameter, and multiplying by -7854; or by multiplying the circumference by the radius, and dividing the pro¬ duct by two. Example 1.—Required the area of a circle, the diameter being 305. Here 305 2 x ‘7854 = 730618350, the area re¬ quired. Example 2.—"What is the area of a circle when the diameter is 1 ? In this case the circumference is 3 1 1416, half of which is 15708, and half of 1 = *5; then 1-5708 x ‘5 = •7854, the area. PBOBLEM XIII. Saving the area of a circle given, to find the diameter. Rule. —Multiply the area by 452, and divide the product by 355, the square root of the product will be the diameter. d 4 56 MENSURATION Or, multiply the square root of the area by 1-12838, and the product will be the diameter. Or, divide the area by *7854, and extract the square root. Example. —Required the diameter of that circle whose area is 122-71875. Here J- 122-71875 X 452 355 _ s/ (392-7 x 113 x 284) 284 = 12-5 diameter. = Ans. = 7 3927 X 113 2840 J 12602528-4 _ 3550 284 284 Or thus, f 122-71875 = 11-078; and 11-078 X 1-12838 = 12-500194 = 12-5 diameter. = Ans. as before. PROBLEM XIY. To find the area of a sector of a circle^. Rule. —Multiply the length of the arc by the-radius of the circle, and half the product will be the area.*^-.^ Example.— Required the area of a sector of a circle whose arc A B C = 26-666, and radius BO = 16-9. Hcrc ABCxBO _26-666 X 169 b 450-6554 = 225*3277 the area of the sector A B 0 O A. = Ans. PROBLEM XT. To find the area of a segment of a circle. Rule. —Multiply the versed sine by *626, to the square of the product add the square of half the chord (or multiply the square of the versed sine by •391876, and to the product add the square of half the chord) ; multiply twice the square root of the sum by f of the versed sine ; and the product will be the area. OP SUPERFICIES. 57 Example. —Required the area of a segment of a circle whose chord A B = 48, and versed sine C D = 18. Here 2 x V ^-391876 X C D= + (^)'J X = 2 x y/ (-391876 x 18* + 24 2 ) x 2 *— 8 = 24 x v/ (-391876 x 324 + 576) = 24 x s/ (126-9'67824 + 576) = 24 x V 702-967824 = 288 X ^ 4-881721 = 2-20946 x 288 = 636-32448 the area. = Ans. The following is a near approximate to the preceding rule: To the cube of the versed sine, divided by twice the length of the chord, add -f of the product of the chord, multiplied by the versed sine; and the sum will be the area of the segment nearly. Take the last example:— Versed sine C D = 18, and chord A B = 48. Here C D 3 A B x 18 . 2 x ABxCD 2 x 48 ‘ + 2 x 48 x 6 = 60-75 + 576 = 636"75 the area as before nearly. Or, the area of a segment may be found by finding the area of a sector having the same radius as the segment; then deducting the area of the triangle leaves the area of the segment. PROBLEM XVI. To find the area of a circular ring or space included between two concentric circles. Rule. —Add the inside and outside diameters to¬ gether, multiply the sum by their difference, and by •7854, and the product will be the area. d 5 68 MENSUBATION Example. —The diameters of two concentric circles, A B and C D, are 10 and 6; required the area of the ring or space contained between them. Here (A B + C D) + (A B — C D) / N X -7854 = (10 + 6) X (10 - 6) JL D \... X -7854 = 16 x 4 X -7854 = 502656 the area. = Ans. PBOBLEM XYII. To find the area of an ellipse. Hole. —Multiply the transverse or longer diameter by the conjugate or shorter diameter, and by ‘7854, and the product will be the area. Example. —Required the area of an ellipse whose longer diameter A B = 12, and shorter diameter C D = 9. Here ABxCDx *7854 = d 12 X 9 X -7854 = 84-8232 the area. = Ans. A c Note. —If half the sum of the two diameters he multiplied hy 3-1416, the product will be the circumference of the ellipse, near enough for most practical purposes. Thus J(AB + CDX 31416) = 4(12 + 9) X 3-1416 = 21 X 1-5708 = 32-9868, the circumference required. OF SOLIDS. 59 MENSURATION OF SOLIDS. By solids are meant all bodies, whether solid, fluid, or bounded space, that can be comprehended within length, breadth, and thickness. PEOBLEM i. To find the convex surface and solid content of a cylindrical cylinder , or any figure of a cubical form. Rule 1.—Multiply the circumference of the base by the height of the cylinder, and the product is the con¬ vex surface. Rule 2.—Multiply the area of the base by the height of the cylinder, and the product is the solid content. Example 1 . —Required the convex surface of the cylinder A B C D, whose base A B = 32 inches, and perpendicular height B C = 6 feet. Here 3*1416 x A B x B C = 3*1416 X 32 X 72 inches = 7238*2464 square or superficial inches, and 7238*2464 -f- 144 = 50*2656 superficial feet. = Ans. Example 2.—Required the solid content, in cubic inches and cubic feet, of the same cylinder as in Ex¬ ample 1. Here (A B 2 x *7854) xBC = (32 2 X *7854) X 72 = 1024 x *7854 x 72 = 57905*9712 cubic inches, and 57905*9712 -r- 1728 = 33*5104 cubic feet. Example 3.—Suppose the cylinder A B C D be in¬ tended to contain a fluid, and that the sides and bottom d 6 60 MENSURATION are each one inch m thickness, how many imperial gallons would it contain ? Here {[(AB-2k) 3 X *7854] x (BC-l™.)} -j- 277-274 = {[(32 - 2) a x -7854] x (72 - 1)} -f- 277-274 = (30 2 x -7854 x 71) -r- 277-274 = (900 x •7854 x 71) 277-274 = 50187-06 cubicin.— 277-274 = 181 imperial gallons. = Arts. Or, since the reciprocal of 1 -f- 277-274 = *003607, therefore 50187-06 cubic in. x -003607 = 181 imperial gallons. = Ans. as before. PROBLEM II. To determine the dimensions of any cylindrical vessel, whereby to contain the greatest cubical contents, bounded by the least superficial surface. Bule. —Multiply the given cubical contents by 2- 5465, and the cube root of the product will equal the diameter, and half the diameter equal the depth. Example 1.—Suppose a cylindrical vessel is to be made so as to contain 600 cubic feet, and of such di¬ mensions as to require the least possible materials by which it is constructed, what must be its depth and diameter ? Here V (600 x 2-5465) = V 1527-9 = 11-5177 feet diameter, and 11-5177 -r- 2 = 5-75885 feet in depth. Note. —If the vessel is to be made with two ends, then the cube root of four times the solidity divided by 3- 1416 will equal both the length and diameter, so as to require the least possible quantity of materials in its construction. Example 2.—Suppose a cylindrical vessel is to be made with two ends, so as to contain 600 cubic feet, as in Example 1; required its depth and diameter, so that it may be constructed with the least possible quantity of materials. OP SOLIDS. 61 Here 7( 6 ^6 4 ) = ^ 764= 9-1418, the depth and diameter in feet, as required. Or thus, V (600 x 4 x -318309) t= V 763 941600 = 91416, the Ans. as before, nearly. PROBLEM in. To find the surface and solid content of a cone or pyramid. Rule 1.—Multiply the circumference of the base by the slant height, and half the product will be the slant surface ; to which add the area of the base, and the pro¬ duct will be the whole surface. Rule 2. —Multiply the area of the base by the per¬ pendicular height, and -£■ of the product will be the solid content. , Example 1.—Required the convex surface of a cone whose base A B = 20 inches, and slant height B D = 29-5. Here \ (31416 xABxBD) = A (3-1416 X 20 X 29-5) = 31416 x 10 X 29 5 = 926 772 square inches, which divided by 144 = 6436 superficial feet. = Ans. Example 2.—Required the solidity of the cone as above, taking the perpendicular C D at 28 inches. Here| (-7854 x AB 2 xCD) = { (-7854 x 20 a X 28) = 2932 16 cubic inches, which divided by 1728 = 1-697 cubic feet. = Ans. PROBLEM IT. To find the surface of the frustum of a cone or pyramid. Rule.—M ultiply the sum of the perimeters of the 62 MENSURATION two ends by the slant height, and half the product will be the slant surface; to which add the areas of the two ends, and the product will be the whole surface. Example.— Required the convex surface of the frustum of a cone A B OD, whose base A B = 20 inches, the slant height B C = 19, and top end C D = 11 inches. Here i [(3-1416 x AB) + (3-1416 X D C) x B C] = | [3-1416 x (AB + D C) x B C] = | [3-1416 x (20 + 11) x 19] = 1-5708 X 31 X 19 = 925-2012 square inches, which divided by 144 = 6425 square feet. = Ans. PROBLEM Y. To find the solid content of the frustum of a cone. Rule. —To the product of the diameters of the two ends, add the sum of their squares ; multiply this sum by the perpendicular height, and by -2618, the product is the solid content. Example 1.—Required the solid content of the frustum of a cone in Problem IV., whose perpendicular E E taken = 18 inches, D C = 11 inches, and A B = 20 inches. Here (AB x D C + A B 2 + D C 2 ) xEEx •2618 = (20 x 11 + 20 2 + ll 2 ) X 18 X -2618 = (220 + 400 + 121) X 18 x -2618 = 741 X 18 x •2618 = 3491-8884 cubic inches, which divided by 1728 = 2-0208 cubic feet. = Ans. Example 2. —Required the content, in imperial gal¬ lons, of the inverted frustum of a cone AB CD, whose inner dimensions are E F = 3] feet deep, D C = 18 inches diameter at bottom, and A B = 22 inches diameter at top. OF SOLIDS. 63 Here $ (A B x D C + A B 2 + D C*) X E F x -2618 } ■— 277-274 = f (18 x 22 + 18 2 + 22 2 ) X 42 x -2618] -r- 277-274 = $(396 + 324 + 484) x 42 x -2618} -j- 277-274 = (1204 X 42 x -2618) ~ 277-274 = 13238-7024 cub. in. -+■ 277’274 = 47‘746 imperial gallons. = Am. Or thus, 13238-7024 X 0-00360654 = 47746 imperial gal- Ions. = Am. as before. FEOBLEM YI. To find the solid content of the frustum of a pyramid. Rule.—T o the sum of the areas of the two ends, add the square root of their product; multiply this sum by the perpendicular height, and of the product is the solid content. Example. —Required the solid content of the frustum of a pyramid ABCD, whose perpendicular height = 24 inches, the area of the base AB = 144 square inches, and area of the top end D C = 64 square inches. Here $[area at A B + area at D C + v/ (area at A B x area at D C)] x perp. height £ -T- 3 = $[144 + 64 + V (144 X 64)] X 24} -r 3 = $208 + V (12 2 X 8 2 )} x 8 = (208 + 12 X 8) x 8 = (208 + 96) x 8 = 304 x 8 = 2432 cubic inches, which -j- 1728 = 1-4074 cubic feet. = Am. PEOBLEM VII. To find the solidity of a wedge. Rule.— To the length of the edge add twice the length of the base ; multiply that sum by the perpen- 64 MENSUBATION dicular height, and by the breadth of the base, and one- sixth of the product will be the solidity. Example.— Required the content in cubic inches of the wedge ABODE, whose base A B C is 12 inches long and 4 inches broad, the length of the edge D E = 10 inches, and perpendicular height r E = 20 inches. Here {(ED +2 x BC) X Er v _ n X AB} -i-6 = K10 + 2 x 12) X 20 x 4} -4-6= {(10 + 24) x 80} = 6 = (34 x 80) - 4 - 6 = 2720 +- 6 = 453-3' cubic inches. = Ans. pboblem vm. To find the convex surface and solid content of a sphere or globe. Rule 1.—Multiply the square of the diameter by 3-1416, the product will be the convex superficies. Rule 2.—Multiply the cube of the diameter by •5236, and the product is the solid content. Example 1 . —Required the convex surface of a sphere, whose diameter A B = 25i inches. Here A B 2 x 3 1416 = 25 5 2 x 3-1416 = 2042-8254 square inches, which -4- 144 = 14-1863 square or superficial feet. = Ans. Example 2.—Required the solid content of a sphere, whose diameter A B = 254 inches. Here A B s X *5236 = 25*5 3 X 5236 = 8682 00795 cubic inches, which divided by 1728 = 5-0243 cubic feet. = Ans. 03? SOLIDS. 65 PROBLEM IX. To find the convex surface and solid content of the segment of a sphere. Btjle 1.—Multiply the height of the segment by the whole circumference of the sphere, and the product is the curved surface. Btjle 2.—Add the square of the height to three times the square of the radius of the base; multiply that sum by the height, and by -5236, and the product is the solid content. Example 1. — The diameter A B of the sphere A B C D = 20 inches, what is the convex surface of that segment of it, whose height ED = 8 inches ? Here 31416 x A B x E D = 3-1416 x 20 X 8 = 502-656 square f inches, which -f- 144 = 3'49 super- A ficial feet. = Ans. c Example 2.—The diameter of the base E G- of the segment E D Gr = 18 inches, and perpendicular E D = 8, what is the solid content ? Here (ED*+ 3EE’) x DE x -5236 = (8 2 + 3x9 ! )x 8 x -5236 == (64 + 243) X 8 X -5236 = 307 x 8 X - 5236 = 1285-9616 cubic inches, which divided by 1728 = -7442 cubic feet. = Ans. Example 3. —Suppose ABCD to be a sugar pan, and that the diameter of the mouth A B is 4 feet, the depth D 0 being 25 inches, how many imperial gallons will it contain ? Here {V C 2 + (half A B) 2 a X3j xDCx -5236 = (25 2 + 24 2 X 3) x 25 x -52 36 = (625 + 576 X 3) x 25 x -5236 66 MENSURATION = (625 + 1728) X 25 X ‘5236 = 2353 X 25 X *5236 = 3080077 cubic inches, which divided by 277‘274 = lir084 imperial gallons. = Ans. problem x. To find the solidity of a spheroid*. Rule. —Multiply the square of the revolving axis by the fixed axis, and by ‘5236, and the product will be the solidity. Example 1.—Required the solid content of the pro¬ late spheroid ABCD, whose fixed axis A C = 50, and revolving axis B D = 30. B Here B D 2 x A C x *5236 = 30 2 X 50 x -5236 = 23562, the solidity. = Ans. i) Example 2. —What is the solid content of an oblate . spheroid, the fixed axis B D = 30, and revolving axis A C = 50 ? Here A C 2 X B D x ’5236 = 50 2 X 30 X *5236 = 39270, the solid content. = Ans. PROBLEM XI. To find the solidity of the segment of a spheroid when the base is circular or parallel to the revolving axis. Rule. —From triple the fixed axis take double the height of the segment; multiply the difference by the square of the height, and by -5236: then, as the square of the fixed axis is to the square of the revolving axis, so is the former product to the solidity. * A spheroid is a solid body generated by an ellipse revolving about one of its axes; when it revolves about the transverse or major axis, the solid thus generated is called a prolate spheroid; when the revolution revolves about the conjugate or minor axis, it is called an oblate spheroid. OP SOLIDS. 67 Example 1. — Required the solid content of the seg¬ ment ABC, whose height B r is 10, the revolving axis BF = 40, and fixed axis BD = 25. HereBD 2 : EF 2 ::(3BD-2Br) X Br 2 x -5236 ; [(3 B D-2 B r) X B r 2 x -5236 x E F 1 ] -f B D 2 . ..'. ft = [(3 x 25 - 2 x 10) x 10 2 x V. •5236 x 40 2 ] 25 2 = [(75-20) » X 100 x -5236 x 1600] -r- 625 = (55 x 52'36 X 64) ~ 25 = (220 x 52 36 x 64) -r- 100 = 22 X 5 236 x 64 = 7372-288. = Ans. Example 2. — "What is the solid content of the seg¬ ment of a spheroid, whose height B r = 20 inches, the revolving axis E E = 25 inches, and fixed axis B D = 50 inches ? Here B D 2 : E E 2 :: (3 B D-2 B r) X B r 2 X '5236 I [(3 BD -2Br) x Br 2 x -5236 X EE 2 ] -i-BD a = [(3 x 50 - 2 x 20) X 20 2 X -5236 x 25 2 ] -f- 50 2 = [(150 - 40) x 400 x -5236 x 625] -j- 2500 = (110 x 52-36 x 2500) -f- 2500 = 110 X 52-36 = 57596 cubic inches, the solid content. = Ans. pboblem xn. To find the convex surface cmd solid content of a cylindric ring. Rule 1 . — To the thickness of the ring, add the inner diameter; multiply the sum by the thickness, and by 9-8696, and the product will be the convex surface. Rule 2. —To the thickness of the ring, add the inner diameter; multiply that sum by the square of the thickness, and by 2-4674, and the product will be the solid content. 68 MEKSTJEATIOF OF SOLIDS. Example 1. —The thickness of a cylindric ring A 0 or D B = 2 inches, and inner diameter =' 18 inches, required the convex superficies. Here (AC + CH)xACx 9-8696 = (2 + 18) X 2 X 9-8696 = 20 X 2 X 9-8696 = 394-784 square inches, which — 144 = 2"742 superficial feet. = Am. Example 2. —Eequired the solid content of the ring as above. Here (AC+CD) X A C 2 x 2-4674 = (18 + 2) X 2 2 x 2-4674 = 20 X 4 x 2-4674 = 197-392 cubic inches, which -f- 1728 = *114 cubic feet. == Ans. Note. —A cubic foot is equal to 1728 cubic inches; or 2200 cylindrical inches ; or 3300 spherical inches { or 6600 conical inches. Also, the cubic foot being considered unity, or 1, A cylinder 1 foot diameter, and 1 foot in length = '7854 A sphere 1 foot in diameter.= ’5236 And a cone 1 foot 1 , diameter at the base and 1 foot in height.= "2618. TIMBER MEASURE. 69 OF TIMBER MEASURE. Timber is chiefly estimated by the square or super¬ ficial foot of 144 inches, or cubic foot of 1728; the calculation of which is performed by duodecimals ; that is, the foot or inch, &c., divided into 12 parts or divisions, thus:—■ 12 fourths make .... 1 third. 12 thirds „ .... 1 second.' 12 seconds „ .... 1 inch. 12 inches „ .... 1 foot. And the several values arising are:— Eeet multiplied by feet give feet, Eeet multiplied by inches give inches, Feet multiplied by seconds give seconds, Inches multiplied by inches give seconds, Inches multiplied by seconds give thirds, Seconds multiplied by seconds give fourths, &c. But this rule is more commonly called Cross Multipli¬ cation, on account of commencing with the left-hand figure of the multiplier. To find the superficial content of timber. Rule 1.—Place the multiplier under the multipli¬ cand, feet under feet, inches under inches, seconds under seconds, &c. 2. —Multiply each denomination of the length by the feet of the breadth, beginning at the lowest, and place each product under that denomination of the multipli¬ cand from which it arises, always carrying one for every 12. 3. —Multiply by the inches, and set each product one place farther to the right hand. 4. —Then multiply by the seconds, and set each pro¬ duct another place toward the right hand, &c. Thus proceed in like manner with all the other deno¬ minations, and their sum will be the content. 70 TIMBER MEASURE. Example 1.—Required the superficial content of a board 12 feet 6 inches long and 1 foot inches broad. A in. 12 , . 6 Multiplied by 1 . 5 . , 6 12 . 6 5 . 2 . 6 6 . , 3 Ans. — 18 . 2 . , 9 When the two ends of a hoard or plank are of differ¬ ent breadths, add the two breadths together, and mul¬ tiply the length by half the sum. Example 2.—A plank is 1 foot 4 inches broad at one end, 11£ inches broad at the other, and 18 feet 9 inches long, what is its superficial content ? Here (16 + 11|) -r- 2- = 27| -r 2 = 13f inches. A in. Then 18 . 9 13f inches = 1 . 1 . 9 18 . 9 1 . 6 . 9 1 . 2 . 0 . . 9 Ans. = 21 , . 5 . 9 . , 9 Superficial measure by the Engineer's Slide Rule. When the length is given in feet, and the breadth in inches, the gauge point is 12; but if the dimensions are all inches, the gauge point is 144. Rule. —Set the breadth upon B to the gauge point upon A, and against the length upon A is the content in square feet upon B. Example 1.—Required the number of square feet contained in a board 11^ inches broad and 18 feet long. Set 11-5 upon B to 12 upon A; and against 18 upon A is 17-3 feet upon B. TIMBER MEASURE. 71 The content of one board being found, the content of any number of the same dimensions may be found by setting 1 upon B to the content of the one found upon A ; and against any number of boards upon B is the whole content upon A. Bind the content of 8 boards, each being 17*8 square feet. Set 1 upon B to 17 3 upon A; and against 8 upon B is 138‘4 feet upon A. Example 2.—If a board is 10 inches broad at one end, and 7 inches at the other, what must be its length to make a square foot P Here (10 + 7) -r- 2 = 17 -f- 2 = 8| inches. Set 8‘5 upon B to 144 upon A; and against 1 upon B is 169 inches long upon A. To find the solidity of timber. The solid content of timber (according to custom) is found by multiplying the length by the square of the girth. Example. —Required the content of a tree in cubic feet, whose girth in the middle is 84 inches, and length 25 feet 6 inches. Here 84 -j- 4 = 21 inches = ~ girth. ft. in. and 21 inches = 1.9 Multiplied by 1 . 9 1 . 9 1.3.9 Then Multiplied by Ans. = Ans. = 3 . ft. in. 25 . 6 3.0.9 76 . 6 1 . 7 . 1 . 6 78 . 1 . 1 . 6 0 . 9 72 TIMBER MEASURE. But a more expeditious method is obtained by means of the following TABLE. i Girth in Area in Feet 1 Girth Inches. Area Feet. 1 Girth Inches. Feet. 6 •25000 121 1-04210 19 250694 61 •27127 12.1 1-08507 194 2-64063 6k •29340 121 1-12891 20 2-77777 61 •31641 13 1 17361" 204 2-91840 7 •34028 13J 1*21918 21 306250 7 \ •36502 13 2 1-26563 214 3-21007 74 •39063 13f 1-31293 22 3-3611 V 7? •41710 14 1*361 'll 224 3-51563 8 "4'4444 14 1 1-41016 23 367361 •47266 141 1-46007 2:14 383507 6 h •50174 143 1-51085 24 4-00000 8f •53168 15 1-56250 2^4 4-16840 9 •56250 Vo \ 1-61502 25 4-34028 91 •59418 1-66840 254 4-51563 91 •62647 153 1-72266 26 4-694'44 9f •66016 16 1-77777 264 4-87674 10 •694'44 161 1*83377 27 5-06250 101 •72960 164 1-89063 274 5-25174 101 •76563 163 1-94835 28 5-4'4444 101 •80252 17 2*00694 284 5-64063 11 •84028 171 206641 29 5-84028 111 •87891 174 2-12674 294 6-04340 111 •91840 171 2*18793 30 6-25000 111 •95877 18 2-25000 31 6-67361 12 1-00000 I84 2-37674 32 7-1'Hll Eule.— Multiply the area corresponding to the girth in inches by the length of the timber in feet; and the product is the solidity in feet and decimal parts. Example.— A piece, of timber 18 feet long and 14 inches square, how many cubic feet does it contain ? Here 1-361' X 18 = 24*5 cubic feet. TIMBER MEASURE. 73 By the Slide Buie. Set the length in feet upon B to 144 upon A ; and against the square, or A girth upon D, is the solid content in feet upon C. Example. —How many cubic feet are contained in a tree 28 feet long and 16 inches ^ girth ? Set 28 upon B to 144 upon A; and against 16 upon D is 49'8 feet upon C. To find the transverse section of the stro ngest lecm that can possibly be cut out of a round piece of timber. Let the circle A B C D be the trans¬ verse section of thepiece of timber given, draw the diameter B D, and divide it into three equal parts, as B l m D, erect the perpendiculars C, meeting the circle in C, draw D C and C B; then draw the chord A B parallel to D C, join A D, and the rectangle ABCD will be a section of the beam as required. To determine the greatest rectangle that can possibly be obtained in a given triangle. Let A B C be a given triangle, bisect any two of its sides, as E F; join E F, to each end of which draw lines at right angles with the .other side A C, and DEFG- will be the rectangle required. 74 • STRENGTH OE MATERIALS. ON THE STRENGTH OF MATERIALS. A knowledge of the strength of materials is one of the most important, and at the same time one of the most difficult subjects that the practical mechanic has to contend with, owing chiefly to the very different qualities of bodies of the same name; hence arise some doubts in selecting experiments whereon to build the data, there being scarcely two experiments made pro¬ ducing the same results. However, the following tables and rules are founded upon a mean of Messrs. Rennie, Barlow, and Telford’s experiments, having found them to agree the best with practice, and my own experi¬ ments on similar bodies. ON THE COHESIYE STRENGTH OE BODIES. The cohesive strength of a body is that force with which it resists separation in the direction of its length, as in the case of ropes, &c.; and no reason can be assigned why the strength should not vary directly as the section of fracture, and is totally independent of the length in position, except so far as the weight of the body may increase the force applied : neglecting this, and supposing the body uniform in all its parts, the strength of bodies exposed to strains in the. direction of their length is directly proportionate to their transverse area, whatever be their figure, length, or position. STRENGTH OF MATERIALS. 75 The following Table contains the result of experiments on the cohe¬ sive strength of various bodies in avoirdupois pounds; also one- third of the ultimate strength of each body, this being considered sufficient, in most cases, for a permanent load. Names of Bodies. Square or rectangular Bar. One Third Round Bar. One Third. WOODS. lbs. lbs. lbs. lbs. Boxwood . 20000 6667 15708 52.16 Ash. 17000 5667 13357 4452 Teak . 15000 5000 11781 3927 Fir. 12000 4000 9424 3141 Beech. 11500 3833 9032 3011 Oak . 11000 3667 8639 2880 METALS. Cast iron. ... 18656 6219 14652 4884 English wrought iron 55872 18624 43881 14627 Swedish ditto 72064 24021 56599 18866 Blistered steel . 133152 44384 104577 34859 Shear ditto . 124400 41467 97703 32568 Cast ditto. 134256 44752 105454 35151 Cast copper . 19072 6357 14979 4993 Wrought ditto .... 33792 11264 26540 8847 Yellow brass. 17968 5989 14112 4704 Cast tin. 4736 1579 3719 1240 Cast lead . 1824 608 1432 477 PROBLEM I. To find the ultimate cohesive strength of square, round, and rectangular bars, of any of the various bodies, as specified in the table. Ktjle.—M ultiply the strength of an inch bar (as in the table), of the body required, by the cross sectional area of square and rectangular bars, or by the square of the diameter of round bars ; and the product will be the ultimate cohesive strength. Example 1 . — A bar of cast iron being 1\ inches square, required its cohesive power. Here 15 x 15 X 18656 lbs. = 41976 lbs. = Ans. e 2 76 STRENGTH OF MATERIALS. Example 2.—Required the cohesive force of a bar of English wrought iron, 2 inches broad, and ■§ of an inch in thickness. Here 2 x -375 X 55872 lbs. = 41904 lbs. = Ans. Example 3.—Requiredtheultimate cohesive strength of a round bar of wrought copper, f of an inch in diameter. Here -75 2 x 26540 lbs. = 14928-75 lbs. = Ans. PROBLEM II. Tlie weight of a body being given , to find the cross sec¬ tional dimensions of a bar or rod capable of sustaining that, weight. Rule. —Eor square and round bars,—Divide the weight given by one-third of the cohesive strength of an inch bar (as specified in the table), and the square root of the quotient will be the side of the square, or diameter of the bar in inches. And if rectangular, divide the quotient by the breaC^R. and the result will be the thickness. Example 1.—What must be the side of a square bar of Swedish iron to sustain a permanent weight of 18000 lbs.? Here %/ = ^ ' 7493 = or nearl y s of au inch square. = Ans. Example 2.—Required the diameter of a round rod of cast copper to carry a weight of 6800lbs. Here = V 1-3619 = 1-16 inches diameter. = Ans. Example 3.—A bar of English wrought iron is to be applied to carry a weight of 2760 lbs.; required the thickness, the breadth being two inches. . i (S) = * (m) = wf = ' m of “ “ ch in thickness. = Ans. STRENGTH OF MATERIALS. 77 A TASLE Showing the circumference of a rope equal to a chain made of iron of a given diameter, and the weight in tom that each is proved to carry; also the weight of afoot of chain made from iron of that dimemion. Circumference Chains. Diameter in Proved, to Weight of a lineal foot 3 s T ' 5 1 s 108 4 2 1-5 4 $ 1 + A 3 2 1 4 2-7 6 4 + A 5 33 6 4 7 1 + A 8 4-6 n 1 9? 5-5 8 i + A 61 9 13 7-2 l + A 15 8-4 101 1 inch. 18 9-4 ON THE TRANSVERSE STRENGTH OF BODIES The transverse strength of a body is that power which it exerts in opposing any force acting in a perpendicular direction to its length, as in the case of beams, levers, &c., for the fundamental principles of which observe the following:— That the transverse strength of beams, &c., is in¬ versely as their lengths, and directly as their breadths and square of their depths, and, if cylindrical, as the cubes of their diameters ; that is, if a beam 6 feet long, 2 inches broad, and 4 inches deep, can carry 2000 lbs., another beam of the same material, 12 feet long, 2 inches broad, and 4 inches deep, will only carry 1000, being inversely as their lengths. Again, if a beam 6 feet long, 2 inches broad, and 4 inches deep, can sup- e 3 78 STRENGTH OP MATERIALS.' port a weight of 2000 lbs., another beam of the same material, 6 feet long, 4 inches broad, and 4 inches'deep, will support double that weight, being directly as their breadths ;—but a beam of that material, 6 feet long, 2 inches broad, and 8 inches deep, will sustain a weight of 8000 lbs.; being as the square of their depths. Prom a mean of experiments made to ascertain the transverse strength of various bodies, it appears that the ultimate strength of an inch square, and an inch round bar of each, 1 foot long, loaded in the middle, and lying loose at both ends, is nearly as follows, in lbs. avoir¬ dupois. Names of Bodies. Square Bar. One Third. Round Bar. One Third. Oak . 800 267 j 628 209 Ash . 1137 379 893 298 "Elm .. 569 190 447 14ft. Pitch pine.. 916 305 719 240 Deal. 566 189 444 148 Past iron . 2580 860 2026 675 Wrought iron .... 4013 1338 3152 1051 PROBLEM i. To find the ultimate transverse strength of any rect¬ angular beam, supported at both ends, and loaded in the middle; or supported in the middle, and loaded at both ends; also, when the weight is between the middle and the end; likewise, when fixed at one end, and loaded at the other. Rule —Multiply the strength of an inch square bar, 1 foot long (as in the table), by the breadth, and square of the depth in inches, and divide the product by the length in feet; the quotient will be the weight in lbs. avoirdupois. STRENGTH OF MATERIALS. 79 Example 1.—"What weight will break a beam of oak, 4 inches broad, 8 inches deep, and 20 feet between the supports ? Here 899 — - X 82 = 40 x 4 x 64 = 10240 lbs. 20 = Am. Note.—W hen a beam is supported in the middle, and loaded at each end, it will bear the same weight as when supported at both ends, and loaded in the middle ; that is, each end will bear half the weight. When the weight is not situated in the middle of the beam, but placed somewhere between the middle and the end,—Divide four times the product of the two distances of the weight from the two ends by the whole length of the beam. The quotient will be the effective length. Example 2.—Eequired the ultimate transverse strength of a pitch pine plank, 24 feet long, 3 inches broad, 7 inches deep, and the weight placed 8 feet from one end. ' Here 4 X 16 x 8 = — = 2E3' effective length. 24 3 \ , 916 x 3 x 7“ _ 916 x 9 x 49 _ 229 x 9 x 49 2D3 7 64 16 = 1QQ9 - 8 - 9 = 6312 lbs. = Am. 16 Again, when a beam is fixed at one end, and loaded at the other, it will only bear \ of the weight as when supported at both ends, and loaded in the middle. Example 3.—What is the weight requisite to break a deal beam, 6 inches broad, 9 inches deep, and pro¬ jecting 12 feet from'the wall? Here 566 x 6 X -- = 283 X 201 = 5730f lbs. 12 x 4 = Ans. The same rules apply as well to beams of a cylindrical form, with this exception, that the strength of a round bar (as in the table) is multiplied by the cube of the diameter, in place of the breadth and square of the depth. E 4 80 STRENGTH OE MATERIALS. Example 4.—Required the ultimate transverse strength of a solid cylinder of cast iron, 12 feet long and 5 inches diameter. Here 2026 X _J>? = 25325 P — 21104 lbs. = Ans. 12 12 Example 5.—What is the ultimate transverse strength of a hollow shaft of cast iron, 12 feet long, 8 inches diameter outside, and containing the same cross sec¬ tional area as a solid cylinder 5 inches diameter ? ■n- {8 3 —s/(8 2 —5 2 ) 3 } X2026_(512- s /39 3 )x2026 ±lere 12 12 _(512 — ^59319) x 2026_(512—243-555) X 1 013 ~ 12 ~ 6 = 268-445 x 1013 = 271934-785 = 4S322 ^ = 6 6 Ans. Note.—W hen a beam is fixed at both ends and loaded in the middle, it will bear one-half more than it will when loose at both ends. And if a beam is loose at both ends, and the weight laid uni¬ formly along its length, it will bear double; but if fixed at both ends, and the weight laid uniformly along its length, it will bear triple the weight. PROBLEM II. To find the breadth or depth of beams intended to support a permanent weight. Rule.—M ultiply the length between the supports, in feet, by the weight to he supported in lbs., and divide the product by one-third of the ultimate strength of an inch bar (as in the table'), multiplied by the square of the depth ; the quotient will be the breadth, or, multiplied by the breadth, the quotient will be the square of the depth, both in inches. STRENGTH OE MATERIALS. 81 Example 1.—Required the breadth of a cast iron beam, 16 feet long, 7 inches deep, to support a weight of 4 tons in the middle. Here 4 tons = 8960 lbs. Then X = 3-4 inches. = Am. 860 x V 301 Example 2 . —What must be the depth of a cast iron beam, 34 inches broad, 16 feet long, to bear a permanent weight of 4 tons in the middle ? Here 8960 x 16 860 x 3-4 /J v 86 8960 86 X 34 " X / 2240 j 2240 V 43 x 17 V 731 X 1*75 = 7 inches. = Ans. = 4 x \/ 3 07 = 4 Note 1 .— When a beam is fixed at both ends, the divisor must be multiplied by 1*5, on account of its being capable of bearing -Cn.'-half more. 2. —When a beam is loaded uniformly throughout, and loose at both ends, the divisor must be multiplied by 2, because it will bear double the weight. 3. —If a beam is fast at both ends, and loaded uniformly through¬ out, the divisor must be multiplied by 3, on account that it will bear triple the weight. Example 3. —Required the breadth of an oak beam, 20 feet long, 12 inches deep, made fast at both ends, to be capable of supporting a weight of 12 tons in the middle. Here 12 tons = 26880 lbs. TW 26880 X 20 _ 2688 X 6 X 2 _ 28 _ 266-6' x 12 2 x 1-5 8 x 144 x 3 “"3 ~ 9| inches. = Am. Again, when a beam is fixed at one end, and loaded at the other, the dividend must be multiplied by 4; because it will only bear one-fourth of the weight. E 5 82 STRENGTH OE MATERIALS. Example 4. — Required the depth of a beam of ash, 6 inches broad, 9 feet projecting from the wall, to carry a weight of 47 cwt. Here 47 cwt. = 5264 lbs. „« /5264 x 9 X 4 /329 x 4 1 x 3’x2 I The “ V 879 x 6 " =V -2274- = 24 X ./JjjJL = 24 X J -1447 = 24 X '38 = 2274 9‘12 inches deep. = Ans. And when the weight is not placed in the middle of a beam, the effective length must be found as in Pro¬ blem I. Example 5. —Required the depth of a deal beam 20 feet long, to support a weight of 63 cwt. 6 feet from one end. Here 4 X 44 - X 6 = ^ = 16*8, effective length of beam, And 63 cwt. = 7056 Tbs. Ans. Beams or shafts exposed to lateral pressure are sub¬ ject to all the foregoing rules; but in the case of water¬ wheel shafts, &c., some allowance must be made for wear, then the divisor may be changed from 675 to 600 for cast iron. Example 6.—Required the diameter of bearings for a water-wheel shaft 12 feet long, to carry a weight of 10 tons in the middle. Here 10 tons = 22400 lbs. Hence J 7056 X 16-8 189 X 6 = 7 784 x 2 . 15 ^ 30 = =2 X 5-477 = 10-22 inches c 15 STRENGTH OF MATERIALS. 83 Then Vi. 2 . X 224Q0 - = V448 = 7’65 inches dia- V 600 meter. = Ans. And when the weight is equally distributed along its length, the cube root of half the quotient will be the diameter, thus: Here = V 224 = 6*07 inches diameter. = Ans. Example 7.— Required the diameter of a solid cylinder of cast iron, for the shaft of a crane, to be capable of sustaining a weight of 10 tons; one end of the shaft to be made fast in the ground, the other to project 6| feet; and the effective leverage of the jib as If to 1. Here 10 tons = 22400 lbs. 22400 X 6-5 X 175 _ 2 X 224 x 13 X 7 • 675 x -25 27 = 1510. And V 1510 = 11*47 inches diameter. = Ans. The strength of cast iron to wrought iron, in this direction, io as 9 is to 14 nearly ; hence, if wrought iron is taken in place of cast iron in the last example, what must be its diameter ? 7 1510 x 9 14 Ans. — V 970 7 = 9 9 inches diameter. = ON TORSION OR TWISTING. The strength of bodies to resist torsion , or wrenching asunder, is directly as the cubes of their diameters; or, if square, as the cube of one side; and inversely as the force applied multiplied into the length of the lever. Hence the rule.—1. Multiply the strength of an inch bar, by experiment (as in the following table), by the 84 STRENGTH OE MATERIALS. cube of the diameter, or of one side in inches; and divide by the radius of the wheel, or length of the lever also in inches ; and the quotient will be the ultimate strength of the shaft or bar in lbs. avoirdupois. 2.—Multiply the force applied in pounds by the length of the lever in inches, and divide the product by one-third of the ultimate strength of an inch bar (as in the table), and the cube root of the quotient will be the diameter, or side of a square bar in inches; that is, capable of resisting that force permanently. The following Table contains the result of experiments on inch bars, of various metals, in lbs. avoirdupois. Names of Bodies. Round Bar. One Third. Square Bar. One Third. Cast iron. 11943 3981 15206 5069 English wrought iron 12063 4021 15360 5120 Swedish ditto 11400 3800 14592 4864 Blistered steel . 20025 6675 25497 84U9 Shear ditto. 20508 6836 26112 8704 Cast ditto . 21111 7037 26880 8960 Yellow brass. 5549 1850 7065 2355 Cast copper . 4825 1608 6144 2048 Tin. 1688 563 2150 717 Lead. 1206 402 1536 512 Example 1. —What weight, applied on the end of a 5 feet lever, will wrench asunder a 3 inch round bar of cast iron ? Hereil^LAS! = ^=5374 Us. avoirdupois. = Ans. Example 2. —Eequired the side of a square bar of wrought iron, capable of resisting the twist of 600 lbs. on the end of a lever 8 feet long. s y 600 X 96 _ s / 15 X 3 _ 8/90_^/90_ J 5120 V 4 J 8 2 inches. = Ans. STRENGTH OE MATERIALS. 85 In the case of revolving shafts for machinery, &c. the strength is directly as the cubes of their diameters and revolutions, and inversely as the resistance they have to overcome; hence, From practice, we find that a 40-horse power steam- engine, making 25 revolutions per minute, requires a shaft (if made of wrought iron) to be 8 inches diameter: now, the cube of 8, multiplied by 25, and divided by 40 = 320; which serves as a constant multiplier for all others in the same proportion. Example 3.—What must be the diameter of a wrought iron shaft for an engine of 65-horse power, making 23 revolutions per minute ? 7 65 x 320 23 Here inches diameter. = Ans. Mr. Eobertson Buchanan, in his Essay on Shafts, gives 400 as a constant multiplier for cast iron shafts that are intended for first movers in machinery; 200 for second movers; and 100 for shafts connecting smaller machinery, &c. Example 1. —The velocity of a 30-horse power steam-engine is intended to be 19 devolutions per minute. Required the diameter of bearings for the fly wheel shaft. Here 400 X 30 19 inches diameter. = Ans. Example 2. —Required the diameter of the bearings of shafts, as second movers for a 30-horse engine ; their velocity being 36 revolutions per minute. 200 x 30 36 86 STRENGTH OE MATERIALS. 12-1/ 36 = X 33 = 5-5 inches diameter. = Arts. 6 6 Note.—W hen shafting is intended to be of wrought iron, use 160 as the multiplier for second movers ; and 80 for shafts con¬ necting smaller machinery. TABLE Of the proportionate length of bearings, or journals for shafts of various diameters. Diameter Length in Inches. Diameter in Inches. Length in Inches. 1 64 8| 14 n 7 9| 2 3 74 10 2i 34 8 10| 24 3J 84 HI 3 M 9 12 34 4 i 94 12| 4 54 10 134 44 64 104 14 5 6| li 144 5J 74 114 154 6 84 12 16 MECHANICAL POWEES. 87 OF THE MECHANICAL POWEES. When power is applied to overcome weight, or force to overcome resistance, the machines employed are called mechanic powers; and the application of such, the science of mechanics. The power and weight are said to balance each other, t>r to be in equilibrio, when the effort of the one to pro¬ duce motion in one direction is equal to the effort of the other to produce it in an opposite direction ; or when ' the weight opposes that degree of resistance which is precisely required to destroy the action of the power. The momentum or quantity of force of any moving body is the result of the quantity of matter multiplied by the velocity with which it is moved; and when the products arising from the multiplication of the particular quantities of matter in any two bodies by their respec¬ tive velocities are equal, their momentums will be so too. And it holds universally true, that when two bodies are suspended upon any machine, so as to act contrary to each other,-if the machine be put in motion, and the perpendicular ascent of one body, multiplied into its weight, be equal to the perpendicular descent of the other multiplied into its weight, those bodies, however unequal they may be in weight, will balance each other in all situations; for, as the whole ascent of the one is performed in the same time as the whole descent of the other, their respective velocities must be as the spaces they move through; and the excess of weight in the one is compensated by the excess of velocity in the other. Upon this principle it is easy to compute the power of any machine, either simple or compound; for it is only finding how much swifter the power moves than the weight, and just so much is the power in¬ creased by the help of the machine. 88 MECHANICAL POWEES. The simple machines, usually called mechanic powers, are six in number, namely, the Lever, the Wheel and Axle, the Pulley, the Inclined Plane, the Wedge, and the Screw. There are three kinds of levers, caused by the dif¬ ferent situations of the weights, props, and powers. 1. —When the weight is at one end, the power at the other, and the prop somewhere between. 2. —When the prop is at one end, the p'ower at the other, and the weight between. And, 3. —When the prop is at one end, the weight at the other, and the power between. Thus, That on any kind of lever there may be a balance between the power and weight, their intensities must be inversely as their distances from the fulcrum, or prop, at which they act, that is, P X AF = W X BP; therefore, Multiply the weight or power given by its distance from the prop, and divide by the distance of the prop from the power or weight; the quotient will be the power or weight required. Examples 1, 2, and 3. Required the power necessary to counterpoise a weight of 80 lbs. on each of the three levers, whose lengths are 60 inches, and in the first and second 10 inches from weight to prop, the third being 10 inches from weight to power. First .. 80 X 10 16 lbs. power. = ls£ Ans. 50 MECHANICAL POWEBS. 89 Second.. 80 x 10 60 = 1333 lbs. power. = 2nd Ans. Third... = 96 lbs. power. = 3rd Ans. Example 4.— What power is necessary to raise a weight of 620 lbs. by a lever of the first order, 72 inches long, and the prop placed 12 inches from the weight F Here 72 — 12 = 60 inches, the distance of prop to power. — 124 lbs. = Ans. Note.—I n a lever of the first kind, when its length is given together with the weight and power, multiply the weight or power by the length of the lever, and divide by the sum of the weight and power, the quotient is the distance of the power or weight from the prop. Example 5.—A weight of 620 lbs. is to be lifted by a power of 124 lbs. applied to the end of a lever of the first order, 72 inches long; required at what distance from the weight the prop must be placed. Herp 124 x 72 _ 124x72 620 + 124 744 = 12 inches. = Ans. Example 6.— A beam 20 feet long, and supported at both ends, bears a weight of 73 cwt. 4 feet 6 inches from one end ; required the proportion of weight upon each support. jj ere whole weight x dist. from nearest support _ whole length ^ 2() ^ ^ = cwt ‘ on the furthest support. = ls£ Ans. And 73 — 16425 = 56 575 cwt. on the nearest sup¬ port. = 2nd Ans. Example 7.— A weight of 300 lbs. is fixed on the end of a lever 6 feet long; required the power, applied 2| feet from the prop, to raise the weight. 90 MECHANICAL POWEKS. Here = ^20 lbs. power. = Ans. WHEEL AND AXLE. Here the velocity of the power is to the velocity of the weight as the diameter of the wheel is to the dia¬ meter of the axle; hence, divide the velocity of the power by the velocity of the weight, and the quotient is the weight that the power is equal to. Example 1.—A power equal to 30 lbs. is applied to the winch of a crane whose length is 15 inches ; the pinion contains 10 teeth, the wheel 120, and the barrel is 9 inches diameter; required the weight raised. Here 15 x 2 = 30, the diameter of the circle described by the winch, or handle. Next 120 10 = 12 revolutions of the pinion for one of the wheel. Then — - X --^ = 1200 lbs. raised by this crane, = Ans. Example 2.—What would be the increase of power, in the last example, if a wheel of 150 teeth, and a pinion of 15, were added to the crane ? Here 150 -r- 15 = 10, that is, the velocity of the weight is diminished, while the velocity of the power is the same. Then ^ ^ 19 ^ = 12000 lbs. raised = Ans., the power here being increased ten times. Example 3. —What power is requisite to raise 42 tons 60 feet high in 10 minutes, the velocity of the power being 20 feet per minute ? Here 60 1 = Ans. :o MECHANICAL POWERS. 91 TO CALCULATE FOR THE DIFFERENT PARTS OF A CRANE, AS RESPECTS MECHANICAL ADVANTAGE. 1. —The number of revolutions of the pinion to one of the wheel, the length of the handle, and the force applied given, to find the diameter of the barrel. Rule.*—M ultiply the diameter of the circle de¬ scribed by the winch, or handle, in inches, by the power applied in lbs., and by the number of revolutions of the pinion to one of the wheel; divide the product by the weight to be raised in lbs., and the quotient is the barrel’s diameter in inches. Example. —Suppose that two men were required to raise a weight of one ton by a crane, and each man to exert a constant force of 33£ lbs. on a handle 16 inches long, the pinion making seven revolutions for one of the wheel, what must be the barrel’s diameter ? Here 16 x 2 = 32 inches, diameter of the circle described by the handle, and 33£ X 2 = 67 lbs. con¬ stant force. Then ^ X ^ = 67 inches. = Ans. 2240 2. — The diameter of the barrel, the length of the handle, and force applied given, to find the number of revolutions of the pinion to one of the wheel. Rule. —Multiply the weight to be raised in lbs. by the diameter of the barrel in inches, and divide the product by the diameter of the circle described by the handle in inches, multiplied by the power applied in lbs., and the quotient is the revolutions of the pinion to one of the wheel. Example.— What must be the number of revolutions of the pinion to one of the wheel, when the power applied is 67 lbs., the length of the handle 16 inches, 92 MECHANICAL POWERS. and the barrel 6 - 7 inches diameter, to counterpoise a weight of one ton, or 2240 lbs. ? Here 2240 X 6 7 _ y revo i u tj ons to one of the 32 x 67 wheel. = Ans. 3. — The diameter of the barrel, the number of re¬ volutions of the pinion to one of the wheel, and the power applied given, to find the length of the handles. Rule. —Multiply the weight to be raised in lbs. by the barrel’s diameter in inches, and divide the product by the power applied in lbs., multiplied by the number of revolutions of the pinion to one of the wheel, and half the quotient is the length of the handles. Example. —It is estimated that the united effort of two men at the handles of a crane is 67 lbs. nearly ; now a crane having a barrel of 6 - 7 inches diameter, and a pinion 7 to 1 of the wheel, what must be the length of handles to raise a weight of 1 ton ? Here 2240 x -- = ^ = 16 inches. = Ans 67 X 7 x 2 2 4. — The diameter of the barrel, the revolutions of the pinion to one of the wheel, and length of handles given , to find the power required. Rule. —Multiply the weight to be raised in lbs. by the diameter of the barrel in inches, and divide the pro¬ duct by the diameter of the circle described by the handle, multiplied by the revolutions of the pinion to one of the wheel, and the quotient is the power re¬ quired. Example. —What power will be required to raise one ton by a crane, whose barrel is 6'7 inches diameter, the pinion 7 to 1 of the wheel, and each handle 16 inches long ? MECHANICAL POWERS. 93 TT 2240 x 6*7 77 1 i± ere ——-=—■ = 67 lbs. power. = Ans. 32 x 7 v Note.— The handles of a crane ought not to he less than 2 feet 11 inches, or 3 feet from the ground, and the jib to stand at an angle of about 45 degrees. To find the thickness of cast iron for a creme post, when fixed at one end, and loaded at the other. Rule. —Multiply the weight that the crane is to lift in lbs. by the leverage of the jib to one of the post, and by the length of the post in feet; divide the pro¬ duct by 168, then subtract the quotient from the cube of the outside diameter, and the cube root of the dif¬ ference is the inside diameter. Example.—W hat thickness must the metal be for a crane post to carry a weight of 10 tons, the diameter of the post being 16 inches, and projecting 6 feet from the ground, the leverage of the jib being as 31 to 1 of the post ? Here 10 tons = 22400 lbs. Then 16’ - 22400 x 8-5 x 6 ) = ^ (4Q98 _ 2800) = V 1298 = ] 0 9 the inside diameter. And — = 2'6 inches in thickness. = Ans. 2 2 the pullet. A single pulley, that only turns on its axis, and does not move out of its place, serves only to change the direction of the power, but gives no mechanical advan¬ tage. The advantage gained is always as twice the number of moveable pulleys, without taking any notice of the fixed pulleys necessary to compose the system of pulleys ; hence, divide the weight to be raised by twice the number of moveable pulleys, and the quotient is the power required to raise the weight, in terms of the same name. 94 MECHANICAL POWERS. Example 1.—"What power is requisite to raise 250 lbs. with a pair of four-shieved blocks, the one block moveable and the other fixed ? Here ■ __ _ 31-25 lbs. power. = Am. 4x24 r Example 2.—What weight will a power of 120 lbs. raise, when applied to a three and four-shieved block, the three being moveable and the other fixed ? Here 3 x 2 X 120 = 720 lbs. raised. = Am. the inclined plane. In the inclined plane, when the power acts parallel thereto, the weight it will support is to the power ap¬ plied as the length of the plane to its perpendicular height; hence, multiply the weight by the perpendicular height of the plane, and divide by its length, the quo¬ tient is the power that will support that weight upon the plane. Example 1.—Required the power, or equivalent weight, capable of supporting a load of 300 lbs. upon an inclined plane 50 feet long and 16 feet high. Here 50 is to 16 as 300 lbs. is to 96 lbs ., the power. = Ans. „ 300 lbs. x 16 ne „ ,, . Or, -—- = 96 lbs. the power. = Am. as before. Example 2.—A power of 120 lbs., with a velocity of 50 feet per minute, is to be applied to move a weight up an inclined plane at the rate of 30 feet per minute; the plane is 25 feet long and 8 feet high; required the weight that the power is equal to. 30 x 8 48 Here ——— = — == perpendicular velocity of the weight. Then, as ^ : 50:: 120 lbs. ; 625 lbs. = Am. 5 MECHANICAL POWEES. 95 The weight multiplied by the length of the base, and divided by the length of the plane, equals the pressure on the plane. The space which a body describes upon an inclined plane, when descending by the force of gravity, is to the space which it would fall freely in the same time, as the height is to the length of the plane, and the spaces being the same, the times will be inversely in this proportion. Again, if two bodies descend from rest down two planes, equally inclined to the horizon, and then, with¬ out any loss of velocity, proceed to descend down two other inclined planes, also equally inclined to the horizon, the lengths of. which are to each other in the same proportion as the lengths of the first two planes, the squares of the times of their whole motion will be in the same proportion as the lengths of the planes. Means of ascertaining 'practically the effect produced by inclined planes. Provide a board or box, a b, capable of holding pebbles, sand, &c., and which, by a screw, c, can be easily raised at one end, as ads, &c. d "When a b lies flat on c b, a carriage placed upon a b will be at rest; but by the screw at c raising a b leisurely, the carriage will, at a certain height, set off by itself, and run down the plane. Then are we in possession of a triangle that solves what force is necessary to drag any load of any kind on a road or level ground; for the 96 MECHANICAL POWERS. hypotenuse a b represents the weight of the carriage, and the perpendicular a c what portion of that weight is necessary to draw the carriage on level ground; thus, Suppose the carriage . . 12 cwt. The line a b .24 feet. Height c a .3 feet. The declivity, then, is as 3 to 24, or In this case it will be found that ^ of the weight of the carriage would drag it on such a road or level ground, namely, 11 cwt.; but if the road were very deep and rough, it might require to be raised perhaps as high as d or s, before the carriage would set off. Now, if c s were half the length of s b, then it would require one-half the weight of the carriage to drag it on level ground, or, in the above case, 6 cwt. This rule is universal, and has been proved by car¬ riages at large, on roads of every description. In estimating the draft up hill, the draft on the level must be added to it. Suppose the hill rises 1 foot in 4, then 4. part of the weight must be added to the draft on level ground. If the weight be, as before, 12 cwt., then ^ would be 3 cwt.; and if its draft on a level were 1^ cwt., then 4 \ cwt. would be the real draft necessary to draw 12 cwt. up a hill rising 1 foot in 4, &c. Example. —Suppose I find that, on an edge railway, a loaded carriage will just move by itself when there is a descent of 3| inches per chain, or about one perpen¬ dicular for 224 horizontal, which is (reckoning the car¬ riage to weigh 1 ton) 10 lbs. = " 2 ?^~ required to move it on a level. Now, from the above data, what force will be required to drag the same weight up a similar road ascending 1 inch per yard, or ^ ? Here -J 6 of a ton is 62‘2 lbs., which added to 10 lbs. as above, amounts to 72 - 2 lbs., the weight required to drag it up an ascent of -g 5 ; and allowing the strength of an ordinary horse MECHANICAL POWERS. 97 to be 140 lbs., he will only be able to drag 1-9, or say 2 tons up an ascending plane of 1 in 36. THE WEDGE. As the wedge is seldom used without being driven, tbe force of the blow is not easily ascertained; of course, in practice it is not worth taking into account with respect to calculation. THE SCREW. The advantage gained by the screw amounts to this, that the circumference of the circle described by the lever or handle is to the pitch of the screw (viz. the distance between two consecutive threads) as the weight to the power. Example. —What power is necessary to raise a .weight of 6000 lbs., the length of the lever being 20 inches and the screw pitch ? Here as 20 X 2 X 3-1416 = 125 7 : *75 :; 6000 lbs. • 35-8 lbs. power required. Note.—T here are few machines but what, on account of the friction of the parts against one another, will require a third part more power to work them, when loaded, than is requisite to con¬ stitute a balance between power and weight. MECHANICAL POWEKS. The following Table shows the estimated power of man or horse as applied to machinery. Application of the Power. Lbs. Avr. at the rate of 220 feet per minute. Or Lbs. Avr. at the rate of one foot per minute. A man is supposed to be capable of) lifting or carrying.J A man is supposed to be capable of | > 27273 I or 6000 turning the winch of a crane with a force equal to.1 When the united efforts of two men'j l 28G36 or 6300 are applied to the winch of a crane, I the handles being at right angles, each man exerts a force equal to .J l 33-409 or 7350 A man is supposed to exert a power' in pumping equal to.j J. 17-336 or 3814 In ringing, a man exerts a force) 33.955 equal to.J or 8570 And in rowing. . 40-955 or 9010 The power of a horse equal to. . 150 or 33000 FALLING BODIES. 99 OF FALLING BODIES. In bodies falling freely by their own weight, their velocities are as the times, and the spaces as the square of the times; therefore, if the times he as the numbers. 1, 2, 3, 4, &e. The velocities will he also.1, 2, 3, 4, &c. The spaces passed through.. 1, 4, 9, 16, &c. And the spaces for each time, as the odd numbers. 1 , 3, 5, 7, &c. It has been ascertained by experiment that a body falling freely from rest will descend through 16^ feet in the first second of time, and will then have acquired a velocity which, being continued uniformly, will carry it through 32j- feet in the next second; conse¬ quently, if the first series of numbers be expressed in seconds, 1" 2 " 3 " Velocities in feet will be.32J- 64f 96|, &c. Spaces in the whole times.16 T ^ 64f 144^, &c. Amd the spaces for each second 16^ 48 T 3 ^ 8 O- 5 -, &c. To find the velocity a falling body will acquire in any given time. Ktjle.— Multiply the time in seconds by 32-19, and the product will be the velocity acquired in feet per second. Example. —Required the velocity in 7 seconds. Here 32-19 X 7 = 225 33 feet, velocity acquired. = Ans. To find the velocity a body will acquire by falling from any given height. Rule.—M ultiply the space in feet by 64-4, and the square root of the product will be the velocity acquired in feet per second. f 2 100 FALLING BODIES. Example. —Required the velocity a hall will acquire iu descending through 201 feet. Here V (64-4 x 201) = ^ 12944-4 = 118-8 feet. = Ans. To find the space through which a bodg will fall in any given time. Rule.— Multiply the square of the time in seconds by 16-095, and the product will be the space in feet. Example. —Required the space fallen through in 7 seconds. Here 16-095 X V = 16 095 x 49 = 788-655 feet. = Ans. Note.—T he velocity acquired by a body in falling from rest, through a given height, is the same whether it fall freely or descend through a plane any way inclined. The diameter of a circle perpendicular to the horizon, and any chord terminating at either extremity of that diameter, are de¬ scribed by a falling body in the same length of time. And the velocities which bodies acquire by descending along chords of the same circle, are as the lengths of those chords. TABLE Of accelerated motion of falling bodies. Time in seconds of the body’s fall. Space fallen through during each second in feet. Whole space fallen through in feet. Velocity acquired at the end of the time. 1 16095 16 095 32-19 2 48-286 64381 64-38 3 80-476 144857 96-57 4 112-667 257-524 12876 5 144-857 402381 160-95 6 177-048 579429 193-14 7 209-238 788-668 225 33 8 241-429 1030097 257-52 9 273-619 1303716 289-71 10 305-810 1609-526 32191 PENDULUMS. 101 ON PENDULUMS. The length of a pendulum that vibrates seconds, or 60, in the latitude of Konigsberg (54° 42' 50" N.) is, ac¬ cording to Bessel, 440-8179 lines of his toise, which is shorter by Trrs- °f a ^ ne than the toise of Peru. The French metre = -513074 of the toise of Peru = 39-370089 English inches, with which we get the length of the second’s pendulum at Konigsberg = 39-14997 English inches, and at London (Lat. 51° 31' 8'4" N.) = 39-13907 English inches; and */ 39-1391 x 60 = 375 367, serves as a constant number for other pendulums; thus, 375 367, divided by the square root of the pendulum’s length, gives the number of vibrations per minute; and divided by the vibrations per minute, gives the square root of the length of the pendulum. Example 1.—Required the number of vibrations a pendulum 25 inches long will make per minute. , T 375-367 375-367 ^ . Here — 25 ~ = —5- = ^^73 vibrations per minute. = Ans. Example 2.—Required the length of a pendulum to ' make 80 vibrations per minute. Here ^ 37 |' 367 J = 4-6921 2 = 22-0157 inches long. = Ans. Table containing the length of pendulums, in English inches, to vibrate seconds in various parts of the world. From the Ency. Metrop., Art. “ Figure of the Earth,” by G. B. Airy, Esq., Astronomer Royal. At Sierra Leone.. 39-01997 in. „ Trinidad. 39 01888 „ „ Madras. 39 026*30 „ „ Jamaica. 39 03503 „ „ Rio Janeiro_ 39 04350 „ At New York .... 39-10120 in. „ Bordeaux. 39-11296 „ „ Paris. 39 12877 „ „ Leith Fort.... 39-15540 „ „ Greenland .... 39-20335 „ A pendulum vibrating half seconds in the latitude of London is 9-8 inches in length; and for quarter seconds, 2 - 5 inches. p 3 102 TELOCITY OF WHEELS. ON THE YELOCITT OE WHEELS, DRUMS, PULLEYS, Ac. When wheels are applied to communicate motion from one part of a machine to another, their teeth act alternately on each other ; consequently, if one wheel contains 60 teeth and another 20, the one containing 20 teeth will make three revolutions, while the other makes but one; and if drums or pulleys are taken in place of wheels, the result will be the same ; because their circumferences, describing equal spaces, render their revolutions unequal: from this the rule is de¬ rived, namely, Multiply the velocity of the driver by the number of teeth it contains, and divide by the velocity of the driven; the quotient will be the number of teeth it ought to contain. Or, multiply the velocity of the driver by its diameter, and divide by the velocity of the driven; the quotient will be the diameter of the driven. If the velocities of driver and driven are given with the distance of their centres, Then the sum of the velocities I { of driven } " ,. , n . . f radius of driven, distance of centres . j radills ofdriveI , Example 1.—If a wheel that contains 75 teeth makes 16 revolutions per minute, required the number of teeth in another to work in it, and make 24 revolutions in the same time. Here 75 X -*- = 50 teeth. = Ans. 24 Example 2.—A wheel, 64 inches diameter, and making 42 revolutions per minute, is to give motion to a shaft at the rate of 77 revolutions in the same time: TELOCITY OE WHEELS. 103 required the diameter of a wheel suitable for that purpose. Here = 34-9 inches. = Ans. 77 Example 3.—Required the number of revolutions per minute made by a wheel or pulley 20 inches diame¬ ter, when driven by another of 4 feet diameter, and making 46 revolutions per minute. Here ^ ^ — = 110*4 revolutions. = Ans. Example 4.—A shaft, at the rate of 22 revolutions per minute, is to give motion, by a pair of wheels, to another shaft at the rate of 15£ ; the distance of the shafts from centre to centre is 45 \ inches; the diameters of the wheels at the pitch lines are required. Here 22+15-5 : 22::45*5m. : — - x45 ’ 5 = 26-69 in. , 22 + 15*5 the radius of the driven wheel; which, doubled, gives 53*38 in., the diameter. = ls£ Ans. Therefore 45*5 in. — 26*69 in. = 18*81 in., the radius of the driver; which, doubled, gives 37*62 in., the dia¬ meter. = 2nd Ans. Example 5.—Suppose a drum to make 20 revolu¬ tions per minute, required the diameter of another to make 58 revolutions in the same time. Here 58 -f* 20 = 2*9, that is, their diameters must be as 2*9 to 1; thus, if the one making 20 revolutions be called 30 inches, the other will be 30 -f- 2*9 = 10*345 inches diameter. Example 6 .—Required the diameter of a pulley, to make 12| revolutions in the same time as one of 32 inches making 26. 32 x 26 H ere — \2r5 — = inches diameter. e 4 104 VELOCITY OE WHEELS. Example 7.—A shaft, at the rate of 16 revolutions per minute, is to give motion to a piece of machinery at the rate of 81 revolutions in the same time; the motion is to be communicated by means of two gearing wheels and two pulleys with an intermediate shaft; the driving wheel contains 54 teeth, and the driving pulley on the axis of the driven wheel is 25 inches diameter; required the number of teeth in the other wheel, and the diameter of the other pulley. Let the driven wheel have a velocity of 36, a mean proportional between the extreme velocities 16 and 81; 16 X 54 then,-= 24, the number of teeth in the driven 36 wheel. = ls£ Ans. And = 11-11 inches, diameter of the driven pulley. = 2nd Ans. Example 8 .—Suppose in the last example the revo¬ lutions of one of the wheels to he given, the number of teeth in both, and likewise the diameter of each pulley, to find the revolutions of the last pulley. Here ^ ^ = 36, velocity of the intermediate 24 shaft. = Ans. Also 36 x 25 81, the velocity of the machine. 1111 TELOCITY OE WHEELS. 105 TABLE For finding the radius of a wheel when the pitch is given, or the pitch of a wheel when the radius is given, that shall contain from 10 to 150 teeth, and any pitch required. N oT Teeth. Radius. Num. Teeth. Radius. Num. Teeth. Radius. Num. of Teeth. Radius. 10 1-618 46 7327 81 12-893 116 18 464 11 1-775 47 7-486 82 13051 117 18623 12 1 932 48 7645 83 13210 118 18782 13 2089 49 7-804 84 13370 119 18 941 14 2-247 50 7963 85 13-531 120 19-101 15 2-405 51 8-122 86 13-691 121 19260 16 2-563 52 8-281 »7 13-849 122 19419 17 2 721 53 8-440 88 14008 123 19578 18 2879 54 8-599 89 14168 124 19737 19 3038 55 8-758 90 14-327 125 19896 20 3196 56 8-917 91 14-486 126 20055 21 3355 57 9-076 92 14-645 127 20214 22 3513 58 9235 93 14804 128 20374 23 3672 59 9 395 94 14-963 129 20533 24 3831 60 9-554 95 15 122 130 20-692 25 3-989 61 9-713 96 15 281 131 20851 26 4148 62 9-872 97 15-440 132 21010 27 4307 63 10 031 98 15-600 133 21 169 28 4-466 64 10190 99 15-759 134 21328 29 4 625 65 10349 100 r 15-918 135 21488 30 4-783 66 10508 101 16077 ! 136 21647 31 4942 67 10667 102 16236 137 21'806 32 5-101 68 10826 103 16-395 138 21-965 33 5 260 69 10985 104 16 554 139 22124 34 5419 70 11144 105 16713 140 22283 35 5578 71 11-303 106 16-873 141 22 4 42 36 5-737 72 11-463 107 17032 142 22602 37 5-896 73 11-622 108 17191 143 22761 38 6055 74 11-781 109 17-350 144 22 920 39 6214 75 11-940 110 17-509 145 23 079 40 6373 76 12099 111 17-668 146 23 238 41 6532 77 12-258 112 17-827 147 23 397 42 6691 78 12-417 113 17-987 148 23 556 43 6-850 79 12576 114 18-146 149 23 716 44 45 7-009 7168 80 12-735 115 18305 150 23-875 106 YELOCITT OE WHEELS. Rule. —Multiply the radius in the table by the pitch given, and the product will be the radius of the wheel required. Or, divide the radius of the wheel by the radius in the table, and the quotient will be the pitch of the wheel required. Example 1.—Required the radius of a wheel to contain 64 teeth, of 3 inch pitch. Here 1019 x 3 = 30 57 inches. = Ans. Example 2.—What is the pitch of a wheel to contain 80 teeth, when the radius is 25 - 47 inches ? Here 25-47 -r- 12*735 = 2 inch pitch. Or, set off upon a straight line seven times the pitch given, divide that, or another exactly the same length, into eleven equal parts ; call each of those divisions four, or each of those divisions will be equal to four teeth upon the radius. Example. —Were it required to find the diameter of a wheel to contain 21 teeth, the construction would be as follows :— ] T~1 2T 3| rr 5| 6 | 7| 111 2 1 3 I 4| *1 6| 71 8 | 9 | 10 | ll| < .. 4.. .8.. 12... 16. .20> Thus, 5 divisions and \ of another equal the radius of the wheel. Regular approved proportions for wheels with flat arms in the middle of the ring, and ribs, or feathers, on each side. The length of the teeth = | the pitch, besides clear¬ ance, or 4 the pitch, clearance included. Thickness of the teeth. ^ the pitch- Breadth on the face.2| „ Edge of the rim._■. u » Rib projecting inside the rim. „ Thickness of the flat arms. f » TELOCITY OF 'WHEELS. 107 Breadth of the arms at the points = 2 teeth and \ the pitch, getting broader towards the centre of the wheel in the proportion of \ inch to every foot in length. Thickness of the ribs, or feathers, ^ the pitch. Thickness of metal round the eye, or centre, £ the pitch. Wheels made with plain arms, the teeth are in the same proportion as above; the ring and the arms are each equal to one cog or tooth in thickness, and the metal round the eye same as above, in feathered wheels. To find the power that a cast won wheel is capable of transmitting at any given velocity. Rule. —Multiply the breadth of the teeth, or face of the wheel in inches, by the square of the thickness of one t^oth, and divide the product by the length of the teeth, the quotient is the strength in horses’ power at a velocity of 136 feet per minute. Example. —Required the power that a wheel of the following dimensions ought to transmit with safety, namely, Breadth of teeth.7-§- inches. Thickness . T4 „ And length. 2 „ Here 7-5 x 1'4 2 2 ^ ^ ^2 ^ ^ = ^ - 35 horses’ power. The strength at any other velocity is found by mul¬ tiplying the power so obtained by any other required velocity, and by ’0044, the product is the power at that velocity. Suppose the wheel as above, at a velocity of 320 feet per minute. Then 7’35 x 320 x - 0044 = 10'3488 horses’ power. f 6 108 WATER WHEELS. ON THE MAXIMUM VELOCITY AND POWER OE WATER WHEELS. Since publishing the first edition of this work, I have endeavoured, as far as possible, to acquire the most im¬ proved practical principles of water wheels as a moving power; and 1 .—Of undershot wheels. The term “undershot” is applied to a wheel when the water strikes at, or below, the centre. And the greatest effect is produced when the periphery of the wheels moves with a velocity of ’57 that of the water; —then, to find the velocity of the water, multiply the square root or the perpendicular height of the fall in feet by 8, and the product is the velocity in feet per second. Example. —Required the maximum velocity of an undershot wheel, when propelled by a fall of water 6 feet in height. Here 8 v/ 6 = 2’45 x 8 = 19’6 feet velocity of water. And 19‘6 x *57 = 11*17 feet per second for the wheel. = Ans. 2.— Of breast and overshot wheels. Wheels that have the water applied between the centre and the vertex are styled breast wheels, and overshot when the water is brought over the wheel and laid on the opposite side ; however, in either case the maximum velocity is -§ that of the water; then, to find the head of water proper for a wheel at any velocity, say, As the square of 16'095, or 259, is to 4, so is the WATEB 'WHEELS. 109 square of the velocity of the water in feet per second, to the head * of water required. Example. —Required the head of water necessary for a wheel of 24 feet diameter, moving with a velocity of 5 feet per second. Here 5 §- = 5 X f = 7*5, velocity of the water in feet. And 259 : 4 :: 7*5 2 : *87 feet, head of water re¬ quired. = Ans. But for wheels of from 15 to 20 feet diameter inclu¬ sive, add one-tenth of the diameter minus 1 foot. An d for wheels of from 20 to 30 feet diameter inclusive, add one-twentieth of the diameter. This additional head is intended to compensate for the friction of water in the aperture of the sluice to keep the velocity as 3 to 2 of the wheel; thus, in place of •87 feet head for a 24 feet wheel, it will be *87 + = *87 -f- 1*2 = 2*07 feet head of water. If the water flow from under the sluice, multiply the square root of the depth in feet by 5*4, and by the area of the orifice also in feet, and the product is the quan¬ tity discharged in cubic feet per second. Again, if the water flow over the sluice, multiply the square root of the depth in feet by 3*6 ; and the pro¬ duct multiplied by the length and depth, also in feet, gives the number of cubic feet discharged per second, nearly. Example 1.—Required the number of cubic feet .per second that will issue from the orifice of a sluice 5 feet long, 9 inches wide, and 4 feet from the surface of the water. Here V 4 x 5*4 X 5 x *75 = 2 X 5*4 x 5 x •75 = 40*5 cubic feet per second. = Ans. * By head is understood the distance between the aperture of the sluice and where the water strikes upon the wheel. 110 "WATER WHEELS. Example 2.— What quantity of water per second will be expended over a wear, dam, or sluice, whose length is 10 feet, and depth 6 inches ? Here * 16 * 62 5 x 60 33000 = 20 horses’ power. N.B.—Only about two-thirds of the above results can be taken as real communicative power to machinery. OE THE CIRCLE OE GYRATION IN WATER WHEELS. The cpntre or circle of gyration is that point or circle in a revolving body into which, if the whole quantity of matter were collected or placed, the same moving force would generate the same angular velocity, WINDMILLS. Ill which renders it of the utmost importance in the erec¬ tion of water wheels, and the motion ought always to be communicated from that centre when it is possible. To find the circle of gyration. Rule.—A dd together twice the weight of the shroud¬ ing, buckets, &c., f- of the weight of the arms, and the weight of the water; multiply the sum by the square of the radius ; divide the product by twice the sum of the weight of the shrouding, arms, &c., added to the weight of the water, and the square root of the quotient is the radius of the circle of gyration from the centre of suspension, nearly. Example. —Required the distance of the centre of gyration from the centre of suspension in a water wheel 22 feet diameter, shrouding, buckets, &c., = 18 tons, arms = 12 tons, and water = 10 tons. Here V [(2 X 18 + | x 12 + 10) X ll 2 -f- [(18 + 12) x 2 + 10]} = VK36 + 8 + 10) X 121 -+ (30 x 2 + 10)} = V[(54 x 121) +• 70} = y/ (6534 +- 70) = V 93'34 = 9*7 feet from the centre of suspension, nearly. Table of angles for windmill sails. The radius is supposed to be divided into six equal )arts, and -J- from the centre is called 1, the extremity )eing denoted by 6. No. Angle with the Plane of Motion. 1 18° 24° 2 19 21 3 18 18 4 16 14 5 121 9 6 7 3 extremity. 112 WINDMILLS. The first column contains the angles according to Smeaton ; but experience has taught us that the angles in the second column are preferable. THE TELOCITY OF THRESHING MACHINES, MILLSTONES, CUTTERS FOR BORING IRON, &C. The drum or beaters of a threshing machine ought to move with a velocity of about 3000 feet per minute; hence, divide 11460 by the diameter of the drum in inches ; or 955 by the diameter of the drum in feet; and the quotient is the number of revolutions required per minute. And The feeding rollers must make half the revolutions of the drum, when their diameters are about 3-| inches. If the machine is driven by horses, their velocity ought to be from 2\ to 3 times round a 24 feet ring per minute. The velocity of millstones ought to be from about 1550 to 1600 feet per minute; hence, divide 500 by the diameter of a millstone, in feet, or 6000 by the dia¬ meter in inches, and the quotient is the number of revolutions required per minute. In boring cast iron, the cutters ought to have a velo¬ city of about 110 inches per minute ; hence, divide 35 by the diameter in inches, the quotient is the number of revolutions of the cutter or boring head per minute. And divide 100'by the diameter in inches, the quo¬ tient is the number of revolutions per minute for turn¬ ing wrought iron in general, which is about 300 feet per minute, and about half that velocity for cast iron. PUMPING ENGINES. 113 OE PUMPS AND PUMPING ENGINES. Pumps are chiefly designated by the names of “ lift¬ ing” and “ force” pumps. Lifting pumps are applied to wells, &c., where the height of the bucket, from the surface of the water, must not exceed 33 feet; this being nearly equal to the pressure of the atmosphere, or the height to which water would be forced up into a vacuum by the pressure of the atmosphere. Eorce pumps are applicable on all other occasions, as raising water to any required height, supplying boilers against the force of the steam, hydrostatic presses, &c. The power required to raise water to any height is as the weight and velocity of the water with an addition of about j of the whole power for friction; hence the rule,—Multiply the perpendicular height of the water, in feet, by the velocity, also in feet, and by the square of the pump’s diameter in inches, and again by 2046 ; divide the product by 165000, and the quotient will be the number of horses’ power required. Example. —Required the power necessary to over¬ come the resistance and friction of a column of water 4 inches diameter, 60 feet high, and flowing with a velocity of 130 feet per minute. Hprc 60 x 130 x 4 2 x 2 046. 165000 .6 x 13 x 16 x 124 _ 154752 100000 100000 nearly. _ 6 x 13 x 16 x 31 25000 -= 1-55 horses’ power, Note. —Hot liquor pumps, or pumps to be employed in raising any fluid where steam is generated, require to be placed in the fluid, or as low as the bottom of it, on account of the steam filling the pipes, and acting as a counterpoise to the atmosphere; and the diameter of the pipes to and from a pump ought not to be less than § of the pump’s diameter. 114 PUMPING- ENGINES. The diameter of a pump and velocity of the water given, to find the quantity discharged in gallons, or cubic feet, in any given time. Eule. —Multiply the velocity of the water, in feet per minute, by the number of minutes in the given time, by the square of the pump’s diameter in inches, and by •034 for imperial gallons ; or, ’0005456 for cubic feet, and the product will be the number of gallons or cubic feet discharged in the given time, nearly. Example. —What is the number of imperial gallons of water discharged per hour by a pump 4 inches dia¬ meter, the water flowing at the rate of 130 feet per minute ? Here 130 x 60 X 4 2 x -034 = 13 x 6 X 16 X 34 = 4243*2 gallons. = Ans. The length of strolce and number of stroikes given, to find the diameter of a pump, and number of horses' power that will discharge a given quantity of water in a given time. Eule 1.—Multiply the number of imperial gallons required in the given time by 353, or the number of cubic feet by 2201, and divide the result by the product of the velocity of the water in inches, and the number of minutes in the given time, the square root of the quotient will be the pump’s diameter in inches. 2.—Multiply the number of gallons in the given time by 60, or the number of cubic feet by 375, and by the perpendicular height of the water in feet, divide the result by the product of 165000 and the number of minutes in the given time, then will the quotient be the number of horses’ power required. Example. —Enquired the diameter of a pump, and number of horses’ power, capable of filling a cistern 20 PUMPING ENGINES. 115 feet long, 12 feet wide, and 6 J feet deep, in 45 minutes, whose perpendicular height is 53 feet; the pump to have an effective stroke of 26 inches, and make 30 strokes per minute. Here 20 X 12 X 6-5 = 1560 cubic feet; and 26 x 30 = 780, the velocity in inches per minute. 2 x 2201 45 a a ao 880-4 _s/880-4 _ 29-67 9 3 3 ^1 = 9-89 inches diameter of pump. = 1st Ans. AUn 1560 x 375 x 53 _ 13 x 53 _ 689 * ' 165000 x 45 165 “ 165 4-18 horses’ power. = 2nd Ans. To find the time a cistern will take in filling , when a known quantity of water is going in , and a known portion of that water is going out , in a given time. Eule. —Divide the content of the cistern, in gallons, by the difference of the quantity going in, and the quan¬ tity going out, and the quotient is the time in hours and parts that the cistern will take in filling. Example. —If 30 gallons per hour run in and 22\ gallons per hour run out of a cistern capable of con¬ taining 200 gallons, in what time will the cistern be filled ? of hours = 26 hours and 40 minutes. = Ans. 116 PUMPING ENGINES. To find the time a vessel will take in emptying itself of water. Mr. Banks ascertained, from very accurate experi¬ ments, that a vessel, 3166 feet long and 2‘705 inches diameter, would empty itself in 3 minutes and 16 seconds, through an orifice in the bottom, whose area is -0141 inches; and another 6‘458 feet long, the diameter and orifice as before, would do the same in 4 minutes and 40 seconds ; hence, from these experi¬ ments a rule is obtained, namely, Multiply the square root of the depth in feet by the area of the falling surface in inches, divide the product by the area of the orifice, multiplied by 3’7, and the quotient is the time required in seconds, nearly. Example. —How long will it require to empty a vessel of water, 9 feet high, and 20 inches diameter, through a hole inch in diameter ? Here ^9 = 3, the square root of the depth, And 20* X *7854 = 400 x *7854 square inches, area of the falling surface, And -75* X '7854 = -5625 X "7854 square inches, area of the orifice; mi . 400 x -7854 X 3 _ 400 x 3 _ ’ *5625 x -7854 x 37 *5625 x 37 16 x 400 _x_3 = 6400 = 576 . 6 seconds . OT> 9 X 3-7 11 1 9 minutes and 37 seconds, very nearly. = Ans. On the pressure of fluids. The side of any vessel containing a fluid sustains a pressure equal to the area of the side, multiplied by half the depth; thus, PUMPING ENGINES. 117 Suppose each side of a vessel to be 12 feet long and 5 feet deep, when filled with water, what pressure is upon each side ? 12 x 5 = 60 feet, the area of the side, 2-5 feet = half the depth, and 62-5 lbs. = the weight of a cubic foot of water. Then, 60 X 25 X 62-5 = 9375 lbs. = Ans. To find the number of imperial gallons contained in a yard of circular pipe, of any given diameter. Rule. —Multiply the square of the diameter of the circular pipe in inches by '102, the product will be the content of the pipe in imperial gallons nearly. Example 1. — Required the number of imperial gallons contained in each yard of a 6| inch circular pipe. Here 6 25 2 X '102 = 39-0625 X *102 = 3-984375 imperial gallons. = Ans. Example 2.— Required the content of a yard of 4 inch circular pipe in imperial gallons. Here 4 2 x -102 = 16 x -102 = 1-632 imperial gallons. == Ans. To find the weight that a given power can raise by one of Bramah's pumps, or hydrostatic presses. Rule. —Multiply the square of the diameter of the ram in inches by the power applied in lbs., and by the effective leverage of the pump handle; divide the product by the square of the pump’s diameter, also in inches, and the quotient is the weight that the power is equal to. Exampl-e. —What weight will a power of 50 lbs. raise by means of an hydrostatic press, whose ram is 118 PUMPING ENGINES. 7 inches diameter, pump and the effective leverage of the pump handle being as 6 to 1 ? Here 7 2 x 50 X 6 _ 7 2 X 8 2 X 50 JX_6 •875 2 7 2 = 64 x 300 = 39200 lbs ., or 8 tons 11 cwt. — Ans. In the following rules for pumping engines the boiler is supposed to be loaded with about 2\ lbs. per square inch, and the barometer attached to the con¬ denser indicating 26 inches on an average, or 13 lbs., making altogether 15^ lbs., from which deducting for friction, leaves a pressure of 10 lbs. nearly upon each square inch of the piston. To find the diameter of a cylinder to work a'pump of a given diameter for a given depth. Rule.—M ultiply the pump’s diameter in inches by the square root of - 3 - of the depth of the pit in fathoms, and the product will be the cylinder’s diameter in inches. Example. —Required the diameter of a cylinder to work a pump 12 inches diameter for a depth of 27 fathoms. Here 12 X ^/| = 12 x n/9=12x 3 = 36 inches diameter. = Ans. To find the diameter of a pump that a cylinder of a given diameter can work at a given depth. Rule. —Multiply the cylinder’s diameter in inches by the square root of 3 times the depth of the pit in fathoms, and divide the product by the depth of the pit in fathoms, the quotient will be the pump’s diameter in inches. Example. —What diameter of a pump will a 36 inch cylinder be capable of working 27 fathoms deep ? PUMPING ENGINES. 119 H 36 X -s/ (3 x 27) _ 86 x V 81 _ 4 x 9 __ 27 27 3 4 x 3 = 12 inches diameter. = Ans. To find the depth from which a pump of a given diameter will work by means of a cylinder of a given diameter. Rule. —Divide three times the square of the cylin¬ der’s diameter in inches by the square of the pump’s diameter also in inches, and the quotient will be the depth of the pit in fathoms. Example. —Required the depth that a cylinder of 36 inches diameter will work a pump of 12 inches diameter. Here = 3 2 x 3 = 27 fathoms. = Ans. 120 RULES EOB CALCULATING LIQUIDS. APPROXIMATE RULES EOR CALCULATING LIQUIDS. To find the number of imperial gallons contained in any square or rectangular cistern. Rule.— Multiply the content of the cistern in cubic feet by 6-232, or the content in cubic inches by ’003607, and the product is the number of gallons nearly. Example.— A cistern is 8 feet long, 4\ feet wide, and 3 feet deep, required its contents in imperial gallons. Here 8 X 4'5 X 3 = 108 cubic feet, And 108 X 6’232 = 673’056 gallons. = Ans. Or, 8 feet = 96 inches; 4j feet = 54 inches; and 3 feet = 36 inches; Then, 96 x 54 X 36 = 186624 cubic inches, And 186624 x -003607 = 673-153 gallons. = Ans. as before, nearly. Any two dimensions of a square or rectangular cistern being given, to find the third, that shall contain any number of imperial gallons required. Rule. —Multiply the number of gallons that the cis¬ tern is required to contain by ‘16046 for feet, or by 277-274 for inches, according as the dimensions are given in feet or inches, divide this product by the pro¬ ducts of the two given dimensions, and the quotient will be the third dimension of the cistern nearly. Example. —Required the depth of a cistern to con¬ tain 800 imperial gallons, the length being 6| feet, and width 4f feet. Here 800 X ’16046 _ 8 x 128 368 _ 1026 944 . H X 4f 4T6 feet deep. = Ans. 13 x 19 247 RULES FOR CALCULATING LIQUIDS. 121 To find the content of a cylinder in imperial gallons. Bule. —Multiply the square of the diameter in feet by the length of the cylinder, also in feet, and by 4-895 ; Or, multiply the square of the diameter in inches by the length in fee^and by -034; Or, multiply the square of the diameter in inches by the length, also in inches, and by -00283', and the pro¬ duct will be the content in gallons, nearly. Example. —How many imperial gallons are contained in a circular well 22 j feet deep, and 3£ feet diameter ? Here 85 2 X 225 x 4-895 = 1349 gallons. = Ans. Or, 3^ feet = 42 inches, And 42 2 X 22-5 X ‘034 = 1349 gallons. = Ans. as before. Also, 22| feet = 270 inches, And 42 2 x 270 X -00283' = 1349 gallons. = Ans. as before. The length of a cylinder given, to find the diameter, or the diameter given, to find the length that shall contain any number of imperial gallons required. Bule. —Multiply the number of gallons that the cylinder is required to contain by -2043, divide the product by the length in feet, and the square root of the quotient is the diameter in feet, and parts of a foot; Or, multiply the number of gallons by -2043, and divide the product by the square of the diameter in feet, and the quotient is the length in feet and parts of a foot;—and If the dimensions are in inches in place of feet, use 353 in place of -2043. Example. —What must be the diameter of a cylinder to contain 5 imperial gallons, when the length is 20 inches ? 122 RULES EOR CALCULATING LIQUIDS. 18*8 - = 9'4 inches diameter. = Ans. 2 The cube of the diameter of a sphere in feet, multi¬ plied by 3'263 = imperial gallons ; Or, the cube of the diameter of a^sphere in inches, multiplied by *001888 = imperial gallons. Note. —The weight of a cubic foot of water = 62 5 lbs. avoir¬ dupois. Weight of a cubic inch = *03617 lbs. avoirdupois. Weight of a column of water 12 inches high and 1 inch square = *434 lbs. avoirdupois. Weight of a cylindrical foot of water = 49*1 lbs. avoirdupois. Weight of a cylindrical inch = *02842' lbs. avoirdupois. Weight of a cylindrical column of water 12 inches high and 1 inch diameter = *341 lbs. avoirdupois. Take, for example, a cylindrical column of water 11 inches diameter and 15 feet high, required its weight. Here ll 2 X 15 X *341 = 618*915 lbs. avoirdupois. 11*2 imperial gallons of water = 1 cwt. 224 imperial gallons of water = 1 ton. 1*8 cubic feet of water.= 1 cwt. 35*84 ditto = 1 ton. 1 ditto = 6*238 imperial gallons. 1 cylindrical'foot of ditto.... = 4*895 imperial gallons. STEAM ENGINES. 123 OF STEAM AND THE STEAM ENGINE. Steam is the visible moist vapour which arises from all bodies that contain juices easily expelled from them by heats not^ sufficient for their combustion. But ste^m, as applicable at present to the steam engine, is highly rarified water, the particles of which are expanded by the absorption of caloric, or the matter of heat. Water rises in vapour at all temperatures, but is confined to the surface of the fluid acted upon until it has attained 212° Fahrenheit, called the boiling point; at that heat steam ascends through it, preventing its elevation to a higher- temperature by carrying the heat off in a latent form. The latent heat of steam at the common pressure of the atmosphere, according to very accurate experiments, is found to be 1000°; and we know that the sensible, or thermometric heat = 212°. Now 212° — 32° = ] 80° and 1000° + 180° = 1180°; therefore, steam at 212° i3 highly rarified water, containing 1180° of heat; hence, to find the latent heat' of steam at any other temperature, subtract the sensible heat from 1180°, and the difference + 32° = the latent heat. Example. —Required the latent heat of steam whose sensible heat is 224°. Here 1180° - 224° = 956°, And 956° + 32° = 988° latent heat. = Am. One cubic inch of water produces about 1700 cubic inches of steam at 212°, or the common pressure of the atmosphere; but the boiling point varies considerably, according to the pressure on the surface of the fluid, and, of course, materially affects the density of the vapour produced; thus, in a vacuum, water boils at about 90° ; under common pressure, at 212°; and when pressed with a column of mercury 5 inches in height, will not boil G 2 124 STEAM ENGINES. until heated to 217°; each inch of mercury producing by its pressure a rise of about 1° in the thermometer. The pressure or force of steam in the boiler (less than the weight upon the safety valve) is generally indicated by a column of mercury' in a bent iron tube, which causes the range of the float to be only half the range of the mercury, 2 inches of mercury bein^f aearly equal to 1 lb. pressure of steam in the boiler, thus:— Each inch of the float indicates a pressure of 1 lb. nearly. —Level of the mercury when there is no pressure of steam. To calculate the effect of a lever and weight upon the safety valve of a steam boiler, fyc. The lever, iu all cases, is supposed to be made, finished, and balanced, by a known weight or weights, on the short end, making that point where it rests, or is attached to the valve, the centre of motion; then that weight, added to the weight of the lever, is the effective weight upon the valve, independent of any other additional weight, thus:— STEAM ENGINES. 125 Then there are three different ways that it may be required to calculate the lever. 1. — When a certain pressure may he required upon the valve, the distance of the weight upon the lever, and distance of the valve from the centre of motion given, to -find what weight will he required upon the lever at that distance. Rule. —From the required pressure on the valve in lbs. subtract the weight of the valve, plus the effective weight of the lever, multiply the remainder by the dis¬ tance between the fulcrum and the valve, divide the product by the distance between the fulcrum and the weight, and the quotient is the weight in lbs. required to be placed upon the lever at that distance. 2. — When a certain pressure upon the valve is re¬ quired, the weight upon the lever and distance of valve from the centre of motion given , to find where that weight must he placed. Rule.— From the required weight upon the valve in lbs. subtract the weight of the valve, plus the effective weight of the lever, multiply the remainder by the dis¬ tance between the fulcrum and ]bhe valve, divide the product by the weight in lbs. upon the lever, and the quotient is the distance in inches from the fulcrum that the weight must be placed. 3. — When the distance of weight, distance of valve from the centre of motion, and weight upon the lever are given, to find what pressure is upon that valve. Rule.— Multiply the weight in lbs. upon the lever by the distance in inches to the fulcrum, divide the product by the distance between the fulcrum and the valve, and the quotient, plus the weight of the valve and effective weight of the lever, equals, the weight upon the valve in lbs. Example 1 . — Suppose the lever A B (as above) to be 24 inches in length, and the valve C placed 5 inches G 3 126 STEAM ENGINES. from the centre of motion A, what weight must be placed upon the lever 20 inches from A, to equal 80 lbs. on the valve C, the weight of the lever being 2 lbs., the weight D, which balances the lever, 4^ lbs., and the weight of the valve 3 lbs. ? Here 2 - 0 lbs. weight of the lever. 4‘5 lbs. to balance ditto. 3-0 lbs. weight of the valve. (80 - 9-5) x 5 _ 70-5 x 5 _ Sum = 9-5 lbs. then v --£0- ^ — 17-625 lbs. = Arts. 4 Example 2. —Suppose, as in the last example, the weight upon the lever equal 17‘625 lbs., it is required at what distance from A the weight must be placed to equal 80 lbs. at 0. ^ ^ ^ hr\.K r Hr\.K vx Ofl ^ (80 - 9-5) X 5 _ 70-5 x 5 _ 70-5 ^<_20 _ ±l6re m525 17-625 70-5 20 inches. = Ans. Example 3. —Suppose, as before, that a weight of 17-625 lbs. is placed upon the lever 20 inches from A, required the pressure at C, the distance from the centre of motion being 5 inches, and the effective weight of the lever at that point equal 6i lbs., also the weight of the valve 3 lbs. v 90 _ - _ __ , n ,. _ T. _ _i_ Here ^ 625 x 20 = 17-625 X 4 = 70 5 lbs. to which 5 Sum = 80 lbs. = Ans. To find the proper diameter for a safety valve. Rule.— Multiply the bottom surface of the boiler, or surface immediately exposed to the action of the fire, in feet, by the multiplier opposite to the pressure m lbs. on each square inch of the safety valve, and the square root of the product is the valve’s diameter in inches at A CONDENSING ENGINE ON THE MOST IMPROVED PRINCIPLE. STEAM' ENGINES. 127 the narrowest part. If the boiler is to have two safety valves, then the square root of half the product equals the diameter of each. Pressure in lbs. Mult ipii ers . per square inch. * 3 . -356 4 . 353 5 . 348 6 . 344 7 . -339 8 . 336 10 . *329 12 . -321 20 . 25 . Multipliers. ... *315 ... -305 ... -293 ... -289 ... -282 ... -275 ... 270 ... 264 Table of the elastic force of steam on a square inch. / \ f N op / N 2i 1-963 o 220 515 3 2-356 222 618 31 2-749 4 2231 7-21 4 3142 2251 227 8-24 4 1 3*534 £ 9-27 5 3-927 h 2281 r 10-3 51 4-320 230 11-3 6 4-712 8 £ 2311 12-3 61 1 5105 233 a 134 § 1 7 71 8 8.1 CD J 5-498 5- 888 6- 283 6-676 1 & |l V e S j 234 235 236 2371 o a s 1 ^ 14- 4 15- 4 16- 5 175 I* C3 1 9 / | ( 7 068 . ) "3 2 Ss§ 239 185 .a ■ rS 91 j a* 7-461 240 * 196 % 10 7-854 ® § 241 s 20-6 J a 101 a 8-247 -g 05 242 S 21 6 .a ra £ 11 o 8-639 to 03 243 22-6 m 111 9032 244 23 7 12 9-424 2451 24-7 15 11-78 252 'g 30-9 20 15-71 'I 261 41-2 25 19-63 •i 269 51 5 30 23-56 276 61-8 35 27-49 C8 283 721 40 31-42 o 289 82-4 45 35-34 oo 2941 92 7 f°\ [39 27 j 300 103 J g 4 128 STEAM ENGINES. Multiply the degrees of heat in either this or the following table by '06, and the product will be the superficial feet of flue plate exposed to the action of the fire for each horse power. And multiply the degrees of heat by '41, and the product will be the areal inches of furnace bar for each horse power. Table of the elastic force of steam on a circular inch. The proportion that various substances bear to each other in producing heats sufficient to raise equal quan- STEAM ENGINES. 129 tities of water to equal temperatures is nearly as fol¬ lows : Coke. 0-375 Culm or Slack.. 1-875 Coal. 1-000 Wood.2875 Hence, multiply the degrees of heat in either of the preceding tables by the following numbers opposite the material by which the steam is to be produced, and the product will be the weight in Ihs. avoirdupois that is required on an average per hour for each horse power : Coke. -024 Slack . '118 Coal . -063 Wood. -180 To find the height of a column of water to supply a steam holler against any pressure of steam required. Rule. —Multiply the pressure in pounds (upon a square inch of the boiler) by 2 - 5, and the product will he the height in feet above the surface of water in the boiler. Example. —Required the length of feed pipe capable of supplying a boiler with water when the pressure of steam is 4 pounds per square inch. Here 2-5x4 = 10 feet above the surface of the water in the boiler. = Ans. Steam Engine is the name of a machine which de¬ rives its moving powers from the elasticity and conden¬ sibility of steam. Steam, to produce a maximum of useful effect as a moving power, requires to be reduced to a certain de¬ termined velocity, and although this maximum velocity has been exhibited to the public by various eminent writers upon the steam engine, still discrepancies exist amongst practical engineers ; and no universally ac¬ knowledged rules have as yet been established: how¬ ever, the following tables may be relied upon as exhibiting the results deduced from the most celebrated g 5 130 STEAM ENGINES. rules, and tested by many engines doing the greatest amount of duty, as proved by accurate trials with indi¬ cators of the most recent and approved construction. Length of Stroke in ft. and in. Number per Minute. Velocity in Feet per Minute. Length of Stroke in ft. and in. Number per Minute. Velocity in Feet per Minute. 2 0 43 172 4 6 241 218J 2 6 38 190 5 0 22 220 3 0 34 204 6 0 19 228 3 6 30 210 7 0 174 245 4 0 27 216 8 0 16 256 N.B.—These are to be considered as the velocities of land engines, or engines whose connecting rods are not less than three times the length of stroke; but marine engines, being generally confined to connecting rods of not more than 2 or 2j times the length of stroke, have their maximum velocities considerably reduced. Hence, the subjoined table will be found pretty correct when the periphery of the wheels moves with a velocity of about 1300 feet per minute, and the floats or paddle boards calculated by the following rules, which I have found, in practice, to produce the greatest satisfaction; namely, economizing of fuel, a steady supply of- steam, without waste, and the vessel propelled quicker than when the surface of the floats was less, and moving at a greater velocity. Table of velocities for marine engines. Length of Stroke in ft. and in. Number per Minute. Velocity in Feet per Minute. Length of Number per Minute. Velocity in Feet per Minute. 2 0 42 168 4 0 24 192 2 3 39J 177® 4 6 214 1934 2 6 36 180 5 0 20 200 2 9 33 1814 5 6 19 209 3 0 31 186 6 0 18 216 3 6 27 189 7 0 15f 2204 STEAM ENGINES. 131 To find the surface of the floats or paddle hoards. Bule 1. —Multiply the number of horses’ power that the engine is equal to by 3f, divide the product by the diameter of the wheel in feet, and the quotient is the area of each float, or paddle board. Bule 2.—Multiply the number of horses’ power by 2^, divide this product by the diameter, and the quotient is the length in feet. Bule 3.—The breadth is 1 foot 10 inches. Example. —Bequired the area, length, and breadth of each paddle board, for a steam vessel with two en¬ gines of 80-horse power each, and wheels of 20 feet diameter. Here ^ ^ ^ = 3§ x 4 = 15 feet area. = ls£ Ans. Then - - -^ Q 2 ^ = 4 X 2*V = 8-^ = length of each board. = 2 nd Ans. And 1 foot 10 inches the breadth. — 3 rd Ans. And when there is only one engine in the vessel, take ■§• of the area, and length found as above. Each wheel, from 12 to 14 feet diameter, ought to have 12 floats ; from 14 to 16 feet diameter, 14 floats ; from 16 to 18 feet diameter, 16 floats; and from 18 to 22 feet diameter, 18 floats, &c. (which is nearly 1 float to each foot of the diameter.) Principles upon which the rule is founded for calcu¬ lating the power of a steam engine. Hitherto it has been customary, in estimating the power of condensing engines, to reckon the force of g 6 132 STEAM ENGINES. the steam at a constant quantity, namely, 2| lbs. per circular inch, totally disregarding any extra pressure in the boiler, or increased weight upon the safety valve.. Hence, in order to form a rule whereby to approxi¬ mate more nearly to the real effective power of the engine, it was necessary first to ascertain the effective force of the steam,—And, To determine this, I recently made a series of experi¬ ments upon engines without any extra lap upon the valves, whereby to work expansively, when I found that, on account of the nature of the valve’s motion, only about three-fourths of the stroke was performed by steam at, or near, the density of the steam in the boiler, the stroke, of course, being terminated expan¬ sively ; hence, the w’hole effective force of the steam thus applied can only be taken at about four-fifths of its original pressure. The benefit arising from the condenser is on an average equal to 26 inches of mercury, or about 13 lbs. per square inch ; consequently, 13 plus four-fifths of the pressure on each square inch of the safety valve, equals the whole effective force on each square inch of the piston’s area. Then about 8i lbs. are expended in overcoming the resistance and friction of a condensing engine, and may be thus estimated: 13 minus 8^ equals 4f, and 4f plus *ths of the weight upon each square inch of the safety valve equals the whole amount of useful effect in giving motion to machinery. The process of calculation may be simplified thus : 4f lbs. per square inch = 3‘73 lbs. per circular inch, by which means the circle only requires to be squared, and the labour of multiplying by ‘7854 is dispensed with. GENERAL RULES. 1.—Multiply the square of the cylinder’s diameter in inches by 3'73 plus J-ths the pressure on each circular STEAM EN&INES. 133 inch of the safety valve, and by the velocity of the pis¬ ton in feet per minute ; divide the product by 33000, and the quotient is the effect of the engine expressed in horses’ power. Example. —Suppose a cylinder 24 \ inches diameter, stroke 4 feet, or 200 feet velocity per minute, and the weight upon the safety valve 3'5 lbs. per circular inch, required the effective power. Here §ths of 3-5 = 2-8, and 3*73 + 2*8 = 6-53 lbs. effective force. Then 24 52 X 6 ' 53 x 200 - 402 x 6 53 x 2 - ’ 33000 2401 x 6 53 _ 2401 X -653 660 66 power. = Ans. 2.—Multiply 33000 by the number of horses’ power required, and divide the product by the velocity of the piston in feet per minute, multiplied by 373 plus fylis the pressure on each circular inch of the safety valve, and the square root of the quotient is the cylin¬ der’s diameter in inches. 2 2 x 330 . 1568 = 24 horses’ Example.— Eequired the diameter of a cylinder for an engine of 30 horses’ power, with a 6-feet stroke, or 228 feet per minute, and steam at 2\ lbs. per circular inch. Here %ths of 2*5 = 2; and 373 + 2 = 573 lbs. effective force. Hence J 33000 x 30 228 x 573 j 247500 __ , z^vtxju __ V 57 x 5 73 yj 326-61 ~~ diameter. = Ans. / 990000 s/ 228 x 5 73 “ v/758 =27-53 inches 134 STEA.H ENGINES. Note. —To obtain four-fifths of the pressure of steam, multiply the original pressure by 4 and divide by 5, the quotient is the pressure required. Or, subtract from the original pressure its fifth part. The above are to be taken as general practical rules for engines not working expansively further than what is compulsory from the nature of the slide valve ; but where engines are worked more expansively, and greater accuracy required, recourse must be had to the following rules for obtaining the uniform force of the steam. Rule 1.—Divide the length of the stroke in inches by the distance (also in inches) that the piston moves before the steam is shut off, and divide the pressure on the boiler in lbs. by the quotient: — 2.—Add 1 to the hyperbolic logarithm of the first quotient, which is the number of times to which the steam is expanded, and multiply the sum by the second quotient, which is the number of lbs. to which the steam is expanded, and the product is the uniform force of the steam acting throughout the whole stroke. Example. —Let the steam in the boiler of an engine equal 45 lbs. per inch, the length of stroke 4 feet, and the steam to be shut off after the piston has moved 16 inches; required an equivalent force of steam in the cylinder. Here 4 feet = 48 inches, and 48 + 16 = 3, the first quotient. Then, 45 + 3 = 15 lbs., the second quotient. And 1 + hyp. log. 3 = 1 + 1-0986123 = 2-0986123. Hence 2-0986123 X 15 lbs. = 31-4791845 lbs. uni¬ form force of the steam. = Ans. PUBLISHED Br SIMPKIN MARSHALL «• CO, LONDON. LITHOGRAPHED BY RH.FRASER. LIVERPOOL AN 80 HORSE MARINE STEAM ENGINE made I,j GEO. FORRESTER SC^ofEPOOL. STEAM ENGINES. 135 HYPERBOLIC LOGARITHMS. No. Logarithms. No. Logarithms. No. Logarithms. No. Logarithms. il •2231436 31 1 1786550 51 1-6582281 71 1-9810015 i* •4054651 31 1-2527630 5.1 1-7047481 n 20149030 il •5596158 31 1-3217558 5| 1-7491999 Ti 20476928 2 •6931472 4 1-3862944 6 1-7917595 8 20794415 21 •8109302 4j 1-4469190 61 1 8325815 8 J 2-1400662 21 •9162907 ■li 1-5040774 65 1-8718022 9 2-1972246 21 1 0116009 4 i 1-5581446 61 1-9095425 9.1 2-2512918 3 1 0986123 5 1-6094379 7 1-9459101 10 2-3025851 THE STEAM WAY. Multiply the square of the cylinder’s diameter by ‘08, and divide the product by 3, the quotient will be the area of port or steam way. Example.— What area of port or steam way is ne¬ cessary for a cylinder 36 inches diameter ? Here 302 * ’° 8 - = 36 X 12 x -08 = 432 x ’08 = 34*6 inches area of steam way. = Ans. oe the slide yalve. When the valve is at the middle of its stroke, the faces ought to cover the apertures on the exhausting side about of an inch; the cover on the steam side being for the purpose of cutting off the steam at any part of the stroke, is, therefore, at the entire discretionary judgment of the engineer. However, we find from practice, that high-pressure engines with short strokes, as locomotives, &c., require no more than will cover the apertures properly; whereas condensing engines, with steam of 2f to 3 lbs. per square inch, will work well with f of an inch cover on the steam side; and marine engines give great satisfaction with If inches cover, when the steam is 46 inches diameter- 45 45 (353 x 15) — Ans. And when the pump’s diameter is given, as above, 353 353 _ 353 we have i X 6 45 x 1-6* length of stroke. " 15 X 2-56 38-4 = ^ = 9 inches 140 STEAM ENGI1TES. t THE AIR PUMP. The Air Pump for a land engine generally requires to be larger in proportion to the cylinder than the air pump for a marine engine, on account of having fre¬ quently to condense with water at a higher temperature; hence, when the stroke of the bucket is half the stroke of the piston, multiply the cylinder’s diameter in inches by *67, and the product is the diameter of air pump. •—Again, multiply the diameter of the cylinder of a marine engine, in inches, by ’575, or ff, and the pro¬ duct is the diameter of air pump. Example.— What diameter of air pump is requisite, for an engine whose cylinder is 28 inches diameter ? Here 28 X '67 X 18'76 inches diameter. = Ans. When the stroke of the bucket is either more or less than half the stroke of the piston, the pump’s diameter will then be obtained by the following Rule. —Square the given diameter, multiply by the length, and divide by the length proposed, extract the square root, and the product will be the diameter. Or, multiply the given diameter by the square root of the quotient of the first length by the altered length. Example. —Suppose an engine with a 4-feet stroke required an air pump 26 inches diameter with a 2-feet stroke, but necessity requires it to be 6 inches nearer the end of the beam, what must be the diameter of air pump, the beam being 11 feet long P Here radius of beam = 66 inches. Then 4 feet : 2 feet :: 66 inches ; 33 inches, first distance of the air pump. And 33 -f 6 = 39 inches, the altered distance of the air pump. # And 66 48 : 39 : 28'36 inches, length of stroke. And / 2 — * 24 = 26 / 24 = 260 / - 24 J 28-36 J 28-36 J 2836 STEAM ENGINES. 141 _ opo / 6 _ 260 n/ (6 X 709) _ 260 ^ 4254 _ s/ 709 709 709 260 7 x 0 ° tf 5 ' 22 = = 24 inches, diameter of pump, nearly. = The Condenser ought to be a little more in capacity than the air pump ; but in the case of marine engines, where the bottom of the condenser and bottom of the cylinder are nearly on a level, care must be taken to make the passage between the valves and condenser large enough to contain the condensing water required for one stroke of the piston, besides leaving a proper communication, otherwise the connexion between the cylinder and condenser will be cut off by water of nearly 100° of heat, on account of the cylinder being twice filled with steam for each effective stroke of the air pump. One-fourtb of the area of air pump will give the area of foot and discharging valves; thus, 24 inches dia¬ meter gives 452'39 inches area, one-fourth of which = 113‘0975 inches, area of valves. The piston rod is about -j- 1 ^ of the cylinder’s diameter; the air pump rod in the same ratio, unless it be made of copper, and then it may be about of the pump’s diameter. THE BEAM. When a beam is applied to an engine its length ought not to be less than three times the length of the stroke, and its breadth half the stroke, or in high pressure engines, J- of the stroke; also its best form is a parabola. To find the thickness of a beam when its length and breadth , together with the diameter of the cylinder , are given. Rule. —Multiply the whole pressure of steam on the piston, in lbs., by half the length of the beam in feet, and divide the product by 70 times the square of the breadth, in inches, and the quotient will be the thickness in inches, nearly. 142 STEAM! engines. Example.— What thickness of beam is requisite for an engine whose cylinder is 25 inches diameter, the length of the beam being 15 feet, length of stroke 5 feet, and the effective pressure on each square inch of the piston = 15 lbs.? Here area of piston = 25 2 x ‘7854 = 490-875 square inches. And length of stroke = 60 inches, which divided by 2, gives 30 inches, the breadth of the beam. „ 490-875 X 15 X 7-5 _ 490 875 x 15 X 15 900 x 140 = *8765625, or ^ of an inch in "Whence 30 2 X 70 . 1402 5 20 X 28 16 490-875 thickness, nearly. = Ans. To find the versed sine of the arc described by the beam of an engine. Rule. —Divide the square of half the length of the stroke, in inches, by the length of the beam, also in inches, and the quotient is the versed sine. Example. —Required the versed sine of the arc de¬ scribed by an engine beam 12 feet in length, the chord of the arc, or length of the stroke, being 4 feet. 24 2 24 2 Here — = —- = 2 2 = 4 inches, the versed sine. 144 12 = Ans. Note. —When the beam is not equal lengths at each end from the centre on which it vibrates, the length is then to be taken equal to twice the radius of that end of which the versed sine is required. TILE PARALLEL MOTION. The beam being given, to find the length of the radius rods . Rule. —Divide the square of the distance between A and B, on the beam, by the distance between B and C, and the quotient is the length of the radius rod d x. TABLE OF PARALLEL MOTIONS. ST !AM ENGINES. 143 Fig. 1, Example. —Suppose a beam 12 feet long, and the stud for the back links 39 inches from the centre, required the length of radius rods. Here radius of beam =72 inches, and 72—39 =33 in. ; then = — = 46-09 inches. = Am. OO 11 11 Note.—T he length of the front and back links equal half the length of the stroke. Fig. 2 , Example. —Suppose b d = 32i, and d a = 35^, to find d F. Here —= 29'5 inches, nearly. = Am. 3525 564 ’ 3 Fig. 3.—As the calculation of this motion is rather tedious, on account of the various angles formed by the side rods, it is considered better to lay it down in the following geometrical form:— Upon the line A on, with the radius of the beam, describe the arc b on t; from on, with half the length of stroke, cut the arc in b and t, draw the line b t and r on equal the versed sine described by the beam ; bisect r on in n, and erect a perpendicular line for the centre of the cylinder. Again, from b ont, with the length of the side rods, cut the perpendicular line; at the bottom, middle, and top stroke of the cross-head, draw the lines b b, on on, 11; from the end of the cross-head, or top of the side rods, with any convenient distance, set off the pin or stud in the side rod for the end of the parallel bar 1, 2, 3, from which, with the distance s t , describe arcs at dJ)d; draw the lines d 1, D 2, &c. Then the length of the crank may be found either by the seventh problem in Geometry, page 31, or the eleventh problem in Mensuration, page 52. THE CONNECTING EOD. The length of connecting rod is in general three times the length of stroke, which determines the per- 144 STEAM ENGINES. pendicular distance between the centre of the beam and centre of fly-wheel shaft. Or , if the engine is erected, the length of connecting rod is the perpendicular dis¬ tance between the centre of the fly-wheel shaft, and centre of the beam. THE FLY-WHEEL. To find the weight of the rim or ring of a fig-wheel proper for a steam engine. Rule. —Multiply the constant number, 1368, by the number of horses’ power that the .engine is equal to; divide the product by the diameter of the wheel, in feet, multiplied by the number of revolutions per minute ; and the quotient is the weight of the ring in cwts., nearly. Example. —Required the weight of the rim of a fly¬ wheel proper for an engine of 20 horses’ power, the wheel to be 16 feet diameter, and make 21 revolutions per minute. m x 20 - 114x5 = 5™ = 81-4 owt., nearly. 16 X 21 7 7 = Ans. Note.—T he fly-wheel of an engine for a corn or flour mill ought to be of such a diameter that the velocity of the periphery of the wheel may exceed the velocity of the periphery of the stones, to prevent, as much as possible, any tendency to back lash, as it is termed. The necessary weight and diameter of the wheel being found, suppose a breadth of rim, and the thick¬ ness to make the weight in cast iron will be found by the following Rule. —Multiply the required weight in lbs. by 4-841, and divide the result by the product of the sum and difference of the diameters, in inches, and the quo¬ tient is the thickness of the ring in inches. STEAM ENGINES. 145 Example. —"What thickness must a ring be to equal 81-4 cwts., when the outer diameter is 16 feet, and the inner diameter 14 feet 8 inches ? Here 814 cwts. = 9116-8 lbs. ml _ 9116-8 X 4-841 _ 9116 8 X 4 841 lhen ’(192 + 176) X (192 - 176) 368 x 16 _ 284-9 x 4-841 137^2 = 7>496 inch near ^ ~ 184 ' 184 " = Ans. And if the ring is to be of a cylindrical form, find the diameter of a circle (by Problem XIII. p. 55, in Mensuration) having the same area as the cross-section of the ring found. Thus, suppose the ring, in the last example, he required to be cylindrical,—Required its cross-sectional diameter to equal 81*4 cwts., the diameter of the wheel being 16 feet. Here 7 496 x 8 = 59-968 inches cross-sectional area of the ring found. * And 1 59 ' 968 x 452 = A 271 . 05 ' 536 = f 76-35 355 \/ 355 = 8*74 inches diameter, nearly. = Ans. Or, as an approximation, multiply the required weight in lbs., by T62 ; divide the product by the diameter of the wheel, in inches, and the square root of the quotient will be the diameter of the cross-section of the ring, in inches, nearly. Thus J ( 911 1 6 8 8 x X 1 ^ ~) = V (284-9 X -27) = 76'923 = 8-78 inches. = Ans. as before, nearly. Sometimes (for various reasons) it is necessary to have the fly-wheel upon a second mover; for instance, H 146 STEAM ENGINES there is a 6-horse engine making 50 revolutions. per minute, having a fly-wheel of 7 ft. diameter, and 9 cwt., but, by the rule, it ought to be 23-46 cwt. Now, a larger wheel cannot be got in, but the same may be put upon a second mover,—required the velocity that will increase its momentum equal to 23 46 cwt. on the first mover. Here 7 feet (diameter) X 3T416 — 2T9912 feet circumference, and 21-9912 x 50 revolutions = 1099-56 feet velocity. Cwt. Cwt. Velocity. Velocity. Then, as 9 : 23-46 :: 1099 56 : 2866-1864, and 2866-1864 21*9912 = 130 revolutions per minute, nearly. = Ans. To find the centrifugal force of a fig-wheel. Rule. —Multiply -6132* by the diameter of the wheel in feet and its weight, and divide the product by the square of the time of one revolution ; the quotient will be the centrifugal force. Example. —Required the centrifugal force of a fly¬ wheel 15 feet diameter, and making 40 revolutions per minute, the weight of the ring being 3 tons. Here 60 -r- 40 = 1*5, time of one revolution. AM ' 6132 X 15 X 3 _ jS132 _ 61j2 _ me4 l*5 a "05 5 tons, the centrifugal force. = Ans. The centre of percussion in a fly-wheel, or wheels in general, is f distant from the centre of suspension nearly. * This number is = - 16*095 length of seconds pendulum - (seepage 101). STEAM ENGINES. 147 Note. —The centrifugal force is that power or tendency which all revolving bodies have to burst, or fly asunder in a direct line. And the centre of percussion in a revolving body is that point where the whole force or motion is collected, or that point which would strike any obstacle with the greatest effect. THE GOVERNOR OR REGULATOR. The length of pendulums given , to find the number of revolutions per minute. Rule. —Divide 375 (see page 101) by the square root of the pendulum’s length, and half the quotient will be the velocity required. Example. —"What number of revolutions ought a governor to make per minute whose pendulums are 24 inches long ? TW, 375 375 x s/ 24 _ 3000 x n/ 24 _ ' */ 24 24 192 12“ x 2 X V6= 1222 x */6 = 1222 x 2-449 = 2449 g 2 ~ = 76, then 76 -f- 2 = 38 revolutions per minute. = Ans. The revolutions per minute of a governor given , to find the length of pendulums. Rule. —Divide 375 by twice the number of revolu¬ tions per minute, and the square of the quotient will be the length required. Example.— When the velocity of a governor is 38 revolutions per minute, what ought to bo the length of pendulums ? Here 38 x 2 = 76, and 93 2 = 24-3049 h 2 inches. = Ans. 148 STEAM ENGINES. OE HIGH-PEESSUEE STEAM ENGINES. High-pressure engines, in general (if in good condi¬ tion), will work when the force of the steam is about 4 lbs. per circular inch,—that is, 4 lbs. on each circular inch of the piston will overcome the resistance and fric¬ tion of the engine itself, divested of machinery, &c. Hence the rule. 1. Prom the pressure in lbs. on each circular inch of the boiler deduct 4 lbs. ; multiply the remainder by the square of the cylinder’s diameter in inches, and by the velocity of the piston in feet per minute ; divide the product by 33000, and the quotient will be the force of the engine expressed in horses’ power. Example.— Suppose a cylinder 8 inches diameter, stroke 2 feet, making 45 revolutions per minute, or 180 feet, and steam 23'5 lbs. per circular inch, required the power. ^ (23-5 - 4) x 8 2 X 180 _ 19 5 X 64 X 6 __ Mere 33000 1100 = 6'8 horses’ power, nearly. = Ans. 1100 v J 2. Multiply 33000 by the number of horses’ power required, and divide the product by the velocity of the piston in feet per minute, multiply by the force of the steam in lbs. on each circular inch of the boiler, minus 4 lbs., and the square root of the quotient is the cylin¬ der’s diameter in inches. Example. —Required the diameter of the cylinder for an engine of 68 horses’ power, when the stroke is 2 feet, and making 45 strokes per minute, the force of the steam being 235 lbs. per circular inch. STEAM ENGINES. 149 Here 7 7 33000 x 6-8 _ 1100 x 6-8 180 x (23-5 - 4) J 6 x 19 5 = s/ 64 = 8 inches diameter. = Ans. Note. —There is always a resistance of steam on the piston of a high-pressure or non-condensing engine equal to the pressure of the atmosphere; but this cannot be taken into account, unless we also take into account the pressure of the atmosphere upon the boiler. 150 MISCELLANIES. MISCELLANIES. Approximate rules for finding the weight of round,square, and rectangular beams, bars, Sfc., of cast and wrought iron. Rule 1. For cylinders. —Multiply the square of the diameter in inches by the length in feet, and by 2 , 65 for wrought iron, or 2‘48 for cast iron, and the product will be the weight in pounds avoirdupois, nearly. 2, or generally. —Multiply the area of the cross section in inches by the length in feet, and by 3’38 for wrought iron, or 3'16 for cast iron, and the product will be the weight in pounds avoirdupois, nearly. Exatwptve 1 .—Required the weight of a round bar of wrought iron 14 feet long and 2\ inches diameter. . Here 2'5 3 x 14 X 2*65 lbs. = 232 lbs. — Ans. Or, if the area of the cross section, 2 , 5 2 X ’7854 = 4 91, had been given, then by the second or general rule 4-91 x 14 x 338 lbs. = 232 lbs. = Ans. as before. Example 2.—The length of a piece of cast iron is 9 \ feet, its breadth 7 inches, and thickness 2\, required its weight. Here 9| X 7 X 2i X 3-16 = 472-815 lbs. = Ans. The dimensions of a cast iron ring being given, to find its weight, nearly. Rule. —Multiply the breadth of the ring added to the inner diameter by -0074, and that again by the breadth and by the thickness, and the product will be its weight in cwts., nearly. Example. —Required the weight of a ring whose dimensions are 8 feet 4 inches interior diameter, 5 inches broad and 4 inches thick. MISCELLANIES. 151 Inches. Here 8 ft. 4 in. = 100 ; then (100 + 5) X 5 x 4 X ’0074 = 105 X 20 X ’0074 = 15-54 cwts., nearly. = Ans. To find the weight of any cast iron hall whose diameter is given. Bttle. —Multiply the cube of the diameter in inches by -1377, and the product will be the weight in avoir¬ dupois pounds, nearly. Example.—B equired the weight of a ball 7 inches diameter. Here 7 3 X '1377 = 343 X *1377 = 47-2311 lbs. — Ans. To find the diameter of a cast iron hall when the weight is given. Btjle. —Multiply the cube root of the weight in pounds by 1-9365, and the product will be the diameter in inches, nearly. Example. —Bequired the diameter of a ball that will weigh 64 pounds. Here V 64 x 1-9365 = 4 X 1-9365 = 7-746 inches diameter. = Ans. 152 MISCELLANIES. TABLE Containing the weight of a square foot of copper and lead, in lbs. avoirdupois, from fa to % an inch in thickness, advancing by fa. Thickness. Copper. Lead. fa 1-45 1-85 fa 2-90 3-70 fa 435 5-54 f 5-80 739 8 + ^ 7-26 924 8 "t" T8 871 11 08 8 + & 10-16 12-93 11-61 14-77 \ + fa 13-07 16-62 \ + fa 1452 18-47 } + * 15-97 1742 20-31 2216 1 + fa 18-87 24 00 l + fa 2032 25-85 l + fa h 21-77 27-70 23-22 29-55 TABLE Of the weight of a square foot of sheet iron in lbs. avoirdupois, the thickness being the number on the wire gauge. — No. 1 is 5 /is °f an inch; No. 4, '/ 4 ; No. 11, */ 8 , 8fc. No. on wire 1 ^ gauge .../ 2 3 4 5 6 7 8 9 10 11 Pounds \10.5 avoir. .../ 12 11 10 9 8 7-5 7 6 5-68 5 No. on wire 1 ^ gauge .../ 1 13 14 15 16 17 18 19 20 21 22 Pounds 1 4 . 62 avoir. .../ Lsi 4 3-95 3 2-5 218 1-93 1-62 1-5 1-37 TABLE Of the weight of a square foot of boilerplate iron, from */ 8 to 1 inch thick, in lbs. avoirdupois. Vs 3 /l6 'A Via 7s 7.6 'A 716 7s 1 V 16 7 « 13/ /16 Vs *7.6 1 in. 5 7-5 10 12-5 15 17 5 20 225 25 275 30 32-5 35 37-5 40 MISCELLANIES. 153 TABLE Of the weight of solid cylinders of cast iron, 12 inches long, in lbs. avoirdupois. Dmr. Inch. Weight, in lbs. Dmr. Inch. Weight, in lbs. Dmr. Inch. Weight, in lbs. Dmr. Inch. Weight, in lbs. Dmr. Inch. Weight, in lbs. $ 1-394 11-193 34 30364 51 81-953 9 200776 7 b 1-898 2\ 12-548 31 32-572 6 89 234 94 223705 1 in. 2-479 2| 13-981 3i 34-857 61 96-825 10 247*872 1 3137 24 15 492 31 37-219 0> 104726 104 273279 1 3-873 2.1 17080 4 39-660 6? 112-936 li 299-925 b 4-686 21 18-745 44-772 7 121-457 114 327811 1 5-577 2 1 20-488 50-194 71 130-288 12 356935 1 6-545 3 22 308 q 55-926 74 139-428 13 418-903 1 7-591 34 24 206 5 61-968 7i 148-878 14 485-829 i; 8714 3? 26181 51 68-320 8 158-638 15 557712 2 9-915 3f 28-234 54 74-981 84 179087 16 634-552 Cubic inches of cast iron multiplied by *263 = lbs. avoirdupois. Circular inches of cast iron multiplied by ’2066 = lbs. avoirdupois. TABLE For finding the weight of malleable iron, copper, and lead pipes, 12 inches long, of various thicknesses, and any diameter required. Thickness. Mall. Iron. Copper. Lead. of an in. •104 •1210 •1539 A •207 •2419 •3078 1 •3108 •3629 •4616 8 •414 •4839 •6155 4 + A •518 •6049 •7694 B + ife •621 •7258 •9232 B 4* & •725 •8468 1 0771 1 •828 •9678 1-2310 Rule. —Multiply the circumference of the pipe in inches by the numbers opposite the thickness required, and by the length in feet; the product will be the weight in avoirdupois lbs. nearly. Example. —Required the weight of a copper pipe 12 feet long, 15 inches in circumference, J and T ' 5 of an inch in thickness. Here 7258 lbs. X 15 X 12 = 130-644 lbs. nearly. = Ans. H 5 154 MISCELLANIES. TABLE Containing the weight of wrought iron bars 12 inches long, in lbs. avoirdupois. Inch. Round. Square. Inch. Round. Square. 4 1 •166 *211 24 16-59 21-13 •373 •475 2 f 18-30 23-29 4 •664 •845 21 2008 25-56 1 1-04 1-32 2 1 21-94 27-94 I 1-50 1-90 3 2396 30 42 l 2-03 2-59 34 28-04 35-70 l 2 65 338 34 32-52 41-41 lj 336 4 28 3f 37 34 47-53 M 4-15 5-28 4 42 48 54-08 n 5-02 639 4 I 47-96 6105 14 lg 5-99 7‘60 4 53-77 6845 7-01 892 H 59-91 76-27 i| 8 13 1035 5 66-38 84-51 1 1 933 11-88 5« 7318 9317 2 10-62 13-52 4 80-32 102-25 2g 11-99 1526 5 I 87-78 111-76 2| 13 44 1711 6 95-58 121 69 . 2g 1498 1907 7 13010 165 63 TABLE Of the proportional dimensions of 6 sided nuts for bolts, from 4 to inches diameter. Diameter of bolts 4 s s 4 8 t 2 1 12 14 Breadth of nuts.. 44 41 i 1i 3 b 1| ift lg 144 22 Breadth over the angles.... / 4i 44 12 n 1ft 141 2 24 2ft Thickness. ft 4b ft i 2 l 12 14 1ft Diameter of bolts n U lg i? U 2 24 24 Breadth of nuts.. 2ft 24 2 tb 22 3ft 34 3f 4 Breadth over 1 the angles .... J 2* 22 32 3ft 34 3? 4ft 4f i Thickness. 1ft 144 l|g 2 22 24 24 n MISCELLANIES. 155 TABLE Of the weight of flat bar iron, 12 inches long, in lbs. avoirdupois. Thickness Vs S /,e Vi Vs Vs Vs Vi 7s 1 inch. / 1 •21 •31 •42 •63 •32 •48 •63 •95 1-27 1-58 l •42 •63 •84 1-26 1-69 2-11 2-53 2-96 l] •52 •79 1-05 1-58 211 264 316 370 4-22 Is •58 •87 116 1-74 2-32 2 90 3-48 4-06 4-64 ij •63 •95 1-27 1-90 2-53 317 3-80 4-44 5-07 • 1 } •74 111 -•48 221 2-95 370 4-43 543 5-91 ’I 2 •84 1-27 1-69 2*53 3-38 4-22 507 592 676 21 •95 1-42 1-90 285 3-80 4-75 5-70 6-65 7'60 ‘g 106 1-58 2-11 3-17 4*22 5-28 6-33 7-40 845 2 : j 116 1-74 2-32 3-49 4-64 5-81 6-97 813 9 29 £ 3 1-27 1-90 2-53 3-80 507 634 7-60 8-87 10-14 § 31 1*37 2-06 2-74 412 5'49 6-86 8-24 1009 10-98 n :s‘ 1-48 2"22 2 95 4-43 5-91 7-39 8-87 1087 11-83 31 1-58 2-38 317 4-75 634 792 9-51 11-65 12-68 4 1-6.9 2-53 3-38 5 07 676 845 1014 11-83 1352 4 i 1-90 2-85 3-80 5-70 7-60 950 1141 13-31 1521 5 211 3-17 4-22 6-34 8-45 1056 12-67 1479 1690 6 2-53 3-80 507 7-60 10-14 12-67 15*21 17-75 2028 Weight of a copper rod 12 inches long and 1 inch diameter = 3‘039 lbs. Weight of a brass rod 12 inches long and 1 inch diameter = 2 - 86 lbs. TABLE Of the specific gravity of water at different temperatures, that at 62° being taken as unity. 70° F. •99913 52° F. 1-00076 68 •99936 50 1-00087 66 •99958 48 1-00095 64 •99980 46 1-00102 62 1 - 44 100107 58 1-00035 42 100111 56 1-00050 40 1-00113 54 1-00064 38 100115 The difference of temperatures between 62° and 38°, where water attains its greatest density, will vary the bulk of a gallon rather less than the third of a cubic inch. 156 MISCELLANIES. TABLE Of the weight of cast iron pipes 12 inches long, in lbs. avoirdupois. THICKNESS IN INCHES. 1 in. iu inches. * * i i 1 * n lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. 1 ’ 6-9 9-9 2 8-8 12 4 16-3 20-4 2J 10-7 14-9 19-4 24-1 3 12 5 17 4 22-4 27-8 336 39-7 46-0 3' 144 199 25-5 31-5 37-9 44-7 51-6 4 162 22-3 28-6 35-2 42-8 496 571 65-1 H 181 24 8 31 7 38-9 467 54-6 62-7 71-2 5 200 27 3 34-9 42-7 51-0 595 68-3 77-4 s* 21-9 29-8 380 464 55-3 64-5 739 83-6 6 23-7 32-2 41-1 50-1 596 69-4 794 89-8 6 * 25 6 34-7 44-2 538 63 9 74-4 85-0 960 7 27'4 37-2 47-3 57-5 68-3 79-3 90-6 102 2 7$ 293 39-7 40-4 61-2 72-6 843 962 108-4 8 311 422 53-5 650 770 893 101-8 114-6 «* 330 447 56-6 68-7 81-3 94 3 1074 120-8 9 349 471 59-6 724 85*7 99-2 1129 127-0 36-8 49*6 62-7 751 890 1042 118-5 133-2 10 386 52 1 65-8 799 943 1091 124 l 139-4 10* 54 6 68-9 82-6 986 114-1 129-7 145 6 11 57-0 720 873 103*0 119-0 13^-2 151-8 11 * 596 751 90-0 107*3 124-0 140-8 1580 12 62 0 78-2 94*7 111-7 1289 146-4 1642 13 84-4 102-2 120-4 138-8 157-5 1766 14 906 1096 129-0 148-8 168-7 1890 15 968 1170 1377 1587 179-8 201-4 16 1245 146-4 1686 1810 2138 17 131-9 1551 178-5 202-1 224-2 18 139-4 163-7 188-4 213-3 238-5 19 172-4 198-3 224 4 250-9 20 181-1 208-2 235 6 263-3 21 218-2 246-7 275-7 22 228-1 257-9 288 1 23 2380 269-0 300-5 24 247-9 2802 312-9 Note.—T he first column is the width of the pipes, expressed in inches and parts of an inch; and the remaining columns are the weights of the pipes, under the different thicknesses in which they are placed. N.B.—Two flanges are generally reckoned equal to one foot of pipe. MISCELLANIES. 157 TABLE Of the weight of cast iron balls in lbs. avoirdupois, from 1 to 12 inches diameter, and advancing by an eighth. Inches. lbs. and parts. Inches. lbs. and parts. Inches. lbs. and parts. 1 •14 41 14-76 81 84-57 H •20 41 15-95 8 | 8835 U •27 5 1721 81 92-25 if •36 51 18*54 81 96-26 U •47 51 19-93 9 100-39 11 •59 51 21-38 91 104-63 If •74 51 2291 91 108-99 U •91 51 24-51 91 113-47 2 110 51 26'18 91 11806 21 1-32 51 2792 91 122-77 21 1-57 6 29 74 9| 127-63 21 1-84 61 3164 91 132-60 21 215 61 33-62 10 13770 21 2-49 61 35-68 101 14293 21 2-86 61 37-81 101 148-29 21 3-27 6 f 4004 101 15378 3 372 61 42 35 • 101 159-40 31 4-20 61 44 75 10 f 165-16 31 4 73 7 4723 10 | 17106 3| 5-29 71 49-80 10 ! 177-10 31 5-90 71 5247 li 183 28 3f 6-56 71 55-23 Ilf 189-60 3f 7-26 71 5809 HI 196-06 3| 801 71 60 04 ill 202-67 4 8 81 71 64 09 HI 209-42 41 9-67 71 6724 HI 216-33 41 10 57 8 7050 HI 223-38 4f 11-53 81 73-86 HI 230-58 41 12-55 81 77-32 12 237-94 4§ 1362 8 f 80-89 TABLE Of the weight of a square foot of millboard in lbs. avoirdupois. Thickness, in inches.. A * \ # Weight, in lbs . •688 1-032 | 1-376 1-72 2-064 158 MISCELLANIES. TABLE Containing some of the properties of various bodies. Names of bodies. Melting and boiling Contracts • in cooling, in parts of an inch, for each foot Expands in heating from 32 to 212 deg. of Fahrenheit, the length being 1 •00000 Power of conducting heat. Cast iron melts 17977° •124 •00111 1-2 Wrought iron"! welding hot J 12780 •137 •00122 11 Copper melts .. 4587 •193 •00172 10 Brass melts .. 3807 •210 •00187 10 Steel red hot .. 1077 •133 •00118 Zinc melts .... 700 •329 •00294 Mercury boils.. 660 •01851 Lead melts.... 594 •319 •00286 25 Bismuth melts . 476 •156 •00139 Tin melts .... 442 •278 •00248 1-7 Water boils . . 212 •04002 TABLE Showing the expansion of water by heat. j Temperature. Expansion. Temperature. Expansion. 12° F. 22 42 52 62 72 82 92 102 112 1-00236 1-00090 1 00022 1-00000 1-00021 1 00083 100180 1 00312 1 00477 1 00672 1 00880 122° F. 132 142 152 162 172 182 192 202 212 1-01116 1-01367 101638 1-01934 1*02245 1 02575 1-02916 103265 1-03634 1-04012 MISCELLANIES. 159 Proportions of cement for cast iron. In mixing cement for cast iron, put one ounce of sal ammoniac to each hundred weight of borings, and use it without allowing it to heat. Multiply the length of any joint in feet by the breadth in inches, by the thickness in eighths, and by -3; the product will be the weight of dry borings, in lbs. avoirdupois, required to make cement to fill that joint nearly. TABLE Of boiling points of water holding various proportions of salt in solution. Parts of Salt. Degrees of Fahrenheit. Degrees of Reaumur. Degrees of Centigrade. Saturated solution 36-37 226-6 86 5 108*1 99 33-34 224-9 857 1072 30-30 223*7 852 106*5 27-28 2225 84*7 105-8 24-25 2214 84-2 105-2 21-22 2202 837 104*6 18-18 2190 831 103*9 15-15 2179 82*6 103-3 12-12 216-7 82 1 102*6 909 2155 81 5 101*9 99 606 2144 811 101*3 Sea water . 303 213-2 80-5 100-7 Common water .. 000 2120 80*0 100*0 To reduce any number of degrees of temperature on Fahrenheit's scale to the number of degrees of an equal temperature on Reaumur's scale ; and also to the number of degrees of an equal temperature on the Centigrade scale or otherwise. The freezing points of water are 32° Fahrenheit, 0° Reaumur, 0° Centigrade; the boiling points of water are 212° E., 80° R., 100° C. j whence 160 MISCELLANIES. FAHRENHEIT AND REAUMUR. To convert degrees of Fahrenheit into degrees of Reaumur. Multiply the difference between 32° and the given degrees of Fahrenheit, by 4, and divide by 9, the result will be degrees of Reaumur. Example 1.—Convert 77° of Fahrenheit into degrees of Reaumur. Here (77° «> 32°) x | - 4 = 20° of Reau¬ mur. = Ans. Example 2 .—Turn 22° of Fahrenheit into degrees of Reaumur. Here (22° ~ 32°) X -i= 10 °^ 4 = 4f° of Reau¬ mur. = Ans. EAHBENHEIT AND CENTIGRADE. To convert degrees of Fahrenheit into Centigrade. Multiply the difference between 32° and the given degrees of Fahrenheit, by 5, and divide by 9, the result is the number of degrees Centigrade. Example 3.—Convert 167° of Fahrenheit into Centi¬ grade. Here (167° ~ 32°) X |= Hg ! x 5 = 75° Centi¬ grade. = Ans. Example 4.—Convert 13° of Fahrenheit into Centi¬ grade. Here (13° co 32°) X |-= = 104 ° Centi¬ grade. = Ans. Note 1.— If the degrees of Fahrenheit are below 0°, take the sum instead of the difference between 32° and their quantity. MISCELLANIES. 161 Thus (Example 5) for 4° of Fahrenheit below 0° we have (4° + 32°) X ^ ^ 4 -= 16° of Eeaumur. = Ans., and Example 6 .—(4° + 32°) X-| = 36 ° 9 X 5 = 20° Centigrade. = Ans. REAUMUR, CENTIGRADE, AND FAHRENHEIT. 1. Above the freezing point.—Multiply the degrees of Eeaumur by 9, and divide by 4; or multiply the Centigrade degrees by 9, and divide by 5 ; the quotient, added to 32°, gives the degrees of Fahrenheit. Example 7.—Convert 20° of Eeaumur into degrees of Fahrenheit. Here 32° + 2 °° X - = 32° + 45° = 77° of Fah- 4 renheit. = Ans. Example 8 .—Convert 75° Centigrade into degrees of Fahrenheit. Here 32° + - 5 ° X 9 = 32° + 135° = 167° of 5 Fahrenheit. = Ans. 2. Below the freezing point.—Multiply and divide the given degrees as before if they are Eeaumur’s degrees and less than 14|°, or Centigrade and less than 17^°, subtract the quotient from 32°, the result will be degrees of Fahrenheit above 0°. But if the given degrees are not less than 14-§°, or 17^° respectively, subtract 32° from the quotient, the remainder will be degrees of Fahrenheit below 0°. Example 9.—Convert 4|° of Eeaumur into Fah¬ renheit. Here 32° - 4 ?° X 9 = 32° - 10° = 22° of Fah- 4 renheit. = Ans. 162 MISCELLANIES. Example 10 .—Convert 10 # 3 Centigrade into Fahren¬ heit degrees. Here 32° - 1Q °° X — = 32° - 19° = 13° of 5 Fahrenheit. = Ans. Example 11.—Turn 16° of Reaumur into degrees Fahrenheit. Here 16 ° X — - 32° = 36° - 32° = 4° below 0°. 4 = Ans. Example 12.—Change 20° Centigrade into degrees Fahrenheit. Here 20 ° X — — 32° = 36° — 32° = 4° of 5 Fahrenheit below zero. = Ans. Note 2.—5° on the Centigrade scale = 4° on Reaumur’s scale. TABLE Showing the quantity and weight of a superficial foot of brick work, from half a brick to two and a half bricks in thickness. Thickness by number. Thickness in inches. Number of bricks. Weight in lbs. avoir. 1 brick 4? 4-58 4019 1 91 9-15 80-37 U 14* 13-72 120-56 2 18| 18-3 160-74 21 2*1 22-875 200-93 Note.—T he weight is independent of mortar. 1 brick weighs 9 lbs. avoirdupois, nearly; 12i = 1 cwt., and 250 = 1 ton. MISCELLANIES. 163 TABLE Of the specific gravities of those bodies chiefly used in machinery, building, 8fC., showing, in avoirdupois ounces and pounds, the weight of a cubic foot of each body; also the weight of a cubic inch, and the number of cubic inches in a pound, with multipliers to each, for finding the weight when the dimensions are given. Names of Bodies. Weight of a Cubic Foot. Weight of a Cubic Inch. Number of Cubic Inches in a Pound. Copper, cast. 8788 lbs. 549-25 5-0 Z 86 lbs. •3178 3-146 Copper, sheet . 8915 557 19 5-159 •3225 3-103 Brass, cast . 8396 524 75 4-859 •3037 . 3-293 Iron, cast . 7271 454-44 4-208 •263 3-802 Iron, bar . 7788 486-75 4-507 •282 3-550 Lead . 11344 709-00 6-565 •4103 2-437 Steel, soft. 7833 489-56 4-533 •2833 3-530 Steel, hard . 7816 488-50 4-523 •2827 3-537 Zinc, cast. 7190 449 37 4-161 •26 3-845 Tin, cast . 7292 455-75 4-222 •2636 3-792 Bismuth . 9880 617-50 5-718 •3574 2-798 Gun Metal . 8784 549-00 5-083 •3177 3-148 Sand . 1520 95 00 •880 •055 18-190 Coal . 1250 7812 •723 •0452 22-118 Brick. 2000 125 00 1157 •0723 13-824 Stone, paving . 2416 15100 1-398 •0873 11-444 Slate . 2672 167-00 1-546 •0967 10-347 Marble . 2742 171-37 1-587 •0991 10 083 White lead . 3160 197-50 1-829 •1143 8-749 Glass . 2880 180-00 1-667 •1042 9-600 Tallow . 945 59 06 •547 •0342 29-257 Cork .. 240 1500 •139 •0087 115-200 Larch. 544 34-00 •315 •0197 50-823 Elm . 671 41-94 •388 •0204 41-204 Pine, pitch . 660 41-25 •382 •024 41-890 Beech. 852 63-25 •493 •0308 32-451 Teak . 745 46 56 •431 •027 37-111 Ash. 845 52-81 •489 •0305 32-720 Mahogany . 1063 66 44 •615 •0384 26-009 Oak. 970 60-32 ■561 •0351 28-503 Oil of Turpentine ... 870 54-37 •503 •0315 31-779 Olive Oil . 915 57-19 •529 •0331 30-216 Linseed Oil . 932 58-25 •539 •0337 29-665 Spirits, proof . 927 57-94 •536 •03352 29-825 Water, distilled . 1000 62-50 •579 •03617 27-648 Water, sea . 1028 64-25 •595 •0372 26-895 Tar. 1015 63-44 •587 •0367 27-239 Vinegar. 1026 64 12 •594 •037 26 947 Mercury . 13568 848 00 7-852 •4907 2-038 164 MISCELLANIES. The ls£, 2nd, 3rd, and 4 th columns require no further explanation than the titles they bear; the fifth column is to find the weight of any number of cubic inches, in avoirdupois pounds, of any of the different bodies required. Example 1.—Suppose a piece of cast iron to be 56§ inches long, 16-^ inches broad, and of an inch in thickness, required its weight. Here 56*75 x 16*5 X *75 = 702*28125 cubic inches, which multiplied by *263 = 184*7 lbs., nearly. — Ans. Example 2.—Required the weight of an imperial «allon of proof spirits. Here 277*274 x *03352 = 9*294 lbs, nearly. = Ans. Example 3.—Required the thickness of metal for a concave copper ball, 8 inches diameter without, so as to sink to its centre in common water. Here 8 3 x *5236 = 268*0832 cubic inches in the ball, the half of which = 134*0416 cubic inches to be immersed, or cubic inches of water to be removed.— Then 134*0416 X *579 (the weight of a cubic inch of water) = 77*6100864 ounces weight of water displaced, or the weight of the copper ball; which divided by 5*159 (the weight of a cubic inch of copper) = 15*0436 cubic inches of copper in the ball. Again, 8 2 X *7854 x 4 = 201*0624 square inches, the superficies of the ball; and 15 0436 -f- 201*0624 = *0748 inches, the required thickness of the copper, nearly. = Ans. Example 4.—Required the weight necessary to counterpoise a float of paving stone proper for a steam- engine boiler, &c., the float being 14 inches diameter and 2\ inches thick. Here 14 2 x *7854 x 2*5 = 384*846 cubic inches. Then 384*846 x *0873 = the weight of the stone in lbs. And, 384*846 x *03617 = the weight in lbs. of water displaced; then, 384*846 X (*0873 - *03617) = 384*846 x *05113 = 19*677 lbs. the counterpoise re¬ quired. = Ans. THE ENGINEEB’s SLIDE EXILE. 165 KTJLES FOB MAKING OB COBBECTING THE GAUGE POINTS ON THE ENGINEEB’S SLIDE BULE. The engineer’s slide rule is an instrument of exten¬ sive use to mechanics, and almost every one who is in possession of the rule, is also, or may be, in ample possession of instructions ; but I am not aware that any information has been given in any other work, for either correcting the old gauge points, or obtaining new ones; hence the following may be found useful:— And first, by making the third column on the rule (or that marked III) the first of our observations, the others are rendered very simple ; thus, The third column is the number of cubic inches con¬ tained in a pound, foot, gallon, &c. The second column is the numbers in the third column expressed in the decimals of a foot, or multiplied by -833. The first column is the third column divided by 1728. The fifth column is the third column divided by •7851. The fourth column is the fifth column expressed in the decimals of a foot, or multiplied by ’833. The seventh column is the third column divided by •5236. And, The sixth column is the seventh column divided by 1728. I 166 DECIMAL APPROXIMATIONS. DECIMAL APPROXIMATIONS FOR FACILITATING CALCULATIONS IN MENSURATION. Lineal feet multiplied by '000189 = miles. „ yards „ •000568 — „ Square inches „ •00694' = square feet. „ yards „ •0002066 = acres. Circular inches „ •005454 = square feet. Cylindrical inches „ • -0004645 = cubic feet. ,, feet „ •02908' = cubic yards. Cubic inches „ •0005787 — cubic feet. „ feet „ •037' — cubic yards. 6*232 = imperial gallons. „ inches „ •0036065 „ Cylindrical feet „ 4-895 = ,, ff „ inches „ •002832 — Cubic inches „ •2630 — lbs. avs. of cast iron. •2817 '= wrought do. n •2833 — „ steel. •3225 — copper. •3037 = „ brass. •2601 — zinc. •4103 — lead. •2636 „ tin. > >» •4908 = „ mercury. Cylindrical inches „ •2065 = „ cast iron. •2212 = wrought iron. >» •2225 — steel. •2533 — copper. •2385 — brass. •2043 zinc. ” •3223 — lead. >» •2071 = it tin. » •3854 = „ mercury. Avoirdupois /is. „ •0089 = cwts. •000446 = tons. DECIMAL EQUIVALENTS. 167 DECIMAL EQUIVALENTS TO FRACTIONAL PARTS OF LINEAL MEASURES. One inch, the integer or whole number. *96875 | + * *9375 i + A *90625 § + * *875 o * *84375 r | + A *8125 S|+A *78125 + A •75 £ I •71875 05 | + 3 ® 5 •6875 i + a •65625 | + A •625 f •59375 | + A •5625 J 4- •53125 o | + A •46875 If + A •4375 ?t + A •40625 £ f + A •375 3 f •34375 \ + A •3125 \ + A *28125 i + A •25 i •21875 i + A •1375 2 i + A •15625 ^ | + A •125 § i •09375 S’ 5 3 •0625 £ X •03125 58 A One foot, or 12 inches, the integer. *9166' 11 inches. •8333' £ 10 „ *75 -3 9,, •6666' g. 8 ,, •5833' % 7 „ •5 3 6 „ •4166' 5 inches. •3333' 34 „ •25 -3 3,, •1666' g. 2 „ •0833' ® 1 „ •07292 fe | „ •0625 % of in. •05208 £ | „ •04167 la i „ •03125 g, f „ •02083 ® J „ •01042 § | „ One yard, or 36 inches, the integer. •972' 35 inches. •944' 34 „ •916' 33 „ •888' o 32 „ •861' Z 31 „ •833' § 30 „ •805' g" 29 „ •777' g 28 „ •75 - 27 „ •722' 26 „ •694' 25 „ •666' 24, „ •638' 23 inches. •611' 22 „ •583' 21 „ •555' 0 20 „ •527' * 10 » •5 g 18 „ •472' S’ 17 „ •444' g 16 „ •416' 01 15 „ •388' 14 „ •361' 13 „ : 333' 12 „ 305' 1 1 inches. *277' 10 „ •25 9 „ •222' o 8 „ *194' Z T „ •166' g 6 „ •138' S’ 5 „ •111' £ 4 „ •083' 58 3 „ •055' 2 „ •027' 1 „ 168 PBICE OF METALS. TABLE Containing the price of metals, or other materials, by the ton, cwt., quarter, or lb. A T 169 170 mniHUIl 1 1 1 I 1 I I 1 1 1 1 I I 1 I 1 ! S I S I I I I I I I I I ! 1 I ! I 1 I I I I I I I S 1 S 1 I I I! I I ! ! I I! 111 rryjiwmH 5 i i |!U K n f f 7 7 I I Min 1 l! IS 1 I 1 1 I lillllli US 11 I i S i i !iillfl{,ljl ;l islil ! I ifiii ills fill I! II Si is 13 i!i!ifil!iii||iiliiiiii'i!il |1111! Ill |! |ll pi p p |l |! p ii ii 17 ii n ii ii ii ii it ii ii a * | x, x x x x 4 s ^ 4 S S 1 S S £ S a | - a . s WITH THEIR LOGARITHMS. 171 ! I; I I *1 i ! fill I I I I I I I I 1 S 11 I I I II I I 11 I 1111 I I 11 i i 11 i mill ! ! I 1 iI ! I II ! I 1 11111 i 1 111111 ii "j* 11 11 i i 11 11 11 ii H j -I * * > > > > *1-- I g I g § g I I II* II i 2 F i r jh i r ii !f t i f ? ?i •: ? II:-! I! I I I Ii ! i l| H 18 |j ! « * ?| I-isl{ fill!? I j.l I« iiilli Hill! Iliisjlj Mi! I? Hi'' ! i l| «l Is Si I|I| ii II! I«»! 1?!?!:!?; HU Hipi! IP 172 173 ! I 1 1 1 i i i u 1 1 1 I i i i 1 i l I I I i i i i ! l 1 1 I I i s i i I 1 i I I I i i i i I l 1 I I ! i I 1 I ! i i TfflFHTf jllliiU ! H!i II |i : il, 11. >i hi !III Iliiilii!H il!l as ; 1!iIIIIIII Mi'¬ ll s ! | ?i- g i 1 i II* ||' ! ! VV\ i -i - Hi III i 3 174 I I I I I I I I 1 I I § I I 11 I I i 11 1 if I 1 I! I I I 11 I 11 I S 1 I 1 I I !I ! I! I 1 11 1 1 I S 1 I 11 II II III ! I I I! -I -1 7 -l* ' min? t ; iji a t Iff I I i I I IIIJl I I Ij I I I 1 I I ill.II 1 I ii 11 11 11 mill 11 i! 111111 iliiii 11 §! 1111 i s stils! 11 iimmwNi -ii -i.*a -ft* > > ? IIIII1- ISI 5 ! 175 1 1 1 I 1 I S ! I 1 fill ISI1111 ii sail .mini II i 11 111 inn II III! 11 Hill 11 1111 1111111 Hill I liliili huh iHirt r ! H!!! >11 !lii ss ~! i I?! I| l! s i i I! M l I l II l| I 1 i s! S| 11 * srf !|i'i; i . f ? !i! I I s I ’ll’! ? I I 111 I I I I I If Ell Mil * - s .3 -S .5 Hill ■s ■S 'S ‘S -s mu 11111 177 178 WING QUANTITIES VARY FROM TIME TO TIME. VARYING QUANTITIES. 179 180 VAEYING QUANTITIES. II II II II II II II II II II II II H II II II II Mtillll ! i J S! 1!! BU lift hum, iiniim^ 111 I 11 I I 11 111 i APPENDIX. TABLE I. CONTAINING THE CIRCUMFERENCES AND AREAS OF CIRCLES, SIDES OF EQUAL SQUARES SQUARES AND CUBES OF THE DIAMETERS, FROM yh TO 100 INCHES, FEET, YARDS, CHAINS, MILES, &C. ADVANCING BY AN fath, AND ALSO THE SIDE OF EQUAL SQUARE, ADVANCING AT AN EQUAL RATIO. 2] CIRCUMFERENCES AND AREAS OF CIRCLES, SIDES OF TABLE I. Dia. or Root. Circum. Area. Side of = square. Square. Cube. h 0 3927 00123 0-1108 0-0156 000195 07854 0-049J 0-2216 0-0625 001563 | 1-1781 01104 0-3323 0-1406 005273 h 1-5708 0-1963 0-4431 . 0-25 0-125 1 1-9635 0-3068 0-5539 0-3906 0-24414 2-3562 0-4418* 0-6647 0-5625 0-42188 i 2-7489 0-6013 0-7754 0-7656 0-66992 1 in. 3-1416 07854 0-8862 1- 1- 4 3-5343 0-9940 0-9970... 1-2656 1-42383 4 3-9270 1-2272 1-1078 1-5625 1-95313 1 4-3197 1-4849 1-2186 1-8906 2-59961 4 4-7124 17671 1-3293 2-25 3-375 i 5-1051 2-0739 1-4401 2-6406 4-29102 f 5-4978 2-4053 1-5509 3-0625 5-35938 4 5-8905 2-7612 1-6617 3-5156 6-59180 2 in. 6-2832 314K 1-7725 4- 8- 4 6-6759 35466 1-8832 4-5156 9-5957 4 7-0686 3-9761 1-9940 50625 11-3906 | 7-4613 4-4301 21048 5-6406 13-3965 4 7-8540 4-9087 2-2156 6-25 15-625 g 8-2467 5-4119 2-3263 6-8906 18-0879 t 8-6394 5-9396 2-4371 7-5625 20-7969 4 9 0321 6-4918 2-5479 8-2656 23-7637 3 in. 9-4248 7-0686 2-6587 9- 27- 4 9-8175 7-6699 2-7695 9-7656 30-5176 4 10-2102 8-2958 2-8802 10-5625 34-3281 I 10-6029 8-9462 2-9910 11-3906 38-4434 4 10-9956 9-6211 3-1018 12-25 42-875 § 11-3883 10-3206 3-2126 13-1406 47-6348 l 11-7810 11-0447 3-3234 14-0625 52-7344 i 12-1737 117932 3-4341 15-0156 58 1855 4 in. 12-5664 12-5664 3-5449 16- 64- 4 12-9591 13-3640 3-6557 17-0156 70-1895 4 13-3518 14-1863 3-7665 18-0625 76-7656 i 137445 150330 3-8772 191406 837402 4 14-1372 15-9043 3-9880 20-25 91-125 14-5299 16-8002 40988 21-3906 98-9316 4 14-9226 17-7205 4-2096 22-5625 107-1719 4 15-3153 18-6655. 4-3204 23-7656 115-8574 5 in. 15-7080 19-6350 4-4311 25- 125- 4 161007 20-6290 4-5419 26-2656 134-6113 4 16-4934 21-6475 4-6527 27-5625 144-7031 i 16-8861 22-6906 4-7635 28-8906 155-2871 4 17-2788 237583 4-8742 30-25 166-375 § 17-6715 24-8505 4-9850 31-6406 177-9785 i 18-0642 25-9672 5-0958 330625 190-1094 4, 18-4569 27-1085 5-2066 34-5156 202-7793 EQUAL SQUARES, SQUARES AND CUBES OF THE DIAMETERS. [3 TABLE I. Dia. or Root. Circum. Area. Side of = square. Square. Cube. 6 in. 18-8496 28-2743 '5-3174 36- 216- ft, 19-2423 29-4647 5-4281 375156 229-7832 ft 19-6350 30-6796 5-5389 390625 244-1406 1 20-0277 31-9191 5-6497 40-6406 259-0840 ft 20-4204 33-1831 5-7605 42-25 274-625 g 20-8131 34-4716 5-8713 43-8906 290-7754 21-2058 35-7847 5-9820 45-5625 307-5469 21-5985 37 1223 60928 47-2656 324-9512 7 in. 21-9912 38-4846 6-2036 49- 343- ft 22-3839 39-8712 6-3144 50-7656 361-7051 ft 22-7766 41-2825 6-4251 52-5625 381-0781 1 23 1693 42-7183 6-5359 54-3906 401-1309 i 23-5620 44-1786 6-6467 56-25 421-875 S 23-9547 45-6635 6-7575 581406 443-3223 t 24-3474 47-1730 6-8683 600625 465-4844 * 24-7401 48-7070 6-9790 62-0156 488-3730 8 in. 25 1328 50-2655 7-0898 64-. 512- ft 25-5255 51-8486 7-2006 660156 536-3770 - 1 25-9182 53-4562 7-3114 68-0625 561-5156 I 26-3109 550883 7-4222 70-1406 587-4277 ft 26-7036 56-7450 7-5329 72-25 614-125 i 27-0963 58-4263 7-6437 74-3906 641-6191 i 27-4890 60-1320 7-7545 76-5625 669-9219 f 27-8817 61-8624 7-8653 78-7656 699-0449 9 in. 28-2743 63-6173 7-9760 81- 729- ft 28-6670 65-3967 80868 83-2656 759-7988 ft 290597 67-2006 8-1976 85-5625 791-4531 1 • 29-4524 690291 8-3084 . 87-8906 823-9746 ft 29-8451 70-8822 8-4192 90-25 857-375 g 30-2378 72-7598 8-5299 92-6406 891-6660 f 30-6305 74-6619 8-6407 95-0625 926-8594 i 31 0232 76-5886 8-7515 97-5156 962-9668 10 in. 31-4159 78-5398 8-8623 100- 1000- ft 31-8086 80-5156 8-9730 102-5156 1037-9707 ft 32 2013 82-5159 9 0838 105-0625 1076-8906 | 32-5940 84-5407 9-1946 107-6406 1116-7715 ft 32-9867 86-5901 9-3054 110-25 1157-625 g 33-3794 88-664 9-4162 1128906 1199-4629 4 33-7721 90-7626 9-5269 115-5625 1242-2969 ft 34-1648 92-8856 9-6377 118-2656 1286-1387 11 in. 34-5575 95-Q332 9-7485 121- 1331- ft 34-9502 ->7-2053 9-8593 123-7656 1376-8926 l 1 35-3429 99-4U-20 9-9701 126-5625 1423-8281 35-7356 101-6232 10-0808 129-3906 1471-8184 ft g 36 1283 103-8689 10-1916 132-25 1520-875 36-5210 106-1392 10-3024 135-1406 1571 0098 | 36-9137 108-4340 10-4132 138-0625 1622-2344 ft 37-3064 110-7534 10-5239 141-0156 1674-5605 4] CIRCUMFERENCES AND AREAS OF CIRCLES, SIDES OF TABLE I. Dia. or Root. Circum. Area. Side of = square. Square. Cube. 12 in. 376991 113-0973 10-6347 144- 1728- i 38-0918 115-4658 . 10-7455 147-0156 1782-5645 i 384845 117-8588 10-8563 1500625 1838-2656 § 38-8772 120-2764 10-9671 153-1406 1895 1152 39-26.99 122-7185 11-0778 156-25 1953-1250 g 39-6626 125-1851 111886 159-3906 2012-3066 | 40-0553 127-6763 11-2994 162-5625 2072-6719 i 40-4480 130-1920 11-4102 165-7656 2134-2324 13 in. 40-8407 132-7323 11-5210 169- 2197- i 41-2334 135-2971 11-6317 172-2656 2260-9863 8 J 41-6261 1378865 11-7425 175-5625 2326-2031 g 42-0188 140-5004 11-8533 178-8906 2392-6621 42-4115 143-1388 11-9641 182-25 2460-3750 g" 42-8042 145-8018 12-0748 185-6406 2529-3535 g 43-1969 148-4893 121856 189-0625 2599-6094 r 43-5896 151-2014 12-2964 192-5156 267M543 14 in. 43-9823 153-9380 12-4072 196- 2744- 4 44-3750 156-6992 12-5180 199-5156 2818 1582 8 i 44-7677 159-4849 12-6287 203-0625 2893 6406 45-1504 162-2952 12-7395 206-6406 2970-4590 45-5531 165-1300 12-8503 210-25 3048-6250 g" 45-9458 167-9893 12-9611 213-8906 3128 1504 g 46-3385 170-8732 130718 217-5625 3209-0469 1 46-7312 173-7816 13-1826 221-2656 3291-3262 15 in. 47-1239 176-7146 13-2934 225- 3375- ft 47-5166 179-6721 13-4042 228-7656 3460 0801 47-9093 182-6542 13-5150 232-5625 3546-5781 g 48-3020 185-6608 13-6257 236-3906 3634-5059 h 48-6947 1886919 13-7365 240-25 3723-8750 g' 49-0874 191-7476 13-8473 244-1406 3814 6973 49-4801 194-8278 13-9581 2480625 3906-9844 £ 49-8728 197-9326 14-0689 2520156 4000-7480 16 in. 50-2655 201-0619 14-1796 256- 4096- £ 50-6582 204-2158 14-2904 260-0156 4192-7520 i 51 0509 207-3942 14-4012 2640625 4291-0156' | 51-4436 210-5972 14-5120 268-1406 4390-8027 % 51-8363 213-8246 14-6227 272-25 44921250 g 52-2290 2170767 14-7335 276-3906 4594-9941 | 52-6217 220-3533 14-8443 280-5625 4699-4219 I 530144 223-6544 14-9551 284-7656 4805-4199 17 in. 53-4071 226-9801 15-0659 289- 4913- 53-7998 230-3303 15-1766 293-2656 5022-1738 £ 54-1925 2337050 15-2874 297-5625 5132-9531 g 54-5852 237 1044 15-3982 301-8906 5245-3496 i 54-9779 240-5282 15-5090 306-25 5359-3750 g 55-3706 243-9766 15-6197 310-6406 5475-0410 4 55-7633 247-4495 15-7305 315-0625 5592-3594 i 56-1560 250-9470 15-8413 319-5156 5711 3418 BQUaL squares, squares and cubes of the diameters. [5 TABLE I. Dia. or Hoot. Circum. Area. Side of = square. Square. Cube. 18 in. 56-5487 254-4690 15-9521 324- 5832- i 56-9414 2580159 16-0629 328-5156 5954-3457 i 573341 261-5867 161736 333-0625 6078-3906 1 57-7268 265-1824 16-2844 337 6406 6204-1465 h 58-1195 268-8025 16-3952 342-25 6331-6250 1 58-5122 272-4473 16-5060 346-8906 6460-8379 l 58-9049 276-1165 16-6168 351-5625 6591-7969 1 59-2976 279-8104 16-7275 356-2656 6724-5137 19 in. 59-6903 283-5287 16-8383 361- 6859- h 60-0830 287-2717 16-9491 365-7656 6995-2676 d 60-4757 291-0599 17-0599 370-5625 7133-3281 1 60-8684 294-8311 17-1706 375-3906 7273-1934 h 61-2611 298-6477 17-2814 380-25 7414-8750 1 61-6538 302-4887 17-3922 385-1406 7558-3848 1 62 0465 306-3544 17-5030 3900625 7703-7344 t 62-4392 310-2446 17-6138 3950156 7850-9355 20 in. 62-8319 314-1593 177245 400- 8000- i 63-2245 3180985 17-8353 405-0156 8150-9395 d 63-6173 3220623 179461 410-0625 8303-7656 § 640100 3260507 180569 415-1406 8458-4902 h 64-4026 3300636 181677 420-25 8615-1250 § 64-7953 334-1010 18-2784 425-3906 8773-6816 1 65-1880 338-1630 18-3892 430-5625 8934-1719 i 65-5807 342-2495 18-5000 435-7656 9096-6074 21 in. 65-9734 346-3606 18-6108 441- 9261- i 66-3661 350-4962 18-7215 446-2656 9427-3613 1 66-7588 354-6564 18-8323 451-5625 95957031 I 67-1515 358-8411 18-9431 456-8906 9766 0371 h 67-5442 3630503 19-0539 462-25 9938-3750 § 67-9369 367-2841 19-1647 467-6406 10112-7285 68-3296 371-5424 19-2754 473-0625 10289-1094 i 68-7223 375-8253 19-3862 478-5156 104675293 22 in. 69-1150 380-1327 19-4970 484- 10648- h 69-5077 384-4647 19-6078 489-5156 10830-5332 d 69-9004 388-8212 19-7185 495 0625 11015-1406 1 70-2931 393-2022 19-8293 500-6406 11201-8340 h 70-6858 3976078 19-9401 506-25 11390-6250 g 71-0785 402 0380 20-0509 511-8906 11581-5254 5 71-4712 406-4926 201617 517-5625 11774-5469 i 71-8639 410-9719 20-2724 523-2656 11969-7012 23 m. 72-2566 415-4756 20-3832 529- 12167* h 72-6493 420-0039 20-4940 534-7656 12366-4551 k 730420 424-5568 20-6048 540-5625 12568-0781 I 73-4347 429-1342 20-7156 546-3906 12771-8809 i 73-8274 4&-7361 20-8263 552-25 12977-8750 g 74-2201 438-3626 20-9371 558 1406 13186 0723 74-6128 443 0137 21 0479 564-0625 13396-4844' 75 0055 447-6892 21-1587 5700156 13609-1230 6] CIRCUMFERENCES AND AREAS OF CIRCLES, SIDES OF TABLE I. Dia. or Root. Circum. Area. Side of = square. Square. Cube. 24 in. 75-3982 452-3893 21-2694 576- 13824- k 75-7909 457 1140 21-3802 582-0156 14041-1270 76 1836 461-8632 21-4910 5880625 14260-5156 | 76-5763 466-6370 • 21-6018 594-1406 14482-1777 h 76-9690 471-4352 21-7126 600-25 14706 125 | 77-3617 476"2581 21-8233 606-3906 14932-3691 77-7544 481 1055 21-9341 612-5625 15160-9219 78-1471 485-9774 220449 618-7656 16391-7949 25 in. 78-5398 490-8739 221557 625- 15625- h 789325 495-7949 22-2665 631-2656 15860-5488 $ 79-3252 500-7404 22-3772 637-5625 16098-4531 i 79-7179 505-7105 22-4880 643-8906 16338-7246 5 r> 80-1106 5107052 22-5988 650-25 16581-375 i 80-5033 515-7243 22-7096 656-6406 16826-4160 f 80-8960 520-7681 22 8203 ' 663 0625 17073-8594 1 81-2887 525 8363 22-9311 669-5156 17323-7168 26 m. 81-6814 530-9292 23 0419 676- 17576- i 820741 536 0465 23-1527 682-5156 17830-7207 82-4668 541T884 23-2635 689-0625 18087-8906 •! 82-8595 546-3549 23-3742 695-6406 18347-5215 83-2522 551-5459 23-4850 702-25 18069-625 8 83-6449 556-7614 23-5958 708-8906 18874-2129 84-0376 5620015 23-7066 715-5625 19141-2969 i 84-4303 567"2661 23-8173 722-2656 19410-8887 27 m. 84-8230 572-5553 23-9281 729- 19683- 85-2157 577 8690 24-0389 735-7656 19957-6426 85-6084 5832072 24-1497 742-5625 20234-8281 | 860011 588-5700 24-2605 749-3906 20514-5684 4 86-3938 593-9574 24-3712 756-25 20796-875 8 , 86-7865 5993692 24-4820 763-1406 21081-7598 87-1792 604-8057 24-5928 770-0625 21369-2344 1 87-5719 610-2666 24-7036 777-0156 21659-3105 28 *». 87-9646 6157522 24-8144 784- 21952- ‘ *» 88-3573 621-2622 24-9251 791-0156 22247-3145 88-7500 626-7968 25 0359 798 0625 22545-2656 89-1427 632-3560 25-1467 805 1406 22845-8652 89-5354 637-9397 25-2575 812-25 23149-125 89-9281 643-5479 25-3682 819-3906 234550566 90-3208 649 1807 25-4790 826-5625 23763-6719 § 90-7135 654-8380 25-5898 833-7656 24074-9824 29 in. 911062 660-5199 25-7006 841- 24389- A 91 4989 666-2263 25-8114 848-2656 24705-7363 91-8916 671-9572 25-9221 855-5625 25025-2031 92-2843 92-6770 677-7127 683-4928 260329 26 1437 862-8906 €70-25 25347-4121 25672-375 8 , . L. 930697 689-2973 26-2545 877-6406 26000-1035 93-4624 695-1265 26-3653 885-0625 26330-6094 93-8551 700-9801 26-4760 892-5156 26663-9043 EQUAL SQUARES, SQUARES AND CUBES OF THE DIAMETERS. [7 TABLE I. Dia. or Root. Circum. Area. bide of = square. Square. Cube. 30 in. 94-2478 706-8583 26-5868 900- 27000- b 946405 712-7611 26-6976 907-5156 27338-9082 4 950332 718-6884 26-8084 915-0625 27680-6406 | 95-4259 724-6403 26-9191 922-6406 28025-2090 h 95-8186 730-6166 27-0299 930-25 28372-625 8 96-2113 736-6176 27 1407 937-8906 28722-9004 i 96-6040 742-6431 27-2515 945-5625 29076-0469 & 96-9967 748-6931 27-3623 953-2656 29432-0762 31 in. 97-3894 754-7676 27 4730 961- 29791- b 977821 760-8667 27-5838 968-7656 30152-8301 4 98 1748 766-9904 276946 976-5625 30517-5781 8 98-5675 773-1386 27 8054 984-3906 30885-2559 h 98-9602 779-3113 27-9161 992-25 31255-875 8 99-3529 785-5086 280269 1000 1406 31629-4473 ! 99-7456 791-7304 28T377 1008-0625 32005-9844 i 100 1383 797-9768 28-2485 10160156 32385-4980 32 in. 100-5310 804-2477 28-3593 1024- 32768- b 100-9237 810-5432 28-4700 ( 10320156 33153-5020 4 101-3164 816-8632 28-5808 10400625 335420156 8 101-7091 823-2077 28-6916 1048T406 33933-5527 h 102-1018 829-5768 28-8024 1056-25 34328T25 8 102-4945 835-9704 28-9132 1064-3906 34725-7441 1 102-8872 842-3886 290239 1072-5625 35126-4219 i 103-2799 848-8313 29-1347 1080-7656 35530T699 33 in. 103-6726 855-2986 29-2455 1089- 35937- 8 104-0653 861-7904 29-3563 1097-2656 36346-9238 4 104-4580 868-3068 29-4670 1105-5625 36759-9531 8 104-8507 874-8477 29-5778 1113-8906 37176-0996 h 105-2434 881-4131 29-6886 1122-25 37595-375 8 105-6361 8880031 29-7994 1130-6406 38017-7910 8 1060288 894-6176 29 9102 11390625 38443-3594 8 106-4215 901-2576 30 0209 11475156 388720918 34 in. 106-8142 9079203 30-1317 1156- 39304- b 107-2068 914-6084 30-2425 1164-5156 39739 0957 4 107 5995 921-3211 30-3533 1173-0625> 40177-3906 8 107-9922 9280584 30-4641 1181-6406 40618-8965 b 108-3849 934-8202 30-5748 1190-25 41063 625 8 1087776 941-6065 30-6856 1198-8906 41511-5879 8 109T703 948-4174 30-7964 1207-5625 41962-7969 8 109-5630 955-2528 30-9072 1216-2656 42417-2636 35 in. 109-9557 962-1128 310179 1225- 42875- b 1103484 968-9973 31 1287 1233-7656 43336-0176 4 110-7411 975-9063 31-2395 1242-5625 43800-3281 8 111-1338 982-8399 31-3503 1251-3906 44267-9433 8 111-5265 989-7980 31-4611 1260-25 44738 875 8 111-9192 996-7807 31-5718 1269 1406 45213-1348 t 112-3119 1003-7879 31-6826 12780625 45690-7344 8 112-7046 1010-6197 31 -7934 1287-0156 46171-6855 8] CIRCUMFERENCES AND AREAS OF CIRCLES, SIDES OF TABLE I. Dia. or Boot. Circum. Area. Side of = square. Square. Cube. 36 in. 113-0973 1017-8760 31-9042 1296- 46656- £ 1134900 1024-9569 320149 1305-0156 47143-6895 \ 113-8827 1032 0623 321257 13140625 47634-7656 § 114-2754 1039 1922 32-2365 13231406 48129-2402 h 114-6681 1046-3467 32-3473 1332-25 48627-125 g 1150608 1053-5257 32-4581 1341-3906 49128-4316 £ 115-4535 10607293 32-5688 1350-5625 49633-1719 i 115-8462 1067 9574 32-6796 1359-7656 50141-3574 116-2389 1075-2101 32-7904 1369- 50653- 116-6316 1082-4873 32-9012 1378-2656 51168-1113 8 .i 1170243 1089-7890 330120 1387-5625 51686-7031 | 1174170 1097-1153 33 0227 1396-8906 52208-7871 j 117-8097 1104-4662 33-2335 1406-25 52734-375 g 118-2024 1111-8415 33-3443 1415-6406 53263-4785 i 118-5951 1119-2415 33-4551 1425-0625 53796-1094 i 118-9878 1126-6659 33-5658 1434-5156 54332-2793 38 in. 119-3805 11341149 33-6766 1444- 54872- £ 119-7732 1141-5885 33-7874 1453-5156 55415-2832 120-1659 1149-0866 33-8982 14630625 55962-1406 f 120-5586 1156-6092 34-0090 1472-6406 56512-5839 120-9513 1164-1564 34T197 1482-25 57066-625 g' 121-3440 1171-7282 34-2305 1491-8906 57624-2754 S | 121-7367 1179-3244 34-3413 1501-5625 58185-5469 i 122-1294 1186-9452 34-4521 1511-2656 58750-4512 39 in. 122-5221 1194-5906 34-5629 1521- 59319* 122-9148 1202-2605 34-6736 1530-7656 59891-2051 123-3075 1209-9550 34-7844 1540-5625 60467 0781 | 123-7002 1217-6739 34-8952 1550-3906 61046-6309 1240929 1225-4175 35 0060 1560-25 61629-875 g 124-4856 1233 1855 35-1167 1570-1406 62216-8223 h 124-8783 1240-9782 35-2275 1580-0625 62807-4844 i 125-2710 1248-7954 35-3383 15900156 63401-8730 40 in. 125-6637 1256-6370 35-4491 1600- 64000- h 1260564 1264-5033 35-5599 1610-0156 64601 -8770 h 126-4491 1272-3941 35-6706 1620-0625 65207-^156 § 126-8418 1280-3095 35-7814 1630-1406 65816-9277 h 127-2345 1288-2493 35-8922 1640-25 66430-125 g 127-6272 1296-2138 36-0030 1650-3906 67047-1191 | 1280199 1304-2027 361137 1660-5625 67667-9219 128-4126 1312-2163 36-2245 1670-7656 68292-5449 41 in. 128-8053 1320-2543 36-3353 1681* 68921- £ 129T980 1328-3169 36-4461 1691-2656 69553-2988 £ T29-5907 1336-4041 36-5569 1701-5625 70189-4531 i 129-9834 1344-5158 36-6676 1711-8906 70829-4746 h 130-3761 1352-6520 36-7784 1722-25 71473-375 g 130-7688 1360-8128 36-8892 1732-6406 72121-1660 | 131T615 1368-9981 370000 17430625 72772-8594 Li_ 131-5542 1377-2080 37-1108 1753-5156*" 73428-4668 EQUAL SQUARES, SQUARES AND CUBES OP THE DIAMETERS. [9 TABLE I. I)ia. or Root. Circum. Area. Side of = square. Square. Cube. 42 in. 131-9469 1385-4424 37-2215 1764- 74088- b 132-3396 1393-7013 37-3323 1774-5156 74751-4707 i 132-7323 1401-9848 37-4431 17850625 75418-8906 1, 133 1250 1410-2928 37-5539 1795-6406 76090-2715 h 133-5177 1418-6254 37-6646 1806-25 76765-625 a. 133-9104 1426-9826 37-7754 1816-8906 77444-9629 134-3031 1435-3642 37-8862 1827-5625 78128-2969 & 134-6958 1443-7704 37-9970 1838-2656 78815-6387 43 135 0885 1452-2012 38-1078 1849- 79507- i 135-4812 1460-6565 38-2185 1859-7656 80202-3926 135-8739 ■1469-1364 38-3293 1870-5625 80901-8281 1 136-2666 1477-6407 38-4401 1881-3906 81605-3183 4 136-6593 1486-1697 38-5509 1892-25 8-2312-875 1 137-0520 1494-7232 38-6616 1903T406 83024-5097 3 137-4447 1503-3012 38-7724 1914 0625 83740-2344 137-8374 1511-9037 38-8832 19250156 84460-0605 44 138-2301 1520-5308 38-9940 1936- 85184- 4 138-6228 1529-1825 39-1048 19470156 85912-0644 J 1390155 1537-8587 39-2155 1958-0625 86644-2656 i, 139-4082 1546-5594 39-3263 1969-1406 87380-6152 J 139-8009 1555-2847 39-4371 1980-25 8812M25 a. 140-1936 1564-0345 39-5479 1991-3906 • 88865-8066 3 140-5863 1572-8089 39-6587 2002-5625 89614-6719 a 140-9790 1581-6078 39-7694 2013-7656 90367-7324 45 *». 141-3717 1590-4313 39-8802 2025- 91125- 4 141-7644 1599-2793 39-9910 2036-2656 91886-4863 3 ■142 1571 1608T518 40-1018 2047-5625 92652-2031 3 142-5498 1617 0489 40-2125 2058-8906 93422-1621 h 142-9425 1625-9705 40-3233 2070-25 94196-375 a 143-3352 1634-9167 40-4341 2081-6406 94974-8535 3 1437279 1643-8874 40-5449 20930625 95757-6094 b 144-1206 1652-8827 40-6557 2104-5156 96544-6543 46 in. 144-5133 1661-9025 40-7664 2116- 97336- 4 144-9060 1670-9469 40-8772 2127-5156 98131-6582 3 145-2987 16800158 40-9880 2139-0625 98931-6406 i 145-6914 1689-1092 41 0988 2150-6406 99735-9589 i 1460841 1698-2272 41-2096 2162-25 100544-625 a 146-4768 1707-3697 ' 41-3203 21738906 101357-6504 3 146-8695 1716-5368 41-4311 2185-5625 102175-0469 4 147-2622 1725-7284 • 41-5419 2197-2656 102996-8261 47 in. 147-6549 1734-9445 41-6527 2209- 103823- 4 1480476 1744-1852 41-7634 2220-7656 104653-5800 3 148-4403 1753-4505 41-8742 2232-5625 105488-5781 1 148-8329 1762-7403 41 -9850 2244-3906 1063280058 h 149-2257 1772-0546 42 0958 2256-25 107171-875 a 149-6183 1781-3935 42-2066 2268-1406 108020-1973 3 1500110 1790-7569 42-3173 2280-0625 108872-9844 150-4037 1800-1449 42-4281 2292-0156 109730 2480 IOJ CIRCUMFERENCES AND AREAS OF CIRCLES, SIDES OF TABLE L Dia. or Root. Ciroum. Area. Side of = square. Square. Cube. j j 48 in. 1507964 1809-5574 42-5389 2304- 110592- j A 151-1891 1818-9944 42-6497 2316-0156 111458-2520 151-5818 1828-4560 42-7604 2328-0625 112329-0156 | 151-9745 1837-9421 42-8712 2340-1406 113204-3027 152-3672 1847-4528 42-9820 2352-25 114084-125 152-7599 1856-9881 430928 2364-3906 114968-4941 | 153-1526 1866-5478 43-2036 2376-5625 115857-4219 153-5453 1876-1321 43-3143 2388-7656 116750-9199 49 in. 153-9380 1885-7410 43-4251 2401- 117649- A 154-3307 1895-3744 43-5359 2413-2656 118551-6738 % 1547234 1905-0323 43-6467 2425-5625 119458-9531 155-1161 1914-7148 43-7575 2437-8906 120370-8496 S A 155-5088 1924-4218 43-8682 2450-25 121287-375 155-9015 1934-1534 43-9790 2462-6406 122208-5410 156-2942 1943-9095 440898 2475-0625 123134-3594 i 156-6869 1953-6902 44-2006 2487-5156 124064-8418 50 in. 157-0796 1963-4951 44-3113 2500- 125000- A 157-4723 1973-3252 44-4221 2512-5156 125939-8457 8 | 157-8650 1983-1794 44-5329 2525-0625 126884-3906 § 158-2577 19930583 44-6437 2537-6406 127833-645 B ft 158-6504 2002-9617 44-7545 2550-25 128787-625 | 159-0431 2012-8896 44-8652 2562-8906 129746-3379 8 | 159-4358 2022-8421 44-9760 2575-5625 130701-7969 ft 159-8285 2032-8191 45-0868 2588-2656 1316780136 51 t«. 160-2212 2042-8206 45-1976 2601- 132651- ft 160-6139 2052-8467 45-3084 2613-7656 133628-7676 ft 161-0066 2062-8974 45-4191 2626-5625 134611-3281 | 161-3993 2072-9725 45-5299 2639-3906 135598-6933 8 i 1617920 2083-0723 45-6407 2652-25 136590-875 8“ 162-1847 2093-1966 45-7515 2665-1406 1375878847 i 162-5774 2103-3454 45-8622 2678-0625 138589-7344 f 162-9701 2113-5187 45-9730 2691-0156 139596-4355 52 in. 163-3628 2123-7166 460838 2704- 140608- ft 163-7555 2133-9391 46-1946 2717-0156 141624-435 ft 164-1482 2144-1861 46-3054 2730-0625 142645-7656 8 164-5409 2154-4576 46-4161 2743-1406 148671-9902 ft 164-9336 2164-7537 46-5269 2756-25 144703-125 1 165-3263 2175-0743 46-6377 2769-3906 145739-1816 * 1657190 2185-4195 46-7485 2782-5625 146780-1719 1 166-1117 2195-7892 46-8592 2795-7656 147826-1074 53 t«. 166-5044 2206-1834 46-9700 2809- 148877- ft 166-8971 2216-6022 47-0808 2822-2656 149932-8613 ft 167-2898 2227-0456 47-1916 2835-5625 150993-7031 § 167-6825 2237-5135 47-3024 2848-8906 152059-5371 ft 168-0752 2248-0059 47-4131 2862-25 153130-375 § 168-4679 2258-5229 47-5239 2875-6406 154206-2285 i 168-8606 22690644 47-6347 2889-0625 155287-1094 * 169-2533 2279-6304 47-7455 2902-5156 156373-0292 EQUAL SQUARES, SQUARES AND CUBES OF THE DIAMETERS. [11 TABLK I. Di». or. Root. Circum. Area. Side of = square. Square. Cube. 54 in. 169-6460 2290-2210 478563 2916- 157464- 170-0387 2300-8362 47-9670 2929-5156 158560-0332 8 i 170-4314 2311-4759 480778 2943-0625 159661 1406 § 170-8241 2322-1401 48-1886 2956-6406 160767-3339 5 J 171-2168 2332-8289 48-2994 2970:25 161878-625 8 171-6095 2343-5422 48-4101 2983-8906 162995-0254 8 | 1720022 2354-2801 48-5209 2997-5625 164116-5469 i 172-3949 2365-0425 48-6317 3011-2656 165243-2011 55 in. 172-7856 2375-8294 48-7425 3025- 166375- JL 173-1803 2386-6409 48-8533 3038-7656 167511-9550 k 173-5730 2397-4770 48-1.640 3052-5625 168654-0781 | 173-9657 2408-3376 49-0748 3066-3906 169801-3809 j 174-3584 2419-2227 49-1856 3080-25 170953-875 | 174-7511 2430-1324 49-2964 3094-1406 172111-5722 i 175-1438 2441-0666 49-4072 3108 0625 173274-4844 1 175-5365 24520253 49-5179 31220156 174442-6230 56 m. 175-9292 2463-0086 49-6287 3136- 175616- i 176-3219 2474-0165 49-7395 3150-0156 176794-6269 i 176-7146 2485 0489 49-8503 3164-0625 177978-5156 | 177-1073 2496-1058 49-9610 3178-1406 179167-6777 i 177-5000 2507 1873 500718 3192-25 180362-125 § 177-8927 2518-2933 50-1826 3206-3906 181561-8691 f 178-2854 2529-4239 50-2934 3220-5625 182766-9219 f 178-6781 2540-5790 50-4042 3234-7656 183977-2949 57 t». 179-0708 2551-7586 50-5149 3249- 185193- 179-4635 2562-9628 50-6257 3263-2656 186414-0488 179-8562 2574-1916 50-7365 3277-5625 187640-4531 | 180-2489 2585-4449 50-8473 3291-8906 188872-2246 h 180-6416 2596-7227 50-9580 3306-25 190109-375 i 181 0343 2608-0250 51-0688 3320-6406 191351-9160 i 181-4270 2619-3520 51-1796 3335-0625 192599-8594 i 181-8197 2630-7034 51-2904 3349-5156 193853-2168 58 in. 182-2124 2642-0794 51-4012 3364- 195112- k 182-6051 2653-4800 51-5119 3378-5156 196376-2207 k 182-9978 2664-9051 51-6227 3393-0625 197645-8906 1 183-3905 2676-3547 51-7335 3407-6406 198921-0214 k 183-7832 2687-8289 51-8443 3422-25 200201-625 § 184-1759 2699-3276 51-9551 3436-8906 201487-7129 1 184-5686 2710-8508 520658 3451-5625 202779-296? 1 184-9613 2722-3987 52-1766 3466-2656 204076-3886 59 tw. 185-3540 2733-9710 52-2874 3481- 205379- i, 185-7467 2745-5679 52-3982 3495-7656 206687-1426 i 186-1394 2757-1893 52-5089 3510-5625 208000-8281 i i, 186-5321 2768-8353 52-6197 3525-3906 2093200683 * 186-9248 2780-5058 52-7305 3540-25 210644-875 § 187-3175 2792-2009 52-8413 3555-1406 211975-2598 1 187-7102 2803-9205 52-9521 3570 0625 213311-2344 L i— 188 1029 2815-6646 530628 3585 0156 214642-8105 121 CIRCUMFERENCES AND AREAS OF CIRCLES, SIDES OF TABLE 1. Dia. or Root. Circum. Area. Side of = square. Square. Cube. 60 in. 1884956 2827-4334 53-1736 3600- 216000- k 188-8883 2839-2266 53-2844 3615-0156 217352-8145 i 189-2810 2851 -0444 53-3952 3630-0625 218711-2656 | 189-6737 2862-8868 53-5060 3645-1406 220075-3652 190-0664 2874-7536 53-6167 3660-25 221445-125 | 190-4591 2886-6450 53-7275 3675-3906 222820-5566 190-8518 2898-5610 53-8383 3690-5625 224201-6719 191-2445 2910-5015 53-9491 3705-7656 225588-4824 61 in. 191-6372 2922-4666 54-0598 3721- 226981- k 1920299 2934-4562 54-1706 3736-2656 228379-2363 i 192-4226 2946-4703 54-2814 3751-5625 229783-2031 | 192-8152 2958-5090 54-3922 3766-8906 231192-9121 % 193-2079 2970-5722 54-5030 3782-25 232608-375 | 193-6006 2982-6600 54-6137 3797-6406 234029-6035 f 193-9933 2994-7723 54-7245 3813-0625 235456-6094 i 194-3860 3006-9091 54-8353 3828-5156 236889-4043 62 in. 194-7787 3019-0705 54-9461 3844- 238328- • i 195-1714 3031-2565 55-0568 3859-5156 239772-4082 i 195-5641 3043-4670 55-1676 3875-0625 241222-6406 § 195-9568 3055-7020 55-2784 3890-6406 242678-7089 % 196-3495 3067-9616 55-3892 3906-25 244140-625 6 196-7422 3080-2457 55-5000 3921-8906 245608-4004 1 197-1349 3092-5544 55-6107 3937-5625 247082-0469 i 197-5276 3104-8876 55-7215 3953-2656 248561-5761 63 197-9203 3117-2453 55-8323 3969- 250047- 198-3130 3129-6276 55-9431 3984-7656 251538-3300 198-7057 3142-0344 56-0539 4000-5625 253035-5781 | 199-0984 3154-4658 56-1646 4016-3906 2545387558 5 i 199-4911 3166-9217 56-2754 4032-25 256047-875 | 199-8838 3179-4022 56-3862 4048-1406 257562-9472 | 200-2765 3191-9072 56-4970 4064-0625 259083-9844 I 200-6692 3204-4368 56-6077 4080-0156 260610-9980 64 in. 201-0619 3216-9909 56-7185 4096- 262144- L 201-4546 3229-5695 56-8293 41120156 263683-0019 8 i 201-8473 3242-1727 56-9401 4128-0625 265228-0156 | 202-2400 3254-8004 57-0509 4144-1406 266779-0527 i 202-6327 3267-4527 57T616 4160-25 268336-125 g 2030254 3280-1295 57-2724 4176-3906 269899-2441 S f 203-4181 3292-8309 57-3832 4192-5625 271468-4219 203-8108 3305-5568 57-4940 4208-7656 273043-6699 65 in. 204-2035 3318-3072 57-6048 4225- 274625- £ 204-5962 3331-0822 577155 4241-2656 276212-4238 204-9889 3343-8818 57-8263 4257-5625 277805-9531 | 205-3816 3356-7058 57-9371 4273-8906 279405-5996 i 205-7743 3369-5545 58-0479 4290-25 281011-375 1 206-1670 3382-4276 58-1586 4306-6406 282623-2910 f 206-5597 3395-3253 58-2694 4323-0625 284241-3594 206-9524 3408-2476 58-3802 4339-5156 285865-5918 EQUAL SQUARES, SQUARES AND CUBES OF MKE DIAMETERS. [13 TABLE I. Dia. or Root. Circum. Area. Side of = square. Square. Cube. 66 in. 207-3451 3421-1944 58-4910 4356- 287496- b 207-7378 3434-1657 58-6018 4372-5156 289132-5957 i 208 1305 3447-1616 58-7125 4389-0625 290775-3906 1 208-5232 3460 1821 58-8233 4405-6406 292424-3964 h 208-9159 3473-2270 58-9341 4422-25 294079-625 | 209-3086 3486-2965 59-0449 4438-8906 295741-0879 1 209-7013 3499-3906 59-1556 4455-5625 297408-7969 i 2100940 3512-5092 59-2664 4472-2656 299082-7637 67 in. 210-4867 3525-6524 59-3772 4489- 300763- b 210-8794 3538-8200 59-4880 4505-7656 302449-5176 i 211-2721 3552 0123 59-5988 4522-5625 304142-3281 S 211-6648 3565-2291 59-7095 4539-3906 305841-4433 i 212-0575 3578-4704 59-8203 4556-25 307546-875 i 212-4502 3591-7362 59-9311 4573 1406 309258-6348 l 212-8429 3605 0267 600419 4590 0625 310976-7344 i 213-2356 3618-3416 60-1527 4607-0156 31270M855 63 in. 213-6283 3631-6811 60-2634 4624- 314432- k 214-0210 3645-0451 60-3742 4641 -0156 316169-1894 i 214-4137 3658-4337 60-4850 46580625 317912-7656 i 214-8064 3671-8469 60-5958 4675-1406 319662-7402 h 215-1991 3685-2845 60-7065 4692-25 321419-125 8 215-5918 3698-7467 60-8173 4709-3906 323181-9316 2 215-9845 3712-2335 60-9281 4726-5625 324951-1719 1 216-3772 3725-7448 61-0389 4743-7656 326726-8574 69 in. 216-7699 3739-2807 61-1497 4761* 328509- 217-1626 3752-8410 61-2604 4778-2656 330297-6113 i 217-5553 3766-4260 61-3712 4795-5625 332092-7031 8 217-9480 3780-0355 61-4820 4812-8906 333894-2871 h 218-3407 3793-6695 61-5928 4830-25 335702-375 8“ 218-7334 3807-3280 61-7035 4847-6406 337516-9785 f 219-1261 3821-0112 61-8143 4865-0625 339338-1094 8 219-5188 3834-7188 61-9251 4882-5156 341165-7793 70 in. 219-9115 3848-4510 62-0359 4900- 343000- 220-3042 3862-2077 62-1467 4917-5156 344840-7832 i 220-6969 3875-9890 62-2574 4935-0625 346688 1406 | 221 0896 3889-7949 62-3682 4952-6406 348542-0839 J 221-4823 3903-6252 62-4790 4970-25 350402-625 § 221-8750 3917-4801 62-5898 4987-8906 352269-7754 I 222-2677 3931-3596 62-7006 5005-5625 354143-5469 222-6604 3945-2636 62-8113 5023-2656 356023-9511 71 in. 223-0531 3959 1921 62-9221 5041- 357911- 8 223-4458 3973-1452 630329 5058-7656 359804-7051 1 223-8385 3987-1229 63-1437 5076-5625 361705-0781 8 224-2312 400M250 63-2544 5094-3906 363612 1308 i 224-6239 4015-1518 63-3652 5112-25 365525-875 8 225-0166 4029-2030 63-4760 5130-1406 367446-3222 | 225-4093 4043-2788 63-5868 5148-0625 369373-4844 225-8020 4057-3792 63-6976 5166 0156 371307 3730 14] CIRCUMFERENCES AND AREAS OF CIRCLES, SIDES OF TABLE I. Dia. or Root. Circum. Area. Side of = square. Square. Cube. 72 in. 226-1947 4071 5041 63-8083 5184- 373248- L 226-5874 4085-6535 63-9191 5202-0156 375195-3770 % 226-9801 4099-8275 64-0299 5220-0625 377149-5156 | 227-3728 4114-0260 64-1407 5238-1406 379110-4277 227-7655 4128-2491 64-2515 5256-25 381078 125 8 “ 2281582 4142-4967 64-3622 5274-3906 383052-6191 228-5509 4156-7689 64-4730 5292-5625 3850339219 1 228-9436 4171-0656 64-5838 5310-7656 387022 0449 73t». 229-3363 4185-3868 64-6946 5329- 389017- A 229-7290 4199-7326 64-8053 5347-2656 391018-7988 % 230-1217 4214-1029 64-9161 5365-5625 393027-4531 230-5144 4228-4978 65-0269 5383-8906 395042-9746 % 230-9071 4242 9172 65-1377 5402-25 397065 375 8 231-2998 4257-3612 65-2485 5420-6406 399094-6660 % 231-6925 4271-8297 65-3592 5439-0625 401130-8594 r 2320852 4286-3227 65-4700 54575156 403173-9668 74 in. 232-4779 4300-8403 65-5808 5476- 405224- A 232-8706 '4315-3825 65-6916 5494-5156 407280-9707 233-2633 4329-9492 65-8023 55130625 4093448906 § 233-6560 4344-5404 65-9131 5531-6406 411415-7714 234-0487 4359-1562 66-0239 5550-25 413493-625 8" 234-4414 4373-7965 66 1347 5568-8906 415578-4629 8 | 234-8341 4388-4613 66-2455 5587-5625 417670-2969 1 235-2267 4403-1507 66-3562 5606-2656 419769-1386 75 in. 235-6194 4417-8647 66-4670 5625- 421875- A 236-0121 4432-6032 66-5778 5643-7656 423987-8926 % 236-4048 4447-3662 66-6886 5662-5625 426107-8281 § 236-7975 4462T538 66-7994 5681-3906 428234-8183 £ 237 1902 4476-9659 66-9104 5700-25 430368-875 8 237-5829 4491-8025 67-0209 • 5719-1406 4325100098 237-9756 4506-6637 67-1317 5738 0625 434658-2344 r 238-3683 4521-5495 67-2425 5757-0156 436813-5605 76 in. 238-7610 4536-4598 67-3532 5776- 438976- A 239-1537 4551-3946 67-4640 5795-0156 441145-5644 % 239-5464 4566-3540 67-5748 5814-0625 443322-2656 § 239-9391 4581-3379 67-6856 58331406 445506 1152 240-3318 4596-3464 677964 5852-25 447697-125 8" 2407245 4611-3794 679071 5871-3906 449895-3066 241-1172 4626-4370 680179 5890-5625 452100-6719 £ 241-5099 4641-5191 68-1287 5909-7656 454313-2324 77 in. 241-9026 4656-6257 68-2395 5929- 456533- A 242-2953 4671-7569 68-3503 5948-2656 458759-9863 242-6880 4686-9126 68-4610 5967-5625 460994-2031 § 243-0807 47020929 68-5718 5986-8906 463235-6621 % 243-4734 4717-2977 68-6826 6006-25 465484-375 8 " 243-8661 4732-5271 68-7937 6025-6406 467740-3535 244-2588 4747-7810 68-9041 6045-0625 470003-6094 244-6515 47630594 690149 6064-5156 472274-1543 EQUAL SQUARES, SQUARES AND CUBES OF THE DIAMETERS. [15 TABLE I. Dia. or Root. Circum. Area. side of — square. Square. Cube. | 78 in. 245 0442 4778-3624 69 1257 6084- 474552- k 245-4369 4793-6900 69-2365 6103-5156 476837-1582 245-8296 48090420 69-3473 6123-0625 479129-6406 246-2223 4824-4187 69-4580 6142-6406 481429-4589 i 246-6150 4839-8198 69-5688 6162-25 483736-625 g 2470077 4855-2455 69-6796 6181-8906 48605M504 f 247-4004 4870-6958 697904 6201-5625 4883730469 i 247-7931 4886-1706 69-9011 6221-2656 490702-3261 79 in. '248 1858 248-5785 248-9712 4901-6699 4917-1938 4932-7423 ' 700119 70 1227 70 2335 6241- 6260-7656 6280-5625 493039- 497734-5781 | 249-3639 4948-3152 70-3443 6300-3906 500093-5058 h . 249-7566 4963-9127 70-4550 6320-25 502459-875 g 250-1493 4979-5348 70-5658 6340-1406 504833-6972 i 250-5420 4995-1814 70-6766 636O-06-25 507214-9844 1 250-9347 5010-8526 70^874 6380-0156 509603-7480 80 in. 251-3274 5026-5482 70-8982 6400- 512000- £ 251-7201 5042-2685 71-0089 64200156 514403-7519 i 252-1128 50580133 7M197 64400625 516815 0156 8 252-5055 5073-7826 712305 6460 1406 519233-8027 i 252-8982 5089-5764 713413 6480-25 521660-125 § 253-2909 5105-3949 71 4520 6500-3906 524093-9941 f 253-6836 5121-2378 71-5628 6520-5625 526535-4219 i 2540763 5137 1053 71-6736 6540-7656 528984-4199 81 in. 254-4690 5152-9974 71-7844 6561- 531441- i 254-8617 5168 9139 71 8952 6581-2656 533905-1738 * 255-2544 5184-8551 720059 6601-5625 536376-9531 | 255-6471 5200-8207 72-1167 6621-8906 538856-3496 J 2560398 5216-8110 72-2275 6642-25 541343-375 8 256-4325 5232-8257 72-3383 6662-6406 543838-0410 i 256-8252 5248-8650 72-4491 6683-0625 546340-3594 i 257-2179 5264-9289 72-5598 6703-5156 548850-3418 82 in. 257-6106 5281 0173 72-6706 6724- 551368- £ 2580033 5297-1302 72-7814 6744-5156 553893-3457 i 258-3960 5313 2677 72-8922 6765-0625 556426-3906 g 258-7887 5329-4297 730029 •6785-6406 558967 1464 £ 259-1814 5345-6162 731137 6806-25 561515-625 g 259-5741 5361-8274 73-2245 6826-8906 564071-8379 1 259-9668 53780630 73-3353 6847-5625 566635-7969 i 260-3595 5394-3232 73-4461 6868-2656 569207-5137 83 i«. 260-7522 5410-6079 73-5568 6889- 571787- 4 261 1449 5426-9172 73-6676 6909-7656 574374-2676 i 261-5376 5443-2511 73-7784 6930-5625 576969-3281 g 261-9303 5459-6094 73-8892 6951-3906 579572 1933 i a 262-3230 5475-9923 73-9999 6972-25 582182-875 g 262 7157 5492-3998 74-1107 6993-1406 584801-3848 263 1084 5508-8318 74-2215 7014-0625 587427-7344 Li 263-5011 5525-2884 74-3323 70350156 590061-9355 K 2 26] CIRCUMFERENCES AND AREAS OF CIRCLES, SIDES OF TABLE I. Dia. or Root. Circum. Area. Side of = square. Square. Cube. J 84 in. 263 8938 5541-7694 74-4431 7056- 592704 264-2865 5558-2751 74-5538 7077-0156 595353-9394 % 264-6792 5574-8053 74-6646 7098-0625 598011-7656 265-0719 5591-3600 74-7754 7119-1406 600677-4902 ^ A 265-4646 5607-9392 74-8862 7140-25 603351-125 265-8573 5624-5430 74-9970 7161-3906 606032-6816 266-2500 564M714 75-1077 7182-5625 608722-1719 & 266-6427 5657-8243 75-2185 7203-7656 611419-6074 85 in. 267-0354 5674-5017 75-3293 7225- 6141^5- A 267-4281 5691-2037 75-4401 7246-2656 616838-3613 8 4 267-8208 5707-9302 75-5508 72675625 619559-7031 f 268-2135 5724-6813 75-6616 7288-8906 6222890371 268-6062 5741-4569 75-7724 7310-25 625026-375 268-9989 5758-2571 75-8832 7331-6406 627771-7285 h 269-3916 57750818 75-9940 7353-0625 630525-1094 z 269-7843 5791-9310 ,76-1047 7374-5156 633286-5293 86 in. 270-1770 5808-8048 76-2155 7396- 636056- 4 270-5697 5825-7031 76-3263 7417-5156 638833-5332 8 i 270-9624 5842-6260 76-4371 7439-0625 641619-1406 f 271-3551 5859-5734 76-5479 7460-6406 644412-8339 % 271-7478 5876-5454 76-6586 7482-25 647214-625 272-1405 5893-5419 76-7694 7503-8906 650024-5254 272-5332 5910-5630 76-8802 7525-5625 652842-5469 i 272-9259 5927-6086 76-9910 7547-2656 655668-7011 87 in. 273-3186 5944-6787 77-1017 7569- 658503- 4 273-7113 5961-7734 77-2125 7590-7656 661345-4551 % 274-1040 5978-8926 77-3233 7612-5625 664196-0781 f 274-4967 5996-0364 77-4341 7634-3906 667054-8808 274-8894 6013-2047 77'5449 7656-25 669921 -875 g 275-2821 6030-3975 77-6556 7678-1406 672797-0722 275-6748 60476149 77'7664 7700-0625 675680-4844 1 276-0675 6064-8569 77-8772 7722-0156 678572-1230 88 in. 276-4602 6082-1234 77'9880 7744- 681472- 4 276-8529 6099-4144 780987 7766-0156 684380-1269 * 277-2456 6116-7300 78-2095 7788-0625 687296-5156 § 277-6382 6134-0701 78-3203 7810-1406 690221-1777 278-0309 6151-4348 78-4310 7832-25 693154-125 § 278-4236 6168-8240 78-5419 7854-3906 696095-3691 278-8163 6186-2377 78-6526 7876-5625 699044-9219 i 279-2090 6203-6760 78-7634 7898-7656 702002 7949 89 in. 279-6017 6221-1389 78-8742 7921* 704969- £ 279-9944 6238-6262 78-9850 7943-2656 707943-5488 8 * 280-3871 6256-1382 79-0958 7965-5625 710926-4531 280-7798 6273-6746 79-2065 7987-8906 713917-7246 281-1725 6291-2356 79-3173 8010-25 716917-375 i" 281-5652 6308-8212 79-4281 8032-6406 719925-4160 -1 281-9579 6326-4313 79-5389 8055-0625 722941-8594 282-3506 6344-0659 79-6496 80775156 725966-7168 EQUAL SQUARES, SQUARES AND CUBES OP THE DIAMETERS. [17 TABLE I. Dia. or Root. Circum. Area. Side of = square. Square. Cube. 90 in. 282-7433 6361-7251 79-7604 8100- 729000- k 283 1360 6379-4089 79-8712 8122-5156 732041-7207 i 283-5287 6397-1171 79-9820 8145-0625 735091-8906 | 2839214 6414-8499 80-0928 8167-6406 738150-5214 h 284-3141 6432-6073 80-2035 8190-25 741217-625 i 284-7068 6450-3892 80-3143 8212-8906 744293-2129 4 2850995 6468 1957 80-4251 8235-5625 747377-2969 l 285-4922 64860267 80-5359 8258-2656 750469-8886 91 in. 285-8849 6503-8822 80-6467 8281- 753571- & 286-2776 6521-7623 80-7574 8303-7656 756680-6426 4 2866703 6539-6669 80-8682 8326-5625 759798-8281 1 287-0630 6557-5981 80-^790 8349-3906 762925-5683 h 2874557 6575-5498 81-0898 8372-25 766060-875 i 287-8484 6593-5280 81-2005 8395-1406 769204-7598 288-2411 6611-5308 81-3113 8418 0625- 772357-2344 § 288-6338 6629-5582 81-4221 8441-0156 775518-3105 92 in. 289-0265 6647-6101 81-5329 8464- 778688- & 289-4192 6665-6865 81-6437 8487-0156 781866-3144 i 289-8119 6683-7875 81-7544 8510-0625 785053-2656 1 290-2046 6701-9130 81-8652 8533 1406 788248-8652 h 290-5973 6720 0630 81-9760 8556-25 791453 125 §' 290-9900 6738-2376 820868 8579-3906 7946660566 1 291-3827 6756-4368 82-1975 8602-5625 797887-6719 i 291-7754 6774-6605 82-3083 8625-7656 801117-9824 93 in. 292 1681 6792-9087 .82-4191 8649- 804357- i 292-5608 68111815 82-5299 8672-2656 807604-7363 i 292-9535 6829-4788 82-6407 8695-5625 810861-2031 § 293-3462 6847 8007 82-7514 8718-8906 814126-4121 h 293-7389 6866 1471 82-8622 8742-25 817400-375 i 294-1316 6884-5180 82-9730 8765-6406 820683 1035 4 294-5243 6902-9135 830838 8789 0625 823974-6094 i 294-9170 6921-3336 83-1946 8812-5156 827274-9043 94 in. 295-3097 6939-7782 83-3053 8836- 830584- k 295-7024 6958-2473 83-4161 8859-5156 833901-9082 i 296-0951 6976-7410 83-5269 8883-0625 837228-6406 I 296-4878 6995-2592 83-6377 8906-6406 840564-2089 h 296-8805 7013-8019 837484 8930-25 843908-625 § 297-2732 7032-3693 83-8592 8953 8906 847261 -9004 I 297-6659 7050-9611 83-9700 8977"i>625 8506240469 i 2980586 7069-5775 840808 9001 2656 853995-0761 95 in. 298-4513 7088-2184 84-1916 9025- 857375- h 298-8440 7106-8839 84-3023 90487656 860763-8301 4 299-2367 7125-5799 84-4131 9072-5625 864161-5781 | 299-6294 7144-2885 84-5239 9096-3906 867568-2558 £ 300-0221 71630276 84-6347 9120-25 870983-875 § 300-4148 7181-7913 84-7454 9144-1406 874408-4472 4 300-8075 7200-5794 84-8562 91680625 877841-9844 i . 301-2002 7219-3922 84-9670 91920156 881284-4980 18] CIRCUMFERENCES AND AREAS OF CIRCLES, SIDES OF TABLE I. Dia. or Root. Circum. Area. Side of = square. Square. Cube. 96 in. 301-5929 7238-2295 85-0778 9216* 884736- i 301-9856 7257-0913 85T886 9240-0156 888196-5019 *£ 302-3783 7275-9777 85-2993 9264-0625 8916660156 1 302-7710 7294-8886 85-4101 9288-1406 895144-5527 % 303-1637 7313-8240 85-5209 9312-25 898632-125 8 ‘ 303-5564 7332-7840 85-6317 9336-3906 902128-7441 f 303-9491 7351-7686 85-7425 9360-5625 905634-4219 i 304-3418 7370-7777 85-8532 9384-7656 909149-1699 97 in. 304-7345 7389-8113 85-9640 9409- 912673- 4 305 1272 7408-8695 86-0748 9433-2656 916205-9238 S £ 305-5199 7427-9522 86-1856 9457-5625 919747-9531 | 305-9126 7447-0595 86-2963 9481-8906 923299-0996 % 306-3053 7466-1913 86-4071 9506-25 926859-375 306-6980 7485-3476 86-5179 9530-6406 930428-7910 % 307 0907 7504-5285 86-6287 95550625 934007-3594 307-4834 7523-7340 86-7395 9579-5156 937595-0918 98 in. 307-8761 7542-9640 86-8502 9604- 941192- i 308-2688 7562-2185 86-9610 9628-5156 944798-0957 *£ 308-6615 7581-4976 87-0718 9653-0625 948413-3906 § 3090542 7600-8012 87-1826 9677 6406 9520378965 309-4469 7620-1293 87-2934 9702-25 955671-625 309-8396 7639-4820 87-4041 9726-8906 959314-5879 310-2323 7658-8593 87-5149 9751-5625 962966-7969 i 3106250 •7678-2611 876257 9776-2656 966628-2637 99 in. 311-0177 7697-6874 87-7365 9801- 970299- £ 311-4104 7717-1383 87-8472 . 9825-7656 973979-0176 £ 311-8031 7736-6137 87-9580 9850-5625 9776(58-3281 § 312-1958 7756-1137 880688 9875-3906 981366-9433 S 4 312-5885 7775-6382 88-1796 9900-25 985074-875 g 312-9812 7795-1872 88-2904 9925-1406 988792-1348 313-3739 7814-7608 88-4011 9950-0625 992518-7344 | * 313-7666 7834-3590 88-5119 9975-0156 996264-6855 100 in. 314-1593 7853-9816 88-6227 10000- 1000000- £ 314-9447 7893-3006 88-8442 10050-0625 1007518-7656 3 315-7301 7932-7178 89-0658 10100-25 1015075 125 ! i 316-5155 7972-2331 89-2874 10150-5625 1022669-1719 101 in. 317-3009 8011-8467 89-5089 10201- 1030301- £ 318-0863 8051-5584 89-7305 10251-5625 1037970-7031 4 318-8717 8091-3682 89-9520 10302-25 1045678-375 r 319-6571 8131-2763 90-1736 103530625 1053424-1093 102 in. 320-4425 8171-2825 90-3951 10404- 1061208- £ 321-2278 8211-3869 90-6167 104550625 1069030-1406 4 322-0132 8251 -5895 90-8383 10506-25 1076890-625 £ 322-7986 8291 -8999 91 0593 10557-5625 J 084789-5468 EQUAL SQUARES, SQUARES AND CUBES OF THE DIAMETERS. [19 TABLE I. Dia. or Root. Circum. Area. Side of =square. Square. Cube. 103 in. 323-5840 8332-2891 91-2814* 10609- 1092727- k 324-3694 8372-7862 91-5029 10660-5625 1100703-0781 h 325-1548 8413-3815 91-7245 10712-25 1108717 875 I 325-9402 8454-0749 91-9460 10764-0625 1116771-4843 104 in. 326-7256 8494-8665 92-1676 10816- 1124864- i 327-5110 8535-7563 92-3892 10868-0625 1132995-5156 h 328-2964 8576-7443 92-6107 10920-25 1141166-125 I 3290818 8617 83U4 92-8323 10972-5625 1149375-9218 105 in. 329-8672 8659-0148 93-0538 11025- 1157625- t 330-6526 8700-2972 93-2754 11077-5625 1165913-4531 i 331-4380 8741-6779 93-4969 11130-25 1174241-375 1 332-2234 87831567 93-7185 11183-0625 1182608-8593 106 in. 3330088 8824-7338 93-9401 11236- 1191016- i 333-7942 8866-4090 94-1616 11289-0625 1199462-8906 h 334-5796 8908-1823 94-3832 11342-25 1207949-625 1 335-3650 8950-0539 94-6047 11395-5625 1216476-2968 107 in. 336 1504 8992 0236 94-8263 11449- 1225043- 336-9358 9034-0915 95-0478 11502-5625 1233649-8281 i 337-7212 9076-2575 95-2694 11556-25 1242296-875 f 338-5066 9118-5218 95-4910 116100625 1250984-2343 108 in. 339-2920 9160-8842 95-7125 11664- 1259712- i 340-0774 9203-3448 '95-9341 11718-0625 1268480-2656 i 340-8628 9245-9035 96 1556 11772-25 1277289-125 1“ 341-6482 9288-5605 96-3772 11826-5625 1286138-6718 109 in. 342-4336 9331-3156 96-5987 11881- 1295029- 343-2190 9374-1689 96-8203 11935-5625 1303960-2031 h 344 0044 9417-1203 970418 119.90-25 1312932-375 1 344-7898 9460 1700 972634 12045-0625 1321945-6093 110 in. 345-5752 9503-3178 • 974850 12100* 1331000- 20 EXAMPLES TO THE PRECEDING TABLE I * Ex. 1. The diameter of a circle being 83J inches, required its circum¬ ference? • Here, in column 1, page 15, we find the diameter 83§ in inches, feet, yards, chains, miles, &c .; the corresponding circumference will be found in column 2; hence, the diameter being 83£ inches, the circumference is 263-5011 inches = Ans. The same result may be found by Prob. IX., page 50; thus 3-14159265 X 83J in. = 263-50108352 inches — Ans. as before. Ex. 2. What is the area of a square whose side is 35 - 25 chains = 35! chains ? Here, in column 1, page 7, we find the side 35.} in inches, feet, yards, chains, miles , &c.; the corresponding square will he found in column 5 ; hence, the side being 351 chains , the area of the square will he 1242'5625 square chaitis — 124-25625 acres = 124 acres 1 rood 1 perch = Ans. The same result may also he found by Proh. I. page 44; thus 35 25 X 35"25 =1242-5625 sq. chains = Ans. as before. Ex. 3. A cellar is to be dug whose length, breadth, and depth are each 10 feet 3 inches (= 10! inches), how many solid feet does it con¬ tain, and what will it cost digging at Is. per solid yard ? Here, in column 1, page 3, we find the side 10.! i n inches, feet, yards, chains , miles, &c.; the corresponding cube will he found in column 6; hence, the side being 10! feet , the solid content or cube will be 1076-890? cubic feet = 39-8848 cubic yards = 1st Ans., and the cost of digging will be 39-8848 shillings = 1/. 19s. 10id. = 2nd Ans. Similarly, by Prob. I. page 44, the area of any side or base of the cube will be 10-25 X 10-25 = 105-0625 sq.feet; and by Prob. 1. page 59, the solid content may be found = 1076-890625 cubic feet, and therefore the rest as before. Ex. 4. The length of a line with which a gardener formed a circular fish-pond was exactly 27 ! yards, what quantity of ground did the fish¬ pond take up. Here, in column 1 ,page 11, we find the diameter 55| (= 27! X 2) in inches, feet, yards, chains, miles, &c.; the corresponding area of the circle will be found in column 3; hence, the diameter being 55^ yards, the area will be 2419-2227 square yards, or half an acre nearly — Ans. The same result may be found as in Ex. 1, page 55, using -78539816 in place of 7854; thus 55i squared x ’78539816 = 55'5 2 X *78539816 = 2419-22268234 square yards = Ans. as before. Ex. 5. If the diameter of a circle be 91 miles 5 furlongs (= 91g miles), what is the side of a square equal in area to that circle? Here, in column 1 ,page 17, we find the diameter 91{j in inches, fed, yards, chains , miles, &c.; the corresponding side of the equal square will be found in column 4; hence, the diameter being 91{j miles, the sides of the equal square will be 81-2005 miles = 81 miles 1 furlong 24 poles 2 feet 2 inches = Ans. The same result may be found as in Ex. 1, page 54, using -8862269 in place of ‘8862 ; thus 91§ X •8P ROO « Q — 81-2005397125 square miles — Ans. as before. * The references to the Problems made use of in the arithmetical compu¬ tations for testing the tables, are taken from Templeton’s “ Millwright and Engineer’s Pocket Companion.” Edition 1852. TABLE II. CONTAINING THE CIRCUMFERENCES AND AREAS OF CIRCLES ; ALSO, THE SIDE OF A SQUARE OF EQUAL AREA, AND THE CONTENTS OF CYLINDERS IN CUBIC YARDS AND IMPERIAL GALLONS, AT 1 FOOT IN DEPTH, FROM 1 TO .50 FEET DIAMETER, ADVANCING BY AN INCH. 22] SIDES OF EQUAL SQUARES l TABLE II. Dia. in feet and Circum. in feet and in square feet . Side of = square in ft. and in. Cubic yards at one foot in depth. Gallons at one foot in depth. ft. in. 0 . 1 •O'. 34 •0055 ft. in* 0. 0£ •0002 •0340 2 0. 64 •0218 0. l| •0008 •1360 3 0. 9$ •0491 .. 0. 23 •0018 •3059 4 1 . og •0873 0. 31 •0032 •5439 5 •1364 o. 4 •0051 •8498 6 1. 6} •1963 0. 5$ •0073 1-2237 7 1 . It) •2673 0. 64 •0099 1-6656 8 2. 4 •3491 0. 7| •0129 2-1754 9 2. 4? •4418 0. 8 •0164 2-7533 10 2. 71 •5454 0. 84 •0202 3-3991 11 2. 101 •6600 0. 9t •0244 4-1129 1A i 3. 1§ •7854 0 . 108 •0291 4-8947 3. 4f •9218 0. Ill •0341 5-7445 2 3. 8 1-0690 1. 03 •0396 6-6622 3 3. Ill 1-2272 1. lj •0455 7-6479 4 4. 2| 1-3963 1. 24 •0517 8-7017 5 4. Si 1-5763 l. sl •0584 9-8234 6 4. 81 1-7671 1. 4 •0654 11-0130 7 4. Ill 1-9689 ] 44 •0729 12-2707 8 5. 2J 2-1817 1 . 5f •0808 13 5963 9 5. 6 2-4053 1 . 63 •0891 14-9900 10 5. 91 2-6398 1 . 7i •0978 16-4516 11 6. o| 2-8852 1 . 8| •1069 17-9812 6. 33 3-1416 1. 94 •1164 19-5787 6. 3 3-4088 1 . 101 •1263 21-2423 2 6. 9f 3-6870 1.11 •1366 22-9778 3 7. 4 3-9761 1.11| •1473 24-7793 4 7 • 4 4-2761 2 of •1584 26-6488 5 r! 71 45869 2. l| •1699 28-5863 6 7 • 10 4-9087 2. 28 •1818 30-5918 7 8. 13 5-2414 2. 31 •1941 32-6652 8 8. 41 5-5851 2. 43 •2069 34-8066 9 8. 78 5-9396 2. 54 •2200 37-0161 10 8. loj 9. 2 6-3050 2. 64 •2335 39-2934 11 6-6813 2. 7 •2475 41-6388 Zft. 1 9. 9. 70686 7-4667 2. 7£ 2. 8| •2618 •2765 44 0522 46-5335 2 9 . Ilf 7-8758 2. 9g •2917 49-0828 3 10 . 24 8-2958 2.108 •3073 51-7001 4 10. 58 8-7266 2. Ill •3232 54-3854 5 10. 8§ 9-1684 3. 03 •3396 57 1386 6 11 . 0 9-6211 3. \\ •3563 59-9599 7 11 • 31 10-0847 3. 24 •3735 62-8491 8 11 . 6? 10-5592 3. 3J •3911 65-8063 9 11 . 9| 11-0447 3. 3f •4091 68-8315 10 12. 01 11-5410 3. 4§ •4274 71-9247 1 11 12. 3g 12-0482 3. 51 •4462 750858 CUBIC YARDS AND IMPERIAL GALLONS. [23 TABLE II. Dia. in feet and inches. Cireum. in feet and Area in square feet. Side of = square in ft. and in. 1 Cubic yards in depth. Gallons at one foot in depth. ft. in. ft. in. ft. in. 4 . 0 12 . 64 12-5664 3. 6i •4654 78-3150 1 12 . 10 130954 3. 7f •4850 81-6121 2 13. li 13-6354 3. 8} •5050 84-9772 3 13. 4? 141863 3. 9| •5254 88-4102 4 13. 7| 14-7480 3. 104 •5462 91-9113 5 13. 10^ 15-3207 3. 11 •5674 95-4803 6 14. If 15-9043 3. 112 •5890 99-1174 7 14. 4 16-4988 4. 0$ '6111 102-8224 8 14. 71 17-1042 4. 18 •6335 106-5954 9 14.114 17-7205 4. 21 '6563 110-4363 10 15. 2} 18-3478 4. 4 •6795 114-3453 11 15. 18-9859 4. 4j •7032 118 3222 5 A 15. 84 19-6350 4. 5i •7272 122-3671 1 15 . Ilf 20-2949 4. 6 •7517 126-4800 2 16. 24 20-9658 4. 7 •7765 130-6609 3 16. 54 21-6475 4. 7£ •8018 134-9097 4 16. 9 22-3402 4. 8| •8274 139-2266 5 17. 0+ 230438 4. 98 •8535 143-6114 6 17- 3f 23-7583 4.10J •8799 1480642 7 17- 64 24-4837 4. Ilf •9068 152-5850 8 17. 98 25-2200 5. 0+ •9341 157 1738 9 18. 0? 25-9672 5. li •9617 161-8305 10 18. 3 i 26-7254 5. 2 •9898 166-5552 11 18. 7 27-4944 5. 2 i 1-0183 171-3480 6 ft. 18. 104 28-2743 5. 3jj 1-0472 176-2086 1 19. 13 290652 5. 4j 1-0765 181-1373 2 19. 44 298669 5. 5| 1-1062 186-1340 3 19. 7f 30-6796 5. 6■ 11363 191-1986 4 19. 10l 31-5032 5. 71 M668 196-3312 5 20 . if 32-3377 5. 8 1-1977 201-5318 6 20. 5 331831 5. 9 1-2290 206-8004 7 20 . 84 34-0394 5. 10 1-2607 212-1370 8 20. 113 34-9066 5. 10* 1-2928 217-5415 9 21. 24 35-7847 5. Ill 1-3254 2230141 10 21. 58 36-6737 6 . Of 1-3583 228-5546 11 21 . 8f 37-5737 6 . li 1-3916 234-1631 7 ft. 21 .llj 38-4846 6 . 2| 1-4254 239-8395 J 1 22 . 3 39-4063 6 . 3? 1-4595 245-5840 2 22. 64 40-3389 6 . 4j 1-4940 251-3964 3 22. 93 41-2825 6 . 54 1-5290 257-2769 4 23. 04 42-2370 6 . 6 1-5643 263-2253 5 23. 33 43-2024 6 . 6* 1-6001 269-2416 6 23. 6| 44-1786 6 . 7| 1-6362 275-3260 . 7 23. 9f 45-1658 6 . 88 1-6728 281-4784 8 24. 1 46T640 6 . 9i 1-7098 287-6987 9 24. 44 47-1730 6 . log 1-7471 293-9870 10 24. 7| 48-1929 6 . Ill 1-7849 300-3433 11 24.104 49-2237 7. 0+ 1-8231 306-7676 24] SIDES OF EQUAL SQUARES! TABLE II. Dia. in fret and inches. Circum. in feet and inches. in square feet. Side of = square ii Cubic yards at one foot in depth. Gallons at one foot in depth. ft. in. 25 . 18 50-2655 ft. in. 7. lh 1-8617 313-2598 1 25 . 4| 51-3181 7-. 2 1-9007 319-8201 2 25 . 7i 52-3817 7. 2 & 1-9401 326-4483 8 25.11 53-4562 7 . 3% 1-9799 333 1445 4 26 . 2£ 545415 7 . 48 2-0201 339-9087 5 26 . 51 55-6378 7 . 54 2-0607 346-7408 6 26 . 8j 56-7450 7 . 62 2-1017 353-6410 7 26.118 57-8631 7. 7l 21431 360-6091 8 27 . 2| 58-9921 7 . 8 l 2-1849 307-6452 9 27 . 4 60 1320 7. 9 a 2-2271 374-7493 10 27 . 9 61-2829 7 . 10 2-2697 381-9214 11 28 . 0i 62-4446 7.107 2-3128 389T614 9 ft. 28. 3i 63-6173 7 . ll| 2-3562 396-4695 28 . 68 64-8008 8 . 08 2-4000 403-8455 2 28 . 98 65-9953 8 . l| 2-4443 411-2895 3 29 . 0| 67-2006 8 . 2 i 2-4889 418-8015 4 29 . 31 68-4169 8. 3} 2-5340 426-3814 5 29 . 7 69-6441 8. 4| 2-5794 434 0294 6 29 . 104 70-8822 8. 0 2-6253 441-7453 7 30 . l| 72-1312 8. 5; 2-6715 449-5292 8 30 . 4| 73-3911 8 . 6- 2-7182 457-3811 9 30 . 78 74-6619 8. 7j 2-7653 465-3010 10 30 . 105 75-9436 8 . 8 : 2-8127 473-2888 11 31 . if 77-2363 8 . 9; 2-8606 481-3446 10/^ 31 . 5 78-5398 8.10; 2-9089 489-4685 31 . 8i 79-8543 8 . 11 2-9576 497-6603 2 31 . Ilf 81-1796 9 . 01 3-0067 505-9200 3 32. 28 82-5159 9 . 1 30561 514-2478 4 32. 5?, 83-8631 9 . 17 3-1060 522-6435 5 32. 8| 85-2212 9 . 2| 31563 53H073- 6 32.114 86-5901 9. 38 3-2070 539-6390 7 33. 3 879700 9. 4 3-2581 548-2387 8 33 . 6i 89-3609 9 . 52 3-3097 556-9063 9 33 . 9} 907626 9. 6| 3-3616 565-6420 10 34 . 08 92-1752 9. 74 3-4139 574-4456 11 34. 3i 93-5987 9. 8| 3-4666 583-3172 11 ft. 34 . 6§ 95-0332 9. 9 3-5197 592-2568 J 1 34. 9£ 96-4785 9 . 97 3-5733 601-2644 2 35. 1 979348 9 . 104 3-6272 610-3400 3 35 . 4i 99-4020 9 . 118 3-6816 619-4835 4 35 . 7i 1008800 10. 04 3-7363 628-6951 5 35 . 108 102-3690 10. 13 37914 637-9746 6 36 . ll 1038689 10 . 24 3-8470 647-3220 7 36 . 48 105 3797 10. 3f 3-9030 656-7375 8 36 . 7 i 106-9014 10 . 4 3-9593 666-2210 9 36 . 11 108-4340 10 . 5 4-0161 675-7724 10 37. 24 109-9776 10. 57 4-0732 685-3918 11 37. 5l 111-5320 10. 63 4-1308 695-0792 CUBIC YARDS AND IMPERIAL GALLONS. [25 TABLE II. Dia. in feet and Circum. in feet and inches. in square feet. Side of = square in [Cubic yards in depth. Gallons at one foot in depth. ft. in. ft. in. ft. in. 12 . 0 37. 88 113-0973 10. 7| 4-1888 704-8346 1 37 .111 114-6736 10 . 81 4-2472 714-6579 2 38. 20 116-2607 10 . 93 4-3060 724-5493 3 38. 50 117 8588 10 . 10? 4-3651 734-5086 4 38 . 9 119-4678 10 .110 4-4247 744-5359 5 39 . 00 121 0877 11 . 0 4-4847 754-6312 6 39 . 3? 122-7185 11 • 00 4"5451 764-7945 7 39 . 68 124-3602 11 • 10 4-6059 775-0257 8 39 . 90 126-0128 11 . 2f 4-6671 785-3250 9 40 . Off 127-6763 11 • 3§ 4-7288 795-6922 10 40 . 34 129-3507 11 . 40 4-7908 806-1274 11 40 . 7 131-0360 11 • 50 4-8532 816-6305 13/it. 40 . 100 132-7323 11 . 6| 4-9160 827-2017 1 41 . 4 134-4394 U • 70 4-9792 837-8408 2 41 . 48 136-1575 11 . 8 50429 848-5480 3 41 • 70 137 8865 11 • 80 5-1069 859-3231 4 41 . 100 139-6263 11 . ol- 51713 870 1662 5 42 . 14 141-3771 ii . 100 5-2362 881-0772 6 42 . 5 143 1388 11 • u§ 5-3014 892-0563 7 42 . 80 144-9114 12 . OA 53671 9031033 8 42 . ll| 146-6949 12 . if 5-4331 914-2183 9 43 . 28 148-4893 12 . 24 5-4996 925-4013 10 43 . 50 150-2947 12. 30 5-5665 936-6523 11 43. 8ff 152-1109 12 . 4 5-6337 9479713 14/it. 43 . Ill 153-9380 12. 40 5-7014 959-3582 J 1 44 . 20 155-7761 12. 5| 5-7695 9708131 2 44 . 60 157-6250 12. 68 5-8380 982-3360 3 44 . 9| 159-4849 12. 70 5-9068 993-9269 4 45. 00 161-3557 12. 88 5-9761 1005-5858 5 45: 30 163-2374 12 . 4 60458 1017-3126 6 45 . 60 165-1300 12 . 10? 6-1159 1029-1074 7 45 . 9| 167-0335 12.1U 6T864 1040-9703 8 46 . 0J 168-9479 13 . 0 6-2573 1052-9011 9 46. 4 170-8732 13 . 00 6-3286 1064-8998 10 46. 7) 172-8094 13. 1£ 6-4003 10769666 11 46 . 108 174-7565 13. 28 6-4725 1089 1013 15/. 47 • 4 176-7146 ,13 . 30 6-5450 1101-3040 7 1 47 . 48 178-6835 13. 4| 6-6179 1113-5747 2 47 . 71 180-6634 13. 5| 6-6912 1125-9134 3 47 . lof 182-6542 13 . 60 6-7650 1138-3201 4 48 . 2 184-6558 13. 70 6-8391 1150-7947 5 48 . 51 186-6634 13 . 8 6-9136 1163-3374 6 48 . 88 188-6919 13 . 80 6-9886 1175-9480 7 48 . Ill 190-7263 13 . 9§ 70639 1188-6266 8 49 . 2ff 192-7716 13 . 108 7-1397 1201-3732 9 49 . 5| 194-8278 13 . 110 7-2158 1214-1877 10 49 . 8| 196-8950 14 . Of 7-2924 1227-0702 11 50 . 0 198-9730 14 . 1? 7 3694 1240-0208 26] SIDES OF EQUAL SQUARES t TABLE II. Dia. in Circnm. in Area Side Of 5= Cubic yards Gallons at feet and feet square in at one foot one foot inches. inches. feet. ft. and in. in depth. in depth. 16 . *0 ft- 50. 3£ 201-0619 3 7-4467 12530393 1 50 . 4 203T618 14. 3 7-5245 1266-1258 2 50. 9* 205-2725 14. 7-6027 1279-2802 3 51 . Off 207-3942 14. 4 7-6813 1292-5027 4 51 . 209-5268 14. 5if 7-7603 1305-7931 5 51 6* 211-6703 14. 6S 7-8396 1319-1515 6 51 . 10 213-8246 14. 7J 79194 1332-5779 7 52 U 215-9899 14 . 84 7-9996 1346-0723 8 52. 4§ 218-1662 14. $ 80802 1359 6346 9 52. 7® 220-3533 14. io| 8-1612 1373-2650 10 52. lOff 222-5513 14. 11 8-2426 1386-9633 11 53. ll 224-7602 14. n* 8-3445 1400-7296 17 ft. 53. 4 226-9801 15. ol 8-4067 1414-5639 53. 8 229-2108 15. ij 8-4893 1428-4661 2 53. lli 231-4525 15. 8-5723 1442-4364 3 54 2| 233-7050 15. 4 8-6557 1456-4746 4 54 . H 235-9685 15. 4| 8-7396 1470-5808 5 54. 238-2429 15. 54 8-8238 1484-7550 6 54. ill 240-5282 15 . 8-9085 1498-9972 7 55 2£ 242-8244 15. 7 8-9935 1513-3073 8 55'. 6 245T315 15. 7 i 90789 1527-6855 9 55. 9i 247-4495 15. K 91648 1542-1316 10 56. 0} 249-7784 15 . !>§ 9-2511 1556-6457 11 56 . 3^ 2521183 15. 104 9-3377 1571-2278 18/i. 56. 254 4690 15. Ilf 9-4248 1585-8778 1 56 . 9| 256-8307 16. 01 9-5122 1600-5959 2 57. o| 259-2032 16. H 9-6001 1615-3819 3 57. 4 261-5867 16. 24 9-6884 1630-2359 4 57. 7* 263-9810 16. 3 9-7771 1645-1579 5 57. 10 1 266-3863 16 . 9-8662 1660-1479 6 58. 1# 268-8025 16. 9-9556 1675-2058 7 58. 48 271-2296 16. 100455 1690-3318 8 58. , 7| 273-6676 16 . 64 101358 1705-5257 9 58. , 10J 276-1165 16 , 74 10-2265 1720-7876 10 59. , 2 2785764 16, 8 ? 10-3176 1736-1175 11 59. 281-0471 16 , 9| 10-4092 1751-5153 19/!!. 59 , , 283-5287 16 . 10 10-5011 1766-9812 1 59 . Ill 2860213 16, . 11 10-5934 1782-5150 2 60, , 2§ 288-5247 16 , . Hf 10-6861 17981168 3 60 , • fit 291-0391 17, • 0 i 10-7792 1813-7866 4 60, 293-5644 17 , ■ 1§ 10 8728 1829-5244 5 61, . 0 296-1106 17, . 24 10-9667 1845-3301 6 61, ■ 298-6477 17 . 3f 110610 1861-2038 7 61 ■ G 1 301-2057 17 ■ n 111558 1877 1456 8 61 • 9$ 303-7746 17 ■ 4 11-2509 1893-1553 \ 9 62 . oi 306-3544 17 . 6 11-3465 1909-2329 ( 10 62 . 3g 308-9451 17 . 6i 11-4424 1925-3786 1 11 62 • H 311-5467 17 . 7| 11-5388 1941-5922 CUBIC YARDS AND IMPERIAL GALLONS. [27 TABLE II. Dia. in feet and Circum. in feat and inches. in square Sect. Side of — square in ft. and in. (Cubic yards at one foot in depth. Gallons at one foot in depth. ft. in. 20 . 0 ft. in. 62.10 3141593 ft. 5 11-6355 1957-8739 1 63. 1* 316-7827 17. 9i | 11-7327 1974-2235 2 63. 4+ 319-4171 17 . 10: i 11-8303 1990-6411 3 63. 7g 322-0623 17. Ill 11-9282 2007-1266 4 63 . ioI 324-7185 18. 0- 12-0266 2023-6802 5 64. If 327-3856 18. 1; 121254 2040-3017 6 64. 4l 330-0636 18. 2 12-2246 2056-9912 .7 64. 8 332-7525 18. 2 { 12-3242 2073-7487 8 64. lli 335-4523 18. 3j 12-4242 2090-5742 9 6.5. 2} 338-1630 18. 4S 12-5246 2107-4677 10 65. 5$ 340-8846 18. 5j 12-6254 2124-4291 11 65 . 8^ 343-6172 18. 6] 12-7266 2141-4585 21 ft. 65 . 118 346-3606 18. 7 i 12-8282 2158-5559 66. 349-1149 18. 8i 12-9302 2175-7213 2 66. 6 351-8802 18. 9A 13-0326 2192-9547* 3 66. 9i 354-6564 18. 10 131354 2210-2560 4 67. 04 357 4434 18 . 103 13-2386 2227-6254 5 67. 38 360-2414 18. lli \ 13-3423 2245-0627 6 67. 6J 3630503 19. OS 13-4463 2262-5680 7 67. 9g 365-8701 19. L 13-5507 2280-1413 8 68. 0i 3687008 19. % 13-6556 2297-7825 9 68. 4 371-5424 19. 3] 137608 2315-4918 10 68. 74 374-3949 19. 4 13-8665 2333-2690 11 68 . 10j 3772584 19. 51 13-9725 2351-1142 22 ft. 69. If 380-1327 19. 6 14-0790 2369-0274 69. 4 3830180 19. 63 141859 2387-0085 2 69. 78 385-9141 19. 73 14-2931 24050577 3 69. 10} 388-8212 19. 81 14-4008 2423-1748 4 70. 1} 391-7392 19. 9, 14-5089 2441-3599 5 70. 5A 394-6680 19 . 10| 14-6173 2459-6130 6 70. 84 397-6078 19 . 11 14-7262 2477-9341 7 70. Ill 400-5585 20. 0 14-8355 2496-3232 8 71. 4 403-5201 20. 1 14-9452 2514-7802 9 7 1 . 58 406-4926 20. 2 150553 2533-3052 10 71. 8} 409-4761 20. 23 \ 151658 2551-8982 11 72. 0 412-4704 20. 3 [ 15-2767 2570-5592 23 ft. 72. 3A 415-4756 20. 4s 15-3880 2589-2882 72. 64 4184918 20. 5t [ 15-4997 2608-0851 2 72. 98 4215188 20. G> 15-6118 2626-9501 3 73. 04 424-5568 20. 7 15-7243 2645-8830 4 73. 3| 427-6057 20. 8 15-8372 2664-8839 5 73. 6} 430-6654 20. 9 15-9506 2683-9527 6 73. 4 4337361 20 . 1(M \ 16-0643 2703-0896 7 74. ll 436-8177 20.10s 161784 2722-2944 8 74. 4+ 439-9102 20 . 11; 16-2930 2741-5673 9 74. 7| 4430137 21. 01 16-4079 2760-9081 10 74 . lOi 446-1280 21. h 16-5233 2780-3168 11 75. lg 449-2532 21 . 2\ 16-6390 2799-7936 281 SIDES EQUAL SQUARES! TABLE II. Dia. in feet and inches. Circum. in feet and inches. in square feet. Side of = square in ft. and in. Cubic yards at one foot Gallons at one foot &. *0 ft. in 75 . 4f 452-3893 ft. in. 21 . 31 16 7552 2819-3384 1 75. 7i 455-5364 21 . 4A 16-8717 2838-9511 2 75 . 11 458-6943 21 . 5 16-9887 2858-6318 3 76. 2.1 461-8632 21 . St 17-1060 2878-3805 4 76 . 5ft 465-0430 21.. 6| 17-2238 2898-1972 5 76 . 81 468-2337 21 . 7« 17-3420 ■29180818 6 76 . Ilf 471-4352 21 . 8J 17-4606 2938-0345 7 77 . 21 474-6477 21 . 9g 17-5795 2958-0551 8 77 . 5g 477 8711 21 . log 17-6989 2978-1437 9 77 . 9 481-1055 21 . 114 178187 29983003 10 78. 01 484-3507 22 . 04 179389 3018-5248 11 78 . 31 487-6068 22 . 1 18-0595 3038-8174 25 A : 78 . 61 490-8739 22 . U 18T805 3059T779 78 . 9| 494 1518 22 . 2l 18-3019 3079-6064 2 79. 0l 497-4407 22 . 3ft 18-4237 3100T029 3 79 . 3f 500-7404 22 . 44 18-5459 3120-6674 4 79 . 7 504-0511 22. 53 18-6686 3141-2998 5 79 101 80 . 11 507-3727 22. 6| 18-7916 31620003 6 510-7052 22. 74 18-9150 3182-7687 7 80 . 41 514-0486 22. 84 190388 3203-6051 8 80 . 78 80 . ioi 517-4029 22 9 19-1631 3224-5095 9 520-7681 22. 9J 19-2877 3245-4818 10 81 1* 524-1442 22.10| 19-4127 3266-5222 11 81 ! 5 527-5312 22 . lift 19-5382 32876305 81 . 84 530-9292 23. 0| 19-6640 3308-8068 81 . Ill 534-3380 23. 13 19-7903 33300511 2 82 . 21 537-7578 23. 24 19-9170 3351-3634 • 3 82 . 5| 541-1884 23. 31 20-0440 33727436 4 82 . 8§ 544-6300 23. 4 201715 3394-1919 5 82 . Hi 5480825 23. 4i 20-2994 3415-7081 6 83. 3 551-5459 23. 51 20-4276 3437-2923 7 83. 6i 555 0202 23. 61 20-5563 3458-9445 8 83. 9| 558-5054 23. 7§ 20-6854 3480-6646 9 84 . 0.4 562-0015 23. 84 20-8149 3502-4528 10 84 . 3| 565-5085 23 . 93 20-9448 3524-3089 11 84 . 6f 5690264 23 . 104 21-0751 3546-2330 27 A j 84 . 9 § 572-5553 23 . 114 21-2058 3568-2251 85. 1 576-0950 24 . 0 21-3369 3590-2852 2 85. 44 579-6457 24. 04 21-4684 3612-4132 3 85. 7$ 583-2072 24. l| 21-6003 3634-6093 4 85 . 104 586-7797 24 . 2ft 21-7326 3656-8733 5 86 . l| 590-3631 24 . 3ft 21-8653 3679-2053 6 86 . 4f 593-9574 24 . 44 21;9984 3701 6053 7 86 . 71 597-5626 24 . 53 22-1319 37240732 8 86 . 11 601-1787 24 . 64 22-2659 3746-6092 9 87 . 24 604-8057 24 . 74 22-4002 ■ 3769 2131 • 10 87 . 5| 608-4436 24 . 8 22-5349 3791-8850 11 87. 8|_ 6120921 24 . 8£ 22-6701 3814-6249 CUBIC YARD® AND IMPERIAL GALLONS. TABLE II. [29 Dia. in feet and Circum. /ee* and inches. in square feet. Side of = square in ft. and in. Cubic yards in depth. Gallons at one foot in depth. ft- in. 28 . 0 ft. in 87 . llg 615-7522 ft. in. 24 . 9$ 22-8056 3837-4328 1 88 . 2; 619-4228 24 . lOg 22-9416 3860 3086 2 88 . 5| 6231044 24 . Ill 230779 3883-2525 3 88 . 9 626-7968 25 . 0l 23-2147 3906-2643 4 89. Oi 630-5002 25 . if 23-3519 3929-3441 5 89 . 3.1 634-2145 25 . 24 23-4894 3952-4918 6 89 . 6g 637-9397 25 . 34 23-6274 3975-7076 7 89 . 9j 641-6758 25 . 4 237658 3998-9914 8 90 . 0; 645-4228 25 . 4f 23-9045 4022-3431 9 90 . 3i 649-1807 25 . 4 240437 4045-7628 10 90 . 7 652-9495 25 . 6| 24 1833 4069-2505 11 90 . 10; 656-7292 25 . 74 24-3233 4092-8061 29 A 91 . 1; 660-5199 25 . 8$ 24-4637 4116-4298 91 . 4s 664-3214 25 . 94 24-6045 41401214 2 91 . 7 668 1339 25 . 104 247457 4163-8810 3 91 . 10; 671-9572 25 . Hi 24-8873 41877086 4 92 . 1^ 675-7915 26 . 0 25-0293 4211-6042 5 92 . 5 679-6367 26 . : 0205 0203 0202 0199 0197 0195 0194 0191 0190 (189 0186 0185 0184 0 J 81 0181 0178 0178 0176 0174 1173 0172 0170 0170 0167 OF NUMBEBS. . R.s. Diff. C. Res. _ 9-4340 9-4868 9-5394 9-5917 9- 6437 9-6954 9- 7468 97 - 9-8995 9-9499 100000 10- 0499 100995 101489 10-1980 10-2470 10-2956 10-3441 10-3923 10-4403 10-4881 10-5357 10-5830 10-6301 10-6771 10-7238 10-7703 10- 8167 10-8628 10-9087 10- 9545 11 - 0000 11-0454 11 0905 1M355 11-1803 11-2250 11 --'694 11-3137 11-3578 11-4018 131 11-4455 132 11-4891 133 11-5326 134 11-5758 135 11-6190 136 11-6619 137 11-7047 •0528 •0526 •0520 •0517 0514 •0512 •0509 •0506 •0504 •0501 •0499 ■0496 •0494 •0491 •0490 •0486 •0485 •0482 •0480 •0478 •0476 •0473 •0471 •0470 •0467 •0465 •0464 •0461 •045& •0458 •0455 •0454 •0451 •0450 •0448 •0444 •0443 ■0441 •0440 •0437 •0436 ■0435 •0432 4-4647 4-4814 4-4979 4-5144 4-5307 4-5468 4 5629 4-5789 4-5947 4-6104 4-6261 4-6416 4-6570 4-6723 4-6875 4-7027 4-7177 4-7326 4-7475 4-7622 4-7769 4-7914 4-8059 4-8203 4-8346 4-8488 4-8629 4-8770 4 8910 4-9049 4-9187 4-9324 4-9461 4-9597 4-9732 4-9866 50000 as?«s 5-0265 50397 50528 5-0658 50788 5-0916 51045 5-1172 0432 51299 •0429 3 njop o 1426 0428 5-1551 Diff. No. Sq. Rts. Diff. C. Rts. •0167 •0165 •0165 •0163 •0161 0161 •0160 •0158 •0157 •0157 •0155 •0154 •0153 •0152 -015-2 •0150 •0149 •0149 •0147 •0147 •0145 •0145 ■0144 •0143 •0142 •0141 •0141 •0140 '013.9 •0138 •0137 -0157 •0130 •0135 •0134 •0134 0133 •0132 •0132 •0131 ■0130 •0130 •0128 •0129 •0127 •0127 •01271 •0125' 11-7047 11-7473 11-7898 11-8322 11-8743 11-9164 11- 9583 120000 120416 120830 12- 1244 121655 12-2066 12-2474 12-2882 12-3288 12-3693 12-4097 12-4499 12-4900 12-5300 12-5698 12-6095 12-6491 12-6886 12-7279 12-7671 12-8062 12-8452 12-8841 12-9228 12- 9615 130000 130384 13- 0767 13-1149 131529 131909 13-2665 13-3041 13-3417 13-3791 13-4164 13-4536 13-4907 15-5277 13-5647 13-6015 ■0426 •0425 •0424 •0421 •0421 ■0419 •0417 •0416 •0414 •0414 ■0411 •0411 •0408 ■0408 •0406 0405 •0404 •0402 •0401 •0400 •0398 ■0397 ■0396 •0395 •0393 ■0392 •0391 •0390 •0389 •0387 •0387 •0385 •0384 0383 •0382 •0380 •0380 •0379 •0377 •0376 •0376 •0374 •0373 •0372 ••0371 •0370 •0370 5-1551 5-1676 51801 5-1925 5-2048 5-2171 5-2293 5-2415 5-2536 5-2656 5-2776 5-2896 5-3015 5-3133 5-3251 5-3368 5-3485 5-3601 5-3717 5-3832 5-3947 5-4061 5-4175 5-4288 5-4401 5-4514 5-4626 5-4737 5-4848 5-4959 5-5069 5-5178 5-5288 5-5397 5-5505 5-5613 5-5721 5-5828 5-5934 5-6041 5-6147 5-62.V2 5-6357 5-6462 5-6567 5-6671 5-6774 5-6877 L o 40 ] SQUARE AND CUBE ROOTS TABLE III. No- 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 Sq. Rts. Diff. C. Rts. Diff. No. 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 Sq. Rts. Diff. C. Rts. Diff. •0088 0088 •0087 0088 •0087 •0086 •0087 •0086 •0086 •0086 •0085 0085 0085 •0085 •0085 •0084 •0084 ■0084 0084 •0083 •0083 •0083 0083 •0083 •0082 0082 •0082 •0082 •0081 0082 •0081 0081 •0080 •0081 •0080 •0080 0080 •0080 •0079 •0080 •0079 •0079 •0078 •0079 •0078 •0078 •0078 •0078 13-6015 13-6382 13-6748 13-7113 13-7477 13-7840 13-8203 13-8564 13-8924 13-9284 13- 9642 140000 14- 0357 14-0712 14-1067 14-1421 14-1774 14-2127 14-2478 14-2829 14-3178 14-3527 14-3875 14-4222 14-4568 14-4914 14-5258 14-5602 14-5945 14-6287 14-6629 14-6969 14-7309 14-7648 14-7986 14-8324 14-8661 14-8997 14-9332 14- 9666 15- 0000 15 0333 15-0665 150997 151327 15-1658 15-1987 15-2315 15-2643 •0367 •0366 •0365 •0364 •0363 •0363 •0361 •0360 •0360 •0358 •0358 •0357 •0355 •0355 •0354 0353 •0353 •0351 •0351 •0349 •0349 •0348 ■0347 •0346 •0346 •0344 •0344 •0343 •0342 •0342 •0340 •0340 0339 0338 0338 •0337 •0336 •0335 0334 •0334 0333 •0332 •0332 •0330 0331 •0329 0328 •0328 5-6980 5-7083 5-7185 5-7287 5-7388 5-7489 5-7590 5-7690 5*7790 5-7890 5-7989 5-8088 5-8186 5-8285 5-8383 5-8480 5-8578 5:8675 5-8771 5-8868 5-8964 5-9059 5-9155 5-9250 5-9345 5-9439 5-9533 5-9627 5-9721 5-9814 5- 9907 6- 0000 6-0092 60185 6-0277 6-0363 6-0459 6-0550 6-0641 6 0732 60822 6-0912 6-1002 61091 61180 6-1269 6-1358 6-1446 6-1534 •0103 0102 •0102 •0101 •0101 •0101 •0100 •0100 •0100 •0099 •0099 •0098 •0099 •0098 •0097 •0098 •0097 •0096 •0097 •0096 •0095 •0096 •0095 •0095 •0094 •0094 •0094 •0094 •0093 •0093 •0093 •0092 •0093 •0092 ■0091 •0091 •0091 •0091 •0091 •0090 •0090 •0090 •0089 ■0089 •0089 •0089 ■0088 •0088 15-2643 15-2971 15-3297 15-3623 15-3948 15-4272 15-4596 15-4919 15-5242 15-5563 15-5885 15-6205 15-6525 15-6844 15-7162 15-7480 15-7797 15-8114 15-8430 15-8745 15-9060 15-9374 15- 9687 16- 0000 16-0312 16-0624 16-0935 16-1245 16-1555 16-1864 16-2173 16-2481 16-2788 16-3095 16-3401 16-3707 16-4012 16-4317 16-4621 16-4924 16-5227 16-5529 16-5831 16-6132 16-6433 16-6733 16-7033 16-7332 16-7631 •0328 ■0326 •0326 ■0325 •0324 •0324 •0323 •032;; •0321 •0322 •0320 •0820 •0319 •0318 •0318 •0317 •0317 •0316 •0315 •0315 •0314 •0313 •0313 •0312 ■0312 •0311 •0310 0310 •0309 •0309 •0308 •0307 •0307 •0306 •0306 •0305 •0305 •0304 •0303 •0303 •0302 •0302 •0301 ■0301 ■(131)0 •0300 •0299 •0299 6*1534 6 1622 6-1710 6-1797 6-1885 6-1972 6-2058 6-2145 6-2231 6-2317 6-2403 6-2488 6-2573 6-2658 6-2743 6-2828 6-2912 6-2996 6-3080 6-3164 6-3247 6-3330 6-3413 6-3496 6-3579 6-3661 6-3743 6-3825 6-3907 6-3988 6-4070 6-4151 6-4232 6-4312 6-4393 6-4473 6-4553 6-4633 6-4713 6-4792 6-4872 6-4951 6-5030 6-5108 6-5187 6-5265 6-5343 6-5421 6-5499 OF NUMBERS. [41 TABLE III. No. Sq. Rts. Diff. C. Rts. Diff. No. Sq. Rts. Diff. C. Rts. Diff. 281 16-7631 •0298 6-5499 •0078 329 18-1384 6-9034 •0070 282 16-7929 6-5577 330 18-1659 •0275 6-9104 283 16-8226 6-5654 •0077 331 181934 •0275 6-9174 •0070 284 16-8523 6-5731 ■0077 332 18-2209 •0275 6-9244 •0070 285 16-8819 •0296 6-5808 •0077 333 182483 •0274 6-9313 •0069 286 16-9115 •0296 6-5885 •0077 334 18-2757 •0274 6-9382 •0069 087 16-9411 6-5962 0077 335 18-3030 •0273 6-9451 •0069 288 16-9706 6-6039 •0077 336 18-3303 *0273 6-9521 •0070 289 170000 6-6115 •0076 337 18-3576 •0273 6-9589 •0068 290 17-0294 6-6191 •0076 338 18-3848 ■0272 6-9658 •0069 291 17-0587 •0293 6-6267 •0076 1339 18-4120 •0272 6-9727 •0069 292 17 0880 6-6343 •0076 340 18-4391 "0271 6-9795 •0068 293 17-1172 •0292 6-6419 •0076 3-11 18-4662 •0271 6-9864 0069 294 17-1464 •0292 6-6494 •0075 342 18-4932 •0270 6-9932 ■0068 295 17-1756 6-6569 •0075 343 18-5203 •0271 70000 •0068 296 17-2047 0291 6-6644 •0075 344 18-5472 •0269 7-0068 •0068 '-“'7 17-2337 66719 •0075 345 18-5742 •0270 70136 •0068 298 17-2627 6-6794 •0075 346 186011 •0269 7 0203 •0067 299 17-2916 •0289 6-6869 •0075 347 18-6279 •0268 7-0271 •0068 300 17-3205 •0289 6-6943 •0074 348 18-6548 •0269 7-0338 "0067 .711 17-3494 0289 6-7018 •0075 349 18-6815 •0267 I 7-0406 •0068 302 17-3781 "0287 6-7092 •0074 350 18-7083 •0268 70473 •0067 303 17-4069 •0288 6-7166 •0074 351 18-7350 •0267 70540 •0067 304 17-4356 •0287 6-7240 •0074 352 187617 •0267 70607 •0067 305 17-4642 0286 6-7313 •0073 353 18-7883 •02661 7-0674 •0067 306 17-4929 •0287 6-7387 •0074 354 18-8149 •0266 7-0740 •0066 307 17-5214 •0285 6-7460 •0073 355 188414 •0265 70807 •0067 308 17-5499 •0285 6-7533 •0073 356 18-8680 •0266 70873 •0066 309 17-5784 "0285 6-7606 •0073 357 18-8944 •0264 7-0940 •0067 310 17-6068 ■0284 6-7679 •0073 358 18-9209 •0265, 7-1006 •0066 311 17-6352 "0284 6-7752 •0073 359 189473 •0264 7-1072 •0066 312 17-6635 •0283 6-7824 •0072 360 18-9737 •0264 71138 0066 313 17-6918 •0283 6-7897 0073 361 190000 •0263 7-1204 •0066 314 17-7200 •0282 6-7969 •0072 362 190263 •0262 7-1269 •0065 315 17-7482 •0282 6 8041 •0072 363 19-0526 *0263 7-1335 •0066 316 177764 •0282 6-8113 •0072 364 19-0788 *0262 7-1400 •0065 317 17-8045 ■0281 6-8185 -0072 365 19-1050 •0262 71466 0066 318 17-8326 •0281 6-8256 •0071 366 191311 •0261 71531 *0065 ;;i:< 178606 •0280 6-8328 •0072 367 19-1572 ■0261 7-1596 •0065 320 17-8885 •0279 6-8399 •0071 368 191833 •026L 7-1661 •0065 321 17-9165 •0280 6-8470 •0071 369 192094 •0261 7-1726 0065 322 17-9444 •0279 6-8541 ■0071 370 19-2354 •02601 7-1791 •0065 17-9722 •0278 6-8612 •0071 371 19-2614 •0260 71855 •0064 324 18-0000 •0278 6-8683 *0071 372 19-2873 •0259 7-1920 "0065 325 18-0278 0278 6-8753 •0070 373 19-3132 •0259 7-1984 •0064 326 180555 •0277 6-8824 "0071 374 19-3391 •02591 7-2048 •0064 327 180831 0276 6-8894 •0070 375 19-3649 •0258 7-2112 •0064 328 181108 *0277 6-8964 •0070 376 193907 •0258 7-2177 0065 829 181384 •0276 6-9034 •0070 377 19-4165 •0258 7-2240 0063 42j SQUARE AND CUBE ROOTS TABLE III. q. Rts. 19-4165 19-4422 19-4679 19-4936 19-5192 19-5448 19-5704 19-5959 19-6214 19-6469 19-6723 19-6977 19-7231 19-7484 19-7737 19-7990 19-8242 19-8494 19-8746 19-8997 19-9249 19-9499 19- 9750 200000 200250 200499 20- 0749 200998 201246 20-1494 20-1742 20-1990 20-2237 20-2485 20 2731 20-2978 20-3224 20-3470 20-3715 20-3961 20-4206 20-4450 20-4695 20*4939 20-5183 20-5426 20-5670 20-5913 20-6155 C. Rts., Diff. No. Sq. Rts.] Diff. C. Rts. Diff. •0257 0257 •0257 •0256 ■0256 •0256 '0255 ■0255 •0255 ■0254 •0254 *0254 •0253 0253 •0253 •0252 •0252 -0252 •0251 •0252 •0250 -0251 0250 •0250 *0249 •0250 •0249 •0248 •0248 •0248 0248 0247 0248 0246 0247 0246 0246 0245 0246 0245 0244 0245 0244 0244 0243 0244 0243 0242 72240 7-2304 7-2368 7-2432 7-2495 7-2558 7-2622 7-2685 7-2748 7-2811 7-2874' 7-2936! 7-2999 7-3061 7-3124 7-3186 7-3248 7-3310 7-3372 7-3434 7-3496 7-3558 7-3619 7-3681 7-3742 7-3803 7-3864 7-3925 7-3986 7-4047 7-4108 7-4169 7-4229 7-4290 7-4350 7-4410 7-4470 7-4530 7-4590 7-4650 7-4710 7-4770 7-4829 7-4889 7-4948 7-5007 7-5067 7-5126 7-5185 •0064 •0064 •0064 •0063 •0063 •0064 •0063 •0063 •0063 •0063 •0062 •0063 •0062 •0063 •0062 •0062 •0062 •0062 •0062 •0062 •0062 •0061 •0062 •0061 •0061 •0061 •0061 •0061 •0061 •0061 •0061 •0060 •0061 •0060 •0060 •0060 •0060 •0060 •0060 •0060 •0060 •0059 •0060 •0059 •0059 •0060 •0059 •0059 20*6155 20-6398 20-6640 20-6882 20-7123 20-7364 20-7605 20-7846 20-8087 20-8327 20 8567 20-8806 20-9045 20-9284 20-9523 20- 9762 21 - 0000 21 0238 21-0476 21-0713 21-0950 211187 21-1424 21-1660 21-1896 21-2132 21-2368 21-2603 21-2838 21-3073 21-3307 21-3542 21-3776 21-4009 21-4243 21-4476 21-4709 21-4942 21-5174 21-5407 21-5639 21-5870 21-6102 21-6333 21-6564 21-6795 21-7025 21-7256 21-7486 •0235 •0235 •0235 •0234 •0235 •0234 •0233 •0234 •0233 •0233 •0233 •0232 •0233 •0232 •0231 •0232 •0231 •0231 •0231 •0230 •0231 •0230 75185 7-5244 7-5302 7-5361 7-5420 7-5478 7-5537 7-5595 7-5654 7-5712 7-5770 7-5828 7 1 7-5944 7-6001 7-6059 7-6117 7-6174 7-6232 7-6289 7-6346 7-6403 7-6460 7-6517 7-6574 7-6631 7-6688 7-6744 7-6801 7-6857 7-6914 7-6970 7-7026 7-7082 7-7138 7-7194 7-7250 7-7306 7-7362 7-7418 7-7473 7-7529 7-7584 7-7639 7-7695 7-7750 7-7805 7-7860 7-7915 •0059 ■0058 •0059 •0059 •0058 •0059 0058 •005.0 •0058 •0058 •005!! •005,'i ■005!! •0057 •0058 ■0058 •0057 •0058 •0057 ■0057 ■0057 •0057 0057 •0057 •0057 •0057 •0056 •0057 ■0050 0057 0056 •0056 •0056 •0056 •0056 •0056 •0056 •0056 ■0056 •0055 •0056 0055 •0055 •0056 •0055 •0055 •0055 •0055 OF NUMBERS. [43 473 21-7486 474 21-7715 475 21-7945 476 21-8174 477 21-8403 478 21-8632 479 21-8861 480 *21-9089 21-9317 21-9545 21- 9773 220000 22- 0227 22-0454 22-0681 220907 221133 22-1359 221585 221811 22-2036 22-2261 22-2486 22-2711 22-2935 22-3159 22-3383 22-3607 22-3830 22-4054 22-4277 22-4499 22-4722 22-4944 22-5167 22-5389 22-5610 22-5832 22-6053 22-6274 22-6495 22-6716 22 6936 516 22-7156 517 22-7376 518 227596 519 22-7816 520 22-8035 521 22 8254 •0229 ■11230 •0229 •0229 •0229 •11 •0228 •0228 0228 •0228 •0228 •0227 •0227 ■0227 •0226 •0226 •0226 •0226 •0226 •0225 •0225 0225 ■0225 •0224 •0224 0224 •0224 •0223 •0224 •0223 •0222 ■0223 •0222 0223 •0222 ■0221 •0222 •0221 •0221 •0221 0221 •0220 •0220 ■0220 •0220 •0220 •021!) ■0219 77915 77970 7*8025 78079 78134 78188 7-8243 78297 78352 7-8406 7-8460 78514 7 78622 74 . 78730 7-8784 7-8837 7 f— 7-8944 7 7-9051 7-9105 79158 79211 7-9264 7-9317 79370 7-9423 7-9476 7-9528 7-9581 7-9634 7-! ‘' 7-9739 7-9791 79843 7- 9896 79948 8- 0(100 8-0052 8-0104 8-0156 80208 80260 80311 80363 80415 8-0466 Diff. No. Sq. Rts. Diff. C. Rts., Diff. •0055 0055 •0054 •0055 •0054 ■1)055 •0054 •0055 •0054 •0054 •0054 •0054 •0054 •0054 ■0054 •0054 •0053 •0054 •0053 •0054 •0053 •0054 •0053 •0053 •0053 •0053 •0053 •0053 •0053 •0052 •0053 •0053 •0052 •0053 •0052 •0052 ■0053 •0052 •0052 ■0042 •0052 ■0052 •0052 ■0052 •0041 •0042 •0042 •0051 23-0217 230434 23-0651 230868 23-1084 23-1301 231517 23 1733 23-1948 23-2164 23-2379 23-2594 23-2809 23-3024 23-3238 23-3452 23-3666 233880 234094 23-4307 23-4521 23-4734 23-4947 23-5160 23-5372 23-5584 23-5797 23-6008 23-6220 23-6432 23-6643 23-6854 23-7065 23-7276 23-7487 23-7697 23-7908 ■ 23-8118 23-8328 23-8537 •0219 •0219 •0218 •0219 •0218 •0218 •0218 •0217 •0217 •0217 •0217 ■0216 •0216 •0217 •0216 •0216 ■0215 •0216 •0215 •0215 •0215 •0214 •0214 •0214 •0214 ■0214 •0214 ■0213 ■0214 •0213 •0213 •0213 •0212 •0212 •0213 ■0211 •0212 •0212 •0211 •0211 •0211 •0211 •0211 •0210 •0211 •0210 •0210 •0209 ;r 80876! 8 09271 8-0978! 8-1028 8-1079 81130 8-1180 8-1231 8-1281 81332 8-1382 81433 81483 81533 81583 8-1633 81683 8 1733 81783 8-1833 8-1882 81932 81982 8-2031 8 2081 8-2130 8-2180 8-2229 8-2278 8-2327 8-2377 8-2426 8-2475 8-2524 8-2573 8-2621 8-2670 8-2719 8-2768 8-2816 8-2865 0051 0051 0051 0051 0052 0051 0051 0051 0051 0051 0050 •0051 ■0051 ■0050 •0051 •0050 •0051 •0050 •0051 •0050 •0050 ■0050 •0050 -oo4i) •0050 •0050 •0050 •0049 •0050 •0050 •0049 •0050 •0049 •0050 •0049 •0049 •0049 •0050 •0049 •0049 •0049 •0049 •0048 •0049 •0049 •0049 •0048 ■004!) L 4 44] SQUARE AND CUBE ROOTS TABLE III. No. 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 5:'3 594 595 596 5! 17 598 599 600 601 602 603 604 605 606 i;<>: 608 609 610 611 612 613 614 615 616 617 Sq. Rts. Diff. C. Rts. Diff. No. 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 045 646 647 648 649 650 651 652 653 654 655 65 6 657 658 |659 660 661 662 663 664 665 Sq. Rts. Diff. C. Rts. Diff. 238537 238747 23-8956 23-9165 239374 23-9583 23- 9792 24 0000 240208 24- 0416 24 0624 24-0832 24 1039 24-1247 24-1454 24-1661 24 1868 24-2074 24-2281 24-2487 24-2693 24-2899 24-3105 24-3311 24-3516 24-3721 24-3926 24-4131 24-4336 24-4540 24-4745 24-4949 24-5153 24-5357 24-5561 24-5764 24-5967 24-6171 24-6374 24-6577 24-6779 24-6982 24-7184 24-7386 24-7588 24-7790 24-7992 24-8193 24-8395 •0210 •0209 •0209 •0209 0209 •0209 0208 0208 •0208 •0208 •0208 •0207 •02(18 0207 •02D7 •0207 '0206 •0207 •0206 •0206 0206 •0206 0206 •0205 •0205 0205 •0205 •0205 •0204 •0205 0204 •0204 •0204 •0204 •0203 0203 •0204 •0203 •0203 •0202 •0203 •0202 •0202 •0202 •0202 •0202 •0201 •0202 8-2865 8-2913 0048 8-2962 0049 8-3010 0048 8 3059 -nnlft 8 3107 -004R 8 ' 3155 ' 0048 8 3203 0048 8 32511 .2*8 83300 -0048 8 3348 -S 8 3396 -0047 8 3443 0047 8-34911 -onlft 8-3539 -ools 8-3587: 0048 8-36341 -oola 8-3682 -0048 8-3730 -0047 8-3777 -oola 8-3825 0048 8-3872 -0047 8-3919 0048 8-3967 -004? 8-4014 .828 8-4061 .228 8-4108 8047 8-4155 0047 8-4202 -0047 8-4249 0047 8-4296 0047 8-4343 .228 8-4390 0047 8-4437 -0047 84484 0047 8-4530 .228 8-4577 -228 8-4623 .0047 8-4670 .228 8-4716 -228 8-4768 228 8-4809 228 8-4856 -228 8-490-2 .228 8-4948 -228 8-4994 2212 8-5040 -228 8-5086 -228 8-5132 0046 24-8395 24-8596 24 8797 24-8998 24-9199 249399 24-9600 24- 9800 25- 0000 25 0200 250400 250599 25-0799 25-0998 25 1197 25-1396 25-1595 25-1794 25-1992 25 2190 25-2389 25-2587 25-2784 25-2982 253180 25-3377 25-3574 25-3772 25-3969 25-4165 25-4362 25-4558 25-4754 25-4951 25-5147 25-5343 25-5539 25-5734 25-5930 25-6125 25-6320 25-6515 25-6710 25-6905 25-7099 25-7294 25-7488 25-7682 25-7876 •0201 •0201 1 •0201 1 0201 •0200 ■0200 1 •0200 1 0200 i •0200! •0200 •0199 •0200 •0199 •0199 0199 •0199 •0199 •0198 •0198 •0199 •0198 •0197 •0198 0198 •0197 •0197 •0198 0197 •0196 •0197 •0196 •0196 •0197 •0196 •0196 •0196 •0195 •0196 •0195 •0195 •0195 •0195 •0195 •0194 •0195 •0194 •0194 •0194 8-5132 8-5178 8-5224 8-5270 8-5316 8-5362 8-5408 8-5453 8-5499 8-5544 8-5590 8-5635 8-5681 8-5726 8-5772 8-5817 8-5862 8-5907 8-5952 8-5997 8-6043 8-6088 8-6132 8-6177 8-6222 8-6267 8-6312 8-6357 8-6401 8-6446 8-6490 8-6535 8-6579 8-6624 8-6668 8-6713 8-6757 8-6801 8-6845 8-6890 8-6934 8-6978 8-7022 8-7066 8-7110 : 8-7154 8-7198 8-7241 | 8-7285 •0046 •0046 •0046 •0046 •0046 •0046 T1045 •0046 •0045 •0046 •0045 •0046 •0045 •0046 0045 •0045 •0045 •0045 0045 •0046 •0045 •0044 •0045 •0045 •0045 •0045 ■0045 •0044 •0045 •0044 0045 ■0044 •0045 •0044 •0045 •0044 •0044 •0044 •0045 ■0044 •0044 0044 •0044 •0044 •0044 ■0044 0043 •0044 OF NUMBERS. [45 TABLE III. No. Sq. Rts. Diff. C. Rts. I 25*7876 25-8070 25-8263 25-8457 25-8650 25-8844 25-9037 25-9230 25-9422 25 9615 25- 9808 260000 26- 0192 260384 26-0576 26 0768 26-0960 26-1151 26-1343 26-1534 26-1725 26-1916 26-2107 26-2298 26-2488 26-2679 26-2869 26-3059 26-3249 26-3439 26-3629 26-3818 26-4008 26-4197 26-4386 26-4575 26-4764 26-4953 26-5141 26-5330 26-5518 26-5707 ... 26-5895 708 26-6083 709 26-6271 710 26-6458 711 26-6646 712 26-6833 713 26-7021 8-7285 8-7329 8-7373 8-7416 8-7460 8-7503 8-7547 8-7590 8-7634 8-7677 8-7721 8-7764 8-7807 8-7850 8-7893 8-7937 8-7980 88023 8-8066 8-8109 8-8152 8-8194 8-8237 8-8280 8-8323 8-8366 8-8408 8-8451 8-8493 8-8536 8-8578 8-8621 8-8663 8-8706 8-8748 8-8790 8-8833 8-8875 8-8917 8-8959 8-9001 8-9043 8-9085 8-9127 8-9169 8-9211 8-9253 8-9295 8*9337 Diff. No. Sq. Rts. Diff. 713 26*7021, 714 26-7208 715 26'7395 716 26-7582 717 26-7769 718 26-7955 719 26-8142 720 26-8328 721 26-8514 722 26-8701 723 26-8887 724 26-9072 725 26-9258 26-9444 26-9629 728 26-9815 27-0000 27-0185 27 0370 27-0555 27-0740 27-0924 27-1109 27-1293 27-1477 27-1662 27-1846 27-2029 27-2213 27-2397 27*2580 27-2764 27-2947 273130 27-3313 27-3496 27*3679 27-3861 27-4044 27-4226 27-4408 27-4591 27-4773 27-4955 27-5136 27-5318 27*5500 27-5681 27-5862 0187 •0187 •0187 •0187 •0186 •0187 •0186 ■0186 •0187 •0186 ■0185 •0186 •0186 •0185 •0186 •0185 •0185 0185 •0185 0185 •0184 ■0185 •0184 •0184 •0185 ■0184 •0183 •0184 ■0184 0183 •0184 •0183 •0183 •0183 •0183 •0183 •0182 •0183 •0182 •0182 0183 •0182 •0182 •0181 •0182 0182 •0181 •0181 8-9337 8-9378 8-9420 8-9462 8-9503 8-9545 8-9587 8*9628 8-9670 8-9711 8*9752 8-9794 8-9835 8-9876 8-9918 8- 9959 9- 0000 90041 9-0082 90123 90164 9 0205 9-0246 90287 90328 9 0369 9-0410 9 0450 9-0491 9-0532 9-0572 9-0613 9 0654 90694 90735 90775 9-0816 9-0856 90896 90937 9-0977 9-1017 9-1057 9-1098 9-1138 91178 9-1218 91258 9-1298 Diff. I- ■0041 46] SQUARE AND CUBE ROOTS TABLE III. No. 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 761 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 308 809 Sq. Rts. Diff. C. Rts. j Diff. No. 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 1142 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 Sq. Rts. Diff. C. Rts. Diff. 0038 •0038 •0039 •0038 •0038 ■0038 0039 •0038 •0038 0038 0038 0038 0038 •0038 0038 •0038 •0038 •0038 •0037 0038 0038 •0038 0037 •0038 •0038 •0037 •0038 •0037 •0038 •0037 0038 •0037 •0038 0037 •0037 •0037 •0037 •0037 •0037 ■0038 •0037 0037 0037 •0037 -0037 •0037 0037 0037 275862 276043 276225 27-6405 27 6586 27-6767 276943 277128 27-7308 27-7489 27-7669 27-7849 27-8029 27-8209 27-8388 27-8568 27-8747 27-8927 27-9106 27-9285 27-9464 27-9643 27- 9821 28- 0000 28-0179 280357 28-0535 28-0713 280891 28-1069 28-1247 28-1425 28-1603 28 1780 28-1957 28-2135 28-2312 28-2489 28-2666 28-2843 28 3019 28-3196 28 3373 28-3549 28-3725 28-3901 28-4077 28-4253 28-4429 •0181 0182 •0180 •0181 •0181 0181 0180 •0180 0181 •0180 •0180 •0180 •0180 •0179 •0180 •0179 0180 •0179 •0179 •0179 0179 0178 0179 •0179 ■0178 •0178 •0178 •0178 •0178 •0178 0178 0178 •0177 •0177 •0178 ■0177 •0177 •0177 ■0177 •0176 •0177 •0177 •0176 0176 •0176 •0176 •0176 •0176 9-1298 91338 91378 91418 9-1458 9 1498; 9-1537 9-1577 9-1617 9-1657 9-1696 9-1736 9-1775 9-1815 9-1855 9-1894 9-1933 9-1973 9-2012 9-2052 9-2091 9-2130 9-2170 9-2209 9-2248 9-2287 9-2326 9-2365 9-2404 9-2443 9-2482 9-2521 9-2560 9-2599 9-2638 9-2677 9-2716 9-2754 9-2793 9-2832 9-2870 9-2909 9 2948 92986 | 9-3025 9-3063 9-3102 9-3140 9-3179 •0040 •0040 •0040 •0040 •0040 •0039 •0040 •0040 •0040 •0039 •0040 •0039 0040 •0040 0039 •0039 •0040 •0039 ■0040 •0039 •0039 •0040 0039 •0039 0039 •0039 •0039 0039 ■0039 •0039 •0039 •0039 •0089 •0039 •0039 •0039 •6031! •0059 ■005.0 •0051! •005.0 ■0039 ■005,1! •0039 0038 •0039 ■0031! •0039 284429 28-4605 284781 28 4956 28-5132 28-5307 285482 28-5657 28-5832 28 6007 28-6182 286356 28 6531 28-6705 28 6880 28-7054 •28-7228 28-7402 28-7576 28-7750 287924 28-8097 28-8271 288444 288617 28-8791 288964 289137 289310 28- 9482 28 9655 289828 29- 0000 29-0172 29-0345 290517 29-0689 290861 29 1033 29-1204 29-1376 29 1548 29-1719 29 1890 29-2062 29-2233 29-2404 29-2575 29-2746 •0176 •0176, 0175 •0176 •0175 •0175 •0175 0175 0175 •0175 •0174 0175 •0174 •0175 •0174 •0174 •0174 •0174 •0174 0174 0173 0174 0173 0173 •0174 0173 •0173 •0173 •0172 0173 0173 •0172 •0172 0173 •0172 •0172 •0172 •0172 •0171 •0172 •0172 •0171 0171 0172 0171 0171 0171 0171 9-3179 9-3217 93255 9 3294 9-3332 9 3370 9-3408 9-3447 9-3485 9-3523 9-3561 9-3599 9-3637 9-3675 9-3713 9 3751 9-3789 9-3827 9 3865 9 3902 93940 9 3978 9-4016 9-4053 9-4091 9-4129 ; 9-4166 l 9-4204 9-4241 9-4279 94316 94354 9-4391 9-4429 9-4466 94503 94541 9-4578 ; 9 4615 9-4652 94690 9-4727 94764 | 9-4801 ! 9 4838 9-4875 9-4912 : 9-4949 9-4986 OF NUMBERS. [47 TABLE III. No. Sq. Rts. Diff. C. Rts. I) iff. No. Sq. Rts. Diff. C. Rts. | Diff. 857 858 29-2746 29-2916 •0170 9-4986 95023 •0037 905 906 i 30-0832 30 0998 •0166 9-6727 9-6763 1 -0036 85a 29-3087 9-5060 0037 907 30-1164! •0166 9 6799 •0036 860 29-3258 0171 9-5097 •0037 908 30-1330 •0166 9-6834 l -0035 861 29-3428 9-5134 •0037 909 30-1496 •0166 96870 ' -0036 862 29-3598 95171 -0037 910 30-1662 •0166 9-6905 •0035 868 293769 9-5207 •0036 911 30-1828 •0166 9-6941 ■0036 864 29-3939 9-5244 •0037 912 30-1993 •0165 9-6976 •0035 865 29-4109 9-5281 •0037 913 30-2159 •0166 97012 •0036 866 29 4279 9-5317 •0036 914 30-2324 •0165 9-7047 •0035 867 29-4449 •0170 9-5354 •0037 915 30-2490 "0166 9-7082 •0035 868 29-4618 9-5391 "0037 916 30-2655 ■0165 97118 •0036 86.9 29-4788 .9-5427 •0036 917 30-2820 "0165 9-7153 0035 870 29-4958 9-5464 •0037 918 30-2985 •0165 9-7188 •0035 871 295127 9-5501 •0037 919 30-3150 "0165 9-7224 •0036 872 29-5296 9-5587 •0036 920 30-3315 "0165 9 7259 0035 878 29-5466 9-5574 •0037 921 30-3480 •0165 9-7294 •0035 874 29-5635 9-5610 •0036 922 30-3645 •0165 97329 •0035 875 29-5804 9-5647 •0037 923 30-3809 •0164 9-7364 •0035 876 29-5973 "0169 9-5683 •0036 .924 30-3974 "0165 9-7400 •0036 877 29-6142 "0169 9-5719 •0036 925 30-4138 "0164 9-7435 0035 878 2.9 6311 ‘0169 9-5756 •0037 926 30-4302 "0164 9-7470 ■0035 879 29-6479 "0168 9-5792 •0036 927 30-4467 •0165 9-7505 •0035 880 29-6648 0169 9-5828 •0036 928 30-4631 •0164 9-7540 •0035 881 29 6816 "0168 9-5865 0037 929 30-4795 •0164 9-7575 ■0035 882 29-6985 "0169 9-5901 •0036 930 30-4959 •0164 9-7610 •0035 8831 29-7153 "0168 9-5937 •0036 931 30-5123 •0164 9-7645 ■0035 884 29-7321 9-5973 ■0036 932 30-5287 "0164 9-7680 •0035 885 29-7489 "0168 9-6010 *0037 933 30-5450 •0163 9-7715 •0035 j •886 297658 "0169 9-6046 •0036 934 30-5614 •0164 97750 •0035 887 29-7825 "0167 9-6082 •0036 935 30-5778 "0164 9-7785 •0035 888 2.9-7993 •0168 9 6118 •0036 936 30-5941 "0163 9-7819 0034 889 29-8161 •0168 9-6154 •0036 937 30-6105 •0164 97854 •0035 890 29-8329 0168 9-6190 •0036 938 30-6268 •0163 9-7889 •0035 891 29-8496 0167 9-6226 •0036 939 30-6431 ■0163 97924 0035 892 2.9-8664 •0168 9-6262 •0036 940 30-6594 •0163 97959 •0035 893 29-8831 ■0167 9-6298 •0036 941 30-6757 •0163 9 7993 •0034 894 29-8998 "0167 9-6334 •0036 942 30-6920 •0163 9-8028 •0035 895 29-9166 •0168 9 6370 •0036 943 30-7083 •0163 9-8063 •0035 896 29-9333 •0167 9-6406 •0036 944 30-7246 "0163 9-8097 ■0034: 897 29-9500 "0167 9-6442 •0036 945 30-7409 "0163 9-8132 •0035| 898 29-9666 •0166 9-6477 *0035 946 30-7571 "0162 9-8167 •0035! 899 29-9833 •0167 9-6513 ■0036 947 30-7734 •0163 9-8201 ■0034! 900 300000 0167 9-6549 •0036 948 30-7896 •0162 9-8236 •0035 901 300167 "0167 9-6585 •0036 949 30-8058 •0162 9-8270 •0034 902 300333 0166 9-6620 •0035 950 30-8221 0163 9-8305 •0035 903 30*0500 •0167 9-6656 "0036 951 30-8383 •0162 9-8339 •00341 904 30-0666 "0166 9-6692 •0036 952 30-8545 •0162 9-8374 •0035 905 300832 •0166 9-6727 •0035 953 30-8707 •0162 9-8408 •00341 l6 48] SQUARE AND CUBE ROOTS OF NUMBERS. TABLE III. No. 953 954 955 956 957 958 959 160 ■ Hil 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 Sq. Rts. Diff. C. Rts. Diff. No. 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 990 1000 Sq. Rts. Diff. •0160 •0160 •0160 •0159 •0160 •0159 •0160 •0159 •0159 •016‘0 •0159 •0159 •0159 0159 0158 0159 •0159 0158 •0159 •0158 •0158 M| - M •0158 C. Rts. 1 Diff. 30-8707 30-8869 30-9031 30-9192 30 9354 30-9516 30- 9677 309839 31- 0000 31-0161 31-0322 31 0483 31-0644 31 0805 31-0966 31-1127 31-1288 31-1448 31-1609 31-1769 31-1929 31-2090 31-2250 31-2410 31-2570 •0162 •0162 •0161 •0162 •0162 •0161 •0162 •0161 •0161 •0161 •0161 •0161 •0161 •0161 •0161 0161 •0160 •0161 •0160 •0160 ■0161 •0160 •0160 •0160 9-8408 9-8443 9-8477 9-8511 9-8546 9-8580 9-8614 9-8648 9*8683 9-8717 9-8751 9-8785 9 8819 9*8854 9-8888 9-8922 9-8956 9-8990 9-9024 9-9058 9-9092 9-9126 9-9160 9-9194 9-9227 •0035 •0034 •0034 •0035 •0034 •0034 •0034 •0035 0034 *0034 •0034 •0034 •0035 ■0034 0034 •0034 •0034 ■0034 •0034 •0034 •0034 •0034 •0034 0033 31-2570 31-2730 31-2890 31-3050 31-3209 31-3369 31-3528 31-3688 31-3847 31-4006 31-4166 31 -4325 31-4484 31-4643 31-4802 31-4960 31-5119 31-5278 31-5436 31-5595 31-5753 31-5911 31-6070 31-6228 9-9227 9-9261 9-9295 9-9329 9-9363 9-9396 9-9430 9-9464 9 9497 9-9531 9-9565 9-9598 9-9632 9-9666 9-9699 9-9732 9-9766 9-9800 9-9833 9-9866 9-9900 9-9933 9-9967 100000 l -0034 •0034 •0034 ■0034 •0033 •0034 ■0034 •0033 ■0034 •0034 0033 •0034 •0034 •0033 •0033 •0034 0034 •0033 0033 •0034 •0033 0034 0033 TO FIND BY THE FOREGOING TABLE THE SQUARE OR CUBE ROOT OF ANY NUMBER NOT EXCEEDING 1000. Rule. —Multiply the difference between the root of the integer part of the given number, and the root of the next higher integer number by the decimal part of the given number, and add the product to the root of the integer number given, the sum is the root required. Example 1.—Required the square root of 53'75? Difference by table = *0684 x *75 = -0513; and the root of 53 = 7-2801; hence, 7 2801 + -0513 = 7-3314, the root required. Example 2.—Required the cube root of the number 734-26? Difference by table = -0041 X ’26 = -001066 ; and the root of 734 = 9-0205; hence, 9 0205 + -001066 = 9 021566 or 9-0216, is the root required. TABLE IV. CONTAINING THE SURFACE AND SOLIDITY OF SPHERES, THE EDGE OR SIDE OF EQUAL CUBES, THE LENGTHS OF EQUAL CYLINDERS, AND THE WEIGHT OF EQUAL VOLUMES OF WATER IN LBS. AVOIRDUPOIS. 50] SURFACE AND SOLIDITY OF SPHERES, SIDE OF = CUBES, LENGTH TABLE IV. Dia. in inches. Surface Solidity in cub. in. Cube in inches. ( Cylinder in j inches. Water in lbs. avoir. 0 in. sq. in. cub. in. in. in k 0-0491 o-ooio 01007 00833 00000 i 0-1963 0-0082 0-2015 01667 0*0003 t 0-4418 00276 0-3022 0-2500 0*0010 k 0-7854 00654 0-4030 0-3333 0-0024 I 1-2-272 0-1278 0-5037 0-4167 0-0046 \ 1-7671 0-2209 0-6045 0-5000 00080 i 2-4053 0-3508 0-7024 0-5833 0*0127 1 in. 3-1416 0-5236 0-8060 0-6667 00189 k 39761 0-7455 0-9067 0-7500 00269 i 4-9087 1-0227 1 0075 0 8333 0-0369 1 5-9396 1-3612 1-1082 09167 0-0491 k 7-0686 1-7671 1-2090 1-0000 0-0637 1 8-2958' 2-2468 1-3097 1-0833 00810 1 9-6211 2-8062 1-4105 1-1667 01012 1 11-0447 3-4515 1-5112 1-2500 0-1245 2«. 12-5664 41888 1-6120 1-3333 0-1511 & 14-1863 50243 1-7127 1-4167 0-1812 i 15*9043 5-9641 1-8135 1-5000 0-2151 | 17-7205 7-0144 1-9142 1-5833 0-2530 4 19-6350 8-1812 2-0150 1-6667 0-2951 § 21-6475 9-4708 2-1157 1-7500 0-3416 i 23-7583 10-8892 2-2165 1-8333 0-3927 i 25-9672 12-4426 2-3172 19166 0-4487 3 in. 28-2743 14*1372 2-4180 20000 0-5099 k 30-6795 15-9790 2-5187 20833 0-5778 i 33 1831 17-9742 2-6195 2-1667 0-6482 1 35-7847 201289 2-7202 22500 0-7256 k 38-4845 22-4493 2-8210 2-3333 0-8096 i 41-2825 24-9415 29217 2-4167 0-8995 44-1786 27*6117 30225 25000 0-9958 i 47-1730 304659 3-1232 2-5833 1-0988 4 in. 50-2655 33-5103 3-2240 2-6667 1-2086 4 53-4562 36*7511 3-3247 2-7500 1-3254 i 56-7450 40-1944 3-4255 2-8333 1-4496 I 60-1320 43-8463 3-5262 29166 1-5813 4 63-6173 477129 3-6270 30000 " 1-7208 1 67-2006 51-8005 3-7277 3-0833 1-8682 f 70-8822 56*1151 3*8285 3-1667 20233 i 74-6619 60-66-28 39292 3-2500 2-1878 5 in. 78-5398 65-4498 4-0300 3-3333 2-3605 4 82-5159 70-4823 4-1307 3-4167 2-5420 4 86-5901 75-7664 4*2315 3-5000 2-7325 i 90*7626 81-8081 4-3322 3-5833 29324 4 950332 87-1137 4*4330 3*6667 31418 99-4020 93-1893 4-5337 3-7500 3-3609 103-8689 99-5410 4-6345 38333 3-5900 ■ i 108-4340 106-1750 4-7352 8-9167 3-8280 OF = CYLINDERS, AND WEIGHT OF = VOLUMES OF WATER. [51 TABLE IV. Dia. in inches. Surface Solidity in cub. in. Cube in inches. Cylinder in inches. Water in 16s. 6 in. 113 0973. 1130973 4-8360 4-0000 4-0789 £ 117-8588 120-3142 4-9367 40833 4-3392 k 122-7185 127-8317 5-0375 4-1667 4-6228 1 127-6763 1356561 5-1382 4-2500 4-8922 £ 132-73-23 143-7933 5-2390 4-3333 5-1860 g 1378865 152-2496 5-3397 4-4167 5-4909 1 143-1388 161-0312 5-4405 4-5000 5-8077 £ 148-4893 170-1440 5-5412 4-5833 6-1863 7 in. 153-9380 179-5944 5-6420 4-6667 6-4771 £ 159-4849 189-3883 5-7427 4-7500 6-8304 k 165-1300 199-5320 5-8435 4-8333 7-1962 1 170-8732 2100316 5-9442 4-9167 7-5749 £ 1767146 220-8932 6 0450 5-0000 7-9666 g 182-6542 232-1230 6-1457 5-0833 8-3716 $ 188-6919 243-7270 6-2465 5-1667 8-7901 £ 194-8278 255-7115 6-3472 5-2500 9-2223 8 in. 201 0619 268-0826 6-4480 5-3333 9-6685 £ 207-3942 280-8463 6-5487 5-4167 10-1288 k 213-8246 294-0089 6-6495 5-5000 10-6036 § 220-3533 307-5764 6-7502 5-5833 11-0933 h 226-9801 321-5551 6-8510 5-6667 11-5970 g 233-7050 335-9510 6-9517 5-7500 12-1159 1 240-5282 350-7703 7-0525 5-8333 12-6507 £ 247-4495 366-0191 71532 5-9167 13-2007 9 in. 254-4690 381-7035 72540 60000 13-7663 £ 261-5867 397*8297 7-3547 60833 14-3479 k 268-8025 414-4039 7-4555 6-1667 14-9457 276-1165 431-4321 7*5562 6-2500 15-5599 h 283-5287 448-9205 7-6570 6-3333 16-1905 g 291-0391 466-8752 7-7577 6-4167 16-8378 i 298-6477 485-3024 78585 6-5000 17-5026 i 306-3544 504-2082 79592 6-5833 18-1845 10 in. 314-1593 523-5988 80600 6-6667 18-8838 £ 322-0623 543-4802 8-1607 6-7500 19-6008 k 330-0636 563-8586 8-2615 6-8333 20-3358 g 338-1630 584-7402 8-3622 6-9167 21-0891 £ 346-3606 606-1310 8-4630 7-0000 21 -8604 g 354-6564 628-0373 8-5637 70833 22-6494 I 363-0503 650-4651 8-6645 7-1667 23-4593 £ 371-5424 673-4206 8-7652 7-2500 24-2872 11 in. 380-1327 696-9100 8-8660 7-3333 25-1344 £ 388-8212 720-9393 8-9667 7-4167 260010 i 397 6078 745-5147 90675 7-5000 26-8873 1 406-4926 770-6423 9-1682 75833 27-7936 £ 415-4756 796-3283 9-2690 7-6667 28-7199 g 424-5568 822-5788 9-3697 77500 29-6662 433-7361 849-3999 9-4705 78333 30-6242 £ 443-0137 876-7979 9-5712 7-9167 31-6221 52] SURFACE AND SOI.IDITY OF SPHERES, SIDE OF = CUBES, &C. TABLE IV. Dia. in inches. Surface in sq. in. Solidity in cud. in. Cube in inches. Cylinder in inches. Water in lbs. 12 in. 452-3893 904-7787 9-6720 80000 32-6312 k 461-8632 933-3486 9-7727 80833 33*6616 i 471-4352 962-5136 9-8735 8-1667 34-7135 | 481-1055 992-2800 9-9742 8"2500 3o-7868 h 490-8739 1022-6539 10-0749 8-3333 36-8824 g 5007404 1053-6413 10-1757 8-4167 38*0000 i 510-7052 1085-2485 10-2764 8-5000 39 1399 i 520-7681 11174815 10-3772 8-5833 40-3024 13 in. 530-9292 1150-3465 10-4779 8-6667 41-4877 h 54H884 1183-8497 10-5787 8-7500 42-6960 4 551-5459 1217-9971 10-6794 88333 43-9276 1 562-0015 1252-7950 10-7802 8-9167 45-1828 h 572-5553 1288-2493 10-8809 9-0000 46-4613 1 583-2072 1324-3664 10-9817 9-0833 47-7436 l 593-9574 1361T523 11-0824 91667 490905 i 604-8057 1398-6131 11-1832 9-2500 50-4416 14 in. 615-7522 1436-7550 11-2839 9-3333 51-8172 k 626-7968 1475-5842 11-3847 9-4167 53-2176 4 637-9397 1515-1067 11-4854 9-5000 54-6430 | 649-1807 1555-3287 11-5862 9-5833 56-0936 h 660-5199 1596-2563 11-6869 9-6667 •57-5697 g 671-9572 1637-8957 11-7877 9-7500. 59-0716 4 683-4928 1680-2530 11-8884 9-8333 60-5990 i 695-1265 1723-3343 11-9892 9-9167 621528 15 in. 706-8583 1767-1459 12-0899 10-0000 63-7329 k 718-6884 1811-6937 12-1907 100833 65-3395 4 730 6166 1856-9840 12-2914 10*1667 66-9729 § 742-6431 19030228 12-3922 10-2500 68-6333 % 754-7676 1949-8164 12-4929 10-3333 70-3210 g 766-9904 1997-3708 12-5937 10-4167 72-0361 4 779-3113 2045-6922 12-6944 10-5000 73-7788 £ 791-7304 2094-7868 12-7952 10-5833 75-5494 16 i«. 804*2477 2144-6606 12-8959 10-6667 77-3481 53 EXAMPLES TO TABLE IV. Ex. 1. Required the surface of a sphere, whose diameter is 4| inches? Here, in column 1, page 50, will he found 4| inches, the diameter; against which, in column 2, will be found 74 6619 square inches, the surface required. Or by mensuration thus:—multiply the square of the diameter by nr. No. 1 of the Table of Useful Numbers, pogre 169 ; Here w x (4£) 2 = 3-1415926 X (5 -£) a = (157079630--3926991) X (5 — J) = 15-3152639 X (5 - £) = 76-5763195 — 1-9144080 = 74-6619115 square inches as before. Ex. 2. Required the solidity, of a sphere, whose diameter is 11 § inches ? Here, in column 1, page 51, will be found Ilf inches, the diameter; against which, in column 3, will be found 770"6423 cubic inches, the solid content required. Or by mensuration thus:—multiply the cube of the diameter by Jtt, No. 8 of the Table of Useful Numbers, page 170 ; Here |tX (11§)* = *5235988 X (11 +£ + |) 3 = (57595868 -f *1308997 +-0654498) x (11 +$ +£) 2 = 5-9559363X (11 +J +£) 2 = (65-5152993 + 1-4889841 + 7444920) X (11 + \ + |) = 677487754 x (11 + $ + £) = 745-2365294 + 16 9371938 + 8 4685969 = 770-6423201 solid inches as before. Ex. 3. What is the length of the edge or side of a cube, equal in volume or solid content to a sphere, whose diameter is 7| inches ? Here, in column \,page 51, will be found 7| inches, the diameter; against which, in column 4, will be found 6'1457 inches, the length required. Or thus by mensuration :—multiply the diameter of the sphere by No 66 of the Table of Useful Numbers, page 175. Here X 71 = ‘805996 X (7+h + h) = 5 641972 + -402998 + '100749 = 6-145719 inches as before. Ex. 4. What is the length of a cylinder of 4| inches diameter, having its volume equal to that of a sphere of the same diameter ? Here, in column 1, page 50, will be found 4| inches , the diameter ; against which, in column 5, will be found 3-1667 inches, the length required. EXAMPLES TO TABLE IV. 54 ] Or thus: — two-thirds of the sphere’s diameter = height of the cylinder, thus, From 4f = 475 Subtract J of 4£ = 15833 3-1667 inches as before. Ex. 5. Required the weight of a quantity of rain water, equal in volume to that of a sphere whose diameter is 15| inches ? Here, in column 1, page 52, will be found 15£ inches, against which, in column 6, will be found 65'3395 lbs. avoirdupois, the weight required. Or thus :—find the solidity of the sphere as example 2, which mul tiply by -0360654 = 10 times No. 89 of the Table of Useful Numbers vage 177. •5235988 = Jit 57595868 = 11 X ^ 63-3554548 = 121 X Jw T9194318 = 1|L x Jir 87'1137498 = 11 X 'f X i* 958-2512478 = 121 X l f X $ir 1197814060 = (143) a x 1317-5954660 = 11 X Qf) a X h* 14493-5501260 = 121 X (If- 1 ) 8 X h™ 1811-693766 = (Ul) *x 456063 = Const. No. inverted. 54-350813 10870163 •108702 9058 725 65-339461 lbs. av. = Ans. as before. TABLE V. CONTAINING THE WEIGHT OF CYLINDRICAL COLUMNS OF WATER, EACH ONE FOOT IN LENGTH, AND OF VARIOUS DIAMETERS, IN LBS. AVOIRDUPOIS. 56J WEIGHT OF TABLE Y. Dia. Weight. Dia. Weight. Dia. Weight. Bin. 3-0592 9 in. 27-5326 15 in. 76-4794 k 3-3194 h 28-3027 £ 77-7594 * 3-5903 i 29-0834 i 79 0500 § 3-8718 I 29-8748 i 80-3512 h 41639 4 30-6768 ■ j 81-6631 i 4-4666 i 31-4894 g 82-9855 i 4-7800 1 32-3126 i 84-3186 5-1039 i 331464 i 85-6623 4 Ml. 5-4385 10 in. 33-9909 16 in. 870166 £ 5-7838 34-8459 £ 88-3816 i 6-1396 i 35-7117 i 89-7571 1 6-5061 i 36-5880 g 911433 i 6-8832 4 374749 i 92-5401 g 7-2709 § 38-3725 g 93-9476 1 7-6692 39-2807 1 95-3656 I 8-0781 40-1995 i 96-7943 5 m. 8-4977 11 in. 41-1289 17 in. 98-2336 h 8-9279 £ 42-0690 £ 99 6835 k . 9-3687 i 430197 * 101-1441 | 9-8202 | 43-9810 i 102-6152 i 10-2822 4 44-9529 i 104-0970 i 10-7549 § 45-9355 g 105-5894 t 11-2382 | 46-9286 1 107 0925 1 11-7322 i 47-9324 f 108-6061 6 in. 12-2367 12 in. 48-9468 18 in. 1101304 i 12-7519 4 49-9719 £ 111-6653 i 13-2777 4 51 0075 i 113-2108 § 13-8141 | 520538 | 114-7670 h 14-3611 i 531107 i 116-3337 g 14-9188 g 54-1783 g 117-9111 1 15-4871 * 55-2564 i 119-4991 160660 * 56-3452 l 121 0978 7 in. 16-6555 13 in. 57-4446 19 in. 122-7070 k 17-2557 4 58-5546 £ 124-3369 i 17-8664 4 59-6752 i 125-9574 § 18-4878 i 60-8065 i 127-5985 i 19-1199 4 61-9484 j 129-2503 g 19-7625 g 63-1009 g 130-9126 | 20-4158 l 64-2640 132-5856 r 21-0796 i 65-4377 g 134-2692 8 272. 21-7542 14 iM. 66-6221 20 in. 135 9635 £ 22-4393 4 678171 £ 137-6683 i 23-1350 i 690227 i 139-3838 i 23-8414 § 70-2389 | 141-1099 4 24-5584 4 71-4658 % 142-8466 i 25-2860 g 72-7033 g 144-5940 260243 73-9514 4 1463519 i 26-7731 4 75-2101 i 1481205 COLUMNS OF WATER. [57 TABLE Y. Dia. Weight. Dia. Weight. Dia. Weight. 21 in. 149-8997 27 in. 247-7934 33 in. 370-1605 ft 151-6895 i ft 250-0931 ft 372-9701 ft 153-4900 ft 252-4034 ft 375-7903 g 155-3011 | 254-7244 g 378-6211 ft 157-1228 ft 257-0559 ft 381-4625 i 158-9551 | 259-3981 g 384-3145 ft 160-7980 ft 261-7509 ft 387-1772 ft 162-6516 ft 264-1143 ft 390 0505 22 in. 164-5158 28 in. 266-4884 34 in. 392-9344 ft 166-3906 ft 268-8731 ft 395-8289 3 168-2760 ft 271-2684 ft 398-7341 3 170-1721 § 273-6743 g 401-6499 ft 1720788 ft 276 0908 ft 404-5763 i 173-9961 i 278-5180 g 407-5133 3 175-9240 t 280-9557 f 410-4609 ft 177-8625 l 283-4042 ft 413-4192 23 m. 179-8117 29 in. 285-8632 35 in. 416-3881 ft 181-7715 ft 288-3328 ft 419-3676 3 183-7419 4 290-8131 ft 422-3577 3 185-7229 g 293-3040 g 425-3585 ft 187-7146 ft 295-8055 ft 428-3699 § 189-7168 g 298-3176 g 431-3919 3 191-7297 ft 300-8404 s 434-4245 ft 193-7532 ft 303-3738 ft 437-4678 24 in. 195-7874 3d M. 305-9178 36 in. 440-5216 ft 197-8321 ft 308-4724 ft 443-5861 3 199-8875 4 311-0377 ft 446-6612 3 201-9535 i 313-6135 g 449-7470 ft 2040302 ft 316-2000 ft 452-8433 § 206-1174 g 318-7971 g 455-9503 3 208-2153 I 321-4049 ft 4590679 ft 210-3238 ft 324-0232 ft 462-1961 25 t». 212-4429 31 in. 326-6522 37 465-3350 ft 214-5727 ft 329-2918 ft 468-4844 3 216-7130 4 331-9420 ft 471-6445 3 218-8640 g 334-6029 g 474-8152 ft 221 0256 ft 337 2744 ft 477-9965 i 223-1978 g 339-9565 g 481 1885 3 225-3807 4 342-6492 4 484-3911 ft 227-5742 ft 345-3525 ft 487 6043 26 in. 229-7783 32 *». 348-0665 38 in. 490-8281 ft 231-9930 ft 350-7910 ft 4940625 3 234-2183 4 353-5262 ft 497-3076 3 236-4543 g 356-2721 g 500-5633 ft 2387009 ft 359 0285 ft 503-8296 g 240-9581 g 361-7956 § 507-1065 3 243-2259 ft 364-5733 1 510-3941 ft 245-5043 ft 367'3616 ft 513-6923 58] WEIGHT OF COLUMNS OF WATER. TABLE y. Dia. Weight. Dia. Weight. Dia. Weight. 39 in. 517-0011 43 in. 628-4911 47 in. 750-8582 b 520-3205 b 632-1504 k 754-8575 i 523-6505 635-8204 \ 758-8673 526-9912 § 639-5010 | 762 8878 530-3425 b 643-1922 766-9189 8 , 5337044 § 646-8940 i 770-9606 537-0769 Y 650-6064 \ 7750130 i 540-4601 i 654-3295 i 7790760 40 in. 543-8539 44 in. 658 0632 48 in. 783-1495 h 547-2582 b 661-8075 k 787-2338 550-6733 i 665-5624 i 791-3286 1 554-0989 | 669-3279 | 795-4341 557 5352 h 673 1041 I 799-5501 8 , 560-9821 § 676 8909 I 803-6768 564*4396 I 680-6883 $ 807-8142 i 567-9077 i 684-4964 i 811-9621 41 in. 571-3865 45 in. 688-3150 49 in. 816-1207 k 574-8758 b 692-1443 b 820-2899 i 578-3758 i 695-984^ i 824-4697 h 581-8864 § 699-8347 i 828-6601 585-4077 h 703-6959 h 832-8612 588-9395 i 707-5677 i 837 0729 592-4820 l 711-4501 3 841-2952 5960351 & 715-3431 i 845-5281 42 in. b i 599-5989 603-1732 606-7582 610-3538 613-9600 617-5769 621-2043 624-8424 46 in. 719-2467 723-1610 727-0859 731-0214 734-9675 738-9242 742-8916 746-8606 50 in. 849-7716 The Editor of the preceding tables begs to state, that in consequence of the numerous errors having been found in former impressions of the five foregoing tables, it was thought necessary to recompute the whole afresh, which has all been carefully done; and it is presumed that in the present edition no errors of any consequence will he found, as the methods employed in the construction of the tables were such as nearly to preclude the possibility of error in the computations. SAMUEL MAYNARD. 8, EarVs Court , Leicester Square, London, September 3rd, 1852. EXAMPLES TO TABLE V. Ex. 1. Required the weight of a cylindrical column of water whose diameter is 47| inches , and length 1 foot ! Here, in column 1, page 58, will be found 47§ inches , against which, in the next column, will be found 762 8878 lbs. avoirdupois, the weight required. Note 1. When the length of the cylinder differs from 1 foot, multiply the tabular number by the length in feet. Note 2. The weight of cylindrical columns of various substances may be found by means of this Table V., and the Table YI. of specific gra¬ vities, page 60, by multiplying the tabular number of Table Y. by the length of the column in feet (when it differs from 1 foot), and by the specific gravity given in page 60, and dividing by 1000. Ex. 2. Required the weight of a cylindrical column of cast iron, whose diameter is 7g inches, and length 115 inches'! Hero 115 inches = 97, feet, the length, and 7271 = the specific gra¬ vity of cast iron, by Table VI., page 60. Then, by Table V., page 56, tabular No., for 7g in. — 17*2557 lbs. av. 155-3013 6in.= i 8-6278 lm. = | 1-4380 Weight of given volume of water = 165 3671 lbs. av. Table VI. page 60, Specific gravity = 7271 1653671 11575697 3307342 11575697 1202384-1841 lbs. av. which, divided by 1000, gives 1202-3842 lbs. av., the weight of the cast -iron column required. Ex. 3. Required the weight of a cast iron cylindrical pipe 26^ inches, outside diameter, and inside diameter 23| in., the length being 6^ feet! lbs. av. The tabular number for ffj^97 Difference = 42-4886 TABLE VI. COMBINING THE SPECIFIC GRAVITIES AND OTHER PROPERTIES OF BODIES. WATER THE STANDARD OF COMPARISON, OR 1000. TABLE VII. OF CIRCUMFERENCES, AREAS, SQUARES, AND CUBES OF FRACTIONAL NUMBERS. 62] CIRCUMFERENCES, AREAS, SQUARES, TABLE VII. OF CIRCUMFERENCES, AREAS, SQUARES, AND CUBES OF FRACTIONAL NUMBERS. Area. Circum. Diara. Square. Cube. 7854 3-1416 1 in. 1 1. •9503 3-4557 1 1-21 1-331 •9940 3-5343 •125 1-26 1-423 11309 37699 ■2 1-44 ]-72f8 1-2271 3-9270 •25 1-56 1-953 1-3273 4-0840 •3 1 69 2-197 1*4848 4-3197 •375 1-89 2-599 1*5393 4-3982 •4 1-96 2-744 17671 4-7124 •5 2-25 3-375 2-0106 50265 •6 ' 2-56 4096 20739 5-1051 •625 2-64 4-291 2-2698 5-3407 •7 2-89 . 4-913 2-4052 5-4978 •75 306 5-359 2-5446 5-6548 •8 3-24 5-832 27610 5-8905 •875 3-51 6-591 2 8352 5-9690 •9 3-61 6-859 3-1416 6-2832 2 in 4 8 34636 6-5973 •1 4-41 9-261 3-5463 6-6759 •125 4-51 9-595 38013 • 6-9115 •2 4-84 10-648 3*9760 7-0686 •25 506 11-390 4-1547 7-2256 •3 5-29 12-167 4-4302 7-4613 •375 5-64 13-396 45239 7-5398 •4 5-76 13-824 4-9087 7 8540 •5 6-25 15-625 5-3093 81681 •6 6-76 17576 5*4119 8-2467 •625 6-89 18087 57255 8-4823 •7 , 7-29 19-683 5-9395 86394 •75 7 56 20-7968 61575 87964 •8 7-84 21-952 6-4918 9 0321 •875 8-26 23-7636 6-6052 9*1106 •9 841 24-389 7-0686 9-4248 Bin. 9 27 75476 97389 •1 9-61 29-791 76699 9-8175 •125 9-76 30-5175 80424 10-0531 •2 10-24 32-768 82957 10-2102 •25 10-56 34-328 85530 10-3672 •3 1089 35-937 89462 106029 •375 11-39 38-443 90792 10*6814 •4 11-56 39 304 9-6211 10-9956 •5 12-25 42-875 101787 11*3097 •6 12-96 46-656 10-3206 11-3883 •625 13-14 47-634 107521 11-6239 •7 13-69 50 653 11 0446 11-7810 ■75 1406 52-734 11-3411 11-9380 •8 14-44 54-872 AND CUBES OF FRACTIONAL NUMBERS. 163 TABLE VII. ATea. Circum. Diam. Square, j Cube. 117932 121737 3-875 1510 58-185 11-9459 12-2522 •9 15-21 59-319 12-5664 12-5664 4 in. 16 64 13-2025 12-8805 •1 16-81 68921 13-3640 12-9591 •125 17-01 ■ 70-189 13-8544 131947 •2 17-64 • 74088 14-1862 13-3518 •25 1806 76-765 14-5220 13-5088 •3 18-49 79-507 15-0331 13-7445 •375 1914 83-740 15-2053 13-8230 •4 19-36 85184 15-9043 14-1372 •5 20-25 91-125 16-6190 14-4513 •6 21-16 97-336 16-8001 14-5299 •625 21-39 98-931 17-3494 14-7655 •7 22-09 103-823 177205 14-9226 •75 22-56 107171 180956 15-0796 •8 2304 110-592 18-6655 15-3153 •875 23-76 115-857 18-8574 15-3938 •9 24-01 117-649 19-6350 15-7080 5 in. 25 125 20-4282 160221 •1 26-01 132-651 20-6290 16-1007 •125 26-26 134-611 21-2372 16-3363 •2 27 04 140-608 21-6475 164934 •25 27-56 144-703 22 0618 16-6504 •3 28-09 148*877 22-6907 168861 •375 28-89 155-287 22-9022 16-9646 •4 29-16 157-464 237583 17-2788 •5 30-25 166-375 24-6301 17-5929 •6 31-36 175-616 24-8515 17-6715 •625 31-64 177 978 25-5176 17-9071 •7 32-49 185 193 25-9672 18-0642 ■75 33-06 190-109 26-4208 18-2212 •8 33-64 195-112 27-1085 18 4569 •875 34-51 202-779 27-3397 18-5354 •9 34-81 205-379 28-2744 18 8496 6 in. 36 216 29-2247 191637 •1 37-21 226-981 29-4647 19-2423 •125 37-51 229-783 30-1907 19-4779 •2 38-44 238 328 30-6796 19-6350 •25 39-06 244-140 311725 19-7920 •3 39-69 250047 31-9192 20-0277 •375 40-64 259 083 32-1699 20-1062 •4 40-96 262 144 33 1831 20-4204 •5 42-25 274-625 34-2120 20-7345 •6 43-56 287-496 34-4717 20-8131 •625 43-89 290-775 35-2566 21-0487 •7 44-89 300-763 357847 21-2058 •75 45-56 307-546 36-3168 21-3628 •8 • 46-24 314-432 37-1224 21-5985 •875 47-26 324-951 373928 21-6770 •9 47-61 328-509 64] CIRCUMFERENCES, AREAS, SQUARES, TABLE VII. Area. Circum. Diam. Square. Cube. 38-4846 21-9912 7 in. 49 343 39-5920 22-3053 •1 50-41 357-911 39-8713 22-3839 ■125 50-76 361-704 407151 22-6195 •2 51-84 373 248 41-2825 22-7766 •25 52-56 381-078 41 -8539 22-9336 •3 53-29 389 017 42-7184 231693 •375 54-39 401-130 43-0089 23-2478 •4 5476 405-224 44-1787 23-5620 •5 56-25 421-875 45-3647 23-8761 •6 57-76 438-976 45-6636 239547 •625 5814 443-3-22 46-5663 24-1903 •7 59-29 456-533 47 1730 24-3474 •75 60-06 465-484 47-7837 24-5044 •8 60-84 474-552 487070 24-7401 •875 6201 ✓ 488-373 49 0168 24-8186 •9 62-41 493*039 50-2656 25-1328 8 in. 64 512 51-5300 25-4469 •1 ‘ 65-61 531-441 51 8486 25-5255 •125 6601 536-376 528102 25-7611 •2 67-24 55 368 53-4562 25-9182 •25 6806 561-515 54-1062 26 1752 •3 68-89 571787 550883 26-3109 •375 70-14 587-427 55-4178 26-3894 •4 70-56 592-704 56-7451 26-7036 •5 72-25 614-125 580881 270177 •6 73-96 636-056 58-4264 27-0963 •625 74-39 641-619 59-4469 27-3319 •7 75-69 658-503 60-1321 27-4890 •75 7656 669-921 60-8213 273460 •8 77-44 681-472 61-8625 27-8817 •875 78-76 699-044 62-2115 27-9652 •9 79-21 704-969 63-6174 28-2744 9 in. 81 729 65-0389 28-5885 •1 82-81 753-571 65-3968 28-6671 •125 83-26 759-798 66-4762 28-9027 • -2 84-64 778-688 672007 29-0598 •25 85-56 791-453 67-9292 29-2168 •3 86-49 804-357 690293 29-4525 •375 87-89 823-974 69-3979 29-5310 •4 88-36 830-584 70-8823 29-8452 •5 90-25 857375 72-3824 30 1593 •6 9216 884-736 72-7599 30-2379 •625 92-64 891-666 73-8982 30-4735 *7 9409 912-673 74-6620 30-6306 •75 9506 926-859 75-4298 30-7876 •8 96-04 941-192 76-5887 310233 •875 97-51 962-096 76-9770 311018 •9 9801 970-299 78'5400 31-4160 10 in. 100 1000 801186 31-7301 •1 10201 1030-301 AND CUBES OF FRACTIONAL NUMBERS. [65 TABLE VII. Area. Circunl. Diam. Square. Cube. 80-5157 31-8087 10-125 102-51 1037 970 817130 320443 •2 104-04 1061-208 82-5160 32-2014 •25 105-06 1076-890 83-2320 32-3580 •3 106 09 1092-727 84-5409 32-5941 •375 107-64 1116-771 84-9488 32-6726 •4 10816 1124*864 86-5903 32-9868 •5 110 25 1157-625 88-2475 333009 •6 112-36 1191016 88-6643 33-3795 •625 112-89 1199-462 89-9204 33 6J 50 •7 114-49 1225043 90-7627 33-7722 •75 115-56 1242-296 91-6090 33-9292 •8 116-64 1259-712 92-8858 34-1649 •875 118-26 1286 138 93-3133 34-2434 •9 118-81 1295-029 95-0334 34-5576 11 in. 121 1331 96-7691 34 8717 •1 123-21 1367-631 972053 34-9503 •125 123-76 1376-892 98-5205 351859 •2 125-44 1404-928 99-4021 35-3430 -25 126-56 1423 828 100-2877 35-5011 •3 127-69 1442-897 101-6234 357357 •375 129-39 1471-818 1020705 35-8142 •4 129-96 1481-044 103-8691 36-1284 •5 132-25 1520-875 105-6834 36-4425 •6 134-56 1560-896 106-1394 36-5211 •625 13514 1571-009 107-5134 36-7567 •7 136-89 1601-613 108-4342 36-9138 •75 13806 1622-234 109-3590 37-0708 •8 139-24 1643032 110-7536 37-3065 •875 141-01 1674-560 111-2204 37-3840 •9 141-61 1685 159 113-0976 37-6992 12 in. 144 1728 114-9904 38 0133 •1 146-41 1771-561 115-4660 380919 •125 147 01 1782-564 116-8989 38-3275 •2 148-84 1815-848 117-8590 38-4846 •25 150-06 1838-265 118-8231 38-6416 •3 151-29 1860-867 120-2766 38-8773 •375 153-14 1895-115 120-7631 38-9558 •4 153-76 1906-624 122-7187 39-2700 •5 156-25 1953-125 124-6901 39-5841 •6 158-76 2000-376 125T854 39-6627 •625 159-39 2012*306 126-6771 39-8983 •7 161-29 2048-383 127-6765 40-0554 •75 162-56 2072-671 128-6799 40-2124 •8 163-84 2097-152 130 1923 40-4481 •875 165-76 2134-232 130-6984 40-5266 •9 166-41 2146-689 132-7236 40-8408 13 in. 169 2197 134-7824 4M549 •1 171-61 2248T91 135-2974 41-2338 •125 172-26 2260-986 136-8480 41-4691 •2 174-24 2299-968 CIRCUMFERENCES, AREAS, SQUARES, TABLE VII. Area. Circum. Diam. | Square. Cube. 137-8867 41-6262 13-25 175-56 2326-203 138-9294 41-7832 •3 176-89 2352-637 140-5007 420189 •375 178-89 2392-661 141-0264 42-0974 •4 179-56 2406-104 143-1391 42-4116 •5 182-25 2460-375 145-2675 42-7257 •6 184-96 2515-456 145-8021 42-8043 •625 185-64 2529-353 147-4177 43 0399 •7 187-69 2571-353 148-4896 43-1970 *75 189-06 2599-609 149-5715 43-3540 •8 190-44 2628-072 151-2017 43-5897 •875 19251 267M54 151-7471 43-6682 •9 193-21 2685-619 153-9384 43-9824 14 in. 196 2744 156-1453 44-2965 •1 198-81 2803-221 156-6995 44-3751 •125 199-51 2818157 158-3680 44-6107 •2 201-64 2863-288 159-4852 44-7676 •25 20306 2893-640 160-6064 44-9248 *•3 204-49 2924-207 162-2956 45-1605 •375 206-64 2970*458 162-8605 45-2390 *4 207-36 2985-984 165-1303 45-5532 •5 210-25 3048-625 1674158 45-8673 •6 21316 3112-136 167-8986 45-9459 •625 213-89 3128150 169-7179 46T815 •7 216-09 3176-523 170-8735 46-3386 •75 217-56 3209046 1720240 46-4656 •8 219-04 3241-792 173-7820 46-7313 •875 221-26 3291-325 174-3636 46-8098 •9 22201 3307-949 176-7150 47-1240 15 in. 225 3375 1790790 47-4381 •1 22801 3442-951 179-6725 475167 •125 22876 3460-079 181-4588 47-7523 •2 23104 3511-808 182-6545 47-9094 •25 232-56 3546-578 183-8542 48-0664 •3 234-09 3581-577 185-6612 48-3021 •375 236-39 3633-505 186-2654 48-3806 •4 237-16 3652-264 188-6923 48*6948 •5 240-25 3723-875 191-1349 49-0089 •6 243-36 3796-416 191-7480 49 0875 •625 244-14 3814-690 193-5932 49-2321 •7 246-49 3869-893 194-8282 49-4802 •75 248-06 3906-984 1960672 49 6372 •8 249*64 3944-312 197-9330 49*8729 •875 252-01 4000-747 198-5569 49-9514 •9 252-81 4019-679 201-0624 50-2656 16 in. 256 4096 203-5835 50-5797 1 259-21 4173-281 204-2162 50-6583 •125 260 01 4192-751 206-1209 50-8939 •2 262-44 4251-528 207-3946 510510 •25 264-06 4291-015 208-6723 51-2080 •3 265-69 4330-747 AND CUBES OF FRACTIONAL NUMBERS. [67 TABLE VII. Area. Circum. Diam. Square. Cube. 210-5976 51-4437 16-375 268-14 4390-802 211-1411 51-5224 •4 268-96 4410-944 213 8251 51-8364 •5 272-25 4492-125 216-4248 52-1505 •6 275-56 4574-296 217 0772 52-2291 •625 276-39 4594-993 2190402 52-4647 •7 278-89 4657-463 220-3587 52-6218 •75 280-56 4699-421 221-6712 52-7788 •8 282-24 4741-632 223-6549 530145 •875 284-76 4805-419 224-3189 53-0930 •9 285-61 4826-809 226-9086 53-4072 17 in. 289 4913 229-6588 53-7213 •1 292-41 5000-211 230-3308 53-7999 •125 293-26 5122173 232-3527 540355 •2 295-84 5088-448 233-7055 54-1926 •25 297-56 5132-953 235 0623 54-3496 •3 299-29 5177717 237-1049 54-5853 •375 301-89 5245-349 237-7877 54-6038 •4 302-76 52(18 024 240-5287 54-9780 •5 306-25 5359-375 243-2855 55-2921 •6 309-76 5451-776 243-9771 55-3707 •625 310-64 5475-040 246-0579 55-6063 *7 . 313-29 5545-233 247-4500 55-7634 •75 315-06 5592-359 248-8461 55-9204 •8 316-84 5639-752 250-9475 56 1561 •875 319-31 5711-341 251 0500 56-2346 •9 320-41 5735-339 254-4696 56-5488 18 in. 324 5832 257-3048 56-8629 •1 327-61 5929-741 258-0161 56-9415 •125 328-51 5954-345 260-1558 571770 •2 331-24 6028-568 261-5872 57 3342 •25 33306 6078-390 2630226 57-4912 •3 334-89 6128-487 265-1829 57-7269 •375 33764 6204-146 265-9050 57-8054 •4 338-56 6229-504 268-8031 581196 •5 342-25 6331-625 2717169 58-4337 •6 345-96 6434-856 272-4479 58-5123 •625 346-89 6460-837 274-6465 58-7479 •7 349-69 6539-203 276 1171 58-9056 •75 351-56 6591-796 277-5917 590620 •8 353-44 6644-672 279-8110 59-2977 •875 356-26 6724-513 280-5527 59-3762 •9 357-21 6751-269 283-5294 59-6904 19 in. 361 6859 286-5217 600045 •1 364-81 6967-871 287-2723 600831 •125 365-76 6995-263 289-5298 60-3187 •2 368-64 7077-888 291 0317 60-4758 •25 370-56 7132-328 292-5536 60-6328 •3 372-49 7189057 294-8312 60-8685 •375 375-39 7273-192 295-5931 60-9470 •4 376-36 7301-384 CIRCUMFERENCES, AREAS, SQUARES, TABLE VII. Area. Circum. Diam. Square. Cube. 298-6483 61-2612 19-5 38025 7414-875 301-7192 61-5753 •6 384-16 7529-536 302-4894 61-6539 •625 38514 7558-384 304-8060 61-8895 •7 38809 7645-373 306-3550 620466 •75 390 06 7703-734 307-9082 62-2036 •8 392 04 7762-392 310-2452 62-4393 •875 395 01 7850-935 3110252 62-5178 •9 39601 7880-599 314-1600 62-8320 20 in. 400 8000 317-3094 63-1461 •1 40401 8120*601 3180992 63-2247 •125 40501 8150-939 320-4746 63-4603 •2 . 40804 8242-408 3220630 63-6174 . -25 41006 8303-765 323-6554 63-7744 •3 41209 8365-427 3260514 64-0101 •375 415-14 8458-489 326-8520 640886 •4 41616 8489-664 3300643 64-4028 •5 420-25 8615-125 333*2923 64-7161 •6 424-36 8741-816 3341018 64-7955 ■625 425-39 8773-681 336-3360 650311 •7 428-49 8869-743 338-1637 65-1882 •75 430-56 8934 171 339-7954 65-3452 •8 432-64 8998-912 342-2503 65-5809 •875 435-76 9196-607 3430705 65-6594 •9 436-81 9129-329 346-3614 65-9736 21 in. 441 9261 349-6679 66-2870 1 445-21 9393-931 350-4970 66-3663 •125 446-26 9427360 352-9901 66-6012 •2 449-44 9528128 354-6571 66-7590 •25 451-56 9595-703 356-3281 66-7916 •3 453-69 9663-597 358-8419 671517 •375 456-89 9766036 359-6817 67-2930 •4 457-96 9800-344 3630511 67-5444 •5 462-25 9938-375 366-4362 678585 •6 466-56 10077696 367-2849 67-9371 •625 4C764 10112-72 369-8370 681727 •7 470-89 10218-313 371-5432 68-3298 •75 473-06 10289-11 373-2534 68-4868 •8 475-24 10360-232 375-8261 68-7225 •875 478-51 10467-52 376-6856 68-8010 •9 479-61 10503-459 380-1336 69-1152 22 in. 484 10648 383-5972 69-4293 •1 488-41 10793-861 384-4655 69-5079 •125 489-51 10830-537 387 0765 69-7435 •2 492-84 10941 048 388-8220 69-9006 •25 495-06 11015142 390-5751 700576 •3 49729 11089-567 393-2031 70-2933 •375 500-64 11201-834 394 0823 70-3718 •4 501-76 11239-424 397-6087 70-6860 •5 506-25 11390-625 401 0509 71 0401 •6 510-76 11543170 AND CUBES OF FRACTIONAL NUMBERS. |69 TABLE VII. Area. Circum. Diam. Square. Cube. 402-0388 71-0787 22-625 511-89 11581-526 4047087 71-3143 •7 515-29 11697-083 406-4935 71-4714 •75 517-56 11774-548 408-2823 71-6284 •8 519-84 11852-352 410-9728 71 -8641 •875 523-26 11969-704 411-8716 71-9426 •9 524-41 12008-989 415-4766 72-2568 23 in. 529 12167 4190972 72-5709 •1 533-61 * 12326-391 420-0049 72-6495 •125 534-76 12366-458 422-7336 72-8851 •2 538-24 12487 168 424-5577 73-0422 •25 540-56 12568-072 426-3858 73 1992 ■3 542-89 12649-337 429*1352 73-4349 •375 546-39 12771-885 430-0536 73-5134 •4 547-56 12812-904 433-7371 73 8276 •5 552-25 12977-875 4374363 74-1417 •6 556-96 13144-256 438-3636 74-2203 •625 558-14 13185-986 4411511 74-4559 7 561-69 13312053 443-0146 74-6130 75 56406 13396-482 444-8819 74-7680 •8 566-44 13481-272 447-6992 75-0057 •875 57001 13609 124 448-6283 75 0882 •9 571-21 13651-919 452-3904 75-3984 24 in. 576 13824 456-1681 75-7125 •1 580-81 13997-541 457-1150 75-7911 •125 58201 14041 126 459-9616 76-0267 •2 585-64 14172-488 461-8642 76-1838 •25 588-06 14260515 4637708 76-3408 •3 ' 590-49 14343-907 466-6380 76-5765 •375 594-14 14482-177 467-5957 76-6528 •4 595-36 14526 784 471-4363 76-9692 •5 600-25 14706-125 475-2926 77 2833 •6 605-16 14886936 476-2592 77 3619 •625 606-39 14932-368 479-1646 77-5975 •7 61009 15069-223 481-1065 77-7546 •75 612-56 15160-921 4830524 77-9116 •8 61504 15252-992 485-9785 78-1473 •875 618-76 15391-794 486 6958 782258 •9 620-01 15438-247 490-8750 78-5400 25 tn. 625 15625 494-8098 78*8541 •1 630-01 15813-251 495-7960 789327 •125 631-26 15860-548 498-7604 79*1683 •2 635-04 16003-008 500-7415 79-3254 •25 637-56 16098-453 502-7266 79-4824 •3 64009 16191-277 505-7117 79-7181 •375 643-89 16338-323 506-7086 797966 •4 645-16 16387064 510-7063 80-8180 •5 650-25 16581-375 514-7196 80-4249 •6 655-26 16777-216 515*7255 80-5035 •625 656-64 16826-415 5187488 80-7391 •7 660-49 16974-593 70] CIRCUMFERENCES, AREAS, SQUARES, TABLE VII. Area. Circum. Diam. Square. Cube. 5207692 80-8962 2575 66306 17073-859 522-7936 81 0532 •8 665-64 17173-512 525-5837 81-2889 •875 669-51 17323-716 526-8541 81-3674 •9 670-81 17373-979 530 9304 81-6816 26 in. 676 17576 5350223 81-9976 1 681-21 17779-581 536-0477 820743 •125 682-51 17830-720 539-1299 82-3099 •2 686-44 17984-728 541 1896 82 4670 •25 68906 18087-890 543 2533 82-6240 •3 691-69 18191-447 546-3561 82-8597 •375 695-64 18347-520 547-3923 82-9382 •4 696-96 18399-744 551-5471 83-2524 •5 702-25 18609-625 555-7176 83-5665 •6 70756 18821 096 556-7627 83-6451 •625 708-89 18874-212 559-9038 83-8807 •7 712-89 19034 163 5620027 84-0378 •75 715-56 19141-296 564-1056 84 1948 •8 718-24 19248-832 567-2674 84-4305 •875 722-26 19410 888 568-3232 84-5090 •9 723-61 19465 109 572-5566 84 8232 27 in. 729 19683 576-8056 851373 •1 734-41 19902-511 5778703 85-2159 •125 735-76 19957 642 581-0703 85-4515 •2 739-84 20123 648 583-2085 85 6086 •25 742-56 2023,-828 585-3507 85-7656 •3 745-29 20346-417 588-5714 860013 •375 749-39 20514-567 589-6469 860798 •4 75076 20570-824 593-9587 86-3940 •5 756-25 20796-875 598-2863 86-7081 •6 761-76 21024-576 599-3706 86-7687 •625 763-14 21081-759 602-6295 870223 •7 767-29 21253-933 604 0470 87-1794 •75 77006 21369-234 606-9885 87-3364 •8 772-84 21484-952 610-2680 875721 •875 77701 21659 309 611-3632 876506 •9 778-41 21717639 615-7536 87-9648 28 M2. 784 21952 6201596 88-2789 •1 789-61 22188-041 621-2636 88-3575 •125 79101 22427-313 624-5814 88-5931 •2 795-24 22425-768 626-7982 887502 •25 798-06 22545-265 6290190 88-9072 •3 800-89 22665-187 632-3574 89 1429 •375 805-14 22845-864 633-4722 89-2214 •4 806-56 22906-304 6379411 89-5316 •5 812-25 23149 125 642-4257 89-8497 •6 817-96 23393656 643-3494 89-9283 •625 819-39 23455-056 . 646-9261 90 1639 •7 823-69 23639-903 6491821 90-3210 •75 826-56 23763-671 651-4421 90-4780 •8 829-44 23887 872 AND CUBES OF FRACTIONAL NUMBERS. [71 TABLE VII. Area. Circum. j Diam. Square. Cube. 654-8395 90-7137 28-875 833-76 24074-981 655-9739 90-7922 •9 835-21 24137 569 660-5214 91-1064 29 in. 841 24389 6650845 91-4205 •1 846-81 24642-171 666-2278 91-4991 •125 848-26 24705-735 669-6634 91-7347 •2 852-64 24897-088 671-9587 91-8918 •25 855-56 25025-203 674-2580 920488 •3 858-49 25153757 677-7143 92-2845 ■375 862-89 25347-411 678-8663 92-3630 •4 864-36 25412-184 683-4943 92-6772 •5 870-25 •25672-375 688-1360 92-9913 ■6 876-16 25934336 689-2989 93-0699 •625 877-64 26600-102 692-7934 93-3055 •7 88209 26198-073 695-1280 93-4626 •75 885-06 26330-609 697 4666 93-6196 •8 88804 26463-592 700-9817 93-8553 •875 892-51 26663-903 702-1554 93-9338 •9 89401 26730-899 706-8600 94-2480 30 in. 900 27000 711-5802 94-5(,21 •1 90601 27270-901 712-7627 94-6407 •125 907-51 27338 907 716-3162 94-8763 •2 91204 27543-608 718-6900 950334 •25 915-06 27689-640 721-0678 95-1904 •3 91809 27818-127 724-6419 95-4261 •375 922-64 28028,-208 725-8352 95 5046 •4 924-16 28094-464 730-6183 95-8188 •5 930-25 2837-2-6-25 735-4171 96-1329 •6 93636 28652-616 736-6193 96-2115 •625 937-89 28722 899 740-2316 96-4471 •7 942-49 •28934-443 742-6447 96*6042 •75 945-56 29076-046 745 0618 967612 •8 948-64 29218112 748-6948 96-9969 •875 95326 2943-2 075 749-9077 97-0754 •9 95481 29503-629 754-7694 973896 31 in. 96] 29791 759 6467 97*7037 •1 967*21 30080231 760-8685 97-7823 •125 96876 30152-829 764-5397 980179 •2 973-44, 30371-328 766-9921 98-1750 •25 976-56 30517-578 769-4485 98-3320 •3 979-69 30664-297 773 1404 98-5677 •375 984-39 30885-255 774-3729 98-6452 •4 985-96 30959-144 779-3131 98-9604 •5 992-25 31255-875 784-2689 99-2745 •6 99856 31554-496 785-5104 99-3531 •625 1000 140 31629-446 789-2406 99 5887 •7 1004-89 31855 013 791-7322 99-7458 •75 1008 06 32005-984 794-2278 99-9028 •8 1011-24 32157-432 797-9786 1001385 •875 101601 32385-497 799-2308 100-2170 •9 1017-61 32461-759 72] CIRCUMFERENCES, AREAS, SQUARES, TABLE VIL Area. Circum. Diam. Square. Cube. 804-2496 100*5312 32 in. 1024 32768 809-2840 100-8453 1 1030-41 33076-161 810-5450 100-9240 •125 1032-01 33153-501 814-3341 1011595 •2 1036-84 33386-248 816-8650 101-3166 -25 104006 33542015 819-3999 101-4736 •3 1043-29 33698-267 823*2096 101-7093 •375 1048 84 33956-314 824-4815 101-7478 •4 1049-76 34012-224 829*5787 102 1020 •5 1056-25 34328125 834-6917 102*4161 *6 1062-76 34645-976 835-9724 102-4947 *625 1064-39 34725-743 839-8203 1027303 •7 1069-59 34965783 842-3905 102-8874 •75 107256 35026-421 844-9647 103 0444 •8 107584 35287-552 848-8333 1032801 •875 1080-76 35530-169 850-9248 103-3586 •9 1082 41 35611*289 855-3006 103*6728 33 in. 1089 3o937 860-4920 103-9869 *1 1095-61 36264-691 861-7904 104-0653 *125 109726 36346-923 865-6992 104-3011 •2 1102-24 36594-368 868-3068 104*4580 •25 1105-56 36759-953 870*92-22 104-6151 •3 1108-89 36926037 874-8477 104-8507 *375 1113-89 37176-099 876-1608 104-9294 •4 1115-56 37259-704 881-4151 105-2436 •5 1122-25 37595-375 886-6851 105-5577 •6 1128-96 37933*056 8880030 105-6361 •625 1130-64 38071-791 891-9709 105-8719 7 1135-69 38272-733 894-6176 106*0288 •75 113906 38443359 897-2723 106-1860 •8 1142-44 38614-472 901-2567 106-4215 •875 114751 38872091 902*5895 106-5002 •9 1149*21 38958-219 907-9224 106-8144 34 in. 1156 39304 913-2709 107-1285 •1 1162*81 39651-821 914-6084 107-2069 *125 1164*51 39739-095 918-6352 107*4272 •2 1169-64 40001-688 921-3211 107-5995 ■25 117306 40177-390 924-0115 107-7568 •3 1176-49 40353-607 928-0584 107-9922 •375 1181-64 40618-896 929-4109 1080710 •4 1183-36 40707-584 934-8223 108-3852 •5 1190*25 41063-625 940-2494 108-6993 •6 119716 41421-736 941-6066 108-7779 •625 1198*89 41511-587 945-6922 1090352 •7 1204-09 41781-923 948-4174 109-1703 *75 1207-56 41962796 951-1508 109-3076 *8 1211-04 42144192 955-2529 109-5630 •875 1216-26 42417-263 956-6250 109-6418 •9 1218-01 42508-549 962-1150 109-9560 35 in. 1225 42875 967-6206 110-2701 1 1232 01 , 43243-551 AND CUBES OF FRACTIONAL NUMBERS. [73 TABLE VII. Area. Circum. Diam. Square. Cube. 968-9973 110-3484 35125 1233-76 43336 017 9731420 110-5843 •2 123904 43614-208 975-9063 110-7411 •25 1242-56 43800-328 978-6790 110-8984 •3 1246 09 43986-977 982-8400 1111338 •375 1251-39 44267-943 984-2318 111-2126 •4 1253-16 44361-864 989-8003 111-5268 •5 1260-25 44738'875 995-8845 111-8409 •6 1267-36 45118016 996-7807 111-9192 •625 1269-14 45123-134 1000 9843 112-1551 *7 1274-49 45499-293 1003-7879 113-3119 *75 1278-06 45690-734 1006-6000 112-4692 •8 1281-64 45882-712 1010-8197 112-7046 *875 1287-01 46171*685 1012-2313 112-7834 •9 1288-81 46268-279 1017-8784 1130976 36 in. 1296 46656 1023-5411 113-4117 *1 1303-21 47045-831 1024-9592 113-4913 •125 130801 47252063 1029-2195 1137259 *2 1310-44 47437-928 10320646 113*8830 *25 131406 47634765 1034-9131 1140400 •3 131769 47832 147 1039-1946 114*2757 *375 1323 14 48129-239 1040-6235 114-3542 •4 1324-96 48228-544 1046-3491 114-6684 •5 1332-25 48627-125 10520904 114-9825 •6 1339-56 49027 896 1053-5281 1150611 •625 1341-39 49128 130 1057-8474 115-2967 •7 1346-89 49430-863 1060-7317 115*4538 •75 1350-56 49633-171 1063-6200 115-6108 •8 1354-24 49836-032 1067-9599 115-8465 •875 135976 50141-356 1069-4084 115 9250 •9 1361-61 50243-409 1075-2126 116-2392 37 in. 1369 • 50653 1081-0324 116*5533 •1 1376-41 51064-811 1082-4898 116-6319 •125 1378-26 51168110 1086 8679 116-8675 •2 1383-84 51478-848 1089-7915 117 0246 •25 138756 51686 703 1092-7191 1171816 •3 1391-29 51895-117 1097-1179 117-4193 •375 1396-89 52208-786 1098-5862 117-4958 •4 1398-76 52313 624 1104-4687 117-8100 •5 1406-25 52734-375 1110-3671 118-1241 •6 1413-76 53157 376 1111 8441 118-2027 •625 1415-64 53263-4*7 1116-2811 118-4383 •7 1421-29 53582-633 1119-2440 118-5954 •75 1425 06 53796-109 1122-2109 118-7524 •8 1428-84 54010 152 1126-6685 118-9881 •875 1434-50 54332-278 1128-1564 119 0666 •9 1436-41 54439-939 11341176 119-3808 38 in. 1444 54872 1140-0926 119-6949 •1 1451-61 55306-341 1141-5919 119-7735 •125 1453-51 55415-282 1146 0870 1200091 •2 1459-24 55742 968 74] CIRCUMFERENCES, AREAS, SQUARES, TABLE VII. Area. Circum. Diam. Square. Cube. 1149-0892 120-1662 38-25 1463-06 55962140 11520954 120-3232 •3 1466-89 56181-887 1156-6119 120 5589 •375 1472-64 56512-583 11581194 120-6374 •4 1474-56 56623 104 1164 1591 120-9516 •5 1482-25 57066-625 1170-2145 121 2657 •6 1489-76 57512-456 1171-7309 121-3443 •625 1491-89 57624-274 1176-2857 121-5799 •7 1497-69 57960-603 1179-3271 121-7370 •75 1501 -56 58185 546 1182*3725 121-8940 •8 1505-44 58411072 1186-9480 122 1297 •875 1511-26 58751 450 1188-4651 122-2082 •9 1513-21 58863-869 1194-5394 122-5224 39 in. 1521 59319 12007273 122-8365 1 1528-81 59776-471 1202-2633 122-9151 •125 1530-76 59891-204 1206-8770 123 1507 •2 1536-64 60236-288 1209-9577 123-3078 •25 1540-56 60466078 1213-0424 123-4648 •3 1544-49 60698'457 12176768 123-7005 ■375 1550-39 61046-629 1219-2243 1237790 •4 1552-36 61162-984 ] 225-4203 1240932 •5 1560-25 61629-875 1231-6328 124-4073 •6 1568T6 62099 136 1233-1884 1244859 •625 1570 14 62216-822 1237 8610 124 7215 •7 1576-09 62570-773 1240-9810 124 8786 •75 158006 62807-484 1244-1210 1250356 •8 1584-04 63044-792 1248-7982 125 2713 •875 159001 63401-872 1250-3646 125-3498 ■9 159201 63521-199 1256-6400 125 6640 40 in. 1600 64000 1262-9310 1259781 1 160801 64481 -201 1264-5062 126-0567 •125 161001 64601 875 1269-2388 126-2923 •2 1616-04 649641,08 1272-3970 126 4494 •25 162006 65207 515 1275-5602 126-6064 •3 1624-09 65450-827 1280-3124 126-8421 •375 1630-14 65816 926 1281 -8984 126-9206 •4 1632 16 65939-264 1288-2523 127-2348 •5 1640-25 66430 125 1294-6219 127-5489 •6 1648-36 66923-416 1296-2168 127-6275 •625 1650-39 67047 110 1301 0071 127 8631 •7 1656-49 67419143 1304-2057 128-0202 •75 1660-56 67667-925 1307-4082 128 1772 •8 1664-64 67917-312 1312-2193 128-4129 •875 1670-76 68292-539 1313-8249 128-4914 •9 1672-81 68417-929 1320-2574 128-8056 41 in. 1681 68921 1326-7055 1291197 •1 1689-21 69426-531 1328-3200 129-1983 •125 1691-26 69553-297 1333 1693 129-4323 •2 1697-44 69934528 1336-4071 129-5910 •25 1701-56 70189453 1339 6489 129-7480 •3 1705-69 70444-997 AND CUBES OF FRACTIONAL NUMBERS. [70- table VII. Area. Circum. Diam. Square. Cube. 1344.5189 129-8939 41-375 171189 70829-473 1346T441 1300622 •4 1713-96 70957944 1352-6551 130-3764 •5 1722-25 71473-375 1359-1818 130-6905 •6 1730-56 71991-296 1360-8159 1307961 •625 1732-64 72121-164 13657242 131 0047 *7 1738-39 7-2511-713 1369 0012 131 1618 •75 174306 72772-S59 1372-2822 131 3188 •8 1747-24 73034-632 1377-2111 131-5545 •875 1753-51 73428-465 1378-8560 131-6320 •9 1755-61 73560-059 1385-4456 131-9472 42 in. 1764 74088 1392*0508 132-2613 •1 1772-41 74618-461 13937045 132-3399 •125 1774-51 74751-469 1398-6717 132-5755 •2 1780-84 75151-448 1401-9880 1327326 •25 1785*06 75418890 1405-3083 132-8896 •3 1789-29 75686-967 1410-2961 133 1253 •375 1795-64 76090-270 1411-9607 133-2038 *4 179776 76225-024 1418-6287 133-5180 •5 1806-25 76765-625 1425-3125 133 8321 •6 181476 77308-776 1426-9859 133-9107 •625 1816-89 77444-961 1432 0119 134 1463 •7 1823-29 77854-483 1435-3675 134*4034 •75 1827-56 78128-296 1438-7271 134-4604 •8 1831-84 78402-752 14437738 134-6961 •875 1838-26 78815-637 1445 4580 134-7746 •9 1840-41 78958-589 1452-2046 135-0888 43 in. 1849 79507 1458-9668 135-4029 •1 1857-61 80062-991 1460-6599 135-4815 •125 1859-76 80-202-391 1465-7448 1357170 -2 1866-24 80621-568 1469-1397 135 8742 •25 1870-56 80901-828 1472-5385 1360332 •3 1874-89 81182-737 1477-6342 136-2669 •375 1881-39 81605-317 1479-3480 136-3454 •4 1883-56 81746-504 1486 1731 136-6596 •5 1892-25 82313-875 14930139 136-9737 •6 1900-96 82881-856 1494-7266 137-0523 •625 1903-14 83024508 1499-8705 137-2879 •7 1909-69 83453-453 15033046 137-4450 •75 1914T6 83740-234 15067427 137 6020 •8 1918-44 84027-672 1511-9072 137-8377 •875 1925-01 84460059 1513-6287 137-9162 •9 1927*21 84604-519 1520-5344 138-2304 44 in. 1936 85184 1527-4537 138-5445 1 1944-81 85766 121 1529-1860 138-6231 •125 194701 85912 063 1534-3888 138-8887 •2 1953-64 86350-888 1537-8622 1390158 •25 195806 86644-265 1541-3396 139-1728 •3 1962-49 86938 307 1546-5530 139-4085 •375 1969-14 87380-614 1548-3061 139-4870 •4 1971-36 87528-384 76] CIRCUMFERENCES, AREAS, SQUARES, TABLE VII. Area. Circum. Diam. Square. Cube. 1555 21883 139-8012 44-5 1980-25 88121125 1562-2882 140-1153 •6 1989 16 88716-536 15640382 140-1939 •625 1991-39 88865-805 1569-2998 140-4295 •7 199809 89314-623 1572-8125 140*5866 •75 2002-56 89614-652 1576-3292 140-7436 •8 2007-04 89915-392 1581-6115 140-9793 •875 2013-76 90367-731 1583-3742 141 0578 •9 2016-01 90518-849 1590-4350 141-3720 45 in. 2025 91125 1597-5114 141-6861 •1 2034-01 91723 851 1599-2830 141-7647 •125 2036-26 91886-485 1604-6036 1420003 •2 2043-04 92345-408 1608-1555 142 1574 •25 2047*56 92652-203 1611-7114 142-3144 •3 2052-09 92959-677 1617-0427 142-5501 •375 2058-89 93422 161 1618-8350 142-6286 •4 2061-16 93576-664 1625-9743 142-9428 •5 2070-25 94196-375 16331293 143-2569 •6 2079-36 94818-818 1634-9205 143-3355 •625 2081-64 94974-852 1640-3020 143-5711 •7 2088-49 95443-993 1643-8912 143-7282 •75 2093-06 95757 609 1647-4864 143 8852 •8 2097-64 96071-912 1652-8865 144 1209 •875 2104-515 96544-653 1654-6885 144-1994 •9 2106-81 96702-579 1661-9064 144-5136 46 in. 2116 97336 1669 1399 144-8277 •1 2125-21 97972-981 1670-9507 144-9063 •125 2127-51 98131-657 1676 3891 1451419 "2 2134-44 98611-128 16800196 145-2990 •25 2139-06 98931-640 1683-6541 145-4560 •3 2143-69 99252-847 1689 1031 145-6917 •375 2150-64 99735-957 1690-9347 145-7702 •4 2152-96 99897 344 1698-2311 146-0844 •5 2162-25 100544-625 1705-5432 146-3985 •6 2171-56 101194-696 17073732 146-4771 •625 2171 89 101357-649 1712-8710 146-7127 •7 2180-89 101847 563 1716-5407 146-8698 •75 2185-56 102175046 1720-2144 147 0268 •8 2190-24 102503-232 1725-7324 147-2625 •875 219726 102996-825 1727-5736 1473410 ■9 219961 103161-705 1734-9486 147-6552 47 in. 2209 103823 1742-3392 147-9693 •1 2218-41 104487-111 1744 1893 148 0479 •125 2220-76 104653579 1749-7455 148-2835 •2 2227-84 105154048 1753-4545 148-4406 •25 2232-56 105488-578 1757-1675 148-5976 •3 2237 29 105823-817 1762-7344 148 8333 •375 2244-39 106328-004 1764-6045 148-9118 •4 2246-76 106496-424 17720587 149-2260 •5 2256-25 -107171-875 1779-5279 149-5361 •6 2265-76 107850-176 AND CUBES OF FRACTIONAL NUMBERS. L77 TABLE VII. Area. Circutn. Diam. Square. Cube. 1781-3976 149-6107 47-625 2268-14 108020196 1787-0127 149-8543 •7 2275-29 108531-333 1790-7610 150-0114 •75 2280-06 108872-984 1794-5133 150-1684 •8 2284-84 109215-352 1800-1491 150 4041 •875 2292-01 109730-246 1802-0296 150-4826 •9 2294-41 109902-239 1809-5616 150-7968 48 in. 2304 110592 1817-1092 151-1109 •1 2313-61 111284-641 1818-9986 151-1895 •125 231601 111458-250 1824-6726 151-4251 •2 2323-24 111980 168 1828-4602 151-5822 •25 2328 06 112329015 18322518 151-7392 ■3 2332-29 112678-587 18379364 151-9749 •375 2340-14 113204-301 1839-8466 152-0534 •4 2342-56 113379-904 1847-4571 1523676 •5 2352-25 114084-125 1855-0833 1526817 •6 2361-96 114791-256 1856-9924 152-7603 •625 2364-39 114968-493 1862-7253 152-9959 •7 2371 69 115501-303 18685521 153T530 •75 2376-56 115857-421 1870-8829 153-3100 •8 238L44 116214-272 1876T365 153-5457 •875 2388-76 116750-918 18780563 153-6242 •9 2391-21 116930-169 1885-7454 153-9384 49 in. 2401 117649 1893-4501 154-2525 •1 2410-81 118370-771 1895-3788 154-3311 •125 2413-26 118551-672 1901-1706 154-5667 •2 2420-64 119095-488 1905-0367 154-7238 •25 2425-56 119458-953 1908-9068 154-8808 •3 2430-49 119823-157 1914-7093 155-1165 •375 2437-89 120370 548 1916-6587 155 1950 •4 2440-36 120553-784 1924-4263 155-5092 •5 2450-29 1-21287-375 1932-2096 155-8233 •6 2460-16 122023-936 1934 1579 155-9019 •625 2462-64 122-208-539 1940-0086 156 1375 •7 2470-09 122763-473 1943-9140 156 2946 •75 2475-06 123134-359 1947-8234 156-4516 •8 2480-04 123505-992 1953-6947 156-6873 •875 2487-51 124064-336 1955-6538 156-7558 •9 2490-01 1-24251-499 1963-5000 157-0800 50 in. 2500 125000 1971-3618 1573941 •1 2510*01 125751-501 1973-3297 157-4727 •125 2512-51 125937-844 1979-2394 157-7083 •2 252004 126506 008 1983-1840 1578654 •25 252506 126884-390 1987-1326 158-0224 •3 2530-09 127263-527 1993-0529 158-2581 •375 2537-64 127833-645 1995-0416 158-3366 •4 2540-16 128024064 2002-9663 158-6508 •5 2550-25 127878 625 2010-9067 1589649 •6 2561-36 129554-216 2012-8943 1590435 •625 256289 129746-336 2018-8628 159-2791 •7 2570-49 130323-843 78] CIRCUMFERENCES, AREAS, SQUARES, TABLE VII. Area. Circum. Diam. Square. Cube. 20228467 159-4362 50-75 257556 130709-797 2026-8346 159-5932 •8 2580-64 131096-512 2032-8238 159-8289 •875 2588-26 131678012 2034 8770 159-9072 •9 2590 81 131872-229 2042-8254 160-2216 51 in. 2601 132651 2050-8443 160-5357 •1 2611-21 133432-831 2052-8515 160-6143 •125 2613-76 133628-766 2058-8784 160-8499 •2 2621-44 134217-728 20620021 161 0070 •25 '2626-56 134611-328 2066-9293 161-1640 •3 2631-69 135005-697 2072-9674 161-3997 •375 2639-39 135598-692 2074-9953 161-4782 •4 2641-96 135796-744 2083-0771 161-7924 •5 2652-25 136590-875 2091 1746 162-1065 •6 2662-56 137388 096 2093-2014 162-1851 •625 2665 14 137587 883 2099-2878 162-4207 •7 2672-89 138188-413 2103-3502 162-5778 •75 267806 138589-734 2107-4166 162-7340 •8 2683-24 138991-832 2113-5236 162-9705 •875 2691 01 139956-434 2115-5612 163-0490 •9 2693-61 139798-359 21237216 163-3632 52 in. 2704 140608 2131-8976 163-6773 •1 2714-41 141420-761 2133-9440 163-7559 •125 2717 07 141624-438 2140-0893 163-9953 •2 2724-84 142236-648 2144-1910 164-1486 •25 273006 142645-765 2148-2967 164-3056 •3 2735-29 143055-667 2154-4626 164-5413 •375 274314 143671 589 2156-5199 164-6198 •4 2745-76 143877-824 2164-7587 164-9340 ■5 2756-25 144703 125 2173-0133 165-2481 •6 2766-76 145531-576 2175-0794 165-3267 •625 2769-39 145739 180 2181-2835 165-5623 •7 2777-29 146363 183 2185-4245 165-7194 •75 2782-56 1467SOT 72 2189-5695 165-8764 •8 2787 84 147197 952 21957943 166-1121 •875 2795-76 147826-106 2197-8712 166 1906 •9 2798-41 148035-889 2206-1886 166-5U48 53 in. 2809 148877 2214-5216 166-8189 •1 2819-61 149721-291 2216-6074 166-8975 •125 2822-26 149932-860 2222 8704 167-1330 •2 2830-24 150568-768 2227 0507 167-2902 -25 2835 56 150993-703 2231-2350 167-4472 •3 294089 151419-437 2237-5107 167-6829 •375 2848-89 152059-535 2239-6152 167-7614 •4 2851-56 152273-304 2248-0111 168 1756 •5 2862-25 153130-375 2256-4227 168-3894 •6 2872-96 153990-656 2258-5281 168-4683 •625 2875-64 154206-227 2264-8701 168-7049 •7 2883-69 154854T53 22690696 168-8610 •75 2889-06 155287-109 2273-2931 1690180 •8 2894-44 155720-872 AND CUBES OF FRACTIONAL NUMBERS. 179 TABLE VII. Area. Circum. Diam. Square, j Cube. 2279-6357 169-2537 53-875 2902-51 156373 028 2281-7519 169-3322 ■9 2905-21 156590-819 2290-2-264 169 6464 54 in. 2916 157464 2298-7165 169-9605 •1 2926-81 158340-421 2300-8415 1700391 •125 2929-51 158560032 2307-2224 170-2747 •2 2937-64 159220-088 2311-4812 170-4318 -25 294306 15966M40 2315-7440 170-5888 •3 2948-49 160103-007 23221455 170-8245 •375 2956-64 160767-332 2324-2813 170-9030 •4 2959-36 160989-184 2332-8343 171-2172 -5 2970-25 161878-625 2341-4030 171-5313 •6 2981T6 162771-336 2343-5477 171 '6099 •625 2983-89 162995-024 2349 9874 171 8455 •7 2992 09 163667-323 2354-2855 172-0026 •75 2997-56 164116-547 2358-5876 172-1596 •8 3003-04 164566-592 2365-0480 172-3953 •875 3011-26 165243-199 2367-2034 172-4738 •9 301401 165469149 2375-8350 1727880 55 in. 3025 166375 2384-4822 1731021 •1 3036-01 167284-151 2386-6465 173-1807 •125 3038-76 167511-953 2393 1452 173-4163 •2 304704 168196-608 2397-4825 173-5734 -25 3052-56 168654-078 2401-8238 173-7304 •3 305809 169112-377 2403-3432 173-9661 •375 3066-39 169801-379 2410-5512 1740446 •4 3069-16 170031-464 2419-2283 174-3588 •5 3080-25 170953-875 2427-9541 174-6729 •6 3091-36 171879-616 2430-1830 174-7515 •625 3094-14 172111-570 2436-6956 174-9771 •7 3102-49 172808-693 2441 0722 175-1442 •75 3108-06 173274-484 2445-4528 175-3092 •8 3113-64 173741 112 2452 0310 175-5369 •875 312201 174442-621 2454-2257 175-6154 •9 3124-81 174676-879 24630144 175-9296 56 in. 3136 175616 2471-8187 176-2437 •1 314721 176558-481 2474-0222 176-3223 •125 315001 176794-625 2480 6387 176-5579 •2 3158-44 177504-328 24850546 176-7150 •25 3164-06 177978-515 2489-4745 176-8720 •3 3169-69 178453-547 2496-1116 177-1077 •375 3178-14 179167-676 2498-3259 177 1862 •4 3180 96 179406-144 2507-1931 . 177-5004 •5 3192-25 180362-125 2516-0760 177-8145 •6 3203-56 181321-496 25182992 177-8931 •625 3206-39 181561-867 2524-9736 178-1287 •7 3214-89 182284-263 2529-4297 178-2858 •75 3220 56 182766-921 2533-8888 178-4428 •8 3226-24 183250-432 2540 5849 178-6785 •875 3234-76 183977-293 2542-8188 ! 178-7570 •9 3237-61 184220009 80 ] CIRCUMFERENCES, AREAS, SQUARES, TABLE VII. Area. Circum. Diara. Square. Cube. 25517646 179-0712 57 in. 3249 185193 2560-7260 179-3853 •1 3260-41 186169-411 2562-9668 179-4639 •125 3263-26 186414-048 2569-7031 179-6995 •2 3271-84 187149-411 2574 1975 179-8566 •25 3277-56 187640-453 2578-6959 180-0136 •3 3283-49 188149-240 2585-4509 180-2493 •375 3291-89 188872-223 2587-7045 180-3278 •4 3294-76 188132-517 2596-7287 180-6420 •5 3306-25 189119-224 2605-7687 180-9561 •6 331776 190109-375 26080311 181 0347 •625 3320-64 191351-914 2614-8243 181-2803 •7 3329-29 191102-976 2619-3580 181-4274 •75 3335 06 192599-859 2623-8957 181-5844 •8 3340-24 192100033 2630-7095 181-8201 •875 3349-51 193833-215 2632-9828 181-8986 •9 3352-41 193100-539 26420856 182-2128 58 in. 3364 195112 2651-2046 182-5269 •1 3375-61 196122-941 2653-4861 1826055 •125 3378-51 196376-219 2660-3382 182-8411 •2 3387-24 197137368 2664-9112 182-9982 •25 3393-06 197645-890 2669-4882 1831552 •3 3398-89 198155-287 2676-3609 183-3909 •375 3407 64 198921 020 2678-6538 183-4694 •4 3410-56 199176-704 2687-8351 183-7836 •5 3422-25 200201-625 2697 0321 184-0977 •6 3433-96 201230056 2699-3338 184T763 •625 3436-89 201487-711 2706-2449 184-4119 •7 3445-69 202262-003 2710-8571 184-5690 •75 3451-56 202779-296 27154733 184-7260 •8 3457-44 203297-472 2722-4050 184-9617 •875 3466-26 204076-387 2724-7175 1850402 •9 3469-21 204336-469 2733-9774 185-3544 59 in. 3481 205379 2743-2529 185-6685 •1 3492-81 206425-071 2745-5743 185-4741 •125 3495-76 206687-141 2752-5442 185-9827 •2 3504-64 207472-688 2757-1957 186-1398 •25 3510-56 208000 828 2761-8512 186-2696 •3 3516-49 208527 857 2768-8418 186-5325 •375 3525-39 209320 066 277M739 186-6110 •4 3528-36 209584 584 2780-5123 186-9252 •5 3540-25 210644-875 2789-8661 187 2393 •6 3552-16 211708-736 2792-2074 187-3179 •625 3555T4 212! )75-258 2799-2362 187-5535 •7 3564-09 212776-173 2803-9270 1877106 •75 3570-06 213311-234 2808-6218 187 8676 •8 3576-04 213847 192 2815-6712 188 1033 •875 3585-01 214642-809 28180230 188-1818 •9 358801 214921-799 2827-4400 188-4960 60 in. 3600 216000 2836-8726 188-8100 •1 3612-01 217081-801 AND CUBES OP FRACTIONAL NUMBERS. [81 TABLE YII. Area. Circum. Diam. Square. Cube. 2839-2332 188-8887 60-125 361501 217352-813 2846-32] 0 189-1243 •2 3624-04 218167-208 2851-0510 189-2814 •25 3630-62 218711-265 28557850 189-4384 •3 3636 09 219256-227 2862-8934 189-6741 •375 3645 14 220075-363 2865-2648 189-7526 •4 3648-16 220348-864 2874-7603 190-0668 •5 3660-25 221445-125 2884-2615 1903809 •6 3672-36 222454 016 2886 6517 190-4595 •625 3675-39 222820-555 2893-798-1 190-6951 •7 3684-49 223668-543 28985677 190-8522 •75 3690-56 224201-672 2903-3410 191-0092 •8 3696-64 2-24755-712 2910-5083 191-2449 •875 3705-76 225588-481 2912-8993 191-3234 •9 3708-81 225866-529 2922-4734 191-6376 61 in. 3721 226981 29320631 191-9517 •1 3733-21 228099-131 2934-4630 1920303 •125 3736-26 228379-235 2941-6685 192-2659 •2 3745-44 229220-928 2946-4771 192-4230 •25 3751-56 229783-203 2957-2897 192-5800 •3 3757-69 230346-397 2958-5159 192-8157 •375 3766-89 231192-911 2960-9265 192-8942 •4 3769-96 231475-544 ' 2970-5790 193-2084 •5 3782-25 232608-375 2980-2474 193-5225 •6 3794-56 233744-896 2982-6669 193-6011 •625 3797-64 234029-602 2989-9314 193-8367 •7 3806-89 234885-113 2994-7792 193-9938 •75 3813-06 235456-609 2999-6300 194-0508 •8 3819-24 236029 032 3006-9161 194-3865 •875 3828-51 236659-403 3009-3464 194-4650 •9 3831-61 ' 237176-659 3019 1776 194-7792 62 in. 3844 238328 3028-8244 195-0533 ■1 3856-41 23.9483-061 3031-2.35 195-1719 •125 3859-51 239772-406 3038-5809 195-4075 •2 3868-84 240641-848 3043-4740 195-5646 •25 387506 241222-640 3048-3651 195-7216 •3 3881-29 241804-367 3055-7091 195-9573 •375 3890-64 242678-707 3058-1591 1960358 •4 3893-76 242970-624 3067-9687 196-3500 •5 3906-25 244140-625 3077-7941 196-6641 •6 3918-76 245314-676 3080-2529 196-7427 •625 3921-89 245608-399 3087-6341 196-9783 •7 3931-29 246491-883 30925615 197-1354 •75 3937-56 247082-047 3097-4919 1972924 •8 3943-84 247673-152 3104-8948 197-5281 •875 3953-26 248561-574 31073644 197-6066 •9 3956-41 248858 189 3117-2526 197-9208 63 in. 3969 250047 3127-1564 198-2349 1 3981-61 251239-591 3129-6349 198-3135 ■125 3984-76 251538-328 31370758 198 5491 •2 3994-24 252435-968 82J CIRCUMFERENCES, AREAS, SQUARES, TABLE VII. Area. Circum. Diara. Square. Cuie. 3142-0417 198-7062 63-25 4000-56 253035-578 3147-0114 198 7632 •3 4006-89 253638 137 3154-4732 1990989 •375 4016-39 254538754 3156-9064 199 1774 •4 4019-56 254840 104 3166-9291 199-4916 •5 4032-25 256047-875 3176-9115 199-8057 •6 4044 96 257259 456 3179-4096 199-8843 •625 4048-14 257562-945 3186-9097 200-1199 •7 4057-69 258474-853 3191-9146 200-2970 •75 4064-06 259083-984 3196-9235 200-4340 ■8 4070-44 259694-072 3204-4442 200-6697 •875 408001 260610-996 3206-9531 200-7482 •9 4083 £1 260917-119 3216-9984 201 0624 64 in. 4096 262144 3227 0593 201-3765 •1 • 4108-81 263374-721 3229-5770 201-4551 •125 411201 263683-000 3237-1360 201-6907 •2 4121-64 264609-288 3242-1782 201-8478 •25 412806 265228 015 3247-2284 202-0048 •3 4134-49 265847-707 3254-8080 202-2405 •375 414414 266779051 3257 3365 202-3190 ■4 4147-36 267089-984 3267-4603 202-6322 •5 4160-25 268336-125 3277-5998 202-9473 •6 4173-16 269586 136 3280 1372 2030259 ■625 4176-39 269899-242 32877550 203-2615 •7 4186 09 270840023 3292-8385 203-4186 •75 4192-56 271468-422 3297-9260 203-5756 •8 419904 272097792 3305-5645 203-8113 •875 4208-76 273043-668 3308-1126 203-8898 •9 421201 273359-449 3318-3150 204-2040 65 in. 4225 274625 3328-5340 204-5181 •1 4238 01 275894-451 3331-0900 204-5917 •125 4241-26 276212-422 33:38-7668 204-8323 •2 4251-04 277105-808 33438875 204-9894 •25 4257-56 277805-953 3349-0162 205-1464 •3 4264 09 278445077 3356-7137 205-3821 •375 4273-89 279405-608 3359-2814 205-4606 •4 4277-16 279726-264 3369-5623 205-7748 •5 4290-25 281011-375 3379-8589 206-0889 •6 4303-36 282300-416 3382-4355 206-1675 •625 4306-64 282623-289 3390 1712 206-4031 •7 4316-49 283593-393 3395-3332 206-5602 •75 432306 284241-359 3400-4992 206-7172 ■8 4329-64 284890-312 3408-2555 206-9529 •875 4339-41 285865-590 3410-8429 207-0314 •9 4342-81 28619H79 3421-2024 2073436 66 t». 4356 287496 3431 -5775 207-6597 •1 4369-21 288804-781 3434-1737 2077383 •125 4372-51 289132-594 3441 9633 207-9739 •2 4382-44 290117-528 3447-1676 208 1310 •25 438906 290775-390 3452-3749 208-2880 •3 4395-69 291434-247 AND CUBES OF FRACTIONAL NUMBERS. TABLE VII. Area. Circum. | Diam. | Square. Cube. 3460-1901 208 5237 66-375 4405-64 292424-395 34627971 2086022 *4 4408*96 292754-944 3473-2351 208-9164 *5 4422*25 294079-625 3483*6888 209-2305 *6 4435-56 295408-296 3486*3047 209*3091 •625 4438-09 295741 086 3494-1640 209*5447 *7 4448 89 296740-936 3499-3987 2097018 *75 4455*56 297408*797 3504*6432 209*8583 •8 4462-24 298077-632 3512-5174 210-0945 •875 4472-26 299082-762 3515-1430 210*1730 •9 4475-61 299418*309 3525-6606 210*4872 67 in. 4489 300763 3536-59-28 210-8013 *1 4502-41 302111711 3538-8283 210-8799 *125 4505-76 302449-516 3546-7407 211*1155 *2 4515-84 303464-448 35520185 211-2726 *25 4522-56 304142-328 3557-3043 211*4296 •3 4529-29 304821-217 3565-2374 211-6653 *375 4539-39 305841-442 3567-8837 211-7438 *4 4542-76 306182 024 3578-4787 212-0580 *5 4556-25 307546-875 35890895 212-3720 *6 4569-76 308915-776 3591-7446 212-5407 •625 4573-14 309258-633 3599*7159 212-6863 •7 4583-29 310288*733 36050350 212-8434 •75 4590-06 310976-734 3610*3581 213*0004 •8 4596-84 311665-752 3618-3500 213-2361 *875 4607-01 312701-184 3621-0160 213-3146 •9 4610*41 313046-839 3631-6896 213*6288 68 in. 4624 314432 3642-3788 213-9429 •1 4637-61 315821-241 36450536 214 0215 *125 4641-01 316169-187 3653-0838 214*2571 *2 4651*24 317214-568 3658*4402 214-4120 *25 4658-06 317912-766 3663-8040 214-5712 *3 4664-89 318611-987 3671 8554 214*8069 *375 4675-14 319662-738 3674*5410 214-8854 *4 4678-56 320013-504 3685*2931 215*1996 *5 4692-25 321419*129 36960060 215-5137 *'6 4705*96 322828 856 3698-7554 215-5923 •625 4709-39 323181-930 3706-8485 215-8279 •7 4719*69 324242*703 3712*2421 215-9850 •75 4726-56 324951-472 3717-6437 2161420 *8 4733-44 325660-672 3725 7535 216-3777 •875 4743-765 326726-977 3728-4587 216-4562 •9 4747-21 327082-769 3739-2894 216-7704 69 in. 4761 328509 3750*1357 2170844 *1 4774-81 329939-371 3752-8498 2171631 *125 4778-26 330297-609 3760-9978 217*3987 •2 4788-64 331373-888 3766*4327 217-5558 *25 4795-56 332092-703 3771-8756 2177128 *3 4802-40 332812*557 3780 1443 217-9485 •375 4812-89 333894-285 3782-7691 218 0270 •4 4816*36 334255-384 841 CIRCUMFERENCES, AREAS, SQUARES, TABLE VII. Area. Circura. | Diam. Square. Cube. 37936783 218 3412 69-5 4830-25 335702-375 3804-6032 218-6553 •6 484416 337153-536 3807 3369 218-7339 •625 4847-64 337516-977 3815-5438 218-9695 •7 485809 338608-873 3821 0200 2191266 •75 4865-06 339338 109 3826-5002 219-2836 •8 487204 340068-392 3834-7277 2195193 •875 4882-51 341165-778 38374722 219-5978 •9 4886-01 341532099 3848-4600 219-9120 70 in. 4900 343000 3859-4952 220-2261 •1 4914-01 344472-101 3862-2167 220-3047 •125 491751 344840-781 3870-4826 220-5403 •2 4920-84 345948-408 3875-9960 220-6974 •25 4935 06 346688-141 3881-5174 2208544 •3 4942-09 347428-927 3889-8039 221-0901 •375 4952-64 348542082 3892-5680 221 1686 •4 4956-16 348913-664 3903-6343 221-4828 •5 4970-25 350402-625 3914-7163 221-7969 •6 4984-36 351895-816 3917-4893 221-8755 •625 498789 352259-774 3925-8140 2221111 •7 4998-49 353393-243 3931-3687 222-2682 •75 5005-56 354413-547 3936-9274 222-4252 •8 5012-64 354894-912 3945-2728 222-6609 •875 5023 26 356023-949 39480565 222-7394 •9 5026-81 356400-829 3959-2014 • 2230536 71 in. 5041 357911 3970-3619 223-3677 •1 5055-21 359425-431 3973-1545 223-4463 •125 5058-76 359804-703 3981-5381 223-6819 •2 5069-44 360944-128 3987 1301 223-8390 •25 5076-56 361705-078 3992-7301 223-9960 •3 5083-69 362467-097 4001-1344 224-2317 •375 5094-39 363612-129 4003-9373 224-3102 •4 5097-96 363994-344 4015-1611 224-6244 •5 5112-25 365525-875 4026-4002 224-9385- •6 5126*56 367061-696 4029-2124 225-0171 •625 513014 367446-320 4037-6550 225-2527 •7 514089 368601-813 4043-2882 225-4098 •75 5148-06 369373-484 4048-9294 225-5668 •8 5155-24 370146-232 40573886 225-8025 •875 516601 371307-371 4060-2116 225-8810 9 5169-61 371694-959 4071-5136 226 1952 72 in. 5184 373248 4082-8332 226-5093 •1 5198-41 374805-361 4085-6631 226-5879 •125 520201 375195-375 4094-1645 226-8235 •2 5212-84 376367-048 4099-8350 226-9806 •25 5220 06 377149-515 41055125 227 1376 •3 5227-29 377933067 41140356 227 3733 •375 523814 379110-425 4116-8793 227-4518 •4 5241-76 379503-424 4128-2587 227-7660 •5 5256-25 381078 125 4139-6524 2280801 •6 5270-76 382657-976 AND CUBES OF FRACTIONAL NUMBERS. [85 TABLE VII. Area. Circum. | Diam. Square. Cube. 4142-5064 228-1587 72-625 5274-39 383052-617 4151-0667 228-3943 •7 5285-29 384240-583 4156-7785 228-5514 •75 5292-56 385033-921 4162-4943 228-7084 •8 5299-84 385828-352 4171-0753 228-9441 •875 5310-76 387022-043 4173-9376 229-0226 •9 5314-41 387420-489 4185-3966 229-3368 73 in. 5329 389017 4196-8712 229-6509 •1 5343-61 390617-891 4199-7424 229-7295 •125 5347-26 391018-797 4208-3614 229-9651 •2 5358-24 392223-168 4214-1107 230*12*22 •25 5365-56 393027-453 4219-8678 230-2792 •3 5372-89 393832-837 4228-5077 230-5149 •375 5383-89 395042-972 4231-3896 230-5934 •4 538756 395446-904 4242-9271 230-9076 •5 5402-25 397065-375 4254-4803 231-2217 •6 5416-96 398688-256 42573711 231-3003 •625 5420-64 399094-664 4266-0493 231-5359 •7 5431-69 400315-553 4271-8396 231-6930 •75 5439-06 401130-359 4277-6399 231*8500 •8 5446-44 401947-272 4286-3327 2320857 •875 545751 403173-964 4289-2343 232-1642 •9 5461-21 403583-419 4300-8504 232-4784 74 in. 5476 405224 4312-4821 232-7925 •1 5490-81 406869-021 4315-3926 232-8711 •125 5494-51 407280-968 4324-1296 2331067 •2 5505-64 408518-488 4329-9572 233-2638 •25 551306 409344-890 4335-7928 233-4208 •3 5520-49 410172-407 4344-5505 233-6565 •375 5531-64 411415-769 4347 4717 233-7350 •4 5535-36 411830-784 4359-1663 234-0492 •5 5550-25 413493 625 4370-8766 234-3633 •6 5565-16 415160-936 4373-8067 234-4419 •625 5568-89 415578-461 4382-6026 234-6775 •7 5580-05 416832-723 4388-4715 234-8346 •75 5587-56 417670-296 4394-3448 234-9916 •8 5595-04 418508-992 4403-1610 235-2273 •875 5606-26 419769-136 4406-1018 235-3058 *9 561001 420189-749 4417-8750 235-6200 75 in. 5625 421875 4429-6638 235-9341 •1 564001 423564-751 4432-6135 236-0127 •125 5643-76 423987-890 4441*4684 236-2483 •2 5655-04 425259008 4447-3745 236-4054 •25 5662-56 426107828 4453*2886 236-5624 •3 567009 426957-777 4462-4260 236-7981 •375 5681-390 428234-816 4465*1246 236-8766 •4 5685 16 428661-064 4476-9763 237-1908 •5 5700-25 430368-875 4488*8437 237-5049 •6 5715-36 432081-216 4491-8130 237-5835 •625 5719-14 432510 007 4500-7268 237-8191 •7 5730-49 433798093 86] CIRCUMFERENCES, AREAS, SQUARES, TABLE VII. Area. Circuvn. Diam. Square. Cube. 4506-6742 237-9762 75-75 5738-06 434658-234 4502-6256 238-1332 •8 5745-64 435519 512 4521-5600 238 3689 •875 5757-01 436813-558 4524-5401 238-4474 •9 5760-81 437245-479 4536-4704 2387616 76 in. 5776 438976 4548-4163 239 0757 *1 5791-21 440711 081 4551-4023 239 1543 •125 5795-01 441145-564 45603787 239-3899 *2 5806-44 442450-728 4566*2626 239-5470 •25 5814-06 443322-265 45723553 239*7040 *3 5821-69 444194-947 4581*3486 ,239-9397 *375 5833-14 445506-113 4584-3583 2400182 •4 5836-96 445943-744 4596 3571 240-3324 •5 5852-25 447697-125 4608-3816 240-6465 •6 5867-56 449455-456 4611-3902 240-7251 •625 587M9 449895 304 4620-4218 240-9607 •7 5882-89 451217-663 4626*4477 241-1178 •75 5890-56 452100-671 4632-4776 241-2748 •8 5898-24 452984-832 4641-5299 241-5105 •875 5909-76 454313-230 4644-5492 241-5987 •9 5913-61 454756-609 4656-6366 241-9032 77 in. 5929 456533 4668*7396 242-2173 1 5944-41 458314-011 4671-7678 242-2959 •125 5948-26 458759-984 4680-8583 242-5315 •2 5959-84 460099-648 <686-9215 242-6886 •25 5967*56 460994-203 4692-9927 242-8456 •3 5975-29 461889-917 4702 1039 2430813 •375 5986-89 463235-660 4705 1429 2431598 •4 5990-76 463684-824 4717-3087 243-4740 •5 6006*25 465484-375 4729*4903 243-7881 •6 6021-76 467288-576 4732-5381 243-8667 •625 6025 64 467740-351 4741-6875 244-1023 *7 6037-29 469097433 4747-7920 244*2594 •75 6045-06 470003-609 4753-9615 244-4164 •8 6052*84 470910-952 4763-0705 244-6521 ■875 6064-51 472274-152 4766-1292 244*7316 •9 6068-41 472729-139 4778-3736 245-0448 78 in. 6084 474552 4790-6336 245-3589 •1 6099-61 476379-541 4793-7012 245-4375 *125 6103-51 476837-156 4802-9094 245-6731 •2 6115-24 478211-768 4809-0512 245-8302 •25 6123*06 479129-640 4815-2010 245-9872 •3 6130-89 480048-687 4824-4299 246-2229 •375 6142*64 481429-457 4827-5082 246*3014 •4 6146-56 481890-304 4839-8311 246*6156 •5 6162-25 483736-625 4852 1697 246-9297 •6 6177-96 485587-656 4855-2568 247-0083 •625 6181-89 486051-148 4864-5241 247 2-139 *7 6193-69 487443-403 4870-7071 247 4010 •75 6201-56 488373047 4876-8973 247-5480 •8 6209-44 489303-872 AND CUBES OP FRACTIONAL NUMBERS. 1.87 TABLE VII. Area. Ciroum. Diam. Square. Cube. 4886 1820 247-7937 78-875 6221-26 490702-324 4889 2799 247-8722 •9 6225-21 491169-069 4901-6814 248 1864 79 in. 6241 493039 4914-0985 248-5005 *1 6256*81 494913-671 4917-2053 248 5791 •125 6260-76 495383078 4926-5314 248-8147 •2 6272-64 496793088 4932-7517 248-9708 •25 6280-56 497734-578 4938-9820 249-1288 •3 6288-49 498677-257 4948-3268 249-3645 •375 6300-39 500093-504 4951-4443 249-4430 •4 6304-36 500566-184 4963-9243 2497572 •5 6320-25 502509-875 4976-4840 250-0713 •6 • 6336-16 504358-336 4979-5456 250-1499 •625 6340-14 504833-695 4988 9314 250-3855 ■7 6352-09 506261-573 4995-1931 -250-5426 •75 6360-06 507214-992 5001-4586 250-6996 •8 6368-04 508169-592 5010-8642 250-9353 •875 6380-01 509603-746 5014-0014 2510138 ■9 6384-04 510082-399 5026-5600 251-3280 80 in. 6400 512000 5039-1342 251-6241 •1 6416-01 513923-401 5042-2803 251-7207 •125 6420-01 514403750 5051-7242 251-9563 •2 6432-04 515849 608 5058 0230 252-1134 •25 6440-06 516815 016 5064-3298 252-2704 •3 6448-09 517781-627 5073-7944 252- 061 •375 6460-14 519233-801 5076-9552 252 5846 •4 6464-16 519718-464 5089-5883 252-8988 •5 6480-25 521660-125 5102-2411 253-2129 •6 6496-36 523606-616 5105-4060 253-2915 •625 6500-39 524093-992 5114 9096 253-5271 •7 6512-49 525557 943 5121-2497 253-6842 •75 652056 526535-422 5127-5938 253 8412 •8 6528-64 527514112 5137*1173 254-0769 •875 6540-76 528984-418 5140-2937 254-1554 •9 6544-81 529475 129 51530094 254-4696 81 in. 6561 531441 5165-7407 254-7837 •1 6577-21 533411-731 5168*9260 254 8623 •125 6581-26 533903 172 5178 4877 2550979 •2 6593-44 535387-328 5184-8651 255-2550 •25 6601-56 536376-953 5191-2505 255*4120 •3 6609-69 537367-797 5200-8329 255-6477 •375 6621-89 538856-347 5204-0285 255-7262 •4 6625 96 539353-144 5216-8231 256-0404 •5 6642-25 541343-375 5229-6330 256-3545 •6 6658-56 543338-496 5232-8371 256-4331 •625 6662-64 543838029 5242-4586 256-6687 •7 6674-89 545338-513 5248-8772 256-8258 •75 668306 546340-359 5255-2998 256-9828 •8 6691-24 547343-432 5264-9411 257-2105 •875 6703-5J 548850-339 5268-1568 257-2970 •9 6707-61 549353-259 88] CIRCUMFERENCES, AREAS, SQUARES, TABLE VII. Area. Circum. Diam. Square Cube. 5281-0296 2576112 82 in. 6724 551368 5293-9180 257-9253 •1 6740-51 553387-661 5297-1426 258-0039 •125 6744-51 553863-343 5306-8221 258-2395 •2 6756-84 555412-248 5313*2780 258-3966 •25 676S06 556426-390 5319-7439 258-5536 •3 6773-29 557441-767 5329-4421 258-7893 •375 6785-64 558967-144 5332-6775 258-8646 •4 6789-76 559476-224 5345-6287 259-1820 •5 6806-25 561515-625 5358-5957 259-4961 •6 6822-76 563559-976 5361-8391 259 5747 •625 6826-89 564071-836 5371-5983 259-8103 •7 6839 29 565609-283 5378-0755 259-9674 •75 6847*56 566635-797 5384-5762 260-1244 •8 6855-84 567663-552 5394-3358 260-3601 •875 6868-26 569207-511 5397-5908 260-4386 •9 6872-41 569722-789 5410-6206 260 7528 83 6889 571787 5423-6660 261-0669 •1 6905-61 573856-191 5426-9299 26M455 •125 6909-76 574374-265 5436-7272 261-3811 •2 6922-24 575930-368 5443-2617 261-5382 •25 6930-56 576969-328 5449-8042 261-6952 •3 6938-89 578009-537 5459-6-22-2 261-9309 •375 6951-39 579572-191 5462-8968 262-0094 •4 6955-56 580093-704 5476-0051 262-3236 •5 6972-25 582182-875 5489-1291 262-6376 •6 6988-96 584277056 5492-4118 262-7163 •625 6993-14 584801-382 5502-2689 262-9519 •7 7005-69 586376-253 5508-8446 263 1090 •75 7014-06 587427-734 5515-4243 263-2640 •8 7022-44 588480-472 5525-3012 263-5057 •875 7035-01 590061-933 5528-5958 263-5802 •9 7039 21 590589-719 5541-7024 263-8944 84 m. 7056 592304 5554-9849 264-2085 •1 7072-81 594823-321 5558-2881 264-2871 •125 707701 595353-937 5568-2032 264-5227 ■2 7089-64 596947688 5574-8162 264-6798 •25 7098 06 598011-765 5581-4372 264-8368 •3 7106-49 599077-107 5591-3730 265-0725 •375 711914 600677-498 5594-6869 265-1510 •4 7123-36 601211-584 56079523 265-4652 •5 7140-25 603351 125 5621-2334 2657793 ■6 715916 605495-736 5624-5554 265-8579 •625 7161-39 606032679 5634-5682 266-0935 •7 717409 607645-423 5641-1845 266-2506 •75 7182-56 608722-172 5647-8428 266-4076 •8 7191-04 609800-192 5657-8357 2666433 •875 7203-76 611419-605 5661-1710 266-7218 •9 720811 611960-049 5674 5150 267-0360 85 in. 7225 614125 56878746 267-3501 •1 7242-01 616295-051 AND CUBES OF FRACTIONAL NUMBERS. TABLE VII. Area. Circum. Diam. Square. Cube. 5691-2170 267-4287 85-125 7246-26 616838-359 5701-2500 267-6643 •2 725904 618470-208 5707-9415 2678214 -25 726756 619559-703 5714-6410 2679784 •3 7276-09 620650-477 5724-6947 268-2141 •375 7288 89 62-2289 035 5728 0478 268-2926 •4 729316 622835-864 5741 4703 268-6068 •5 7310-25 625026-375 5754-9185 268-9209 •6 7327-36 627222-016 57582697 268-9997 •625 7331-64 627771-726 5768-3624 269-2350 •7 7344-49 629422-793 57750952 269-3922 •75 735306 630525T09 5781-8320 269-5492 •8 7361-64 631628-712 5791-9445 269 7849 •875 7374-51 633286-527 5795-3173 269-8634 •9 7378-81 633839-779 5808-8184 270-1776 86 in. 7396 636056 5822-3351 270-4917 •1 741321 638277-381 5825-7168 270-5703 •125 7417-51 638833-531 5835-8675 270-8059 •2 7430-44 640503 928 5842-6376 270-9630 •25 743906 641619-140 5849-4157 271 1200 •3 7447-69 642735-647 5859-5871 271 '3557 •375 7460-64 644412-832 5862-9795 271-4342 •4 7464-96 644972-544 5876-5591 271 -7484 •5 7482-25 647214-625 5890 1541 272 0665 •6 7499-56 649461 -896 5893-5549 272-1411 •625 7503-89 650024-523 5903-7654 272-3767 •7 7516-89 651714-263 5910-5767 272-5338 •75 7525-56 652842-547 5917-3920 272-6908 •8 7o34-24 653912-032 5927-6224 272-9265 •875 7547-26 655668-699 5931-0344 273-0050 •9 7551-61 656234-909 5944-6926 273-3190 87 in. 7569 658503 5958-3644 273-6333 •1 7586-41 660776-311 5961-7873 2737119 •125 7590-76 661345-453 59720559 2739875 •2 7603-84 663054-818 5978-9045 274-1046 •25 7612-56 664196078 5985-7691 274-2616 •3 7621-29 665338-617 5996-0504 274-4973 •375 7634-39 667054-878 5999-4821 274-5758 •4 7638 76 667627-624 6013-2187 274-8900 •5 7656-25 669921-875 6026-9711 275-2041 •6 7673-76 672221-376 6030-4108 275-2827 •625 7678-14 672797-070 6040-7391 275-5183 •7 7691-29 674526 133 6047-6290 275-6754 •75 7700-06 675680-484 6054-5149 275-8324 •8 7708 84 676836-152 6064-8710 2760681 •875 7722-01 678572-121 6068-3224 2761466 •9 7726-41 679151-439 6082-1376 276-4608 88 in. 7744 681472 6095-9684 276-7749 •1 7761-61 683797 841 6099-4287 276-8535 •125 776601 684380-125 6109-8150 277-0891 •2 7779-24 686128-968 90 ] CIRCUMFERENCES, AREAS, SQUARES, TABLE VII. ■Area. Circum. Diam. Square. Cube. 61167422 277-2462 88-25 778806 687296-516 6123-6774 277-4032 •3 7796-89 688465-387 6134 0844 2776389 •375 7810T4 690221-175 61375554 277-7174 •4 7814-56 690807-104 6151-4491 2780316 •5 7832-25 693154-122 6165-5585 278 3457 •6 7849-96 695506-456 6169-8376 278-4243 •625 7854-39 696095-367 6179-2837 278-6599 •7 7867-69 697864 103 6186-2521 278-8170 •75 7876-56 699044-922 6193-2245 278-9750 •8 7885-44 700227-072 6203-6905 279-2097 •875 7898-76 702002-793 6207-1811 279-2882 •9 7903-21 702955-369 6221 1534 279-6024 89 in. 7921 704969 62351413 279-9165 •1 7938-81 707349-971 6238-6408 279-9951 •125 7943-26 707943-547 6249-1450 280-2307 •2 7956-64 709732-288 6256-1507 280-3878 •25 7965-56 710926-453 6263-1644 280-5448 •3 7974-49 712121-957 6273-6893 280-7805 •375 7987-89 713907-722 6277-1995 2808590 •4 7992*36 714516-984 6291-2035 281 1732 •5 8010-25 716917-375 6305-3168 281-4873 •6 8028-16 719232 136 6308-8351 281-5659 •625 8032-64 719925-414 6319-3990 281-8825 •7 8046-09 721734-273 6326-4460 281-9586 •75 8055 06 723051-859 6333-4970 282 1156 •8 806404 724150-792 63440847 282-3513 •875 8077-51 725966-714 6347-6813 282-4298 •9 8082-01 726572 699 63617400 2827440 90 in. 8100 729000 6375-8850 283-0581 •1 8118-01 731432-701 6379-4238 283 1367 •125 8122-51 732041-718 63900458 283-2733 •2 813604 733870-808 6397-1300 283-5294 •25 8145 06 735091-890 6404-2222 283-8664 •3 815409 736314-327 6414-8649 283-9221 •375 8167-64 738150-519 6418-4144 2840006 •4 8172-16 738763-264 6432 6223 284-3148 •5 8190-25 741217-625 6446-8474 284-6289 •6 8208-36 743677*416 6450-4039 284-7075 •625 8212-89 744293-210 6461-0852 284-9431 •7 8226-49 746142-643 6468-2107 285-1002 •75 8235-56 747377-297 6475-3402 285-2572 •8 8244-64 748613-312 6486 0418 285-4929 •875 8258 26 750469-886 6489-6109 285-5714 •9 8262-81 751089-429 6503-8974 285-8856 91 in. 8281 753571 6518 1995 286 1997 •1 8299-21 756058-530 65217775 286-2783 •125 8303-76 756680-640 6532 5173 286-5139 •2 8317-44 758550-528 6539-6801 286-6710 .25 8326-562 759798-828 6546-8909 286-8290 •3 8335-69 761048-497 AND CUBES OF FRACTIONAL NUMBERS. [91 TABLE VII. Area. Circum. Diam. Square. 1 Cube. 65576114 287-0637 91-375 8349-39 762925-566 656i-2081 287-1422 •4 8353-96 763551-944 6575-5651 2874564 •5 837225 766060-875 6589-9458 287-7705 •6 8390-56 768575-296 6593 5431 2878491 •625 8395-14 769204-757 6604-3222 288 0847 •7 8408-89 770915-213 6611-5462 288-2418 •75 8418-06 772357-234 6618-7542 288-3988 •8 8427-24 773620-632 6629-5736 288-6345 ■875 8441 -01 775518-308 6633 1820 2887130 •9 8445-61 776151-559 66476256 2890272 92 in. 8464 778688 6662 1848 289-3413 -1 8482-41 781229-961 6665-7021 289-4199 •125 848701 781866-312 6676-5597 289-6555 •2 8500-84 783777-448 6683-8010 289-8125 •25 851006 785053-265 6691 0161 289-9696 •3 8519 29 786330-467 6701-9286 290-2053 •375 853314 788248-863 6705-5567 290-2838 •4 853776 788889-024 67200787 290-5980 •5 8556-25 791453125 6734-6165 2909120 •6 8574-76 794022-776 6738-2530 290-5907 •625 8579-39 794666-054 6749 1699 291-2263 •7 8593"29 796597-983 6756-4525 291-3834 •75 8602-56 797887-672 6763*7391 291-5404 •8 8611-84 799178-752 6774-6763 2917761 •875 8625-76 801117-980 67783240 291-8546 •9 8630-41 801765085 6792-9246 292-1688 93 in. 8649 804357 6807-5408 292-4829 •1 8667-61 807964-491 6811 1974 292-5615 •125 8672-26 807604-734 6822-1730 292-7971 -2 8686-24 809557 568 6829-4927 292 9542 •25 8695-56 810861-203 6836-8206 293-1112 •3 8704-89 812166-237 6847-8167 2933469 •375 8718-89 814126-410 6851-4840 293-4354 •4 8723-56 814780-504 6866 1631 2937396 •5 874225 817400-375 6880-8579 2940537 •6 8760-96 820025-856 6884-5338 2941323 •625 8765-64 820683101 6895-5685 294-3679 •7 8779-69 822656-953 6902-3497 294-5350 •75 8789-06 823974-902 6910-2947 294-6820 •8 8798-44 825293-672 6921-3336 294-9170 •875 8812-51 827274-906 6925-0367 294-9962 •9 8817-21 827936019 6939-7944 295-3104 94 in. 8836 830584 6954-5677 295-6245 •1 8854-81 833237-621 6958-2473 295-7024 •125 8859-51 833901-908 6969-3568 295-9387 •2 8873-64 835896-386 6976-7410 2960951 •25 8883 06 837228*640 6984T614 296-2436 •3 8892-49 838561-807 6995-2755 296-4739 •375 8906-64 840564-208 6998-9821 296-5670 •4 8911-36 841232-384 92 ] CIRCUMFERENCES, AREAS, SQUARES, TABLE VII. Area. Circum. Diam. j Square. Cube. 70138183 296-8812 945 8930-25 843908-625 7028-6702 297-1953 •6 8949-16 846590-536 7032-3853 297-2739 •625 8953-89 847261-898 7043-5025 297-5095 •7 896809 849278-123 7050-9775 297-6666 •75 8977-56 850264047 7058-4180 297-8236 •8 8987-04 851971-392 7069-5940 2980593 •875 9001-26 853995-074 7073-3202 298-1378 •9 900601 854670-349 7088-2350 298-4520 95 in. 9025 857375 7103-1654 298-7661 •1 9044-01 860885-351 7106-9005 298 8447 •125 9048-76 860763-828 71181116 2990723 .-2 9063-04 862801-408 7125-5885 299-2374 •25 907256 864161-578 71330734 299-3944 •3 9082-09 865523 177 7144-3052 299-6301 •375 9096-39 867568-253 71480510 299-7086 •4 9101-16 868250-664 71630443 300 0228 •5 9120-25 870983-875 7178-0533 300-3369 •6 9139-36 873722-816 7181-8077 300-4155 •625 9144-14 874408-445 7193-0780 300-6511 •7 9158-49 876467 493 7200-5962 300-8082 •75 916806 877841-984 7208-1184 300-9652 •8 917764 879217-912 7219-4090 301-2009 •875 9192-01 881284-495 7223-1745 301-2794 •9 9196-81 881974-079 7238-2464 301-5936 96 in. 9216 884736 7253-3339 301-9077 •1 9235-21 887503-681 7257-1083 301 -9863 •125 924801 888965-499 7268-4371 302-2219 •2 9254-44 890277-128 7275-9926 302-3790 •25 926406 891666-015 7283-5561 302-5360 •3 9273-69 893056-347 7294-9056 302-7717 •375 9288-14 894944-550 7298-6917 302-8502 •4 9292-96 895841-344 7313-8411 303 1644 •5 9312-25 898632 125 7329-0072 303-4785 ■6 9331-56 901428 696 7332-8008 303-5571 •625 9336-39 902128-742 7344-1890 303-7927 •7 9350-89 904231-063 7361-2873 303-9498 •75 9360-56 905634-422 7359-3864 304-1068 •8 9370-24 907039-232 7370-7949 304-3425 •875 9384-76 909149 167 7374-5996 3044210 •9 9389-61 909853-209 7389-8286 304-7352 97 in. 9409 912673 74050732 3050493 1 9428-41 915498-611 7408-8868 305 1279 •125 9433-26 916205-921 7420-3335 305-3635 •2 9447-84 918330048 7427-9675 305-5206 •25 9457-56 919747 953 7435-6095 305-6776 •3 9467-29 921167-317 7447 0769 305-9133 •375 9481-89 923299097 7450-9013 305-9918 •4 9486-76 924010-424 7466-2087 306-3060 •5 9506-25 926859-375 7481-5319 306-6201 •6 9525-76 929714-176 I AND CUBES OF FRACTIONAL NUMBERS. [93 TABLE VII. Area. Circum. Diam. Square. Cube. 74583648 306-6987 97625 9530-640 930428788 7496-8706 306-9363 •7 9545-29 932574-833 7504-5468 307-0914 •75 9555-06 934007 359 7512-2253 307 2484 •8 9564-84 935441-352 75237515 3074841 •875 9579-51 937595 089 7527-5956 307-5626 ■9 9584-41 938313 739 7542-9816 307-8768 98 in. 9604 941192 7558-3832 308-1909 •1 9623-61 944076-141 7562-2362 308-2695 •125 9628-51 944789-093 7573-8006 308-5051 •2 9643-24 946966*168 7581-5132 308-6662 •25 9653-06 948413-390 7589-2338 308-81.92 •3 9662-89 949862-087 7600-8189 309-0549 •375 9677-64 952037-894 7604-6826 309-1334 •4 9682-56 952763-904 7620-1471 309-4476 •5 9702-25 955671-625 7635-6-273 309-7617 •6 9720-06 958858-256 7639-4995 309-8403 •625 9726-89 959314-585 7651-1933 310-0759 •7 9741-69 961504-803 7658-8771 310-2330 •75 9751-56 962966-797 7666-6349 310-3960 •8 9761-44 964430-272 7678-2790 310-6257 •875 9776-26 966628-261 7682-1623 310-7042 •9 9781-21 967361-669 7697-7054 3110184 99 in. 9801 970299 7713-2641 311-3325 •1 9820-81 973242-271 7717 1563 311-4111 •125 9825-76 973979-015 7728-8386 311-6467 •2 9840-54 976911-488 7736-6297 311-8038 •25 9850-56 977688-328 7744-4288 311-9608 •3 9860-49 979146-657 7756T318 312-1965 •375 9875-39 981366-941 7760-0347 312-2750 •4 9880-36 982107-784 7775-6563 312-5892 •5 9900-25 985074-875 7791-2936 312-9033 •6 9920-16 988047-936 7795-2051 312-9819 •625 9925-14 988792-J32 78069466 313-2175 •7 9940-09 991026-973 7814-7790 313-3746 *75 9950-06 992518-734 7822-6054 313-5116 •8 9960-04 994011-992 7834-3772 313-7630 •875 9975-01 996254-683 7838-2998 313-8458 •9 9980-01 997022-999 7854-0000 314-1600 100 in. 10000 1000000 A concise Method of Verifying Dates, in accordance with the Julian and Gregorian Calendars; or for Finding the Day of the Week, corresponding to any proposed Date of the Month after Christ, without limitation. By SAMUEL MA\ N ARD, Editor of Keith's and Bonnycastle's Mathematical -Works, & 95 EXAMPLES FOR PRACTICE ON THE PRECEDING TABLE. (1.) Find the Dominical letter for the year 1727, Old Style. Here, 1727 -L- 700 leaves a remainder of 327 years, that is, 3 centuries and 27 years; then, on the same horizontal line with 27 at the left hand of the table, among the “ Remaining years of the given date less than one hundred,” and opposite the 27 in the first vertical column of letters will be found the Literal Index C; enter this C in the vertical column under 3. 0. S. (3 being the number of centuries of the tabular date 327), it will be found on the top line of letters under its respective tabular cen¬ turies ; then, on the same horizontal line, in the first vertical column of letters will be found A, the Dominical letter required. (2.) Required the day of the week; 1st of January, 1800, and 12 th of February , 1852, New Style in both cases?* Here, 1800 — 400 leaves a remainder of 200 years, that is, 2 cen¬ turies and 0 years, which denotes a common year, on the line of 0 in the table ; the years less than one hundred will be found as in Ex. 1, the Literal Index A, enter this Index A under the tabular centuries 2, N. S., and it will be found on the fourth line of letters ; then, on the same horizontal line, in the fourth vertical column of months will be found the given month, January (along with October), over which, in the same column and on the same horizontal line with 1, the given day of the month, will be found Wednesday, the day of the week required. Again, for 1852, we have 1852 -j- 400, remainder 252, and - 52 in the table gives as above the Literal Index F, this Index F will be found again on the last line but one of letters in column 2, N. S., the given year being a leap year: proceed on this horizontal line to the first ver¬ tical column of months, where will be found F, (along with August,) viz. February, leap year, the given month; then, as in the preceding case, will be found Thursday, the day of the week required. (3.) Required the days of the week N. S. corresponding to the dates of the fallowing registrations:—Samuel was born December 1 6th, 1789; Hannah on 14 th July, 1791; Rebecca on IDA October, 1800; Ann on 2 5th December, 1813; Alfred, on 23 rd February, 1821: Augustus on 20 th May, 1823; Sarah on 30th January, 1826; Newton on 6th December, 1832; and Mary Ann on 26th December, 1834. A ns. Samuel on a Wednesday ; Hannah on a Thursday; Rebecca on a Saturday; Ann on a Saturday ; Alfred on a Friday ; Augustus on a Tuesday; Sarah on a Monday; Newton on a Thursday; and Mary Ann on a Friday. (4.) What day of the month did the last Friday in January, Fe¬ bruary, August, and December fall on in the year 1844, N. S.? Am. 26th of January , 23 rd of February, 30 th of August, and 27 th of December. (5.) An elderly lady, speaking of her age, says, she was born in the year 1760, N. S., but does not know on what day of the month ; she only recollects hearing her father say it was the second Wednesday in February; required the day of the month she was born ? Am. 13 th. (6.) In what years of the 19 the leap years required are 1812, 1840, 1868, and 1896; in all these instances 17$ January falls on a Friday. With regard to 16$ December, all years before 1752 are to be taken oy Old Style; these years are 1702, 1713, 1719, 1724, 1730, 1741, and 1747, 6. S.; the years after 1751 are to be taken by New Style, which, in the 18$ century, are 1761, 1767, 1772. 1778, 1789, and 1795. In the 19$ century the years are 1801, 1807, 1812, 1818, 1829, 1835, 1840, 1846, 1857, 1863, 1868, 1874, 1885, 1891, and 1896, in all which cases 16$ December falls on a Wednesday. N.B. Those who may be desirous of acquainting themselves with further particulars on the perpetuity of the Civil and Ecclesiastical Calendars, may with advantage consult Bonnycastle’s Arithmetic and Key, by Samuel Maynard , Edition 1851, and Key, 1852. and all sub¬ sequent editions. THE END. Gilbert & Rivington, Printers, St. John’s Square, London. _ DUE DATE 1st MAY 1 n 1990 201-6503 Printed in USA LONDON: SIM THIN. MARSHALL l& C? THE MILLWRIGHT AND ENGINEER’S POCKET COMPANION; COMPRISING DECIMAL ARITHMETIC, TABLES OF SQUARE AND CUBE ROOTS, PRACTICAL GEOMETRY, MENSURATION, STRENGTH OF MATERIALS, MECHANIC POWERS, WATER WHEELS, PUMPS AND PUMPING ENGINES, STEAM ENGINES, TABLES OF SPECIFIC GRAVITY, &c. &c. TO WHICH IS ANNEXED, AN APPENDIX OF A SEEIES OE MATHEMATICAL TABLES; CONTAINING THE CIRCUMFERENCES AND AREAS OF CIRCLES, SUPERFICIES, AND SOLIDITY OF SPHERES, &C. &C. By WILLIAM TEMPLETON, AUTHOR OP “ THE ENGINEER’S COMMON PLACE BOOK OP PRACTICAL REFERENCE.” WITH LITHOGRAPHIC AND OTHER ILLUSTRATIONS. Corrected by SAMUEL MAYNARD. .ELEVENTH EDITION, CAREFULLY REVISED; TO WHICH IS ADDED A NEW TABLE OF FRACTIONAL NUMBERS, LONDON: SIMPRIN, MARSHALL, AND CO. AND SOLD BY G. & R. W. HEBERT, CHEAPSIDE. J. THOMSON, MANCHESTER : AND OLIYER AND BOYD, EDINBURGH. 1856. V LONDON! GILBERT AND RIVINGTON, PRINTERS, st. John’s square. _ TZ4 1 502 . Templet cm. Lillwr^Ue^me^ fsM «""P ion am I 0114441039