# Full text of "The Mechanical engineer's pocket-book ; a reference book of rules, tables, data, and formulĂ¦, for the use of engineers, mechanics, and students"

## See other formats

GIFT OF Consulting Engineer Sv <&/*, t r\^ MM. 3T3., , Consulting Engineer The Publishers and the Author will be grateful to any of the readers of this volume who will kindly call their attention to any errors of omission or of commis- sion that they may find therein. It is intended to make our publications standard works of study and reference, and, to that end, the greatest accuracy is sought. It rarely happens that the early editions of works of any size are free from errors; but it is the endeavor of the Publishers to have them removed immediately upon being discovered, and it is therefore desired that the Author may be aided in his task of revision, from time to time, by the kindly criticism of his readers. JOHN WILEY & 13 & 45 EAST NINETEENTH STREET. gnilli WORKS OF WILLIAM KENT PUBLISHED BY JOHN WILEY & SONS. The Mechanical Engineers' Pocket-Book. A Reference Book of Rules, Tables, Data, and Formulae, for the Use of Engineers, Mechanics, and Students, xxxii-f- noo pages, i6mo, morocco, $5.00. Steam-Boiler Economy. A Treatise on the Theory and Practice of Fuel Economy in the Operation of Steam-Boilers. xiv + 458 P a S es > T 3 6 figures, 8vo, cloth, $4.00. THE MECHANICAL ENGINEER'S POCKET-BOOK A REFERENCE-BOOK OF RULES, TABLES, DATA, AND FORMULAE, FOR THE USE OF ENGINEERS, MECHANICS, AND STUDENTS. BY WILLIAM KENT, A.M., M.E., Dean and Professor of Mechanical Engineering in the L. C. Smith College of Applied Science, Syracuse University, Member Amer. Soc'y Mechl Engrs. and Amer. Inst. Mining Engrs. SEVENTH EDITION, REVISED AND ENLARGED TEXTH THOUSAND. TOTAL ISSTO FORTY-FIVE THOUSAND. NEW YORK : JOHN WILEY & SONS. LONDON: CHAPMAN & HALL, LIMITED. 1906. COPYRIGHT, 1895, 1902, BY WILLIAM KENT. PRESS OF BRAUNWORTH & CO. BOOKBINDERS AND PRINTERS BROOKLYN. N. Y. PREFACE. MORE than twenty years ago the author began to follow the advice given by Nystrom : " Every engineeer should make his own pocket-book, as he proceeds in study and practice, to suit his particular business." The manuscript pocket-book thus begun, however, soon gave place to more modern means for disposing of the accumulation of engi- neering facts and figures, viz., the index rerum, the scrap- book, the collection of indexed envelopes, portfolios and boxes, the card catalogue, etc. Four years ago, at the re- quest of the publishers, the labor was begun of selecting from this accumulated mass such matter as pertained to mechanical engineering, and of condensing, digesting, and arranging it in form for publication. In addition to "this, a careful examination was made of the transactions of engi- neering societies, and of the most important recent works on mechanical engineering, in order to fill gaps that might be left in the original collection, and insure that no impor- tant facts had been overlooked. Some ideas have been kept in mind during the prepara- tion of the Pocket-book that will, it is believed, cause it to differ from other works of its class. In the first place it was considered that the field of mechanical engineering was so great, and the literature of the subject so vast, that as little space as possible should be given to subjects which especially belong to civil engineering. While the mechan- ical engineer must continually deal with problems which belong properly to civil engineering, this latter branch is so well covered by Trautwine's " Civil Engineer's Pocket- book " that any attempt to treat it exhaustively would not only fill no "long-felt want," but would occupy space which should be given to mechanical engineering. Another idea prominently kept in view by the author has been that he would not assume the position of an "au- thority " in giving rules and formulae for designing, but only that of compiler, giving not only the name of the originator of the rule, where it was known, but also the volume and page from which it was taken, so that its lii 288901 17 PREFACE. derivation may be traced when desired. When different formulae for the same problem have been found they have been given in contrast, and in many cases examples have been calculated by each to show the difference between them. In some cases these differences are quite remark- able, as will be seen under Safety-valves and Crank-pins. Occasionally the study of these differences has led to the author's devising a new ^formula, in which case the deriva tion of the formula is given. Much attention has been paid to the abstracting' of data of experiments from recent periodical literature, and numer- ous references to other data are given. In this respect the present work will be found to differ from other Pocket- books. The author desires to express his obligation to the many persons who have assisted him in the preparation of the work, to manufacturers who have furnished their cata- logues and given permission for the use of their tables, and to many engineers who have contributed original data and tables. The names of these persons are mentioned in their proper places in the text, and in all cases it has been endeavored to give credit to whom credit is due. The thanks of the author are also due to the following gentle- men who have given assistance in revising manuscript or proofs of the sections named : Prof. De Volson Wood, mechanics and turbines ; Mr. Frank Richards, compressed air ; Mr. Alfred R. Wolff, windmills ; Mr. Alex. C. Humphreys, illuminating gas ; Mr. Albert E. Mitchell, locomotives ; Prof. James E. Denton, refrigerating-ma* chinery ; Messrs. Joseph Wetzler and Thomas W. Varley, electrical engineering ; and Mr. Walter S. Dix, for valuable contributions on several subjects, and suggestions as to their treatment. WILLIAM KENT. PASSAIC, N. J M April^ 1895. FIFTH EDITION, MARCH, 1900. Some typographical and other errors discovered in the fourth edition have been corrected. New tables and some additions have been made under the head of Compressed Air. The new (1899) code of the Boiler Test Committee of the American Society of Mechanical Engineers has been substituted for the old (1885) code. W. K. PREFACE TO FOURTH EDITION. IN this edition many extensive alterations have been made. Much obsolete matter has been cut out and fresh matter substi- tuted. In the first 170 pages but few changes have been found necessary, but a few typographical and other minor errors have been corrected. The tables of sizes, weight, and strength of materials (pages 172 to 282) have been thoroughly revised, many entirely new tables, kindly furnished by manufacturers, having been substituted. Especial attention is called to the new matter on Cast-iron Columns (pages 250 to 253). In the remainder of the book changes of importance have been made in more than 100 pages, and all typographical errors reported to date have been corrected. Manufacturers' tables have been revised by reference to their latest catalogues or from tables furnished by the manufac- turers especially for this work. Much new matter is inserted under the heads of Fans and Blowers, Flow of Air in Pipes, and Compressed Air. The chapter on Wire-rope Transmission (pages 917 to 922) has been entirely rewritten. The chapter on Electrical Engineering has been improved by the omission of some matter that has become out of date and the insertion of some new matter. It has been found necessary to place much of the new matter of this edition in an Appendix, as space could not conveniently be made for it in the body of the book. It has not been found possi- ble to make in the body of the book many of the cross-references which should be made to the items in the Appendix. Users of the book may find it advisable to write in the margin such cross-refer- ences as they may desire. The Index has been thoroughly revised and greatly enlarged. The author is under continued obligation to many manufacturers who have furnished new tables and data, and to many individual engineers who have furnished new matter, pointed out errors in the earlier editions, and offered helpful suggestions. He will be glad to receive similar aid, which will assist in the further improvement of the book in future editions. WILLIAM KENT. PASSAIC, N. J., September^ 1898. SIXTH EDITION. DECEMBER, 1902. THE chapter on Electrical Engineering has been thoroughly revised, much of the old matter cut out and new matter sub- stituted. Fourteen new pages have been devoted to the sub- ject of Alternating Currents. The chapter on Locomotives has been revised. Some new matter has been added under Cast Iron, Specifications for Steel, Springs, Steam-engines, and Friction and Lubrication. Slight changes and corrections in the text have been made in nearly a hundred pages. IV& PBEFACE. SEVENTH EDITION, OCTOBER 1904. AN entirely new index has been made, with about twice as many titles as the former index. The electrical engineering chapter has been further revised and some new matter added. Four pages on Coal Handling Machinery have been inserted at page 911, .and numerous minor changes have been made. W. K. SYRACUSE, N. Y. CONTENTS. (For Alphabetical Index see page 1093.) MATHEMATICS. Arithmetic. PAGE Arithmetical and Algebraical Signs 1 Greatest Common Divisor 2 Least Common Multiple 2 Fractions 2 Decimals 3 Table. Decimal Equivalents of Fractions of One Inch 3 Table. Products of Fractions expressed in Decimals 4 Compound or Denominate Numbers 5 Reduction Descending and Ascending 5 Ratio and Proportion 5 Involution, or Powers of Numbers 6 Table. First Nine Powers of the First Nine Numbers 7 Table. First Forty Powers of 2 7 Evolution. Square Root 7 CubeRoot 8 Alligation. 10 Permutation 10 Combination < 10 Arithmetical Progression 11 Geometrical Progression 11 Interest 13 Discount 13 Compound Interest 14 Compound Interest Table, 3, 4, 5, and 6 per cent 14 Equation of Payments 14 Partial Payments 15 Annuities 16 Tables of Amount, Present Values, etc., of Annuities 16 Weights and Measures. Long Measure 17 Old Land Measure 17 Nautical Measure 17 Square Measure 18 Solid or Cubic Measure 18 Liquid Measure 18 The Miners' Inch 18 Apothecaries' Fluid Measure 18 Dry Measure * 18 Shipping Measure 19 Avoirdupois Weight 19 Troy Weight 19 Apothecaries' Weight 19 To Weigh Correctly on an Incorrect Balance 19 Circular Measure 20 Measure of time , 20 V y: CONTENTS. Board and Timber Measure ] 20 Table. Contents in Feet of Joists, Scantlings, and Timber 20 French or Metric Measures 21 British and French Equivalents 21 Metric Conversion Tables 23 Compound Units. of Pressure and Weight 27 of Water, Weight, and Bulk f 27 of Work, Power, and Duty P 27 of Velocity 27 of Pressure per unit area 27 Wire and Sheet Metal Gauges , 28 Twist-drill and Steel-wire Gauges 28 Music- wire Gauge 29 Circular- mil Wire Gauge 30 NewU. S. Standard Wire and Sheet Gauge, 1893 30 Decimal Gauge 32 Algebra. Addition, Multiplication, etc 33 Powers of Numbers 33 Parentheses, Division 34 Simple Equations and Problems 34 Equations containing two or more Unknown Quantities 35 Elimination 35 Quadratic Equations 35 Theory of Exponents.. 36 Binomial Theorem.... 36 Geometrical Problems of Construction 37 of Straight Lines 37 of Angles 38 of Circles , 39 of Triangles 41 of Squares and Polygons 42 oftheEllipse 45 of the Parabola , 48 of the Hyperbola 49 of the Cycloid... 49 of the Tractrix or Schiele Anti-friction Curve 50 oftheSpiral 50 of the Catenary . 51 of the Involute 52 Geometrical Propositions 53 ' Mensuration, Plane Surfaces. Quadrilateral, Parallelogram, etc 54 Trapezium and Trapezoid c . . .. 54 Triangles 54 Polygons. Table of Polygons. . 55 Irregular Figures 55 Properties of the Circle 57 Values of ir and its Multiples, etc 57 Relations of arc, chord, etc 58 Relations of circle to inscribed square, etc 58 Sectors and Segments 59 Circular Ring 59 The Ellipse 59 The Helix 60 TheSpiral 60 Mensuration, Solid Bodies. Prism ... 60 Pyramid 60 Wedge , 61 The Prismoidal Formula > Rectangular Prismoid. 61 Cylinder 61 Cone * > > > *1 CONTENTS. Vll PAGE Sphere 61 Spherical Triangle J Spherical Polygon j Spherical Zone "2 Spherical Segment 62 Spheroid or Ellipsoid Polyedron 62 Cylindrical Ring 62 Solids of Revolution 62 Spindles J Frustrum of a Spheroid " Parabolic Conoid t Volume of a Cask 64 Irregular Solids 64 Plane Trigonometry. Solution of Plane Triangles 65 Sine, Tangent, Secant, etc t Signs of the Trigonometric Functions t Trigonometrical Formulae ,. C Solution of Plane Right-angled Triangles C Solution of Oblique-angled Triangles 68 Analytical Geometry. Ordinates and Abscissas 69 Equations of a Straight Line, In tersections, etc C Equations of the Circle 70 Equations of the Ellipse . 70 Equations of the Parabola 70 Equations of the Hyperbola 70 Logarithmic Curves . 71 Differential Calculus. Definitions 72 Differentials of Algebraic Functions 72 Formulae for Differentiating 73 Partial Differentials 73 Integrals.. . . 73 Formulae for Integration 74 Integration between Limits 74 Quadrature of a Plane Surface 74 Quadrature of Surfaces of Revolution 75 Cubature of Volumes of Revolution 75 Second, Third, etc., Differentials , 75 Maclaurin's and Taylor's Theorems 76 Maxima and Minima.. 76 Differential of an Exponential Function 77 Logarithms.. 77 Differential Forms which have Known Integrals 78 Exponential Functions 78 Circular Functions 78 The Cycloid 79 Integral Calculus 79 Mathematical Tables. Reciprocals of Numbers 1 to 2000 80 Squares, Cubes, Square Roots, and Cube Roots from 0.1 to 1600 86 Squares and Cubes of Decimals 101 Fifth Roots and Fifth Powers 102 Circumferences and Areas of Circles, Diameters 1 to 1000 103 Circumferences and Areas of Circles, Advancing by Eighths from ^ to 100 108 Decimals of a Foot Equivalent to Inches and Fractions of an Inch 112 Circumferences of Circles in Feet and Inches, from 1 inch to 32 feet 11 inches in diameter 113 Lengths of Circular Arcs, Degrees Given 114 Lengths of Circular Arcs, Height of Arc Given . 115 Areas of the Segments of a Circle 116 viii CONTENTS. PAGE Spheres 118 Contents of Pipes and Cylinders, Cubic Feet and Gallons 120 Cylindrical Vessels, Tanks, Cisterns, etc 121 Gallons in a Number of Cubic Feet 122 Cubic Feet in a Number of Gallons 122 Square Feet in Plates 3 to 32 feet long and 1 inch wide 123 Capacities of Rectangular Tanks in Gallons 125 Number of Barrels in Cylindrical Cisterns and Tanks 126 Logarithms 127 Table of Logarithms 129 Hyperbolic Logarithms 156 Natural Trigonometrical Functions 159 logarithmic Trigonometrical Functions 162 MATEKIAL.S. Chemical Elements 165 Specific Gravity and Weight of Materials 163 Metals, Properties of 164 The Hydrometer 165 Aluminum 166 Antimony 166 Bismuth 166 Cadmium 167 Copper 167 Gold .- 167 Iridium 167 Iron 167 Lead 167 Magnesium 168 Manganese 168 Mercury 1 68 Nickel 168 Platinum 168 Silver 168 Tin 168 Zinc 168 Miscellaneous Materials. Order of Malleability, etc., of Metals 169 Formulae and Table for Calculating Weight of Rods, Plates, etc 169 Measures and Weights of Various Materials 169 Commercial Sizes of Iron Bars 170 Weights of Iron Bars 171 of Flat Rolled Iron 172 of Iron and Steel Sheets 174 of Plate Iron 175 of Steel Blooms 176 of Structural Shapes 177 Sizes and Weights of Carnegie Deck Beams 177 " Steel Channels 178 " " ZBars 178 " " Pencoyd Steel Angles 179 * " " Tees... 180 ' Channels 10 " Roofing Materials 181 " * Terra-cotta 181 " * Tiles 181 " " Tin Plates ...181 " Slates 183 " " PineShingles 183 ' " Sky-light Glass . 184 Weights of Various Ropf-coverings 184 Cast-iron Pipes or Columns 185 " " " 12- ft. lengths 186 * " Pipe-fittings 187 " " Water and Gas-pipe 188 and thickness of Cast-iron Pipes 189 Safe Pressures on Cast Iron Pipe 189 CONTENTS. IX PAGE Sheet-iron Hydraulic Pipe.. 191 Standard Pipe Flanges 192 Pipe Flanges and Cast-iron Pipe 193 Standard Sizes of W rough t-iron Pipe 194 Wrought-iron Welded Tubes ... 196 Riveted Iron Pipes 197 Weight of Iron for Riveted Pipe , 197 Spiral Riveted Pipe 198 Seamless Brass Tubing 198, 199 Coiled Pipes 199 Brass, Copper, and Zinc Tubing 200 Lead and Tin-lined Lead Pipe , 201 Weight of Copper and Brass Wire and Plates 202 Round Bolt Copper 203 44 Sheet and Bar Brass 203 Composition of Rolled Brass 203 Sizesof Shot 204 Screw-thread, U. S. Standard 204 Limit-gauges for Screw-threads 205 Size of Iron for Standard Bolts 206 Sizes of Screw-threads for Bolts and Taps 207 Set Screws and Tap Screws 208 Standard Machine Screws 209 Sizes and Weights of Nuts 209 Weight of Bolts with Heads 210 Track Bolts 210 Weights of Nuts and Bolt-heads 211 Rivets 211 Sizes of Turnbuckles 211 Washers 212 Track Spikes 212 Railway Spikes 212 Boat Spikes 212 Wrought Spikes 213 Wire Spikes ... 213 Cut Nails 213 Wire Nails , 214, 215 Iron Wire, Size, Strength, etc 216 Galvanized Iron Telegraph Wire 217 Tests of Telegraph Wire 217 Copper Wire Table, B. W. Gauge 218 " Edison or Circular Mil Gauge.... 219 " 4t B.&S.Gauge 220 Insulated Wire 221 Copper Telegraph Wire 221 Electric Cables 221,222 Galvanized Steel-wire Strand 223 Steel-wire Cables for Vessels 223 Specifications for Galvanized Iron Wire 224 Strength of Piano Wire 224 Plough-steel Wire 224 Wires of different metals 225 Specifications for Copper Wire 225 Cable-traction Ropes 226 Wire Ropes 226, 227 Plough-steel Ropes 227, 228 Galvanized Iron Wire Rope 228 Steel Hawsers 223, 229 Flat Wire Ropes 2*9 Galvanized Steel Cables ... 230 Strength of Chains and Ropes 230 Notes on use of Wire Rope 231 ' Locked Wire Rope 231 Crane Chains 232 Weights of Logs, Lumber, etc 232 Sizes of Fire Brick 233 Fire Clay, Analysis 234 Magnesia Bricks 235 Asbestos 235 X CONTENTS. Strength of Materials. _ , . _. PAQK Stress and Strain 236 ElasticLimit . 236 Yield Point 237 Modulus of Elasticity 237 Resilience 238 Elastic Limit and Ultimate Stress 238 Repeated Stresses 238 Repeated Shocks 240 Stresses due to Sudden Shocks 241 Increasing Tensile Strength of Bars by Twisting 241 Tensile Strength 242 Measurement of Elongation 243 Shapes of Test Specimens 243 Coinpressive Strength 244 Columns, Pillars, or Struts 246 Hodgkinson's Formula 246 Gordon's Formula. 247 Moment of Inertia 247 Radius of Gyration 247 Elements of Usual Sections 248 Strength of Cast-iron Columns 250 Transverse Strength of Cast iron Water-pipe 251 Safe Load on Cast-iron Columns 252 Strength of Brackets on Cast-iron Columns 252 Eccentric Loading of Columns 254 Wrought-iron Columns 255 Built Columns . . . . 256 Phoenix Columns 257 Working Formulae for Struts 259 Merriman's Formula for Columns 2GC Working Strains in Bridge Members 263 Working Stresses for Steel 263 Resistance of Hollow Cylinders to Collapse 264 Collapsing Pressure of Tubes or Flues 265 Formula for Corrugated Furnaces 266 Transverse Strength 266 Formulae for Flexure of Beams 267 Safe Loads on Steel Beams 269 Elastic Resilience 270 Beams of Uniform Strength 271 Properties of Rolled Structural Shapes 272 " SteellBeams 273 Spacing of Steel I Beams 276 Properties of Steel Channels 277 " T Shapes 278 ** ' Angles 279a " Zbars 280 Size of Beams for Floors 280 Flooring Material.. 281 Tie Rods for Brick Arches 281 Torsional Strength 281 Elastic Resistance to Torsion 282 Combined Stresses , 282 Stress due to Temperature 283 Strength of Flat Plates 283 Strength of Unstayed Flat Surfaces 284 Unbraced Heads of Boilers 285 Thickness of Flat Cast-iron Plates ... 286 Strength of Stayed Surfaces 286 Spherical Shells and Domed Heads 286 Stresses in Steel Plating under Water Pressure 287 Thick Hollow Cylinders under Tension 287 Thin Cylinders under Tension 289 Hollow Copper Balls 289 Holding Power of Nails, Spikes, Bolts, and Screws 289 Cut versus Wire Nails 29(1 Strength of Wrought-iron Bolts.: 293 CONTENTS. xi PAGE Initial Strain on Bolts 292 Stand Pipes and their Design , 292 Riveted Steel Water-pipes 295 Mannesmann Tubes 296 Kirkaldy's Tests of Materials 296 Cast Iron 296 Iron Castings 297 Iron Bars, Forgings, etc 297 Steel Rails and Tires 298 Steel Axles, Shafts, Spring Steel 299 Riveted Joints 299 Welds 300 Copper, Brass, Bronze, etc 300 Wire, Wire-rope , 301 Ropes, Hemp, and Cotton 301 Belting, Canvas 302 Stones, Brick, Cement * 302 Tensile Strength of Wire 303 Watertown Testing-machine Tests 303 Riveted Joints 303 Wrought-iron Bars, Compression Tests 304 Steel Eye-bars ^ 304 Wrought-iron Columns ... 305 Cold Drawn Steel 305 American Woods 306 Shearing Strength of Iron and Steel 306 Holding Power of Boiler-tubes 307 Chains, Weight, Proof Test, etc 307 Wrought-iron Chain Cables , 308 Strength of Glass 308 Copper at High Temperatures 309 Strength of Timber 309 Expansion of Timber 311 Shearing Strength of Woods 312 Strength of Brick, Stone, etc . ... 312 " Flagging 313 " ** Lime and Cement Mortar 313 Moduli of Elasticity of Various Materials 314 Factors of Safety 314 Properties of Cork 316 Vulcanized India-rubber 316 XylolithorWoodstone 316 Aluminum, Properties and Uses 317 Alloys. Alloys of Copper and Tin, Bronze 319 Copper and Zinc, Brass 321 Variation in Strength of Bronze 321 Copper-tin-zinc Alloys. Liquation or Separation of Metals. Alloys used in Brs Jrass, Foundries 325 Copper-nickel Alloys 326 Copper-zinc-iron Alloys 326 Tobin Bronze 326 Phosphor Bronze 327 Aluminum Bronze 328 Aluminum Brass 329 Caution as to Strength of Alloys 329 Aluminum hardened 330 Alloys of Aluminum, Silicon, andiron 330 Tungsten-aluminum Alloys 331 Aluminum-tin Alloys 331 Manganese Alloys 331 Manganese Bronze 331 German Silver , 332 Alloys of Bismuth 332 Fusible Alloys , 333 bearing Metal Alloys , , ,,...,.., 333 Xll CONTENTS. PAGE Alloys containing Antimony 03 336 White-metal Alloys 336 Type-metal 336 Babbitt metals 336 Solders 338 Ropes and Chains. Strength of Hemp, Iron, and Steel Ropes , 333 FlatRopes t 339 Working Load of Ropes and Chains ... 339 Strength of Ropes and Chain Cables 340 Rope for Hoisting or Transmission 340 Cordage, Technical terms of 341 Splicing of Ropes 341 Coal Hoisting 343 Manila Cordage, Weight, etc 344 Knots, how to make ., 344 Splicing Wire Ropes 346 Springs. Laminated Steel Springs 847 Helical Steel Springs- 347 Carrying Capacity of Springs 349 Elliptical Springs .. 352 Phosphor-bronze Springs 352 Springs to Resist Torsional Force 352 Helical Springs for Cars, etc .1 353 Riveted Joints. Fairbairn's Experiments 354 Loss of Strength by Punching .'. 354 Strength of Perforated Plates 354 Hand vs. Hydraulic Riveting 355 Formulae for Pitch of Rivets 357 Proportions of Joints 358 Efficiencies of Join ts 359 Diameter of Rivets 360 Strength of Riveted Joints 361 Riveting Pressures 362 Shearing Resistance of Rivet Iron 363 Iron and Steel. Classification of Iron and Steel 364 Grading of Pig Iron 365 Influence of Silicon Sulphur, Phos. and Mn on Cast Iron 365 Tests of Cast Iron 369 Chemistry of Foundry Iron 370 Analyses of Castings , 373 Strength of Cast Iron 374 Specifications for Cast Iron 374 Mixture of Cast Iron with Steel 375 Bessemerized Cast Iron 375 Bad Cast Iron , 375 Malleable Cast Iron 375 Wrought Iron ; 377 Chemistry of Wrought Iron 377 Influence of Rolling on Wrought Iron 377 Specifications for Wrought Iron 378 Stay-bolt Iron 378 Formulae for Unit Strains in Structures 379 Permissible Stresses in Structures 381 Proportioning Materials in Memphis Bridge 382 Tenacity of Iron at High Temperatures 382 Effect of Cold on Strength of Iron 383 Expansion of Iron by Heat 385 Durability of Cast Iron 885 Corrosion of Iron and Steel 386 Preservative Coatings ; Paints, etc 387 CONTENTS Xlll PAGE Non-oxidizing Process of Annealing 387 Manganese Plating of Iron 389 Steel. Relation between Chemical and Physical Properties 389 Variation in Strength 391 Open-hearth 392 Bessemer 392 Hardening Soft Steel 393 Effect of Cold Rolling 393 Comparison of Full-sized and Small Pieces 393 Treatment of Structural Steel 394 Influence of Annealing upon Magnetic Capacity 396 Specifications for Steel 397 Chemical Requirements 397 Kinds of Steel used for Different Purposes 397 Castings, Axles, Forgings 397 Tires, Rails, Splice-bars, Structural Steel 398 Boiler-plate and Rivet Steel 399 May Carbon be Burned out of Steel ? 402 Recalescerice of Steel 402 Effect of Nicking r. Bar 402 Electric Conductivity 403 Specific Gravity 403 Occasional Failures 403 Segregation in Ingots 404 Earliest Uses for Structures 405 Steel Castings 405 Manganese Steel 407 Nickel Steel 407 Aluminum Steel 409 Chrome Steel 409 Tungsten Steel 409 Compressed Steel 410 Crucible Steel 410 Effect of Heat on Grain 412 1 ' Hammering, etc 412 Heating and Forging 412 Tempering Steel 413 MECHANICS. Force, Unit of Force 415 Inertia 415 Newton's Laws of Motion 415 Resolution of Forces 415 Parallelogram of Forces 416 Moment of a Force 416 Statical Moment, Stability 417 Stability of a Dam 417 Parallel Forces 417 Couples 418 Equilibrium of Forces 418 Centre of Gravity 418 Moment of Inertia 419 Centre of Gyration 420 Radius of Gyration 420 Centre of Oscillation 421 Centre of Percussion 422 The Pendulum 422 Conical Pendulum 423 Centrifugal Force 423 Acceleration 423 Falling Bodies 424 Value of g 424 Angular Velocity 425 Height due to Velocity 425 Parallelogram of Velocities 426 Mass 427 XIV CONTENTS. PAGE Force of Acceleration, 427 Motion on Inclined Planes : 428 Momentum 428 Vis Viva 428 Work, Foot-pound 428 Power, Horse-power 429 Energy 429 Work of Acceleration ....... 430 Force of a Blow . . , , 430 Impact of Bodies . . . . . . .... 431 Energy of Recoil of Guns 431 Conservation of Energy 432 Perpetual Motion 432 Efficiency of a Machine 432 Animal-power, Man-power 433 Work of aHorse . ... 434 Man-wheel 434 Horse-gin 434 Resistance of Vehicles 435 Elements of Machines, The Lever 435 The Bent Lever 436 The Moving Strut 436 The Toggle-joint 436 The Inclined Plane 437 The Wedge 437 The Screw 437 The Cam 438 The Pulley 438 Differential Pulley 439 Differential Windlass 439 Differential Screw 439 Wheel and Axle 439 Toothed-wheel Gearing 439 Endless Screw , 440 Stresses in Framed Structures. Cranes and Derricks 440 Shear Poles and Guys 442 King Post Truss or Bridge 442 Queen Post Truss 442 Burr Truss 443 Pratt or Whip pie Truss 443 Howe Truss 445 Warren Girder . 445 Roof Truss 446 HEAT. Thermometers and Pyrometers 448 Centigrade and Fahrenheit degrees compared 449 Copper-ball Pyrometer 451 Thermo-electric Pyrometer 451 Temperatures in Furnaces 451 Wiborgh Air Pyrometer , 453 Seeger's Fire-clay Pyrometer , 453 Mesur6 and Nouel's Pyrometer 453 Uehling and Steinbart's Pyrometer 453 Air-thermometer 454 High Temperatures judged by Color.... 454 Boiling-points of Substances 455 Melting-points 455 Unit of Heat 455 Mechanical Equivalent of Heat 456 Heat of Combustion 456 Specific Heat 457 Latent Heat of Fusion 459, 461 Expansion by Heat 460 Absolute Temperature 461 Absolute Zero ; 461 CONTENTS. XY PAGE Latent Heat 461 Latent Heat of Evaporation 462 Total Heat of Evaporation 462 Evaporation and Drying 462 Evaporation from Reservoirs 463 Evaporation by the Multiple System 463 Resistance to Boiling 463 Manufacture of Salt 464 Solubility of Salt and Sulphate of Lime 464 Salt Contents of Brines 464 Concentration of Sugar Solutions 465 Evaporating by Exhaust Steam 465 Drying in Vacuum 466 Radiation of Heat 467 Conduction and Convection of Heat 468 Rate of External Conduction 469 Steam-pipe Coverings . . 470 Transmission through Plates 471 in Condenser Tubes 473 * ** Cast-iron Plates 474 ** from Air or Gases to Water 474 ' from Steam or Hot Water to Air 475 ' through Walls of Buildings 478 Thermodynamics 478 PHYSICAL PROPERTIES OF GASES. Expansion of Gases 479 Boyle and Marriotte's Law.... 479 Law of Charles, Avogadro's Law 479 Saturation Point of Vapors 480 Law of Gaseous Pressure , 480 Flow of Gases 480 Absorption by Liquids 480 AIR. Properties of Air 481 Air-manometer 481 Pressure at Different Altitudes 481 Barometric Pressures .... 482 Levelling by the Barometer and by Boiling Water 482 To find Difference in Altitude 483 Moisture in Atmosphere 483 Weight of Air and Mixtures of Air and Vapor 484 Specific Heat of Air 484 Flow of Air. Flow of Air through Orifices 484 Flow of Air in Pipes 485 Effect of Bends in Pipe 488 Flow of Compressed Air 488 Tables of Flow of Air 489 Anemometer Measurements 491 Equalization of Pipes 491 Loss of Pressure in Pipes 493 Wind. Force of the Wind 493 Wind Pressure in Storms 495 Windmills 495 Capacity of Windmills 497 Economy of Windmills 498 Electric Power from Windmills 499 Compressed Air. Heating of Air by Compression 499 Loss of Energy in Compressed Air 499 Volumes and Pressures...., , 500 CONTENTS. Loss due to Excess of Pressure *.... 501 Horse-power Required for Compression 501 Table for Adiabatic Compression 502 Mean Effective Pressures 502 Mean and Terminal Pressures 503 Air-compressors. 503 Practical Results 505 Efficiency of Compressed-air Engines 506 Requirements of Rock-drills , 506 Popp Compressed-air System 507 Small Compressed-air Motors 507 Efficiency of Air-heating Stoves 507 Efficiency of Compressed-air Transmission 508 Shops Operated by Compressed Air 509 Pneumatic Postal Transmission t 509 Mekarski Compressed-air Tramways 510 Compressed Air Working Pumps in Mines 511 Fans and Blowers. Centrifugal Fans 511 Best Proportions of Fans 512 Pressure due to Velocity 513 Experiments with Blowers 514 Quantity of Air Delivered 514 Efficiency of Fans and Positive Blowers 516 Capacity of Fans and Blowers 517 Table of Centrifugal Fans 518 Engines, Fans, and Steam-coils for the Blower System of Heating 519 Sturtevant Steel Pressure-blower 519 Diameter of Blast-pipes 519 Efficiency of Fans 520 Centrifugal Ventilators for Mines 521 Experiments on Mine Ventilators 522 Disk Fans 524 Air Removed by Exhaust Wheel , 525 Efficiency of Disk Fans 525 Positive Rotary Blowers 526 Blowing Engines . . . . 526 Steam-jet Blowers <> 527 Steam -jet for Ventilation 527 HEATING AND VENTILATION. Ventilation 528 Quantity of Air Discharged through a Ventilating Duct 530 Artificial Cooling of Air , 531 Mine-ventilation 531 Friction of Air in Underground Passages 531 Equivalent Orifices 533 Relative Efficiency of Fans and Heated Chimneys 533 Heating and Ventilating of Large Buildings 534 Rules for Computing Radiating Surfaces *. 536 Overhead Steam-pipes 537 Indirect Heating-surface 537 Boiler Heating-surface Required 538 Proportion of Grate-surface to Radiator-surface 538 Steam-consumption in Car-heating 538 Diameters of Steam Supply Mains 539 Registers and Cold-air Ducts 539 Physical Properties of Steam and Condensed Water 540 Size of Steam-pipes for Heating 540 Heating a Greenhouse by Steam 541 Heating a Greenhouse by Hot Water 542 Hot- water Heating , 542 Law of Velocity of Flow 542 Proportions of Radiating Surfaces to Cubic Capacities 543 Diameter of Main and Branch Pipes 543 Rules for Hot-water Heating w 544 Arrangements of Mains ,..,..,.... tt .,.,,..,.,...... 544 CONTENTS. XVll PAGE Blower System of Heating and Ventilating 6 ;u 545 Experiments with Radiators . . , 545 Heating a Building to 70 F. ., 545 Heating by Electricity 546 WATER. Expansion of Water .... 547 Weight of Water at different temperatures 547 Pressure of Water due to its Weight 549 Head Corresponding to Pressures , 549 Buoyancy 550 Boiling-point 550 Freezing-point 550 Sea-water 549,550 Ice and Snow 550 Specific Heat of Water 550 Compressibility of Water 551 Impurities of Water 551 Causes of Incrustation 551 Means for Preventing Incrustation 552 Analyses of Boiler-scale 552 Hardness of Water 553 Purifying Feed-water 554 Softening Hard Water 555 Hydraulics. Flow of Water. Fomulae for Discharge through Orifices 555 Flow of Water from Orifices 555 Flow in Open and Closed Channels 557 General Formulae for Flow , . 557 Table Fall ofFeet per mile, etc 558 Values of Vr for Circular Pipes 559 Kutter's Formula 559 Molesworth's Formula 562 Bazin's Formula ... 563 IV Arcy's Formula 563 Older Formulae 564 Velocity of Water in Open Channels 564 Mean, Surface and Bottom Velocities 564 Safe Bottom and Mean Velocities 565 Resistance of Soil to Erosion 565 Abrading and Transporting Power of Water 565 Grade of Sewers 566 Relations of Diameter of Pipe to Quantity discharged 566 Flow of Water in a 20-inch Pipe 566 Velocities in Smooth Cast-iron Water-pipes 567 Table of Flow of Water in Circular Pipes 568-573 Loss of Head . . 573 Flow of Water in Riveted Pipes 574 Fractional Heads at given rates of discharge 577 Effect of Bend and Curves 578 Hydraulic Grade-line 578 Flow of Water in House-service Pipes 578 Air-bound Pipes 579 VerticalJets 579 Water Delivered through Meters 579 Fire Streams 579 Friction Losses in Hose 580 Head and Pressure Losses by Friction , 580 Loss of Pressure in smooth 2^-inch Hose 580 Rated capacity of Steam Fire-engines 580 Pressures required to throw water through Nozzles 581 The Siphon 581 Measurement of Flowing Water 582 Piezometer 582 Pitot Tube Gauge ... . 583 The Venturi Meter... 583 Measurement of Discharge by means of Nozzles 584 XV111 CONTENTS. PAGE Flow through Rectangular Orifices ....................... . ...... t ........ 584 Measurement of an Open Stream ......................................... 584 Miners' Inch Measurements ..... ............ ............................ 585 Flow of Water over Weirs .............................................. 586 Francis's Formula for Weirs ............................................. 586 Weir Table ............................................................... 587 Bazin's Experiments .................... .. ............................... 587 Water-power* Power of a Fall of Water .................................................. 588 Horse-power of a Running Stream ..... ., > ............................... 589 Current Motors. ... .................... ...... . ....................... 589 Horse-power of Water Flowing in a Tube... 7 . .......................... 589 Maximum Efficiency of a Long Conduit .................. ............... 589 Mill-power ............. . ............................................. ,.,.. 689 Value of Water-power ........................... , ..................... , . 590 The Power of Ocean Waves ................. ...... ....................... 599 Utilization of Tidal Power ............................................... 600 Turbine Wheels. Proportions of Turbines ..... ........................................ ..... 591 Tests of Turbines ............... .......................................... 596 Dimensions of Turbines ......... ....................................... 597 The Pelton Water-wheel .................................................. 597 Pumps. Theoretical capacity of a pump ....... . ................................. 601 Depth of Suction .......... ............................................... 602 Amount oi Water raised by a Single-acting Lift-pump ................... 602 Proportioning the Steam-cylinder of a Direct-acting Pump .............. 602 Speed of Water through Pipes and Pump -passages .................... 602 Sizes of Direct-acting Pumps ................... ................... ----- 603 The Deane Pump .............................. .......................... 603 Efficiency of Small Pumps .............................. ....... .. ..... 603 The Worthington Duplex Pump .......................................... 604 Speed of Piston ............... ,. ......................................... 605 Speed of Water through Valves. ........... . .......................... .. 605 Boiler-feed Pumps ............... . ....................................... 605 Pump Valves ........... ................................................. 606 Centrifugal Pumps ............ ....................... .................. 606 Lawrence Centrifugal Pumps .......................................... 607 Efficiency of Centrifugal and Reciprocating Pumps ...................... 608 Vanes of Centrifugal Pumps ............................................. 609 The Centrifugal Pump used as a Suction Dredge ........................ 609 Duty Trials of Pumping Engines... ............................... 609 Leakage Tests of Pumps ............................................... 611 Vacuum Pnmps .... ............ . ....................................... 612 The Pulsometer... ...................................... ................ 612 TheJetPump ........................ .................................. 614 The Injector ................................................ . .............. 614 Air-lift Pump ................................................ .............. 6i4 The Hydraulic Ram ............................ .......................... 614 Quantity of Water Delivered by the Hydraulic Ram . ............ , ........ 615 Hydraulic Pressure Transmission. Energy of Water under Pressure .......................... . ............ 616 Efficiency of Apparatus .................................................. 616 Hydraulic Presses -------- ................... ........................... 617 Hydraulic Power in London ............. ..... .......................... 617 Hydraulic Riveting Machines ............................................. 618 Hydraulic Forging ................. . . . ............ ....................... 618 The Aiken Intensifler ......... , .......................... ................ 619 Hydraulic Engine ..................................... ................... 61S FUEL. Theory of Combustion Total Heat of Combustion CONTENTS. XIX PACK Analyses of Gases of Combustion 622 Temperature of the Fire 622 Classification of Solid Fuel 623 Classification of Coals 624 Analyses of Coals 624 Western Lignites 631 Analyses of Foreign Coals 631 Nixon's Navigation Coal 632 Sampling Coal for Analyses 632 .Relative Value of Fine Sizes 632 Pressed Fuel... 632 Relative Value of Steam Coals , ... 633 Approximate Heating Value of Coals 634 Kind of Furnace Adapted for Different Coals 635 Downward-draught Furnaces 635 Calorimetric Tests of American Coals 636 E vaporati ve Power of Bituminous Coals 636 Weathering of Coal , 637 Coke 637 Experiments in Coking 637 Coal Washing 633 Recovery of By-products in Coke manufacture 638 Making Hard Coke 638 Generation of Steam from the Waste Heat and Gases from Coke-ovens. 638 Products of the Distillation of Coal 639 Wood as Fuel 639 Heating Value of Wood 639 Composition of Wood 640 Charcoal 640 Yield of Charcoal from a Cord of Wood 641 Consumption of Charcoal in Blast Furnaces 641 Absorption of Water and of Gases by Charcoal 641 Composition of Charcoals 642 Miscellaneous Solid Fuels 642 Dust-fuel Dust Explosions 642 Peat or Turf 643 Sawdust as Fuel 643 Horse-manure as Fuel 643 Wet Tan-bark as Fuel.... 643 Straw as Fuel 643 Bagasse as Fuel in Sugar Manufacture 643 Petroleum. Products of Distillation 645 Lima Petroleum 645 Value of Petroleum as Fuel 645 Oil vs. Coal as Fuel 646 Fuel Gas* Carbon Gas 646 Anthracite Gas , 647 Bituminous Gas 647 Water Gas 64& Producer-gas from One Ton of Coal 649 Natural Gas in Ohio and Indiana 649 Combustion of Producer-gas 650 Use of Steam in Producers 650 Gas Fuel for Small Furnaces *. 651 Illuminating Gas, Coal-gas 651 Water-gas 652 Analyses of Water-gas and Coal gas 653 Calorific Equivalents of Constituents 654 Efficiency of a Water-gas Plant 654 Space Required for a Water-gas Plant 656 Ruel-value of Illuminating-gas 666 XX CONTENTS. PAGE Flow of Gas in Pipes 657 Service for Lamps 658 STEAM. Temperature and Pressure > 659 Total Heat 659 Latent Heat of Steam 659 Latent Heatof Volume 660 Specific Heat of Saturated Steam 660 Density andVolume 660 Superheated Steam 661 Regnault's Experiments 661 Table of the Properties of Steam .- 662 Flow of Steam. Napier's Approximate Rule .... 669 Flow of Steam in Pipes , 669 Loss of Pressure Due to Radiation 671 Resistance to Flow by Bends 672 Sizes of Steam-pipes for Stationary Engines 673 Sizes of Steam-pipes for Marine Engines 674 Steam Pipes. Bursting-tests of Copper Steam-pipes 674 Thickness of Copper Steam-pipes.. , 675 Reinforcing Steam-pipes , 675 Wire-wound Steam- pipes 675 Riveted Steel Steam-pipes.. 675 Valves in Steam-pipes 675 Failure of a Copper Steam-pipe 676 The Steam Looj> 676 Loss from an Uncovered Steam-pipe , 676 THE STEAM BOILER. The Horse-power of a Steam -boiler .... 677 Measures for Comparing the Duty of Boilers 678 Steam-boiler Proportions 678 Heating-surface 678 Horse-power, Builders' Rating 679 Grate-surface 680 AreasofFlues 680 Air-passages Through Grate-bars 681 Performance of Boilers , ,...., 681 Conditions which Secure Economy , 682 Efficiency of a Boiler .. ...683 Tests of Steam-boilers 685 Boilers at the Centennial Exhibition 685 Tests of Tubulous Boilers 686 High Rates of Evaporation , 687 Economy Effected by Heating the Air..., 687 Results of Tests with Different Coals 688 Maximum Boiler Efficiency with Cumberland Coal . , 689 Boilers Using Waste Gases 689 Boilers for Blast Furnaces 689 Rules for Conducting Boiler Tests , 690 Table of Factors of Evaporation 695 Strength of Steam-boilers. Rules for Construction... 700 Shell-plate Formulae 701 Rules for Flat Plates 701 Furnace Formulae 702 Material for Stays 703 Loads allowed on Stays 703 Girders 703 Rules for Construction of Boilers in Merchant Vessels iu U. 8 705 CONTENTS. Xxi PAGB U. S. Rule for Allowable Pressures 706 Safe-working Pressures 707 Rules Governing Inspection of Boilers in Philadelphia 708 Flues and Tubes for Steam Boilers 709 Flat-stayed Surfaces 4. 709 Diameter of Stay-bolts. 710 Strength of Stays 710 Stay-bolts in Curved Surfaces . 710 Boiler Attachments, Furnaces, etc* Fusible Plugs 710 Steam Domes 711 Height of Furnace. 711 Mechanical Stokers 711 The Hawley Down-draught Furnace 712 Under-feed Stokers 712 Smoke Prevention 712 Gas-fired Steam-boilers 714 Forced Combustion .....i 714 Fuel Economizers. 715 Incrustation and Scale 716 Boiler-scale Compounds.. 717 Removal of Hard Scale 718 Corrosion in Marine Boilers 719 TJseofZinc 720 Effect of Deposit on Flues 720 Dangerous Boilers 720 Safety Valves. Rules for Area of Safety-valves ..... 721 Spring-loaded Safety-valves 724 The Injector. Equation of the In jector 725 Performance of Injectors ............. 726 Boiler-feeding Pumps 726 Feed-water Heaters. Strains Caused by Cold Feed-water 727 Steam Separators* Efficiency of Steam Separators 728 Determination of Moisture in Steam* Coil Calorimeter. 729 Throttling Calorimeters 729 Separating Calorimeters 730 Identification of Dry Steam 730 Usual Amount of Moisture in Steam 731 Chimneys* Chimney Draught Theory 731 Force or Intensity of Draught. . 732 Rate of Combustion Due to Height of Chimney 733 High Chimneys not Necessary 734 Heights of Chimneys Required for Different Fuels 734 Table of Size of Chimneys 734 Protection of Chimney from Lightning 736 Some Tall Brick Chimneys 737 Stability of Chimneys . 738 Weak Chimneys 739 Steel Chimneys 740 Sheet-iron Chimneys 741 THE STEAM ENGINE* Expansion of Steam , 742 Mean and Terminal Absolute Pressures 743 CONTENTS. Condensers, Air-pumps, Circulating-pumps, etc. PAGE The Jet Condenser 839 Ejector Condensers 840 The Surface Condenser 840 Condenser Tubes 840 Tube-plates 841 Spacing of Tubes 841 Quantity of Cooling Water 841 Air-pump 841 Area through Valve-seats 842 Circulating-pump. . . . 843 Feed-pumps for Marine Engines 843 An Evaporative Surface Condenser 844 Continuous Use of Condensing Water 844 Increase of Power by Condensers 846 Evaporators and Distillers 817 GAS, PETROLEUM, AND 1IOT-AIU ENGINES. Gas-engines 847 Efficiency of the Gas-engine 848 Tests of the Simplex Gas-engine 848 A 320-H.P. Gas-engine 848 Test of an Otto Gas-engine 849 Temperatures and Pressures Developed 849 Test of the Clerk Gas-engine 849 Combustion of the Gas in the Otto Engine 849 Use of Carburetted Air in Gas-engines 849 The Otto Gasoline-engine . 850 The Priestman Petroleum-engine 850 Test of a 5-H.P. Priestman Petroleum-engine 850 Naphtha-engines 851 Hot-air or Caloric Engines 851 Test of a Hot-air Engine 851 LOCOMOTIVES. Resistance of Trains 851 Inertia and Resistance at Increasing Speeds 853 Efficiency of the Mechanism of a Locomotive 854 Size of Locomotive Cylinders 855 Size of Locomotive Boilers 855 Qualities Essential for a Free-steaming Locomotive 855 Wootten's Locomotive 855 Grate -surf ace, Smokestacks, and Exhaust-nozzles for Locomotives .... 856 Exhaust Nozzles 856 Fire-brick Arches 857 Size. Weight, Tractive Power, etc 857 Leading American Types 858 Steam Distribution for High Speed 858 Speed of Railway Trains 859 Formulae for Curves 859a Performance of a High-speed Locomotive 859a Locomotive Link-motion 859a Dimensions of Some American Locomotives 859-862 Indicated Water Consumption 862 Locomotive Testing Apparatus 863 Waste of Fuel in Locomotives 863 Advantages of Compounding 863 Counterbalancing Locomotives 864 Maximum Safe Load on Steel Rails 865 Narrow-gauge Railways 865 Petroleum -burning Locomotives 865 Fireless Locomotives 866 SHAFTING. Diameters to Resist Torsional Strain 867 Deflection of Shafting 868 Horse-power Transmitted by Shafting 869 Table for Laying Out Shafting 871 CONTEXTS. XXV PULLETS. PAGE Proportions of Pulleys 873 Convexity of Pulleys 874 Cone or Step Pulleys 874 BELTING. Theory of Belts and Bands 876 Centrifugal Tension 876 Belting Practice, Formulae for Belting 877 Horse-power of a Belt one mch wide 878 A. F. Nagle's Formula : 878 Width of Belt for Given Horse-power 879 Taylor's Rules for Belting 880 Notes on Belting 882 Lacing of Belts 883 Setting a Belt on Quarter-twist 883 To Find the Length of Belt 884 To Find the Angle of the Arc of Contact 884 To Find the Length of Belt when Closely Rolled 884 To Find the Approximate Weight of Belts 884 Relations of the Size and Speeds of Driving and Driven Pulleys 884 Evils of Tight Belts 885 Sag of Belts 885 Arrangements of Belts and Pulleys 885 Care of Belts 886 Strength of Belting 886 Adhesion, Independent of Diameter 886 Endless Belts 886 Belt Data 886 Belt Dressing 887 Cement for Cloth or Leather 887 Rubber Belting 887 GEARING. Pitch, Pitch-circle, etc 887 Diametral and Circular Pitch 888 Chordal Pitch 889 Diameter of Pitch-line of Wheels from 10 to 100 Teeth 889 Proportions of Teeth 889 Proportion of Gear-wheels 891 Width of Teeth 891 Rules for Calculating the Speed of Gears and Pulleys 891 Milling Cutters for Interchangeable Gears 892 Forms of the Teeth. The Cycloidal Tooth 892 The Involute Tooth 894 Approximation by Circular Arcs 896 Stepped Gears 897 Twisted Teeth 897 Spiral Gears 897 Worm Gearing 897 Teeth of Bevel-wheels. 898 Annular and Differential Gearing 898 Efficiency of Gearing 899 Strength of Gear Teeth. Various Formulas for Strength 900 Comparison of Formulae 903 Maximum Speed of Gearing 905 A Heavy Machine-cut Spur-gear 905 Fractional Gearing 905 Frictional Grooved Gearing 906 HOISTING. Weight and Strength of Cordage 906 Working Strength of Blocks 906 XXVI CONTENTS. PAGE Efficiency of Chain-blocks 907 Proportions of Hooks 907 Power of Hoisting Engines. 908 Effect of Slack Rope on Strain in Hoisting 908 Limit of Depth for Hoisting ' 908 Large Hoisting Records 908 Pneumatic Hoisting 909 Counterbalancing of Winding-engines 909 Cranes. Classification of Cranes ."-. 911 Position of the Inclined Brace in a Jib Crane 912 A Large Travelling-crane ; 912 A 150-ton Pillar Crane 912 Compressed-air Travelling Cranes .' 912 Coal-handling Machinery. Weight of Overhead Bins 912a Supply-pipes from Bins 912a Types of Coal Elevators 912a Combined Elevators and Conveyors ; 912a Coal Conveyors 912a Weight of Chain 9126 Weight of Flights 912c Horse-power of Conveyors 912c Bucket Conveyors 912c Screw Conveyors 912d Belt Conveyors Ql2d Capacity of Belt Conveyors 9l2d Wire-rope Haulage. Self-acting Inclined Plane 913 Simple Engine Plane 913 Tail-rope System 913 Endless Rope System 914 Wire-rope Tramways ^ 914 Suspension Cableways and Cable Hoists 915 Stress in Hoisting-ropes on Inclined Planes 915 Tension Required to Prevent Wire Slipping on Drums 916 Taper Ropes of Uniform Tensile Strength 916 Effect of Various Sized Drums on the Life of Wire Ropes 917 WIRE-ROPE TRANSMISSION. Elastic Limit of Wire Ropes 917 Bending Stresses of Wire Ropes - 918 Horse-power Transmitted 919 Diameters of Minimum Sheaves 919 Deflections of the Rope 920 Long-distance Transmission 921 ROPE DRIVING. Formulae for Rope Driving 922 Horse-power of Transmission at Various Speeds 924 Sag of the Rope Between Pulleys 925 Tension on the Slack Part of the Rope 925 Miscellaneous Notes on Rope-driving 926 FRICTION AND LUBRICATION. Coefficient of Friction 928 Rolling Friction 928 Friction of Solids 928 Friction of Rest . . 928 Laws of Unlubricated Friction 928 Friction of Sliding Steel Tires 928 Coefficient of Rolling Friction 929 Laws of Fluid Friction 929 Angles of Repose 929 CONTENTS. XXV11 PAGE Friction of Motion 929 Coefficient of Friction of Journal 930 Experiments on Friction of a Journal 931 Coefficients of Friction of Journal with Oil Bath 932 Coefficients of Friction of Motion and of Rest 932 Value of Anti-friction Metals 932 Cast-iron for Bearings 933 Friction of Metal Under Steam-pressure 933 Morin's Laws of Friction . . . 933 Laws of Friction of well-lubricated Journals 934 Allowable Pressures on Bearing-surface 935 Oil-pressure in a Bearing 937 Friction of Car-journal Brasses 937 Experiments on Overheating of Bearings 938 Moment of Friction and Work of Friction 938 Pivot Bearings 939 The Schiele Curve. ^ 939 Friction of a Flat Pivot-bearing 939 Mercury-bath Pivot 940 Ball Bearings 940 Friction Rollers. . . 940 Bearings for Very High Rotative Speed 941 Friction of Steam-engines 941 Distribution of the Friction of Engines 941 Lubrication. Durability of Lubricants. 942 Qualifications of Lubricants 943 Amount of Oil to run an Engine 943 Examination of Oils 943 Penna. R. R. Specifications 944 Soda Mixture for Machine Tools 945 Solid Lubricants , 945 Graphite, Soapstone, Fibre-graphite, Metaline 945 THE FOUNDRY. Cupola Practice 946 Charging a Cupola > 948 Charges in Stove Foundries 949 Results of Increased Driving 949 Pressure Blowers 950 Loss of Iron in Melting 950 Use of Softeners. . 950 Shrinkage of Castings 951 Weight of Castings from Weight of Pattern 952 Moulding Sand 952 Foundry Ladles 952 THE MACHINE SHOP. Speed of Cutting Tools 953 Table of Cutting Speeds 954 Speed of Turret Lathes 954 Forms of Cutting Tools 955 Rule for Gearing Lathes -. 955 Change-gears for Lathes 956 Metric Screw-threads 956 Setting the Taper in a Lathe 956 Speed of Drilling Holes 956 Speed of Twist-drills 957 Milling Cutters 957 Speed of Cutters 958 Results with Milling-machines 959 Milling with or Against Feed 960 Milling-machine vs. Planer 960 Power Required for Machine Tools 960 Heavy Work on a Planer 960 Horse-power to run Lathes 961 XXX CONTENTS. Electrical Resistance. Laws of Electrical Resistance 1027 Electrical Conductivity of Different Metals and Alloys 1028 Conductors and Insulators 1028 Resistance Varies with Temperature 1028 Annealing 1029 Standard of Resistance of Copper Wire 1029 Direct Electric Currents. Ohm's Law 1029 Series and Parallel or Multiple Circuits 1030 Resistance of Conductors in Series and Parallel 1030 Internal Resistance 1031 Electrical, Indicated, and Brake Horse-power 1031 Power of the Circuit 1031 Heat Generated by a Current 1031 Heating of Conductors 1032 Fusion of Wires 1032 Heating of Coils 1032 Allowable Carrying Capacity of Copper Wires 1033 Underwriters' Insulation 1033 Copper-wire Table 1034, 1035 Electric Transmission, Direct Currents. Section of Wire Required for a Given Current 1033 Weight of Copper for a Given Power 1036 Short-circuiting 1036 Economy of Electric Transmission 1036 Wire Table for 110, 220, 500, 1000, and 2000 volt Circuits 1037 Efficiency of Long-distance Transmission 1038 Table of Electrical Horse-powers 1039 Cost of Copper for Long-distance Transmission 1040 Systems of Electrical Distribution 1041 Electric Lighting. Arc Lights 1042 Incandescent Lamps 1042 Variation in Candle-power and Life 1042 Specifications for Lamps 1043 Special Lamps 1043 Nernst Lamp 1043 Electric Welding 1044 Electric Heaters 1044 Electric Accumulators or Storage-batteries. Description of Storage-batteries 1045 Sizes and Weights of Storage-batteries 1048 General Rules for Storage-cells 1048 Electrolysis 1048 Electro-chemical Equivalents 1049 Efficiency of a Storage-cell 1048 Electro-magnets. Units of Electro-magnetic Measurements 1050 Lines of Loops of Force 1050 The magnetic Circuit 1051 Permeability 1052 Tractive or Lifting Force of a Magnet 1053 Magnet Windings 1053 Determining the Polarity of Electro-magnets 1054 Determining the Direction of a Current 1054 Dynamo-electric Machines. Kinds of Dynamo-electric Machines as regards Manner of Winding. . . 1055 Moving Force of a Dynamo-electric Machine 1055 Torque of an Armature : . 1056 Electro-motive Force of the Armature Circuit 1056 Strength of the Magnetic Field 1057 Dynamo Design 1058 COKTEOTS. Alternating Currents. PAGE Maximum, Average, and Effective Values 1061 Frequency 1061 Inductance, Capacity, Power Factor 1062 Reactance, Impedance, Admittance 1063 Skin Effect Factors 1063 Ohm's Law Applied to Alternating Currents 1064 Impedance Polygons 1066 Capacity of Conductors 1066 Self-inductance of Lines and Circuits 1066 Capacity of Conductors 1067 Single-phase and Polyphase Currents 1068 Measurement of Power in Polyphase Circuits 1069 Alternating-current Generators 1070 Transformers, Converters, etc 1070 Synchronous Motors 1071 Induction Motors 1072 Calculation of Alternating-current Circuits 1072 Weight of Copper Required in Different Systems. .' 1074 Electrical Machinery. Direct-current Generators and Motors 1074-1076 Alternating-current Generators 1077 Induction Motors 1077 Symbols Used in Electrical Diagrams 1078 APPENDIX. Strength of Timber. Safe Load on White-oak Beams 1079 Mathematics. Formula for Interpolation 1080 Maxima and Minima without the Calculus 1080 Riveted Joints. Pressure Required to Drive Hot Rivets 1080 Heating and Ventilation. Capacities for Hot-blast or Plenum Heating with Fans and Blowers. . 1081 Water-wheels. Water-power Plants Operating under High Pressure 1G81 Formulae for Power of Jet Water-wheels 1082 Gas Fuel. Composition Energy, etc., of Various Gases 1082 Steam-boilers. Rules for Steam-boiler Construction 1083 Boiler Feeding 1083 Feed-water Heaters 1083 The Steam-engine. Current Practice in Engine Proportions 1084 Work of Steam-turbines 1085 Relative Cost of Different Sizes of Engines 1085 Gearing:. Efficiency of Worm Gearing 1086 Hydraulic Formulae. Flow of Water from Orifices, etc 1087 Tin and Terne Plate. Penna. R. R. Co.'s Specifications 1088 LIST OF AUTHORITIES 1089 NAMES AND ABBREVIATIONS OF PERIODICALS AND TEXT-BOOKS FREQUENTLY REFERRED TO IN THIS WORK. Am. Mach. American Machinist. App. Cyl. Mech. Appleton's Cyclopaedia of Mechanics, Vols. I and n. Bull. I. & S. A. Bulletin of the American Iron and Steel Association (Philadelphia). Burr's Elasticity and Resistance of Materials. Clark, E. T. D. D. K. Clark's Rules, Tables, and Data for Mechanical En- gineers. Clark, S. E. D. K. Clark's Treatise on the Steam-engine. Col. Coll. Qly. Columbia College Quarterly. Eugg. Engineering (London). Eng. News. Engineering News. Engr. The Engineer (London). Fairbairn's Useful Information for Engineers. Flynn's Irrigation Canals and Flow of Water. Jour. A. C. I. W. Journal of American Charcoal Iron Workers' Association. Jour. F. I. Journal of the Franklin Institute. Kapp's Electric Transmission of Energy. Lanza's Applied Mechanics. Merriman's Strength of Materials. Modern Mechanism. Supplementary volume of Appleton's Cyclopaedia of Mechanics. Proc. Inst. C. E. Proceedings Institution of Civil Engineers (London). Proc. Inst. M. E. Proceedings Institution of Mechanical Engineers (Lon- don). Peabody's Thermodynamics. Proceedings Engineers' Club of Philadelphia. Rankine, S. E. Rankine's The Steam Engine and other Prime Movers. Rankine's Machinery and Millwork. Rankine, R. T. D. Rankine's Rules, Tables, and Data. Reports of U. S. Test Board. Reports of U. S. Testing Machine at Watertown, Massachusetts. Rontgen's Thermodynamics, Seaton's Manual of Marine Engineering. Hamilton Smith, Jr.'s Hydraulics. The Stevens Indicator. Thompson's Dynamo-electric Machinery. Thurston's Manual of the Steam Engine. Thurstou's Materials of Engineering. Trans. A. I. E. E. Transactions American Institute of Electrical Engineers. Trans. A. I. M. E. Transactions American Institute of Mining Engineers. Trans. A. S. C. E. Transactions American Society of Civil Engineers. Trans. A. S. M. E. Transactions American Soc'ty of Mechanical Engineers Trautwine's Civil Engineer's Pocket Book. The Locomotive (Hartford, Connecticut). Unwin's Elements of Machine Design. Weisbach's Mechanics of Engineering. Wood's Resistance of Materials. Wood's Thermodynamics. xxzii MATHEMATICS. a Alpha /3 Beta y Gamma 6 Delta e Epsilon Zeta H I K A M Eta Theta Iota Kappa Lambda Mu N v H f O -o n TT p p 2 <r * Nu Xi Omicron Pi Rho Sigma T Y * X * n T V X w Tau Upsilon P& Chi Psi Omega Greek Letters. B r A E Z Arithmetical and Algebraical Signs and Abbreviation*. angle. L right angle. perpendicular to. sin., sine, cos., cosine, tang., or tan., tangent, sec., secant, versin., versed sine, cot., cotangent. cosec., cosecant, covers., co- versed sine. In Algebra, the first letters of the alphabet, a, 6, c, d, etc., are gener- ally used to denote known quantities, and the last letters, w, x, y, z, etc., imknown quantities. Abbreviations and Symbols com- monly used. d, differential (in calculus). /, integral (in calculus). J *, integral between limits a and 6. A, delta, difference. 2. sigma, sign of summation. IT, pi, ratio of circumference of circle to diameter = 3.14159. g, acceleration due to gravity = 32.16 ft. per sec. per sec. Abbreviations frequently used in this Book. L., 1., length in feet and inches. B., b., breadth in feet and inches. D., d., depth or diameter. H., h., height, feet and inches. T., t., thickness or temperature. V.,v., velocity. F., force, or factor of safety. f., coefficient of friction. E M coefficient of elasticity. R., r., radius. W., w., weight. P., p., pressure or load. H.P., horse-power. I.H.P., indicated horse-power. B.H.P., brake horse-power, h. p., hif 1 - 4- plus (addition). 4- positive. - minus (subtraction). negative. i: plus or minus. =F minus or plus. z= equals. x multiplied by. ab or a.b = a x b. ^- divided by / divided by. - = a/6 = a -f- b. 15-16 = .2 = , .002 =^. V* square root. V cube root. V 4th root. : is to, :: so is, : to (proportion). 2 : 4 x 3 : 6, as 2 is to 4 so is 3 to 6. : ratio; divided by. 2 : 4, ratio of 2 to 4 = 2/4. /. therefore. > greater than. < less than, n square. O round. degrees, arc or thermometer. ' minutes or feet. 11 seconds or inches. ' " '" accents to distinguish letters, as a', a", a'". i 2< 3< ;,' <V r ad a sub 1, a sub 6, etc. ( ) C ] { } vincula, denoting that the numbers enclosed are to be taken together ; as, (a -f 6)c = 4 + 3 x 5 = 35. a 8 , a 8 , a squared, a cubed. a n , a raised to the_nth power. a3 = |/2 ? a f = |/ a 3. a-* = -,a-2 = -L a a a 10 = 10 to the 9th power = 1,000 000 - 000. sin. a = the sine of a. sin. J a= the arc whose sine is a. sin. a-* = -; sin. a. log. = logarithm. log. or hyp. log. = hyperbolic loga- rithm. . . p., high pressure. . p., intermediate pressure. 1. p., low pressure. A.W. G., American Wire Gauge (Brown & Sharpe). B.W.G., Birmingham Wire Gauge. r. p. m., or revs, permin., revolutions per minute. MATHEMATICS. ARITHMETIC. The user of this book is supposed to have had a training in arithmetic as well as in elementary algebra. Only those rules are given here which are apt to be easily forgotten. GREATEST COMMON MEASURE. OR GREATEST COMMON DIVISOR OF TWO NUMBERS. Rule. Divide the greater number by the less ; then divide the divisor by the remainder, and so on, dividing always the last divisor by the last remainder, until there is no remainder, and the last divisor is the greatest common measure required. LEAST COMMON MULTIPLE OF TWO OR MORE NUMBERS. Rule. Divide the given numbers by any number that will divide the greatest number of them without a remainder, and set the quotients with the undivided numbers in a line beneath. Divide the second line as before, and so on, until there are no two numbers that can be divided ; then the continued product of the divisors and last quotients will give the multiple required. FRACTIONS. To reduce a common fraction to its lowest terms. Divide both terms by their greatest common divisor: 39/52 = 3/4. To change an improper fraction to a mixed number. Divide the numerator by the denominator; the quotient is the whole number, and the remainder placed over the denominator is the fraction: 39/4 = 9%. To change a mixed number to an improper fraction. Multiply the whole number by the denominator of the fraction; to the prod- uct add the numerator; place the sum over the denominator: 1% = 15/8. To express a whole number in the form of a fraction with a given denominator. Multiply the whole number by the given denominator, and place the product over that denominator: 13 = 39/3. To reduce, a compound to a simple fraction, also to multiply fractions. Multiply the numerators together for a new numerator and the denominators together for a new denominator: To reduce a complex to a simple fraction. The numerator and denominator must each first be given the form of a simple fraction; then multiply the numerator of the upper fraction by the denominator of the lower for the new numerator, and the denominator of the upper by the numerator of the lower for the new denominator: To divide fractions. Reduce both to the form of simple fractions, invert the divisor, and proceed as in multication: 3 3 5 3 v 4 12 3 _^ 1M= _ 4 ._ = _ X _ = _ == _. Cancellation of fractions. In compound or multiplied fractions, divide any numerator and any denominator by any number which will divide them both without remainder, striking out the numbers thus divided and setting down the quotients in their stead. To reduce fractions to a common denominator. Reduce each fraction to the form of a simple fraction ; then multiply each numera- DECIMALS. tor by all the denominators except its own for the new numerators, and all the denominators together for the common denominator: 1 1 3_21 14 18 2' 3' 7 " 42' 42' 42* To add fractions. Reduce them to a common denominator, then add the numerators and place their sum over the common denominator: _ 2 ,3 ' 7 __ 42 ~ 42 ~~ To subtract fractions. Reduce them to a common denominator, subtract the numerators and place the difference over the common denomi- nator: 1 3_7-6_ 1 2 7~ 14 ~14* DECIMALS. To add decimals. Set down the figures so that the decimal points are one above the other, then proceed as in simple addition: 18.75+ .012 = 18.762. To subtract decimals. Set down the figures so that the decimal points are one above the other, then proceed as in simple subtraction: 18.75 - .012 = 18.738. To multiply decimals. Multiply as in multiplication of whole numbers, then point off as many decimal places as there are in multiplier and multiplicand taken together: 1.5 X .02 = .030 = .03. To divide decimals. Divide as in whole numbers, and point off in the quotient as many decimal places as those in the dividend exceed those in the divisor. Ciphers must be added to the dividend to make its decimal places at least equal those in the divisor, and as many more as it is desired to have in the quotient: 1.5 -*- .25 = 6. 0.1 -f- 0.3 = 0.10000 *- 0.3 = 0.3333 -f- Decimal Equivalents of Fractions of One Incb. 1-64 .015625 17-64 .265625 33-64 .515625 49-64 .765625 1-32 .03125 9-32 .28125 17-32 .53125 25-32 .78125 3-64 .046875 19-64 .296875 35-84 .546875 51-64 .796875 1-16 .0625 5-16 .3125 9-16 .5625 13-16 .8125 5-64 .078125 21-64 .328125 37-64 .578125 53-64 .828125 3-32 .09375 11-32 .34375 19-32 .59375 27-32 .84375 7-64 .109375 23-64 .359375 39-64 .609375 55-64 .859375 1-8 .125 3-8 .375 5-8 .625 7-8 .875 9-64 .140625 25-64 .390625 41-64 .640625 57-64 .890625 5-32 .15625 13-32 .40625 21-32 .65625 29-32 .90625 11-64 3-16 .171875 .1875 27-64 7-16 .421875 .4375 43-64 11-16 .671875 .6875 59-64 15-16 .921875 .9375 13-64 .203125 29-64 .453125 45-64 .703125 61-64 .953125 7-32 .21875 15-32 .46875 23-32 .71875 31-32 .96875 15-64 .234375 31-64 .484375 47-64 .734375 63-64 .984375 1-4 .25 1-2 .50 3-4 .75 1 1. To convert a common fraction into a decimal. Divide the numerator by the denominator, adding to che numerator as many ciphers prefixed by a decimal point as are necessary to give the number of' decimal places desired in the result: % = 1.0000 -=-3 = 0.3333 -f. To convert a decimal into a common fraction. Set down the decimal as a numerator, and place as the denominator 1 with as many ciphers annexed as there are decimal places in the numerator; erase the 0*0 fe joo ARITHMETIC. TH O T- 1> ~ GO g CO t- i-i s s C CO CO to 10 rf< CO CO CO -^1 8 g g O O O O CO 00 COMPOUKD NUMBERS. 5 decimal point in the numerator, and reduce the fraction thus formed to It* lowest terms: To reduce a recurring decimal to a common fraction. Subtract the decimal figures that do not recur from the whole decimal in- cluding one set of recurring figures; set down the remainder as the numer- ator of the fraction, and as many nines as there are recurring figures, fol- lowed by as many ciphers as there are non-recurring figures, in the denom- inator. Thus: .79054054, the recurring figures being 054. Subtract 79 J17 = (reduced to its lowest terms) ^ COMPOUND OR DENOMINATE NUMBERS. Reduction descending. To reduce a compound number to a lower denomination. Multiply the number by as many units of the lower denomi- nation as makes one of the higher. 3 yards to inches: 3 X 36 = 108 inches. .04 square feet to square inches: .04 X 144 = 5.76 sq. in. If the given number is in more than one denomination proceed in steps from the highest denomination to the next lower, and so on to the lowest, adding in the units of each denomination as the operation proceeds. 3 yds. 1 ft. 7 in. to inches: 3x3 = 9, -f 1 = 10, 10 X 12 = 120, -f 7 = 127 in. Reduction ascending* To express a number of a lower denomi- nation in terms of a higher, divide the number by the numb r of units of the lower denomination contained in one of the next higher; the quotient is in the higher denomination, and the remainder, if any, in the lower. 127 inches to higher denomination. 127 -*- 12 = 10 feet + 7 inches ; 10 feet -*- 3 = 3 yards + 1 foot. Ans. 3 yds. 1 ft. 7 in. To express the result in decimals of the higher denomination, divide the given number by the number of units of the given denomination contained in one of the required denomination, carrying the result to as many places of decimals as may be desired. 127 inches to yards: 127 -*- 36 = 3f = 3.5277 -f yards. RATIO AND PROPORTION. Ratio is the relation of one number to another, as obtained by dividing one by the other. Ratio of 2 to 4, or 2 : 4 = 2/4 = 1/2. Ratio of 4 to 2, or 4 : 2 = 2. Proportion is the equality of two ratios. Ratio of 2 to 4 equals ratio of 3 to 6, 2/4 = 3/6; expressed thus, 2 : 4 : : 3 : 6; read, 2 is to 4 as 3 is to 6. The first and fourth terms are called the extremes or outer terms, the second and third the means or inner terms. The product of the means equals the product of the extremes: 2 : 4 : : 3 : 6; 2 X 6 = 12; 3 X 4 = 12. Hence, given the first three terms to find the fourth, multiply the second and third terms together and divide by the first. 4 v 3 2 : 4 : : 3 : what number ? Ans, = 6, 6 ARITHMETIC. Algebraic expression of proportion. a : b : : c : d; =- = %;ad be . be . ad ad = be; from which a ; d ; & = ; c = -= . d a c b Mean proportional between two given numbers, 1st and 2d, is such a number that the ratio which the first bears to it equals the ratio which it bears to the second. Thus, 2 : 4 : : 4 : 8; 4 is a mean proportional between 2 and 8. To find the mean proportional between two numbers, extract the square root of their product. Mean proportional of 2 and 8 = V% x 8 = 4. Single Rule of Three ; or, finding the fourth term of a proportion when three terms are given. Rule, as above, when the terms are stated in their proper order, multiply the second by the third and divide by the first. The difficulty is to state the terms in their proper order. The term which is of the same kind as the required or fourth term is made the third; the first and second must be like each other in kind and denomination. To deter- mine which is to be made second and which first requires a little reasoning. If an inspection of the problem shows that the answer should be greater than the third term, then the greater of the other two given terms should be made the second term otherwise the first. Thus, 3 men remove 54 cubic feet of rock in a day; how many men will remove in the same time 10 cubic yards ? The answer is to be men make men third term; the answer is to be more than three men, therefore make the greater quantity, 10 cubic yards, the second term ; but as it is not the same denomination as the other term it must be reduced, = 270 cubic feet. The proportion is then stated: 3 X 270 54 : 270 : : 3 : x (the required number) ; x = = 15 men. The problem is more complicated if we increase the number of given terms. Thus, in the above question, substitute for the words " in the same time " the words " in 3 days." First solve it as above, as if the work were to be done in the same time; then make another proportion, stating it thus: If 15 men do it in the same time, it will take fewer men to do it in 3 days; make 1 day the 2d term and 3 days the first term 3:1 : : 15 men : 5 men. Compound Proportion, or Double Rule of Three. By this rule are solved questions like the one just given, in which two or more stat- ings are required by the single rule of three. In it as in the single rule, there is one third term, which is of the same kind and denomination as the fourth or required term, but there may be two or more first and second terms. Set down the third term, take each pair of terms of the same kind separately, and arrange them as first and second by the same reasoning as is adopted in the single rule of three, making the greater of the pair the second if this pair considered alone should require the answer to be greater. Set down all the first terms one under the other, and likewise all the second terms. Multiply all the first terms together and all the second terms together. Multiply the product of all the second terms by the third term . and divide this product by the product of all the first terms. Example: If 3 men remove 4 cubic yards in one day, working 12 hours a day, how many men working 10 hours a day will remove 20 cubic yards in 3 days ? Yards 4 Days 3 Hours 10 Products 120 20 1 : : 3 men. 12 240 : : 3 : 6 men. Ans. To abbreviate by cancellation, any one of the first terms may cancel either the third or any of the second terms; thus, 3 in first cancels 3 in third, making it 1, 10 cancels into 20 making the latter 2, which into 4 makes it 2, which into 12 makes it 6, and the figures remaining are only 1 : 6 : : 1 : 6. INVOLUTION, OR POWERS OF NUMBERS. Involution is the continued multiplication of a number by itself a given number of times. The number is called the root, or first power, and the products are called powers. The second power is called the square and POWERS OF HUMBERS. the third power the cube. The operation may be indicated without being performed by writing a small figure called the index or exponent to the right of and a, little above the root; thus, 3 3 = cube of 3, = 27. To multiply two or more powers of the same number, add their exponents; thus, 22 x 2 3 ' = 2 5 , or 4 X 8 = 32 = 2 5 . To divide two powers of the same number, subtract their exponents; thus, 2 3 -r- 2 2 = 2 1 = 2; 2 2 -f- 2 4 = 2~ 2 = = -. The exponent may thus be nega- tive 2 3 -t- 2 3 = 2 = 1, whence the zero power of any number = 1. The first power of a number is the number itself. The exponent may be frac- tional, as 2*, 23, which means that the root is to be raised to a power whose exponent is the numerator of the fraction, and the root whose sign is the denominator is to be extracted (see Evolution). The exponent may be a deeimal, as 2' 5 , 2 1 * 6 ; read, two to the five-tenths power, two to the one and five-tenths power. These powers are solved by means of Logarithms (which see). First Nine Powers of the First Nine Numbers. 1st 3d 3d 4th 5th 6th 7th 8th 9th Pow'r Pow'r Power. Power. Power. Power. Power. Power. Power. 1 1 1 j 1 1 1 1 1 2 4 8 16 32 64 128 256 512 3 9 27 81 243 729 2187 6561 19683. 4 16 64 256 1024 4096 16384 65536 262144 5 25 125 625 3125 15625 78125 390625 1953125 6 36 216 1296 7776 46656 279936 1679616 10077696 7 49 343 2401 16807 117649 823543 5764801 40353607 8 64 512 4096 32768 262144 2097152 16777216 134217728 9 81 729 6561 59049 531441 4782969 43046721 387420489 The First Forty Powers of 2. h o L* c L L oJ "3 1 o "3 i 1 o 1 o 1 > fi p. > PH > PH l 9 512 18 262144 27 134217728 36 68719476736 2 10 1024 19 524288 28 268435456 37 137438953472 4 11 2048 20 1048576 29 536870912 38 274877906944 8 12 4096 21 2097152 30 1073741824 39 549755813888 16 13 8192 22 4194304 31 2147483048 40 1099511627776 32 14 16384 23 8388608 32 4294967296 64 15 32768 24 16777216 33 8589934592 128 16 65536 25 33554432 34 17179869184 256 17 131072 26 67108864 35 34350738368 EVOLUTION. Evolution is the finding of the root (or extracting the root) of any number the power of which is given. I/ V * the The sign tf indicates that the square root is to be extracted : cube root, 4th root, ?ith root. A fractional exponent with 1 for the numerator of the fraction is also used to indicate that the operation of extracting the root is to be performed; thus, 2*, 2* = V2, Vs. When the power of a number is indicated, the involution not being per- formed, the extraction of any root of that power may also be indicated by 8 ARITHMETIC. dividing the index of the power by the index of the root, indicating the division by a fraction. Thus, extract the square root of the 6th power of 2: |/2 = 2$ _ 2 f _ g 3 _ 8< The 6th power of 2, as in the table above, is 64 ; |/64 ss 8. Difficult problems in evolution are performed by logarithms, but the square root and the cube root may be extracted directly according to the rules given below. The 4th root is the square root of the square root. The 6th root is the cube root of the square root, or the square root of the cube root ; the 9th root is the cube root of the cube root etc. To Extract tlie Square Root. Point off the given number into periods of two places each, beginning with units. If there are decimals, point these off likewise, beginning at the decimal point, and supplying as many ciphers as may be needed. Find the greatest number whose square is less than the first left-hand period, and place it as the first figure in the quotient. Subtract its square from the left-hand period, and to the remainder annex the two figures of the second period for a dividend. Double the first figure of the quotient for a partial divisor ; find how many times the latter is contained in the dividend exclusive of the right-hand figure, and set the figure representing that number of times as the second figure in the quotient, and annex it to the right of the partial divisor, forming the complete divisor. Multiply this divisor by the second figure in the quotient and subtract the product from the divi- dend. To the remainder bring down the next period and proceed as before, in each case doubling the figures in the root already found to obtain the trial divisor. Should the product of the second figure in the root by the completed divisor be greater than the dividend, erase the second figure both from the quotient and from the divisor, and substitute the next smaller figure, or one small enough to make the product of the second figure by the divisor less than or equal to the dividend. 3.141 5926536 1 [1.77245 -f 27T274 34712515 1 2489 3542 8692 7084 35444 160865 1141776 354485 1908936 1772425 To extract the square root of a fraction, extract the root of numerator /4 2 and denominator separately. \/ - -, or first convert the fraction into a p 9 3 decimal, j/|= 4/.4444 + = .6666 + . T 9 To Kxtract the Cube Root. Point off the number into periods of 3 figures each, beginning at the right hand, or unit's place. Point off deci- mals in periods of 3 figures from the decimal point. Find the greatest cube that does not exceed the left-hand period ; write its root as the first figure in the required root, Subtract the cube from the left-hand period, and to the remainder bring down the next period for a dividend. Square the first figure of the root; multiply by 300, and divide the product into the dividend for a trial divisor ; write the quotient after the first figure of the root as a trial second figure. Complete the divisor by adding to 3CO times the square of the first figure, 30 times the product of the first by the second figure, and the square of the second figure. Multiply this divisor by the second figure; subtract the product from the remainder. (Should the product be greater than the remainder, the last figure of the root and the complete divisor are too large ; CUBE ROOT. substitute for the last figure the next smaller number, and correct the trial divisor accordingly.) To the remainder bring down the next period, and proceed as before to find the third figure of the root that is, square the two figures of the root already found; multiply by 300 for a trial divisor, etc. If at any time the trial divisor is greater than the dividend, bring down an- other period of 3 figures, and place in the root and proceed. The cube root of a number will contain as many figures as there are periods of 3 in the number. Shorter Methods of Extracting the Cube Root,- 1, From Went worth's Algebra: 300 x 30x x 2 = 1,881, 365,963,625 1 12345 1 300 881 728 64 153365 300 x 30 x 122 = 12 x 3 = 43200 1080 300 x 30 x 123 x 4 = 4,2 = 442891 132867 1089 J 20498963 4538700 1 14760* 16| 4553476 ) 18213904 _ I 4 ! 7 !! 25J85059625 300 x 12342 = 456826800 30 x 1234 x 5 = 185100 52= 25 457011925 2285059625 After the first two figures of the root are found the next trial divisor is found by bringing down the sum of the 60 and 4 obtained in completing the preceding divisor; then adding the three lines connected by the brace, and annexing two ciphers. This method shortens the work in long examples, as is seen in the case of the last two trial divisors, saving the labor of squaring 123 and 1234. A further shortening of the work is made by obtaining the last two figures of the root by division, the divisor employed being three times the square of the part of the root already found ; thus, after finding the first three figures: 3 x 123 = 45387|20498963|45.1-f 181548 ~ 234416 226935 74813 The error due to the remainder is not sufficient to change the fifth figure of the root. 2. By Prof. H. A. Wood (Stevens Indicator, July, 1890): I. Having separated the number into periods of three figures each, count- ing from the right, divide by the square of the nearest root of the first period, or first two periods ; the nearest root is the trial root. II. To the quotient obtained add twice the trial root, and divide by 3. This gives the root, or first approximation. III. By using the first approximate root as a new trial root, and proceed- ing as before, a nearer approximation is obtained, which process may be repeated until the root has been extracted, or the approximation carried as far as desired. 10 ARITHMETIC. EXAMPLE. Required the cube root of 20. The nearest cube to 20 is 3*. 32 = 9)20.0 2.2 6_ 3)871 2.7 IstT. R. 7.29)20.000 3)8.143 2.714, 1st ap. cube root, 2.714 2 = 7.365796)20.0000000 2.7152534 5.428 3)8.1432534 2.7144178 2d ap. cube root. REMARK. In the example it will be observed that the second term, or first two figures of the root, were obtained by using for trial root the root of the first period. Using, in like manner, these two terms for trial root, we obtained four terms of the root ; and these four terms for trial root gave seven figures of the root correct. In that example the last figure should be 7. Should we take these eight figures for trial root we should obtain at least fifteen figures of the root correct. To Extract a Higher Root than the Cube, The fourth root is the square root of the square root ; the sixth root is the cube root of the square root or the square root of the cube root. Other roots are most con- veniently found by the use of logarithms. ALLIGATION shows the value of a mixture of different ingredients when the quantity and value of each is known. Let the ingredients be a, 6, c, d, etc., and their respective values per unit w>> x, y, z, etc. A = the sum of the quantities = a-\-b-\-c-}-d, etc. P mean value or price per unit of A. AP = aw -f bx -f- cy + dz, etc. _ aw -\-bx-\-cy-\-dz A PERMUTATION shows in how many positions any number of things may be arranged in a vrow; thus, the letters a, b, c may be arranged in six positions, viz. abc, acb, 'cab, cba, bac, bca. Rule. Multiply together all the numbers used in counting the things; thus, permutations of 1, 2, and 3 = 1X2X3 = 6. In how many positions can 9 things in a row be placed ? 1X2X3X4X5X6X7X8X9 = 362880. COMBINATION shows how many arrangements of a few things may be made out of a greater number. Rule : Set down that figure which indicates the greater number, and after it a series of figures diminishing by 1, until as many are set down as the number of the few things to be taken in each combination. Then beginning under the last one set down said number of few things ; then going backward set down a series diminishing by 1 until arriving under the first of the upper numbers. Multiply together all the upper numbers to form one product, and all the lower numbers to form another; divide the upper product by the lower one. GEOMETRICAL PROGRESSION. 11 How many combinations of 9 things can be made, taking 3 in each com- bination ? 9X8X7 _ 504 _ 84 1X2X3" 6 ARITHMETICAL PROGRESSION, in a series of numbers, is a progressive increase or decrease in each succes- sive number by the addition or subtraction of the same amount at each step, as 1, 2, 3, 4, 5, etc., or 15, 12, 9, 6, etc. The numbers are called terms, and the equal increase or decrease the difference. Examples in arithmetical pro- gression may be solved by the following formulae : Let a = first term, I = last term, d = common difference, n = number of terms, s = sum of the terms: I = a -f (u l)d, _2s _ ~~ n ~ ' = d + a) , - d) + Sds OE01TIKTRICAI, PROGRESSION, in a series of numbers, is a progressive increase or decrease in each sue. cessive number by the same multiplier or divisor at each step, as 1, 2, 4, 8, 16. etc., or 243, 81, 27, 9, etc. The common multiplier is called the ratio. Let a = first term, I = last term, r ratio or constant multiplier, n =: number of terms, m = any term, as 1st, 2d, etc., s = sum of the terms: ' a -r- (r - I) _ (r-l)sr~ - l l = ar*-l> -p- -7nr~ iog Z = log a + (n - 1) log r, f( - l) n ~ x - a( - a) n - J = 0. m = af"* - a * log w = log a -f- (m - 1) log r. n - 1 / n 1/~~^ yj n - y a n = n-l- n-l.- ' ARITHMETIC. =: o. log I - log a 1 logr " log I log a '' log (s - a) - log (* - " log a = log I - (n 1) log r. log I log a logr = n __! . lg [ft + (r l)s] log a log r log Z - log [?/ - (r - l)s] log r Population of the United States. (A problem in geometrical progression.^ Tear. 1860 1870 1880 1890 1900 1905 1910 Population. 81,443,821 39,818,449* 50,155,783 62,622,250 76,295,220 Est. 83,577,000 " 91554,000 Increase in 10 Annual Increase, Years, per cent. per cent. 26.63 25.96 24.86 21.834 2.39 2.33 2.25 1.994 Est. 1.840 44 1.840 Est. 20.0 Estimated Population in Each Year from 1870 to 1909. (Based on the above rates of increase, in even thousands.) 1870. . . . 39,818 1880.. . 50,156 1890.. 62,622 1900. .. 76,295 1871 .... 40,748 1881.. . 51,281 1891. 63,871 1901. .. 77,699 1872. .. 41,699 1882 . . 52,433 1892. . 65,145 1902. .. 79,129 1873.... 42,673 1883.. . 53,610 1893. . 66,444 1903. .. 80,585 1874.... 43,670 1884.. . 54,813 1894. 67,770 1904. .. 82,067 1875... 44,690 1885.. 56,043 1895. 69,122 1905. .. 83,577 1876.... 45,373 1886.. . 57,301 1896. 70,500 1906. .. 85,115 1877.... 46,800 1887.. . 58,588 1897. 71,906 1907. .. 86,681 1878 .. 47,893 1888.. . 59,903 1898. 73,341 1908. .. 88,276 1879.... 49,011 1889.. . 61,247 1899. 74,803 1909. .. 89,900 The above table has been calculated by logarithms as follows : log r = log I - log a -*- (n - 1), log m = log a -f- (w* 1) lo S r Pop. 1900. . . . 76,295,2-20 log = 7.8824988 = log I " 1890 . . . 62,022,250 log = 7.7967285 = log a cliff. = .0857703 n = 11, n - 1 = 10; diff. * 10 = .00857703 = log r, add log for 1890 7.7967285 = log a log for 1891 = 7.80530553 No. = 63,871 . . . add again .00857703 log for 1892 7.81388256 No. = 65,145 . . . Compound interest is a form of geometrical progression ; the ratio be- ing 1 plus the percentage. * Corrected by addition of 1,260,078, estimated error of the census of 1870, Census Bulletin No, 16, Dec, 12, 1890, DISCOUNT. 13 INTEREST AND DISCOUNT. Interest is money paid for the use of money for a given time; the fao tors are : p, the sum loaned, or the principal: t, the time in years; r, the rate of interest; t, the amount of interest for the given rate and time; a = p + * = the amount of the principal with interest at the end of the time. Formulae : i = interest = principal X time X rate per cent = i = ^~; a = amount = principal -f- interest = p -{- ^55; _ 100* - pt> , ,. 100* t = time = . pr If the rate is expressed decimally as a per cent, thus, 6 per cent = .06, the formulae become pt 1 pr 7 tr Rules for finding Interest. Multiply the principal by ;the rate per annum divided by 100, and by the time in yc *i ; and fractions of a year. If the time is given in days, interest = Principal X rate X no. of days oo5 X 100 In banks interest is sometimes calculated on the basis of 360 days to a year, or 12 months of 30 days each. Short rules for interest at 6 per cent, when 360 days are taken as 1 year: Multiply the principal by number of days.and divide by 6000. Multiply the principal by number of months and divide by 200. The interest of 1 dollar for one month is ^ cent. Interest of 100 Dollars for Different Times and Rates. Time. 2# 3# 4# 6# 6 8 10* lyear $2.00 $3.00 $4.00 $5.00 $6.00 $8.00 $10.00 1 month .16f .25 .33$ .41f .50 .66| .83$ 1 day = 3 $s year .0055f .0083$ .0111$ .0138| .0166 .0222* .0277$ 1 day = 3$ 5 year .005479 .008219 .010959 .013699 .016438 .0219178 .0273973 Discount is interest deducted for payment of money before it is due. True discount is the difference between the amount of a debt pay- able at a future date without interest and its present worth. The present worth is that sum which put at interest at the legal rate will amount to the debt when it is due. To find the present worth of an amount due at future date, divide the amount by the amount of $1 placed at interest for the given time. The dis- count equals the amount minus the present worth. What discount should be allowed on $103 paid six months before it is due, interest being 6 per cent per annum ? 103 T = $100 present worth, discount = 8.00. 1 + 1 X .06 X | Bank discount is the amount deducted by a bank as interest on money loaned on promissory notes. It is interest calculated not on the act- ual sum loaned, but on the gross amount of the note, from which the dis- count is deducted in advance. It is also calculated on the basis of 360 days in the year, and for 3 (in some banks 4) days more than the time specified "in the note. These are called days of grace, and the note is not payable till tfce last of these days. In some States days of grace have been abolished. 14 ARITHMETIC. What discount will be deducted by a bank in discounting a note for $108 payable 6 months hence ? Six months = 182 days, add 3 days grace = 185 days ,103 X 185 6000 = $3.176. Compound Interest. In compound interest the interest is added to the principal at the end of each year, (or shorter period if agreed upon). Let p = the principal, r = the rate expressed decimal^, n = no of years, and a the amount : a = amount = p (1 + r) n ; r = rate = p = principal = no . ot years _ n = Compound Interest Table. (Talue of one dollar at compound interest, compounded yearly, at 3, 4, 5, and 6 per cent, from 1 to 50 years.) 3# 4* 6* w 05 1 F 3* W 5* w i 1.03 3.04 1.05 1.06 16 1.6047 1.8730 2.1829 2.5403 2 1.0609 1.0816 1.1025 1.1236 17 1.6528 1.9479 2.2920 2.6928 3 1.0927 1.1249 1.1576 1.1910 18 1.7024 2.0258 2.4066 2.8543 4 .1255 1.1699 1.2155 1.2625 19 1.7535 2.1068 2.5269 3.0256 5 .1593 1.2166 1.2763 1.3382 20 1.8061 2.1911 2.6533 3.2071 6 .1941 1.2653 1.3401 1.4185 21 1.8603 2.2787 2.7859 3.3995 7 .2299 1.3159 .4071 1.5036 22 1.9161 2.3699 2.9252 3.6035 8 .2668 1.3686 .4774 1.5938 23 1.9736 24647 3.0715 3.8197 9 .3048 .4233 .5513 1.6895 24 2.0328 2.5633 3.2251 40487 10 .3439 .4802 .6289 1.7908 25 2.0937 2.6658 3.3863 4.2919 11 1.3842 .5394 .7103 1.8983 30 2.4272 3.2433 4.3219 5.7435 12 1.4258 .6010 .7958 2.0122 35 2.8138 3.9460 5.5159 7.6862 13 1.4685 .6651 .8856 2.1329 40 3.2620 4.8009 7.0398 10.2858 14 1.5126 .7317 .9799 2.2609 45 3.7815 5.8410 8.9847 13.7648 15 1.5580 1.8009 2.0789 2.3965 50 4.3838 7.1064 11.4670 18.4204 At compound interest at 3 per cent money will double itself in 23J^ years, at 4 per cent in 17% years, at 5 per cent in 14.2 years, and at 6 per cent in 11. 9 years. EQUATION OF PAYMENTS. By equation of payments we find the equivalent or average time in which one payment should be made to cancel a number of obligations due at dif- ferent dates ; also the number of days upon which to calculate interest or discount upon a gross sum which is composed of several smaller sums pay- able at different dates. Rule. Multiply each item by the time of its maturity in days from a fixed date, taken as a standard, and divide the sum of the products by the sum of the items: the result is the average time in days from the standard date. A owes B $100 due in 30 days, $200 due in 60 days, and $300 due in 90 days. In how many days may the whole be paid in one sum of $600 ? 100 x 30 -f- 200 x 60 -f 300 x 90 = 42,000 ; 42,000 -f- 600 = 70 days, ana. A owes B $100, $200, and $300, which amounts are overdue respectively 30, 60, and 90 days. If he now pays the whole amount, $600, how many days' interest should he pay on that sum ? Ans. 70 days. ANNUITIES. 15 PARTIAL PAYMENTS. To compute Interest on notes and bonds when partial payments have been made: United States Rule. Find the amount of the principal to the time of the first payment, and, subtracting the payment from it, find the amount of the remainder as a new principal to the time of the next payment. If the payment is less than the interest, find the amount of the principal to the time when the sum of the payments equals or exceeds the interest due, and subtract the sum of the payments from this amount. Proceed in this manner till the time of settlement. Note. The principles upon which the preceding rule is founded are: 1st. That payments must be applied first to discharge accrued interest, and then the remainder, if any, toward the discharge of the principal. 26. That only unpaid principal can draw interest. Mercantile Method. When partial payments are made on short notes or interest accounts, business men commonly employ the following method : Find the amount of the whole debt to the time of settlement ; also find the amount of each payment from the time it was made to the time of set- tlement. Subtract the amount of payments from the amount of the debt; the remainder will be the balance due. ANNUITIES. An Annuity is a fixed sum of money paid yearly, or at other equal times agreed upon. The values of annuities are calculated by the principles of compound interest. 1. Let i denote interest on $1 for a year, then at the end of a year the amount will be 1 + i. At the end of n years it will be (1 + i). 2. The sum which in n years will amount to 1 is n or (l + i)~ w , or the present value of 1 due in n years. (1 I i) n 1 3. The amount of an annuity of 1 in any number of years n is v ~ . - . 4. The present value of an annuity of 1 for any number of years n is 5. The annuity which 1 will purchase for any number of years n is 6. The annuity which would amount to 1 in n years is - Amounts, Present Values, etc., at 5% Interest. Years (1) (2) (3) (4) (5) (6) (l+i) n (l_f i)-n (1 4- i)n - 1 l-(l+9- i i i i i-d+9-" (1 + 9" -1 1 1.05 .952381 1. .952381 1.05 1. 2 1.1025 .907029 2.05 1.859410 .537805 .487805 3 1.157625 .863838 3.1525 2.723248 .367209 .317209 4 1.215506 .822702 4.310125 3.545951 .282012 .232012 5 1.276282 .783526 5.525631 4.329477 .230975 .180975 6... .340096 .746215 6.801913 5.075692 .197017 .147018 7 .407100 .710681 8.142008 5.786373 .172820 .122820 8 .477455 .676839 9.549109 6.463213 .154722 .104722 9 1. 5513">8 .644609 11.026564 7.107822 .140690 .090690 10 .628895 .613913 12.577893 7.721735 .129505 .078505 16 ARITHMETIC. jTzgfc^ SSfeSS S8S3& gSS&S ?* v * o *Q 8iO C7 Tf i> O rf ic rf O CO CJ t- *5 " !- t- C* O I- Ci rf O OS O O* CO T-I CO <M GC 1C <?* O O 1O i-~> 00 CO T 00 T-. CO GO-* 0OOSt>t- C040*OTr-t COCOCOCOCi T-ITH ^ CO <7^ > I T-< T I T-1 T^ ! OCOOC005 T^C< iSS $g OcOCOrt* i>COOO o FH ce- ^ o 5 5 3 ^TM iOT-icoooi."- T-IO;OTJ<O SS ^33^ 2fc'aa: *- -<*COOil O5 !7J TT C oocot".T?QO Tj<e>?Tfcoi> j>oconco ^9- O JO 00 CO O GO OS CO CO CO O GO -rt CO 5^0500**: coo sigs'feS ^'dois^ oscoi>o;kO ost-Tj-Nu Ot-OOWO T-t-JOr-iG ?^c?^ g^^S o * tO T I T-I W CO W Th CO QC t ost-cocjnj oio^^oo ftGOCOlC Ot^-O1<?CO 31-TCO OSC7OOO s ^ I 5co wrt c ft O CO T-H C 5 ^^^^g| glTSS^S 3R OTfTfOOCO T-QOGOO5Tti C^GOCOOSC? lO OS I- ri iO iO TJ< <* C? r-t ift O tO t- CO ^' ^ ^J ,_; i>I co o' r-' o' T-i co j - T-I co w ?t^i2? COOtOOSO O5CO W-T-I - WEIGHTS AND MEASUBES. TABLES FOR CALCULATING SIN It ING -FUNDS AND PRESENT VALUES. Engineers and others connected with municipal work an d*indus trial enter- prises often find it necessary to calculate payments to sinking-funds which will provide a sum of money sufficient to pay off a bond issue or other debt at the end of a given period, or to determine the present value of certain annual charges. The accompanying tables were computed by Mr. John W. Hill, of Cincinnati, Eng'g News, Jan. 25, 1894. Table I (opposite page) shows the annual sum at various rates of interest required to net $1000 in from 2 to 50 years, and Table II shows the present value at various rates of interest of an annual charge of $1000 for from 5 to 50 years, at five-year intervals and for 100 years. Table II. Capitalization of Annuity of $1000 for from 5 to 1OO Years. 5 10 15 20 25 Rate of Interest, per cent. w 3 SM 4 * 5 6* 6 4,645 88 8,752.17 12,381.41 15,589.215 18,424.67 4,579.60 8,530.13 11,937.80 14,877.27 17,413.01 4,514.92 8,316.45 11,517.23 14,212.12 16,481.28 4,451.68 8,110.74 11,118.06 13,590.21 15,621.93 4,389.91 7,912.67 10,739.42 13,007.88 14,828.12 4,329.45 7,721.73 10,379.53 12,462.13 14,093.86 4,268.09 7,537.54 10,037.48 11,950.26 13,413.82 4,212.40 7,860.19 9,712.30 11,469.96 12,783.38 30 35 40 45 50 100 20,930.59 23,145.31 25,103.53 26,833.15 28,362.48 36,614.21 19,600.21 21,487.04 23,114.36 24,518.49 25,729.58 31,598.81 18,391.8517,291.86 20,000.43 18,664.37 21,354.8319,792.65 22,495.2320,719.89 23,455.21i21,482.08 27,655.3624,504.96 16,288.77 17,460.89 18,401.49 19,156.24 19,761.93 21,949.21 15,372.36 16,374.36 17,159.01 17,773.99 18,255.86 19,847.90 14,533.63 15,390.48 16,044.92 16,547.65 16,931.97 18,095.83 13,764.85 14,488.65 15,046.31 15,455.85 15,761.87 16,612.64 WEIGHTS AND MEASURES. Long Measure. Measures of Length. 12 inches = 1 foot. 3 feet = 1 yard. 1760 yards, or 5280 feet = 1 mile. Additional measures of length in occasional use : 1000 mils = 1 inch ; 4 inches = 1 hand; 9 inches = 1 span; 2*4 feet = 1 military pace; 2 yards = 1 fathom; 5V yards, or 16^> feet = 1 rod (formerly also called pole or perch). Old Land Measure. 7.92 inches = 1 link; 100 links, or 66 feet, or 4 rods =i 1 chain; 10 chains, or 220 yards = 1 furlong; 8 furlongs = 1 mile; 10 square chains = 1 acre. Nautieal Measure* 6 8 ut 6 e f mlles r 1>15156 Stat " \ = 1 nautical mile, or knot.* 3 nautical miles = 1 league. 60 Ta'tuTe milll' P 69 ' 168 [ = 1 degree (at the ec l uator )- 360 degrees = circumference of the earth at the equator. *The British Admiralty takes the round figure of 6080 ft. which is the length of the *' measured mile 1 ' used in trials of vessels. The value varies from 6080.26 to 6088.44 ft. according to different measures of the earth's di- ameter. There is a difference of opinion among writers as to the use of the word " knot" to mean length or-a distance some holding that it should be used only to denote a rate of speed. The length between knots on the log line is 1/120 of a nautical mile, or 50.7 ft., when a half-minute glass is used; so that a speed of 10 knots is equal to 10 nautical miles per hour. 18 ARITHMETIC. Square Measure. Measures of Surface. 144 square incites, or 183.35 circular \ _ ^ square f oot . 9 square feet = 1 square yard . 30]4 square yards, or 272> square feet = 1 square rod. 10 sq. chains, or 160 sq. rods, or 4840 sq. | * yards, or 43560 sq. feet, f - - 1 e< 640 acres ' = 1 square mile. An acre equals a square whose side is 208.71 feet. Circular Inch; Circular Mil. A circular inch is the area of a circle 1 inch in diameter = 0.7854 square inch. I square inch = 1.2732 circular inches. A circular mil is the area of a circle 1 mil, or .001 inch in diameter. 1000 2 or 1,000,000 circular mils = 1 circular inch. 1 square inch = 1,273,239 circular mils. The mil and circular mil are used in electrical calculations involving the diameter and area of wires. Solid or Cubic Measure. Measures of Volume. 1728 cubic inches = 1 cubic foot. 27 cubic feet = 1 cubic yard. 1 cord of wood = a pile, 4x4x8 feet = 128 cubic feet. 1 perch of masonry = 16^X 1MX1 foot =24^ cubic feet. Liquid Measure. 4 gills = 1 pint. 2 pints = 1 quart. j U. S. 231 cubic inches. 4 quarts = 1 gallon J Eng 2?7 274 cubic inchegi 31 K gallons = 1 barrel. 42 gallons = 1 tierce. 2 barrels, or 63 gallons = 1 hogshead. 84 gallons, or 2 tierces = 1 puncheon. 2 hogsheads, or 126 gallons = 1 pipe or butt. 2 pipes, or 3 puncheons = 1 tun. A gallon of water at 62 F. weighs 8.3356 Ibs. The U. S. gallon contains 281 cubic inches; 7.4805 gallons = 1 cubic foot. A cylinder 7 in. diam. and 6 in. high contains 1 gallon, very nearly, or 230.9 cubic inches. The British Imperial gallon contains 277.274 cubic inches = 1.20032 U. S. gallon, or 10 Ibs. of water at 62 F. Tlie Miner's Inch. (Western U. S. for measuring flow of a stream of water). The term Miner's Inch is more or less indefinite, for the reason that Cali- f ornip, water companies do not all use the same head above the centre of the aperture, and the inch varies from 1.36 to 1.73 cubic feet per minute each; but the most common measurement is through an aperture 2 inches high and whatever length is required, and through a plank 1 inches thick. The lower edge of the aperture should be 2 inches above the bottom of the measuring-box, and the plank 5 inches high above the aperture, thus mak- ing a 6-inch head above the centre of the stream. Each square inch of this opening represents a miners inch, which is equal to a flow of H cubic feet per minute. Apothecaries' Fluid Measure. 60 minims = 1 fluid drachm. 8 drachms = 1 fluid ounce. In the U. S. a fluid ounce is the 128th part of a U. S. gallon, or 1 805 cu. ins. It contains 456.3 grains of water at 39 F, In Great Britain the fluid ounce is 1.732 cu. ins. and contains i ounce avoirdupois, or 437.5 grains of water at 62 F. Dry Measure, U. S. 2 pints = 1 quart. 8 quarts = 1 peck. 4 pecks = 1 bushel. The standard U. S. bushel is the Winchester bushel, which is in cylinder WEIGHTS AND MEASURES. 19 form, 18>3 inches diameter and 8 inches deep, and contains 3150.42 cubic inches. A struck bushel contains 2150.42 cubic inches = 1.2445 cu. ft.; 1 cubic foot - 0.80356 struck bushel. A heaped bushel is a cylinder 18^ inches diam- eter and 8 inches deep, with a heaped cone not less than 6 inches high. It is equal to 1^ struck bushels. The British Imperial bushel is based on the Imperial gallon, and contains 8 such gallons, or 2218.192 cubic inches = 1.2837 cubic feet. The English quarter = 8 Imperial bushels. Capacity of a cylinder in U. S. gallons = square of diameter, in inches X height in inches X .0034. (Accurate within 1 part in 100,000.) Capacity of a cylinder in U. S. bushels = square of diameter in inches X height in inches X .0003652. Shipping Measure* Register Ton. For register tonnage or for measurement of the entire internal capacity of a vessel : 100 cubic feet = 1 register ton. This number is arbitrarily assumed to facilitate computation. Shipping Ton. For the measurement of cargo : 1 U. S. shipping ton. 40 cubic feet = 42 cubic feet = 31. 16 Imp. bushels. 32.143 U. S. " 1 British shipping ton. 32.719 Imp. bushels. 33.75 U. S. Carpenter's Rule. Weight a vessel will carry = length of keel X breadth at main beam X depth of hold in feet -4-95 (the cubic feet allowed for a ton). The result will be the tonnage. For a double-decker instead of the depth of the hold take half the breadth of the beam. Measures of Weight. Avoirdupois, or Commercial Weight. 16 drachms, or 437.5 grains = 1 ounce, oz. 16 ounces, or 7000 grains = 1 pound, Ib. 28 pounds =1 quarter, qr. 4 quarters = 1 hundredweight, cwt. = 112 Ibs. 20 hundred weight = 1 ton of 2240 pounds, or long ton. 2000 pounds = 1 net, or short ton. 2204.6 pounds = 1 metric ton. 1 stone = 14 pounds ; 1 quintal = 100 pounds. The drachm, quarter, hundredweight, stone, and quintal are now seldom used in the United States. Tr*y Weight. 24 grains = 1 pennyweight, dwt. 20 pennyweights = 1 ounce, oz. = 480 grains. 12 ounces = 1 pound, Ib. = 5760 grains. Troy weight is used for weighing gold and silver. The grain is the same in Avoirdupois, Troy, and Apothecaries' weights. A carat, used in weighing diamonds = 3.168 grains = .205 gramme. Apothecaries 9 Weight. 20 grains = 1 scruple, 3 3 scruples = 1 drachm, 3 = 60 grains. 8 drachms = 1 ounce, = 480 grains. 12 ounces = 1 pound, Ib. = 5760 grains. To determine whether a balance has unequal arms. After weighing an article and obtaining equilibrium, transpose the article and the weights. If the balance is true, it will remain in equilibrium ; if untrue, the pan suspended from the longer arm will descend. To weigh correctly on an incorrect balance. First, by substitution. Put the article to be weighed in one pan of the balance and 20 ARITHMETIC. counterpoise it by any convenient heavy articles placed on the other pan. Remove the article to be weighed and substitute for it standard weights until equipoise is again established. The amount of these weights is the weight of the article. Second, by transposition. Determine the apparent weight of the article as usual, then its apparent weight after transposing the article and the weights. If the difference is small, add half the difference to the smaller of the apparent weights to obtain the true weight. If the difference is 2 per cent the error of this method is 1 part in 10,000. For larger differences, or to obtain a perfectly accurate result, multiply the two apparent weights together and extract the square root of the product. Circular Measure* 60 seconds, " = 1 minute, '. 60 minutes, ' = 1 degree, . 90 degrees = 1 quadrant. 360 " = circumference. Time. 60 seconds = 1 minute. 60 minutes = 1 hour. 24 hours = 1 day. 7 days = 1 week. 365 days, 5 hours, 48 minutes, 48 seconds = 1 year. By the Gregorian Calendar every year whose number is divisible by 4 is a leap year, and contains 366 days, the other years containing 365 days, ex- cept that the centesimal years are leap years only when the number of the year is divisible by 400. The comparative values of mean solar and sidereal time are shown by the following relations according to Bessel : 365.24222 mean solar days = 366.24222 sidereal days, whence 1 mean solar day = 1.00273791 sidereal days; 1 sidereal day = 99726957 mean solar day; 24 hours mean solar time = 24 h 3 m 56 8 .555 sidereal time; 24 hours sidereal time = 23 h 56 m 4 8 .091 mean solar time, whence 1 mean solar day is 3 m 55 8 .91 longer than a sidereal day, reckoned in mean solar time. BOARD AND TIMBER MEASURE. Board Measure. In board measure boards are assumed to be one inch in thickness. To obtain the number of feet board measure (B. M.) of a board or stick of square timber, multiply together the length in feet, the breadth in feet, and the thickness in inches. To compute the measure or surface in square feet. When all dimensions are in feet, multiply the length by the breadth, and the pro- duct will give the surface required. When either of the dimensions are in inches, multiply as above and divide the product by 12. When all dimensions are in inches, multiply as before and divide product by 144. Timber Measure. To compute the vol ume of round timber. When all dimen- sions are in feet, multiply the length by one quarter of the product of the mean girth and diameter, and the product will give the measurement in cubic feet. When length is given in feet and girth and diameter in inches, divide the product by 144 ; when all the dimensions are in inches, divide by 1728. To compute the volume of square timber. When all dimen- sions are in feet, multiply together the length, breadth, and depth; the product will be the volume in cubic feet. When one dimension is given in inches, divide by 12; when two dimensions are in inches, divide by 144; when all three dimensions are iu inches, divide by 1728. WEIGHTS AKD MEASURES. Contents in Feet of Joists, Scantling, and Timber. Length in Feet. Size. 12 14 16 18 20 22 24 26 28 30 Feet Board Measure. 2X 4 8 9 11 12 13 15 16 17 19 20 2X6 12 14 16 18 20 22 24 26 28 30 2X8 16 19 21 24 27 29 32 35 37 40 2 X 10 20 23 27 30 33 37 40 43 47 50 2 X 12 24 28 32 36 40 44 48 52 56 60 2 X 14 28 33 37 42 47 51 56 61 65 70 3X8 24 28 32 36 40 44 48 52 56 60 3 X 10 30 35 40 45 50 55 60 65 70 75 3 X 12 36 42 48 54 60 66 72 78 84 90 3X14 42 49 56 63 70 77 84 91 98 105 4X4 16 19 21 24 27 29 32 35 37 40 4X6 24 28 32 36 40 44 48 52 56 60 4X8 32 37 43 48 53 59 64 69 75 80 4 X 10 40 47 53 60 67 73 80 87 93 100 4 X 12 48 56 64 72 80 88 96 104 112 120 4 X 14 56 65 75 84 93 103 112 121 131 140 6X6 36 42 48 54 60 66 72 78 84 90 6X8 48 56 64 72 80 88 96 104 112 120 6 X 10 60 70 80 90 100 110 120 130 140 150 6X 12 72 84 96 108 120 132 144 156 168 180 6X 14 84 98 112 126 140 154 168 182 196 210 8X* 8 64 75 85 96 107 117 128 139 149 160 8 X 10 80 93 107 120 133 147 160 173 187 200 8 X 12 96 112 128 144 160 176 192 208 224 240 8 X 14 112 131 149 168 187 205 224 243 261 280 10 X 10 100 117 133 150 167 183 200 217 233 250 10 X 12 120 140 160 180 200 2^0 240 260 280 300 10 X 14 140 J63 187 210 233 257 280 303 327 350 12 X 12 144 168 192 216 240 264 288 312 336 360 12 X 14 168 196 224 252 280 308 336 364 392 420 14 X 14 196 229 261 294 327 359 392 425 457 490 FRENCH OR METRIC MEASURES. The metric unit of length is the metre = 39.37 inches. The metric unit of weight is the gram = 15.432 grains. The following prefixes are used for subdivisions and multiples; Milli = T0 ^ Centi = T fo, Deci = j^, Deca = 10, Hecto = 100, Kilo = 1000, Myria = 10,000. FRENCH AND BRITISH (AND AMERICAN) EQUIVALENT MEASURES. Measures of Length. FRENCH. BRITISH and U. S. 1 metre = 39.37 inches, or 3.28083 feet, or 1.09361 yards. .8048 metre = 1 foot. 1 centimetre = .3937 inch. 54 centimetres = 1 inch. 1 milimetre = .03937 inch, or 1/25 inch, nearly. 25.4 millimetres = 1 inch. 1 kilometre = 1093.61 yards, or 0.62137 mile. 22 ARITHMETIC. Measures of Surface. FRENCH. BRITISH and U. S. isquaremetre = j '; .836 square metre = 1 square yard. .0929 square metre = 1 square foot. 1 square centimetre = .155 square inch. 6.452 square centimetres = 1 square inch. 1 square millimetre = .00155 sq. in. = 1973.5 circ. mils. 645.2 square millimetres = 1 square inch. 1 centiare = 1 sq. metre = 10.764 square feet. 1 are = 1 sq. decametre = 1076.41 " 4 * 1 hectare = 100 ares = 107641 " " = 2.4711 acres. 1 sq. kilometre = .386109 sq. miles = 247.11 " 1 sq. myriametre =38.6109" " Of Volume. FRENCH. BRITISH and U. S. 1 onbip metre - 4 35 ' 314 cubic feet - 1 1.308 cubic yards. .7645 cubic metre = 1 cubic yard. .02832 cubic metre = 1 cubic foot. 1 cubic decimetre = {;<, ^ic inches, 28.32 cubic decimetres = 1 cubic foot. 1 cubic centimetre = .061 cubic inch. 16.387 cubic centimetres = 1 cubic inch. 1 cubic centimetre = 1 millilitre = .061 cubic inch. 1 centilitre = = .610 " " 1 decilitre = = 6.102 " 1 litre = 1 cubic decimetre = 61.023 " " = 1.05671 quarts, U. S. 1 hectolitre or decistere = 3.5314 cubic feet = 2.8375 bushels, " 1 stere, kilolitre, or cubic metre = 1.308 cubic yards = 28.37 bushels, " Of Capacity. FRENCH. BRITISH and U. S. f 61. 023 cubic inches, ( 1 niihin rlppimpfr^ - J - 08531 Cubic foot, 1 .2642 gallon (American), [2.202 pounds of water at 62 F. 28.317 litres = 1 cubic foot. 4.543 litres = 1 gallon (British). 3.785 litres = 1 gallon (American). Of Weight. FRENCH. BRITISH and U. S. 1 gramme = 15.432 grains. .0648 gramme = 1 grain. 28.35 gramme = 1 ounce avoirdupois. 1 kilogramme = 2.2046 pounds. .4536 kilogramme = 1 pound. 1 tonne or metric ton = j - 1000 kilogrammes = 1 Mr. O. H. Titmann, in Bulletin No. 9 of the U. S. Coast and Geodetic Sur- vey, discusses the work of various authorities who have compared the yard and the metre, and by referring all the observations to a common standard has succeeded in reconciling the discrepancies within very narrow limits. The following are his results for the number of inches in a metre according to the eomparisc/zis of the authorities named: 1817. Hassler ......................... 39.36994 inches. 1818. Kater ........................... 39.36990 " 1835. Baily ........................... 39.36973 " 1866. Clarke .......................... 39.36970 " 1885. Comstock ...................... 39.36984 " The mean of these is ......... ... 39.36982 " METKIC WEIGHTS AND MEASURES. 23 METRIC CONVERSION TABLES. The following tables, with the subjoined memoranda, were published in 1890 by the United States Coast and Geodetic Survey, office of standard weights and measures, T. C. Mendenhall, Superintendent. Tables for Converting U. S. Weights and Pleasures Customary to Metric. LINEAR. Inches to Milli- metres. Feet to Metres. Yards to Metres. Miles to Kilo- metres. J _ 25.4001 0.304801 0.914402 1.60935 2 = 50.8001 0.609601 1.828804 3.21869 O 76.2002 0.914402 2.743205 4.82804 4 = 101.6002 1.219202 3.657607 6.43739 5 = 127.0003 1.524003 4.572009 8.04674 6 = 152.4003 1.828804 5.486411 9.65608 7 = 177.8004 2.133604 6.400813 11.26543 8 = 203.2004 2.438405 7.315215 12.87478 9 = 228.6005 2.743205 8.229616 14.48412 SQUARE. Square Inches to Square Centi- metres. Square Feet to Square Deci- metres. Square Yards to Square Metres. Acres to Hectares. 1 6.452 9 290 0.836 0.4047 2 = 12.903 1&581 1.672 0.8094 3 = 19.355 27.871 2.508 1.2141 4 = 25.807 37.161 3.344 1.6187 5 = 32.258 46.452 4.181 2.0234 6 = 38.710 55.742 5.017 2.4281 7 = 45.161 65.032 5.853 2.8328 8 = 51.613 74.323 6.689 3 2375 9 = 58.065 83.613 7.525 3.6422 CUBIC. Cubic Inches to Cubic Centi- metres. Cubic Feet to Cubic Metres. Cubic Yards to Cubic Metres. Bushels to Hectolitres. 1 = 16.387 0.02832 0.765 0.35242 2 = 32.774 0.05663 1.529 0.70485 o 49.161 0.08495 2.294 1.05727 4 = 65.549 0.11327 3.058 1.40969 5 = 81.936 0.14158 3.823 1.76211 6 = 98.323 0.16990 4.587 2.11454 7 = 114.710 0.19822 5.352 2.46696 8 = 131 097 0.22654 6.116 2.81938 9 = 147.484 0.25485 6.881 3.17181 ARITHMETIC. CAPACITY. Fluid Drachms to Millilitres or Fluid Ounces to Quarts to Litres. Gallons to Litres. Cubic Centi- Millilitres. metres. 1 = 3.70 29.57 0.94636 3.78544 2 = 7.39 59.15 1.89272 7.57088 3 = 11.09 88.72 2.83908 11.35632 4 = 14.79 118.30 3.78544 15.14176 5 = 18.48 147.87 4.73180 18.92720 6 = 22.18 177.44 5.67816 22.71264 7 = 25.88 207.02 6.62452 26.49808 8 = 29.57 236.59 7.57088 30.28352 9 = 33.38 266.16 8.51724 34.06896 WEIGHT. Grains to Milli- grammes. Avoirdupois Ounces to Grammes. Avoirdupois Pounds to Kilo- grammes. Troy Ounces to Grammes. 1 = 64.7989 28.3495 0.45359 31.10348 2 = 129.5978 56.6991 0.90719 62.20696 3 = 194.3968 85.0486 1.36078 93.31044 4 = 259.1957 113.3981 1.81*37 124.41392 5 = 323.9946 141.7476 2.26796 155.51740 6 = 388.7935 170.0972 2.72156 186.62089 7 = 453.5924 198.4467 3.17515 217.72437 518.3914 226.7962 3.62874 248.82785 9 = 583.1903 255.1457 4.08233 279.93133 1 chain = 20.1169 metres. 1 square mile 259 hectares. 1 fathom = 1.829 metres. 1 nautical mile = 1853.27 metres. 1 foot = 0.304801 metre. 1 avoir, pound = 453.5924277 gram. 15432.35639 grains = 1 kilogramme. Tables for Converting 17. S. Weights and Measures Metric to Customary. LINEAR. 1 = 2 = 3 = 4 = 5 = Metres to Inches. Metres to Feet. Metres to Yards. Kilometres to Miles. 39.3700 78.7400 118.1100 157.4800 196.8500 3.28083 6.56167 9.84250 13.12333 16.40417 1.093611 2.187222 3.280833 4.374444 5.468056 0.62137 1.24274 1.86411 2.48548 3.10685 II II II II 0i>ooca 1 236.2200 275.5900 314.9600 354.3300 19.68500 22.96583 26.24667 29.52750 6.561667 7.655278 8.748889 9.842500 3.72822 4.34959 4.97096 5.59233 METRIC CONVERSION TABLES. SQUARE. Square Centi- metres to Square Inches. Square Metres to Square Feet. Square Metres to Square Yards. Hectares to Acres. 1 = 0.1550 10.764 1.196 2.471 2 0.3100 21.528 2.392 4.942 3 = 0.4650 32.292 3.588 7.413 4 = 0.6200 43.055 4.784 9.884 5 = 0.7750 53.819 5.980 12.355 6 = 0.9300 64.583 7.176 14.826 7 = 1.0850 75.347 8.372 17.297 8 = 1.2400 86.111 9.568 19.768 9 = 1.3950 96.874 10.764 22.239 CUBIC. Cubic Centi- metres to Cubic Inches. Cubic Deci- metres to Cubic Inches. Cubic Metres to Cubic Feet. Cubic Metres to Cubic Yards. 1== 0.0610 61.023 35.314 1.308 2 = 0.12^0 122.047 70.629 2.616 3 = 0.1831 183.070 105.943 3.924 4 = 0.2441 244.093 141.258 5.232 5 = 0.3051 305.117 176.572 6.540 6 = 0.3661 366.140 211.887 7.848 7 = . 0.4272 427.163 247.201 9.156 8 = 0.4882 488.187 282.516 10.464 9 = 0.5492 549.210 317.830 11.771 CAPACITY. Millilitres or Cubic Centi- litres to Fluid Drachms. Centilitres to Fluid Ounces. Litres to Quarts. Dekalitres to Gallons. Hektolitres to Bushels. 1 = 0.27 0.338 1.0567 2.6417 2.8375 2 0.54 0.676 2.1134 5.2834 5.6750 3 = 0.81 1.014 3.1700 7.9251 8.5125 4 = 1.08 1.352 4.2267 10.5668 11.3500 5 = 1.35 1.691 5.2834 13.2085 14.1875 6 = 1.62 2.029 6.3401 15.8502 17.0250 7 1.89 2.368 7.3968 18.4919 i 19.8625 8 = *.16 2.706 8.4534 21.1336 ! 22.7000 9 = 2.43 3.043 9.5101 23.7753 25.5375 26 ARITHMETIC. WEIGHT. 1 = 2 = 3 = 4 = 5 = Milligrammes to Grains. Kilogrammes to Grains. Hectogrammes (100 grammes) to Ounces Av. Kilogrammes to Pounds Avoirdupois. 0.01543 0.03086 0.04630 0.06173 0.07716 15432.36 30864.71 46297.07 61729.43 77161.78 3.5274 7.0548 10.5822 14.1096 17.6370 2.20462 4.40924 6.61386 8.81849 11.02311 6 = 8 = 9 = 0.09259 0.10803 0.12346 0.13889 92594.14 108026.49 123458.85 138891 .21 21.1644 24.6918 28.2192 31.7466 13.22773 15.43235 17.63697 19.84159 WEIGHT (Continued). Quintals to Pounds Av. Milliers or Tonnes to Pounds Av. Grammes to Ounces, Troy. W^CdO*-* II II II II II 220.46 440.92 661.38 881.84 1102.30 2204 6 4409.2 6613.8 8818.4 11023.0 0.03215 0.06430 0.09645 0.12860 0.16075 6 = 7 = 8 = 9 = 1322.76 1543.22 1763.68 1984.14 13227.6 15432.2 17636.8 19841.4 0.19290 0.22505 0.25721 0.28936 i The only authorized material standard of customary length is the Troughton scale belonging to this office, whose length at 59.62 Fahr. con- forms to the British standard. The yard in use in the United States is there- fore equal to the British yard. The only authorized material standard of customary weight is the Troy pound of the mint. It is of brass of unknown density, and therefore not suitable for a standard of mass. It was derived from the British standard Troy pound of 1758 by direct comparison. The British Avoirdupois pound was also derived from the latter, and contains 7000 grains Troy. The grain Troy is therefore the same as the grain Avoirdupois, and the pound Avoirdupois in use in the United States is equal to the British pound Avoirdupois. The metric system was legalized in the United States in 1866. By the concurrent action of the principal governments of the world an International Bureau of Weights and Measures has been established near The International Standard Metre is derived from the Metre des Archives, and its length is defined by the distance between two lines at Centigrade, on a platinum-iridium bar deposited at the International Bureau. The International Standard Kilogramme is a mass of platinum-iridium deposited at the same place, and its weight in vacua is the same as that of the Kilogramme des Archives. Copies of these international standards are deposited in the office of standard weights and measures of the U. S. Coast and Geodetic Survey. The litre is equal to a cubic decimetre of water, and it is measured by the quantity of distilled water which, at its maximum density, will counterpoise the standard kilogramme in a vacuum; the volume of such a quantity of water being, as nearly as has been ascertained, equal to a cubic decimetre. WEIGHTS AND MEASURES COMPOUND UNITS. 27 COMPOUND UNITS. Measures of Pressure and Weight. f 144 Ibs. per square foot. 2.0355 ins. of mercury at 32* F. 1 Ib. per square inch. = i 2.0416 " " " " 62 F. 2.309 ft. of water at 62 F. [ 27.71 ins. " " " 62 F. 1 ounce per sq. In. = { f 2116.3 Ibs. per square foot. I 33.947 ft. of water at 62 F. 1 atmosphere (14.7 Ibs. per sq. in.) = -j 30 ins. of mercury at 62 F. - 29.922 ins. of mercury at 32 F. 1.76 . . . .760 millimetres of mercury at 32 F. .03609 Ib. or .5774 oz. per sq. in. 1 inch of water at 62 F. =< 5.196 Ibs. per square foot. .0736 in. of mercury at 62 F. 1 < ( j 1 foot of water at F. = \ 1 inch of water at 38 F. = 5 ;gg}^.I? r ( .491 Ib. or 7.86 oz. per sq. in. 1 inch of mercury at 62 F. t=4 1.132ft. of water at 62 F. f 13. 58 ins. " " * 62 F. Weight of One Cubic Foot of Pure Water. At 32 F. (freezing-point) ............................ 62.418 Ibs. " 39.1 F. (maximum density) .......... . ............ 62.425 " " 62 F. (standard temperature) ....... ............... 62.355 " u 212 F. (boiling-point, under 1 atmosphere) ........ 59.76 * American gallon = 231 cubic ins. of water at 62 F. = 8.3356 Ibs. British = 277.274 " " " " " " =10 Ibs. Measures of Work, Power, and Duty. Work. /The sustained exertion of pressure through space. Unit of work. One foot-pound, i.e., a pressure of one pound exerted through a space of one foot. Horse-power. The rate of work. Unit of horse- power = 33,000 ft.- Ibs. per minute, or 550 ft. -Ibs. per second = 1,980,000 ft. -Ibs. per hour. Heat unit = heat required to raise 1 Ib. of water 1 F. (from 39 to 40). 33000 Horse-power expressed in heat units = ~^g~ = 42.416 heat units per min- ute = .707 heat unit per second = 2545 heat units per hour. 1 Ib. of f ue! per H. P. per hour= 1,000,000 ft.-lbs. per Ib. of fuel = 1.98 Ibs. of fuel per H. P. per hour. 5280 22 Velocity. Feet per second = ^^ = 15 x miles per hour. Gross tons per mile = ^ = Ibs. per yard (single rail.) French and British Equivalents of Compound Units. FRENCH. BRITISH. ramme per square millimetre = 1.422 Jbs. per square inch. *' 1 g 1 ki . ilogramme per square *' = 1422.32 1 " J* centimetre = 14.223 " * " j 1.0335 kg. per sq. cm. 1 atmosphere = 14.7 0.070308 kilogramme per square centimetre = 1 Ib. per square inch. 1 gramme per litre = 0.062428 Ib. per cubic foot. 1 kilogrammetre = 7.2330 foot-pounds. 28 ARITHMETIC. WIRE AND SHEET-METAL, GAUGES COMPARED. Number of Gauge. Birmingham (or Stubs' Iron) Wire Gauge. American or Brown and Sharpe Gauge. Roebling's and Washburn & Moen's Gauge. 2| 4> <M fit! QQ &0 <S CQ British Imperial Standard Wire Gauge. (Legal Standard in Great Britain since March 1, 1884.) U. S. Standard Gauge for Sheet and Plate Iron and Steel. (Legal Standard since July 1, 1893.) Number of Gauge. inch. inch. inch. inch. inch. millim. inch. 0000000 .49 .500 12.7 .5 7/0 000000 .46 .464 11.78 .469 6/0 00000 .43 .432 10.97 .438 5/0 0000 .454 .46 .393 .4 10.16 .406 4/0 000 .425 .40964 .362 .372 9.45 .375 3/0 00 .38 .3648 .331 .348 8.84 .344 2/0 .34 .32486 .307 .324 8.23 .313 1 .3 .2893 .283 .227 .3 7.62 .281 1 2 .284 .25763 .263 .219 .276 7.01 .266 2 3 .259 .22942 .244 .212 .252 6.4 .25 3 4 .238 .20431 .225 .207 .232 5.89 .234 4 5 .22 .18194 .207 .204 .212 5.38 .219 5 6 .203 .16202 .192 .201 .192 4.88 .203 6 7 .18 .14428 .177 .199 .176 4.47 .188 7 8 .165 .12849 .162 .197 .16 4.06 .172 8 9 .148 .11443 .148 .194 .144 3.66 .156 9 10 .134 .10189 .135 .191 .128 3.25 .141 10 11 .12 .09074 .12 .188 .116 2.95 .125 11 12 .109 .08081 .105 .185 .104 2.64 .109 12 13 .095 .07196 .092 .182 .092 2.34 .094 13 14 .083 .06408 .08 .180 .08 2.03 .078 14 15 .072 .05707 .072 .178 .072 1.83 .07 15 16 .065 .05082 .063 .175 .064 1.63 .0625 16 17 .058 .04526 .054 .172 .056 1.42 .0563 17 18 .049 .0403 .047 .168 .048 1 . 22 .05 18 19 .042 .03589 .041 .164 .04 1,02 .0438 19 20 .035 .03196 .035 .161 .036 .91 .0375 20 21 .032 .02846 .032 .157 .032 .81 .0344 21 22 .028 .02535 .028 .155 .028 .71 .0313 22 23 .025 .02257 .025 .153 .024 .61 .0281 23 24 .022 .0201 .023 .151 .022 .56 .025 24 25 .02 .0179 .02 .148 .02 .51 .0219 25 26 .018 .01594 .018 .146 .018 .46 .0188 26 27 .016 .01419 .017 .143 .0164 .42 .0172 27 28 .014 .01264 .016 .139 .0148 .38 .0156 28 29 .013 .01126 .015 .134 .0136 .35 .0141 29 30 .012 .01002 .014 .127 .0124 .31 .0125 30 31 .01 .00893 .0135 .120 .0116 .29 .0109 31 32 .009 .00795 .013 .115 .0108 .27 .0101 32 33 .008 .00708 .011 .112 .01 .25 .0094 33 34 .007 .0063 .01 .110 .0092 .23 .0086 34 35 .005 .00561 .0095 .108 .0084 .21 .0078 35 36 004 .005 .009 .106 .0076 .19 .007 36 37 .00445 .0085 .103 .0068 .17 .0066 37 38 .00390 .008 .101 .006 .15 .0063 38 39 .00353 .0075 .099 .0052 .13 39 40 .00314 .007 .097 .0048 .12 40 41 .095 .0044 .11 41 42 .092 .004 .10 42 43 .088 .0036 .09 43 44 .085 .0032 .08 44 45 .081 .0028 .07 45 46 .079 .0024 .06 46 47 .077 .002 .05 47 48 .075 .0016 .04 48 49 .072 .0012 .03 49 50 .069 .001 .025 50 WIRE GAUGE TABLES. EDISON, OR CIRCULAR ftllL GAUGE, FOR ELEC- TRICAL WIRES. Gauge Num- ber. Circular Mils. Diam- eter in Mils. Gauge Num- ber. Circular Mils. Diam- eter in Mils. Gauge Num- ber. Circular Mils. Diam- eter in Mils. 3 3,000 54.78 70 70,000 264.58 190 190,000 435.89 5 5,000 70.72 75 75,000 273.87 200 200.000 447.22 8 8,000 89.45 80 80,000 282.85 220 220,000 469.05 12 12,000 109.55 85 85,000 291.55 240 240,000 489.90 15 15,000 122.48 90 90,000 300.00 260 260,000 509.91 20 20,000 141.43 95 95,000 308.23 280 280,000 529.16 25 25,000 158.1? 100 100,000 316.23 300 300,000 547.73 30 30,000 173.21 110 110,000 331.67 320 320,000 565.69 35 35,000 187.09 120 120,000 346.42 340 340,000 583.10 40 40,000 200.00 130 130,000 360.56 360 360,000 600.00 45 45,000 212.14 140 140,000 374.17 50 50,000 223.61 150 150,000 387.30 55 55,000 234.53 160 160,000 400.00 60 60,000 244.95 17'0 170,000 412.32 65 65,000 254.96 180- 180,000 424.27 TWIST DRILL AND STEEL WIRE GAUGE. (Morse Twist Drill aud Machine Co.) No. Size. No. Size. No. Size. No. Size. No. Size. No. Size. inch. inch inch. inch. inch. inch. 1 .2280 11 .1910 21 .1590 31 .1200 41 .0960 51 .0670 y .2210 12 .1890 22 .1570 32 .1160 42 .0935 52 .0635 3 .2130 13 .1850 23 .1540 33 .1130 43 .0890 53 .0595 4 .2090 14 .1620 24 .1520 34 .1110 44 .0860 54 .0550 5 .2055 15 .1800 25 .1495 35 .1100 45 .0820 55 .0520 6 .2040 16 .1770 26 .1470 36 .1065 46 .0810 56 .0465 7 .2010 17 .1730 27 .1440 37 .1040 47 .0785 57 .0430 8 .199J 18 .1695 28 .1405 38 .1015 48 .0760 58 .0420 9 .1960 19 .1660 29 .1360 39 .0995 49 .0730 59 .0410 10 .1935 20 .1610 30 .1285 40 .0980 50 .0700 60 .0400 STUBS' STEEL WIRE GAUGE. (For Nos. 1 to 50 see table on page 28.) No. Size. No. Size. No. Size. No. Size. No. Size. No. Size. inch. inch. inch. inch. inch. inch. Z .413 P .323 F .257 51 .066 61 .038 71 .026 Y .404 O .316 E .250 52 .063 62 .037 72 .024 X .397 N .302 D .246 53 .058 63 .036 73 .023 W .386 M 295 C .242 54 .055 64 .035 74 .022 V .377 L !&o B .238 55 .050 65 .033 75 .020 U .368 K .281 A .234 56 .045 66 .032 76 .018 T .358 J .277 1 (See 57 .042 67 .031 77 .016 S .348 I .272 to 1 page 58 .041 68 .030 78 .015 K .339 H .266 50 ( 28 59 .040 69 .029 79 *.014 Q .332 G .261 60 .039 70 .027 80 .013 The Stubs' Steel Wire Gauge is used in measuring drawn steel wire or drill rods of Stubs' make, and is also used by many makers of American drill rods, 30 AKITHMETIC. THE: EDISON OR CIRCULAR MIL, WIRE GAUGE. (For table of copper wires by this'gauge, giving weights, electrical resist ances, etc., see Copper Wire.) Mr. C. J. Field (Stevens Indicator, July, 1887) thus describes the origin of the Edison gauge: The Edison company experienced inconvenience and loss by not having a wide ^nough range nor sufficient number of sizes in the existing gauges. This was felt more particularly in the central-station work in making electrical determinations for the street system. They were compelled to make use of two of the existing gauges at least, thereby introducing a complication that was liable to lead to mistakes by the contractors and linemen. In the incandescent system an even distribution throughout the entire system and a uniform pressure at the point of delivery are obtained by cal- culating for a given maximum percentage of loss from the potential as delivered from the dynamo. In carrying this out, on account of lack of regular sizes, it was often necessary to use larger sizes than the occasion demanded, and even to assume new sizes for large underground conductors. It was also found that nearly all manufacturers based their calculation for the conductivity of their wire on a variety of units, and that not one used the latest unit as adopted by the British Association and determined from Dr. Matthiesseif s experiments ; and as this was the unit employed in the manufacture of the Edison lamps, there was a further reason for construct- ing a new gauge. The engineering department of the Edison company, knowing the requirements, have designed a gauge that has the widest range obtainable and a large number of sizes which increase in a regular and uniform manner. The basis of the graduation is the sectional area, and the number of the wire corresponds. A wire of 100,000 circular mils area ts No. 100 ; a wire of one half the size will be No. 50 ; twice the size No. .200. In the older gauges, as the number increased the size decreased. With this gauge, however, the number increases with the wire, and the number multiplied by 1000 will ^ive the circular mils. The w r eight per mil-foot, 0.00000302705 pounds, agrees with a specific gravity of 8.889, which is the latest figure given for copper. The ampere capacity which is given was deduced from experiments made in the com- pany's laboratory, and is based on a rise of temperature of 50 F. in the wire. In 1893 Mr. Field writes, concerning gauges in use by electrical engineers: The B. and S. gauge seems to be in general use for the smaller sizes, up to 100,000 c. m., and in some cases a little larger. From between one and two hundred thousand circular mils upwards, the Edison gauge or its equivalent is practically in use, and there is a general tendency to designate all sizes above this in circular mils, specifying a wire as 200,000, 400,000, 500,- 000, or 1,000,000 c. m. In the electrical business there is a large use of copper wire and rod and other materials of these large sizes, and in ordering them, speaking of them, specifying, and in every other use, the general method is to simply specify the circular milage. I think it is going to be the only system in the future for the designation of wires, and the attaining of it means practically the adoption of the Edison gauge or the method and basis of this gauge as the correct one for wire sizes. THE U. S. STANAR GAUGE FOR SHEET AND PL. ATE IRON AN STEEL., 1893. There is in this country no uniform or standard gauge, and the same numbers in different gauges represent different thicknesses of sheets or plates. This has given rise to niHch misunderstanding and friction between employers and workmen and mistakes and fraud between dealers and con- sumers. An Act of Congress in 1893 established the Standard Gauge for sheet iron and^teel which is given on the next page. It is based on the fact that a cubic foot of iron weighs 480 pounds. A sheet of iron 1 foot square and 1 inch thick weighs 40 pounds, or 640 ounces, and 1 ounce in weight should be 1/640 inch thick. The scale has been arranged so that each descriptive number represents a certain number of ounces in weight and an equal number of 640ths of an inch in thickness. The law enacts that on and after July 1, 1893, the new gauge shall be used in determining duties and taxes levied on sheet and plate iron and steel; and that in its application a variation of 2^4 per cent either way may be allowed. GAUGE FOE SHEET AND PLATE IKON AND STEEL. 31 U. S. STANDARD GAUGE FOR SHEET AND PliATE IRON AND STEEL., 1893. fliM .2 a S SH- 5 "o 03 Si "ft 2 - fe s Number oi Gauge. 'i fl-S fl 5 73 r2 3 lf.N eusia ^SQ 03 %$ * 28 | fls-1 g-g-M gl 1 <3 ^ -111 ||I{ sill. iili ft*! }l| III S3* i ^.s '~S !'- 0000000 1-2 0.5 12.7 320 20. 9.072 97.65 215.28 000000 15-32 0.46875 11.90625 300 18.75 8.505 91.55 201.82 00000 7-16 0.4375 11.1125 280 17.50 7.938 85.44 188.37 0000 13-32 0.40625 10.31875 260 16.25 7.371 79.33 174.91 000 3-8 0.375 9.525 240 15. 6.804 73.24 161.46 00. 11-32 0.31375 8.73125 220 13.75 6.237 67.13 148.00 5-16 0.3125 7.9375 200 12.50 5.67 61.03 134.55 1 9-32 0.28125 7.14375 180 11.25 5.103 54.93 121.09 2 17-64 0.265625 6.746875 170 10.625 4.819 51.88 114.37 3 1-4 0.25 6.35 160 10. 4.536 48.82 107.64 4 15-64 0.234375 5.953125 150 9.375 4.252 45.77 100.91 5 7-32 0.21875 5.55625 140 8.75 3.969 42.72 94.18 6 13-64 0.203125 5.159375 130 8.125 3.685 39.67 87.45 7 3-16 0.1875 4.7625 120 7.5 3.402 36.62 80.72 8 11-64 0.171875 4.365625 110 6.875 3.118 33.57 74.00 9 5-32 0.15625 3.9S875 100 6.25 2.835 30.52 67.27 10 9-64 0.140625 3.571875 90 5.625 2.552 27.46 60.55 11 1-8 0.125 3.175 80 5. 2.268 24.41 53.82 1<2 7-64 0.109375 2.778125 70 4.375 1.984 21.36 47.09 13 3-32 0.09375 2.38125 60 3.75 1.701 18.31 40.36 14 5-64 0.078125 1.984375 50 3.125 1.417 15.26 33.64 15 9-128 0.0703125 1.7859375 45 2.8125 1.276 13.73 30.27 16 1-16 0.0625 1.5875 40 2.5 1.134 12.21 26.91 17 9-160 0.05625 1 .42875 36 2.25 1.021 10.99 24.22 18 1-20 0.05 1.27 32 2. 0.9072 9.765 21.53 19 7-160 0.04375 1.11125 28 1.75 0.7938 8.544 18.84 20 3-80 0.0375 0.9525 24 1.50 0.6804 7.324 16.15 21 11-320 0.034375 0.873125 22 1.375 0.6237 6.713 14.80 22 1-32 0.03125 0.793750 20 1.25 0.567 6.103 13 46 23 9-320 0.028125 0.714375 18 1.125 0.5103 5.493 12.11 24 1-40 0.025 0.635 16 1. 0.4536 4.882 10.76 25 7-320 0.021875 0.555625 14 0.875 0.3969 4.272 9.42 26 3-160 0.01875 0.47625 12 0.75 0.3402 3.662 8.07 27 11-640 0.0171875 0.4365625 11 0.6875 0.3119 3.357 7.40 28 1-64 0.015625 0.396875 10 0.625 0.2835 3.052 6.73 29 9-640 0.0140625 0.3571875 9 0.5625 0.2551 2.746 6.05 30 1-80 0.0125 0.3175 8 0.5 0.2268 2.441 5.38 81 7-640 0.0109375 0.2778125 7 0.4375 0.1984 2.136 4.71 32 13-1280 0.01015625 0.25796875 gi^ 0.40625 0.1843 1.9R3 4.37 33 3-320 0.009375 0.238125 6 0.375 0.1701 1.831 4.04 34 11-1280 0.00859375 0.21828125 5^ 0.34375 0.1559 1.678 3 70 35 5-640 0.0078125 0.1984375 5 0.3125 0.1417 1.526 3.36 36 9-1280 0.00703125 0.17859375 41^ 0.28125 0.1276 1.373 3.03 37 17-2560 0.006640625 0.168671875 4/4 0.265625 0.1205 1.297 2.87 38 1-160 0.00625 0.15875 4 0.25 0.1134 1.221 2.69 MATHEMATICS. Tlie Decimal Gauge. The legalization of the standard sheet-metaj gauge of 1893 and its adoption by some manufacturers of sheet iron have only added to the existing confusion of gauges. A joint committee of the American Society of Mechanical Engineers and the American Railway Master Mechanics' Association in 1895 agreed to recommend the use of the decimal gauge, that is, a gauge whose number for each thickness 3s the number of thousandths of an inch in that thickness, and also to recommend *'the abandonment and disuse of the various other gauges now in use, as tending to confusion and error. 1 " A notched gauge of oval form, shown in the cut below, has come into use as a standard form of the decimal gauge. In 1904 The Westinghouse Electric & Mfg. Co. abandoned the use of gaug numbers in referring to wire, sheet metal, etc. Weight of Sheet Iron and Steel. Thickness by Decimal Oauge uge 00 c 1 Weight per Square Foot 03 E to I Weight per Square Foot o 1 in Pounds. . o s in Pounds. fi be 2 !. a 1 JL | DB-p * ' I | 5 < So> 3 O *! gfe ^ 1 |a 2 |o |;> "3 gg H 2 M ^ ft ft c ^ 1 & o & ft i3 O 02 ft o i/ (-1 Q 3 02 H < 5 1 02 0.002 1/500 0.05 0.08 0.082 0.060 1/16 - 1.52 2.40 2.448 0.004 1/250 0.10 0.16 0.163 0.065 13/200 1.65 2.60 2.652 0.006 3/500 0.15 0.24 0.245 0.070 7/100 1.78 2.80 2.856 0.008 1/125 0.20 0.32 0.326 0.075 8/40 1.90 3.00 3.060 0.010 1/100 0.25 0.40 0.408 0.080 2/25 2.03 3.20 3.264 0.012 3/250 0.30 0.48 0.490 0.085 17/200 2.16 8.40 3.468 0.014 7/500 C.36 0.56 0.571 0.090 9/100 2.28 3.60 3.672 0.016 1/64 -f 41 0.64 0.653 0.095 19/200 2.41 3.80 3.876 0.018 9/500 0.46 0.72 0.734 0.100 1/10 2.54 4.00 4.080 0.020 1/50 0.51 0.80 0.816 0.110 11/100 2.79 4.40 4.488 0.022 11/500 0.56 0.88 0.898 0.125 1/8 3.18 5.00 5.100 0.025 1/40 0.64 1.00 1.020 0.135 27/200 3.43 5.40 5.508 0.028 7/250 0.71 1.12 1.142 0.150 3/20 3.81 6.00 6.120 0.032 1/32 + 0.81 1.28 1.306 0.165 33/200 4.19 6.60 6.732 0.036 9/250 0.91 1.44 1.469 0.180 9/50 4.57 7.20 7.344 0.040 1/25 1.02 1.60 1.632 0.200 1/5 5.08 8.00 8.160 0.045 9/200 1.14 1.80 1.836 0.220 11/50 5.59 8.80 8.976 0.050 1/20 1.27 2 00 2.040 0.240 8/25 6.10 9 60 9.792 0.055 11/200 1.40 2.20 2.244 0.250 1/4 6.35 10.00 10.200 ALGEBRA. 33 ALGEBRA. Addition. Add a and ft. Ans. a-\-b. Add a, 6, and -c. Ans. a-\-b c. Ad'l 2a and 3. Ans. a. Add 2ab t Sab, c, 3c. Ans. - ab 4c. Subtraction. Subtract a from 6. Ans. b a. Subtract a from 6. Aus. b -f a. Subtract b -f c from a. Ans. a b c. Subtract 3a 2 6 9c from 4a 2 6 -f- c. Ans. 2 6 4- lOc. RULE: Change the signs of the subtrahend and proceed as in addition. Multiplication. Multiply a by 6. Ans. ab. Multiply ab bya-f b. Ans. a 2 6 + a6 2 . Multiply a -f 6 by a -\-b. Ans. (a-f 6)(a + 6) = a 2 -f 2a6 + 6 2 . Multiply a by b. Ans. a6. Multiply a by 6. Ans. ab. Like signs give plus, unlike signs minus. Powers of numbers. The product of two or more powers of any number is the number \\ith an exponent equal to the sum of the powers: a 2 x a 3 = o 5 ; a* IP x ab = a 3 6 3 ; - 7ab x 2ac = - 14 a 2 6c. To multiply a polynomial by a monomial, multiply each term of the poly- nomial by the monomial and add the partial products: (6a 36) x 3c = 18ac - 96c. To multiply two polynomials, multiply each term of one factor by each term of the other and add the partial products: (5a 66) x (3a 46) = 15a 2 - 38a6 + 246 2 . The square of the sum of two numbers sum of their squares -f twice their product. \ The square of the difference of two numbers the sum of their squares twice their product. The product of the sum and difference of two numbers = the difference of their squares: (a-f6) 2 = a 2 +2a6-f 6 2 ; (a - 6) 2 =a 2 - 2a6-f 6 2 ; (a + 6) x (a- 6) = a 2 -6 2 . The square of half the sums of two quantities is equal to their product pi us the square of half their difference: ( a "^ J = ab -f- ( a ~ J The square of the sum of two quantities is equal to four times their prod- ucts, plus the square of their difference: (a -f 6; 2 = 4a6 + (a 6) 2 The sum of the squares of two quantities equals twice their product, plus the square of their difference: a 2 -j- 6 2 = 2ab -{- (a 6) 2 . The square of a trinomial = the square of each term -f twice the product of each term by each of the terms that follow it: (a +6 + c) 2 = a 2 -f-6" -f- c 2 -4- 2ab -f 2ac +26f; (a - 6 - c) 2 = 2 + 6 2 + c 2 - 2ab - Sac -f 2bc. The square of (any number -f ^) = square of the number -4- the number -f- 14; =*the number X (the number -f 1) -f- J4; (a-f-U) = a*-l-a- r .J4, - a (a+l) + %. (4^) 2 =4 2 -4- 4-4^^= Tiie product of any number -f y> by any other number + *4 = product of the numbers + half their sum -f J4- ( a + ^) X 6 + ^) = a6 -f ^(a-f- 6)-f- J4. 4> X 6^ = 4 X G -f- 1^(4 + 6) + H = 24 + 5 + H = 29^. Square, cube, 4tli poiver, etc., of a binomial a 4 6. (a + 6) 2 = a 2 -f- 26 -f 6 2 ; (a + 6) 3 = a* + 3a 2 6 -f 3a6 2 + 6 3 ; 4a6 3 + 6 4 . In each case the number of terms is one greater than the exponent of the power to which the binomial is raised. 2. In the first term the exponent of a is the same as the exponent of the power to which the binomial is raised, and it decreases by 1 in each succeed- ing term. 3. 6 appears in the second term with the exponent 1, and its exponent increases by 1 in each succeeding term. 4. The coefficient of the first term is 1. 5. The coefficient of the second term is the exponent of the power to which the binomial is raised. 6. The coefficient of each succeeding term is found from the next pre- ceding term }yy multiplying its coefficient by the exponent of a, and divid- ing the product by a number greater by 1 than the exponent of 6. (See Binomial Theorem, below.) 34 ALGEBRA." Parentheses* When a parenthesis is preceded by a plus sign it may be removed wuhout changing the value of the expression: a -f b -f (a -f ft) 2a -f- 2b. When a parenthesis is preceded by a minus sign it may be removed if we change the signs of all ihe terms within the parenthesis: 1 (a b c) \ a -{- ft + c. When a parenthesis is within a parenthesis remove the inner one first: a |ft ( c (d e) i a I ft \c d -f e ! | = a [6 c -f- "d e] = a b -\- c d -f- e. A multiplication sign, X, has the effect of a parenthesis, in that the oper- ation indicated by it must be performed before the operations of addition or subtraction, a -f- b X a + b a -f- ab -f- ft; while (a -f- b) X (a + b) = a 2 -f 2ab -f ft 2 , and (a -f b) X a + b = a 2 -f- aft + ft. Division. The quotient is positive when the dividend and divisor have like signs, and negative when they have unlike signs: abc -*- b = ac; abc -. b = ac. To divide a monomial by a monomial, write the dividend over the divisor with a line between them. If the expressions have common factors, remove the common factors: a 2 bx ax a 4 a 3 a?bx-r-aby = - = ; = = CK aby y a 3 a 5 -- To divide a polynomial by a monomial, divide each term of the polynomial by the monomial: (Sab 12ac) -* 4a = 2b 3c. To divide a polynomial by a polynomial, arrange both dividend and divi- sor in the order of the ascending or descending powers of some common letter, and keep this arrangement throughout the operation. Divide the first term of the dividend by the first term of the divisor, and write the result as the first term of the quotient. Multiply all the terms of the divisor by the first term of the quotient and subtract the product from the dividend. If there be a remainder, consider it as a new dividend and proceed as before: (a 2 ft 2 ) -*- (a + b). a 2 - ft 2 | a + b. a* -\-fib | a"- b. -ab- ft 2 . - ab- ft 2 . The difference of two equal odd powers of any two numbers is divisible by their difference and also by their sum: (a 3 - ft 3 ) -t- (a - b) = a 2 -f ab + ft 2 ; (a 3 - ft 3 ) -*- (a -f- ft) = a 2 - ab + ft 2 . The difference of two equal even powers of two numbers is divisible by their difference and also by their ^um: (a 2 ft 2 ) -*- (a ft) = a + b. The sum of two equal even powers of two numbers is not divisible by either the difference or the sum of the numbers; but when the exponent of each of the two equal powers is composed of an odd and an even factor, the sum of the given power is divisible by the sum of the powers expressed by the even factor. Thus x* -f- y* is not divisible by x -f y or by x y^ but is divisible by x* + ?/ 2 . Simple equations. An equation is a statement of equality between two expressions; as, a -f- ft = c -f- d. A simple equation, or equation of the first degree, is one which contains only the first power of the unknown quantity. If equal changes be made (by addition, subtraction, multiplication, or division) in both sides of an equation, the results will be equal. Any term may be changed from one side of an equation to another, pro- vided its sign be changed: a -\- b = c -}- d; a = c -\- d ft. To solve an equation having one unknown quantity, transpose all the terms involving the unknown quantity to one side of the equation, and all the other terms to the other side; combine like terms, and divide both sides by the coefficient of the unknown quantity. Solve 8x - 29 = 26 - 3x. 8x -f 3x = 29 + 26; llx = 55; x = 5, ans. Simple algebraic problems containing one unknown quantity are solved by making x = the unknown quantity, and stating the conditions of the problem in the form of an algebraic equation, and then solving the equa- tion. What two numbers are those whose sum is 48 and difference 14 ? Let x = the smaller number, x + 14 the greater, x + x -f- 14 = 48. 2x = 34, x = 17; a; + 14 = 31, ans. Find a number whose treble exceeds 50 as much as its double falls short of 40. Let x = the number. 3x - 50 = 40 - 2x; 5x = 90; x - 18, ans. Prov- ing, 54 - 50 = 40 - 36. ALGEBRA, 35 Equations containing tfwo unknown quantities. It one equation contains two unknown quantities, x and ?/, an indefinite number of pairs of values of x and y may be found that will satisfy the equation, but if a second equation be given only one pair of values can be found that will satisfy both equations. Simultaneous equations, or those that may be satis- fied by the same values of the unknown quantities, are solved by combining the equations so as to obtain a single equation containing only one unknown quantity. This process is called elimination. Elimination by addition or subtraction. Multiply the equation by such numbers as will make the coefficients of one of the unknown quanti- ties equal in the resulting equation. Add or subtract the resulting equa- tions according as they have unlike or like signs. Solve J 2 * + 3 2/ = 7. Multiply by 2: 4x + Vy = U 7 } 4 X _ ty - 3. Subtract: 4x - 5y = 3 \\y = 11; y = 1. Substituting value of ?/ in first equation, 2x -f- 3 = 7; x = 2. Elimination by substitiLtion. From one of the equations obtain the value of one of the unknown quantities in terms of the other. Substitu- tute for this unknown quantity its value in the other equation and reduce the resulting equations. j2o; + 3/ = 8. (1). From (1) we find x = -. TQ l3x+7y = 7. (2). Substitute this value in (2): s( ~ ) + 7y = 7; = 24 - 9y -f 14y = 14, whence y = - 2. Substitute this value in (1): 2x - 6 = 8; x = 7. Elimination by comparison. From each equation obtain the value of one of the unknown quantities in terms of the other. Form an equation from these equal values, and reduce this equation. -9y=il. (1). From (1) we find x = 11 Solve-! I 3x - 4y = 7. (2). From (2) we find x- l (. <* Equating these values of x, ~\ } ' ' * - - ?/ ; IQy = - 19; y = - 1. Substitute this value of ?/ in (1):~2# -J- 9 = 11; x = 1. If three simultaneous equations are given containing three unknown quantities, one of the unknown quantities must be eliminated between two pairs of the equations; then a second between the two resulting equations. Quadratic equations. A quadratic equation contains the square of the unknown quantity, but no higher power. A pure quadratic contains the square only; an affected quadratic both the square and the first power. To solve a pure quadratic, collect the unknown quantities on one side, and the known quantities on the other; divide by the coefficient of the un- known quantity and extract the square root of each side of the resulting equation. Solve 3# 2 - 15_= 0. 3tf 2 = 15; a; 2 = 5; x = |/5 A root like ^5, which is indicated, but which can be found only approxi- mately, is called a surd. Solve 3o; 2 + 15 = 0. 3x* = - 15; x* = - 5; x = V- 5. The square root of 5 cannot be found even approximately, for the square of any number positive or negative is positive; therefore a root which is in- dicated, but cannot be found even approximately, is called imaginary. To solve an affected quadratic.]. Convert the equation into the form a 2 ^ 2 2abx = c, multiplying or dividing the equation if necessary, so as to make the coefficient of x' 2 a square number. 2. Complete the square of the first member of the equation, so as to con- vert it to the form of a% 2 2abx + fc 2 , which is the square of the binomial ax b, as follows: add to each side of the equation the square of the quo- tient obtained by dividing the second term by twice the square root of the first term. 3. Extract the square root of each side of the resulting equation. Solve 3x 2 - 4x = 32. To make the coefficient of x"* a square number, multiply by 3: 9a* 2 - 12# = 96; 12x H- (2 x 3x) = 2; 2 2 = 4. Complete the square: 9# 2 - 12x -j- 4 = 100 Extract the root: 3x 2 = 36 ALGEBRA. 10, whence x 4 or 2 2/3. The square root of 100 is either -f 10 or - 10, since the square of - 10 as well as -f 10 2 = 100. Problems involving quadratic equations have Apparently two solutions, as a quadratic has two roots. Sometimes botli will be true solutions, but gen- erally one only will be a solution and the other be inconsistent with the conditions of the problem. The sum of the squares of two consecutive positive numbers is 481. Find the numbers. Let x = one number, a; -f 1 the other, x* -f (x + I) 2 = 481. 2x"* -f 2x -f 1 = 481. re 2 -f- x = 240. Completing the square, # 2 -f x + 0.25 = 240.25. Extracting the root we obtain x -f- 0.5 = !5.5; x = 15 or 16. The positive root gives for the numbers 15 and 16. The negative root 16 is inconsistent with the conditions of the problem. Quadratic equations containing two unknown quantities require different meihods for their solution, according to the form of the equations. For these methods reference must be made to works on algebra. n - Theory of exponents. \a when n is a positive integer is one of n n - equal factors of a. \ a m means a is to be raised to the with power and the u th root extracted. (y ~a; means that the nth root of a is to be taken and the result raised to the with power. . -- / , -- \7 y a m = Vy<* / = a n . When the exponent is a fraction, the numera- tor indicates a power, and the denominator a root. l = T* = a 3 ; i = VVr.3 = a 1 ' 5 . To extract the root of a quantity raised to an indicated power, divide the exponent by the index of the required root; as, n, . 3,- 6 \a m =z a ' r a 6 = a 3 = a 2 . Subtracting 1 from the exponent of a is equivalent to dividing by a : a 2 -i =a = a; a 1 - 1 ^ a = - =1; a - 1 = a - 1 = - ; a - 1 - = a - a = A number with a negative exponent denotes the reciprocal of the number with the corresponding positive exponent. A factor under the radical sign whose root can be taken may, by having the root taken, be removed from under the radical sign: |/o2/7 = |/o2 x |/b = a tyb. A factor outside the radical sign may be raised to the corresponding power and placed under it: Binomial Theorem. To obtain any power, as the nth, of an ex- pression of the form x -{- a (a + * = i- f W- i + '""-r"-^ + etc. The following laws hold for any term in the expansion of (a -f x) n . The exponent of x is less by one than the number of terms. The exponent of a is n minus the exponent of x. The last factor of the numerator is greater by one than the exponent of a, The last factor of the denominator is the same as the exponent of x. In the rth term the exponent of x will be r - 1. The exponent of a will be n (r 1), or n r + 1. The last factor of the numerator will be n r + 2. The last factor of the denominator will be = r 1. Hence the rth term = L- 2> - -- ' + ^ - , + i x r-i GEOMETRICAL PROBLEMS. 37 GEOMETRICAL PROBLEMS. f E 1. To bisect a straight line, or an arc of a circle (Fig. l}. From the ends A, B, as centres, de- scribe arcs intersecting at C and Z>, and draw a line through C and D which will bisect the line at E or the arc at F. 2. To draw a perpendicular to a straight line, or a radial line to a circular arc. Same as in Problem 1. C D is perpendicular to the line A B, and also radial to the arc. 3. To draw a perpendicular to a straight line from a given point in that line (Fig. 2). With any radius, from the given point A in the line B C, cut the line at B and C. With a longer radius describe arcs from B and (7, cutting each other at Z), arid draw the perpendicular D A. 4. From the end A. of a given line A D to erect a perpendic- ular A E (Fig. 3). From any centre j<', above A D, describe a circle passing through the given point A, and cut- ting the given line at D. Draw D F and produce it to cut the circle at E, and draw the perpendicular A E. Second Method (Fig. 4). From the given point A set off a distance A E equal to three parts, by any scale ; and on the centres A and E, with radii of four and five parts respectively, describe arcs intersecting at C. Draw the perpendicular A C. NOTE. This method is most useful on very large scales, where straight edges are inapplicable. Any multiples of the numbers 3, 4, 5 may be taken with the same effect as 6, 8, 10, or 9, 12, 15. 5. To draw a perpendlcula o a, straight line from an lar to a straight line' from any point -without it (Fig. 5.) From the point A, with a sufficient radius cut the given line at F and G, and from these points describe arcs cut- ting at E. Draw the perpendicular AE. A B FIG. 6. 6. To draw a straight line parallel to a given line, at a given distance apart (Fig. 6). From the centres A, B, in the given line, with the given distance as radius, describe arcs C, D, and draw the par- allel lines C D touching the arcs. 38 GEOMETRICAL PROBLEMS. G 7. 'jfo divide a straight line into a number of equal parts (Fig. 7). To divide the line A B into, say, five parts, draw the line A C at an angle from ^4; set off five equal parts; draw B 5 and draw parallels to it from the other points of division in A C. These parallels divide A B as required. NOTE. By a similar process a line may be divided into a number of un- equal parts; setting off divisions on A O, proportional by a scale to the re- quired divisions, and drawing parallel cutting A B. The triangles All, A22, A83, etc., are 'similar triangles. FIG. 8. 8. Upon a straight line to draw an angle equal to a given angle (Fig. 8). Let A be the given angle and F G the line. From the point A with any radius describe the arc D E. From F with the same radius describe I H. Set off the arc / H equal to D K, and draw F H. The angle F is equal to A, as required. 9. To draw angles of 60 and 30 (Fig. 9). From f\ with any radius FL describe an arc IH ; and from 7, with the same radius, cut the arc at H and draw F H to form the required angle I F H. Draw the perpendicular H K to the base line to form the angle of 30- F H K. 1O. To draw an angle of 45 (Fig. 10). Set off the distance FT, draw the perpendicular I H equal to IF, and .loin HFto form the angle at F. The angle at H is also 45. 11. To bisect an angle (Fig. 11). Let A C B be the angle; with G as a centre draw an arc cutting the sides at A, B. From A and B as centres, describe arcs cutting each other at D. Draw C D, dividing the angle into two equal "parts. FIG. 1 12. Through two given points to describe an arc of a circle with a given radius (Fig. 12). From the points A and B as centres, with the given radius, de< scribe arcs cutting at C , and from Cwith the same radius describe an arc A B. GEOMETRICAL PROBLEMS. 39 FIG. 13. FIG. 14. 13. To find I lie centre of a circle or of an arc of a circle (Fig. 13). Select three points, A, B, (7, in the circumference, well apart; with the same radius describe arcs from these three points, cutting each other, and draw the two lines, D E, F G, through their intersections. The point O, where they cut, is the centre of the circle or arc. To describe a circle passing through three given points. Let A, B, G be the given points, and proceed as in last problem to find the centre O, from which the circle may be described. 14. To 'describe an arc of a circle passing through three given points when the centre is not available (Fig. 14). From the extreme points A, B, as centres, describe arcs A H< B G. Through the third point O draw A E, B F, cutting the arcs. Divide A F and B E into any num- ber of equal parts, and set off a series of equal parts of the same length on the upper portions of the arcs beyond the points E F. Draw straight lines, B L, B M, etc., to the divisions in A F, and A I, A K, etc., to the divisions in E G. The successive intersections N, O, etc., of these lines are points in the circle required between the given points A and C. which may be drawn in ; similarly the remaining part of the curve B C may be described. (See also Problem 54.) 15. To draw a tangent to a circle from a given point in the circumference (Fig. In). Through the given point A, draw the radial line A (7, and a perpendicular to it, F #, which is the tangent re- quired. 16. To draw tangents to a circle from a point without it (Fig. 16). From A. with the radius A C, describe an arc B C Z>, and from C. with a radius equal to the diameter of the circle, cut the arc at B D. Join B <7, C D, cutting the circle at E F, and draw A E, A F, the tangents. NOTE. When a tangent is already drawn, the exact point of contact may be found by drawing a perpendicular to it from the centre. 17. Between two inclined lines to draw a series of cir- cles touching these lines and touching each other (Fig. 1?). Bisect the inclination of the given lines A B, CD, by the line NO. From a point P in this line draw the perpendicular P B to the line A B, and FIG. 15. 40 GEOMETRICAL PROBLEMS. A on P describe the circle B D, touching the lines and cutting the centre line at E. From E d raw E F perpendicular to the centre line, cutting A B at F, and from F describe an arc E G, cut- ting A B at G. Draw G H parallel to B P, giving H, the centre of the next circle, to be described with the radius C JJ E, and so on for the next circle IN. Inversely, the largest circle may be described first, and the smaller ones in succession. This problem is of fre- quent use in scroll-work. 18. Between two inclined lines to draw a circular seg- ment tangent to tlie lines and passing through a point !' on tlie line /' C which bisects the angle of the lines (Fig. 18). Through .Fclraw D A at right angles to F C ; bisect the angles A and D, as in Problem 11, by lines cutting at C, and from C with radius (7-Fdiaw the arc H F G required. 19. To draw a circular arc that will he tangent to two given lines A Jl and C 1) in- clined to one another, one tangential point E being given (Fig. 19). Draw the centre line G F. From ldraw E Fat right to angles A B ; then F is the centre of the circle required. 20. To describe a circular arc joining two circles, and touching one of them at a given point (Fig. 20). To join the circles .4 B, F G, by an arc touching one of them at F, draw the radius E f\ and produce it both ways. Set off F H equal to the radius A C of the other circle; join C H and bisect it with the perpendicular LI, cutting E F at L On the centre /, with radius IF, de- scribe the arc F A as required. 21. To draw a circle with a given radius It that will be tangent to two given circles A. and J* (Fig. 21) From centre of circled with radius equal R plus radius of A, and from centre of B with radius equal to R + radius of B, draw two arcs cutting each other in (7, which will be the centre of the circle re- quired. 22. To construct an equi- lateral triangle, the sides heing given (Fig. aa). On the ends of one side, A, B, with A B as radius, describe arcs cutting at C, and draw AC, CB. GEOMETRICAL PROBLEMS. B- C- FIG. 23. 23. To construct a triangle of unequal sides (Fig. 28). On either end of the base A Z>, with the side B as radius, describe an arc; and with the side C as radius, on the other end of the base as a centre, cut the arc at E. Join A E, D E. 24. To construct ft square on a given straight line A JK (Fig. 24). With A B as radius and A and B as centres, draw arcs A D and B C, intersecting at E. Bisect EB at F. With E as centre and E F as radius, cut the arcs A D and B C in D and C. Join A C, C Z>, and D J5 to form the square. 25. To construct a rect- angle witli given base ./<; /' and height -E Jf (Fig. 25). On the base E Fdraw the perpendiculars EH, F G equal to the height, and join Q H. 26. To describe a circle about a triangle (Fig. 26). Bisect two sides A B, A C of the tri- angle at E F, and from these points draw perpendiculars cutting at K. On the centre K, with the radius K A, draw the circle A B C. 27. To inscribe a circle in a triangle (Fig. 27). Bisect two of the angles A, <7, of the triangle by lines cutting at D ; from D draw a per- pendicular D Eto any side, and with D E as radius describe a circle. When the triangle is equilateral, draw a perpendicular from one of the angles to the opposite side, and from the side set off one third of the per- pendicular. 28. To describe a circle about a square, and to in- scribe a square in a circle (Fig. 28). To describe the circle, draw the diagonals A B, C D of the square, cut- ting at E. On the centre E. with the radius A E, describe the circle. To inscribe the square. Draw the two diameters, A 5, CD, at right angles, and join the points A, B, C /), to form the square. NOTE. In the same way a circle may be described about a rectangle. GEOMBTEICAL PROBLEMS. 29. To inscribe a circle in a square (Fig. 29). To inscribe the Circle, draw the diagonals A B, CD Of the square, cutting at E\ draw the perpendicular E F to one side, and with the radius E F describe the circle. 30. To describe a square about a circle (Fig. 30). Draw two diameters A B, CD at right angles. With the radius of the circle and A, B, C and D as centres, draw the four half circles which cross one another in the corners of the square. 9 31. To inscribe a pentagon in a circle (Fig. 31). Draw diam- eters AC, B D at right angles, cutting at o. Bisect A o at E, and from E, with radius E B, cut A C at F ; from B, with radius B F, cut the circumfer- ence at Gr, H, and with the same radius step round the circle to /and K; join the points so found to form the penta gon. 32. To construct a penta- gon on a given line A B (Fig. 82). Frotti B erect a perpendicular B C half the length of A B\ join A C and prolong it to D, making CD = B C. Then B D is the radius of the circle circumscribing the pentagon. From A and B as centres, with B Das radius, draw arcs cutting each other in O, which is the centre of the circle. 33. To construct a hexagon upon a given straight line (Fig. 33). From A and B, the ends of the given line, with radius A B, de- scribe arcs cutting at g ; from g, with the radius g A, describe a circle ; with the same radius set off the arcs A G, G F, and B D, D E. Join the points so found to form the hexagon. The side of a hexagon = radius of its circum- scribed circle, 34. To inscribe a hexagon in a circle (Fig. 34). Draw a diam- eter A CB. From A audl? as centres, with the radius of the circle A (7, cut the circumference at D, E, F, G, and drawyl D, D E, etc., to form the hexa- gon. The radius of the circle is equal to the side of the hexagon ; therefore the points D, E, etc., may also be found by stepping the radius six times round the circle. The angle between the diameter and the sides of a hexagon and also the exterior angle between a side and an adjacent side Erolonged is 60 degrees; therefore a exagon may conveniently be drawn by the use of a 60-degree triangle. GEOMETKICAL PROBLEMS. 43 m l 35. To describe a hexagon about a circle (Fig. 35). Draw a diameter AD B, and with the radius A D, on the centre A, cut the circum- ference at C ; join A C, and bisect it with the radius D E ; through E draw FG, parallel to A O, cutting the diam- eter at F, and with the radius D F de- scribe the circumscribing circle F H. Within this circle describe a hexagon by the preceding problem.. A more convenient method is by use of a 60- degree triangle. Four of the sides make angles of 60 degrees with the diameter, and the other two are par- allel to the diameter. 36. To describe an octagon on a given straight line {Fig. 36). Produce the given line A B both ways, and draw perpendiculars A E, B F', bisect the external angles A and B by the lines A H, B C, which make equal to A B. Draw C D and H G par- allel to A E, and equal to A B ; from the centres G, D, with the radius A B, cut the perpendiculars at E, F, and draw E F to complete the octagon. 37. To convert a square into an octagon (Fig. 37). Draw the diagonals of the square cutting at e ; from the corners A, B, C, D, with A e as radius, describe arcs cutting the sides at gn, /fc, /im, and ol, and join the points so found to form the octagon. Adjacent sides of an octa- gon make an angle of 135 degrees. 38. To inscribe an octagon in a circle (Fig. 38). Draw two diameters, A C, B D at right angles; bisect the arcs A B t B (7, etc., at ef\ etc., and join A e, e B, etc., to form the octagon. 39. To describe an octagon about a circle (Fig. 39). Desci itr a square about the given circle A B , draw perpendiculars h k, etc. . to the diagonals, touchiDg the circle to form the octagon. 4O. To describe a polygon of any number of sides upon ** given straight line (Fig. 40). Produce the given line A B, and on A, 44 GEOMETRICAL PROBLEMS. with the radius A B, describe a semi- circle; divide the semi-circumference into as many equal parts as there are to be sides in the polygon say, in this example, five sides. Draw lines from A through the divisional points D, 6, and c, omitting one point a ; and on the centres J5, D, with the radius A B, cut A b at E and A c at F. Draw D E, E F, F B to complete the polygon. 41. To Inscribe a circle within a polygon (Figs. 41, 42). When the polygon has an even number of sides (Fig. 41), bisect two opposite sides at A and B; draw A B, and bisect it at C by a diagonal D E, and with the radius C A describe the circle. When the number of sides is odd (Fig. 42), bisect two of the sides at A and B : and draw lines A E, B D to the opposite angles, intersecting at (7; from C, with the radius C A, describe the circle. 42. To describe a circle without a polygon (Figs. 41, 42). Find the centre (J as before, and with the radius C D describe the circle. 43. To inscribe a polygon of any number of sides with* in a circle (Fig. 43). Draw the diameter A B and through the centre E draw the perpendicular EC, cutting the circle at F. Divide E F into four equal parts, and set off three parts equal to those from F to C. Divide the diameter A B into as many equal parts as the polygon is to have sides ; and from C draw CD, through the second point of division, cutting the circle at D. Then A D is equal to one side of the polygon, and by stepping round the circumference with the length A D the polygon may be com- pleted. TABLE OF POLYGONAL ANGLES. Number of Sides. Angle at Centre. 1 Number of Sides. Angle at Centre. Number of Sides. Angle at Centre. No. Degrees. No. Degrees. No. Degrees. 3 120 9 40 15 24 4 90 10 36 16 22 5 72 11 32 T " T 17 21 T 3 7 6 60 12 80 18 20 7 8 S 13 14 If 19 20 19 18 GEOMETRICAL PROBLEMS. 45 C In this table the angle at the centre is found by dividing 360 degrees, the number of degrees in a circle, by the number of sides in the polygon; and by setting off round the centre of the circle a succession of angles by means of the protractor, equal to the angle in the table due to a given number of sides, the radii so drawn will divide the circumference into the same number of parts. 44. To describe an ellipse when the length and breadth are given (Fig. 44). A B, transverse axis; C D, conjugate axis; F G, foci. The sum of the distances from C to .Fand G, also the sum of the distances from F and G to any other point in the curve, is equal to the transverse axis. From the centre C, with A E as radius, cut the axis AB at .Fand G, the foci ; fix a couple of pins into the axis at F and G, and loop on a thread or cord upon them equal in length to the axis A B, so as when stretched to reach to the extremity C of the con- jugate axis, as shown in dot-lining. Place a pencil inside the cord as at H, and guiding the pencil in this way, keeping the cord equally in tension, carry the pencil round the pins .F, G, and so describe the ellipse. NOTE. This method is employed in setting off elliptical garden - plots, walks, etc. 2d Method (Fig. 45). Along the straight edge of a slip of stiff paper mark off a distance a c equal to A C, half the trans verse axis; and from the same point a distance a b equal to C D, half the conjugate axis. Place the slip so as to bring the point 6 on the line A B of the transverse axis, and the point c on the line D E ; and set off on the drawing the position of the point a. Shifting the slip so that the point b travels on the transverse axis, and the point c on the conjugate axis, any number of points in the curve may be found, through which the curve'may be traced. 3d Method (Fig. 46). The action of the preceding method may be em- bodied so as to afford the means of describing a large curve continuously by means of a bar m fc, with steel points m, 7, fc, riveted into brass slides adjusted to the length of the semi- axis and fixed with set-screws. A rectangular cross E G, with guiding- slots is placed, coinciding with the two axes of the ellipse A C and B H. By sliding the points k, I in the slots, and carrying round the point m, the curve may be continuously described. A pen or pencil may be fixed at m. 4th Method (Fig. 47). Bisect the transverse axis at C, and through C draw the perpendicular D E, making C D and C E each equal to half the conjugate axis. From D or E, with the radius A C, cut the transverse axis at F, F', for the foci. Divide A C into a number of parts at the FIG. 45. 46 GEOMETRICAL PROBLEMS. FIG. 48. P 2 ' & et v c * ^ th fche radhls ^ 7 on F and *" as centres, describe aics, and with the radius B I on the same centres cut these arcs as shown. Repeat the operation for the other divisions of the transverse axis. The series of intersections thus made are points in the curve, through which the curve may be traced. 5th Method (Fig. 48). On the two axes A B, D E &s diameters, on centre C, describe circles; from a number of points a, 6, etc., in the circumference AFB, draw radii cutting the inner circle at a', b', etc. From a, b, etc., draw perpendiculars to AB; and from a', b', etc., draw parallels to A B, cut- ting the respective perpendiculars at n, o, etc. The intersections are points in the curve, through which the curve may be traced. 6th Method (Fig. 49). When the transverse and conjugate diameters are given, A B, C D, draw the tangent EF parallel to A B. Produce CD, and on the centre G with the radius of half A B, describe a semicircle HDK; from the centre G draw any number of straight lines to the points E, r, etc., in the line E F, cutting the circumference at I, m, n, etc. ; from the centre O of the ellipse draw straight lines to the points E, r, etc. ; and from the points I, m, n, etc., draw parallels to G C, cutting the lines O E, Or, etc., at Z,, M, N, etc. These are points in the circumference of the ellipse, and the curve may be traced through them. Points in the other half of the ellipse are formed by ex- tending the intersecting lines as indi- cated in the figure. 45. To describe an ellipse approximately by means of circular arcs. First. With arcs of two radii (Fig. 50j. Find the differ- ence of the semi-axes, and set it off from the centre O to a and c on O A and OC; draw ac, and set off half a c to d ; draw d i parallel to a c; set off O e equal to O d; join e i, and draw the parallels e m, d m. From m, with radius m C, describe an arc through C ; and from i describe an arc through Z); from d and e describe arcs through A and B. The four arcs form the ellipse approximately. NOTE. This method does not apply satisfactorily when the conjugate axis is less than two thirds of the trans- verse axis. 2d Method (by Carl G. Earth, Fig. 51). -In Fig. 51 a & is the major and c d the minor axis of the ellipse to be approximated. Lay off b e equal to the semi-minor axis c 0, and use a e as radius for the arc at each extremity of the minor axis. Bisect e o at / and lay off e g equal to e /, and use g b as radius for the arc at each extremity of the major axis. FIG. 51. GEOMETRICAL PROBLEMS. 47 The method is not considered applicable for cases in which the minor axis is less than two thirds of the major. 3d Method : With arcs of three radii ^~- T-^^ (Fig. 52). On the transverse axis A B V ^- -^.-^-^Cl 7^... _f draw the rectangle B G on the height l\ / B^~ ^ OC; to the diagonal A C draw the perpendicular G H D\ set off OK equal to O C, and describe a semi- circle on A K, and produce O Cto I/; set off M equal to C L, and from D describe an arc with radius D M ; from A, with radius O -L, cut A B at JV; from H, with radius HJV, cut arc a 6 at a. Thus the five centres D, a, 6, H, H' are found, from which the arcs are described to form the ellipse. This process works well for neaily all proportions of ellipses. It is used in striking out vaults and stone bridges. 4th Method (by F. R,. Honey, Figs. 53 and 54). Three radii are employed. With the shortest radius describe the two arcs which pass through the ver- tices of the major axis, with the longest the two arcs which pass through the vertices of the minor axis, and with the third radius the four arcs which connect the former. A. simple method of determining the radii of curvature is illustrated in Fig. 53. Draw the straight lines a f and a c, forming any angle at a. With a as a Centre, and with radii a b and a c, re- spectively, equal to the semi- minor and semi-major axes, draw the arcs b e and c d. Join ed, and through b and c re- spectively draw b g and c / parallel to e d, intersecting a c at g, and af at/; af is the radius of curvature at the ver- tex of the minor axis; and a g vertex of the major axis. the radius of curvature at the Lay off d h (Fig. 53) equal to one eighth of b d. Join e h, and draw c k and 6 I parallel to e h. Take a k for the longest radius (= R), a I for the shortest radius (= rl and the arithmetical mean, or one half the sum of the semi-axes, for the third radius (= p), and employ these radii for the eight-centred oval as follows: Let a b and c d (Fig. 54) be the major and minor axes. Lay off a e equal to r, and af equal to p: also lay off c g equal to R, and c h equal to p. With g as a centre and g h as a radius, draw the arc h 7c; with the centre e and radius e f draw the arc / fc, intersecting hk at k. Draw the line g k and produce it, making g I equal to R. Draw ke and produce it, making k m equal to jp. With the centre g and radius g c ( R) draw the arc c I ; with the centre k and radius k I (= p) draw the arc I m, and with the centre e and radius e m (=r) draw the arc m a- The remainder of the work is symmetrical with respect to the axes. 48 GEOMETRICAL PROBLEMS. E A G 2 rxj F \ J O \ ~\rc J o \ Y / o \ D B b FIG . 55. ' 46. The Parabola. A parabola (D A C, Fig. 55) is a curve such that every point in the curve is equally distant from the directrix KL&ud the focus F. The focus lies in the axis A B drawn from the vertex or head of the curve A, so as to divide the figure into two equal parts. The vertex A is equidistant from the directrix and the focus, or A e = A F. Any line parallel to the axis is a diameter. A straight line, as EG or DC, drawn across the figure at right angles to the axis is a double ordinate, and either half of it is an ordinate. The ordinate to the axis E F G, drawn through the focus, is called the parameter of the axis. A segment of the axis, reckoned from the vertex, is an abscissa of the axis, and it is an abscissa of the ordi- nate drawn, from the base of the ab- scissa. Thus, A B is an abscissa of the ordinate B C. Abscissae of a parabola are as the squares of their ordinates. To describe a parabola when an abscissa and its ordi- nate are given (Fig. 55). Bisect the given ordinate B Cat a, draw A a, and then a b perpendicular to it, meeting the axis at b. Set off A e, A F, each equal to B b; and draw KeL perpendicular to the axis. Then K L is the directrix and F is the focus. Through F and any number of points, o, o, etc., in the axis, draw double ordinates, n o n, etc , and from the centre F r with the radii Fe, o e, etc., cut the respective ordinates at E } G, n, n, etc. The curve may be traced through these points as shown. -iL 2d Method : By means of a square and a cord (Fig. 56). Place a straight- edge to the directrix EN, and apply to it a square LEG. Fasten to the end G one end of a thread or cord equal in length to the edge E G, and attach the other end to the focus F', slide the square along the straight- edge, holding the cord taut against the e(ige of the square by a pencil D, by which the curve is described. 3d Method: When the height and the base are given (Fig. 5?). Let A B be the given axis, and C D & double ordinate or base; to describe a para- bola of which the vertex is at A. Through A draw E F parallel to CD, and through C and D draw C E and D F parallel to the axis. Divide B C and B D into any number of equal parts, say five, at 'a, b, etc., and divide C E and DF into the same number of parts. Through the points a, ft, c, d in the base C D on each side of the axis draw perpendiculars, and through a, 6, c, d in C E and D F draw lines to the vertex A, cutting the perpendicu- lars at e. /, g, h. These are points in the parabola, and the curve C A D may be traced as shown, passing through then;. FIG. 56. A __ V t- e ^x i^i ' ^ 3 ./ \^ . : ^/ t/ g x> 1 C d cbaBabcd FIG. 57. GEOMETRICAL PROBLEMS. 49 FIG. 58. FIG. 59. 47. The Hyperbola (Fig. 58). A hyperbola is a plane curve, such that the difference of the distances from any point of it to two fixed points is equal to a given distance. The fixed points are called the foci. To construct a hyperbola. Let F' and F be the foci, and F' F the distance between them. Take a ruler longer than the distance F r F, and fasten one of its extremities at the focus F'. At the other extremity, H, attach a thread of such a length that the length of the ruler shall exceed the length of the thread by a given distance A B. Attach the other ex- tremity of the thread at the focus F. Press a pencil, P, against the ruler, and keep the thread constantly tense, while the ruler is turned around F' as a centre. The point of the pencil will describe one branch of the curve. 2d Method: By points (Fig. 59). From the focus F' lay off a distance F' N equal to the transverse axis, or distance between the two branches of the curve, and take any other distance, as F'H, greater than F'N. With F' as a centre and F'H as a radius describe the arc of a circle. Then with Fa,s a centre and N H as a radius describe an arc intersecting the arc before described at p and q. These will be points of the hyperbola, for F' q Fq is equal to the trans- verse axis A B. If, with F as a centre and F' H as a radius, an arc be described, and a second arc be described with F' as a centre and NH as a radius, two points in the other branch of the curve will be determined. Hence, by changing the centres, each pair of radii will determine two points in each branch. The Equilateral Hyperbola, The transverse axis of a hyperbola is the distance, on a line joining the foci, between the two branches of the curve. The conjugate axis is a line perpendicular to the transverse axis, drawn from its centre, and of such a length that the diagonal of the rect- angle of the transverse and conjugate axes is equal to the distance between the foci. The diagonals of this rectangle, indefinite!} 7 prolonged, are the asymptotes of the hyperbola, lines which the curve continually approaches, but touches only at an infinite distance. If these asymptotes are perpen- dicular to each other, the hyperbola is called a rectangular or equilateral hyperbola. It is a property of this hyperbola that if the asymptotes are taken as axes of a rectangular system of coordinates (see Analytical Geom- etry), the product of the abscissa and ordinate of any point in the curve is equal to the product of the abscissa and ordinate of any other point ; or, if p is the ordinate of any point and v its abscissa, and p^ and v t are the ordi- iiate and abscissa of any other point, pv=p* v\ ; or pv = a constant. 48. The Cycloid K J5_ / (Fig. 60). If a circle Ad be rolled along a straight line 46, any point of the circumference as A will describe a curve, which is called a cycloid . The circle is called the generating circle, and A the generat- ing point. To draw a cycloid. Divide the circumference of the generating circle into an even number of equal parts, as A 1, 12, etc., and set off these distances on the base. Through the points 1, 2, 3, etc., on the circle draw horizontal lines, and on them set off distances la = A\, 26 = A2, 'ic = A3, etc. The points A, a, 6, c, etc., will be points in the cycloid, through which draw the curve. 50 GEQMETKICAL PROBLEMS. 49. The Epicycloid (Fig. 61) is generated by a point D in one circle D C rolling upon the circumference of another circle A C B, instead of on a flat surface or line; the former being the generating circle, and the latter the fundamental circle. The generat- ing circle is shown in four positions, in which the generating point is succes- sively marked D, D', D", D"'. A D'" B is the epicycloid. 50. The Hypocycloid (Fig. 62) is generated by a point in the gener- ating circle rolling on the inside of the fundamental circle. When the generating circle radius of the other circle, the hypocycloid becomes a straight line. 51. Tlie Traetrix or Schiele's anti-friction curve (Fig. 63). R is the radius of the shaft, C, 1,2, etc.. the axis. From O set off on R a small distance, o a; with radius R and centre a cut the axis at 1, join a 1, and set off a like small distance a 6; from b with radius R cut axis at 2, join 6 2, and so on, thus finding points o, a, 6, c, d, etc., through which the curve is to be drawn. FIG. 63. 52. The Spiral. The spiral is a curve described by a point which moves along a straight line according to any given law, the line at the same time having a uniform angular motion. The line is called the radius vector. If the radius vector increases directly as the measuring angle, the spires, or parts described in each revolution, thus gradually increasing their dis- tance from each other, the curve is known as the spiral of Archimedes (Fig. 64). This curve is commonly used for cams. To describe it draw the radius vector in several different directions around the centre, with equal angles between them; set off the distances 1, 2, 3, 4, etc., corresponding to the scale upon which the curve is drawn, as shown in Fig. 64. In the common spiral (Fig. 64) the pitch is uniform; that is, the spires are equidistant. Such a spiral is made by rolling up a belt of uniform thickness. To construct a spiral with four centres (Fig. 65). Given the pitch of the spiral, construct a square abont the centre, with the sum of the four sides equal to the pitch. Prolong the sides in one direction as shown; the corners are the centres for each arc of the external angles, forming a quadrant of a spire. Fig. 65. GEOMETRICAL PROBLEMS. 51 FIG. 53. To find the diameter of a circle into which a certain number of rings will fit on its inside (Fig. 66). For instance, what is the diameter of a circle into which twelve J^-inch rings will fit, as per sketch ? Assume Uiat we have found the diameter of the required circle, and have drawn the rings inside of it. Join the centres of the rings by straight lines, as shown : we then obtain a regular polygon with 12 sides, each side being equal to the di- ameter of a given ring. We have now to find the diameter of a circle cir- cumscribed about this polygon, and add the diameter of one ring to it; the sum will be the diameter of the circle into which the rings will fit. Through the centres A and D of two adjacent rings draw the radii CA and CD; since the polygon has twelve sides the angle A C D = 30 and A C B = 15. One half of the side A D is equal to A B. We now give the following pro- portion : The sine of the angle A C B is to A B as 1 is to the required ra- dius. From this we get the following rure : Divide A B by the sine of the angle A CB ; the quotient will be the radius of the circumscribed circle ; add to the corresponding diameter the diameter of one ring ; the sum will be the required diameter F G. 54. To describe an arc of a circle which is too large to be drawn by a beam compass, by means of points in the arc, radius being given. Suppose the radius is 20 feet and it is desired to obtain five points in an arc whose half chord is 4 feet. Draw a line equal to the half chord, full size, or on a smaller scale if more con- venient, and erect a perpendicular at one end, thus making rectangular axes of coordinates. Erect perpendiculars at points 1, 2, 3, and 4 feet from the first perpendicular. Find values of y in the formula of the circle. #2 -f 2/2 = ^2 D y substituting for x the values 0, 1, 2, 3, and 4, etc.. and fov_R* the_squajre of _the radius, or 400. The values will be y = V R* a 2 = ^400, ^399, ^396, ^391, ^384; = 20, 19.975, 19.90, 19.774, 19.596. Subtract the smallest, or 19.596, leaving 0.404, 0.379, 0.304, 0.178, feet. Lay off these distances on the five perpendiculars, as ordinates from the fcuUf chord, and the positions of five points on the arc will be found. Through these the curve may be drawn. (See also Problem 14.) 55. The Catenary is the curve assumed by a perfectly flexible cord when its ends are fastened at two points, the weight of a unit length being constant. The equation of the catenary is - ~ iV \ |, in which e is the base of the Naperian system of log- arithms. To plot the catenary. Let o (Fig. 67) be the origin of coordinates. Assigning to a any value as 3, the equation becomes FIG. 67. To find the lowest point of the curve. '0 -o 1 GEOMETRICAL PROBLEMS. Thenput* = 1; .-. y = l\e* + e 3 ) = | (1.396 -f 0.717) = 8.17. = ? (1.948 -I- 0.513) = 3.69. Fut x = 3, 4, 5, etc., etc., and find the corresponding values of y. For each value of y we obtain two symmetrical points, as for example p 'and p l . In this way, by making a successively equal to 2, 3, 4, 5, 6, 7, and 8, the curves of Fig. 67 were plotted. In each case the distance from the origin to the lowest point of the curve is equal to a ; for putting x o, the general equation reduces to y a. For values of a = 6, 7, and 8 the catenary closely approaches the parabola. For derivation of the equation of the catenary see Bowser's Analytic Mechanics. For comparison of the catenary with the parabola, see article by F. R. Honey, Amer. Machinist, Feb. 1, 1894. 56. The Involute is a name given to the curve which is formed by the end of a string which is unwound from a cylinder and kept taut ; con- sequently the string as it is unwound will always lie in the direction of a tangent to the cylinder. To describe the involute of any given circle, Fig. 68, take any point A on its circum- ference, draw a diameter A B, and f rom B draw B b perpendicular to AB. Make Bb equal in length to half the circumference of the circle. Divide Bb and the semi-circumference into the same number of equal parts, say six. From each point of division 1, 2, 3, etc., on the circumference draw lines to the centre C of the circle. Then draw 1 a perpendicular to C 1 ; 2a 2 perpendicular to C2; and so on. F IG 68. Make la equal to b b, ; 2a 2 equal to b 6 2 ; 3 a 3 equal to b b 2 ; and so on. Join the points A, a^, a 2 , a 3 , etc., by a curve; this curve will be the required involute. 57. Method of plotting angles without using a protrac- tor. The radius of a circle whose circumference is 360 is 57.3 (more ac- curately 57.296). Striking a semicircle with a radius 57.3 by any scale, spacers set to 10 by the same scale will divide the arc into 18 spaces of 10 each, and intermediates can be measured indirectly at the rate of 1 by scale for each 1, or interpolated by eye according to the degree of accuracy required. The following table shows the chords to the above-mentioned radius, for every 10 degrees from up to 1JO. By means of one of these, Angle. Chord. 1 0.999 10 9.988 20 19.899 30 29.658 40 39.192 50.. .. 48.429 Angle. Chord. 60. 57.296 70 65.727 80 73.658 90 81.029 100 87.782 110 93.869 a 10 point is fixed upon the paper next less than the required angle, and the remainder is laid off at the rate of 1 by scale for each degree. GEOMETRICAL PROPOSITIONS. 53 GEOMETRICAL PROPOSITIONS. In a right-angled triangle the square on the hypothenuse is equal to the sum of the squares on the other two sides. If a triangle is equilateral, it is equiangular, and vice versa. If a straight line from the vertex of an isosceles triangle bisects the base, it bisects the vertical angle and is perpendicular to the base. If one side of a triangle is produced, the exterior angle is equal to the sum of the two interior and opposite angles. If two triangles are mutually equiangular, they are similar and their cor- responding sides are proportional. If the sides of a polygon are produced in the same order, the sum of the exterior angles equals four right angles. (Not true if the polygon has re- entering angles ) In a quadrilateral, the sum of the interior angles equals four right angles. In a parallelogram, the opposite sides are equal ; the opposite angles are equal; it is bisected by its diagonal, and its diagonals bisect each other. If three points are not in the same straight line, a circle may be passed through them. If two arcs are intercepted on the same circle, they are proportional to the corresponding angles at the centre. If two arcs are similar, they are proportional to their radii. The areas of two circles are proportional to the squares of their radii. If a radius is perpendicular to a chord, tt bisects the chord and it bisects the arc subtended by the chord. A straight line tangent to a circle meets it in only one point, and it is perpendicular to the radius drawn to that point. If from a point without a circle tangents are drawn to touch the circle, there are but two; they are equal, and they make equal angles with the chord joining the tangent points. If two lines are parallel chords or a tangent and parallel chord, they Intercept equal arcs of a circle. If an angle at the circumference of a circle, between two chords, fa sub- tended by the same arc as an angle at the centre, between two ntcJii, the angle at the circumference is equal to half the angle at the centre. If a triangle is inscribed in a semicircle, it is right-angled. If two chords intersect each other in a circle, the rectangle of the seg- ments of the one equals the rectangle of the segments of the other. And if one chord is a diameter and the other perpendicular to it, the rectangle of the segments of the diameter is equal to the square on half the other chord, and the half chord is a mean proportional between the seg- ments of the diameter. If an angle is formed by a tangent and chord, it is measured by one half of the arc intercepted by the chord; that is, it is equal to half the angle at the centre subtended by the chord. Degree of a Railway Curve. This last proposition is useful in staking out railway curves. A curve is designated as one of so many degrees, and the degree is the angle at the centre subtended by a chord of 100 ft. To lay out a curve of n degrees the transit is set at its beginning or " point of curve, 1 ' pointed in the direction of the tangent, and turned through Y%n degrees; a point 100 ft. distant in the line of sight will be a point in the curve. The transit is then swung y z n degrees further and a 100 ft. chord is measured from the point already found to a point in the new line of sight, which is a second point or " station " in the curve. The radius of a 1 curve is 5729.05 ft., and the radius of a curve of any degree is 5729.05 ft. divided by the number of degrees. 54 MENSURATION. MENSURATION. PLANE SURFACES. Quadrilateral. A four-sided figure. Parallelogram. A quadrilateral with opposite sides parallel. Varieties. Square : four sides equal, all angles right angles. Rectangle: opposite .sides equal, all angles right angles. Rhombus: four sides equal, opposite angles equal, angles not right angles. Rhomboid: opposite sides equal, opposite angles equal, angles not right angles. Trapezium. A quadrilateral with unequal sides. Trapezoid. A quadrilateral with only one pair of opposite sides parallel. Diagonal of a square = 4/2 x side 2 = 1.4142 X side. IMag. of a rectangle = |/sum of squares of two adjacent sides. Area of any parallelogram = base X altitude. Area of rhombus or rhomboid = product of two adjacent sides X sine of angle included between them. Area of a trapezium = half the product of the diagonal by the sum of the perpendiculars let fall on it from opposite angles. Area of a trapezoid = product of half the sum of the two parallel sides by the perpendicular distance between them. To find the area of any quadrilateral figure. Divide the quadrilateral into two triangles; the sum of the areas of the triangles is the area. Or, multiply half the product of the two diagonals by the sine of the angle at their intersection. To find the area of a quadrilateral inscribed in a circle. From half the sum of the four sides subtract each side severally; multi- ply the four remainders together; the square root of the product is the area. Triangle. A three-sided plane figure. Varieties. Right-angled, having one right angle; obtuse-angled, having one obtuse angle ; isosceles, having two equal angles and two equal sides? equilateral, having three equal sides and equal angles. The sum of the three angles of every triangle = 180. The sum of the two acute angles of a right-angled triangle = 90. Hypothenuse of a right-angled triangle, the side opposite the right angle, = |/sum of the squares of the other two sides. If a and 6 are the two sides and c the hypothenuse, c 2 = a 2 + b 2 ; a = f'c 2 - 6- = \/(c -f b)(c b). To find the area of a triangle : RULE 1. Multiplj 7 the base by half the altitude. RULE 2. Multiply half the product of two sides by the sine of the included angle. RULES. From half the sum of the three sides subtract each side severally; multiply together the half sum and the three remainders, and extract the square root of the product. The area of an equilateral triangle is equal to one fourth the square of one of its sides multiplied by the square root of 3, = , a being the side; or 4 a 2 X .433013. Hypothenuse and one side of right-angled triangle given, to find other side, Required side = ^hyp 2 given side 2 . If the two sides are equal, side = hyp -t- -1.4142; or hyp X .7071, Area of a triangle given, to find base: Base = twice area -f- perpendicular height Area of a triangle given, to find height: Height = twice area -H base. Two sides and base given, to find perpendicular height (in a triangle in which both of the angles at the base are acute). RULE. As the base is to the sum of the sides, so is the difference of the sides to the difference of the divisions of the base made by drawing the per- pendicular. Half this difference being added to or subtracted from half the base will give the two divisions thereof. As each side and its opposite PLANE SURFACES. 55 t fvision of the base constitutes a right-angled triangle, the perpendicular is ascertained by the rule perpendicular = Vhyp 2 base 2 . Polygon. A plane figure having three or more sides. Regular or irregular, according as the sides or angles are equal or unequal. Polygons are named from the number of their sides and angles. To find the area of an Irregular polygon. Draw diagonals dividing the polygon into triangles, and mid the sum of the areas of these triangles. To find the area of a regular polygon : RULE. Multiply the length of a side by the perpendicular distance to the centre; multiply the product by the number of sides, and divide it by 2. Or, multiply half the perimeter by the perpendicular let fall from the centre on one of the sides. The perpendicular from the centre is equal to half of one of the sides of the polygon multiplied by the cotangent of the angle subtended by the half Side. The angle at the centre = 360 divided by the number of sides. TABLE OF REGULAR POLYGONS. Radius of Cir- cumscribed 'd 1 t Circle. ^ls 3 fcJO II </j a) II | Is B ri s i 2 II o > ss^ 4> || o5 o 02 dS ^3 r2 ^ II ->-> W O <D (D J> c a OJ 0*0) <x> 5f .5 .^5 'So 1^ d rt c * < P* 00 ? ^ < <J 3 Triangle .4330127 o .5773 .2887 1.732 120 60 4 Square 1. I'AU .7071 .5 1.4142 90 90 6 Pentagon 1.7204774 1.238 .8506 .6882 1.1756 72 108 (5 7 Hexagon Heptagon 2 5980762 3.6339124 1,155 1.11 1. 1.1524 .866 1.0383 18677 60 5126' 120 128 4-7 8 Octagon 4.8284271 1.083 1.3066 1.2071 .7653 45 135 9 Nonagon 6.1818242 1.064 1.4619 1.3737 .684 40 140 10 Decagon 7.6942088 1.051 1.618 1.5388 .618 36 144 11 Undecagon 9.3656399 1.042 1.7747 1.T028 .5634 32 43' 1473-11 12 Dodecagon 11.1961524 1.037 1.9319 1.866 .5176 30 150 To find the area of a regular polygon, when the length of a side only is given : RULE. Multiply the square of the side by the multiplier opposite to the name of the polygon in the table. To find the area of an ir- regular figure (Fig. 69). Draw or- dinates across its breadth at equal distances apart, the first and the last ordinate each being one half space from the ends of the figure. Find the average breadth by adding together the lengths of these lines included be- tween the boundaries of the figure, and divide by the number of the lines added ; multiply this mean breadth by the length. The greater the number of lines the nearer the approximation. 1 1 2 3 4 5 6 7 8 9 j 10 t-j Length. 4 FIG. 69. In a figure of very irregular outline, as an indicator-diagram from a high- ipeed steam-engine, mean lines may be substituted for the actual lines of the figure, being so traced as to intersect the undulations, so that the total area of the spaces cut off may be compensated by that of the extra spaces in- closed. 56 MENSURATION 2d Method: THE TRAPEZOIDAL RULE. Divide the figure into any suffi- cient number of equal parts; add half the sum of the two end ordinates to the sum of all the other ordinates; divide by the number of spaces (that is, one less than the number of ordinates) to obtain the mean ordinate, and multiply this by the length to obtain the area. 3d Method: SIMPSON'S RULE. Divide the length of the figure into an.y even number of equal parts, at the common distance D apart, and draw or- dinates through the points of division to touch the boundary lines. Add together the first and last ordinates and call the sum A; add together the even ordinates and call the sum B; add together the odd ordiuates, except the first and last, and call the sum C, Then, area of the figure = x D. o 4th Method: DURAND'S RULE. Add together 4/W the sum of the first and last ordinates, 1 1/10 the sum of the second and the next to the last (or the penultimates), and the sum of all the intermediate ordinates, Multiply the sum thus gained by the common distance between the ordinates to obtain the area, or divide this sum by the number of spaces to obtain the mean or- dinate. Prof. Duraucl describes the method of obtaining his rule in Engineering News, Jan. 18, 1891. He claims that it is more accurate than Simpson's rule>> and practically as simple as the trapezoidal rule. He thus describes its ap- plication for approximate integration of differential equations. Any deft- nite integral may be represented graphically by an area. Thus, let Q = J*u dx be an integral in which u is some function of #, either known or admitting of computation or measurement. Any curve plotted with x as abscissa and u as ordinate will then represent the variation of u with x. and the area be- tween such curve and the axis Xwill represent the integral in question, no matter how simple or complex may be the real nature of the function u, Substituting in the rule as above given the word ' volume " for " area ** and the word '* section " for " ordinate," it becomes applicable to the deter- mination of volumes from equidistant sections as well as of areas from equidistant ordinates. Having approximately obtained an area by the trapezoidal rule, the area by Durand's rule may be found by adding algebraically to the sum of the ordinates used in the trapezoidal rule (that is, half the sum of the end ordi- nates -f- sum of the other ordinates) 1/10 of (sum of penultimates sum of first and last) and multiplying by the common distance between the ordi- 5th 'Method Draw the figure on cross-section paper. Count the number of squares that are entirely included within the boundary; then estimate ruling o the cross-section paper the more accurate the result 6th Method.- Use a planimeter. 7th Method With a chemical balance, sensitive to one milligram, draw the figure on paper of uniform thickness and cut it out carefully; weigh the piece cut out, and compare its weight with the weight per square inch of the oaper as tested by weighing a piece of rectangular shape. THE CIRCLE. 57 THE CIRCLE. Circumference = diameter x 3.1416, nearly; more accurately, 3.14159265359. Approximations, ^ = 3.143; ~ = 3.1415929. 7 Ho The ratio of circum. to diam. is represented by the symbol n- (called Pi). Multiples of TT. ITT= 3.14159265359 277 = 6.28318530718 377 = 9.42477796077 47r = 12.56637061436 577=15.70796326795 6i7=18.84955592I54 777 = 21.99 11 4857513 877 = 25.13274122872 977 = 28.27433388231 Multiples of -. '77 = .7853982 " x 2=1.5707963 " x 3 = 2.3561945 " x 4=3.1415927 " x 5 = 3.9269908 " x 6 = 4.7123890 " x 7 = 5.4977871 x 8 = 6. 2831853 " x 9 = 7.0685835 Ratio of diam. to circumference = reciprocal of 77 = 0.3183099. procal of ^77 = 1.27324. - = 2.22817 77 ~jr = 0.261799 i a 77 Multiples of -. - = 2.54648 77 ~- = 0.0087266 = .31831 - = 2.86479 ? = 114.5915 77 77 77 = .63662 - = 3.18310 772 = 9.86960 77 77 = .95493 = 3.81972 = 0.1 01321 77 77 77^ = 1.27324 77 -^77 = 1.570796 VTT = 1 772453 5 j -y/l = 0.564189 = 1.59155 -77 = 1.047197 7T 77 3 = 1.90986 *77= 0.523599 Log 77= 0.49714987 77 6 Log \v-= 1.895090 4 Diam. in ins. = 13.5405 Varea in sq. ft. Area in sq. ft. = (diam. in inches) 2 x .0054542. D = diameter, R = radius, C = circumference, A A A = area. C =77Z>;= 277/2; = ~; = 2*77.4; = 3.545^ ; = Z> 2 x .7854 ; = = 4 x .7854 ; = 7 ~. >=-; =0.31831(7; ;=2V-r; = 1.12838*^1; R = x. ; = 0.159155C; = V - ; = 0.564189 ^A. 27r' w ' Areas of circles are to each other as the squares of their diameters. To find the length of an arc of a circle : RULE 1. As 360 is to the number of degrees in the arc, so is the circum- ference of the circle to the length of the arc. RULE 2. Multiply the diameter of the circle by the number of degrees in the arc, and this product by 0.0087266. 58 MENSURATION. Relations of Arc, Chord, Chord of Half the Arc, Versed Sine, etc. Let R = radius, D = diameter, Arc length of arc, Cd = chord of the arc, ch = chord of half the arc, F = versed sine, or height of the arc, Sch Cd . Vcd* + 4F* x 10 T a Arc = __ (very nearly), = 15C ^ + 33Fa + 2ch < Deai>lv ' 2ch x 10F , ArC = GQD-27V 4 ' Uea y ' Chord of the arc = 2 Vch*-V*; = VD* - (D- 2F)~2; = Sch - 3 Arc. = 2V J R2-(#-F) 2 ; _ 2 V(lT_ F) x F. Chord of half the arc, ch = \^ Cd* -f 4 F a ; = i ch* Diameter = -==; Versed sine 1 . (or -(D + v Z) 2 - Cd 2 ), if F is greater than radius Half the chord of the arc is a mean proportional between the versed sine and diameter minus versed sine: Y^Cd |/F x (D - F) Length of the Chord subtending an angle at the centre = twice the sine of halt the angle. (See Table of Sines, p. 15?..) Length of a Circular Arc. Huyghens's Approximation. Let C represent the length of the chord ot the arc and c the length of the chord of half the arc; the length of the arc 3 ' Professor Williamson shows that when the arc subtends an angle of 30, the radius being 100,000 feet (nearly 19 miles), the error by this formula is about two inches, or 1/600000 part of the radius. When the length of the arc is equal to the radius, i.e., when it subtends an angle of 57. 3, the error is less than 1/7680 part of the radius. Therefore, if the radius is 100,000 feet, the 100000 error is less than =13 feet. The error increases rapidly with the increase of the angle subtended. In the measurement of an arc which is described with a short radius the error is so small that it may be neglected. Describing an arc with a radius of 12 inches subtending an angle of 30, the error is 1/50000 of an inch. For 57. 3 the error is less than 0".0015. In order to measure an arc when it subtends a large angle, bisect it and measure each half as before in this case making B = length of the chord of half the arc, and b = length of the chord of one fourth the arc ; then T _ 166 - 2B ~F Relation of the Circle to its Equal, Inscribed, and Cir- cumscribed Squares. Diameter of circle x .88623 _ . , f , , miarp Circumference of circle x .28209 f = . equal squaie. Circumference of circle x 1.1284 = perimeter of equal square. THE ELLIPSE. 59 Diameter of circle x .7071 J Circumference of circle x .22508 v = side of inscribed square. Area of circle x .90031n- diameter j Area of circle x 1.273-2 = area of circumscribed square. Area of circle x .63662 = area of inscribed square. Side of square x 1.4142 = diarn. of circumscribed circle. x 4.4428 = circum. " " x 1.1284 = d jam. of equal circle. " x 3.5449 = circum. " " Perimeter of square x 0.88623 = Square inches x 1.2732 = circular inches. Sectors and Segments. To find the area of a sector of a circle. RULE 1. Multiply the arc of the sector by half its radius. RULE 2. As 360 is to the number of degrees in the arc, so is the area of the circle to the area of the sector. RULE 3. Multiply the number of degrees in the arc by the square of the radius and by .008727. To find the area of a segment of a circle: Find the area of the sector which has the same arc, and also the area of the triangle formed by the chord of the segment and the radii of the sector. Then take the sum of these areas, if the segment is greater than a semi- circle, but take their difference if it is less. Another Method ; Area of segment = ^>#(arc - sin A), in which A is the central angle, R the radius, and arc the length of arc to radius J To find the area of a segment of a circle when its chord and height only are given. First find radius, as follows : 1 P square of half the chord , radms = L height - + he ' 2. Find the angle subtended by the arc, as follows: half chord -f- radius = sine of half the angle. Take the corresponding angle from_a table of sines, and double it to get the angle of the arc. 3. Find area of the sector of which the segment is a part; area of sector = area of circle x degrees of arc -f- 360. 4. Subtract area of triangle under the segment]: Area of triangle = half chord x (radius height of segment). The remainder is the area of the segment. When the chord, arc, and diameter are given, to find the area. From the length of the arc subtract the length of the chord. Multiply the remainder by the radius or one-half diameter; to the product add the chord multiplied by the height, and divide the sum by 2. Given diameter, d, and height of segment, h. When h is from to Yd, area = h ^1.766d/i -1>; _ _ " " " " Y 4 d to y%d, area =-. h Vo.017d 2 + 1.7d/i- A a (approx.). Greatest error 0.23#, when h y^d. To-find the chord: From the diameter subtract the height ; multiply the remainder by four times the height and extract the square root. When the chords of the arc and of half the arc and the rise are given: Tc the chord of the arc add four thirds of the chord of half the arc; multiply the sum by the rise and the product by .40426 (approximate). Circular King:. To find the area of a ring included between the cir- cumferences of two concentric circles; Take the difference between the areas of the two circles; or, subtract the square of the less radius from the square of the greater, and multiply their difference by 3.14159. The area of the greater circle is equal to nRV; and the area of the smaller, wr 2 . Their difference, or the area of the ring, is TrCR 2 - r 2 ). Tlie Ellipse. Area of an ellipse = product of its semi-axes x 3.14159 = product of its axes x .785398. The Ellipse. Circumference (approximate) = 3.1416 V i , D and d being the two axes. Trautwine gives the following as more accurate: When the longer axis D is not more than five times the length of the shorter axis, d, 60 MENSUKATIOJT. Circumference = 3.1416 _ /i O.O When D is more than 5d, the divisor 8.8 is to be replaced by the following : ForD/d = 6 7 8 9 10 1:3 14 16 18 20 30 40 50 Divisor* = 9 9.3 9.3 9.35 9.4 9.5 9.6 9.68 9.75 9.8 9.92 9.98 10 / AI A^ A^ 2*1/48 \ An accurate formula is O = (o + 6) (l + + - + ~~ + ~^ . . . ) , in which A = -r-y. Ingenieurs Taschenbuch, 1896. Carl G. Barth (Machinery, Sept., 1900) gives as a very close approximation to this formula _, , ... 64 - 3^4* <*="< + > 64^1035- ^rea o/ a segment of an ellipse the base of which is parallel to one of the axes of the ellipse. Divide the height of the segment by the axis of which it is part, and ftnd the area of a circular segment, in a table of circu- lar segments, of which the height is equal to the quotient; multiply the area thus found by the product of the two axes of the ellipse. Cycloid. A curve generated by the rolling of a circle on a plane. Length of a cycloidal curve ~ 4 X diameter of the generating circle. Length of the base = circumference of the generating circle. Area of a cycloid = 3 X area of generating circle. Helix (Screw). A line generated by the progressive rotation of a point around an axis and equidistant from its centre. Length of a helix. -To the square of the circumference described by the generating-point add the square of the distance advanced in one revolution, and take the square root of their sum multiplied by the number of revolu- tions of the generating point. Or, V(c 2 + h"*)n = length, n being number of revolutions. Spirals. Lines generated by the progressive rotation of a point around a fixed axis, with a constantly increasing distance from the axis. A plane spiral is when the point rotates in one plane. A conical spiral is when the point rotates around an axis at a progressing distance from its centre, and advancing in the direction of the axis, as around a cone. Length of a plane spiral line. When the distance between the coils is uniform. RULE. Add together the greater and less diameters; divide their sum by 2; multiply the quotient by 3.1416, and again by the number of revolutions. Or, take the mean of the length of the greater and less circumferences and multiply it by the number of revolutions. Or, length = irn -~ -, d and d' being the inner and outer diameters. Length of a conical spiral line. Add together the greater and less diam- eters; divide their sum by 2 and multiply the quotient by 3.1416. To the square of the product of this circumference and the number of revolutions of the spiral add the square of the height of its axis and take the square root of the sum. Or, length = + h*. SOLID BODIES. Xlie Prism. To find the surface of a right prism : Multiply the perim- eter of the base by the altitude for the convex surface. To this add the areas of the two ends when the entire surface is required. Volume of a prism = area of its base X its altitude. The pyramid. Convex surface of a regular pyramid = perimeter of its base X half the slant height. To this add area of the base if the whole surface is required. Volume of a pyramid = area of base X one third of the altitude. SOLID BODIES. 61 To find the surface of a frustum of a regular pyramid : Multiply half the slant height by the sum of the perimeters of the two bases for the convex surface. To this add the areas of the two bases when the entire surface is required. To find the volume of a frustum of a pyramid : Add together the areas of the two bases and a mean proportional between them, and multiply the sum by one third of the altitude. (Mean proportional between two numbers = square root of their product.) Wedge. A wedge is a solid bounded by five planes, viz.: a rectangular base, two trapezoids, or two rectangles, meeting in an edge, and two tri- angular ends. The altitude is the perpendicular drawn from any point in the edge to the plane of the base. To find the volume of auiedge: Add the length of the edge to twice the length of the base, and multiply the sum by one sixth of the product of the height of the wedge and the breadth of the base. Rectangular prismoid. A rectangular prismoid is a solid bounded by six planes, of which the two bases are rectangles, having their corre- sponding sides parallel, and the four upright sides of the solids are trape- zoids. To find the volume of a rectangular prismoid : Add together the areas of the two bases and four times the area of a parallel section equally distant from the bases, and multiply the sum by one sixth of the altitude. Cylinder. Convex surface of a cylinder = perimeter of base X altitude. To this add the areas of the two ends when the entire surface is required. Volume of a cylinder = area of base X altitude. Cone. Convex surface of a cone = circumference of base X half the slant side. To this add the area of the base when the entire surface is required. Volume of a cone = area of base X one third of the altitude. To find the surface of a frustum of a cone: Multiply half the side by the sum 6t the circumferences of the two bases for the convex surface; to this add the areas of the two bases when the entire surface is required. To find the volume of a frustum of a cone : Add together the areas of the two bases and a mean proportional between them, and multiply the sum by one third of the altitude. Or, Vol. = 0.2618a(b a + c 3 + be) ; a = altitude : 6 and c, diams. of the two bases. Sphere. To find the surface of a sphere : Multiply the diameter by the ciicumference of a great circle; or, multiply the square of the diameter by 3.14159. Surface of sphere = 4 X area of its great circle. *' *' " = convex surface of its circumscribing cylinder. Surfaces of spheres are to each other as the squares of their diameters. To find the volume of a sphere : Multiply the surface by one third of the radius; or, multiply the cube of the diameter by 7r/6; that is, by 0.5236. Value of TT/O to 10 decimal places = .5235987756. The volume of a sphere = 2/3 the volume of its circumscribing cylinder. Volumes of spheres are to each other as the cubes of their diameters. Spherical triangle. To find the area of a splierical triangle : Com- pute the surface of the quadrantal triangle, or one eighth of the surface of the sphere. From the sum of the three angles subtract two right angles; divide the remainder by 90, and multiply the quotient by the area of the quadrantal triangle. Spherical polygon. To find the area of a spherical polygon: Com- pute the surface of the quadrantal triangle. From the sum of all the angles subtract the product of two right angles by the number of sides less two; divide the remainder by 90 and multiply the quotient by the area of the quadrantal triangle. The prismoid. The prismoid is a solid having parallel end areas, and may be composed of any combination of prisms, cylinders, wedges, pyra- mids, or cones or frustums of the same, whose bases and apices lie in the end areas. Inasmuch as cylinders and cones are but special forms of prisms and pyramids, and warped surface solids may be divided into elementary forms of them, and since frustums may also be subdivided into the elementary forms, it is sufficient to say that all prismoids may be decomposed into prisms, wedges, and pyramids. If a formula can be found which is equally applicable to all of these forms, then it will apply to any combination of them. Such a formula is called MENSURATION-. The Prismoidal Formula. Let A area of the base of a prism, wedge, or pyramid; %, Am = the two end and the middle areas of a prismoid, or of any of its elementary solids; h = altitude of the prismoid or elementary solid; V its volume; For a prism, A t , Am and A z are equal, = A ; V=-x6A = liA. For a wedge with parallel ends, A^ = 0, Am = -A l ; V ^(A l -f ZAJ = - For a cone or pyramid, A% = 0, Am = -.A^ V = ~(A } + AJ - ~. 4 O O The prismoidal formula is a rigid formula for all prismoids. The only approximation involved in its use is in the assumption that the given solid may be generated by a right line moving over the boundaries of the end areas. The area of the middle section is never the mean of the two end areas if the prismoid contains any pyramids or cones among its elementary forms. When the three sections are similar in form the dimensions of the middle area are always the means of the corresponding end dimensions. This fact often enables the dimensions, and hence the area of the middle section, to be computed from the end areas. Polyedrons. A polyedron is a solid bounded by plane polygons. A regular pblyedrou is one whose sides are all equal regular polygons. To find the surface of a regular polyedron. Multiply the area of one of the faces by the number of faces ; or, multiply the square of one of the edges by the surface of a similar solid whose edge is unity. A TABLE OF THE REGULAR POLYEDRONS WHOSE EDGES ARE UNITY. Names. No. of Faces. Surface. Volume. Tetraedron .......................... 4 3.7320508 0.1178513 Hexaedron. .' ......................... 6 6.0000000 3 .0000000 Octaeclron ........................... 8 3. 4641016 0.4714045 Dodecaedron ......................... 12 20.6457288 7.6631189 Icosaedron ........................... 20 8.GCO-J540 2.1816950 To find the volume of a regular polyedron. Multiply the surface by one third of the perpendicular let fall from the centre on one of the faces ; or, multiply the cube of one of the edges by the solidity of a similar polyedron whose edge is unity. Solid of revolution. The volume of any solid of revolution is equal to the product of the area of its generating surface by the length of the path of the centre of gravity of that surface. The convex surface of any solid of revolution is equal to the product of the perimeter of its generating surface by the length of path of its centre of gravity. Cylindrical ring. Let d = outer diameter ; d' inner diameter ; - (d d') = thickness = t\ -irf 2 = sectional area ; ~(d-\-d') = mean diam- eter = M ; TT t = circumference of section ; irM mean circumference of ring; surface = TT t X * M; = ^ 772 (d 2 - d' 2 ); = 9.86965 1 M- = 2.46741 (d 2 -d' 2 ); volume = 7 TT t* M TT; = 2.46741*2 M. 4 Spherical zone. Surface of a spherical zone or segment of a sphere = its altitude X the circumference of a great circle of the sphere. A great circle is one whose plane passes through the centre of the sphere. Volume of a zone of a sphere. To the sum of the squares of the radii of the ends add one third of the square of the height ; multiply the sum by the height and by 1.5708. Spherical segment. Volume of a spherical segment with one base. SOLID BODIES. 63 i Multiply half the height of the segment by the area of the base, and the cube of the height by .5236 and add the two products. Or, from three times the diameter of the sphere subtract twice the height of the segment; multi- ply the difference by the square of the height and by .5236. Or, to three times the square of the radius of the base of the segment add the square of its height, and multiply the sum by the height and by .5236. Spheroid or ellipsoid. When the revolution of the spheroid is about : the transverse diameter it is prolate, and when about the conjugate it is oblate. Convex surface of a segment of a spheroid. Square the diameters of the : spheroid, arid take the square root of half their sum ; then, as the diameter ; from which the segment is cut is to this root so is the height of the segment to the proportionate height of the segment to the mean diameter. ; Multiply the product of the other diameter and 3.1416 by the proportionate ? height. Convex surface of a frustum or zone of a spheroid. Proceed as by t previous rule for the surface of a segment, and obtain the proportionate I height of the frustum. Multiply the product of the diameter parallel to the base of the frustum and 3.1416 by the proportionate height of the frustum. Volume of a spheroid is equal to the product of the square of the revolving axis by the fixed axis and by .5236. The volume of a spheroid is two thirds i. of that of the circumscribing cylinder. Volume of a segment of a spheroid.]. When the base is parallel to the i revolving axis, multiply the difference between three times the fixed axis and twice the height of the segment, by the square of the height and by ; .5236. Multiply the product by the square of the revolving axis, and divide | by the square of the fixed axis. 2. When the base is perpendicular to the revolving axis, multiply the ! difference between three times the revolving axis and twice the height of the segment by the square of the height and by .5236. Multiply the product by the 'length of the fixed axis, and divide by the length of the revolving axis. Volume of the middle frustum of a spheroid.]. When the ends are circular, or parallel to the revolving axis : To twice the square of the middle diameter add the square of the diameter of one end ; multiply the sum by the length of the frustum and by .2618. 2. When the ends are elliptical, or perpendicular to the revolving axis: To twice the product of the transverse and conjugate diameters of the middle section add the product of the transverse and conjugate diameters of one end ; multiply the sum by the length of the frustum and by .2618. Spindles. Figures generated by the revolution of a plane area, when the curve is revolved about a chord perpendicular to its axis, or about its double ordinate. They are designated by the name of the arc or curve from which they are generated, as Circular, Elliptic, Parabolic, etc., etc. Convex surface of a circular spindle, zone, or segment of it Rule: Mul- tiply the length by the radius of the revolving arc; multiply this arc by the central distance, or distance between the centre of the spindle and centre of the revolving arc ; subtract this product from the former, double the remainder, and multiply it by 3.1416. Volume of a circular spindle. Multiply the central distance by half the area of the revolving segment; subtract the product from one third of the cube of half the length, and multiply the remainder by 12.5664. Volume of frustum or zone of a circular spindle. From the square of half the length of the whole spindle take one third of the square of half the length ofj the frustum, and multiply the remainder by the said half length of the frustum ; multiply the central distance by the revolving area which generates the frustum ; subtract this product from the former, and multi- ply the remainder by 6.2832. Volume of a segment of a circular spindle. Subtract the length of the segment from the half length of the spindle ; double the remainder and ascertain the volume of a middle frustum of this length ; subtract the result from the volume of the whole spindle and halve the remainder. Volume of a cycloidal spindle = five eighths of the volume of the circum- scribing cylinder. Multiply the product of the square of twice the diameter of the generating circle and 3.927 by its circumference, and divide this pro- duct by 8. Parabolic -conoid. Volume of a parabolic conoid (generated by the revolution of a parabola on its axis). Multiply the area of the base by half the height. 64 MENSURATION. Or multiply the square of the diameter of the base by the height and by Volume of a frustum of a parabolic conoid. Multiply half the sum of the areas of the two ends by the height. Volume of a parabolic spindle (generated by the revolution of a parabola on its base). Multiply the square of the middle diameter by the length and by .4189. The volume of a parabolic spindle is to that of a cylinder of the same height and diameter as 8 to 15. Volume of the middle frustum of a parabolic spindle. Add together 8 times the square of the maximum diameter, 3 times the square of the end diameter, and 4 times the product of the diameters. Multiply the sum by the length of the frustum and by .05236. This rule is applicable for calculating, the content of casks of parabolic form. Casks. To find the volume of a cask of any form. Add together 39 times the square of the bung diameter, 25 times the square of the head diameter, and 26 times the product of the diameters. Multiply the sum by the length, and divide by 31,773 for the content in Imperial gallons, or by 26,470 for U. S. gallons. This rule was framed by Dr. Hutton, on the supposition that the middle third of the length of the cask was a frustum of a parabolic spindle, and each outer third was a frustum of a cone. To find the ullage of a cask, the quantity of liquor in it when it is not full. 1. For a lying cask : Divide the number of wet or dry inches by the bung diameter in inches. If the quotient is less than .5, deduct from it one fourth part of what it wants of .5. If it exceeds .5, add to it one fourth part of the excess above .5. Multiply the remainder or the sum by the whole content of the cask. The product is the quantity of liquor in the cask, in gallons, when the dividend is wet inches; or the empty space, if dry inches. 2. For a standing cask : Divide the number of wet or dry inches by the length of the cask. If the quotient exceeds .5, add to it one tenth of its excess above .5; if less than .5, subtract from it one tenth of what it wants of .5. Multiply the sum or the remainder by the whole content of the cask. The product is the quantity of liquor in the cask, when the dividend is wet inches; or the empty space, if dry inches. Volume of cask (approxiimite) U. S. gallons = square of mean diam. X length in inches X .0034. Mean diam. = half the sum of the bung and. head diams. Volume of an irregular solid. Suppose it divided into parts, resembling prisms or other bodies measurable by preceding rules. Find the content of each part; the sum of the contents is the cubic contents of the solid. The content of a small part is found nearly by multiplying half the sum of the areas of each end by the perpendicular distance between them. The contents of small irregular solids may sometimes be found b)' im- mersing them under water in a prismatic or cylindrical vessel, and observ- ing the amount by which the level of the water descends when the solid is withdrawn. The sectional area of the vessel being multiplied by the descent of the level gives the cubic contents. Or, weigh the solid in air and in water; the difference is the weight of water it displaces. Divide the weight in pounds by 62.4 to obtain volume in cubic feet, or multiply it by 27.7 to obtain the volume in cubic inches. When the solid is very large and a great degree of accuracy is not requisite, measure its length, breadth, and depth in several ( itferent places, and take the mean of the measurement for each dimension, and multiply the three means together. When the surface of the solid is very extensive it is better to divide it into triangles, to find the area of each triangle, and to multiply it by the mean depth of the triangle for the contents of each triangular portion; the contents of the triangular sections are to be added together, The mean depth of a triangular section is obtained by measuring the depth at each angle, adding together the. three measurements, and taking one third of the sum. PLANE TRIGONOMETRY, 65 PLANE TRIGONOMETRY. Trigonometrical functions. Every triangle has six parts three angles and three sides. When any three of these parts are given, provided one of them is a side, the other parts may be determined. By the solution of a triangle is meant the deter- mination of the unknown parts of a triangle when certain parts are given. The complement of an angle or arc is what remains after subtracting the angle or arc from 90. In general, if we represent any arc by A, its complement is 90 A. Hence the complement of an arc that exceeds 90 is negative. Since the two acute angles of a right-angled triangle are together equal to a right angle, each of them is the complement of the other. The supplement of an angle or arc is what remains after subtracting the angle or arc from 180. If A is an arc its supplement is 180 A. The sup- plement of an arc that exceeds 180 is negative. The sum of the three angles of a triangle is equal to 180. Either angle is the supplement of the other two. In a right-angled triangle, the right angle being equal to 90, each of the acute angles is the complement of the other. In all right-angled triangles having the same acute angle, the sides have to each other the same ratio. These ratios have received special names, as follows: If A is one of the acute angles, a the opposite side, b the adjacent side, and c the hypothenuse. The sine of the angle A is the quotient of the opposite side divided by the a hypothenuse. Sin. A = - The tangent of the angle A is the quotient of the opposite side divided by the adjacent side. Tang. A = The secant of the angle A is the quotient of the hypothenuse divided by c the adjacent side. Sec. A = jf The cosine, cotangent, and cosecant of an angle are respec- tively the sine, tangent, and secant of the complement of that angle. The terms sine, cosine, etc., are called trigonometrical functions. In a circle whose radius is unity, the sine of an arc, or of the angle at the centre measured by thai arc, is the perpendicular let fall from one extrem- ity of the arc upon the diameter passing through the other extremity. The tangent of an arc is the line which touches the circle at one extrem- ity of the arc, and is limited by the diameter ( produced) passing through the other extremity. The secant of an arc is that part of the produced diameter which is intercepted beticeen the centre and the tangent. The versed sine of an arc is that part of the diameter intercepted between the extremity of the arc and the foot of the sine. In a circle whose radius is not unity, the trigonometric functions of an arc will be equal to the lines here defined, divided by the radius of the circle. It 1C A (Fig. 70) is an angle in the first quadrant, and C F= radius, FG , Oft KF The sine of the angle = = r - r . Cos = ,-= r = - Rad. Had. Had. IA CI ' DL Tan S- = Rad/ Secant = Rad/ Cot = Ral." CL ._ . GA Rad/ If radius is 1, then Rad. in the denominator is omitted, and sine = F G, etc. The sine of an arc = half the chord of twice the arc. The sine of the supplement of the arc is the same as that of the arc itself. Sine of arc B D F = F G = sin arc FA. Fi. 70, 66 PLANE TRIGONOMETRY* The tangent of the supplement is equal to the tangent of the arc, but with a contrary sign. Tang. B D F = B M. The secant of the supplement is equal to the secant of the arc, but with a contrary sign. Sec. B D F = CM. Signs of the functions in the four quadrants. If we divide a circle into four quadrants by a vertical and a horizontal diame- ter, the upper right-hand quadrant is called the first, the upper left the sec- ond, the lower left the third, and the lower right the fourth. The signs of the functions in the four quadrants are as follows: First quad. Second quad. Third quad. Fourth quad. Sine and cosecant, + + Cosine and secant, + Tangent and cotangent, + + The values of the functions are as follows for the angles specified: Ajigle..,.., , 30 45 60 90 120 135 150 180 270 360 1 1 y o Vs 1 1 Sine 1 (I -1 2 Vg 2 2 ^2 2 V3 1 1 1 1 i/3 Cosine 1 _ -1 1 2 ^2 2 Vo 2 Tangent 1 1 V* 00 -VI 1 V8 j V3 00 Cotangent 1 1 *3 Va ~yo 1 ~ | 7 3 00 00 Secant 1 2 ^2 8 00 -2 i'o 2 -1 OC 1 1/3 o g 1/3 Cosecant QO 2 y 'o 71 1 V2 2 oc -1 CO l^ersed sine 2- ^3 4/o 1 i j 3 V*+i 2+1/3 2 2 2 2 ^ 2 TRIGONOMETRICAL. FORJNLULJE. The following relations are deduced from the properties of similar tri- angles (Radius = 1): cos A : sin A :: 1 : tan A, whence tan A = sin A : cos A :: 1 : cot A, cos A : 1 :: 1 : sec A, cos -4' cos^l cotan A = -. - sin A sec A = sin A : 1 tan A : 1 1 : cosec A, " cosec A = _ cos A 1 sin A* The sum of the square of the sine of an arc and the square of its c'osine equals unity. Sin 2 A -\- cos 2 A = 1. Also, 1 -ftan 2 ^ = sec* A: 1 + cot 2 A = cosec 5 A. Functions ot the sum and difference of two angles : Let the two angles be denoted by A and B, their sum A -f B = C, and their difference A - B by D. ) = sin u4 cos B -\- cos A sin B; (1) TRIGONOMETRICAL FORMULAE. 67 cos (A -f- J5) = cos A cos B sin .4 sin P; . . . . . (2) sin ( A B) = sin A cos 5 cos A sin 5; ..... (3) cos (A B) = cos A cos . + sm -4 sin P ...... (4) From these four formulae by addition and subtraction we obtain sin (A + B) -f sin (A - B) = 2 sin A cos B\ ..... (5) sin U + B) - sin U - 5) = 2cos ^ sin 5; ..... (6) cos (A + B) 4- cos (4 - B) = 2 cos ^. cos B; ..... (7) cos (A B) - cos (A + 5) = 2 sin A sin 5 ...... (8) If we put A 4- B = C, and ^ - P = A then ^ = ^74- >) and B = Y%(C - Z>), and we have sin C 4 sin D = 2 sin }(C 4- D) cos ^(C f - D); .... (9) sin C - sin D = 2 cos ^7 -f D) sin y 2 (C - D); . . . . (10) cosC-f cosZ) = 2cos^(C+D)cos^(<7 - D); . . . . (11) cos D - cos (7 = 2 sin ^((7 + D) sin ^(C - D) ..... (12) Equation (9) may be enunciated thus: The sum of the sines of any two angles is equal to twice the sine of half the sum of the angles multiplied by the cosine of half their difference. These formulae enable us to transform a sum or difference into a product. The sum of the sines of two angles is to their difference as the tangent of half the sum of those angles is to the tangent of half their difference. sin A 4 sin B _ 2 sin \fljA + B) cos y%(A - B) _ tan %(A -f- B) siu A - sin B ~ 2 cos %>(A + B) sin %>(A - B)~ tan %(A - B)' The sum of the cosines of two angles is to their difference as the cotangent of half the sum of those angles is to the tangent of half their difference. cos A + cos B _ 2 cos y 2 (A + B) cos \^(A - B) _ cot ^(A + B) eos B cos A ~~ 2 sin %(A + B) sin y^(A - B) ~ tan fflA - W The sine of the sum of two angles is to the sine of their difference as the sum of the tangents of those angles is to the difference of the tangents. sin (A + B) _ tan A + tan B m sin (A - B) tan A - tanJ5' ' = tan ^ | t tan .1 + tang cos A cos B sin (A - B) cos A cos B cos (A 4- B) cos ^1 cos B cos (A B) cos A cos .# = tan A - tan B; = 1 tan JL tan . = 1 + tan ^1 tan 5; tan (A-B) = cot U + JB) = 1 tan A tan J?' tan A - tan B ^ 1 4~ tan .4 tan B' cot ^4 cot j? 1 m cot #4- cot A ' cot U - B) = y cot J5 cot A ' Functions of t \vice an angle : sin 2 A = 2 sin A cos A ; tan 2A = 2 tan A 1 - tan 2 A* cos 2A = cos 2 .4 sin 2 A\ cot 2.4 = cot 2 A I 2 cot .4 Functions of naif an angle : sin \4>A = 1 + cos J. ' cos \&A = 68 PLANE TRIGONOMETRY. Solution oi Plane Right-angled Triangles. Let A and B be the two acute angles and C the right angle, and a, 6, and c the sides opposite these angles, respectively, then we have 1. sin A = cosB ; 3. tan J. = cot = ; 2. cos A = s'mB s" -: 4. cot jt = tan B = - c a 1. In any plane right-angled triangle the sine of either of the acute angles is equal to the quotient of the opposite leg divided by the hypothenuse. 2. The cosine of either of the acute angles is equal to the quotient of the adjacent leg divided by the hypothenuse. 3. The tangent of either of the acute angles is equal to the quotient of the opposite leg divided by the adjacent leg. 4. The cotangent of either of the acute angles is equal to the quotient of the adjacent leg divided by the opposite leg. 5. The square of the hypothenuse equals the sum of the squares of the other two sides. Solution of Oblique-angled Triangles. The following propositions are proved in works on plane trigonometry. In any plane triangle Theorem 1. The sines of the angles are proportional to the opposite sides. Theorem 2. The sum of any two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their differ- ence. Theorem 3. If from any angle of a triangle a perpendicular be drawn to the opposite side or base, the whole base will be to the sum of the other two sides as the difference of those two sides is to the difference of the segments of the base. CASE I. Given two angles and a side, to find the third angle and the other two sides. 1. The third angle = 180 sum of the two angles. 2. The sides may be found by the following proportion : The sine of the angle opposite the given side is to the sine of the angle op- posite the required side as the given sde is to the required side. CASE II. Given two sides and an angle opposite one of them, to find the third side and the remaining angles. The side opposite the given angle is to the side opposite the required angle as the sine of the given angle is to the sine of the required angle. The third angle is found by subtracting the sum of the other two from 180, and the third side is found as in Case I. CASE III. Given two sides and the included angle, to find the third side and the remaining angles. The sum of the required angles is found by subtracting the given angle from 180. The difference of the required angles is then found by Theorem II. Half the difference added to half the sum gives the greater angle, and half the difference subtracted from half the sum gives the less angle. The third side is then found by Theorem I. Another method : Given the sides c, b, and the included angle A, to find the remaining side a and the remaining angles B and G. From either of the unknown angles, as B, draw a perpendicular B e to the opposite side. Then Ae = ccosA, Be = csinA, eC=b - Ae, B e-t- e C = ten C. Or, in other words, solve B <?, A e and B e C as right-angled triangles. CASE IV. Given the three sides, to find the angles. Let fall a perpendicular upon the longest side from the opposite angle, dividing the given triangle into two right-angled triangles. The two seg- ments of the base may be found by Theorem III. There will then be given the bypothenuse and one side of a right-angled triangle to find the angles. For areas of triangles, see Mensuration. r V' ANALYTICAL GEOMETRY. 69 ANALYTICAL GEOMETRY. Analytical geometry is that branch of Mathematics which has for its object the determination of the forms and magnitudes of geometrical magnitudes by means of analysis. Ordinates and abscissas. In analytical geometry two intersecting lines YY', XX' are used as coordinate axes^ XX' being the axis of abscissas or axis of Jf, and YY' the axis of ordinates or axis of Y. A. the intersection, is called the origin of co- ordinates. The distance of any point P from the axis of Y measured parallel to the axis of X is called the abscissa of the point, as AD or' CP, Fig. 71. Its distance from the axis of X, measured parallel to the axis of Y, is called the ordinate, as AC or PD. The abscissa and ordinate taken together are called the coor- dinates of the point P. The angle of intersec- tion is usually taken as a right angle, in which JT IG 7^ case the axes of X and Fare called rectangu- lar coordinates. The abscissa of a point is designated by the letter x and the ordinate by y. The equations of a point are the equations which express the distances of the point from the axis. Thus x = a,y = b are the equations of the point P. Equations referred to rectangular coordinates. The equa- tion of a line expresses the relation which exists between the coordinates of every point of the line. Equation of a straight line, y = ax b, in which a is the tangent of the angle the line makes with the axis of X, and b the distance above A in which the line cuts the axis of Y. Every equation of the first degree between two variables is the equation of ft straight line, as Ay 4- Bx 4- C = 0, which can be reduced to the form y = ax b. Equation of the distance between two points: in which x'y'^ x"y" are the coordinates of the two points. Equation of a line passing through a given point : y - y' = a(x #'), in which x'y' are the coordinates of the given point, a, the tangent of the angle the line makes with the axis of x, being undetermined, since any num- ber of lines may be drawn through a given point. Equation of a line passing through two given points : Equation of a line parallel to a given line and through a given point; y - y' = a(x - x'\ Equation of an angle V included between two given lines: a' a im % v =T+tt in which a and a' are the tangents of the angles the lines make with the axis of abscissas. If the lines are at right angles to each other tang V = oo, and 1 + a'a - 0. Equation of an intersection of two lines, whose equations are y = ax -f b, and y = a'x + &', b - b' ab' - a'b x = and y = . a - a" * a - a ; ' 70 ANALYTICAL GEOMETRY. Equation of a perpendicular from a given point to a given line: Equation of the length of the perpendicular Pi p _ y' - ax' - b yTT& The circle. Equation of a circle, the origin of coordinates being at the centre, and radius = R : If the origin is at the left extremity of the diameter, on the axis of X: y* = 2Rx - it* 2 . If the origin is at any point, and the coordinates of the centre are x'y' : (x - x')* + (y- 2/') 2 = &. Equation of a tangent to a circle, the coordinates of the point of tangency being x"y" and the origin at the centre, Tlie ellipse. Equation of an ellipse, referred to rectangular coordi- nates with axis at the centre: A i y * 4. #2^ = ^apa, in which A is half the transverse axis and B half the conjugate axis. Equation of the ellipse when the origin is at the vertex of the transverse axis: The eccentricity of an ellipse is the distance from the centre to either focus, divided by the semi-transverse axis, or The parameter of an ellipse is the double ordinate passing through the focus. It is a third proportional to the transverse axis and its conjugate, or 2B* %A : 2B :: 2B : parameter; or parameter = . Any ordinate of a circle circumscribing an ellipse is to the corresponding ordinate of the ellipse as the semi-transverse axis to the semi-conjugate. Any ordinate of a circle inscribed in an ellipse is to the corresponding ordi- nate of the ellipse as the semi-conjugate axis to the semi-transverse. Equation of the tangent to an ellipse, origin of axes at the centre : A*yy" -f B^xx" = A*B*, y"x" being the coordinates of the point of tangency. Equation of the normal, passing through the point of tangency, and per- pendicular to the tangent: The normal bisects the angle of the two lines drawn from the point of tangency to the foci. The lines drawn from the foci make equal angles with the tangent. Tlae parabola. Equation of the parabola referred to rectangular coordinates, the origin being at the vertex of its axis, y* = 2px, in which 2p is the parameter or double ordinate through the focus. ANALYTICAL GEOMETRY. 71 The parameter is a third proportional to any abscissa and its corresponding ordinate, or x :y :iy:2p. Equation of the tangent: yy" - p(x -f x"), y''x' f being coordinates of the point of tangency. Equation of the normal: The sub-normal, or projection of the normal on the axis, is constant, and equal to half the parameter. The tangent at any point makes equal angles with the axis and with the line drawn from the point of tangency to the focus. The hyperbola. Equation of the hyperbola referred to rectangular coordinates, origin at the centre: A*y* - B*x* = - -4 2 B 2 , in which A is the semi-transverse axis and B the semi-conjugate axis. Equation when the origin is at the right vertex of the transverse axis: Conjugate and equilateral hyperbolas. If on the conjugate axis, as a transverse, and a focal distance equal to \fA* -\- # 2 , we construct the two branches of a hyperbola, the two hyperbolas thus constructed are called conjugate hyperbolas. If the transverse and conjugate axes are equal, the hyperbolas are called equilateral, in which case y* # 2 = A* when A is the transverse axis, and # a 2/ 2 = B* when B is the trans- verse axis. The parameter of the transverse axis is a third proportional to the trans- verse axis and its conjugate. 2A : 2B : : 2B : parameter. The tangent to a hyperbola bisects the angle of the two lines drawn from the point of tangency to the foci. The asymptotes of a tiyperbola are the diagonals of the rectangle described on the axes, indefinitely produced in both directions. In an equilateral hyperbola the asymptotes make equal angles with the transverse axis, and are at right angles to each other. The asymptotes continually approach the hyperbola, and become tangent to it at an infinite distance from the centre. Conic sections, Every equation of the second degree between two variables will represent either a circle, an ellipse, a parabola or a hyperbola. These curves are those which are obtained by intersecting the surface of a cone by planes, and for this reason they are called conic sections. Logarithmic curve. A logarithmic curve is one in which one of tho coordinates of any point is the logarithm of the other. The coordinate axis to v hich the lines denoting the logarithms are parallel is called the axis of logarithms, and the other the axis of numbers. If y is the axis of logarithms and x the axis of numbers, the equation of the curve is y = log x. If the base of a system of logarithms is a, we have a y = x, in which y is the logarithm of x. Each system of logarithms will give a different logarithmic curve. If y = 0, x = 1. Hence every logarithmic curve will intersect the axis of numbers at a distance from the origin equal to 1. 72 DIFFERENTIAL CALCULUS. DIFFERENTIAL CALCULUS. The differential of a variable quantity is the difference between any two of its consecutive values; hence it is indefinitely small. It is expressed by writing d before the quantity, as dx, which is read differential of x. The term - is called the differential coefficient of y regarded as a func- Q.X tion of x. The differential of a function is equal to its differential coefficient mul- tiplied by the differential of the independent variable; thus, -J^dx = dy. The limit of a variable quantity is that value to which it continually approaches, so as at last to differ from it by less than any assignable quan- tity. The differential coefficient is the limit of the ratio of the increment of the independent variable to the increment of the function. The differential of a constant quantity is equal to 0. The differential of a product of a constant by a variable is equal to the constant multiplied by the differential of the variable. If u = Av, du = Adv. In any curve whose equation is y=f(x), the differential coefficient = tan a; hence, the rate of increase of the function, or the ascension of dx the curve at any point, is equal to the tangent of the angle which the tangent line makes with the axis of .abscissas. All the operations of the Differential Calculus comprise but two objects: 1. To find the rate of change in a function when it passes from one state of value to another, consecutive with it. 2. To find the actual change in the function : The rate of change is the differential coefficient, and the actual change the differential. Differentials of algebraic functions. The differential of the sum or difference of any number of functions, dependent on the same variable, is equal to the sum or difference of their differentials taken sepa- rately : If u = y -{- z w, du dy -\- dz dw. The differential of a product of two functions dependent on the same variable is equal to the sum of the products of each by the differential of the other : d(uv) du , dv d(tti>) - vdu + udv. _-=_ + _. The differential of the product of any number of functions is equal to the sum of the products which arise by multiplying the differential of each function by the product of all the others: d(uts) = tsdu -f usdt 4- uids. The differential of a fraction equals the denominator into the differential of the numerator minus the numerator into the differential of the denom- inator, divided by the square of the denominator : /u\ vdu udv dt = d {J = . If the denominator is constant, dv = 0, and dt 5- = . v v If the numerator is constant, du = 0, and dt = The differential of the square root of a quantity is equal to the differen tial of the quantity divided by twice the square root of the quantity: If v u^, or v = 4/w, dv = ; 2 Vu DIFFEREHTIAL CALCULUS. 73 The differential of any power of a function is equal to the exponent multi- plied by the function raised to a power less one, multiplied by the differen- tial of the function, d(u n } = nu n - 1 du. Formula* for differentiating algebraic functions. 1. d (a) = 0. 2. d (ax) = adx. ry A i n yn\ _ ~)ix m dX. dx ydx - xdy 5. d (xy) = xdy + ydx. To find the differential of the form u = (a + bx n ) m : Multiply the exponent of the parenthesis into the exponent of the varia- ble within the parenthesis, into the coefficient of the variable, into the bi- nomial raised to a power less 1, into the variable within the parenthesis raised to a power less 1, into the differential of the variable. du = d(a 4 bx n ) m = mnb(a + bx 1l ) m ~ 1 x n ~ l dx. To find the rate of change for a given value of the variable : Find the differential coefficient, and substitute the value of the variable in the second member of the equation. EXAMPLE. If x is the side of a cube and u its volume, u = x 9 , - = &e 2 . Hence the rate of change in the volume is three times the square of the edge. If the edge is denoted by 1, the rate of change is 3. Application. The coefficient of expansion by heat of the volume of a body is three times the linear coefficient of expansion. Thus if the side of a cube expands .001 inch, its volume expands .003 cubic inch. 1.001 s = 1.003003001. A partial differential coefficient is the differential coefficient of a function of two or more variables under the supposition that only one of them has changed its value. A partial differential is the differential of a function of two or more vari- ables under the supposition that only one of them has changed its value. The total differential of a function of any number of variables is equal to the sum of the partial differentials. If uf(xy\ the partial differentials are -^dx, ~dy. dx dy Itu = x* + y*-z,du = ^dx 4- d ~dy 4 ^dz\ = 2xdx + 3y*dy-dz. ax ay dz Integrals. An integral is a functional expression derived from a differential. Integration is the operation of finding the primitive function from the differential function. It is indicated by the sign /, which is read ** the integral of." ThusfZxdx = x"* ; read, the integral of 2xdx equals x-. To integrate an expression of the form mx m ~ 1 dx or x m dx, add 1 to the exponent of the variable, and divide by the new exponent and by the differ- ential of the variable: f3x"*dx = x 3 . (Applicable in all cases except wheu 1. Forjx dx see formula 2 page 78.) The integral of the product of a constant by the differential of a vari- able is equal to the constant multiplied by the integral of the differential: fax m dx = a/x m dx = a - x m + l . J m-f 1 The integral of the algebraic sum of any number of differentials is equal to the algebraic sum of their integrals: du = 2ax*dx - bydy - z*dz; fda = ao; 3 - y* - . & 6 O Since the differential of a constant is 0, a constant connected with a vari- able by the sign + or - disappears in the differentiation; thus d(a + x) = dx m = mx m ~ l dx. Hence in integrating a differential expression we must 74 DIFFERENTIAL CALCULUS. annex to the integral obtained a constant represented by C to compensate for the term which may have been lost in differentiation. Thus if we have dy = adx\ Jdy = afdx. Integrating, y = ax C. The constant (7, which is added to the first integral, must have such a value as to render the functional equation true for every possible value that may be attributed to the variable. Hence, after having found the first integral equation and added the constant C, if we then make the variable equal to zero, the value which the function assumes will be the true valus of C. An indefinite integral is the first integral obtained before the value of the constant C is determined. A particular integral is the integral after the value of Chas been found. A definite integral is the integral corresponding to a given value of the variable. Integration "between limits e Having found the indefinite inte- gral and the particular integral, the next step is to find the definite integral, and then the definite integral between given limits of the variable. The integral of a function, taken between two limits, indicated by given values of a?, is equal to the difference of the definite integrals correspond- ing to those limits. The expression /W / / dy = a I dx Jx' J is read: Integral of the differential of ?/, taken between the limits x' and x"' the least limit, or the limit corresponding to the subtractive integral, being placed below. Integrate du = Qx^dx between the limits x = 1 and x = 3, u being equal tc 81 when x = 0. fdu = fQx' 2 dx = 3# 3 + <?; C = 81 when x = 0, then -. = 3 du = 3(3)3 _j_ gl, minus 3(1) 3 -f- 81 = 78. x = 1 Integration of particular forms. To integrate a differential of the form du - (a-f- bx n ) m x n ~ *dx. 1. If there is a constant factor, place it without the sign of the integral, and omit the power of the variable without the parenthesis and the differ ential; 2. Augment the exponent of the parenthesis by 1, and then divide this quantity, with the exponent so increased, by the exponent of the paren- thesis, into the exponent of the variable within the parenthesis, into the co- efficient of the variable. Whence J. (m -f l)nb Tlie differential of an arc is the hypothenuse of a right-angle triangle of which the base is dx and the perpendicular dy. If z is an arc, dz = Vdx* + d?/ 2 Quadrature of a plane figure. T/ie differential of the area of a plane surf ace is equal to the ordinate into the differential of the abscissa. da = ydx. To apply the principle enunciated in the last equation, in finding the area of any particular plane surface : Find the value of y in terms of x. from the equation of the bounding line; substitute this value in the differential equation, and then integrate between the required limits of x. Area of the parabola, Find the area of any portion of the com- mon parabola whose equation is yi = 2px' t whence y = ^2px. DIFFEKENTIAL CALCULUS. 75 Substituting this value of y in the differential equation ds = ydx gives P / ds = I \/2pxdx = |/^p / x^dx = ^ xl -f C\ Tf we estimate the area from the principal vertex, x = 0. y = 0, and (7=0; and denoting the particular integral by s', s' = r y. o That is, the area of any portion of the parabola, estimated from the ver- tex, is equal to % of the rectangle of the abscissa and ordinate of the extreme point. The curve is therefore quadrable. Quadrature of surfaces of revolution. The differential of a surface of revolution is equal to the circumference of a circle perpendicular to the axis into the differential of the arc of the meridian curve. ds = Ziry^d in which y is the radius of a circle of the bounding surface in a plane per- pendicular to the axis of revolution, and x is the abscissa, or distance of the plane from the origin of coordinate axes. Therefore, to find the volume of any surface of revolution: Find the value of y and dy from the equation of the meridian curve in terms of x and dx, then substitute these values in the differential equation, and integrate between the proper limits of x. By application of this rule we may find: The curved surface of a cylinder equals the product of the circumference of the base into the altitude*. The convex surface of a cone equals the product of the circumference of the base into half the slant height. The surface of a sphere is equal to the area of four great circles, or equal to the curved surface of the circumscribing cylinder. ubature of volumes of revolution. A volume of revolution is a volume generated by the revolution of a plane figure about a fixed line called the axis. If we denote the volume by F", dV iry^ dx. The area of a circle described by any ordinate y is iry*; hence the differ- ential of a volume of revolution is equal to the area of a circle perpendicular to the axis into the differential of the axis. The differential of a volume generated by the revolution of a plane figure about the axis of Y is irx*dy. To find the value of Ffor any given volume of revolution : Find the value of ?/ 2 in terms of x from the equation of the meridian curve, substitute this value in the differential equation, and then integrate between the required limits of x. By application of this rule we may find: The volume of a cylinder is equal to the area of the base multiplied by the altitude. The volume of a cone is equal to the area of the base into one third the latitude. The volume of a prolate spheroid and of an oblate spheroid (formed by ihe revolution of an ellipse around its transverse and its conjugate axis re- spectively) are each equal to two thirds of the circumscribing cylinder. If the axes are equal, the spheroid becomes a sphere and its volume = 2 1 yrR* x D = ~irD s i -R being radius and D diameter. o o The volume of a paraboloid is equal to half the cylinder having the same base and altitude. The volume of a pyramid equals the area of the base multiplied by one third the altitude. Second, third, etc., differentials, The differential coefficient being a function of the independent variable, it may be differentiated, and iv e thus obtain the second differential coefficient: d() = d . Dividing by dx, we have for the second differential coeffl- \dx/ dx 76 DIFFEBEOTIAL CALCULUS. cient -r-^, which is read: second differential of u divided by the square of the differential of x (or dx squared). d 3 u The third differential coefficient ^ is read: third differential of u divided by dx cubed. The differentials of the different orders are obtained by multiplying the differential coefficients by the corresponding powers of dx; thus ^ dx 3 = third differential of u. Sign of the first differential coefficient. If we have a curve whose equation is y = /x, referred to rectangular coordinates, the curve will recede from the axis of X when - is positive, and approach the axis when it is negative, when- the curve lies within the first angle of the coordinate axes. For all angles and every relation of y and x the curve will recede from the axis of X when the ordinate and first differential co- efficient have the same sign, and approach it when they have different signs. If the tangent of the curve becomes parallel to the axis of X at any point -^ = 0. If the tangent becomes perpendicular to the axis of X at any dx dy point =co. dx Sign of the second differential coefficient. -The second dif- ferential coefficient has the same sign as the ordinate when the curve is convex toward the axis of abscissa and a contrary sign when it is concave. Maclaurin's Theorem. For developing into a series any function of a single variable as u = A -f- Bx -f Ox* -\- Dx 3 -f- Ex 4 , etc., in which A, B, <7, etc., are independent of x: In applying the formula, omit the expressions x = 0, although the coeffi- cients are always found under this hypothesis. EXAMPLES : 1 J ___ * . ^ _ ^ . X* etc a -f- x ~~ a a 2 ^ a 3 a 4 ^ ' a n + i ' Taylor's Theorem. For developing into a series any function of the sum or difference of two independent variables, as u' = f(x y): . du , d^u y* . d 3 u y 3 ' in which u is what u' becomes when y = 0, is what becomes when dx dx y = 0. etc. Maxima and minima. To find the maximum or minimum value of a function of a single variable: 1. Find the first differential coefficient of the function, place it equal to 0, and determine the roots of the equation. 2. Find the second differential coefficient, and substitute each real root, in succession, for the variable in the second member of the equation. Each root which gives a negative result will correspond to a maximum value of the function, and each which gives a positive result will correspond to a minimum value. EXAMPLE. To find the value of x which will render the function y a maximum or minimum in the equation of the circle, y* + x z = R*' t -^ = - ; making - - = gives x = 0. dx y y DIFFERENTIAL CALCULUS, 77 dM The second differential coefficient is: -=-^ -- When x = 0, ;; ^ R-, hence -^-| = , which being negative, y is a maxi- mum for R positive. In applying the rule to practical examples we first find an expression for the function which is to be made a maximum or minimum. 2. If in such expression a constant quantity is found as a factor, it may- be omitted in the operation; for the product will be a maximum or a mini- mum when the variable factor is a maximum or a minimum. 3. Any value of the independent variable which renders a function a max- imum or a minimum will render any power or root of that function maximum or minimum; hence we may square both members of an eo tion to free it of radicals before differentiating. By these rules we may find: The maximum rectangle which can be inscribed in a triangle is one whose altitude is half the altitude of the triangle. The altitude of the maximum cylinder which can be inscribed in a cone is one third the altitude of the cone. The surface of a cylindrical vessel of a given volume, open at the top, is a minimum when the altitude equals half the diameter. The altitude of a cylinder inscribed in a sphere when its convex surface is a maximum is r |/2. r = radius. The altitude of ajcylinder inscribed in a sphere when the volume is a maximum is 2r -*- V3. (For maxima and minima without the calculus see Appendix, p. 1070.) Differential of an exponential function. If u = a x . . , ............ (1) then du = da x = a x k dx t (2) in which fc is a constant dependent on a. The relation between a and k is eft = e\ whence a = e^ t ..... (3) in which e 2.7182818 . . . the base of the Naperian system of logarithms. logarithms. The logarithms in the Naperian system are denoted by Z, Nap. log or hyperbolic log, hyp. log, or log e ; and in the common system always by log. k Nap. log a, log a = k log e ....... (4) The common logarithm of e, = log 2.7182818 . . . = .4342945 . . . , is called the modulus of the common system, and is denoted by M. Hence, if we have the Naperian logarithm of a number we can find the 1 common logarithm of the same number by muliiplying by the modulus. Reciprocally, Nap. log com. log x 2 3025851. If in equation (4) we make a = 10, we have 1 = k log e, or - = log e = M. That is, the modulus of the common system is equal to 1, divided by the Naperiau logarithm of the 'common base. From equation (2) we have du da x = = kdx. u a x If we make a =s 10, the base of the common system, x = log u t and That is, the differential of a common logarithm of a quantity is equal to the differential of the quantity divided by the quantity, into the modulus. If we make a =? e, the base of the Naperian system, x becomes the Nape- 73 DIEFEBENTIAL CALCULUS. rian logarithm of w, and k becomes 1 (see equation (3)); hence M = 1, and du du d(Nap. log u) = dx ; = . a**' That is, the differential of a Naperian logarithm of a quantity is equal to the differential of the quantity divided by the quantity; and in the Naperian system the modulus is 1. Since k is the Naperian logarithm of a, du = a x I a dx. That is, the differential of a function of the form a x is equal to the function, into the Naperian logarithm of the base a, into the differential of the exponent. If we have a differential in a fractional form, in which the numerator is the differential of the denominator, the integral is the Naperian logarithm of the denominator. Integrals of fractional differentials of other forms are given helow: Differential forms which have known integrals; ex- ponential functions. (I = Nap. log.) 1. / a x I a dx = a x -f- C\ o f*dx /\ * / = / dxx ~ L = lx + Cl J J 3. / (xy x ~ l dy -f y x ly x dx) = y x -f C\ 4. C dX = l(x + |/a; 2 a 2 ) + C; J yx* a* 5. C _ d L = l(x a + y x i 2ax) 4- C; J MX* 2ax r-^= =i(=-t J x\/a* + x* \fVa + a;-f /> _ %adx fa - A/C&~~X / - - = Zf _ 1 J x y^- & \ a + -zrr i @. " + Circular functions. Let 2 denote an arc in the first quadrant, y tts sine, x its cosine, v its versed sine, and t its tangent; and the following nota- tion be employed to designate an arc by any one of its functions, viz., sin ~ 1 y denotes an arc of which y is the sine cos" 1 x u " " " " x is the cosine, tan" 1 f " ' " " " t is the tangent DIFFERENTIAL CALCULUS. 79 <read "atv whose sine is ?/," etc.), we have the following differential forms which have known integrals (r = radius): cos z dz = sin z-\-C\ sin z dz = cos z -f C; / *# _ _i / dv _ C; = ver-sin ~" 1 v -f- (7; / |/r 2 - 2/ 2 /- rcte _ i = cos * x -f- C; |/r 2 - * 2 f- rj, J cos 2 rd_v y-&^+& = sin z dz = ver-sin 2; -f- C; =: tan -f C; In "W-f'Cj /, :r=- = sin ~~ * -f- O; |/a 2 - w 2 /~ dtC __- = cos- 1 -4-C; |/a 2 - w* <* / U = ver-sin ~ J - -f (7; |/^aw - ti 2 /adit _ _ i, a2 + w2 a The cycloid. If a circle be rolled along a straight line, any point of the circumference, as P, will describe a curve which is called a cycloid. The circle is called the generating circle, and Pthe generating point. The transcendental equation of the cycloid is x ver-sin- l ~ \'%ry - 2/ 2 , ydx and the differential equation is dx = 4/0..- _ == 1' The area of the cycloid is equal to three times the area of the generating circle. The surface described by the arc of a cycloid when revolved about its base is equal to 64 thirds of the generating circle. The volume of the solid generated by revolving a cycloid about its base is equal to five eighths of the circumscribing cylinder. Integral calculus. In the integral calculus we have to return from the differential to the function from which it was derived A number of differential expressions are given above, each of which has a known in- tegral corresponding to it, and which being differentiated, will produce the given differential. In all classes of functions any differential expression may be integrated when it is reduced to one of the known forms; and the operations of the integral calculus consist mainly in making such transformations of given differential expressions as shall reduce them to equivalent ones whose in- tegrals are known. For methods of making these transformations reference must be made to Uie text-books on differential and integral calculus. 80 MATHEMATICAL TABLES. RECIPROCALS OF NUMBERS. No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. 1 1.00000000 64 .01562500 127 .00787402 190 .00526316 253 .00395257 2 .50000000 5 .01538461 8 .00781250 1 .00523560 4 .00393701 3 .33333333 6 .01515151 9 .00775191 2 .00520833 5 .00392157 4 .25000000 7 .01492537 130 .00709231 3 .00518135 6 .00390625 5 .20000000 8 .01470588 1 .00763359 4 .00515464 ri .00389105 6 .16666667 o .01449275 2 .00757576 5 .00512820 8 .00387597 r- .14285714 70 .01428571 3 .00751880 6 .00510204 9 . 00386 100 8 .12500000 1 .01408451 4 .00746269 7 .00507614 260 .00384015 9 .11111111 2 .01388889 5 .00740741 8 .00505051 1 .00383142 10 .10000000 3 .01369863 6 .00735294 9 .00502513 2 .00381079 11 .09090909 4 .01351351 7 .00729927 200 .00500000 3 .00380228 1-2 .08338333 ^ .01333533 8 .00724638 1 .00497512 4 .00378188 13 .0769:2308 6 .01315789 <j .00719424 2 .00495049 .00377358 14 .07142857 7 .01298701 140 .00714286 3 .00492611 6 .00375940 15 .06666667 8 .01282051 1 .00709220 4 .00490196 7 .00374532 16 .06250000 9 .01265823 2 .00704225 f. .00487805 8 .00373134 17 .05882353 80 .01250000 t .00699301 6 .00485437 9 .00371717 18 . 05555556 1 .01234568 4 .00694444 7 .00483092 270 .0037(1370 19 .05263158 .01219512 5 .00689655 8 .00480769 j .00309004 20 .05000000 8 .01204819 6 .00681931 9 .00478469 < .00367647 1 .04761905 4 .01190476 r- .00680272 210 .00476190 j .00300300 2 .04545455 5 .01176471 8 .00675676 11 .00473934 L .00364963 3 .04347826 6 .01162791 r .00671141 12 .00471698 5 .00363636 4 .04166667 7 .01149425 150 .00606667 13 .00469484 ( .00302319 5 .04000000 8 .01136364 1 .00662252 14 .00467290 7 .00361011 6 .03846154 c .01123595 o .00657895 15 .00465116 h .00359712 7 .03703704 90 .01111111 3 .00653595 16 .00462963 9 .00358423 8 .03571429 1 .01098901 4 .00649351 17 .00460829 280 .00357143 9 .03448276 5 .01086956 5 .00645101 18 .00458716 ] .00355872 30 .01333333 jj .01075269 6 .00641026 19 .00456621 \ .00354610 1 .03225806 4 .01063830 7 .00636943 220 .00454545 ; .00353357 2 .03125000 r .01052632 8 .00682911 1 .00452489 i. .00352113 3 .03030303 6 .01041667 9 .00628931 c .00450450 5 .00350877 4 .02941176 7 .01030928 160 .00625000 J .00448430 6 .00349350 5 .02857143 8 .01020408 1 .00621118 4 .00446429 7 .00348432 6 .02777778 g .01010101 2 .00617284 5 .00444444 8 .00347222 7 .02702703 100 .01000000 ? .00613497 6 .00442478 1 .00346021 8 .02631579 1 .00990099 4 .00609756 7 .00440529 290 .00344828 9 .02564103 o .00980392 5 . 00606061 8 .00438596 .00343613 40 .02500000 3 .00970874 6 .00602410 c .00436681 o .00342406 1 .02439024 4 .00961538 7 .00598802 230 .00434783 3 .00341297 2 .02380952 5 .00952381 8 .00595238 1 .00432900 L .00340136 3 .02325581 6 .00943396 9 .00591716 2 .00431034 5 .00338983 4 .02272727 7 .00934579 170 '.00588235 c .00429184 ( .00337S38 5 .02222222 8 .00925926 1 .00584795 4 .00427350 7 .00330700 6 .02173913 9 .00917431 o .00581395 ^ .00425532 8 .00335570 7 .02127660 110 .00909091 3 '.00578035 6 .00423729 9 .00334448 8 .02083333 11 .00900901 4 .00574713 n .00421941 300 .00333333 9 .02040816 12 .00892857 5 .00571429 8 .00420168 .00332226 50 .02000000 13 .00884956 6 .00568182 c .00418410 jj .00331120 1 .01960784 14 .00877193 7 .00564972 240 .00416667 i .00330033 2 .01923077 15 .00869565 8 .00561798 1 .00414938 4 .00328947 3 .01886792 16 .00862069 c .00558659 2 .00413223 5 .00327809 4 .01851852 17 .00854701 180 .00555556 3 .00411523 6 .00320797 5 .01818182 18 .00847458 1 .00552486 4 .00409836 r .00325733 6 .01785714 19 .00840336 r .00549451 ft .00408163 8 .00324670 7 .01754386 120 .00833333 3 .00546448 e .00406504 9 .003.23625 8 .01724138 1 .00826446 4 .00543478 7 .00404858 310 .00322581 9 .01694915 c .00819672 r .00540540 8 .00403226 11 .00321543 60 .01666667 3 .00813008 6 .00537634 9 .00401606 12 .00320513 1 .01639344 4 .00806452 7 .00534759 250 .00400000 13 .00319489 2 .01612903 i .00800000 8 .00531914 1 .00398406 14 .00318471 3 .01587302 6 .00793651 c .00529100 2 .00396825 15 .00317460 RECIPROCALS OF NUMBERS. 81 No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. 316 .00316456 381 .00262467 446 .0022-1215 511 .00195695 576 .00173611 17 .00315457 2 .00261780 7 .00223714 12 .00195312 7 .00173310 18 .00314465 3 .00261097 8 .00223214 13 .00194932 8 .00173010 19 .00313480 4 .00260417 9 .00222717 14 .00194552 9 .00172712 320 .00312500 5 .00259740 450 .00222222 15 .00194175 580 .00172414 1 .00311526 6 .00259067 1 .00221729 16 .00193798 1 .00172117 2 .00310559 7 .00258398 g .00221239 17 .00193424 2 .00171821 3 .00309597 8 .00257732 3 .00220751 18 .00193050 3 .00171527 4 .00308642 9 .00257069 4 .00220264 19 .00192678 4 ,001?J00 5 .00307692 390 .00256410 5 .002197'80 520 .00192308 5 .00170940 6 .00306748 1 .00255754 6 .00219298 1 .00191939 6 .00170648 .00305810 2 .00255102 7 .00218818 2 .00191571 7 .00170358 8 .00304878 3 .00254453 8 .00218341 3 .00191205 8 .00170068 9 .00303951 4 .00253807 9 .00217865 4 .00190840 9 .00169779 330 .00303030 5 .00253165 460 .00217391 5 .00190476 590 .00169491 1 .00302115 6 .00252525 1 .00216920 6 .00190114 1 .00169205 2 .00301205 7 .00251889 2 .00216450 7 .00189753 2 .00168919 3 .00300300 8 .00251256 3 .00215983 8 .00189394 3 .00168634 4 .00299401 g .00250627 4 .00215517 9 .00189036 4 .00168350 .00298507 400 .0025000'! 5 .00215054 530 .00188679 R .00168007 ii .00297619 1 .00249377 6 .00214592 1 .00188324 6 .00167785 7 .00296736 2 .00248756 7 .00214133 2 .0018797'0 7 .00167504 8 .00295858 3 .00248131) 8 .00213675 3 .00187617 8 .00167224 9 .00294985 4 .00247525 9 .00213220 4 .00187-266 9 .00166945 340 .00294118 5 .00246914 470 .00212760 5 .00186916 600 .00166667 1 .00293255 6 .002-16305 1 .00212314 6 .00186567 1 .00166389 c .00292398 7 .00245700 2 .00211864 7 .00186220 2 .00166113 C .00291545 8 .00245098 g .00211416 8 .00185874 .00165837 4 .00290698 9 .00244490 4 .00210970 9 .00185528 4 .00165563 5 .00289855 410 .00243902 5 .00210526 540 .00185185 5 .00165289 6 .00289017 11 .00243309 6 .00210084 1 .00184^43 6 .001C5016 r< .00288184 12 .00242718 7 .00209644 .00184502 7 .00164745 8 .00287356 13 .00242131 8 .00200205 .00184162 8 .00164474 9 .00286533 14 .00241546 9 .00208768 .00183823 9 .00164204 350 . .00285714 15 .0021096-1 480 .00208333 .00183486 610 .00163934 ] .00284900 16 .00240385 1 .00207900 .00183150 11 .0016361)6 .00784091 17 .00239808 f .00207469 .0018-2815 12 .001 64399 f .00283286 18 .00-J39234 3 .00207039 .0018248-2 13 .00163132 4 .00288486 19 .00238663 4 .00206612 .00182149 14 .00162866 5 .00281690 420 .00238095 e^ .00206186 55 .00181818 15 .00162602 6 .00280899 1 .00237530 6 .00205761 .00181488 16 .00162338 7 .00280112 2 .00236967 r- .00205339 .00181159 17 .00162075 8 .00279330 3 .00286407 8 .00-204918 .00180832 18 .00161812 9 .00278551 4 .00235849 9 .00204499 .00180505 19 .00161551 360 .00277778 5 .00235294 490 .C0204082 .00180180 620 .00161-290 1 .00277008 6 .00234742 1 .00-203666 .CO 179856 1 .00161031 2 .00276243 7 .0023419!? .00203252 .00179533 f .00160772 .00275482 8 .00283645 c .0020-2840 8 .00170211 3 .00160514 4 .00274725 9 .00233100 4 .00202429 9 .00178891 3 .00160256 f .00273973 430 .00232558 5 .00202020 560 .00178571 c .00160000 6 .00273224 1 .00232019 6 .00201613 1 .00178253 I .00159744 r .00272480 g .00231481 7 .00-201207 2 .00177936 1 .00159490 8 .00271739 3 .00230947 8 .00200803 3 .00177620 8 .00159-236 9 .00271003 4 .00230415 t .00200401 4 .00177305 9 .00158982 370 .00270270 5 .00229885 500 .00200000 5 .00176991 630 .00158730 1 .00269542 6 .00229358 1 .00199601 6 .00176678 1 .0015847-9 2 .00268817 7 .00228833 o .00199203 7 .00176367 c .00158228 : .00268096 8 .00228310 '. .00198807 8 .00176056 \ .00157978 4 .00267380 9 .00227790 i .00198413 9 .00175747 i .00157729 5 .00266667 440 .00227273 r .00198020 570 .00175439 f .00157480 6 .00265957 1 .00226757 \ .00197628 1 .00175131 I .00157233 7 .00265252 2 .00226244 7 .00197239 2 .00174825 f t .00156986 8 .00264550 3 .00225734 8 .001968^0 3 .00174520 8 .00156740 9 .00263852 4 .00225225 ( .00196464 4 .00174216 9 .00156494 380 .00263158 5 ! .00224719 510 .00196078 5 .00173913 640 .00156250 MATHEMATICAL TABLES. No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. No. Reci?iro- cal. No. Recipro- cal. 641 .00156006 706 .00141643 771 .00129702 1 836 .00119617 901 .00110988 2 .00155763 7 .00141443 2 .00129534 7 .00119474 2 .00110865 3 .001555-21 8 .00141243 3 .00129366 8 .00119332 3 .00110742 4 .00155279 9 .00141044 4 .00129199 9 .00119189 4 .00110619 5 .00155039 710 .00140845 5 .00129032 840 .00119048 5 .00110497 . 6 .00154799 11 .00140647 6 .00128866 1 .00118906 6 .00110375 7 .00154559 12 .00140449 7 .00128700 2 .00118765 7 .00110254 8 .00154321 13 .00140252 8 .00128535 3 .00118(524 8 .00110132 9 .00154083 14 .00140056 9 .00128370 4 .00118483 9 .00110011 650 .00153846 15 .00139860 780 .00128205 5 .00118343 910 .00109890 1 .00153610 16 .00139665 1 .00128041 6 .00118203 11 .00109769 2 .00153374 17 .00139470 2 .00127877 7 .00118064 12 .00109649 g .00153140 18 .00139276 g .00127714 8 .00117924 13 .00109529 4 .00152905 19 .00139082 4 .00127551 9 .00117786 14 .00109409 5 .00152672 720 .00138889 e .00127388 850 .00117647 15 .00109290 6 .00152439 1 .00138696 6 .00127226 1 .00117509 16 .00109170 .00152207 2 .00138504 7 .00127065 2 .00117371 17 .00109051 8 .00151975 .00138313 8 .00126904 3 00117233 18 .00108932 9 .00151745 4 .00138121 c .00126743 4 .00117096 19 .00108814 660 .00151515 5 .00137931 790 .00120582: 5 .00116959 920 .00108696 1 .00151286 6 .00137741 1 .001264221 6 .00116822 1 .00108578 .00151057 r .00137552 2 .00126263 7 .00116686 2 .00108460 3 .00150830 8 .00137363 e .00126103 8 .00116550 3 .00108342 i .00150602 9 .00137174 L .00125945 9 .00116414 4 .00108225 t .00150376 730 .00136986 f .00125786 860 .00116279 i .00108108 ( .00150150 .00136799 ( .00125628 ] .00116144 6 .00107991 j .00149925 2 .00136612 1 .00125470 2 .00116009 ij .00107875 8 .00149701 .00136426 8 .00125313 3 .00115875 8 .00107759 ] .00149477 i .00136240 9 .00125156 t .00115741 c .00107643 670 .00149254 5 .00136054 800 .00125000 5 .00115607 930 .00107527 .00149031 1 .00135870 .00124844 6 .00115473 .00107411 .00148809 ' .00135685 2 .00124688 7 .00115340 < .00107296 .00148588 j .00135501 < .00124533 8 .00115207 ] .00107181 i .00148368 .00135318 .00124378 ( .00115075 i .00107066 .00148148 74i .00135135 | .00124224 870 .00114942 5 .0010695* .00147929 .00134953 I .00124069 ; .00114811 .0010683^ .00147710 .0013477 ' .00123916 J .00114679 ' .00106724 .0014749: .00134589 .00123762 .00114547 I 001066 1C .00147275 i .00134409 .00123609 t .00114416 .00106496 68 .00147059 .00134228 8li .00123457 j .00114286 941 .0010638? .00146843 .00134048 11 .00123305 1 .00114155 .0010627C .00146628 .00133869 12 .00123153 1 .00114025 .0010615" .00146413 .00133690 13 .00123001 .00113895 .00106044 .00146199 .0013351 14 .00122850 .00113766 i .00105935 .00145985 75 .0013333 lo .00122699 881 .00113636 .0010582C .00145773 .00133156 16 .00122549 .00113507 i .0010570* .00145560 .0013297 r .00122399 .00113379 i .0010559, .00145349 .0013280 18 .00122249 .00113250 .0010548? .00145137 .0013262 19 .00122100 < .00113122 .00105374 69 .00144927 .00132450 82C .00121951 .00112994 95< .0010526c .00144718 .0013227o .00121803 i .00112867 . 00105 15i .00144509 .00132100 .00121654 r .00112740 .0010504$ .00144300 .00131926 .00121507 | .00112613 .0010493^ .00144092 .00131752 i .00121359 .00112486 <. .00104822 .00143885 76' .00131579 j .0012121? 891 .00112360 1 .00104712 .00143678 .00131406 i .00121065 ; .00112233 < .00104602 .00143472 .00131234 r .00120919 2 .00112108 ' .00104493 .00143266 .00131062 | .00120773 j .00111982 8 .00104384 .00143061 t .00130890 .00120627 i .00111857 j .00104275 70 .00142857 .00130719 i 831 .00120482 5 .00111732 960 .0010416; .00142653 ( .00130548 .00120337 6 .00111607 .00104058 .00142450 1 .00130378 < .00120192 \ .00111483 j .00103950 .00142247 8 .00130208 j .00120048 8 .00111359 j .00103842 ^ .00142045 j .00130039 i .00119904 9 .00111235 ^ .00103734 ; .00141844 770 .00129870 5 .00119760 900 .00111111 t ,00103627 RECIPROCALS OF NUMBERS. 83 No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. 966 00103520 1031 000969932 1096 000912409 1161 .000861326 1226 .000815661 7 00103413 2 000968992 7 000911577 2 .000860585 .000814996 8 00103306 3 000968054 8 000910747 3 .000859845 8 .000814332 9 00103199 4 000967118 9 000909918 4 .000859106 9 .000813670 970 00103093 5 000966184 1100 .000909091 5 .000858369 1230 .000813008 1 .00102987 6 000965251 1 .000908265 6 .000857633 1 .000812348 2 00102881 7 .000964320 2 .000907441 7 .000856898 2 .000811688 3 .00102775 8 .000963391 3 .000906618 8 .000856164 3 .000811030 4 .00102669 9 000962464 4 .000905797 9 .000855432 4 .000810373 fc .00102564 1040 .000961538 5 .000904977 1170 .000854701 5 .000809717 6 .00102459 1 .000960615 6 .000904159 1 .000853971 6 .000809061 7 .00102354 2 .000959693 n .000903342 2 .000853242 7 .000808407 8 .00102250 3 .000958774 8 .000902527 3 .000852515 8 .000807754 9 .00102145 4 .000957854 c .000901713 4 .000851789 9 .000807102 980 .00102041 g .000956938 1110 .000900901 5| .000851064 1240 .000806452 I .00101937 I .000956023 11 000900090 6 .000850340 1 .000805802 2 .00101833 7 .000955110 12 .000899281 7 .000849618 2 .000805153 3 .00101729 8 .000954198 13 .000898473 8 .000848896 3 .000804505 4 .00101626 9 .000953289 14 .000897666 9 .000848176 4 .000803858 5 .00101523 1050 .000952381 15 .000896861 1180 .000847457 5 .000803213 6 .00101420 1 .000951475 16 .000896057 1 .000846740 6 .000802568 .00101317 .000950570 17 .000895255 2 .000846024 7 .000801925 8 .00101215 \ .000949668 18 .000894454 3 .000845308 8 .000801282 c .00101112 i .000948767 19 .000893655 4 .000844595 9 .000800640 990 .00101010 5 .000947867 1120 .000892857 5 .000843882 1250 .000800000 .00100908 6 .000946970 3 .000892061 6 .000843170 1 .000799360 < .00100806 \ .000946074 < .000891266 7 .000842460 2 .000798722 < .00100705 8 .000945180 j .000890472 S .000841751 3 .000798085 c .00100604 c .000944287 <. .000889680 9 .000841043 4 .000797448 t .00100502 1060 .000943396 5 .000888889 1190 .000840336 5 .000796813 ( .00100J02 .000942507 6 .000888099 1 ,000839631 6 .000796178 " ( .00100301 .000941620 \ .000887311 2 .000838926 "t .000795545 8 .00100200 { .000940734 8 .000886525 3 .000838222 8 .000794913 < .00100100 t .000939850 9 .000885740 4 .000837521 9 .000794281 1000 .00100000 5 .000938967 1130 .000884956 5 .000836820 1260 .000793651 .000999001 < .000938086 .000884173 6 .000836120 1 .000793021 2 .000998004 7 .000937207 < .000883392 7 .000835422 * .000792393 j .000997009 8 .000936330 < .000882612 8 .000834724 3 .000791766 t ,000996016 < .000935454 i .000881834 9 .000834028 4 .000791139 5 .000995025 1070 .000934579 5 .000881057 1200 000833333 5 .000790514 ( .000994036 .000933707 6 .000880282 1 .000832639 6 .000789889 .000993049 2 .000932836 r .000879508 2 .000831947 7 .000789266 I .000992063 j .000931966 8 .000878735 3 .000831255 8 .000788643 I .000991080 c .000931099 j .000877963 4 .000830565 9 .000788022 1010 .000990099 I .000930233 1140 .000877193 5 .000829875 1270 .000787402 11 .000989120 6 .000929368 .000876424 6 .000829187 1 .000786782 12 .000988142 t .000928505 i .000875657 7 .000828500 2 .000786163 13 .000987167 8 .000927644 .000874891 8 .000827815 3 .000785546 14 .000986193 c .000926784 t .000874126 9 .000827130 i .000784929 15 .000985222 1080 .000925926 5 .000873362 1210 .000826446 5 .000784314 16 .000984252 .000925069 6 .000872600 11 .000825764 6 .000783699 1" .000983284 j .000924214 \ .000871840 12 .000825082 .000783085 18 .000982318 j .000923361 8 .000871080 13 .000824402 8 .000782473 19 .000981354 t .000922509 c .000870322 14 .000823723 9 .000781861 1020 .000980392 i .000921659 1150 .0008695G5 15 .000823045 1280 .000781250 ; .000979432 ( .000920810 .000868810 16 .000822368 1 .000780640 000978474 1 .000919963 < .000868056 17 .000821693 2 .000780031 < .000977517 8 .000919118 j .000867303 18 .000821018 3 .000779423 t .000976562 9 000918274 t .000866551 19 .000820344 i .000778816 t .000975610 1090 .000917431 5 .000865801 1220 .000819672 5 .000778210 ( .000974659 ] .000916590 6 .000865052 1 .000819001 6 .000777605 \ .000973710 .000915751 1 .000864304 2 .000818331 j .000777001 8 .000972763 < .000914913 8 .000863558 3 .000817661 8 .000776397 9 .000971817 i .000914077 ( .000862813 4 .000816993 9 .000775795 1030 .000970874 5 .000913242 1160 .000862069 5 .000816326 1290 .000775194 8"4 MATHEMATICAL TABLES, No. Recipro- cal. No. ! Recipro- cal. No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. 1291 .000774593 1356 .000737463 1421 .000703730 1486 .000672948 1551 .000644745 2 .00077391)4 7 000736920 2 .000703235 7 .000672495 2 .000644330 3 .000773395 8 .000736377 3 .000702741 8 .000672043 3 .000643915 4 .000772797 ( .000735835 4 .000702247 9 .000671592 4 .000643501 5 .000772:201 1360 .000735294 5 .000701754 1490 .000671141 5 .000643087 6 .000771605 1 .000734754 6 .000701262 1 .000670691 6 .000642673 7 .000771010 2 .000734214 7 .000700771 2 .000670241 .000642261 8 .000770416 f .000733676 8 .000700280 3 .000669792 8 .000641848 9 1.000769823 4 .000733138 9 .000699790 4 .000669344 9 .000641437 1300 .000769231 r .00073-2601 1430 .000699301 5 .000668896 1560 .000641026 1 .000768639 e .000732064 1 .000698812 6 .000668449 1 .000640615 2 .000768049 .000731529 2 .000698324 7 .000668003 2 .000640205 3 .000767459 8 .000730994 3 .000697'837 8 .000667557 3 .000639795 4 .000766871 9 .000730460 4 .000697350 9 .000667111 4 .000639386 5 .000766283 1370 .000729927 5 .000690864 1500 .000666667 5 .000638978 6 .000765697 1 .000729395 6 .000696379 1 .000666223 6 .000638570 7 .000765111 c .000728863 7 .000695894 2 .000665779 7 .000638162 8 .000764526 3 .000728332 8 .000695410 3 .000665336 8 .000637755 9 .000763942 4 .000727802 9 .000694927 4 .000664894 9 .000637349 1310 .000763359 s .000727273 1440 .000694444 5 .000664452 1570 .000636943 11 .000762776 6 .000726744 1 .000693962 6 .000664011 1 .000636537 12 .000762195 7 .000726216 2 .000693481 .000663570 2 .000636132 13 .000761615 8 .000725689 3 .000693001 8 .000663130 3 ! . 000635728 14 .000761035' 9 .000725163 4 .000692521 9 .000662691 4 .000635324 15 . 000760456 ! 1380 .000724638 51.000692041 1510 . 000662252 5 .000634921 16 .000759878! 1 .000724113 6 .000691563 11 .000661813 6 .000634518 17 .000759301 2 .000723589 7 .000691085 12 .000661376 7 .000634115 18 .000758725; g .000723066 8 .000690608 13 .000660939 8 .000633714 19 .000758150; 4 .000722543 9 .000690131 14 .000660502 9 .000633312 1320 .000757576! .000722022 1450L 000689655 15 .000660066 1580 .000632911 1 .000757002 6 .000721501 1 .000689180 16 .000659631 1 .000632511 2 .000756430 7 .000720980 a .000688705 17 .000659196 2 .000632111 3 .000755858 8 .000720461 3 .000688231 18 .000658761 3 !. 000631712 4 .000755287 9 .000719942 4 .000687758 19 .000658328 4 .000631313 5 .000754717 1390 .000719424 5 .000687285 1520 .000657895 5 .000630915 6 .000754148 1 .000718907 6 .000686813 1 .000657462 .000630517 7 .000753579 2 000718391 .000686341 2 .000657030 7 .000630120 8 .000753012 3 .000717875 g .000685871 3 .000656598 8 .000629723 9 .000752445 4 .000717360 9 .000685401 4 .000656168 9 .000629327 1330 .000751880 5 .000716846 1460 .000684932 5 .0006557381 1590 .000628931 1 .000751315 6 .000716332 1 .000684463 6 .000655308 1 .000628536 2 .000750750 7 .000715820 2 .000683994 7 .000654879 o .000628141 3 .000750187 8 .000715308 3 .000683527 8 .000654450 3 .000627746 4 .000749625 9 .000714796 41.000683060 9 .000654022 4 .000627353 5 .000749064 1400 .000714286 5 ! . 000682594 1530 .000653595 5 .000626959 6 .000748503 1 .000713776 6 .000682128 1 .000653168 6 .000626566 7 .000747943 2 .000713267 7 .000681663 2 .000652742' 7 .000626174 8 .000747384 3 .000712758 8 .000681199 3 .000652316' 8 .000625782 9 . 000746826 I 4 .000712251 9 .000680735 4 .000651890! 9 .000625391 1340 .000746269; 5 .000711744 1470 .000680272 5 . 000651466 ' 1600 .000625000 1 .000745712 6 .000711238 1 .000679810 6 .000651042 2 .000624219 2 .000745156 7 .000710732 2 .000679348 7 .000650618 4 .000623441 3 .000744602! 8 .000710227 3 .000678887 8 .000650195 6 .000622665 4 .000744048! 9 .000709723 4 .000678426 9 .000649773 8 .000621890 5 .0007434941 1410 .000709220 5 .000677966 1540 .000649351 1610 .000621118 6 .000742942 11 .000708717 6 .000677507 1 .000648929 2 .000620347 7 .000742390 12 .000708215 7 .000677048 2 .000648508 4 .000619578 8 .000741840 13 .000707714 8 .000676590 3 .000648088 6 .000618812 9 .000741290 14 000707214 9 .0006-76138 4 .000647668 8 .000618047 1350 .000740741J 15 000706714 1480 000675676 5 .000647249 1620 .000617284 1 .000740192; 16 .000706215 1 .00^675219 6 .000646830 2 .000616523 2 000739645 17 .000705716 2 .0006?4"64 .000646412 4 .000615763 3 .000789098 18 .000705219 3 .000674309 8 .000645995 6 .000615006 4 .000738552 19 .000704722 4 .000673854 9 .000645578 8 .000614250 5 .000738007! 1420 .000704225 5 .0006^3*01 1550 .000645161 1630 .000613497 RECIPROCALS OF NUMBERS. 85 No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. No. Recipro- cal. 1632 .000612745 1706 .000586166 1780 .000561798 1854 .000539374 1928 .000518672 4 .000611995 8 .000585480 2 .000561167 6 .000538793 1930 .000518135 6 .000611247 1710 .000584795 4 .000560538 8 .000538213 2 .000517599 8 .000610500 12 .000584112 6 .000559910 1860 .000537634 4 .000517063 1640 .000609756 14 .000583430 8 .000559284 2 .000537057 6 .000516528 2 .000609013 16 .000582750 1790 .000558659 4 .000536480 8 .000515996 4 .000608272 18 .000582072| 2 .000558035 6 .000535905 1940 .000515464 6 .000607533 1720 .000581395! 4 .000557413 8 .000535332 2 .000514933 8 .000606796 2 .000580720 6 .000556793 1870 .000534759 4 .000514403 1650 .000606061 4 .000580046 8 .000556174 2 .000534188 6 .000513874 2 .000305327 6 .000579374 1800 .000555556 4 .000533618 8 .000513347 4 .000604595 8 .000578704 2 .000554939 6 .000533049 1950 .000512820 6 .000603865 1730 .000578035 4 .000554324 8 .000532481 2 .000512295 8 .000603136 2 .000577367 6 .000553710 1880 .000531915 4 .000511770 1660 .000602410 4 .000576701 8 .000553097 2 .000531350 6 .000511247 o .000601685 6 .000576037 1810 .000552486 4 .000530785 8 .000510725 4 .OOD600962 8 .000575374 12 .000551876 6 .000530222 1960 .000510204 6 .000600240 1740 .000574713 14 .000551268 8 .000529661 2 .000509684 8 .000599520 2 .000574053 16 . 000550661 1890 .000529100 4 .000509165 1670 .000598802 4 .000573394 18 .000550055 2 .000528541 6 .000508647 2 .000598086 6 .000572737 1820 .000549451 4 .000527983 8 .000508130 4 .000597371 8 .000572082 2 .000548848 6 .000527426 1970 .000507614 6 .000596658 1750 .000571429 4, .000548246 8 .000526870 2 .000507099 8 .000595947 2 .000570776 6 .000547645 1900 .000526316 4 .000506585 1680 .000595238 4 .000570125 8 000547046 2 .000525762 6 .000506073 2 .000594530 6 .000569476 1830 .000546448 4 .000525210 8 .000505561 4 .000593824 8 .000568828 2 .000545851 6 .000524659 1980 .000505051 6 .000593120 1760 .000568182 4 .000545255 8 .000524109 2 .000504541 8 .000592417 2 .000567537 6 i .000544662 1910 .000523560 4 .000504032 1690 .000591716 4 .000566893 8 .000544069 12 000523012 6 .000503524 2 .000591017 6 .000566251 1840 .000543478 14 .000522466 8 .OOOoOSOlS 4 .000590319 8 .000565611 2 .000542888 16 .000521920 1990 .00050213 6 .000589622 1770 .000564972 4 .000542299 18 .000521376 2 .000502008 8 .000588928 2 000564334 C .000541711 1920 .000520833 4 .000501504 1700 .000588235 4 .000563698 8 .000541125 2 .000520291 6 .000501002 2 .000587544 6 000563063 1850 .000540540 4 .000519750 8 .000500501 4 .000586854 8 .000562430 2 .000539957 6 .000519211 2000 000500000 Use of reciprocals. Reciprocals may be conveniently used to facili- tate computations iu long division. Instead of dividing as usual, multiply the dividend by the reciprocal of the divisor. The method is especially- useful when many different dividends are required to be divided by the same divisor. In this case find the reciprocal of the divisor, and make a small table of its multiples up to 9 times, and use this as a multiplication- table instead of actually performing the multiplication in each case. EXAMPLE. 9871 and several other numbers are to be divided by 1638. The reciprocal of 1638 is .000610500. Multiples of the reciprocal : .0006105 .0012210 .0018315 .0024420 .0030525 The table of multiples is made by continuous addition of 6105. The tenth line is written to check the accuracy of the addition, but it is not afterwards used. Operation: Dividend 9871 Take from table 1 ... 7... .0006105 0.042735 00.48S40 005.4945 .0042735 .0048840 .0054945 10. .0061050 Quotient.. 6.0262455 Correct quotient by direct division 6.0262515 The result will generally be correct to as many figures as there are signifi- cant figures in the reciprocal, less one, and the error of the next figure will in general not exceed one. In the above example the reciprocal has six sig nificant figures, 610500, and the result is correct to five places of figures. 86 MATHEMATICAL TABLES. SQUARES, CUBES, SQUARE ROOTS AND CUBE ROOTS OF NUMBERS FROM .1 TO 1600. No. Square. Cube. Sq. Root. Cube Root. No. Square. Cube. Sq. Root. Cube Root. .1 .01 .001 .3162 .4642 3.1 0.61 29.791 1.761 1.458 .15 .0225 .0034 .3873 .5313 .2 10.24 32.768 1.789 1.474 .2 .04 -008 .4472 .5848 .3 10.89 35.937 1.817 1.489 .25 .0625 0156 .500 .6300 .4 11.56 39.304 1.844 1.504 .3 .09 027 .5477 .6694 .5 12.25 42.875 1.871 1.518 .35 .1225 0429 .5916 .7047 .6 12.96 46.656 1.897 1.533 .4 .16 064 .6325 .7368 7 13.69 50.653 1.924 1.547 .45 .2025 .0911 .6708 .7663 '.8 14.44 54.872 1.949 1.560 .5 .25 125 .7071 .7937 .9 15.21 59.319 1.975 1.574 .55 .3025 .1664 .7416 .8193 4. 16. 64. 2. 1.5874 .6 .36 216 .7746 .8434 .1 16.81 68.921 2.025 1.601 .65 .4225 .2746 .8062 .8662 .2 17.64 74.088 2.049 1.613 .7 .49 .343 .8367 .8879 .3 18.49 79.507 2.074 1.626 .75 .5625 .4219 .8660 .9086 .4 19.36 85.184 2.098 1.639 .8 .64 .512 .8944 .9283 .5 20.25 91.125 2.121 1.651 .85 .7225 .6141 .9219 .9473 .6 21.16 97.336 2.145 1.663 .9 .81 .729 .9487 .9655 .7 22.09 103.823 2.168 1.675 .95 .9025 .8574 .9747 .9830 .8 23.04 110.592 2.191 1.687 1. 1. 1. 1. 1. .9 24.01 117.649 2.214 .698 1.05 1.1025 1.158 1.025 1.016 5. 25. 125. 2.2361 .7100 1.1 1.21 1.331 1.049 1.032 .1 26.01 132 651 2.258 .721 1.15 1.3225 1.521 1.072 1.048 .2 27.04 140.608 2.280 .732 1.2 1.44 1.728 1.095 1.063 .3 28.09 148.877 2.302 .744 1.25 1.5625 1.953 1.118 1.077 .4 29.16 157.464 2.324 .754 1.3 1.69 2.197 1.140 1.091 .5 30.25 166.375 2.345 .765 .35 1.8225 2.460 1.162 1.105 .6 31.36 175.616 2.366 .776 .4 1.96 2.744 1.183 1.119 7 32.49 185.193 2 . 387 .786 .45 2.1025 3.049 1.204 1.132 '.8. 33.64 195.112 2.408 .797 .5 2.25 3.375 1.2247 1.1447 .9 34.81 205.379 2.429 .807 .55 2.4025 3.724 1.245 1.157 6. 36. 216. 2.4495 .8171 .6 2.56 4.096 1.265 1.170 .1 37.21 226.981 2.470 .827 .65 2.7225 4.492 1.285 1.182 o 38.44 238.328 2.490 .837 1.7 2.89 4.913 1.304 1.193 .3 39.69 250.047 2.510 .847 1.75 3.0625 5.359 1.323 1.205 4 40.96 262.144 2.530 .857 1.8 3.24 5.832 1.342 1.216 .5 42.25 274.625 2.550 .866 1.85 3.4225 6.332 1.360 1.228 .6 43.56 287.496 2.569 .876 1.9 3.61 6.859 1.378 1.239 .7 44 89 300.763 2.588 .885 1.95 3.8025 7.415 1.396 1.249 .8 46.24 314.432 2.608 .895 2. 4. 8. 1.4142 1.2599 .9 4? 61 328.509 2.627 .904 .1 4.41 9.261 1.449 1.281 7. 49. 343. 2.6458 1.9129 .2 4.84 10.648 1.483 1.301 .1 50.41 357.911 2.665 1.922 .3 5.29 12.167 1.517 1.320 .2 51.84 373.248 2.683 1.931 .4 5.76 13.824 1.549 1.339 .3 53.29 389.017 2.702 1.940 .5 6.25 15.625 1.581 1.357 .4 54.76 405.224 2.720 1.949 .6 6.76 17.576 1.612 1.375 .5 56.25 421.875 2.739 1.957 .7 7 29 19.683 1.643 1.392 .6 57.76 438.976 2.757 1.966 .8 7^84 21.952 1.673 1.409 .7 59.29 456.533 2.775 1.975 .9 8.41 24.389 1.703 1.426 .8 60.84 474.552 2 793 1.983 3. 9. 27. 1.7321 1.4422 .9 62.41 493.039 2.81J 1.992 SQUARES, CUBES, SQUARE AKD CUBE ROOTS. 87 No. Square. Cube. Sq. Root. Cube Root. No. Square. Cube. Sq. Root. Cube Root. 8. 64. 512. 2.8284 2. 45 2025 91125 6.7082 3.5569 .1 65.61 531.441 2.846 2 008 46 2116 97336 6.7823 3.5830 .2 67.24 551.368 2.864 2.017 47 2:209 103823 6.8557 3.6088 .3 68.89 571.787 2.881 2 0^5 48 2304 110592 6.9282 3.6342 .4 70.56 592.704 2.898 2.033 49 2401 117649 7. 3.6593 .5 72.25 614.125 2.915 2.041 50 2500 125000 7.0711 3.6840 .6 73.96 636.056 2.933 2.049 51 2601 132651 7.1414 3.7084 .7 75.69 658.503 2.950 2.057 52 2704 140608 7 2111 3.7325 .8 77.44 681.472 2.966 2.065 53 2809 148877 7.2801 3.7563 9 79.21 704.969 2.983 2.072 54 2916 157464 7.3485 3.7798 9. 81. 729. 3. 2.0801 55 3025 166375 7.4162 3.8030 .1 82.81 753.571 3.017 2.088 56 3136 175616 7.4833 3.8259 .2 84.64 778.688 3.033 2.095 57 3249 185193 7.5498 3.8485 .3 86.49 804.35? 3.050 2.103 58 3364 195112 7.6158 3.8709 .4 88.36 830.584 3.066 2.110 59 3481 205379 7.6811 3.8930 .5 90.25 857.375 3.082 2.118 60 3600 216000 7.7460 3.9149 .6 92.16 884.736 3.098 2.125 61 3721 226981 7.8102 3.9365 .7 94.09 912.673 3.114 2 133 62 3844 238328 7.8740 3.9579 .8 96.04 941.192 3.130 2.140 63 3969 250047 7.9373 3.9791 .9 98.01 970.299 3.146 2.147 64 4096 262144 8. 4. 10 100 1000 3.1623 2.1544 65 4225 274625 8.0623 4.0207 11 121 1331 3.3166 2.2240 66 4356 287496 8.1240 4.0412 12 144 1728 3.4641 2.2894 67 4489 300763 8.1854 4.0615 13 169 2197 3.6056 2.3513 68 4624 314432 8.2462 4.0817 14 196 2744 3.7417 2.4101 69 4761 3vJ8509 8.3066 4.1016 15 225 3375 3.8730 2.4662 70 4900 343000 8.3666 4.1213 16 256 4096 4. 2.5198 71 5041 357911 8.4261 4.1408 17 289 4913 4.1231 2.5713 72 5184 373248 8.4853 4.1602 18 324 5832 4.2426 2.6207 73 5329 389017 8.5440 4.1793 19 361 6859 4.3589 2.6684 74 5476 405224 8.6023 4.1983 20 400 8000 4.4721 2./144 75 5625 421875 8.6603 4.2172 21 441 9261 4.5826 2 7589 76 5776 438976 8.7178 4.2358 22 484 10648 4.6904 2.8020 77 5929 456533 8.7750 4.2543 23 529 12167 4.7958 2.8439 78 6084 474552 8.8318 4.2727 24 576 13824 4.8990 2.8845 79 6241 493039 8.8882 4.2908 25 625 15625 5. 2.9240 80 6400 512000 8.9443 4.3089 26 676 17576 5.0990 2.9625 81 6561 531441 9. 4.3267 27 729 19683 5.1962 3. 8-.' 6724 551368 9.0554 4.3445 28 784 21952 5.2915 3 0366 83 6889 571787 9.1104 4.3621 29 841 24389 5.3852 3.0723 84 7056 592704 9.1652 4.3795 30 900 27000 5.4772 3.1072 85 7225 614125 9.2195 4.3968 31 961 29791 5.5678 3.1414 86 7396 636056 9.2736 4.4140 32 1024 32768 5.6569 3.1748 87 7569 658503 9 3276 4.4310 33 1089 35937 5.7446 3.2075 88 7744 6S1472 9.3808 4.4480 34 1156 39304 5.8310 3.2396 89 7921 704969 9.4340 4.4647 35 1225 42875 5.9161 3.2711 90 8100 729000 9.4868 4.4814 36 1296 46656 6. 3.3019 91 8281 753571 9.5394 4.4979 37 1369 50653 6.0828 3.3322 92 8464 778688 9.5917 4.5144 38 1444 54872 6.1644 3.3620 93 8649 804357 9 6437 4.5307 39 1521 59319 6.2450 3.3912 94 8836 830584 9.6954 4.5468 40 1600 64000 6.3246 3 4200 95 9025 857375 9 7468 4.5629 41 1681 689'>1 i 6. 4031 3.4482 96 9216 884736 9.7980 4.5789 42 1764 74088 6.4807 3.4760 97 9409 912673 9.8489 4.5947 43 1849 79507 6.5574 3.5034 98 9604 941192 9.8995 4.6104 44 1936 85184 6.6332 3.5303 99 9801 970-299 9.9499 4.6261 88 MATHEMATICAL TABLES. No. Square. Cube. Sq. Root. Cube Root. No. Square. Cube, Sq. Root. Cube Root. 100 10000 1000000 10. 4.6416 155 24025 3723875 12.4499 5.3717 101 10201 1030301 10.0499 4.6570 156 24336 3796416 12.4900 5.3832 10:3 10404 1061208 10.0995 4.6723 157 24649 3869893 12.5300 5.3947 103 10609 1092727 10.1489 4.6875 158 24964 3944312 12.5698 5.4061 104 10816 1124864 10.1980 4.7027 159 25281 4019679 12.6095 5.4175 105 11025 1157625 10.2470 4.7177 160 25600 4096000 12.6491 5.4288 106 11236 1191016 10.2956 4.7326 161 25921 4173281 12.6886 5.4401 107 11449 1225043 10.3441 4.7475 162 26244 4251528 12.7279 5 4514 108 11664 1259712 10.3923 4.7622 163 26569 4330747 12.7671 5.4626 109 11881 1295029 10.4403 4.7769 164 26896 4410944 12.8062 5.4737 110 12100 1331000 10.4881 4.7914 165 27225 4492125 12.8452 5.4848 111 12321 1367631 10.5357 4.8059 166 27556 4574-296 12.8841 5.4959 112 12514 1404928 10.5830 4.8203 167 27889 4657463 12.9228 5.5069 113 12769 1442897 10.6301 4.8346 168 28224 4741632 12.9615 5.5178 114 12996 1481544 10.6771 4.8488 169 28561 4826809 13.0000 5.5288 115 132-25 1520875 10.7238 4.8629 170 28900 4913000 13.0384 5.5397 116 13456 156089(5 10.7703 4.8770 171 29241 500021 1 13.0767 5.5505 117 13689 1601613 10.8167 4.8910 172 29584 5088448 13.1149 5.5613 118 13924 1643032 10.8628 4.9049 173 29929 5177717 13.1529 5.5721 119 14161 1685159 10.9087 4.9187 174 30276 5268024 13.1909 5.5828 120 14400 1728000 10.9545 4.9324 175 30625 5359375 13.2288 5.5934 121 14641 1771561 11.0000 4.9461 176 30976 5451776 13.2665 5.6041 122 14884 1815848 11.0454 4.9597 177 31329 5545233 13.3041 5.6147 123 15129 1860867 11.0905 4.9732 178 31684 5639752 13.3417 5.6252 124 15376 1906624 11.1355 4.9866 179 32041 5735339 13.3791 5.6357 125 15625 1953125 11.1803 5.0000 180 32400 583-2000 13.4164 5. 6402 126 15876 2000376 11.2250 5.0133 181 32761 5929741 13.4536 5.6567 127 16129 2018383 11.2694 5 0265 182 33124 6028568 13.4907 5.6671 128 16384 2097152 11.3137 5.0397 183 33489 6128487 13.5277 5.6774 129 16641 2146689 11.3578 5.0528 184 33856 6229504 13.5647 5.6877 130 16900 2197000 11.4018 5.0658 185 342-25 6331625 13.6015 5.6980 131 17161 2248091 11.4455 5.0788 186 34596 6434856 13.6382 5.7083 132 17424 2299968 11.4891 5.0916 187 34969 6539203 13.6748 5.7185 133 17689 2352637 11.5326 5.1045 188 35344 6644672 13.7113 5.T287 134 17956 2406104 11.5758 5.1172 189 35721 6751269 13.7477 5.7388 135 18225 2460375 11.6190 5.1299 190 36100 6859000 13.7840 5.7489 136 18496 2515456 11.6619 5.1426 191 36481 6967871 13.8203 5.7590 137 18769 2571353 11.7047 5.1551 192 36864 7077888 13.8564 5.7690 138 19044 2628072 11.7473 5.1676 193 37249 7189057 13.8924 5 7790 139 19321 2685619 11.7898 5.1801 194 37636 7301384 13.9284 5.7890 140 19600 2744000 11.8322 5.1925 195 38025 7414875 3.9642 5.7989 141 19881 2803221 11.8743 5.2048 196 38416 7529536 14.0000 5.8088 142 20164 2863286 11.9164 5.2171 197 38809 7645373 14.0357 5.8186 143 20449 29-24207 11.9583 5.2293 198 39204 7762392 14.0712 5.8285 144 20736 2985984 12.0000 5.2415 199 39601 7880599 14.1067 5.8383 145 21025 3048625 12.0416 5.2536 200 40000 8000000 14.1421 5.8480 146 21316 3112136 12.0830 5.2656 201 40401 8120601 14.1774 5.8578 147 21609 31765-23 12.1244 5.2776 202 40804 824-2408 14.2127 5.8675 148 21904 3241792 12.1655 5.2896 203 41209 8365427 14.2478 5.8771 149 22201 3307949 12.2066 5.3015 204 41616 8489664 14.2829 5.8868 150 22500 3375000 12.2474 5.3133 205 42025 8615125 14.3178 5.8964 151 2-2801 3442951 12.2882 5.3251 206 42436 8741816 14.3527 5.9059 152 23104 3511808 12.3288 5.3368 207 42849 8869743 14.3875 5.9155 153 23409 3581577 12.3603 5 . :>, 185 208 43264 8998912 14.4222 5.9250 154 23716 3652264 1? *M>~ 5.3C,0! 209 43681 9129329 14.4568 5.9345 SQUARES, CUBES, SQUARE AtfD CUBE ROOTS. 89 No. Square. Cube. Sq. Root. Cube Root. No. Square. Cube. Sq. Root. Cube Root. 210 44100 9261000 14.4914 5.9439 265 70225 18609625 16.2788 6.4232 211 44521 9393931 14.5258 5.9533 266 70756 18821096 16.3095 6.4312 212 44944 9528128 14.5602 5.9627 267 71289 19034163 16.3401 6.4393 213 45369 9663597 14.5945 5.9721 268 71824 19248832 16.3707 6.4473 214 45796 9800344 14.6287 5.9814 269 72361 19465109 16.4012 6.4553 215 46225 9938375 14.6629 5 9907 270 72900 19683000 16 4317 6.4633 216 46656 10077696 14.6969 6.0000 271 73441 19902511 16.4621 6.4713 217 47089 10218313 14.7309 6.0092 272 73984 20123648 16.4924 6.4792 218 47524 10360232 14.7648 6 0185 273 74529 20346417 16.5227 6.4872 219 47961 10503459 14.7986 6 0477 274 75076 20570824 16.5529 6.4951 220 48400 10648000 14.8324 6.0368 275 75625 20796875 16.5831 6.5030 221 48841 10793861 14.8661 6.0459 276 76176 21024576 16.6132 6.5108 222 49284 10941048 14.8997 6 0550 277 76729 21253933 16.6433 6.5187 223 49729 11089567 14.9332 6.0641 278 77284 21484952 16.6733 6.5265 224 50176 112394^4 14.9666 6.0732 279 77841 21717639 16.7033 6.5343 2-25 50625 11390625 15.0000 6.0822 280 78400 21952000 16.7332 6.5421 226 51076 11543176 15.0333 6.0912 281 78961 22188041 16.7631 6.5499 227 51529 11697083 15.0665 6.1002 282 79524 22425768 16.7929 6.5577 228 51984 11852352 15.0997 6.1091 283- 80089 22665187 16.8226 6.5654 229 52441 12008989 15.1327 6.1180 284 80656 22906304 16.8523 6.5731 230 52900 12167000 15.1658 6.1269 285 81225 23149125 16.8819 6.5808 231 53361 12326:391 15.1987 6.1358 :286 81796 23393656 16.9115 6.5885 232 53824 12487168 15.2315 6.1446 287 82369 23639903 16.9411 6.5962 233 54289 12649337 15.2643 6.1534 288 82944 23887872 16.9706 6.6039 234 54756 12812904 15.2971 6.1622 289 83521 24137569 17.0000 6.6115 235 55225 12977875 15.3297 6.1710 290 84100 24389000 17.0294 6.6191 236 55696 13144256 15.3623 6.1797 2.)1 84681 2464-J171 17.0587 6.6267 237 56169 13312053 15 3948 6.1885 292 85264 24897088 17.0880 6.6343 238 56644 13481272 15.4272 6.1972 293 85849 25153757 17.1172 6.6419 239 57121 13651919 15.4596 6.2058 294 86436 25412184 17.1464 6.6494 240 57600 13824000 15.4919 6.2145 295 87025 2567:2375 17.1756 6.6569 241 58081 13997521 15.5242 6.2231 296 87616 25934336 17.2047 6.6644 242 58564 14172488 15.5563 6.2317 297 88-^09 26198073 17.2337 6.6719 243 59049 14348907 15.5885 6.2403 r) 98 88804 26463592 17.2627 6.6794 244 59536 14526784 15.6205 6.2488 299 89401 26730899 17.2916 6.6869 245 60025 14706125 15.6525 6.2573 300 90000 27000000 17.3205 6.6943 246 60516 14886936 15.6844 6.2658 301 90601 27270901 17.3494 6.7018 247 61009 15069^3 15.7162 6.2743 30.2 91204 27543608 17.3781 6.7092 248 61504 15252992 15.7480 6.2828 303 91809 27818127 17.4069 6.7166 249 62001 15438249 15.7797 6.2912 304 92416 28094464 17 4356 6.7240 250 62500 15625000 15.8114 6.2996 305 93025 28372625 17.4642 6.7313 251 63001 15813-J51 15.8430 6.3080 306 93636 28652616 17.4929 6.7387 252 63504 16003008 15.8745 6.3164 307 94249 28934443 17.5214 6.7460 253 64009 16194;77 15.9060 6.3247 308 94864 29218112 17.5499 6.7533 254 64516 16387064 15.9374 6.3330 309 95481 29503629 17.5784 6.7606 255 65025 16581375 15.9687 6.3413 310 96100 29791000 17.6068 6.7679 256 65536 16777216 16.0000 6.3496 311 96721 30080231 17.6352 6 7752 257 66049 16974593 16.0312 6.3579 312 97344 30371328 17.6635 6.7824 258 66564 17173512 16.0624 6.3661 313 97969 30664297 17.6918 6.7897 259 67081 17373979 16.0935 6.3743 314 98596 30959144 17.7200 6.7969 260 67600 17576000 16.1245 6.3825 315 99225 31255875 17.7482 6.8041 261 68121 17779581 16.1555 6.3907 316 99856 31554496 17.7764 6.8113 262 68644 17984728 16.1864 6.3988 317 100489 31855013 17.8045 6.8185 2G3 69169 18191447 16.2173 6.4070 318 101124 32157432 17.8326 6.8256 264 69696 18399744 16.2481 6.4151 319 101761 324617o9 17.8606 6.8328 90 MATHEMATICAL TABLES. No. Square. Cube. Sq. Root. Cube Root. No. Square. Cube. Sq. Root. Cube Root. 320 102400 32768000 17.8885 6.8399 375 140625 52734375 19.3649 7.2112 321 103041 33076161 17.9165 6 8470 376 141376 58157376 19.3907 7.2177 322 103684 33386248 17.9444 6.8541 377 142129 53582633 19.4165 7.2240 323 104329 33698267 17.9722 6.8612 378 142884 54010152 19.4422 7.2304 324 104976 34012224 18.0000 6.8683 379 143641 54439939 19.4679 7.23G8 325 105625 34328125 18.0278 6.8753 380 144400 54872000 19.4936 7.2432 326 106276 34645976 18.0555 6.8824 381 145161 55306341 19.5192 7.2495 327 106929 34965783 18.0831 6.8894 382 145924 55742968 19.5448 7.2558 328 107584 35287552 18.1108 6.8964 383 146689 56181887 19.5704 7.2622 329 108241 35611289 18.1384 6.9034 384 147456 50623104 19.5959 7.2685 330 108900 35937000 18.1659 6.9104 385 148225 57066625 19.6214 7.2748 331 109561 36264691 18.1934 6.9174 386 148996 57512456 19.6469 7.2811 332 110224 36594368 18.2209 6.9244 387 149769 57960603 19.6723 7.2874 333 110889 36926037 18.2483 6.9313 388 150544 58411072 19.6977 7.2936 334 111556 37259704 18.2757 6.9382 389 151321 58863869 19.7231 7.2999 335 112225 37595375 18.3030 6.9451 390 152100 59319000 19.7484 7 3061 336 112896 37933056 18.3303 6.9521 391 152881 59776471 19.7737 7.3124 337 113569 38272753 18.3576 6.9589 392 153664 60236288 19.7990 7.3186 338 114244 38614472 18.3848 6.9658 393 154449 60698457 19.8242 7.3248 339 114921 38958219 18.4120 6.9727 394 155236 61162984 19.8494 7.3310 340 115600 39304000 18.4391 6.9795 395 156025 61629875 19.8746 7.3372 341 116281 39651821 18.4662 6.9864 396 156816 62099136 19.8997 7.3434 342 116964 40001688 18.4932 6 9932 397 157609 62570773 19.9249 7.3496 343 117649 40353607 18.5203 7.0000 398 158404 63044792 19.9499 7.3558 344 118336 40707584 18.5472 7.0068 399 159201 63521199 19.9750 7.3619 345 119025 41063625 18.5742 7.0136 400 160000 64000000 20 0000 7.3681 346 119716 41421736 18.6011 7.0203 401 160801 64481201 20 0250 7.3742 347 120409 41781923 18.6279 7.0271 402 161604 64904808 20.0499 7.3803 348 121104 42144192 18.6548 7.0338 403 162409 65450827 20 0749 7.3864 349 121801 42508549 18.6815 7.0406 404 163216 65^39264 20.0998 7.3925 350 122500 42875000 18.7083 7.0473 405 164025 66430125 20.1246 7.3986 351 123201 43243551 18.7350 7.0540 406 164836 66923416 20.1494 7.4047 352 123904 43614208 18.7617 7.0607 407 165649 67419143 20.1742 7.4108 353 124609 43986977 18.7883 7.0674 408 166464 67917312 20.1S90 7.4169 354 125316 44361864 18.8149 7.0740 409 167281 68417'929 20.2237 7.4229 355 126025 44738875 18.8414 7.0807 410 168100 68921000 20.2485 7.4290 356 126736 45118016 18.8680 7.0873 411 168921 69426531 20.2731 7.4350 357 127449 45499293 18.8944 7.0940 412 169744 69934528 20.2978 7.4410 358 128164 45882712 18 9209 7.1006 413 170569 70444997 20.3224 7.4470 359 128881 46268279 18.9473 7.1072 414 171396 70957944 20.3470 7.4530 360 129600 46056000 18.9737 7.1138 415 172225 71473375 20.3715 7.4590 361 130321 47045881 19.0000 7.1204 416 173056 71991296 20.3961 7.4650 362 131044 47437928 19.0263 7.1269 417 173889 72511713 20.4206 7.4710 363 131769 47832147 19.0526 7.1335 418 174724 73034632 20.4450 7.4770 364 132496 48228544 19.0788 7.1400 419 175561 73500059 20.4695 7.4829 365 133225 48627125 19.1050 7.1466 420 176400 74088000 20.4939 7.4889 366 133956 49027896 19.1311 7.1531 421 177241 74618461 20.5183 7.4948 367 134689 49430863 19.1572 7.1596 422 178084 75151448 20.5426 7.5007 368 135424 49836032 19.1833 7.1661 423 178929 75686967 20.5670 7.5067 369 136161 50243409 19.2094 7.1726 424 179776 76225024 20.5913 7.5126 370 136900 50653000 19.2354 7.1791 425 180625 76765625 20.6155 7.5185 371 137641 51064811 19.2614 7.1855 426 181476 77308776 20.6398 7.5244 372 138384 51478848 19.2873 7.1920 427 182329 77854483 20.6640 7.5302 373 139129 51895117 19.3132 7.1984 428 183184 78402752 20.6882 7.5361 374 139876 52313624 19.3391 7.2048 429 184041 78953589 20.7123 7.5420 SQUARES, CUBES, SQUARE AND CUBE ROOTS. 91 No. Square. Cube. Sq. Root. Cube Root. No. Square. Cube. Sq. Root. Cube Root. 430 184900 79507000 20.7364 7.5478 485 235225 114084125 22.0227 7.8568 431 185761 80062991 20.7605 7 . 5537 486 236196 114791256 22.0454 7.8622 432 186624 80621568 20.7846 7.5595 487 237169 115501303 22.0681 7.8676 433 187489 SI 182737 20.8087 7.5654 488 238144 116214.272 22.0907 7.8730 434 188356 81746504 20.8327 7.5712 489 239121 116930169 22.1133 7.8784 435 189225 82312875 20.8567 7.5770 490 240100 117649000 22.1359 7.8837 436 190096 82881856 20.8806 7.5828 491 241081 ,118370771 22.1585 7.8891 437 190969 83453453 20.9045 7.5886 492 242064 1119095488 22.1811 7.8944 438 191844 84027672 20.9284 7.5944 493 243049 1119823157 22.2036 7.8998 439 192721 84604519 20.9523 7.6001 494 244036 120553784 22.2261 7.9051 440 193600 85184000 20.9762 7.6059 495 245025 121287375 22.2486 7.9105 441 194481 85766121 21.0000 7.6117 496 246016 122023936 22.2711 7.9158 442 195364 86350888 21.0238 7.6174 497 247009 122763473 22.2935 7.9211 443 196249 86938307 21.0476 7.6232 498 248004 123505992 22.3159 7.9264 444 197136 87528384 21.0713 7.6289 499 249001 124251499 22 3383 7.9317 445 198025 88121125 21.0950 7.6346 500 250000 125000000 22.3607 7.9370 446 198916 88716536 21.1187 7.6403 501 251001 125751501 22.3830 7.9423 447 199809 89314623 21.1424 7.6460 502 252004 126506008 22.4054 7.9476 448 200704 8991539-3 21.1660 7.6517 503 253009 127263527 22.4277 7.9528 449 201601 90518849 21.1896 7.6574 504 254016 128024064 22.4499 7.9581 450 202500 91125000 21.2132 7.6631 505 255025 128787625 22.4722 7.9634 451 203401 91733851 21.2368 7.6688 506 256036 129554216 22.4944 7.9686 452 204804 92345408 21.2603 7.6744 507 257049 130323843 22.5167 7.9739 453 205209 92959677 21.2838 7.6800 508 258064 131096512 22.5389 7.9791 454 206116 93576664 21.3073 7.6857 509 259081 131872229 22.5610 7.9843 455 207025 94196375 21.3307 7.6914 510 260100 132651000 22.5832 7.9896 456 207936 94818816 21.3542 7.6970 511 261121 133432831 22.6053 7.9948 457 208849 95443993 21.3776 7.7026 512 262144 134217728 22.6274 8.0000 458 209764 96071912 21.4009 7.7082 513 263169 135005697 22.6495 8.0052 159 210681 96702579 21.4243 7.7138 514 264196 135796744 22.6716 8.0104 460 211600 97336000 21.4476 7.7194 515 265225 136590875 22 6936 8.0156 iQl 212521 97972181 21.4709 7.7250 516 266256 137388096 22.7156 8.0208 462 213444 98611128 21.4942 7.7306 517 267289 1138188413 22.7376 8.0260 463 214369 99252847 21.5174 7.7362 518 268324 138991832 22.7596 8.0311 464 215296 99897344 21.5407 7.7418 519 269361 139798359 22.7816 8.03G3 465 216225 100544625 21.5639 7 . 7473 520 270400 140608000 22.8035 8.0415 466 217156 101194696 21 5870 7.7529 521 271441 1141420761 22.8254 8.0466 467 218089 101847563 21.6102 7.7584 522 272484 142236648 22.8473 8.0517 468 219024 102503232 21.6333 7.7639 523 273529 143055667 22.8692 8.0569 469 219961 103161709 21.6564 7.7695 524 274576 143877824 22.8910 8.0620 470 220900 103823000 21.6795 7.7750 525 275625 144703125 22.9129 8.0671 471 221841 104487111 21.7025 7.7805 526 276676 145531576 22.9347 8.0723 472 222784 105154048 21.7256 8.7860 527 277729 146363183 22.9565 8.0774 473 223729 105823317 21.7486 7.7915 528 278784 147197952 22.9783 8.0825 474 224676 106496424 21.7715 7.7970 529 279841 148035889 23.0000 8.0876 475 225625 107171875 21.7945 7.8025 530 280900 148877000 23.0217 8.0927 476 226576 107850176 121.8174 7. 8079 1 531 281961 149721291 23.0434 8.0978 477 227529 108531333 21.8403 7. 8184 1 582 283024 150568768 23.0651 8.1028 478 228484 109215352 21 8632 7.8188 533 284089 151419437 23.0368 8.1079 479 229441 109902239 21.8861 7.8243 534 285156 152273304 23.1084 8.1130 480 230400 110592000 21 9089 7.8297 535 286225 153130375 23.1301 8.1180 481 231361 111284641 21.9317 7.8352 536 287296 153990656 23.1517 8.1231 482 232324 111980168 21.9545 7.8406 537 288369 154854153 23.1733 8.1281 483 233-89 112678587 21.9773 7.8460 538 289444 155720872 23.1948 8.1332 484 234256 113379904 22.0000 7.8514 539 290521 156590819 23.2164 8.1382 92 MATHEMATICAL TABLES. No. Square. Cube. Sq. Root. Cube Root. No. Square. Cube. Root. Cube Root. 540 291600 157464000 23.2379 8.1433 595 354025 210644875 24.3926 8.4108 541 292681 158340421 23.2594 8.1483 596 355216 211708736 24.4131 8.4155 542 293764 159220088 23.2809 8.1533 597 356409 212776173 24.4336 8.4202 543 294849 160103007 23.3024 8.1583 598 357604 213847192 24.4540 8.4249 544 295936 160989184 23.3238 8.1633 599 358801 214921799 24.4745 8.4296 545 297025 161878625 23.3452 8.1683 6CO 360000 216000000 24.4949 8 4343 546 298116 162771336 23.3666 8.1733 601 301201 217081801 24.5153 8.4390 547 299209 163667323 23.3880 8.1783 602 362404 218167208 24.5357 8.4437 548 300304 164566592 23.4094 8.1833 603 363609 219256227 24.5561 8.4484 549 301401 165469149 23.4307 8.1882 604 364816 220348864 24.5764 8.4530 550 302500 166375000 23.4521 8.1932 605 366025 221445125 24.5967 8.4577 551 303601 167284151 23.4734 8.1982 606 367236 22254501 ( 24.6171 8.4623 552 304704 168196H08 23.4947 8.2031 607 368449 223648543 24.6374 8.4670 553 305809 169112377 23.5160 8.2081 608 369664 224755712 24.6577 8.4716 554 306916 170031464 23.5372 8.2130 609 370881 225866529 24.6779 8.4763 555 308025 170953875 23.5584 8.2180 610 372100 226981000 24 6982 8.4809 556 309136 171879616 23.5797 8.2229 611 373321 228099131 24.7184 8.4856 557 310249 172808693 23.6008 8.2278 612 374544 229220928 24 7386 8.4902 558 311364 173741112 .23.6220 8.2327 613 375769 230346397 24.7588 8.4948 559 312481 174676879 23.6432 8.2377 614 376996 231475514 24.7790 8.4994 560 313600 175616000 23.6643 8.2426 615 378225 232608375 24.7992 8.5040 561 314721 176558481 23.6854 8.2475 616 379456 233744896 24.8193 8.5086 562 315844 177504328 23.7065 8 2524 617 380689 234885113 24.8395 8.5132 563 316969 178453547 23.7276 8.2573 618 381924 236029032 24.8596 8.5178 564 318096 179406144 23.7487 8.2621 619 383161 237176659 24.8797 8.5224 565 319225 180362125 23.7697 8.2670 620 384400 238328000 24.8998 8.5270 566 320356 181321496 23.7908 8.2719 621 385641 239483061 24.9199 8.5316 567 321489 182284263 23.8118 8.2768 622 3*6884 240641848 24 . 9399 8.5362 568 322624 183250432 23.8326 8.2816 623 388129 241804367 24.9600 8.5408 569 323761 184220009 23.8537 8.2865 624 389376 242970624 24.9800 8.5453 570 324900 185193000 23.8747 8.2913 625 390625 244140625 25.0000 8.5499 571 326041 186169411 23.8956 8.2962 626 391876 245314376 25.0200 8.5544 572 327184 187149248 23.9165 8.3010 627 393129 246491883 25.0400 8.5590 573 328329 188132517 23.9374 8.3059 628 394384 247673152 25.0599 8.5635 574 329476 189119224 23.9583 8.3107 629 395641 248858189 25.0799 8.5681 575 330625 190109375 23.9792 8.3155 630 396900 250047000 25.0998 8.5726 576 331776 191102976 24.0000 8.3203 631 398161 251239591 25.1197 8.5772 577 332929 19-2100033 24.0208 8.3251 632 399424 252435968 25.1396 8.5817 578 334084 193100552 24.0416 8.3300 633 400689 253636137 25.1595 8.5862 579 335241 194104539 24.0624 8.3348 634 401956 254840104 25.1794 8.5907 580 336400 195112000 24.0832 8.3396 635 403225 256047875 25.1992 8.5952 581 337561 196122941 24.1039 8.3443 636 404496 257259456 25.2190 8 5997 582 338724 197137368 24.1247 8.3491 637 405769 258474853 25.2389 8.6043 583 339889 198155287 24.1454 8.3539 638 407044 259694072 25.2587 8.6088 584 341056 199176704 24.1661 8.3587 639 408321 260917119 25.2784 8.6132 5S5 342225 200201625 24.1868 8.3634 640 409600 262144000 25.2982 8.6177 586 343396 201230056 24.2074 8.3682 641 410881 263374721 25.3180 8.6222 587 344569 202262003 24.2281 8.3730 642 412164 264609288 25.3377 8.6267 588 345744 203297472 24.2487 8.3777 643 413449 265847707 25.3574 8.6312 589 346921 204336469 24.2693 8.3825 644 414736 267089984 25.3772 8.6357 590 348100 205379000 24.2899 8.3872 645 416025 268836125 25.3969 8.6401 591 349281 206425071 24 3105 8.3919 646 417316 269586136 25.4165 8 6446 592 350464 207474688 24.3311 8.3967 647 418609 270840023 25.4362 8.6490 593 351649 208527857 24.3516 8.4014 648 419904 272097792 25.4558 8.G535 594 352836 209584584 24.3721 8.4061 649 421201 273359449 25.4755 8.6579 SQUARES, CUBES, SQUARE AND CUBE ROOTS. 93 No. Square. Cube. Sq. Root. Cube Root. No. Square. Cube. Sq. Root. Cube Root. 650 422500 274625000 25.4951 8.6624 705 497025 350402625 26.5518 8.9001 651 423801 275894451 25.5147 8.66(58 706 498436 351895816 26.5707 8.9043 652 425104 277167808 25.5343 8.6713 707 499849 353393243 26.5895 8.9085 653 426409 278445077 25.5539 8.6757 708 501264 354894912 26.6083 8.9127 654 427716 279726264 25.5734 8.6801 709 502681 356400829 26.6271 8.9169 655 429025 281011375 25.5930 8.6845 710 504100 357911000 26.6458 8.9211 656 430336 282300416 25.6125 8.6890 711 505521 359425431 26.6646 8.9253 65? 431649 283593393 25.6320 8.6934 712 506944 360944128 J6.6833 8.9295 658 432964 284890312 25.6515 8.6978 713 508369 362467097 26.7021 8.9337 659 434281 286191179 25.6710 8.7022 714 509796 363994344 26.7208 8.9378 660 435600 287496000 25.6905 8.7066 715 511225 365525875 26.7395 8.9420 661 436921 288804781 25.7099 8.7110 716 512656 367061696 26.7582 8.9462 662 438244 290117528 25 7294 8.7154 717 514089 368601813 26.7769 8.9503 663 439569 291434247 25.7488 8.7198 718 515524 370146232 26.7955 8.9545 664 440896 292754944 25.7682 8.7241 719 516961 371694959 26.8142 8.9587 665 442225 294079625 25.7876 8.7285 720 518400 373248000 26.8328 8.9628 666 443556 295408296 25.8070 8.7329 721 519841 374805361 26.8514 8.9670 667 444889 296740963 25.8263 8.7373 722 521284 376367048 26.8701 8.9711 668 446224 298077632 25.8457 8.7416 723 522729 377933067 26.8887 8.9752 669 447561 299418309 25.8650 8.7460 724 524176 379503424 26.9072 8.9794 670 448900 300763000 25.8844 8.7503 725 525625 381078125 26.9258 8.9835 671 450241 302111711 25.9037 8.7547 726 527076 382657176 26.9444 8.9876 672 451584 303464448 25.9230 8.7590 727 528529 384240583 26.9629 8.9918 673 452929 304821217 25.942". 8.7G34 728 529984 385828352 26.9815 8.9959 674 454276 306182024 25.9615 8.7677 729 531441 387420489 27.0000 9.0000 675 455625 307546875 25.9808 8.7721 730 532900 389017000 27 0185 9.0041 676 456976 308915776 26.0000 8.7764 731 534361 390617891 27.0370 9.0082 677 458329 310288733 26.0192 8.7807 732 535824 392223168 27.0555 9.0123 678 459684 311665752 26.0384 8.7850 733 537289 393832837 27.0740 9.0164 679 461041 313046839 26.0576 8.7893 734 538756 395446904 27.0924 9.0205 680 462400 314432000 26.0768 8.7937 735 540225 397065375 27.1109 9.0246 681 463761 315821241 26.0960 8.7980 736 541696 398688256 27.1293 9.0287 682 465124 317214568 26.1151 8.8023 737 543169 400315553 27.1477 9.0328 683 466489 318611987 26.1343 8.8066 738 544644 401947272 27.1662 9.0369 684 467856 320013504 26.1534 8.8109 739 546121 403583419 27.1846 9.0410 685 469225 321419125 26.1725 8.8152 740 547600 405224000 27.2029 9.0450 686 470596 322828856 26.1910 8.8194 741 549801 406869021 27.2213 9.0491 687 471969 324242703 26.2107 8.8237 742 550564 408518488 27.2397 9.0532 688 473344 325660672 26.2298 8.8280 743 552049 410172407 27.2580 9.0572 689 474721 327082769 26.2488 8.8323 744 553536 411830784 27.2764 9.0613 690 476100 328509000 26.2679 8.8366 745 555025 413493625 27.2947 9.0654 691 477481 329939371 26.2869 8.8408 746 556516 415160936 27.3130 9.0694 692 478864 331373888 26.3059 8.8451 747 558009 416832723 27.3313 9.0735 693 480249 332812557 26.3249 8.8493 748 559504 418508992 27.3496 9.0775 694 481636 334255384 26.3439 8.8536 749 561001 420189749 27.3679 9.0816 695 483025 335702375 26.3629 8.8578 750 562500 421875000 27.3861 9.0856 696 484416 337153536 26.3818 8.8621 751 564001 423564751 27.4044 9.0896 697 485809 338608873 26.4008 8.8663 752 565504 425259008 27.4226 9.0937 698 487204 340068392 26.4197 8.8706 753 567009 426957777 27.4408 9.0977 699 488601 341532099 26.4386 8.8748 754 568516 428661064 27.4591 9.1017 700 490000 343000000 26.4575 8.8790 755 570025 430368875 27.4773 9.1057 701 491401 344472101 26.4764 8.8833 756 571536 432081216 27.4955 9.1098 702 492804 345948408 26.4953 8.8875 757 573049 433798093 27.5136 9.1138 703 494209 347428927 26.5141 8.8917 758 574564 435519512 27.5318 9.1178 704 495616 348913664 26.5330 8.8959 759 576081 437245479 27.55001 9.1&18 MATHEMATICAL TABLES. No. Square. Cube. Sq. Root. Cube Root. No. Square. Cube. Sq. Root. Cube Root. 760 577600 438976000 27.5681 9.1258 815 664225 541343375 28.5482 9.3408 761 579121 440711081 27.5862 9.1298 81 6 j 665856 543338496 28.5657 9.3447 762 580644 442450728 27.6043 9.1338 817 667489 545338513 28.5832 9.3485 763 582169 444194947 27.6225 9.1378 818 669124 547343432 28.6007 9.3523 764 583696 445943744 27.6405 9.1418 819 670761 549353259 28.6182 9.3561 765 585225 447697125 27.6586 9.1458 820 672400 551368000 28.6356 9.3599 766 586756 449455096 27.6767 9.1498 821! 674041 553387661 28.6531 9.3637 767 588289 451217663 27.6948 9.1537 822 67'5684 555412248 28 . 6705 9.3675 768 589824 452984832 27.7128 9.1577 823 677329 557441767 28.6880 9.3713 769 591361 454756609 27.7308 9.1617 824 678976 559476224 28.7054 9.3751 770 592900 456533000 27.7489 9.1657 825 680625 561515625 28.7228 9.3789 771 594441 458314011 27.7669 9.1696 826 682276 563559976 28.7402 9.3827 772 595984 460099648 27.7849 9.1736 827 683929 565609283 28.7576 9.3865 773 597529 461889917 27.8029 9.1775 828 685584 567663552 28.7750 9.3902 774 599076 463684824 27.8209 9.1815 829 687241 569722789 28.7924 9.3940 775 600625 465484375 27.8388 9.1855 830 688900 571787000 28.8097 9.3978 776 602176 467288576 27.8568 9.1894 831 690561 573856191 28.8271 9.4016 777 603729 469097433 27.8747 9.1933 832 692224 575930368 28.8444 9.4053 778 779 605284 606841 470910952 472729139 27.8927 27.9106 9.1973 9.2012 883 834 693889 695556 578009537 580093704 28.8617 28.8791 9.4091 9.4129 780 608400 474552000 27.9285 9.2052 835 697'225 582182875 28.8964 9.4166 781 609961 476379541 27.9464 9.2091 836 698896 584277056 28.9137 9.4204 782 611524 478211768 27.9643 9.2130 837 700569 586376253 28.9310 9.4241 783 613089 480048687 27.9821! 9.2170 838 702244 588480472 28.9482 9.4279 784 614656 481890304 28.0000 9.2209 839 703921 590589719 28.9655 9.4316 785 616225 483736625 28.0179 9.2248 840 705600 592704000 28.9828 9.4354 786 617796 485587656 28.03571 9.2287 841 707281 594823321 29.0000 9.4391 787 619369 487443403 28.0535 9.2326 842 708964 596947688 29.0172 9.4429 788 620944 489303872 28.0713! 9.2365 843 710649 599077107 29.0345 9.4466 789 622521 491169069 28.0891 9.2404 844 712336 601211584 29.0517 9.4503 790 624100 493039000 28.1069 1 9.2443 845 714025 603351125 29.0689 9.4541 791 625681 494913671 28.1247. 9.2482 846 715716 605495736 29.0861 9.4578 792 627264 496793088 28.14251 9.2521 847 717409 607645423 29.1033 9.4615 793 628849 498677257 28.1603 9.2560 848 719104 609800192 29.1204 9.4652 794 630436 500566184 28.1780 9.2599 849 720801 611960049 29.1376 9.4690 795 632025 502459875 28.1957 9.2638 850 722500 614125000 29.1548 9.4727 796 633616 504358336 28.2135 9.2677 851 724201 616295051 29.1719 9.4764 797 635209 50626157'3 28.2312! 9.2716 852 725904 618470208 29.1890 9.4801 798 636804 508169592 28.2489 9.2754 853 727609 620650477 29.2062 9.4838 799 638401 510082399 28.2666 9.2793 854 729316 622835864 29.2233 9.4875 800 640000 512000000 28.2843 9.2832 855 731025 625026375 29.2404 9.4912 801 641601 513922401 28.3019 9.2870 856 732736 627222016 29.2575 9.4949 802 643204 515849608 28.3196 9.2909 857 734449 629422793 29.2746 9.4986 803 644809 517781627 28.3373 9.2948 858 736164 631628712 29.2916 9.5023 804 646416 519718464 28.3549 9.2986 859 737881 633839779 29.3087 9.5060 805 648025 521660125 28.3725 9.3025 860 739600 636056000 29.3258 9.5097 806 649636 523606616 28.3901 9.3063 861 741321 C38277381 29.3428 9.5134 807 651249 525557943 28.4077 9.3102 862 743044 640503928 29.3598 9.5171 808 652864 5275141 12 '28. 4253 9.3140 863 744769 642735647 29.3769 9.5207 809 654481 529475129 28.4429 9.3179 864 746496 644972544 29.3939 9.5244 810 656100 531441000 28.4605 9.3217 865 748225 647214625 29.4109 9.5281 811 657721 53341173128.4781 9.3255 866 749956 649461896 29.4279 9.5317 812 659344 53538732828.4956 9.3294 867 751689 651714363 29.4449 9.5354 813 660969 53736779728.5132 9.3332 868 753424 653972032 29.4618 9.5391 814 662596 53935314428.5307 9.3370 869 755161 656234909 29.4788 9.5427 SQUARES, CUBES, SQUARE AND CUBE ROOTS. 95 No. Square. Cube. Sq. Root, Cube Root. No. Square. Cube. Sq. Root. Cube Root. 870 756900. 658503000 29.4958 9.5464 925 855625 791453125 30.4138 9.7435 871 758641 060776311 29.5127 9.5501 926 857476 794022776,30.4302 9.7470 872 760384 663054848 29.5296 9.5537 927 859329 796597983 '30. 4467 9.7505 873 762129 665338617 29.5466 9.5574 928 861184 799178752 30.4631 9 7540 874 763876 667627624 29.5635 9.5610 929 863041 801765089 30.4795 9.7575 875 765625 669921875 29.5804 9.5647 930 864900 804357000 30.4959 9.7610 876 767376 672221376 29.5973 9.5683 931 866761 806954491 30.5123 9.7645 877 769129 674526133 29.6142 9.5719 932 868624 809557568 30.5287 9.7680 878 770884 676836152 29.6311 9.5756 933 870489 812166237 30.5450 9.7715 879 772641 679151439 29.6479 9.5792 934 872356 814780504 30.5614 9.7750 880 774400 681472000 29.6648 9.5828 935 874225 817400375 30.5778 9.7785 881 776161 683797841 29.6816 9.5865 936 876096 820025856 30.5941 9.7819 882 777924 686128968 29.6985 9.5901 937 877969 822656953 30.6105 9.7854 883 779689 688465387 29.7153 9.5937 938 879844 825293672 30.6268 9.7880 884 781456 690807104 29.7321 9.5973 939 881721 827936019 30.6431 9.7924 885 783225 693154125 29.7489 9.6010 940 883600 830584000 30.6594 9.7959 886 784996 695506456 29.7658 9.6046 941 885481 833237621 30.6757 9.799S 887 786769 697864103129.7825 9.6082 942 887364 835896888 30.6920 9.8028 888 788544 70022707229.7993 9.6118 943 889249 838561807 30.7083 9.8063 889 790321 702595369 29.8161 9.6154 944 891136 841232384 30.7246 9.8097 890 792100 704969000 29.8329 9.6190 945 893025 843908625 30.7409 9.8132 891 793881 707347971 29.8496 9.6226 946i 894916 846590536 30.7571 9.8167 892 795664 709732288 29.8664 9.6262 947 1 896809 849278123 30.7734 9.8201 893 797449 712121957 29.8831 9.6298 948 898704 851971392 30.7896 9.8236 894 799236 714516984 29.8998 9.6334 949, 900601 854670349 30.8058 9.8270 895 801025 716917375 29.9166 9.6370 950 902500 857375000 30.8221 9.8305 896 802816 719323136 29.9333 9.6406 951 904401 860085351 30.8383 9.8339 897 804609 721734273 29.9500 9.6442 952 906304 862801408 30.8545 9.8374 898 806404 724150792 29.9666 9.6477 953 908209 865523177 30.8707 9.8408 899 808201 726572699 29.9833 9 6513 954 910116 868250664 30.8869 9.8443 900 810000 729000000 30 0000 9.6549 955' 912025 870983875 30.9031 9.8477 901 811801 731432701 30.0167 9.6585 956 913936 873722816 30.9192 9.8511 902 I 813604 733870808 30.0333 9.6620 957 i 915849 876467493 30.9354 9.8546 9031 815409 736314327 30.0500 9.6656 958 917764 879217912 30.9516 9.8580 904 817216 738763264 30.0666 9.6692 959 919681 881974079 30.9677 9.8614 905 819025 741217625 30.0832 9.6727 960 921600 884736000 30.9839 9.8648 906 820836 743677416 30.0998 9.6763 961 923521 887503681 31.0000 9.8683 907 822649 746142643 30.1164 9.6799 962 925444 890277128 31.0161 9.8717 908 824464 748613312 30.1330 9.6834 963 927369 893056347 31.0322 9.8751 909 826281 751089429 30.1496 9.6870 964 929296 895841344 31.0483 9.8785 910 828100 753571000 30.1662 9.6905 965 931225 898632125 31.0644 9.8819 911 829921 756058031 30.1828 9.6941 966 933156 901428696 31.0805 9.8854 912 831744 758550528 30.1993 9.6976 967 935089 904231063 31.0966 9.8888 913 833569 761048497 30.2159 9.7012 968 937024 907039232 31.1127 9.8922 914 835396 763551944 30.2324 9.7047 969 938961 909853209 31.1288 9.8956 915 837225 766060875 30.2490 9.7082 970 940900 912673000 31.1448 9.8990 916 839056 768575296 30.2655 9.7118 971 942841 915498611 31.1609 9.9024 917 840889 771095213 30.2820 9.7153 972 944784 918330048 31.1769 9.9058 918 842724 773620632 30.2985 9.7188 973 946729 921167317 31.1929 9.9092 919 844561 776151559 30.3150 9.7224 974 948676 924010424 31.2090 9.9126 920 846400 778688000 30.3315 9.7259 975 950625 926859375 31.2250 9.9160 921 848241 781229961 30.3480 9.7294 976 952576 929714176 31.2410 9.9194 922 850084 783777448 30.3645 9.7329 977 954529 932574833 31.2570 9 9227 923 851929 786330467 30.3809 9.7364 978 956484 935441352 31.2730 9.9261 924 853776 ! 788889024 30.3974 9.7400 979 958441 938313739 31.2890 9.9295 96 MATHEMATICAL TABLES. No. 980 981 982 983 984 Square. Cube. Sq. Root. Cube. Root. No. Square. Cube. Sq. Root. Cub6 Root, 960400 962361 964324 966289 968256 941192000 944076141 946966168 949862087 952763904 31.3050 31.3209 31.3369 31.3528 31.3688 9.9329 9.9363 9.9396 9.9430 9.9464 1035 1036 1037 1038 1039 1071225 1073296 1075369 1077444 1079521 1108717875 1111934656 1115157653 1118386872 1121622319 32.1714 32.1870 32.2025 32.2180 32.2335 10.1153 10.1186 10.1218 10.1251 10.1283 985 986 987 988 989 970225 972196 974169 976144 978121 955671625 958585256 961504803 964430272 967361669 31.3847 31.4006 31.4166 31.4325 31.4484 9.9497 9.9531 9.9565 9.9598 9.9632 1040 1041 1042 1043 1044 1081600 1083681 1085764 1087849 1089936 1124864000 1128111921 1131366088 1134626507 1137893184 32.2490 32.2645 32.2800 32.2955 32.3110 10.1316 10.1348 10.1381 10.1413 10.1446 990 991 992 993 994 980100 982081 984064 986049 988036 970299000 973242271 976191488 979146657 982107784 31.4643 31.4802 31.4960 31.5119 31.5278 9.9666 9.9699 9.9733 9.9766 9.9800 1045 1046 1047 1048 1049 1092025 1094116 1096209 1098304 1100401 1141166125 1144445336 1147730823 1151022592 1154320649 32.3265 32.3419 32.3574 32.3728 32.3883 10.1478 10.1510 10.1543 10.1575 10.1607 995 996 997 998 999 990025 992016 994009 996004 998001 985074875 988047936 991026973 994011992 997002999 31.5436 31.5595 31.5753 31.5911 31.6070 9.9833 9.9866 9.9900 9 9933 9.9967 1050 1051 1052 1053 1054 1102500 1104601 1106704 1108809 1110916 1157625000 1160935651 1164252608 1167575877 1170905464 32.4037 32.4191 32.4345 32.4500 32.4654 10.1640 10.1672 10.1704 10.1736 10.1769. 1000 3001 1002 1003 1004 1000000 1002001 1004004 1006009 1008016 1000000000 1003003001 1006012008 1009027027 1012048064 31.6228 31.6386 31.6544 31.6702 31.6860 10.0000 10.0033 10.0067 10.0100 10.0133 1055 1056 1057 1058 1059 1113025 1115136 1117249 1119364 1121481 1174241375 1177583616 1180932193 1184287112 1187648379 32.4808 32.4962 32.5115 32.5269 32.5423 10.1801 10.1833 10.1865 10.1897 10.1929 1005 1006 1007 1008 1009 1010025 1012036 1014049 1016064 1018081 1015075125 1018108216 1021147343 1024192512 1027243729 31.7017 31.7175 31.7333 31.7490 31.7648 10.0166 10.0200 10.0233 10.0266 10.0299 1060 1061 1062 1063 1064 1123600 1125721 1127844 1129969 1132096 1191016000 1194389981 1197770328 1201157047 1204550144 32.5576 32.5730 32.5883 32.6036 32.6190 10 1961 10.1993 10.2025 10.2057 10.2089 1010 1011 1012 1013 1014 1020100 1022121 1024144 1026169 1028196 1030301000 1033364331 1036433728 1039509197 1042590744 31.7805 31.7962 31.8119 31.8277 31.8434 10.0332 10.0365 10.0398 10.0431 10.0465 1065 1066 1067 1068 1069 1134225 1136356 1138489 1140624 1142761 1207949625 1211355496 1214767763 1218186432 1221611509 32.6343 32.6497 32.6650 32.6803 32.6956 10.2121 10.2153 10.2185 10.2217 10.2249 1015 1016 1017 1018 1019 1030225 1032256 1034289 1036324 1038361 1045678375 1048772096 1051871913 1054977832 1058089859 31.8591 31.8748 31.8904 31.9061 31.9218 10.0498 10.0531 10.0563 10.0596 10.0629 1070 1071 1072 1073 1074 1144900 1147041 1149184 1151329 1153476 1225043000 1228480911 1231925248 1235376017 1238833224 32.7109 32.7261 32.7414 32.7567 32.7719 10.2281 10.2313 10.2345 10.2376 10.2408 1020 1021 1022 1023 1024 1040400 1042441 1044484 1046529 1048576 1061208000 1064332261 1067462648 1070599167 1073741824 31.9374 31.9531 31.9687 31.9844 32.0000 10.0662 10.0695 10.0728 10.0761 10.0794 1075 1076 1077 1078 1079 1155625 1157776 1159929 1162084 1164241 1242296875 1245766976 1249243533 1252726552 1256216039 32.7872 32.8024 32.8177 32.8329 32.8481 10.2440 10.2472 10.2503 10.2535 10.2567 1025 1026 1027 1028 1029 1050625 1052676 1054729 1056784 1058841 1076890625 1080045576 1083206683 1086373952 1089547389 32.0156 32.0312 32.0468 32.0624 32.0780 10.0826 10.0859 10.0892 10.0925 10.0957 1080 1081 1082 1083 1084 1166400 1168561 1170724 1172889 1176056 1259712000 1263214441 1266723368 1270238787 1273760704 32.8634 32.8786 32.8938 32.9090 32.9242 10.2599 10.2630 10.2662 10.2693 10.2725 1030 1031 1032 1033 1034 1060900 1062961 1065024 1067089 1069156 1092727000 1095912791 1099104768 1102302937 1105507304 32.0936 32.1092 32.1248 32.1403 32.1559 10.0990 10.1023 10.1055 10.1088 10.1121 1085 1086 1087 1088 1089 1177225 1179396 1181569 1183744 1185921 1277289125 1280824056 1284365503 1287913472 1291467969 32.9393 32.9545 32.9697 32.9848 33.0000 10.2757 10.2788 10.2820 10.2851 10.2883 SQUARES, CUBES, SQUARE AKD CUBE ROOTS. 9? No. Square. Cube. Sq. Root. Cube Root. No. Square. Cube. Sq. Root. Cube Root. 1090 1188100 1295029000 33.0151 10.2914 1145 1311025 1501123625 33.8378 10.4617 1091 1190281 1298596571 33.0303 10.2946 1146 1313316 1505060136 33.8526 10.4647 1092 1192464 1302170688 33.0454 10.2977 1147 1315609 1509003523 33.8674 10.4678 1093 1194649 1;05751357 33.0606 10.3009 1148 1317904 1512953792 33.8821 10.4708 1094 1196836 1309338584 33.0757 10.3040 1149 1320201 1516910949 33.8969 10.4739 1095 1199025 1312932375 33.0908 10.3071 1150 1322500 1520875000 33.9116 10.4769 1096 1201216 1316532736 33.1059 10.3103 1151 1324801 1524845951 33.9264 10.4799 1097 1203409 1320139673 33.1210 10.3134 1152 1327104 1528823808 33.9411 10.4830 1098 1205604 1323753192 33.1361 10.3165 1153 1329409 1532808577 33.9559 10.4860 1099 1207801 1327373299 33.1512 10.3197 1154 1331716 1536800264 33.9706 10.4890 1100 1210000 1331000000 33.1662 10.3228 1155 1334025 1540798875 33.9853 10.4921 1101 1212201 1334633301 33.1813 10.3259 1156 1336336 1544804416 34.0000 10.4951 1103 1214404 1338273208 33.1964 10.3290 1157 1338649 1548816893 34.0147 10.4981 1103 1216609 1341919727 33.2114 10.3322 1158 1340964 1552836312 34.0294 10.5011 1104 1218816 1345572864 33.2264 10.3353 1159 1343281 1556862879 34.0441 10.5042 1105 1221025 1349232625 33.2415 10.3384 1160 1345600 1560896000 34.0588 10.5072 1106 1223236 1352899016 33.2566 10.3415 1161 1347921 1564936281 34.0735 10.5102 1107 1225449 1356572043 33.2716 10.3447 1162 1350244 1568983528 34.0881 10.5132 1108 1227G64 1360251712 33.2866 10.3478 1163 1352569 1573037747 34.1028 10.5162 1109 1229881 1363938029 33.3017 10.3509 1164 1354896 1577098944 34.1174 10.5192 1110 1232100 136?631000'33.3167 10.3540 1165 1357225 1581167125 34.1321 10 5223 1111 1234321 1371330631 33.3317 10.3571 1166 1359556 1585242296 34.1467 10.5253 1112 1236544 137503692833.3467 10.3602 1167 1361889 1589324463 34.1614 10.5283 1113 1238769 137874989733.3617 10.3633 1168 1364224 1593413632 34.1760 10.5313 111* 1240996 1382469544 33.3766 10.3664 1169 1366561 1597509809 34.1906 10.5343 1115 1243225 1386195875 33.3916 10 3695 1170 1368900 1601613000 34.2053 10.5373 1116 1245456 138992889633.4066 10.3726 1171 1371241 1605723211 34.2199 10.5403 1117 1247689 139366861338.4215 10.3757 1172 1373584 1609840448 34.2345 10.5433 1118 1249924 139741503233.4365 10.3788 1173 1375929 1613964717 34.2491 10.5463 1119 1252161 1401 168159|33. 4515 10.3819 1174 1378276 1618096024 34.2637 10.5493 1120 1254400 140492800033.4664 10.3850 1175 1380625 1622234375 34.2783 10.5523 11<J1 1256641 1408694561 33.4813 10.3881 1176 1382976 1626379776 34.2929 10.5553 1122 1258884 141246784833.4963 10.3912 1177 1385329 1630532233 34.3074 10.5583 1123 1261129 141624786733.5112 10.3943 1178 1387684 1634691752 34.3220 10.5612 1124 1263376 142003462433.5261 10.3973 1179 1390041 1638858339 34.3366 10.5642 1125 1265625 1423828125 33 5410 10.4004 1180 1392400 1643032000 34.3511 10.5672 1126 1267876 142762837633.5559 10.4035 1181 1394761 1647212741 34.3657 10.5702 1127 1270129 143143538333.5708 10.4066 1182 1397124 1651400568 34.3802 10.5732 1128 1272384 143524915233.5857 10.4097 1183 1399489 1655595487 34.3948 10 5762 1129 1274641 1439069689 33.6006 10.4127 1184 1401856 1659797504 34.4093 10.5791 | 1130 1276900 144289700033.6155 10.4158 1185 1404225 1664006625 34.4238 10.5821 1131 1279161 1446731091 33.6303 10.4189 1186 1406596 1668222856 34.4384 10.5851 1132 1281424 1450571968 33.6452 10.4219 1187 1408969 1672446203 34.4529 10.5881 1133 1283689 1454419637 33.6601 10.4250 1188 1411344 1676676672 34.4674 10.5910 1134 1285956 1458274104 33.0749 10.4281 1189 1413721 1680914269 34.4819 10.5940 1135 1288225 1462135375 33.6898 10.4311 1190 1416100 1685159000 34.4964 10.5970 1136 1290496 1466003456 33.7046 10.4342 1191 1418481 1689410871 34.5109 10.6000 1137 1292769 1469S78353 33.7174 10.4373 1192 1420864 1693669888 34.5254 10.6029 1138 1295044 1473760072 33.7342 10.4404 1193 1423249 1697936057 34.5398 10.6059 1139 1297321 1477648619 33.7491 10.4434 1194 1425636 1702209384 34.5543 10.6088 1140 1299600 1481544000 33.7639 10.4464 1195 1428025 1706489875 34.5688 10.6118 1141 1301881 1485446221 33.7787 10.4495 1196 1430416 1710777536 34.5832 10.6148 1142 1304164 1489355288 33.7935 10.4525 1197 1432809 1715072373 34.5977 10 6177 1143 1306449 1493271207 33.8083 10.4556 1198 1435204 1719374392 34.6121 10.6207 1144 1308736 1497198984 33.8231 10.4586 1199 1437601 1723683599 34.6266 10.6230 98 MATHEMATICAL TABLES. No. 1200 1201 1202 1203 1304 Square. Cube. Sq. Boot, Cube Root. No. Square. Cube. Sq. Root. Cube Root. 1440000 1442401 1444804 1447209 1449616 172800000034.6410 1732323601 34.6554 173665440834.6699 174099242734.6843 174533766434.6987 10.6266 10.6295 10.6325 10.6354 10.6384 1255 1256 1257 1258 1259 1575025 1577536 1580049 1582564 1585081 1976656375 35.4260 1981385216 35.4401 1986121593 35.4542 199086551235.4683 199561697935.4824 10.7865 10.7894 10.7922 10.7951 10.7980 1205 1206 1207 1208 1209 1452025 1454436 1456849 1459264 1461681 1749690125 1754049816 1758416743 1762790912 1767172329 34.7131 34.7275 34.7419 34.7563 34.7707 10.6413 10.6443 10.6472 10.6501 10.6530 1260 1261 1262 1263 1264 1587600 il 590 121 1592644 1595169 1597696 2000376000 35.4965 2005142581 35.5106 200991672835.5246 201469844735.5387 2019487744,35.5528 10.8008 10.8037 10.8065 10.8094 10.8122 1210 1211 1212 1213 1214 1464100 1466521 1468944 1471369 1473796 1771561000 1775956931 1780360128 1784770597 1789188344 34.7851 34.7994 34.8138 34.8281 34.8425 10.6560 10.8590 10.6619 10.6648 10.6678 1265 1266 1267 1268 1269 1600225 1602756 1605289 1607824 1610361 2024284625 2029089096 2033901163 2038720832 2043548109 35.5668 35.5809 35.5949 35.6090 35.6230 10.8151 10.8179 10 8208 10.8236 10.8265 1215 1216 1217 1218 1219 1476225 1478656 1481089 1483524 1485961 1793613375 1798045696 1802485313 1806932232 1811386459 34.8569 34.8712 34.8855 34.8999 34.9142 10.6707 10.6736 10.6765 10.6795 10.6824 1270 1271 1272 1273 1274 1612900 1615441 1617984 1620529 1623076 2048383000 2053225511 2058075648 2062933417 2067798824 35.6371 35.6511 35.6651 35.6791 35.6931 10.8293 10.8322 10.8350 10 8378 10.8407 1220 1221 1222 1223 1224 1488400 1490841 1493284 1495729 1498176 1815848000 1820316861 1824793048 1829276567 1833767424 34.9285 34.9428 34.9571 34.9714 34.9857 10.6853 10.6882 10.6911 10.6940 10.6970 1275 1276 1277 1278 1279 1625625 1628176 1630729 1633284 1635841 2072671875 2077552576 2082440933 2087336952 2092240639 35.7071 35.7211 35.7351 35.7491 35.7631 10.8435 10.8463 10.8492 10.8520 10.8548 1225 1226 12:27 1228 1229 1500625 1503076 15055-*) 1507984 1510441 1838265625 1842771176 1847284083 1851804352 1856331989 35.0000 35.0143 35.0286 35.0428 35.0571 10.6999 10.7028 10.7057 10.7086 10.7115 1280 1281 1282 1283 1284 1638400 1640961 1643524 1646089 1648656 2097152000 2102071041 2106997768 2111932187 2116874304 35.7771 35.7911 35.8050 35.8190 35.8329 10.8577 10.8605 10.8633 10.8661 10.8690 1230 1231 1232 1233 1234 1512900 1515361 1517824 1520289 1522756 1860867000 1865409391 1869959168 1874516337 1879080904 35.0714 35.0856 35.0999 35.1141 35.1283 10.7144 10.7173 10.7202 10.7231 10.7260 1285 1286 1287 1288 1289 1651225 1653796 1656369 1658944 1661521 2121824125 2126781656 2131746903 2136719872 2141700569 35.8469 35.8608 35.8748 35.8887 35.9026 10.8718 10.8746 10.8774 10.8802 10.8831 1235 1236 1237 1238 1239 1525225 1527696 1530169 1532644 1535121 1883652875 1888232256 1892819053 1897413272 1902014919 35.1426 35.1568 35.1710 35.1852 35.1994 10.7289 10.7318 10.7347 10.7376 10.7405 1290 1291 1292 1293 1294 1664100 1666681 1669264 1671849 1674436 2146689000 2151685171 2156689088 2161700757 2166720184 35.9166 35.9305 35.9444 35.9583 35.9722 10.8859 10.8887 10.8915 10.8948 10.8971 1240 1241 1242 1243 1244 1537600 1540081 1542564 1545049 1547536 1906624000 1911240521 1915864488 1920495907 1925134784 35.2136 35.2278 35.2420 35.2562 35.2704 10.7434 10.7463 10.7491 10.7520 10.7549 1295 1296 1297 1298 1299 1677025 1679616 1682209 1684804 1687401 2171747375 2176782336 2181825073 2186875592 2191933899 35.9861 36.0000 36.0139 36.0278 36.0416 10.8999 10.9027 10.9055 10.9083 10.9111 1245 1246 1247 1248 1249 1550025 1552516 1555009 1557504 1560001 1929781125 1934434936 1939096223 1943764992 19^8441249 35.2846 35.2987 35.3129 35.3270 35.3412 10.7578 10.7607 10.7635 10.7664 10.7693 1300 1301 1302 1303 1304 1690000 1692601 1695204 1697809 1700416 2197000000 2202073901 2207155608 2212245127 2217342464 36.0555 36.0694 36.0832 36.0971 36.1109 10.9139 10.9167 10.9195 10.9223 10.9251 1250 1261 1252 1253 1254 1562500 1565001 1567504 1570009 1572516 19531 25000 1957816251 1962515008 1967221277 1971935064 35.3553 35.3695 35.3836 35.3977 35.4119 10.7722 10.7750 10.7779 10 7808 10.7837 1305 1306 1307 1308 1309 1703025 1705636 1708249 17108G4J 1713481! 2222447625 2227560616 2232681443 2237810112 2242946629 36.1248 36.1386 36.1525 36.1663 36.1801 10.9279 10.9307 10.9335 10.9363 10.939 SQUARES, CUBES, SQUARE AKD CUBE ROOTS, 1)9 STo. Square. Cube. Sq. Root. Cube Root. No. Square. Cube. Sq. Root. Cube Root. 310 311 312 313 314 17161002248091000 1718721 2253243231 1721344 2258403328 1723969 2263571297 1726596 2268747144 36.1939 36. 2077 36.2215 36.2353 36.2491 10.9418 10.9446 10.9474 10.9502 10.9530 1365 1366 1367 1368 1369 1863225 1865956 1868689 1871424 1874161 2543302125 2548895896 2554497863 2560108032 2565726409 36.9459 36.9594 36.9730 36.9865 37.0000 11.0929 11.0956 11.0983 11.1010 11.1037 315 31(5 317 318 319 1729225 2273930875 1731856 2279122496 17344892284322013 1737124 2289529432 1739761 2294744759 36.2629 36.2767 36.2905 36.3043 36.3180 10.9557 10.9585 10.9613 10.9640 10.9668 1370 1371 1372 1373 1374 1876900 1879641 1882384 1885129 1887876 2571353000 2576987811 2582630848 2588282117 2593941624 37.0135 37.0270 37.0405 37.0540 37.0675 11.1064 11.1091 11.1118 11.1145 11.1172 3-20 321 322 823 324 1742400 1745041 1747684 1750329 1752976 2299968000 2305199161 2310438248 2315685267 2320940224 36.3318 36.3456 36.3593 36.3731 36.3868 10.9696 10.9724 10.9752 10.9779 10.9807 1375 1376 1377 1378 1379 1890625 1893376 1896129 1898884 1901641 2599609375 2605285376 2610969633 2616662152 2622362939 37.0810 37 0945 37.1080 37.1214 37.1349 11.1199 11.1226 11.1253 11.1280 11.1307 325 326 38? 32S 329 1755625 1758276 1760929 1763584 1766241 2326203125 2331473976 2336752783 2342039552 2347334289 36.4005 36.4143 36.4280 36.4417 36.4555 10.9834 10.9862 10.9890 10.9917 10.9945 1380 1381 1382 1383 1384 1904400 1907161 1909924 1912689 1915456 2628072000 2633789341 2639514968 2645248887 2650991104 37.1484 37.1618 37.1753 37.1887 37.2021 11.1334 1 1 . 1361 11.1387 11.1414 11.1441 330 331 332 333 33 1 1768900 1771561 1774224 1776889 1779556 2352637000 2357947691 2363266368 2368593037 2373927704 36.4692 36.4829 36.4966 36.5103 36.5240 10.9972 11.0000 11.0028 11.0055 11.0083 1385 1386 1387 13S8 1389 1918225 1920996 1923769 1926544 1929321 2656741625 2662500456 2668267603 2674043072 2679826869 37.2156 37.2290 37.2424 37.2559 37.2693 11.1468 11.1495 11.1522 11.1548 11.1575 335 836 337 388 339 1782225 1784896 1787569 1790244 1792921 2379270375 2384621056 2389979753 231)5346472 2400721219 36.5377 36.5513 36.5650 36.5787 36.5923 11.0110 11.0138 11.0165 11 0193 11.0220 1390 1391 1392 1393 1394 1932100 1934881 1937664 1940449 1943236 2685619000 2691419471 2697228288 2703045457 2708870984 37.2827 37.2961 37.3095 37.3229 37.3363 11.1602 11.1629 11.1655 11.1682 11.1709 340 341 312 343 344 1795600 1798281 1800964 1803649 1806336 2406104000 2411494821 2416893688 2422300607 2427715584 36.6060 36.6197 36.6333 36.6469 36.6606 11.0247 11.0275 11.0302 11.0330 11.0357 1395 1396 1397 1398 1399 1946025 1948816 1951609 1954404 1957201 2714704875 2720547136 2726397773 2732256792 2738124199 37.3497 37.3631 37.3765 37.3898 37.4032 11.1736 11.1762 11.1789 11.1816 11.1842 345 340 347 34H 349 1809025 1811716 1814409 1817104 1819801 2433138625 2438569736 2444008923 2449456192 2454911549 36.6742 36.6879 36.7015 36.7151 36.7287 11.0384 11.0412 11.0439 11.0466 11.0494 1400 1401 1402 1403 1404 1960000 1962801 1965604 1968409 1971216 2744000000 2749884201 2755776808 2761677827 2767587264 37.4166 37.4299 37.4433 37.4566 37.4700 11.1869 11.1896 11.192? 11.1949 11.1975 350 351 352 353 354 1822500 1825201 1827904 1830609 1833316 2460375000 2465846551 2471326208 2476813977 2482309864 36.7423 36.7560 36.7696 36.7831 36.7967 11.0521 11.0548 11.0575 11.0603 11.0630 1405 1406 1407 1408 1409 1974025 1976836 1979649 1982464 1985281 2773505125 2779431416 2785366143 2791309312 2797260929 37.4833 37.4967 37.5100 37.5233 37.5366 11.2002 11.2028 11.2055 11.2082 11.2108 355 356 357 35S 359 1836025 1838736 1841449 1844164 1846881 2487813875 2493326016 2498846293 2504374712 2509911279 36.8103 36.8239 36.8375 36.8511 36.8646 11.0657 11.0684 11.0712 11.0739 11.0766 1410 1411 1412 1413 1414 1988100 1990921 1993744 1996569 1999396 2803221000 2809189531 2815166528 2821151997 2827145944 37.5500 37.5633 37.5766 87.5899 37.6032 11.2135 11.2161 11.2188 11.2214 11.2240 360 361 362 :!t)3 364 1849600 1852321 1855044 1857769 1860496 2515456000 2521008881 2526569928 2532139147 2537716544 36.8782 36.8917 36.9053 36.9188 36.9324 11.0793 11.0820 11.0847 11.0875 11.0902 1415 1416 1417 1418 1419 2002225 2005056 2007889 2010724 2013561 2833148375 2839159296 2845178713 2851206632 2857243059 37.6165 37.6298 37.6431 37.6563 37.6696 11.2267 11 2293 11.2320 11.2346 11.2373 100 MATHEMATICAL TABLES. No. 1420 1421 1422 1423 1424 Square, Cube. Sq. Root. Cube Root. No. Square. Cube. Sq. Root. Cube Root. 2016400 2019241 2022084 2024929 2027776 286328800037.6829 2869341461 37.6962 287540344837.7094 288147396737.7227 2887553024 37.7359 11.2399 11.2425 11.2452 11.2478 11.2505 1475 1476 1477 1478 1479 2175625 2178576 2181529 2184484 2187441 3209046875 3215578176 3222118333 3228667352 3235225239 38.4057 38.4187 38.4318 38.4448 38.4578 11.3832 11.3858 11.3883 11.3909 11.3935 1425 1426 1427 1428 1429 2030625 2033476 2036329 2039184 2042041 289364062537.7492 289973677637.7624 290584148337.7757 291195475237.7889 291807658937.8021 11.2531 11.2557 11.2583 11.2610 11.2636 1480 1481 1482 1483 1484 2190400 2193361 2196324 2199289 2202256 3241792000 3248367641 3254952168 3261545587 3268147904 38.4708 38.4838 38.4968 38.5097 38.5227 11.3960 11.3986 11.4012 11.4037 11.4063 1430 1431 1432 1433 1434 2044900 2047761 2050624 2053489 2056356 292420700037.8153 2930345991 37.8286 2936493568 37.8418 294264973737.8550 2948814504 ' 37. 8682 11.2662 11.2689 11.2715 11.2741 11.2767 1485 1.486 1487 1488 1489 2205225 2208196 2211169 2214144 2217121 3274759125 3281379256 3288008303 3294646272 3301293169 38.5357 38.5487 38.5616 38.5746 38.5876 11.4089 11.4114 11.4140 11.4165 11.4191 1435 1436 1437 1438 1439 2059225 2062096 2064969 2067844 2070721 295498787537.8814 2961169856 37.8946 296736045337.9078 297355967237.9210 2979767519 37.9342 11.2793 11.2820 11.2846 11.2872 11.2898 1490 1491 1492 1493 1494 2220100 2223081 22.20064 2229049 2232036 3307949000 3314613771 3321287488 3327970157 3334661784 38.6005 38.6135 38.6264 38.6394 38.6523 11.4216 11.4242 11.4268 11.4203 11.4319 1440 1441 1442 1443 1444 2073600 2076481 2079364 2082249 2085136 298598400037.9473 299220912137.9605 299844288837.9737 300468530737.9868 301093638438.0000 11.2924 11.2950 11.2977 11.3003 11.3029 1495 1496 1497 1498 1499 2235025 2238016 2241009 2244004 2247001 3341362375 3348071936 3354790473 3361517992 3368254499 38.6652 38.6782 38.6911 38.7040 38.7169 11.4344 11.4370 1 1 . 4395 11.4421 11.4446 1445 1446 1447 1448 1449 2088025 2090916 2093809 2096704 2099601 3017196125 3023464536 3029741623 3036027392 3042321849 38.0132 38.0263 38.0395 38.0526 38.0657 11.3055 11.3081 11.3107 11.3133 11.3159 1500 2250000 1501 2253001 1502 ; 2256004 1503 2259009 1504 2262016 3375000000 3381754501 3388518008 3395290527 3402072064 38.7298 38.7427 38.7556 38.7685 38.7814 11.4471 11.4497 11.4522 11.4548 11.4573 1450 1451 1452 1453 1454 2102500 2105401 2108304 2111209 2114116 3048625000 3054936851 3061257408 3067586677 3073924664 38.0789 38.0920 38.1051 38.1182 38.1314 11.3185 11.3211 11.3237 11.3263 11.3289 1505 1506 1507 1508 1509 2265025 2268036 2271049 2274064 2277081 3408862625 3415662216 3422470843 3429288512 3436115229 38.7943 38.8072 38.8201 38.8330 38.8458 11.4598 11.4624 11.4649 11.4675 11.4700 1455 1456 1457 1458 ,1459 2117025 2119936 2122849 2125764 2128681 3080271375 3086626816 3092990993 3099363912 3105745579 38.1445 38.1576 38.1707 38.1838 38.1969 11.3315 11.3341 11.3367 11.3393 11.3419 1510 1511 1512 1513 1514 2280100 2283121 2286144 2289169 2292196 3442951000 3449795831 3456649728 3463512697 3470384744 38.8587 38.8716 38 8844 38.8973 38.9102 11.4725 11.4751 11.4776 11.4801 11.4820 1460 1461 1462 1463 1464 2131600 2134521 2137444 2140369 2143296 3112136000 3118535181 3124943128 3131359847 3137785344 38.2099 38.2230 38.2361 38.2492 38.2623 11.3445 11.3471 1 1 . 3496 11.3522 11.3548 1515 1516 1517 1518 1519 2295225 2298256 2301289 2304324 2307361 3477265875 3484156096 3491055413 3407963832 3504881359 38.9230 38.9358 38.9487 38.9615 38.97'44 11.485-3 11.4877 11.4902 11.4927 11.4953 1465 1466 1467 1468 1469 2146225 2149156 2152089 2155024 2157961 3144219625 3150662696 3157114563 3163575232 3170044709 38.2753 38.2884 38.3014 38.3145 38.3275 11.3574 11.3600 11.3626 11.3652 11.3677 1520 1521 1522 1523 1524 2310400 2313441 2316484 2319529 2322576 3511808000 3518743761 3525688648 3532642667 3539605824 38.9872 39.0000 39.0128 39.0256 39.0384 11.4978 11.5003 11.5028 11.5054 11.5079 1470 1471 1472 1473 1474 2160900 2163841 2166784 2169729 2172676 3176523000 3183010111 3189506048 3196010817 3202524424 38.3406 38.3536 38.3667 38 . 3797 88.3SWT 11.3703 11.3729 11.3755 11.3780 11.3806 1525 1526 1527 1528 1529 2325625 2328676 2331729 2334784 2337841 3546578125 3553559576 3560550183 3567549952 357455889 39.0512 39.0640 39.0768 39.0896 39.1024 11.5104 11.5129 11.5154 11.5179 11.5204 SQUARES, CUBES, SQUARE AND CUBE R<JOTS. 101 No. Square. Cube. Sq. Root. Cube Root. No. Square. Cube. Sq. Root. Cube Root. 1530 1531 1532 1533 1534 2340900 2343961 2347024 2350089 2353156 3581577000 3588604291 3595640768 3602080437 3609741304 39.1152 39.1280 39.1408 39.1535 39.1663 11.5230 11.5255 11.5280 11.5305 11.5330 1565 1566 1567 1568 1569 2449225 3833037125 2452356,3840389496 24554893847751263 2458624 3855123432 2461761 3862503009 39.5601 39.5727 39.5854 39.5980 39.6106 11.6102 11.6126 11.6151 11.6176 11.6200 1535 1536 1537 1538 1539 2356225 2359296 2362369 2365444 2368521 3616805375 3623878656 3630961153 3638052872 3645153819 39.1791 39.1918 39.2046 39.2173 39.2301 11.5355 11.5380 11.5405 11.5430 11.5455 1570 1571 1572 1573 1574 2464900 3869893000 2468041 3877292411 24711 84 384701 248 24743293892119517 24774763899547224 39.6232 39.6358 39.6485 39.6611 39.6737 11.6225 11.6250 11.6274 11.6299 11.6324 1540 1541 1542 1543 1544 2371600 2374681 2377764 2380849 2383936 3652264000 3659383421 3666512088 3673650007 2680797184 39.2428 39.2556 39.2683 39.2810 39.2938 11.5480 11.5505 11.5530 11.5555 11.5580 1575 1576 1577 1578 1579 2480625 3906984375 2483770 3914430976 2486929 3921887033 2490084 :^9^9352552 249324 lj 3936827539 39.6863 39.6989 39.7115 39.7240 39.7366 11.6348 11.6373 11.6398 11.6422 11.0447 1545 1546 1547 1548 1519 2387025 2390116 2393209 2396304 2399401 3687953625 3695119336 3702294323 3709478592 3716672149 39.3065 39.3192 39.3319 39.3446 39.3573 11.5605 11.5630 11.5655 11.5680 11.5705 1580 1581 1582 1583 1584 2496400 3944312000 2499561 3651805941 2502724 3959309368 2505889 3960822287 25090563974344704 39.7492 39.7618 39.7744 39.7869 39.7995 11.6471 11.6496 11.6520 11.6545 11,6570 1550 1551 1552 1553 1554 2402500 2405601 2408704 2411809 2414916 3723875000 3731087151 3738308608 3745539377 3752779464 39.3700 39.3827 39.3954 39.4081 39.4208 11.5729 11.5754 11.5779 11.5804 11.5829 1585 1586 1587 1588 1589 2512225 3981876625 2515396 3989418056 2518569 3996969003 2521744 4004529472 2524921 4012099469 39.8121 11.6594 39.824611.6619 39.837211.6643 39.849711.6608 39.862311.6692 1555 1556 1557 1558 1559 2418025 2421136 2424249 2427364 2430481 3760028875 3767287616 3774555693 3781833112 3789119879 39.4335 11.5854 39.4462 11.5879 39.4588 11.5903 39.4715 11.5928 39.4842 11.5953 1590 1591 1592 1593 1594 2528100 4019679000 2531281 4027268071 2534464 4034866688 2537649 4042474857 2540836 4050092584 39.874811.6717 39.887311.6741 39.8999 11.6765 39.9124 11.6790 39.924911.6814 1560 1561 1562 1563 1564 2433600 2436721 2439844 2442969 2446096 3796416000 3803721481 3811036328 3818360547 3825694144 39 496811.5978 39.5095111.6003 39.5221 11.6027 39. 5348' 11. 6052 39.547411.6077 1595 1590 1597 1598 1599 2544025 4057719875 2547216 4065356736 2550409 4073003173 2553604 4080659192 2556801 4088324799 39.937511.7839 39.950011.6863 39. 9625 jll. 6888 39.9750 11.6912 39.9875 11.6936 1600 2560000 4096000000 40.000011.6961 SQUARES AND CUBES OF No. Square. Cube. No. Square. Cube. No. Square. Cube. .1 .01 .001 .01 .0001 .000 001 .001 .00 00 01 .000 000 001 .2 .04 .008 .02 .0001 .000 008 .002 .00 00 04 .000 000 008 3 .09 .027 .03 .0009 .000 027 .003 .00 00 09 .000 000 027 4 .16 .064 .04 .0016 .000 064 .004 .00 00 16 .000 000 064 .5 .25 .125 .05 .0025 .000 125 .005 00 00 25 .000 000 125 .6 .36 .216 .06 .0036 .000 216 .006 .00 00 36 .000 000 216 .7 .49 .343 .07 .0049 .000 343 007 .00 00 49 .OOO.OOQ 343 .8 .64 .512 .08 .0064 .000 512 .008 ; .00 00 64 .000' 000 512 .9 .81 .729 .09 .0081 .000 729 009 > .00 0'.) 81 .000 COu 729 1.0 1.00 1.000 .10 .0100 .001 000 .010 .^X) 01 00 .000 '001 000 1.2 1.44 1.728 .12 .0144 .001,728 .012 ,00 M *4 .ooa qo 1 . 728 Note that the square has twice as many xlecirf al places ; an'i times as many decimal places, as the root. 102 MATHEMATICAL TABLES. FIFTH ROOTS AND FIFTH (Abridged from TRAUTWINE.) II Power. II Power. II Power. ji Power. t, . II Power. .10 .000010 3.7 693.440 9.8 90392 21.8 4923597 40 102400000 .15 .000075 3.8 792.352 9.9 95099 22.0 5153632 41 115856201 .20 .000320 3.9 902.242 10.0 100000 22.2 5392186 42 130691232 .25 .000977 4.0 1024.00 10 2 110408 22.4 5639493 43 147008443 .30 .002430 4.1 1158.56 10.4 121665 22.6 5895793 44 164916224 .35 .005252 4.2 1306.91 10.6 133823 22.8 6161327 45 184528125 .40 .010240 4.3 1470.08 10.8 146933 23 6436343 46 205962976 .45 .018453 4.4 1649.16 11.0 161051 23^2 6721093 47 229345007 .50 .031250 4.5 1845.28 11.2 176234 23.4 7015834 48 254803968 .55 .050328 4.6 2059.63 11.4 192541 23.6 7320825 49 282475249 .60 .077760 4.7 2293.45 11.6 210084 23.8 7636332 50 312500000 .65 .116029 4.8 2548.04 11.8 228776 24.0 7962624 51 345025251 .70 168070 4.9 2824.75 12.0 248832 24.2 8299976 52 380204032 .75 ,237305 5.0 3125.00 12.2 270271 24.4 8648666 53 418195498 .80 .327680 5.1 3450.25 12.4 293163 24.6 9008978 54 459165024 .85 .443705 5.2 3802 04 12. G 317580 24.8 9381200 55 503284375 .90 .590490 5.3 4181.95 12.8 343597 25.0 9765625 56 550731776 .95 .773781 5.4 4591.65 13.0 371293 25.2 10162550 57 601692057 1.00 1.00000 5.5 5032.84 13.2 400746 25.4 10572278 58 656356768 1.05 1.27628 5.6 5507.32 13.4 432040 25.6 10995116 59 714924299 1.10 1.61051 5.7 6016.92 13 6 465259 25.8 11431377 60 777600000 1.15 2.01135 5.8 6563.57 13.8 500490 26.0 11881376 61 844596301 1.20 2.48832 5.9 7149.24 14.0 537824 26.2 12345437 62 916132832 1.25 3.05176 6.0 7776.00 34.2 577353 26.4 1282388G 63 992436543 1.30 3.71293 6.1 8445.96 14.4 619174 26.6 13317055 64 1073741824 1.35 4.48403 6.2 9161.33 'A 6 663383 26.8 13825281 65 1160290625 1.40 5.37824 6.3 9924.37 14.8 710082 27.0 14348907 66 1252332576 1.45 6.40973 6 4 10737 15.0 759375 27.2 14888280 67 1350125107 1.50 7.59375 6.5 11603 15 2 811368 27.4 15443752 68 1453933568 1.55 8.94661 6.6 12523 15.4 866171 27.6 16015681 69 1564031349 1.60 10.4858 6.7 13501 15. G 923896 27.8 16604430 70 168070000C 1.65 12.2298 6.8 14539 15.8 984658 28.0 17210368 71 1804229351 1.70 14.1986 6.9 15G40 16.0 1048576 28.2 17633868 72 1934917632 1.75 16.4131 7.0 16807 16.2 1115771 28.4 18475309 73 2073071593 1.80 18.8957 7.1 18042 16.4 118G367 28.6 19135075 74 2219006G24 1.85 21.6700 7.2 19349 16.6 1260493 28.8 19813557 75 2373046875 1.90 24.7610 7.3 20731 16 8 1338278 29.0 20511149 76 2535525376 1.95 28.1951 7.4 22190 17.0 1419857 29.2 21228253 77 2706784157 2.00 32.0000 7.5 23730 17.2 1505366 29.4 21965275 78 2887174368 2.05 36.2051 7.6 25355 17.4 1594947 29.6 22722628 79 3077056399 2.10 40.8410 7.7 27'068 17.6 1688742 29 8 3500728 80 3276800000 2.15 45.9401 7.8 28872 17.8 17'86899 30.0 24300000 81 3486784401 2 20 51.5363 7.9 30771 18.0 1889568 30.5 26393634 82 3707398432 2.25 57.6650 8.0 32768 18.2 199G903 31.0 28G29151 83 3939040643 2.30 64.3634 8.1 34868 18.4 2109061 31.5 31013642 84 4182119424 2.35 71.6703 8.2 37074 18.6 2226203 32.0 33554432 85 4437053125 2.40 79.6262 8.3 39390 18.8 2348493 32.5 36259082 86 4704270176 2.45 88.2735 8.4 41821 19.0 2476099 33.0 39135393 87 4984209207 2.50 97.6562 8.5 44371 19.2 2609193 33.5 42191410 88 5277319168 2.55 107.820 8.6 47043 19.4 2747949 34.0 45435424 89 5584059449 2.60 118.814 8.7 49842 19.6 2892547 31.5 4S875980 90 5904900000 2.70 143.489 8.8 52773 19.8 3043168 35.0 52521875 91 6240321451 2.80 172.104 8.9 55841 20.0 3200000 35.5 56382167 92 6590815232 2.90 205.111 9.0 59049 20.2 3363232 36.0 G0466176 93 6956883693 3 00 243.000 9.1 62403 20.4 3533059 36 5 64783487 94 7339040224 3,10 286.292 9.2, 05998- 20.6 3709677 37.0 69343957 95 7737809375 3 f*0 335.544 a. 3 '58509 20.8 3893289 37.5 74157715 96 815372G976 3^30 391.354 9.4 V3390 21.0 4084101 38.0 79235168 97 8587340257 ft. 40 454 . 54 ,9,5, 77378 .21.2 4282322 38.5 84587005 98 90392079G8 3..M) 525 19 96-1 81237- , 21.4 4488166 39.0 90224199 99 950990U499 Sieol OC4.'662 9";7j 8,5873--' 21.6 4701850 39.5 96158012 CIRCUMFERENCES AND AREAS OF CIRCLES, 103 CIRCUMFERENCES AND AREAS OF CIRCLES. Piam. Circum. Area. Diam. Circum. Area. Diam. Circum. Area. 1 3.1416 0.7854 65 204.20 3318.31 129 405.27 13069.81 2 6.2832 3.1416 66 207.34 3421 . 19 130 408.41 13273.23 3 9.4248 7.0686 67 210.49 3525.65 131 411.55 13478.22 4 12.5664 12.5664 68 213.63 3631.68 132 414.69 13684 78 5 15.7080 19.635 69 216.77 3739.28 133 417.83 13892.91 6 18.850 28 274 70 219.91 3848.45 134 420.97 14102.61 7 21.991 38.485 71 223.05 3959.19 135 424.12 14313.88 8 25.133 50.266 72 226.19 4071.50 136 427.26 14526.72 9 28.274 63.617 73 229.34 4185.39 137 430.40 14741.14 ilO 31.416 78.540 74 232.48 4300 84 138 433.54 14957.12 11 34.558 95.033 75 235.62 4417.86 139 436.68 15174.68 12 37.699 113.10 76 238.76 4536.46 140 439.82 15393.80 13 40.841 132.73 77 241.90 4656.63 141 442.96 15614.50 14 43.982 153.94 78 245.04 4778.36 142 446.11 15836.77 15 47.124 176.71 79 248.19 4901.67 143 449.25 16060.61 16 50.265 201.06 80 251.33 5026.55 144 452.39 16286.02 17 53.407 226.98 81 254.47 5153.00 145 455.53 16513.00 18 56.549 254.47 82 257.61 5281.02 146 458.67 16741.55 10 59.690 283.53 83 260.75 5410.61 147 461.81 16971.67 20 62.832 314.16 84 263.89 5541.77 148 464.96 17203.36 21 65.973 346.36 85 267.04 5674 50 149 468.10 17436.62 22 69.115 380.13 86 270.18 5808.80 150 471.24 17671.46 23 72.257 415.48 8?' 273.32 5944.68 151 474.38 17907 86 24 75.398 452.39 88 276.46 6082.12 152 477.52 18145.84 25 78.540 490.87 89 279.60 6221.14 153 480.66 18385.39 26 81.681 530.93 90 282.74 6361.73. 154 483.81 18626.50 27 84.823 572.56 91 285.88 6503.88' 155 486.95 18869.19 28 87.965 615.75 92 289.03 6647.61 156 490.09 19113.45 29 91.106 660.52 93 292.17 6792.91 157 493.23 19359.28 30 94.248 706.86 94 295.31 6939.78 158 496.37 19606.68 31 97.389 754 . 77 95 298.45 7088.22 159 499.51 19855.65 32 100.53 804.25 96 301.59 7238.23 160 502.65 20106.19 33 103.67 855.30 97 304.73 7389.81 161 505.80 20358.31 34 106.81 907.92 98 307.88 7542.96 162 508.94 20611.99 35 109.96 962.11 99 311.02 7697.69 163 512.08 20867.24 36 113.10 1017.88 100 314.16 7853.98 164 515.22 21124.07 37 116.24 1075.21 101 317.30 8011.85 165 518.36 21382.46 38 119.38 1134.11 102 320.44 8171.28 166 521.50 21642.43 39 122.52 1194.59 103 323.58 8332.29 167 524.65 21903 97 40 125.66 1256.64 104 326.73 8494.87 168 527.79 22167 08 41 128.81 1320 25 105 329 87 8659.01 169 530.93 22431.76 42 131.95 1385.44 106 333.01 8824.73 170 534.07 22698.01 43 135.09 1452.20 107 336.15 8992.02 171 537.21 22965.83 44 138.23 '1520.53 108 339.29 9160.88 172 540.35 23235.22 45 141.37 1590.43 109 342.43 9331.32 173 543.50 23506.18 46 144.51 1661.90 110 345.58 9503.32 174 546.64 23778.71 47 147.65 1734.94 111 348.72 9676.89 175 549.78 24052.82 48 150.80 1809.56 112 351.86 9852.03 176 552.92 24328.49 49 153 94 1885.74 113 355.00 10028.75 177 556.06 24605.74 50 157.08 1963.50 114 358.1.4 10207.03 178 559.20 24884.56 51 160.22 2042.82 115 361.28 10386 89 179 562.35 25164.94 53 163.36 2123.72 116 364.42 10568.32 180 565.49 25446.90 53 166.50 2206.18 117 367.57 10751.32 181 568.63 25730.43 54 169.65 2290 22 118 370.71 10935.88 182 571.77 26015.53 55 172.79 2375.83 119 373.85 11122.02 183 574.91 26302.20 56 175.93 2463 01 120 376.99 11309.73 184 578.05 26590.44 57 179.07 2551.76 121 380.13 11499.01 185 581.19 26880.25 58 182.21 2642.08 122 383.27 11689.87 186 584.34 27171.63 59 185.35 2733.97 123 386.42 11882.29 187 587.48 27464.59 60 188.50 2827.43 124 389.56 12076.28 188 590.62 27759.11 61 191.64 2922.47 125 392.70 12271.85 189 593.76 28055.21 6-4 194.78 3019.07 126 395.84 12468.98 190 596.90 28352 87 63 197.92 3117.25 127 398.98 12(567.69 191 600.04 28652.11 64 201.06 3216.99 128 402.12 12867.96 192 603.19 28952.92 104 MATHEMATICAL TABLES. Diam. Circum. Area. Diam. Circum. Area. Diam. Circum. Area. 193 606.33 29255.30 260 816.81 53092.92 327 1027.30 83981.84 194 609.47 29559.25 261 819.96 53502.11 328 1030.44 84496.28 195 612.61 29864.77 262 823.10 53912.87 329 1033.58 85012.28 196 615.75 30171.86 263 826.24 54325.21 330 1036.73 85529.86 197 618.89 30480.52 264 829.38 54739.11 331 1039.87 86049.01 198 622.04 30790.75 265 832.52 55154.59 332 1043.01 86569.73 199 625.18 31102.55 266 835.66 55571.63 333 1046.15 87092.02 200 628.32 31415.93 267 838.81 55990.25 334 1049.29 87615.88 201 631.46 31730.87 268 841 . 95 56410.44 335 1052.43 88141.31 202 634.60 32047.39 269 845.09 56832.20 336 1055.58 88668.31 203 637.74 32365.47 270 848.23 57255.53 337 1058.72 89196.88 204 640.88 32685.13 271 851.37 57680.43 338 1061.86 89^27.03 205 644.03 33006.36 272 854.51 58106.90 339 1065.00 90258.74 206 647.17 33329.16 273 57.65 58534.94 340 1068.14 90792.03 207 650.31 33653.53 274 860.80 58964.55 341 1071.28 91326.88 208 653.45 33979.47 275 863.94 59S95.74 342 1074.42 91863.31 209 656.59 34306.98 276 867.08 59828.49 343 1077.57 92401 . 31 210 659.73 34G36.06 277 870.22 60262.82 344 1080.71 92940.88 211 662.88 34966.71 278 873.36 60698.71 345 1083.85 93482.02 212 066.02 35298.94 279 876.50 61136.18 346 1086.99 94024.73 213 669.16 35632.73 280 879.65 61575.22 347 1090.13 94569.01 214 672.30 35968.09 281 882.79 62015.82 348 1093.27 95114.86 215 675.44 36305.03 282 885.93 62458.00 349 1096.42 95662.28 216 678.58 36643.54 283 889.07 62901.75 350 1099.56 96211.28 217 681.73 36983.61 284 892.21 63347.07 351 1102.70 96761.84 218 684.87 37325.26 285 895.35 63793.97 352 1105.84 97313.97 219 688.01 37668.48 286 898.50 64242.43 353 1108.98 97867.68 220 691.15 38013.27 287 901.64 64692.46 354 1112.12 98422.96 221 694.29 38359.63 288 904.78 65144.07 355 1115.27 98979.80 222 697.43 88707.56 289 907.92 65597.24 356 1118.41 99538.22 223 700.58 39057.07 290 911.06 66051.99 357 1121.55 100098.21 224 703.72 39408.14 291 914.20 66508.30 358 1124.69 100659.77 225 706.86 39760.78 292 917.35 66966.19 359 1127.83 101222.90 226 710.00 40115.00 293 920.49 67425.65 360 1130.97 101787.60 227 713.14 40470.78 294 923.63 67886.68 361 1134.11 102353.87 228 716.28 40828.14 295 926.77 68349.28 362 1137.26 102921.72 229 719.42 41187.07 296 929.91 68813.45 363 1140.40 103491.13 230 722.57 41547.56 297 933.05 69279.19 364 1143.54 104062.12 231 725.71 41909.63 298 936.19 69746.50 365 1146.68 104634.67 232 728.85 42273.27 299 939.34 70215.38 306 1149.82 105208.80 233 731.99 42638.48 300 942.48 70685.83 367 1152.96 105784.49 234 735.13 43005.26 301 945.62 71157.86 368 1156.11 106361.76 235 738.27 43373.61 3J33 948.76 71631.45 369 1159.25 106940.60 236 741.42 43743.54 303 951.90 72106.62 370 1162.39 107521.01 237 744.56 44115.03 304 955.04 72583.36 371 1165.53 108102.99 238 747.70 44488.09 305 958.19 73061.66 372 1168.67 108686.54 239 750.84 44862.73 306 961.33 73541.54 373 1171.81 109271.66 240 753.98 45238.93 307 964.47 74022.99 374 1174.96 109858.35 241 757.12 45616.71 308 967.61 74506.01 375 1178.10 110446.62 242 760.27 45996.06 309 970.75 74990.60 376 1181.24 111036.45 243 763.41 46376.98 310 973.89 75476.76 377 1184.38 111627.86 244 766.55 46759.47 311 977.04 75964.50 378 1187.52 112220.83 245 769.69 47143.52 312 980.18 76453.80 379 1190.66 112815.38 246 772.83 47529.16 313 983.32 76944.67 380 1193.81 113411.49 247 775.97 47916.36 314 986.46 77437.12 381 1196.95 114009.18 248 779.11 48305.13 315 989.60 77931.13 382 1200.09 114608.44 249 782.26 '48695.47 316 992.74 784-26.72 383 1203.23 115209.27 250 785.40 49087.39 317 995.88 78923.88 384 1206.37 115811.67 251 788.54 49480.87 318 999.03 79422.60 385 1209.51 116415.64 252 791.68 49875.92 319 1002.17 79922.90 386 1212.65 117021.18 253 794.82 50272.55 320 1005.31 80424.7? 387 1215.80 117628.30 254 797.96 50670.75 321 1008.45 80928.21 388 1218.94 118236.98 255 801.11 51070.52 322 1011.59 81433 22 389 1222.08 118847.24 256 804.25 51471.85 323 1014.73 81939.80 390 1225.22 119459.06 257 807.39 51874.76 324 1017.88 82447.96 391 1228.36 120072.46 258 810.53 52279.24 325 1021.02 82957.68 392 1231.50 120687.42 259 813.67 52685.29 326 1024.16 83468.98 393 1234.65 121303.96 CIRCUMFERENCES AND AREAS OF CIRCLES. Diam. Circum. Area. Diam. Circum. Area. Diam. Circum. Area. 394 1237.79 121922.07 461 1448.27 166913.60 528 1658.76 218956.44 395 1240.93 122541.75 462 1451.42 167638.53 529 1661.90 219786.61 396 1244.07 123163.00 463 1454.56 168365.02 530 1665.04 220618.34 397 1247.21 123785.82 464 1457.70 169093.08 531 1668.19 221451.65 398 1250.35 124410.21 465 1460.84 169822.72 532 1671.33 222286.53 399 1253.50 125036.17 466 1463.98 170553.92 533 1674.47 223122.98 400 1256.64 125663.71 467 1467.12 171286.70 534 1677.61 223961.00 401 1259.78 126292.81 468 1470.27 172021.05 535 1680.75 224800.59 402 1262.92 126923.48 469 1473.41 172756.97 536 1683.89 225641.75 403 1266.06 127555.73 470 1476.55 173494.45 537 1687.04 226484.48 404 1269.20 128189.55 471 1479.69 174233.51 538 1690.18 227328.79 405 1272.35 128824.93 472 1482.83 174974.14 539 1693.32 228174.66 406 1275.49 129461.89 473 1485.97 175716.35 540 1696.46 229022.10 407 1278.63 130100.42 474 1489.11 176460.12 541 1699.60 229871.12 408 1281.77 130740.52 475 1492.26 177205.46 542 1702.74 230721.71 409 1284.91 131382.19 476 1495.40 177952.37 543 1705.88 231573.86 410 1288.05 132025.43 477 1498.54 178700.86 544 1709.03 232427.59 411 1291.19 132670.24 478 1501.68 179450.91 545 1712.17 233282.89 412 1294.34 133316.63 479 1504.82 180202.54 546 1715.31 234139.76 413 1297.48 133964.58 480 1507.96 180955.74 547 1718.45 234998720 414 1300.62 134614.10 481 1511.11 181710.50 548 1721.59 235858.21 415 1303.76 135265.20 482 1514.25 182466.84 549 1724.73 236719.79 416 1306.90 135917.86 483 1517.39 183224.75 550 1727.88 237582.94 417 1310.04 136572.10 484 1520.53 183984.23 551 1731.02 238447.67 418 1313.19 137227.91 485 1523.67 184745.28 552 1734.16 239313.96 419 1316.33 137885.29 486 1526.81 185507 90 553 1737.30 240181.83 420 1319.47 138544.24 487 1529.96 186272.10 554 1740.44 241051.26 421 1322.61 139204.76 488 1533.10 187037.86 555 1743.58 241922.27 422 1325.75 139866.85 489 1536.241 187805.19 556 1746.73 242794.85 423 1328.89 140530.51 490 1539.38 188574.10 557 1749.87 243668.99 424 1332.04 141195.74 491 1542.52 189344.57 558 1753.01 244544.71 425 1335.18 141862 54 492 1545.66 190116.62 559 1756.15 245422.00 426 1338.32 142530.92 493 1548.81 190890.24 560 1759.29 246300.86 427 1341.46 143200.86 494 1551.95 191665.43 561 1762.43 247181.30 428 1344.60 143872.38 495 1555 09 192442.18 562 1765.58 248063.30 429 1347.74 144545.46 496 1558.23 193220.51 563 1768.72 248946.87 430 1350.88 145220.12 497 1561.37 194000.41 564 1771.86 249832.01 431 1354.03 145896.35 498 1564.51 194781.89 565 1775.00 250718 73 432 1357.17 146574.15 499 1567.65 195564.93 566 1778.14 251607.01 433 1360.31 147253.52 500 1570.80 196349.54 567 1781.28 252496.87 434 1363.45 147934.46 501 1573.94 197135.72 568 1784.42 253388.30 435 1366.59 148616.97 502 1577.08 197923.48 569 1787.57 254281.29 436 1369.73 149301.05 503 1580.22 198712.80 570 1790.71 355175.86 437 1372.88 149986.70 504 1583.36 199503.70 571 1793.85 256072.00 438 1376.02 150673.93 505 1586 50 200296.17 572 1796.99 256969.71 439 1379.16 151362.72 506 1589.65 201090.20 573 1800.13 257868.99 440 1382.30 152053.08 507 1592.79 201885.81 574 1803.27 258769.85 441 1385.44 152745.02 508 1595.93 202682.99 575 1806.42 259672.27 442 1388.58 153438.53 509 1599.07 203481.74 576 1809.56 260576.26 443 1391.73 154133.60 510 1602.21 204282.06 577 1812.70 261481.83 444 1394.87 154830.25 511 1605.35 205083.95 578 1815 84 262388.90 445 1398.01 155528.47 512 1608.50 205887.42 579 1818.98 263297.67 446 1401.15 156228.26 513 1611.64 206692.45 580 1822.12 264207.94 447 1404.29 156929.62 514 1614.78 207499.05 581 1825.27 265119.79 448 1407.43 157632.55 515 1617.92 208307.23 582 1828.41 266033.21 449 1410.58 158337,06 516 1621.06 209116.97 583 1831.55 266948.20 450 1413.72 159043.13 517 1624.20 209928.29 584 1834.69 267864.76 451 1416.86 159750.77 518 1627.34 210741.18 585 1837.83 268782.89 452 1420.00 160459.99 519 1630.49 211555.63 586 1840.97 269702.59 453 1423.14 161170.77 520 1633.63 212371.66 587 1844.11 270623.86 454 1426.28 161883.13 521 1636.77 213189.26 588 1847.26 271546.70 455 1429.42 162597.05 522 1639.91 214008.43 589 1850.40 272471.12 456 1432.57 163312.55 523 1643.05 214829.17 590 1853.54 273397.10 457 1435.71 164029.62 524 1646.19 215651.49 591 1856.68 274324.60 458 1438.85 164748.26 525 1649.34 216475.37 592 1859.82 275253.78 459 1441.99 165468.47 526 1652.48 217300.82 593 1862.96 276184.48 460 1445.13 166190.25 527 1655.62 _218127.85 594 1866.11 277116.75 106 MATHEMATICAL TABLES. Diarn. Circum. Area. Diam- Circum. Area. Diana- JCircum. Area. 595 1869.25 278050.58 663 2082.88 345236.69 731 ! 2296.50 419686.15 596 1872.39 278985.99 664 2086.02 346278.91 732 2299.65 420835.19 597 1875.53 279922.97 665 2089.16 347322.70 733 2302.79 421985.79 598 1878.67 280861.52 666 2092.30 348368.07 734 2305.93 423137.97 599 1881.81 281801 65 667 2095.44 349415.00 735 2309.07 424291.72 600 1884.96 282743.34 668 2098.58 350463.51 736 2312.211 425447.04 601 1888.10 283686.60 669 2101.73 351513.59 737 2315.35 426603.94 602 1891.24 284631.44 670 2104.87 352565.24 738 2318.50 427762.40 603 1894.38 285577.84 671 2108.01 353618.45 739 2321.64 428922.43 604 1897.52 286525.82 672 2111.15 354673 24 740 2324.78 430084.03 605 1900.6(5 287475.36 673 2114.29 355729.60 741 2327.92 431247.21 606 1903.81 288426.48 674 2117.43 356787.54 742 8981.06 432411.95 60? 1906.95 289379.17 675 2120.58 357847.04 743 2334.20 433578.27 608 1910.09 290333.43 676 2123.72 358908.11 744 2337.34 434746.16 609 1913.23 291289.26 677 2126.86 359970.75 745 2340.49 435915.62 610 1916.37 292246.66 678 2130.00 361034.97 746 2343.63 437086.64 611 1919 51 293205.63 679 2133.14 362100.75 747 2346.77 438259.24 612 1922.65 294166.17 680 2136.28 363168.11 748 2349.91 439433.41 613 1925.80 295128.28 681 2139.42 364237.04 749 2353.05 440609 16 614 1928.94 296091.97 682 2142.57 365307.54 750 2356.19 441786.47 615 1932.08 297057.22 683 2145.71 366379.60 751 2359.34 442965.35 616 1935.22 298024.05 684 2148.85 367453.24 752 2362.48 444145.80 617 1938.36 298992.44 685 2151 99 368528.45 753 2365.62 445327.83 618 1941.50 299962.41 686 2155.13 369605.23 754 2368.76 446511.42 619 1944.65 300933.95 687 2158.27 370683.59 755 2371.90 447696.59 620 1947.79 301907.05 688 2161.42 371763.51 756 2375.04 448883.32 6-21 1950.93 302881.73 689 2164.56 372845.00 757 2378.19 450071.63 622 1954.07 303857.98 690 2167.70 373928.07 758 2381.33 451261.51 623 1957.21 304835.80 691 2170.84 375012.70 759 2384.47 452452.96 624 1960.35 305815.20 692 2173.98 376098.91 760 2387.61 453645.98 625 1963.50 306796.16 693 2177.12 377186.68 761 2390.75 454840.57 626 1966.64 307778.69 694 2180.27 378276.03 762 2393.89 456036.73 627 1969.78 308762.79 695 2183.41 379366.95 763 S397.04 457234.46 628 1972.92 309748.47 696 2186.55 380459.44 764 SJ400.18 458433.77 629 1976.06 310735.71 697 2189.69 381553.50 765 2403.32 459634.64 630 1979.20 311724.53 698 2192.83 382649.13 766 2406.46 460837.08 631 1982.35 312714.92 699 2195.97 383746.33 767 2409.60 462041.10 632 1985.49 313706.88 700 2199.11 384845.10 768 2412.74 463246.69 633 1988.63 314700.40 701 2202.26 385945.44 769 2415.88 464453.84 634 1991.77 315695.50 7'02 2205.40 387047.36 770 2419.03 465662.57 635 1994.91 316692.17 703 2208.54 388150.84 771 2422.17 466872.87 636 1998.05 317690.42 704 2211.68 389255.90 772 2425.31 468084.74 637 2001.19 318690.23 705 2214.82 390362.52 773 2428.45 469298.18 638 2004.34 319691.61 706 2217.96 391470.72 774 2431.59 470513.19 639 2007.48 320694.56 707 2221.11 392580.49 775 2434.73 471729.77 640 2010.62 321699.09 708 2224,25 393691.82 776 2437.88 472947.92 641 2013.76 322705.18 709 2227.39 394804.73 777 2441.02 474167.65 642 2016.90 323712.85 710 2230.53 395919.21 778 2444.16 475388.94 643 2020.04 324722.09 711 2233.67 397035.26 779 2447.30 476611.81 644 2023.19 325732.89 712 2236.81 398152.89 780 2450.44 477836.24 645 2026.33 326745.27 713 2239.96 399272.08 781 2453.58 479062.25 646 2029.47 327759.22 714 2243.10 400392.84 782 2456.73 480289.83 647 2032.61 328774.74 715 2246.24 401515.18 788 2459.8? 481518.97 648 2035.75 329791.83 716 2249.38 402639.08 784 2463.01 482749.69 649 2038.89 330810.49 717 2252.52 403764.56 785 2466.15 483981.98 650 2042. 04 331830.72 718 2255.66 404891.60 786 2469.29 485215.84 651 2045.18 332852.53 719 2258.81 406020.22 787 2472.43 486451.28 652 2048.32 333875.90 720 2261.95 407150.41 788 2475.58 487688.28 653 2051.46 334900.85 721 2265.09 408282.17 789 2478.72 488926.85 654 2054.60 335927.36 722 2268.23 409415.50 790 2481.86 490166.99 655 2057.74 336955.45 723 2271.37 410550.40 791 2485.00 491408.71 656 2060.88 337985.10 724 2274.51 411686.87 792 2488.14 492651.99 657 2064.03 339016.33 725 2277.65 412824.91 793 2491.28 493896.85 658 2067.17 340049.13 726 2280.80 413964.52 794 2494.42 495143.28 659 2070.31 341083,50 727 22S3.94 415105.71 795 2497.57 496391.27 660 2073.45 342119.44 728 2287.08 416248.46 796 2500.71 497640.84 661 2076.59 343156.95 729 2290.22 417392.79 797 2503.85 498891.98 662 2079.73 344196.03 730 2293.36 418538.68 798 1 2506.99 500144.69 CIRCUMFERENCES AND AREAS OF CIRCLES. 107 Diam. Circum. Area. Diam. Circum. Area. Diam. Circum. Area. 799 2510.13 501398.97 867 2723.76 590375.16 935 2937.39: 686614.71 800 2513.27 502654.82 868 2726.90 591737.83 936 2940.53 688084.19 801 2516.42 503912.25 869 2730.04 593102.06 937 2943.67 689555.24 802 2519.56 505171.24 870 2733.19 594467.87 938 2946 81 691027.86 803 2522.70 506431.80 871 2736.33 595835.25 939 2949.96 692502.05 804 2525.84 507693.94 872 2739.47 597204.20 940 2953.10 693977.82 805 2528.98 508957.64 873 2742.61 598574.72 941 2956.24 695455.15 806 2532.12 510222.92 874 2745.75 599946.81 942 2959.38 696934.06 807 2535.27 511489.77 875 2748.89 601320.47 943 2962.52 698414.53 808 2538.41 512758.19 876 2752.04 602695.70 944 2965.66 699896.58 809 2541.55 514028.18 877 2755.18 604072.50 945 2968.81 701380.19 810 2544.69 515299.74 878 2758.32 605450.88 946 2971.95 702865.38 811 2547.83 516572.87 879 2761.46 606830.82 947 2975.09 704352.14 812 2550.97 517847.57 880 2764.60 608212.34 948 2978.23 705840.47 813 2554.11 519123 84 881 2767.74 609595.42 949 2981.37 707330.37 814 2557.26 520401.68 882 2770.88 610980.08 950 2984.51 708821.84 815 2560.40 521681.10 883 2774.03 612366.31 951 2987.65 710314.88 816 2563.54 522982.08 884 2777.17 613754.11 952 2990.80 711809.50 817 2566.68 524244.63 885 2780.31 615143.48 953 2993.94 713305.68 818 2569.82 525528.76 886 2783.45 616534.42 954 2997.08 714803.43 819 2572 96 526814.46 887 2786.59 617926.93 955 3000.22 716302.76 820 2576.11 528101.73 888 2789.73 619321.01 956 3003.36 717803.66 821 2579.25 529390.56 889 2792.88 620716.66 957 3006.50 719306.12 822 2582.39 530680.97 890 2796.02 622113.89 958 3009.65 720810.16 823 2585.53 531972.95 891 2799.16 623512.68 959 3012.79 722315.77 824 2588.67 533266.50 892 2802.30 624913.04 960 3015.93 723822.95 825 2591.81 534561.62 893 '2805.44 626314.98 961 3019.07 725331.70 826 2594.96 535858.32 894 2808.58 627718.49 962 3022.21 726842.02 827 2598.10 537156.58 895 2811.73 629123.56 963 3025.35 728353.91 828 2601.24 538456.41 896 2814.87 630530.21 964 3028.50 729867.37 829 2604.38 539757.82 897 2818.01 631938.43 965 3031.64 731382.40 830 2607.52 541060.79 898 2821.15 633348.22 966 3034.78 732899.01 831 2610.66 542365.34 899 2824.29 634759.58 967 3037.92 734417.18 832 2613.81 543671.46 900 2827.43 636172.51 968 8041.06 735936.93 833 2616.95 544979.15 901 2830.58 637587.01 969 3044.20 737458.24 834 2620.09 546288.40 902 2833.72 639003.09 970 3047.34 738981.13 835 2623.23 547599.23 903 2836.86 640420.73 971 3050.49 740505.59 836 2626.37 548911.63 904 2840.00 641839.95 972 3053.63 742031.62 837 2629.51 550225.61 905 2843.14 643260.73 973 3056.77 743559.22 838 2632.65 551541.15 906 2846.28- 644683.09 974 3059.91 745088.39 839 2635.80 552858.26 907 2849.42 646107.01 975 3063.05 746619.13 840 2638.94 554176.94 908 2852.57 647532.51 976 3066.19 748151.44 841 2642.08 555497.20 909 2855.71 648959.58 977 3069.34 749685.32 842 2645.22 556819.02 910 2858.85 650388.22 978 3072.48 751220.78 843 2648.36 558142.42 911 2861.99 651818.43 979 3075.62 752757.80 844 2651.50 559467.39 912 2865.13 653250.21 980 3078.76 754296.40 845 2654.65 560793.92 913 2868.27 654683.56 981 3081.90 755836.56 846 2657.79 562122.03 914 2871.42 656118.48 982 3085.04 757378.30 847 2660.93 563451.71 915 2874.56 657554.98 983 3088.19 758921.61 848 2664.07 564782.96 916 2877.70 658993.04 984 3091.33 760466.48 849 2667.21 566115.78 917 2880.84 600432.68 985 3094.47 762012.93 850 2670.35 567450.17 918 2883.98 661873.88 986 3097.61 763560.95 851 2673.50 568786.14 919 2887.12 663316.66 987 3100.75 765110.54 852 2676.64 570123.67 920 2890.27 664761.01 988 3103.89 766661.70 853 2679.78 571462.77 921 2893.41 666206.92 989 3107.04 768214.44 854 2682.92 572803.45 922 2896.55 667654.41 990 3110.18 769768.74 855 2686.06 574145.69 923 2899.69 669103.47 991 3113.32 771324.61 856 2689.20 575489.51 924 2902.83 670554.10 992 3116.46 772882.06 857 2692.34 576834.90 925 2905.97 672006.30 993 3119.60 774441.07 H58 2695.49 578181.85 926 2909.11 673460.08 994 3122.74 776001.66 859 2698.63 579530.38 927 2912.26 674915.42 995 31.25.88 777563.82 860 2701.77 580880.48 928 2915.40 676372.33 996 3129.03 779127.54 861 2704.91 582232.15 929 2918.54 677830.82 997 3132.17 780692.84 862 2708.05 583585.39 930 2921.68 679290.87 998 3135.31 782259.71 863 2711.19 584940.20 931 2924.82 680752.50 999 3138.45 783828.15 864 2714.34 586296.59 932 2927.96 682-215.69 1000 3141.59 785398 16 865 2717.48 587654.54 933 2931.11 683680.46 866 2720 62 589014.07 934 2934.25 685146.80 108 MATHEMATICAL TABLES. CIRCUMFERENCES AND AREAS OF CIRCLES Advancing: by Eighths. Diam. Circum. Area. Diam. Circum. Area. Diam. Circum. Area. 1/64 .04909 .00019 2 % 7.4613 4.4301 6 H 19.242 29.465 1/32 .09818 .00077 7/16 7.6576 ' 4.6664 H 19.635 30.680 3/64 .14726 .00173 H 7.8540 4.9087 % 20.028 31.919 1/16 .19635 .00307 9/16 8.0503 5.1572 20.420 33.183 3/33 .29452 .00690 % 8.2467 5.4119 % 20.813 34.472 Ys .39270 .0122? 11/16 8.4430 5.6727 M 21 206 35.785 5/32 .49087 .01917 n 8.6394 5.9396 % 21.598 37.122 3/16 .58905 .02761 13/16 8.8357 6.2126 7. 21.991 38.485 7/32 .08722 .03758 Vs 9.0321 6.4918 ix 22.384 39.871 15/16 9.2284 6.7771 /4 22.776 41.282 y .78540 .04909 ax 23.169 42.718 pa .88357 .06213 3. 9.4248 7.0686 i^ 23.562 44.179 5/16 .98175 .07670 1/16 9.6211 7.3662 % 23.955 45 664 11/33 1.0799 .09281 Hi 9.8175 7.6699 % 24.347 47.173 % 1.1781 .11045 3/16 10.014 7.9798 7X 24.740 48.707 13/32 1.2763 .12962 y* 10.210 8.2958 8. 25.133 50.265 7/16 1.3744 .15033 5/16 10.407 8.6179 /^ 25.525 51.849 15/32 1.4726 . 17257 % 10.603 8.9462 f4 25.918 53.456 7/16 10.799 9.2806 % 26.311 55.088 ^ 1 5708 .19635 H 10.996 9.6211 y* 26.704 56.745 17/32 1.6690 .22166 9/16 11.192 9.9678 % 27.096 58.426 9/16 1.7671 .24850 % 11.388 10.321 M 27.489 60.132 19/32 1.8653 .27688 11/16 11.585 10.680 H 27.882 61.862 % 1.9635 .30680 H 11.781 11.045 9. 28.274 63.617 21/32 2.0617 .33824 13/16 11.977 11.416 /^j 28.667 65.397 11/16 2.1598 .37122 % 12.174 11.793 ^4 29.060 67.201 23/32 2.2580 .40574 15/16 12.370 12.177 % 29.452 69.029 4. 12.566 12.566 L 20.845 70.882 % 2.3562 .44179 1/16 12.763 12.962 % 30.238 72.760 25/32 2.4544 .47937 H 12.959 13.364 3 30.631 74.662 13/16 2.5525 .51849 3/16 13.155 13.772 /^O 31.023 76.589 27/32 2.6507 .55914 H 13.352 14.186 10. 31.416 78.540 K 2.7489 .60132 5/16 13.548 14.607 H 31.809 80.516 29/32 2.8471 .64504 % 13.744 15.033 $ 32.201 82 516 15/16 2.9452 .69029 7/16 13.941 15.466 32.594 84.541 31/32 3.0434 .73708 H 14.137 15.904 i^ 32.987 86.590 9/16 14.334 16.349 % 33.379 88.664 I. 3.1416 .7854 % 14.530 16.800 % 33.772 90.703 1/16 3.3379 .8860 11/16 14.726 17.257 % 34.165 92.886 y& 3.5343 .9940 H 14.923 17.721 11 34.558 95.033 3/16 3.7306 1.1075 13/16 15.119 18.190 34.950 97.205 k 3.9270 1.2272 % 15.315 18.665 M 35.343 99.402 5/16 4.1233 1.3530 15/16 15 512 19.147 78 35.736 101.62 % 4.3197 1.4849 5. 15.708 19.635 ^ 36.128 103.87 7/16 4.5160 1.6230 1/16 15.904 20.129 % 36.521 106.14 H 4.7124 1.7671 H 16.101 20.629 % 36.914 108.43 9/16 4.9087 1.9175 3/16 16.297 21.135 % 37.306 110.75 % 5.1051 2.0739 M 16.493 21.648 12 37.699 113.10 11/16 5.3014 2.2365 5/16 16.690 22.166 /^ 38.092 115.47 H 5.4978 2.4053 % 16.886 22.691 M 38.485 117.86 13/16 5.6941 2.5802 7/16 17.082 23.221 a| 38.877 120.28 % 5.8905 2.7612 y% 17.279 23.758 L^ 39.270 122.72 15/16 6.0868 2.9483 9/16 17.475 24.301 Kg 39.663 125.19 % 17.671 24.850 % 40.055 127.68 2. 6.2832 3.1416 11/16 17.868 25.406 % 40.448 130.19 1/16 6.4795 3.3410 H 18.064 25.967 13. 40.841 132.73 M 6.6759 3.5466 13-16 18.261 26.535 ix 41.233 135.30 3/16 6.8722 3.7583 % 18.457 27.109 M 41.626 137.89 k 7.0686 3.9761 15-16 18.653 27.688 % 42.019 140.50 5/16 7.2649 4.2000 fi 18.850 28.274 ^ 42.412 143.14 CIRCUMFERENCES AND AREAS OF CIRCLES. 109 Diam. Circum . Area. Diam. Circum u Area. Diam. Circum. Area. n% 42.804 145.80 21% 68.722 375.83 301/6 94.640 712.76 % 43.197 148.49 22. 69.115 380.13 M 95.033 718.69 % 43.590 151.20 /^ 69.508 384.46 % 95.426 724 64 14. 43.982 153.94 ix 69.900 388.82 95.819 730.62 44.375 156.70 % 70.293 393.20 % 96.211 736.62 IX 44.768 159.48 12 70.686 397.61 M 96.604 742.64 az 45.160 162.30 % 71.079 402.04 To 96.997 748.69 i/ 45.553 165.13 'M 71.471 406.49 31 97.389 754.77 % 45.946 167.99 % 71.864 410.97 /^ 97.782 760.87 SX 46.338 170.87 23 72.257 415.48 J4 98.175 766.99 Yi 46.731 173.78 H 72.649 420.00 % 98.567 773.14 15 47.124 176.71 & 73.042 424.56 /^ 98.960 779.31 ^ 47.517 179.67 3X 73.435 429.13 % 99.353 785.51 | 47.909 ' 182.65 ix: 73.827 433.74 % 99.746 791.73 ax 48.302 185.66 ft/. 74.220 438.36 % 100.138. 797.98 ix 48.695 188.69 M 74.613 443.01 32. 100.531 804.25 % 49.087 191.75 % 75.006 447.69 I/, 100.924 810.54 % 49.480 194.83 24. 75.398 452.39 % 101.316 816.86 % 49.873 197.93 /^ 75.791 457.11 % 101.709 823.21 1 i 50.265 201.06 IX: 76.184 461.86 \fa 102.102 829.58 50.658 204.22 a 76.576 466.64 % 102.494 835.97 ix 51.051 207.39 IX 76.969 471.44 % 102.887 842.39 ty 51.444 210.60 % 77.362 476.26 % 103.280 848.83 /^ 51.836 213.82 % 77.754 481.11 33. 103.673 855.30 5X 52.229 217.08 % 78.147 485.98 YB 104.065 861.79 sx 52.622 220.35 25. 78.540 490.87 104.458 868.31 % 53.014 223.65 78.933 495.79 % 104.851 874.85 17 53.407 226.98 ' 24 79.325 500.74 /^ 105.243 881.41 /6 53.800 230.33 % 79.718 505.71 % 105.636 888.00 54.192 233.71 Lj 80.111 510.71 M 106.029 894.62 % 54.585 237.10 ?B 80.503 515.72 Ys 106.421 901.26 HJ 54.978 240.53 3X 80.896 520.77 34 106.814 907.92 % 55.371 243.98 % 81.289 525.84 i^ 107.207 914.61 % 55.763 247.45 26. 81.681 530.93 /4 107.600 921.32 % 56.156 250.95 i^ 82.074 536.05 % 107.992 928.06 18 56.549 254.47 IX 82.467 541.19 ix. 108.385 934.82 /^ 56.941 258.02 a2 82.860 546.35 % 108.778 941.61 /4 57.334 261.59 /^3 83.252 551.55 !% 109.170 948.42 % 57.727 265.18 5X. 83.645 556.76 78 109.563 955.25 V& 58.119 268.80 M 84.038 562.00 35. 109.956 962.11 % 58.512 272.45 72 84.430 567.27 ^ 110.348 969.00 ax 58.905 276.12 27. 84.823 572.56 y 110.741 975.91 % 59.298 279.81 85.216 577.87 111.134 982.84 19. 59.690 283.53 \A 85.608 583.21 Xsjj 111.527 989.80 ii 60.083 287.27 s/. 86.001 588.57 % 111.919 996. 7'8 60.476 291.04 /"* 86.394 593.96 M 112.312 1003.8 % 60.868 294.83 % 86.786 599.37 % 112.705 1010.8 v& 61.261 298.65 M 87.179 604.81 36. 113.097 1017.9 % 61.654 302.49 xo 87.572 610.27 ^ 113.490 1025.0 $4 62.046 306.35 28 87.965 615.75 H 113.883 1032.1 7X 62.439 310.24 88.357 621.26 % 114.275 1039.2 20. 62.832 314.16 IX 88.750 626.80 114.668 1046.3 H 63.225 318.10 a^. 89.143 632.36 % 115.061 1053.5 IX 63.617 322.06 ^> 89.535 637.94 M 115.454 1060.7 5s 64.010 326.05 5? 89.928 643.55 % 115.846 1068.0 l^ 64.403 330.06 % 90.321 649.18 37 116.239 1075.2 7& 64.795 334.10 7X 90.713 654.84 116.632 1082.5 % 65.188 338.16 29. 91.106 660.52 ix 117.024 1089.8 % 65.581 342.25 ^ 91.499 666.23 % 117.417 1097.1 21. 65.973 346.36 H 91.892 671.96 ix 117.810 1104.5 ^ 66.366 350.50 a| 92.284 677.71 % 118.202 1111.8 \A 66.759 354.66 VZ 92.677 1683.49 ax 118.596 1119.2 $ 67.152 358.84 % 93.070 689.30 % 118.988 1126.7 ^ 67.544 363.05 % 93.462 695.13 38. 119.381 1134.1 7& 67.937 367.28 so 93.855 700.98 ^ 119.773 1141.0 s % 68.330 371.54 30. 94.248 706.86 | 120.166 1149.1 MATHEMATICAL TABLES. Diam. Circum. Area, Diam. Circum. Area. Diam. Circum. Area 38% 120.559 1156.6 46% 146.477 1707.4 54% 172.395 2365.0 120.951 1164.2 % 146.869 1716.5 55. 172.788 2375.8 % 121.344 1171.7 % 147.262 1725.7 H 173.180 2386.6 M 121.737 1179.3 47 147.655 1734.9 H 173.573 2397.5 /o 122.129 1186.9 /4 148.048 1744.2 % 173.966 2408 3 89 122.522 1194.6 /4 148.440 1753.5 174.358 2419.2 x*6 122.915 1202.3 % 148.833 1762.7 % 174.751 2430.1 % 123.308 1210.0 \& 149.226 1772.1 M 175.144 2441.1 % 123.700 1217.7 % 149.618 1781.4 Vs 175.536 2452.0 $& 124.093 1225.4 M 150.011 1790.8 56 175.929 2463.0 % 124.486 1233.2 xo 150.404 1800.1 */8 ' 176.322 2474.0 % 124.878 1241.0 48 150.796 1809.6 M 176.715 2485.0 % 125.271 1248.8 151.189 1819.0 % 177.107 2496.1 40. 125.664 1256.6 /4 151.582 1828.5 177.500 2507.2 K 126.056 1264.5 J^B 151.975 1837.9 % 177.893 2518.3 H 126.449 1272.4 L 152.367 1847.5 <X 178.285 2529.4 % 126.842 1280.3 % 152.760 1857.0 % 178.678 2540.6 ^ 127.235 1288.2 3x 153.153 1866.5 57 179.071 2551.8 % 127.627 1296.2 % 153.545 1876.1 Vs 179.463 2563.0 $4 128.020 1304.2 49 153.938 1885.7 IX 179.856 2574.2 % 128.413 1312.2 3^ 154.331 1895.4 % 180.249 2585.4 41. 128.805 1320.3 x4 154.723 1905.0 IX 180.642 2596.7 H 129.198 1328.3 % 155.116 1914.7 % 181.034 2608.0 J4 129.591 1336.4 i^ 155.509 1924.4 34 181.427 2619.4 i 129.983 1344.5 % 155.902 1934.2 7 /8 181.820 2630.7 130.376 1352.7 M 156.294 1943.9 58. 182.212 2642.1 ax 130.769 1360.8 7X 156.687 1953.7 182.605 2653.5 M 131.161 1309.0 50. 157.080 1963.5 IX- 182.998 2664.9 % 131.554 1377.2 157.472 1973.3 % 183.390 2676.4 42. 131.947 1385.4 \A 157.865 1983.2 x"l3 183.783 2687.8 ! 132.340 1393.7 % 158.258 1993.1 % 184.176 2699.3 M 132.732 1402.0 L/j 158.650 2003.0 34 184.569 2710.9 % 133.125 1410.3 % 159.043 2012.9 % 184.961 2722.4 IX 133.518 1418.6 M 159.436 2022.8 59. 185.354 2734.0 K^ 133.910 1427.0 % 159.829 2032.8 Ys 185.747 2745.6 M 134.303 1435.4 51 160.221 2042.8 x4 186.139 2757.2 % 134.696 1443.8 H 160.614 2052 .8 a2 186.532 2768.8 43 135.088 1452.2 M 161.007 2062.9 x-4 186.925 2780.5 ^ 135.481 1460.7 | 161.399 2073.0 % 187.317 2792.2 i 135.874 1469.1 161.792 2083.1 M 187.710 2803.9 % 136.267 1477.6 % 162.185 2093.2 % 188.103 2815.7 136.659 1486.2 94 162.577 2103.3 60. 188.496 2827.4 K/ 137.052 1494.7 7X 162.970 2113.5 Ys 188.888 2839.2 % 137.445 1503.3 53. 163.363 2123.7 M 189.281 2851.0 137.837 1511.9 163.756 2133.9 % 189.674 2862.9 44.? 138.230 1520.5 14 164.148 2144.2 /^ 190.066 2874.8 138.623 1529.2 % 164.541 2154.5 % 190.459 2886.6 IX 139.015 1537.9 jx 164.934 2164. H M 190.852 2898.6 % 139.408 1546.6 % 165.326 2175.1 /o 191.244 2910.5 IX 139.801 1555.3 ax 165.719 2185.4 61 191.637 2922.5 KX 140 194 1564.0 % 166.112 2195.8 *6 192.030 2934.5 3X 140.586 1572.8 53. 106.504 2206.2 H 192.423 2946.5 7X 140.979 1581.6 166.897 2216.6 % 192.815 2958.5 45. 141.372 1590.4 IX 167.290 2227.0 193.208 2970.6 141.764 1599.3 % 167.683 2237.5 % 193.601 2982.7 IX 142.157 1608.2 x^> 168.075 2248.0 M 193.993 2994.8 KX 142.550 1617.0 RX 168.468 2258.5 % 194.386 3006.9 IX 142.942 1626.0 ax 168.861 2269.1 62 194.779 3019.1 KX 143.335 1634.9 xo 109.253 2279.6 x6 195.171 3031.3 3X 143.728 1643.9 54 169.646 2290.2 y. 195.564 3043.5 % 144.121 1652.9 170.039 2300.8 % 195.957 3055.7 46 144.513 1661.9 IX 170.431 2311.5 /12 196.350 3068.0 144.906 1670.9 a/j 170.824 2322.1 % 196.742 3080.3 IX 145.299 1680.0 IX 171.217 2332.8 a^ 197.135 3092.6 az 145.691 1689.1 % 171.609 2343.5 yQ 197.528 3104.9 H 146.084 1698.2 M 172.002 2354.3 63 197.920 3117.2 CIRCUMFERENCES AND AREAS OF CIRCLES. Ill Diam. Circum. Area. Diam. Circum. Area. Diam. Circum. Area. 63^ 198.313 3129.6 71 % 224.231 4001.1 79% 250.149 4979.5 i% 198.706 3142.0 224.624 4015.2 M 250.542 4995.2 % 199.098 3154.5 % 225.017 4029.2 % 250.935 5010.9 ix 199.491 3166.9 ax 225.409 4043.3 80. 251.327 5026.5 % 199.884 3179.4 % 225.802 4057.4 /^ 251.720 5042.3 3X 200.277 3191.9 72 * 226.195 4071.5 /4 252.113 5058.0 % 200.669 3204.4 Y 226.587 4085.7 ^8 252.506 5073.8 64. 201.062 3217.0 (? 226.980 4099.8 1^2 252.898 5089.6 H 201.455 3229.6 % 227.373 4114.0 % 253.291 5105.4 3 201.847 3242.2 ix 227.765 4128.2 ax 253.684 5121.2 % 202.240 3254.8 % 228.158 4142.5 YH 254.076 5137.1 ix 202.633 3267.5 ax 228.551 4156.8 81. 254.469 5153.0 % 203.025 3280.1 7X 228.944 4171.1 ^ 254.862 5168.9 ax 203.418 3292.8 73. 229.336 4185.4 ! 255.254 5184.9 VB 203.811 3305.6 ^ 229.729 4199.7 % 255.647 5200.8 65. 204.204 3318.3 230.122 4214.1 ^ 256.040 5216.8 H 204.596 3331.1 % 230.514 4228.5 % 256.433 5232.8 8 204.989 3343.9 V& 230.907 4242.9 M 256.825 5248.9 % 205.382 3356.7 % 231.300 4257.4 TO 257.218 5264.9 205.774 3369.6 ax 231.692 4271.8 82 257.611 5281.0 % 206.167 3382.4 % 232.085 4286.3 258.003 5297.1 M 206.560 3395.3 74. 232.478 4300.8 4 258.396 5313.3 % 206.952 3408.2 232.871 4315.4 % 258.789 5329.4 66 207.345 3421.2 /4 233.263 4329.9 1^ 259.181 5345.6 YB 207.738 3434.2 9s 233.656 4344.5 % 259.574 5361.8 /4 208.131 3447.2 LX 234.049 4359.2 ax 259.967 5378.1 &x 208.523 3460.2 5^j 234.441 4373.8 % 260.359 5394.3 Via 208.916 3473.2 ax 234.834 4388.5 83 260.752 5410 6 % 209.309 3486.3 % 235.227 4403.1 H 261.145 5426.9 M 209.701 3499.4 75. 235.619 4417.9 xl 261.538 5443.3 % 210.094 3512.5 236.012 4432.6 % 261.930 5459.6 67. 210.487 3525 7 M 236.405 4447.4 % 262.323 5476.0 K 210.879 3538.8 % 236.798 4462.2 % 262.716 5492.4 H 211.272 3552.0 /^ 237.190 4477.0 ax^ 263.108 5508.8 % 211.665 3565.2 % 237.583 4491.8 xo 263.501 5525.3 212.058 3578.5 M 237.976 4506.7 84 263.894 5541.8 % 212.450 3591.7 /a 238.368 4521.5 /^ 264.286 5558.3 M 212.843 3C05.0 76 238.761 4536.5 IX- 264.679 5574.8 % 213.236 3618.3 ^6 239.154 4551.4 % 265.072 5591.4 48 213.628 3631.7 /4 239.546 4566.4 IX 265.465 5607.9 M 214.021 3645.0 % 239.939 4581.3 5^ 265.857 5624.5 M 214.414 3658.4 Hi 240.332 4596.3 ax 266.250 5641.2 214.806 3671.8 % 240.725 4611.4 TO 266.643 5657.8 /^ 215.199 3685.3 M 241.117 4626.4 85 267.035 5674.5 % 215.592 3698.7 % 241.510 4641.5 267.428 5691.2 ^4 215.984 3712.2 77. 241.903 4656.6 /4 267.821 5707.9 % 216.377 3725.7 242.295 4671.8 a| 268.213 5724.7 69. 216.770 3739.3 /4 242.688 4686.9 ix 268.606 5741.5 Ml 217.163 3752.8 % 243.081 4702.1 &x 268.999 5758.3 H 217.555 3766.4 Vi> 243.473 4717.3 ax 269.392 5775.1 % 217.948 3780.0 ^i 243.866 4732.5 % 269.784 5791.9 H 218.341 3793.7 ax 244.259 4747.8 86. 270.177 5808.8 % 218.733 3807.3 % 244.652 4763.1 270.570 5825.7 M 219.126 3821.0 78 245.044 4778.4 M 270.962 5842.6 % 219.519 3834.7 /^ 245.437 4793.7 % 271.355 5859.6 so. 219.911 3848.5 /4 245.830 4809.0 ^ 271.748 5876.5 H 220.304 3862.2 % 246.222 4824.4 % 272.140 5893.5 220.697 3876.0 i^ 246.615 4839.8 ax 272.533 5910.6 % 221.090 3889.8 % 247.008 4855.2 TO 272.926 5927.6 /^ 221.482 3903.6 M 247.400 4870.7 87 273.319 5944.7 % 221.875 3917.5 7^ 247.793 4886.2 273.711 5961.8 M 222.268 3931.4 79. .248.186 4901.7 IX 274.104 5978.9 % 222.660 3945.3 248.579 4917.2 % 274.497 5996.0 71. 223.053 3959.2 M 248.971 4932.7 /^ 274.889 6013.2 H 223.446 3973.1 % 249.364 4948.3 KX 275.282 6030.4 i 223.838 3987.1 ^ 249.757 4963.9 M 275.675 6047.6 112 MATHEMATICAL TABLES. Diam. Circum. Area. Diam. Circum. Area. Diam. Circum. Area. 87% 276.067 6064.9 92. 289.027 6647.6 96^ 301.986 7257.1 88. 276.460 6082.1 X 289.419 6665.7 y 302.378 7276.0 YB 276.853 6099.4 M 289.812 6683.8 % 302.771 7294.9 1? 277.246 6116.7 % 290.205 6701.9 r 303.164 7313.8 % 277.638 6134.1 /^ 290.597 67^0.1 % 303.556 7332.8 /^3 278.031 6151.4 % 290.990 6738.2 M 303.949 7351.8 % 278.424 6168.8 M 291.383 6756 . 4 % 304.342 7370.8 M 278.816 6186.2 sn 291.775 0774. 7 97 304.734 7389.8 7A 279.209 6203.7 93 292.168 6792.9 H 305.127 7408.9 89. 279.602 6221 . 1 H 292.561 6811.2 M 305.520 7428.0 H 279.994 6238.6 H 292.954 6829.5 % 305.913 7447.1 Y4. 280.387 6256.1 8 293.316 6847.8 iz 306.305 7466.2 g 280.780 6273.7 293.739 6866.1 5^ 306.698 7485.3 281.173 6291.2 % 294.132 6884.5 H 307.091 7504 5 % 281.565 6 08.8 M 294.524 6902.9 Vs 307.483 7523.7 M 281.958 6326.4 % 294.917 6921.3 98 307.876 7543.0 % 282.351 6344.1 94. 295.310 6939.8 YB 308.269 7562.2 90, 282.743 6361.7 ^ 295.702 6958.2 M 308.661 7581.5 x^J 283.136 6379.4 ! 296.095 6976.7 % 309.054 7600.8 ^4 283.529 6397.1 % 296.488 6995.3 /^ 309.447 7620.1 % 283.921 6414.9 Yi 296.881 7013.8 5X 309.840 7639.5 /^ 284.314 6432.6 % 297.273 703^.4 M 310.232 7658.9 % 284.707 6450.4 M 297.666 7051 .0 % 310.625 7678.3 % 285.100 6468.2 *2 298.059 7069.6 99.^ 311.018 7697.7 To 285.492 6486.0 95 298.451 7088.2 311.410 7717.1 91 285.885 6503.9 298.844 7106.9 IX 311.803 7736.6 /^ 286.278 6521.8 M 299.237 7125.6 3X 312.196 7756.1 M 286.670 6539.7 % 299.629 7144.3 \ 312.588 7775.6 2 287.063 6557 . 6 \^ 300.022 7163.0 % 312.981 ?795.2 L 287.456 6575.5 % 300.415 7181.8 3/ 313.374 7814.8 % 287.848 6593.5 M 300.807 7200.6 % 313.767 7834.4 M 288.241 6611.5 % 301.200 7219.4 100. 314.159 7854.0 I/B 288.634 6629.6 96 301.593 7238.2 DECIMALS OF A FOOT EQUIVALENT TO INCHES AND FRACTIONS OF AN INCH. Inches. Ys H % K YB H VB .01042 .02083 .03125 .04167 .05208 .06250 .07292 1 .0833 .0938 .1042 .1146 .1250 .1354 .1458 .1563 2 .1667 .1771 .1875 .1979 .2083 .2188 .2292 .2396 3 .2500 .2604 .2708 .2813 .2917 .3021 3125 ,3229 4 .3333 .3438 .3542 .3646 .3750 .3854 .3958 .4063 5 .4167 .4271 .4375 .4479 .4583 .4688 .4793 .4896 6 .5000 .5104 .5208 .5313 .5417 .5521 .5625 .5729 7 .5833 .5938 .6042 .6146 .6250 .6354 .6458 .6563 8 .6667 .6771 .6875 .6979 .7083 .7188 .7292 .7396 9 .7500 .7604 .7708 .7813 .7917 .8021 .8125 8229 10 .8333 .8438 .8542 .8646 .8750 .8854 .8958 .9063 11 .9167 .9271 .9375 .9479 .9583 .9688 .9792 .9896 CIRCUMFERENCES OF CIRCLES. 113 M O O W CO O - Vf\?*NflCvx si -rx xx e$\r>XTH\l>\-N-Kl>\aSWS .. _ . _ >i iOOiOOJOOT-i-i lCOiOCOaOOr-ir-lCO^ h i e* o o* c* o QO i- -5P gp r-i -3" t^ o co o o eo CD OB g} *o GO i-t o oo e 3! fc: QS? 5r ff ^ T-.T-iT-i(?4^0<^5COCOrfT}i-riOiOOiO35COJ.-J.-J>QOOOQ00500JOO M JO & 30 O O T O rT ' 1 -" r^coio?oacooT-icoiooaDO' ' i ICC ^ ( O 114 MATHEMATICAL TABLES. LENGTHS OF CIRCULAR ARCS. (Degrees being given. Radius of Circle = 1 .) FORMULA. Length of arc - <OA X radius X number of degrees. loU RULE. Multiply the factor in table for any given number of degrees by the radius. EXAMPLE. Given a curve of a radius of 55 feet and an angle of 78 20'. What is the length of same in feet ? Factor from table for 78 1.3613568 Factor from table for 20' .0058178 Factor 1.3671746 1.3671746 X 55 = 75.19 feet. Degrees. 1 .0174533 61 1.0646508 121 2.1118484 1 .0002909 2 .0349066 62 1.0821041 122 2.1293017 2 .0005818 3 .0523599 63 1.0995574 123 2.1467550 3 .0008727 4 .0698132 64 1.1170107 124 2.1642083 4 .0011636 5 .0872665 65 1.1344640 125 2.1816616 5 .0014544 6 .1047198 66 1.1519173 126 2.1991149 6 .0017453 7 .1221730 67 1.1693706 127 2.2165682 7 .0020362 8 .1396263 68 1.1868239 128 2.2340214 8 .0023271 9 .1570796 69 1.2042772 129 2.2514747 9 .0026180 10 .1745329 70 1.2217305 130 2.2689280 10 .0029089 11 .1919862 71 1.2391838 131 2.2863813 11 .0031998 12 .2094395 72 1.2566371 132 2.3038346 12 .0034907 13 .2268928 73 1.2740904 133 2.3212879 13 .0037815 14 .2443461 74 1.2915436 134 2.3387412 14 .0040724 15 .2617994 75 1.3089969 135 2.3561945 15 .0043633 16 .2792527 76 1.3264502 136 2.3736478 16 .0046542 17 .2967060 77 1.3439035 137 2.3911011 17 .0049451 18 .3141593 78 1.3613568 138 2.4085544 18 .0052360 19 .3316126 79 1.3788101 139 2.4260077 19 .0055269 20 .3490659 80 1.3962634 140 2.4434610 20 .0058178 21 .3665191 81 1.4137167 141 2.4609142 21 .0061087 22 .3839724 82 1.4311700 142 2.4783675 22 .0063995 23 .4014257 83 1.4486233 143 2.4958208 23 .0066904 24 .4188790 84 1.4660766 144 2.5132741 24 .0069813 25 .4363323 85 1.4835299 145 2.5307274 25 .0072722 26 .4537856 86 1.5009832 146 2.5481807 26 .0075631 27 .4712389 87 1.5184364 147 2.5656340 27 .0078540 28 .4886922 88 1.5358897 148 2.5830873 28 .0081449 29 .5061455 89 1.5533430 149 2.6005406 29 .0084358 30 .5235988 90 1.5707963 150 2.6179939 30 .0087266 31 .5410521 91 1.5882496 151 2.6354472 31 .0090175 32 .5585054 92 1.6057029 152 2.6529005 32 .0093084 33 .5759587 93 1.6231562 153 2.6703538 33 .0095993 34 .5934119 94 1.6406095 154 2.6878070 34 .0098902 35 .6108652 95 1.6580628 155 2.7052603 35 .0101811 36 .6283185 96 1.6755161 156 2.7227136 36 .0104720 37 .6457718 97 1.6929694 157 2.7401669 37 .0107629 38 .6632251 98 1.7104227 158 2.7576202 38 .0110538 39 .6806784 99 1.7278760 159 2.7750735 39 .0113446 40 .6981317 100 1.7453293 160 2.7925268 40 0116355 41 .7155850 101 1.7627825 161 2.8099801 41 .0119264 42 .7330383 102 1.7802358 162 2.8274334 42 .0122173 43 .7504916 103 1.7976891 163 2.8448867 43 .0125082 44 .7679449 104 1.8151424 164 2.8623400 44 .0127991 45 .7853982 105 1.8325957 165 2.8797933 45 .0130900 46 .8028515 106 1.8500490 166 2.8972466 46 .0133809 47 .8203047 107 1.8675023 167 2.9146999 47 .0136717 48 .8377580 108 1.8849556 168 2.9321531 48 .0139626 49 .8552113 109 1.9024089 169 2.9496064 49 .0142535 50 .8726646 110 1.9198622 170 2.9670597 50 .0145444 51 .8901179 111 1.9373155 171 2.9845130 51 .0148358 52 .9075712 112 1.9547688 172 3.0019663 52 .0151262 53 .9250245 113 1.9722-221 173 3.0194196 53 .0154171 54 .9424778 114 1.9896753 174 3.0368729 54 .0157080 55 .9599311 115 2.0071286 175 3.0543262 55 .0159989 56 .9773844 116 2.0245819 176 3.0717795 56 .0162897 57 .9948377 117 2 042o:;:.:> 177 3.0892328 57 .0165806 58 1.0122910 118 2.0594885 178 3.1066861 58 .0168715 59 1.0297443 119 2.0769418 179 3.1241394 59 .0171624 60 1.0471976 120 2.0943951 180 3.1415927 60 .0174533 LENGTHS OF CIRCULAR ARCS. 115 LENGTHS OF CIRCULAR ARCS. (Diameter = 1. Given tlie Chord and Height of the Arc.) RULE FOR USE OP THE TABLE. Divide the height by the chord. Find in the column of heights the number equal to this quotient. Take out the corre- sponding number from the column of lengths. Multiply this last number by the length of the given chord; the product will be length of the arc. If the arc is greater than a semicircle, first find, the diameter from the formula, Diam. (square of half chord -*- rise) -f rise; the formula is true whether the arc exceeds a semicircle or not. Theji find the circumference. From the diameter subtract the given height of arc, the remainder will be height of the smaller arc of the circle; find its length according to the rule, arid subtract it from the circumference. Hgts. Lgths. Hgts. Lgths. Hgts. Lgths. Hgts. Lgths. Hgts. Lgths. .001 1.00002 .15 1.05896 .238 1.14480 .326 1.26288 .414 1.40788 .005 1.00007 .152 1.06051 .24 1.14714 .328 1.26588 .416 1.41145 .01 1.00087 .154 1.06209 .242 1.14951 .33 1.26892 .418 1.41503 .015 1.00061 .156 1.06368 .244 1.15189 .332 1.27196 .42 1.41861 .02 1.00107 .158 1.06530 .246 1.15428 .331 1.27502 .422 1.42221 .025 1.00167 .16 1.06693 .248 1.15C70 .336 1.27810 .424 1.42583 .03 1.00240 .162 1.06858 .25 1.15912 .338 1.28118 .426 1.42945 .035 1.00327 .164 1.07025 .252 1.16156 .34 1.28428 .428 1.43309 .04 1.00426 .166 .07194 .254 1.16402 .342 1.28739 .43 1.43673 .045 1.00539 .168 .07365 .256 1.16650 .344 1.29052 .432 .44039 .05 1.00665 .17 .07537 .258 1.16899 .346 1.29366 .434 .44405 .055 1.00805 .172 .07711 .26 1.17150 .348 1.29681 .436 .44773 .06 1.00957 .174 .07888 .262 1.17403 .35 1.29997 .438 .45142 .065 1.01123 .176 .08066 .264 1.17657 .352 1.30315 .44 .45512 .07 1.01302 .178 .08246 .266 1.17912 .354 1.30634 .442 .45883 .075 1.01493 .18 1.08428 .268 1.18169 .356 1.30954 .444 .46255 .08 1.01698 .182 1.08611 .27 1.18429 .358 1.31276 .446 .46628 .085 1.01916 .184 1.08797 .272 1.18689 .36 1.31599 .448 .47002 .09 1.02146 .186 1.08984 .274 1.18951 .362 1.31923 .45 .47377 .095 1.02389 .188 1.09174 .276 1.19214 .364 1.32249 .452 .47753 .10 1.02646 .19 1.09365 .278 1.19479 .366 1.32577 .454 .48131 .102 1.02752 .192 1.09557 .28 1.19746 .368 1.32905 .456 .48509 .104 1.02860 .194 1.09752 .282 1.20014 .37 1.33234 .458 .48889 .106 1.02970 .196 1.09949 .284 1.20284 .372 1.33564 .46 .49269 .108 1.03082 .198 1.10147 .286 1.20555 .374 1.33896 .462 .49651 .11 1.03196 .20 1.10347 .288 1.20827 .376 1.34229 .464 .50033 .112 1.03312 .202 1.10548 .29 1.21102 .378 1.34563 .466 .50416 .114 1.03430 .204 1.10752 .292 1.21377 .38 1.34899 .468 .50800 .116 1.03551 .206 1.10958 .294 1.21654 .382 1.35237 .47 .51185 .118 1.03672 .208 1.11165 .296 1.21933 .384 1.35575 .472 .51571 .12 1.03797 .21 1.11374 .298 1.22213 .386 1.35914 .474 .51958 ,122 1.03923 .212 1.11584 .30 1.22495 .388 1.36254 .476 .52346 .124 1.04051 .214 1.11796 .302 1.22778 .39 1.36596 .478 .52736 .126 1.04181 .216 1.12011 .304 1.23063 .392 1.36939 .48 .53126 .128 1.04313 .218 1.12225 .306 1.23349 .394 1.37283 .482 .53518 .13 1.04447 .22 1.12444 .308 1.23636 .396 1.37628 .484 .53910 .132 1.04584 .222 1.12664 .31 1.23926 .398 1.37974 .486 .54302 .134 1.04722 .224 1.12885 .312 1.24216 .40 1.38322 .488 .54696 .136 1.04862 .226 1.13108 .314 1.24507 .402 1.38671 .49 .55091 .138 1.05003 .228 1.13331 .316 1.24801 .404 1.39021 .492 .55487 -14 1.05147 .23 1.13557 .318 1.25095 .406 1.39372 .494 .55854 .142 1.05293 .232 1.13785 .32 1.25391 .408 1.39724 .496 .56282 .144 1.05441 .234 1.14015 ,322 1.25689 .41 1.40077 .498 .56681 146 1.05591 .236 1.14247 .324 1.25988 .412 1.40432 .50 1.57080 .148 1.05743 116 MATHEMATICAL TABLES. AREAS OF THE: SEGMENTS OF A (Diameter = 1; Rise or Height in parts of Diameter being given.) RULE FOR USB OF THE TABLE. Divide the rise or height of the segment by the diameter. Multiply the area in the table corresponding to the quotient, thus found by the square ot the diameter. If the segment exceeds a semicircle its area is area of circle area of seg ment whose rise is (diam. of circle rise of given segment) Given chord and rise, to find diameter. Diam = (square of half chord *- rise) ~\- rise The half chqrd is a mean proportional between the two parts into which the chord divides the diameter which is perpendicular to it. Rise -5- Diam. Area, Rise Diam Area Rise -5- Diam. Area. Rise Diam Area. Rise Diam Area. .001 .00004 .054 .01646 .107 .04514 .16 .08111 .213 .12235 .002 .00012 .055 .01691 .108 .04576 .161 .08185 .214 .12317 .003 .00022 .056 .01737 .109 .04638 .162 .08258 .215 . 12399 .004 .00034 .057 .01783 .11 .04701 .163 .08332 .216 .12481 .005 .00047 .058 .01830 .111 .04763 .164 .08406 .217 .12563 .006 .00062 .059 .01877 .112 .04826 .165 .08480 .218 .12646 .007 .00078 .06 .01924 .113 .04889 .166 .08554 .219 .12729 .008 .00095 .061 .01972 .114 .04953 .167 .08629 .22 .12811 .009 .00113 .062 .02020 .115 .05016 .168 .08704 .221 .12894 .01 .00133 .063 .02068 .116 .05080 .169 .08779 .222 .12977 .011 .00153 .064 .02117 .117 .05145 .17 .08854 .223 .13060 .012 .00175 .065 .02166 .118 .05209 .171 .08929 .224 .13144 .013 .00197 .066 .02215 .119 .05274 .172 .09004 .225 .13227 .014 .0022 .067 ,02265 .12 .05338 .173 .09080 .226 .13311 .015 .00244 .068 .02315 .121 .05404 .174 .09155 .227 .13395 .016 .00268 .069 .02366 .122 .05469 .175 .09231 .228 .13478 .017 .00294 .07 .02417 .123 .05535 .176 .09307 .229 .13562 .018 .0032 .071 .02468 .124 .05600 .177 .09384 .23 .13646 .019 .00347 .072 .02520 .125 .05666 .178 .09460 .231 .13731 .02 .00375 .073 .02571 .126 .05733 .179 .09537 .232 .13815 .021 .00403 .074 .02624 .127 .05799 .18 .09613 .233 .13900 .02-2 .00432 .075 .02676 .128 .05866 .181 .09690 .234 .13984 .023 .00462 .076 .02729 .129 .05933 .182 .09767 .235 .1406S .024 .00492 .077 .02782 .13 .06000 .183 .09845 .236 .14154 .025 .00523 .078 .02836 .131 .06067 .184 .09922 .237 .14239 .026 .00555 .079 .02889 .132 .06135 .185 .10000 .238 .14324 .027 .00587 .08 .02943 .133 .06203 .186 .10077 .239 .14409 .028 .00619 .081 .02998 .134 .06271 .187 .10155 .24 .14494 .029 .00653 .082 .03053 .135- .06339 .188 .10233 .241 .14580 .03 .00687 .083 .03108 .136 .06407 .189 .10312 .242 .14666 .031 .00721 .084 .03163 .137 .06476 .19 . 10390 .243 . 14751 .032 .00756 .085 .03219 .138 .06545 .191 .10469 .244 .14837 .033 .00791 .086 .03275 .139 .06614 .192 .10547 .245 .14923 .034 .00827 .087 .03331 .14 .06683 .193 .10626 .246 -.15009 .035 .00864 .088 .03387 .141 .06753 .194 .10705 .247 .15095 ,036 .00901 .089 .03444 .142 .06822 .195 .10784 .248 .15182 .037 .00938 .09 .03501 .143 .06892 .196 .10864 .249 . 15268 038 .00976 .091 .03559 .144 .06963 .197 .10943 .25 .15355 .039 .01015 .092 .03616 .145 .07033 .198 .11023 .251 .1,5441 .04 .01054 .093 .03674 .146 .07103 .199 .11102 .252 .15528 .041 .01093 .094 .03732 .147 .07174 .2 .11182 .253 .15615 .042 .01133 .095 .03791 .148 .07245 .201 .11262 .254 .15702 .043 .01173 .096 .03850 .149 .07316 .202 .11343 .255 .15789 .044 .01214 .097 .03909 .15 .07'387 .203 .11423 .256 .15876 .045 .01255 .098 .03968 .151 .07459 .204 .11504 .257 .15964 .046 .01297 .099 .04028 .152 .07531 .205 .11584 .258 .16051 .047 .01339 .1 .04087 .153 .07603 .206 .11665 .259 .16139 .048 .01382 .101 .04148 .154 .07675 .207 .11746 .26 .16226 .049 .01425 .102 .04208 .155 .07747 .208 .11827 .261 .16314 .05 .01468 .103 .04269 .156 .07819 .209 .11908 .262 .16402 .051 .01512 .104 .04330 .157 .07892 .21 .11990 .263 .16490 .052 .01556 .105 .04391 .158 .07965 .211 .12071 .264 .16578 .053 .01601 .106 .04452 .159 .08038 .212 .12153 .265 .16666 AREAS OF THE SEGMENTS OF A CIRCLE. 117 Rise -5- Diam Area. Rise -i- Diam. Area. Rise -r- Diara. Area. Rise -f- Diam. Area Rise Diam Area. .266 .16755 .313 .21015 .36 .25455 .407 .30024 .454 .34676 .267 .16843 .314 .21108 .361 .25551 .408 .30122 .455 .34776 .268 .16932 .315 .21201 .362 .25647 .409 .30220 .456 .34876 .269 .17020 .316 .21294 .363 .25743 .41 .30319 .457 .34975 .27 .17109 .317 .21387 .364 .25839 .411 .30417 .458 .35075 .271 .17198 .318 .21480 .365 .25936 .412 .30516 .459 .35175 272 .17287 .319 .21573 .366 .26032 .413 .30614 .46 .35274 i273 .17376 .32 .21667 .367 .26128 .414 .30712 .461 .35374 .274 .17465 .321 .21760 .368 .26225 .415 .30811 .462 .35474 .275 .17554 ,32~i .21853 .369 .26321 .416 .30910 .463 .35573 .276 .17644 .323 .21947 .37 .26418 ,417 .31008 .4C4 .35673 .277 .17733 .324 .22040 .371 .26514 .418 .31107 .465 .35773 .278 .17823 .325 ,22134 .372 .26611 .419 .31205 .466 .35873 .279 .17912 .326 22228 .373 .26708 .42 .31304 .467 .35972 .28 .18002 .327 ,22322 .374 .26805 .421 .31403 .468 .36072 .281 .18092 .328 .82415 .375 .26901 .422 .31502 .469 .36172 .282 .18182 .329 .22509 .376 .26998 .423 .31600 .47 .36272 .283 .18272 .33 .22603 .377 .27095 .424 .31699 .471 .36372 .284 .18362 .331 .22697 .378 .27192 .425 .31798 .472 .36471 .285 .18452 .332 .22792 .379 .27289 .426 .31897 .473 .36571 .286 .18542 .333 .28886 .38 .27386 .427 .31996 .474 .36671 .287 .18633 .334 .25J980 .381 .27483 .428 .32095 .475 .36771 .288 .18723 .335 .28074 .382 .27580 .429 .32194 .476 .36871 .289 .18814 .336 .23169 .383 .27678 .43 .32293 .477 .36971 .29 .18905 .337 .23263 .384 .27775 .431 .32392 .478 .37071 .291 .18996 .338 .23358 .385 .27872 .432 .32491 .479 .37171 .292 .19086 .339 .33453 .386 .27969 .433 .32590 .48 .37270 .293 .19177 .34 .513547 .387 .28067 .434 .32689 .481 .37370 .294 .19268 .341 .^8642 .388 .28164 .435 .32788 .482 .37470 .295 .19360 .342 .'23737 .389 .28262 .436 .32837 .483 .37570 .296 .19451 .343 ,23832 .39 .28359 .437 .32987 .484 .37670 .297 .19542 .344 ,,23927 .391 .28457 .438 .33086 .485 .37770 .298 .19634 .345 .4022 .392 .28554 .439 .33185 .486 .37870 .299 .19725 .346 .24117 .393 .28652 .44 .33284 .487 .37970 .3 .19817 .347 ,24212 .394 .28750 .441 .33384 .488 .38070 .301 .19908 .34S ,24307 .395 .28848 .442 .33483 .489 .38170 .302 .20000 .349 ,24403 .396 .28945 .443 .33582 .49 .38270 .303 .20092 .35 ,24498 .397 .29043 .444 .33682 .491 .38370 .304 .20184 .351 .24593 .398 .29141 .445 .33781 .492 .38470 .305 .20276 .352 .24689 .399 .29239 .446 .33880 .493 .38570 ,306 .20368 ,353 .24784 .4 .29337 .447 .33980 .494 .38670 .307 .20460 .854 .24880 .401 .29435 .448 .34079 .495 .38770 ,308 .20553 855 .24976 .402 .29533 .449 .34179 .496 .38870 .309 .20845 .356 .25071 .403 .29631 .45 .34278 .497 .38970 .31 .20738 .357 .25167 .404 .29729 .451 .34378 .498 .39070 11 .20830 .358 .25263 .405 .29827 .452 .34477 .499 .39170 ^312 ,99^3 .359 .25359 .406 .29926 .453 .34577 .5 .39270 Fof rrttes for finding the area of a segment see Mensuration, page 59. MATHEMATICAL TABLES. SPHERES. (Some errors of 1 in the last figure only. From TRAUTWINE.) Diam. Sur- face. Vol- ume. Diam. Sur- face. Vol- ume. Diam. Sur- face. Vol- ume. 1-32 .00307 .00002 3 M 33.183 17.974 9 Vs 306.36 504.21 1-16 .01227 .00013 5-16 34.472 19.031 10. 314.16 523.60 3-32 .02761 .00043 35.784 20.129 322.06 543.48 t* .04909 .07670 .00102 .00200 7-16 y* 37.122 38.484 21.268 22.449 ?! 330.06 338.16 563.86 584.74 3-16 .11045 .00345 9-16 39.872 23.674 L 346.36 606.13 7-32 .15033 .00548 % 41.283 24.942 % 354.66 628.04 .19635 .00818 11-16 42.719 26.254 4 363.05 650.46 9-32 .24851 .01165 M 44.179 27.611 7 371.54 673.42 5-16 .30680 .01598 13-16 45.664 29.016 11. 380.13 696.91 11-32 .37123 .02127 Vs 47.173 30.466 388.83 720.95 .44179 .02761 15-16 48.708 31.965 M 397.61 745.51 13-32 .51848 .03511 4. 50.265 33.510 % 406.49 770.64 7-16 .60132 .04385 53.456 36.751 x^2 415.48 796.33 15-32 .69028 .05393 /4 56.745 40.195 % 424.50 822.58 L .78540 .06545 % 60.133 43.847 54 433.73 849.40 9-16 .99403 .09319 i^ 63.617 47.713 Xo 443.01 876.79 % 1.2272 .12783 % 67.201 51.801 12. 452.39 904.78 11-16 1.4849 .17014 %: 70.883 56.116 ^ 471.44 962.52 1.7671 .22089 % 74.663 60.663 i^ 490.87 1022.7 13-16 2.0739 .28084 5. 78.540 65.450 a/ 510.71 1085.3 Vs 2.4053 .35077 82.516 70.482 13. 530.93 1150.3 15-16 2.7611 .43143 M 86.591 75.757 551.55 1218.0 1. 3.1416 .52360 % 90.763 81.308 i^ 572.55 1288.3 1-16 3.5466 .62804 Y& 95.033 87.113 % 593.95 1361.2 3.9761 .74551 5^ 99.401 93.189 14. 615.75 1436.8 3-16 4.4301 .87681 M 103.87 99.541 637.95 1515.1 M 4.9088 1.0227 % 108.44 106.18 Lj 660.52 1596.3 5-16 5.4119 1.1839 6. 113.10 113.10 3^ 683.49 1680.3 5.9396 1.3611 117.87 120.31 15. 706 85 1767.2 7-16 6.4919 1.5553 J4 122.72 127.83 y* 730.63 1857.0 7.0686 1.7671 % 127.68 135.66 754.77 1949.8 9-16 7.6699 1.9974 L 132.73 143.79 % 779.32 2045.7 % 8.2957 2.2468 % 137.89 152.25 16. 804.25 2144.7 11-16 8.9461 2.5161 M 143.14 161.03 829.57 2246.8 H 9.6211 2.8062 % 148.49 170.14 L 855.29 2352.1 13-16 10.321 3.1177 7. 153.94 179.59 ax 881.42 2460.6 VB 11.044 3.4514 K 159.49 189.39 17. 907.93 2572.4 15-16 11.793 3.8083 165.13 199.53 /4 934.83 2687.6 2. 12.566 4.1888 % 170.87 210.03 i^ 962.12 2806.2 1-16 13.364 4.5939 i 176.71 220.89 ax 989.80 2928.2 14.186 5.0243 % 182.66 232.13 18. 1017.9 3053.6 3-16 15.033 5.4809 54 188.69 243.73 /4 1046.4 3182.6 14 15.904 5.9641 % 194.83 255.72 L/j 1075.2 3315.3 5-16 16.800 6.4751 8. 201.06 268.08 % 1104.5 3451.5 17.721 7.0144 207.39 280.85 19. 1134.1 18681.4 7-16 18.666 7.5829 IX- 213.82 294.01 i/ 1164.2 J3735.0 /^ 19.635 8.1813 % 220.36 307.58 Jl2 1194.6 3882.5 9-16 20.629 8.8103 L 226.98 321.56 3 1225.4 4033.7 % 21.648 9.4708 R/. 233.71 335.95 20. 1256.7 4188.8 11-16 22.691 10.164 3/ 240.53 350.77 /4 1288.3 4347.8 23.758 10.889 /o 247.45 360.02 1Z 1320.3 4510.9 13-16 24.850 11.649 9. 254.47 381.70 3^ 1352.7 4677.9 % 25.967 12.443 261.59 397.83 21. 1385.5 4849.1 15-16 27.109 13.272 /4 268.81 414.41 /4 1418.6 5024.3 3. 28.274 14.137 % 270.12 431.44 ^ 1452.2 5203.7 1-16 29.465 15.039 Y% 283.53 448.92 % 1486.2 5387.4 % 30.680 15.979 % 1291.04 466.87 22. 1520.5 5575.3 3-16 .31.919 16.957 % i 298. 65 485.31 M 1555.3 15767.6 SPHERES. SPHERES (Continued.) 119 Diam. Sur- face. Vol- ume. Diam. Sur- face. Vol- ume Diam. Sur- face. Vol. ume. 22 % 159C.4 5964.1 40 54 5153.1 34783 70 Yz 15615 183471 n 1626.0 6165.2 41. 5281.1 36087 .71. 15837 187402 23. 1661.9 6370.6 54 5410.7 37423 Yz 16061 191389 /4 1698.2 6580.6 42. 5541.9 38792 72. 16286 195433 54 1735.0 6795.2 ^ 5674.5 40194 Yz 16513 199532 M 1772.1 7014.3 43. 5808.8 41630 73. 16742 203689 24. 1809.6 7238.2 54 5944.7 43099 Yz 16972 207903 54 1847.5 7466.7 44. 6082.1 44602 74. 17204 212175 /4 1885.8 7700.1 M 6221.2 46141 K 17437 216505 % 1924.4 7938.3 45. 6361.7 47713 75. 17672 220894 25. 1963.5 8181.3 H 6503.9 49321 y* 17908 225341 54 2002.9 8429.2 46. 6647.6 50965 76. 18146 229848 ^2 2042.8 8682.0 fcf 6792.9 52645 Yz 18386 234414 M 2083.0 8939.9 47. 6939.9 54362 77. 18626 239041 26. 2123.7 9202.8 ^ 7088.3 56115 Yz 18869 243728 M 2164.7 9470.8 48. 7238.3 57906 78. 19114 248475 2206.2 9744.0 K 7389.9 59734 Y* 19360 253284 M 2248.0 10022 49. 7543.1 61601 79. 19607 258155 27. 2290.2 10306 K 7697.7 63506 54 19856 263088 M 2332.8 10595 50. 7854.0 65450 80. 20106 268083 2375.8 10889 K 8011.8 67433 H 20358 273141 M 2419.2 11189 51. 8171.2 69456 81. 20612 278263 28. 2463.0 11494 H 8332.3 71519 54 20867 283447 ix 2507.2 11805 52. 8494.8 73622 82. 21124 288696 Jij 2551.8 12121 54 8658.9 75767 54 21382 294010 M 2596.7 12443 53. 8824.8 77952 83. 21642 299388 29. 2642.1 12770 54 8992.0 80178 54 21904 304831 54 2687.8 13103 54. 9160.8 82448 84. 22167 310340 /^ 2734.0 13442 K 9331.2 84760 54 22432 315915 M 2780.5 13787 55. 9503.2 87114 85. 22698 321556 30. 2827.4 14137 54 9676.8 89511 H 22966 327264 54 2874.8 14494 56. 9852.0 91953 86. 23235 333039 54 2922.5 14856 54 10029 94438 54 23506 338882 a/ 2970.6 15224 57. 10207 96967 87. 23779 344792 81. 3019.1 15599 ^ 10387 99541 K 24053 350771 ^4 3068.0 15979 58. 10568 102161 88. 24328 356819 3117.3 16366 54 10751 104826 54 24606 362935 34 3166.9 16758 59. 10936 107536 89. 24885 369122 32. 3217.0 17157 K 11122 110294 54 25165 375378 54 3267.4 17563 60. 11310 113098 90. 25447 381704 % 3318.3 17974 Y% 11499 115949 54 25730 388102 M 3369.6 18392 61. 11690 118847 91. 26016 394570 33. 3421.2 18817 54 11882 121794 54 26302 401109 ix 3473.3 19248 62. 12076 124789 92. 26590 407721 Yz 3525.7 19685 H 12272 127832 54 2(5880 414405 M 3578.5 20129 63. 12469 130925 93. 27172 421161 34. 3631.7 20580 H 12668 134067 54 27464 427991 54 3685.3 21037 64. 12868 137259 94. 27759 434894 I/, 3739.3 21501 54 13070 140501 54 28055 441871 35. " 3848.5 22449 65. 13273 143794 95. 28353 448920 N 3959.2 23425 54 13478 147138 54 28652 456047 36. 4071.5 24429 66. 13685 150533 96. 28953 463248 34 4185.5 25461 H 13893 153980 54 29255 470524 37. 4300.9 26522 67. 14103 157480 97. 29559 477874 & 4417.9 27612 54 14314 161032 54 29865 485302 38. 4536.5 28731 68. 14527 164637 98. 30172 492808 54 4656.7 29880 Ya 14741 168295 54 30481 500388 39. 4778.4 31059 69. 14957 172007 99. 30791 508047 34 4901.7 32270 ^ 15175 175774 54 31103 515785 40. 5026.5 33510 70. 15394 179595 100. 31416 523598 120 MATHEMATICAL TABLES. CONTENTS IN CUBIC FEET AND U. S. GALLONS OF PIPES AND CYLINDERS OF VARIOUS DIAMETERS AND ONE FOOT IN LENGTH. 1 gallon = 231 cubic inches. 1 cubic foot = 7.4805 gallons. For 1 Foot in For 1 Foot in For 1 Foot in a Length. jd t- . Length. a Length. Diameter Inches. Cubic Ft. also Area in Sq. Ft. U.S. Gals., 231 Cu. In. Diametei Inches Cubic Ft. also Area in Sq. Ft. U.S. Gals., 231 Cu. In. Diamete] Inches, Cubic Ft. also Area in Sq. Ft. U.S. Gals., 231 Cu. In. H .0003 .0025 fA .2485 1.859 19 1.969 14.73 5-18 .0005 .004 .2673 1.999 1014 2.074 15.51 % .0008 .0057 714 .28<57 2.145 20 2.182 16.32 7^16 .001 .0078 .3068 2.295 20^ 2.292 17.15 H ,0014 .0102 7% .3276 2.45 21 2. -105 17.99 9-16 .0017 .0129 8 .3491 2.611 2H/2 2.521 18.86 % .0021 .0159 8J4 .3712 2.777 22 8.640 19.75 11-16 .00-20 .0193 gL .3941 2'. 9 48 221^ 2.761 20.UO H .0031 .0230 8% .4176 3.125 23 2.885 21.58 13-16 .0030 .0269 9 .4418 3.305 23^ 3.012 22.53 % .0042 .0312 9J4 .4667 3.491 24 3.142 23.50 15-16 .0048 .0359 9^ .4922 3.682 25 3.409 25.50 1 .0055 .0408 9% .5185 3.879 26 3.087 27.58 .0085 .0638 10 .5454 4.08 27 3.970 29.74 % .0123 .0918 IOM .5730 4.286 28 4.276 31.99 1% .0167 .1249 10^ .6013 4.498 29 4.587 34.31 2 4 .0218 .1632 10% .6303 4.715 30 4.909 36.72 2J4 .0276 .2066 11 .66 4.937 31 5.241 39.21 giz .0341 .2550 11^4 .6903 5.164 32 5.585 41.78 2% .0412 .3085 11^ .7213 5.396 33 5.940 44.43 3 .0491 .3672 11% .7530 5.633 34 6.305 47.16 314 .0576 .4309 12 .7854 5.875 35 6.681 49.98 .0608 .4998 12*6 .8522 6.375 36 7.069 52.88 3M .0767 .5788 13 .9218 6.895 37 7.467 55.86 4 .0873 .6528 13fc .994 7.436 38 7.876 58.92 VA .0985 .7369 14 1.069 7.997 39 8.296 62.06 41 .1104 .8263 14^ 1 147 8.578 40 8.727 65.28 4% .1231 .9200 15 1.227 9.180 41 9.168 68.58 5 .1364 1.020 15J4 1.310 9.801 42 9.621 71.97 5^ .1503 1.125 16 1.396 10.44 43 10.085 75.44 5^ .1650 1.234 16J4 1.485 11.11 44 10.559 78.99 5% .1803 1.349 17 1.576 11.79 45 11.045 82.62 6 .1963 1.469 17^ 1.670 12.49 46 11.541 86.33 6^ .2131 1.594 18 1.788 13.22 47 12.048 90.13 6^ .2304 1.724 18J* 1.867 13.96 48 12.566 94.00 To find the capacity of pipes greater than the largest given in the table, look in the table for a pipe of one half the given size, and multiply its capac- ity by 4; or one of one third its size, and multiply its capacity by 9, etc. To find the weight of water in any of the given sizes multiply the capacity in cubic feet by 62*4 or the gallons by 8^, or, if a closer approximation is required, by the weight of a cubic foot of water at the actual temperature in the pipe. Given the dimensions of a cylinder in inches, to find its capacity in U. S. gallons: Square the diameter, multiply by the length and by .0034. If d ~ diameter, I = length, gallons = d * X 54 - = .0034cW. CAPACITY OF CYLINDRICAL VESSELS. 121 CYLINDRICAL VESSELS, TANKS, CISTERNS, ETC. Diameter in Feet and Indies, Area in Square Feet, and U. S. Gallons Capacity for One Foot in Depth. 1 gallon = 231 cubic inches = 1 C u b Ot = 0.13368 cubic feet. Diam. Area. Gals. Diam. Area. Gals. Diam. Area. Gals. Ft. In. Sq. ft. 1 foot depth. Ft. In. Sq. ft. 1 foot depth. Ft. In. Sq. ft. 1 foot depth. 1 .785 5.87 5 8 25.22 188.66 19 283.53 2120.9 1 1 .922 6.89 5 9 25.97 194.25 19 3 291.04 2177.1 2 1.069 8.00 5 10 26.73 199.92 19 6 298.65 2234.0 ' 3 1.227 9.18 5 11 27.49 205.67 19 9 306.35 2291.7 4 1.396 10.44 6 28.27 211.51 20 314.16 2350.1 5 1.576 11.79 6 3 30.68 229.50 20 3 322.06 2409.2 6 1.767 13.22 6 6 33.18 248.23 20 6 330.06 2469.1 7 1.969 14.73 6 9 35.78 267.69 20 9 338.16 2529.6 8 2.182 16.32 7 38.48 287.88 21 346.36 2591.0 9 2.405 17.99 7 3 41.28 308.81 21 3 354.66 2653.0 10 2.640 19.75 7 6 44.18 330.48 21 6 363.05 2715.8 11 2.885 21.58 7 9 47.17 352.88 21 9 371.54 2779.3 2 3.142 23.50 8 50.27 376.01 22 380.13 2843.6 2 1 3.409 25.50 8 3 53.46 399.88 22 3 388.82 2908.6 2 2 3.687 27.58 8 6 56.75 424.48 22 6 397.61 2974.3 2 3 3.976 29.74 8 9 60.13 449.82 22 9 406.49 3040.8 2 4 4.276 31.99 9 63.62 475.89 23 415.48 3108.0 2 5 4.587 3431 9- 3 6720 502.70 23 3 424.56 3175.9 2 6 4.909 36.72 9 6 70.88 53024 23 6 433.74 3244.6 2 7 5.241 39.21 9 9 74.66 558.51 23 9 443.01 33140 2 8 5.585 41.78 10 78.54 587.52 24 452.39 3384.1 2 9 5.940 44.43 10 3 82.52 617.26 24 3 461.86 3455.0 2 10 6.305 47.16 10 6 86.59 647.74 24 6 471.44 3526.6 2 11 6.681 49.98 10 9 90.76 678.95 24 9 481.11 3598.9 3 7.069 52.88 11 95.03 710.90 25 490.87 3672.0 3 1 7.467 55.86 11 3 99.40 743.58 25 3 500.74 3745.8 3 2 7.876 58.92 11 6 103.87 776.99 25 6 510.71 38203 3 3 8.296 62.06 11 9 108.43 811.14 25 9 520.77 3895.6 3 4 8.727 65.28 12 113.10 846.03 26 530.93 3971.6 3 5 9.168 68.58 12 3 117.86 881.65 26 3 541.19 4048.4 3 6 9.621 71.97 12 6 122.72 918.00 26 6 551.55 4125.9 3 7 10.085 75.44 12 9 127.68 955.09 26 9 562.00 4204.1 3 8 10.559 78.99 13 132.73 992.91 27 572.56 4283.0 3 9 11.045 82.62 13 3 137.89 1031.5 27 3 583.21 4362.7 3 10 11.541 86.33 13 6 143.14 1070.8 27 6 593.96 4443.1 3 11 12.048 90.13 13 9 148.49 1110.8 27 9 604.81 4524.3 4 12.566 94.00 14 153.94 1151.5 28 615.75 4606.2 4 1 13.095 97.96 14 3 159.48 1193.0 28 3 626.80 4688.8 4 2 13.685 102.00 14 6 165.13 1235.3 28 6 637.94 4772.1 4 3 14.186 106.12 14 9 170.87 1278.2 28 9 649.18 4856.2 4 4 14.748 110.32 15 176.71 1321.9 29 660.52 4941.0 4 5 15.321 114.61 15 3 182.65 1366.4 29 3 671.96 5026.6 4 G 15.90 118.97 15 6 188.69 1411.5 29 6 683.49 5112.9 4 7 16.50 123.42 15 9 194.83 1457.4 29 9 695.13 5199.9 4 8 17.10 127.95 16 201.06 1504.1 30 706.86 5287.7 4 9 17.72 132.56 16 3 207.39 1551.4 30 3 718.69 5376.2 4 10 18.35 137.25 16 6 21382 1599.5 30 6 730.62 5465.4 4 11 18.99 142.02 16 9 220.35 1648.4 30 9 742.64 5555.4 ft 19.63 146.88 17 226.98 1697.9 31 754.77 5646.1 5 1 20.29 151.82 17 3 233.71 1748.2 31 3 766.99 5737.5 5 2 20.97 156.83 17 6 240.53 1799.3 31 6 779.31 5829.7 5 3 21.65 161.93 17 9 247.45 1851.1 31 9 791.73 5922.6 5 4 22.34 167.12 18 254.47 1903.6 32 80425 6016.2 5 5 23.04 172.38 18 3 261.59 1956.8 32 3 816.86 6110.6 5 6 23.76 177.72 18 6 268.80 2010.8 32 6 829.58 6205.7 5 7 24.48 183.15 18 9 276.12 2065.5 32 9 842.39 6301.5 122 MATHEMATICAL TABLES. GALLONS AND CUBIC FEET. United States Gallons in a given Number of Cubic Feet. 1 cubic foot = 7.480519 U. S. gallons; 1 gallon = 231 cu. in. = .13368056 cu. ft. Cubic Ft. Gallons. Cubic Ft. Gallons. Cubic Ft. Gallons. 0.1 0.75 50 374.0 8,000 59,844.2 0.2 1.50 60 448.8 9,000 67,324.7 0.3 2.24 70 523.6 10,000 74,805.2 0.4 2.99 80 598.4 20,000 149,610.4 0.5 3.74 90 673.2 30,000 224,415.6 0.6 4.49 100 748.0 40,000 299,220.8 0.7 5.24 200 1,496.1 50,000 374,025.9 0.8 5.98 300 2,244.2 60,000 448,831.1 0.9 6.73 400 2,992.2 70,000 523,636.3 1 7.48 500 3,740.3 80,000 598,441.5 2 14.96 600 4,488.3 90,000 673,246.7 3 22.44 700 5,236.4 100,000 748,051.9 4 29.92 800 5,984.4 200,000 1,496,103.8 5 37.40 900 6,732.5 300,000 2,244,155.7 6 44.88 1,000 7,480.5 400,000 2,992,207.6 7 52.36 2,000 14,961.0 500,000 - 3,740,259.5 8 59.84 3,000 22,441.6. 600,000 4,488,311.4 9 67.32 4,000 29,922.1 700,000 5,236,363.3 10 74.80 5,000 37,402.6 800,000 5,984,415.2 20 149.6 6,000 44,883.1 900,000 6,732,467.1 30 224.4 7,000 52,363.6 1,000,000 7,480,519.0 40 299.2 Cubic Feet in a given Number of Gallons. Gallons. Cubic Ft. Gallons. Cubic Ft. Gallons. Cubic Ft. 1 .134 1,000 133.681 1,000,000 133,680.6 2 .267 2,000 267.361 2,000,000 267,361.1 3 .401 3,000 401.042 3,000,000 401,041.7 4 .535 4,000 534.722 4,000,000 534,722.2 5 .668 5,000 668.403 5,000,000 668,402.8 6 .802 6,000 802.083 6,000,000 802,083.3 7 .936 7,000 935.764 7,000,000 935,763.9 8 1.069 8,000 1,069.444 8,000,000 1,069,444.4 9 1.203 9,000 1,203.125 9,000,000 1,203,125.0 10 1.337 10,000 1,336.806 10,000,000 1,336,805.6 NUMBER OF SQUARE FEET IK PLATES. 123 NUMBER OF SQUARE FEET IN PLATES 3 TO 32 FEET LONG, AND 1 INCH WIDE. For other widths, multiply by the width in inches. 1 sq. in. .0069$ sq. ft. Ft. and In. Long. Ins. Long. Square Feet. Ft. and Ins. Long. Ins. Long. Square Feet. Ft. and Ins. Long. Ins. Long. Square Feet. 8. 36 .25 7.10 94 .6528 13.8 152 .056 37 .2569 11 95 .6597 9 153 .063 2 38 .2639 8. 96 .6667 10 154 .069 3 39 .2708 1 97 .6736 11 155 .076 4 40 .2778 2 98 .6806 13.0 156 .083 5 41 .2847 3 99 .6875 1 157 .09 6 42 .2917 4 100 .6944 2 158 .097 7 43 .2986 5 101 .7014 3 159 .104 8 44 .3056 6 102 .7083 4 160 .114 9 45 .3125 103 .7153 5 161 .118 10 46 .3194 8 104 .7222 6 162 1.125 11 47 .3264 9 105 .7292 7 163 1.132 4. 48 .3333 10 106 .7361 8 164 1.139 1 49 .3403 11 107 .7431 9 165 1.146 2 50 .3472 9. 108 .75 10 166 1.153 a 51 .3542 1 109 .7569 11 167 1.159 4 . 52 .3611 2 110 .7639 14.0 168 1.167 5 53 .3681 3 111 .7708 1 169 1.174 6 54 .375 4 112 .7778 2 170 1.181 7 55 .3819 5 113 .7847 3 171 1.188 8 56 .3889 6 114 .7917 4 172 1.194 9 57 .3958 7 115 .7986 5 173 1.201 10 58 .4028 8 116 .8056 6 174 1.208 11 59 .4097 9 117 .8125 7 175 1.215 5. 60 .4167 10 118 .8194 8 176 1.222 1 61 .4236 11 119 .8264 9 177 1.229 2 62 .4306 10.0 120 .8333 10 178 1.236 3 63 .4375 1 121 .8403 11 179 1.243 4 64 .4444 2 122 .8472 15.0 180 1.25 5 65 .4514 3 123 .8542 1 181 1.257 6 66 .4583 4 124 .8611 2 182 1.264 7 67 .4653 5 125 .8681 3 183 1.271 8 68 .4722 6 126 .875 4 184 1.278 9 69 .4792 7 127 .8819 5 185 1.285 10 70 .4861 8 128 .8889 6 186 1.292 11 71 .4931 9 129 .8958 7 187 1.299 6. 72 .5 10 130 .9028 8 188 1.306 1 73 .5069 11 131 .9097 9 189 1.313 2 74 .5139 11.0 132 .9167 10 190 1.319 3 75 .5208 1 133 .9236 11 191 1.326 4 76 .5278 2 134 .9306 16.0 192 1.333 5 77 .5347 3 135 .9375 1 193 1.34 6 78 .5417 4 136 .9444 2 194 1.347 7 79 .5486 5 137 .9514 3 195 1.354 8 80 .5556 6 138 .9583 4 196 1 361 9 81 .5625 7 139 .9653 5 197 1.368 10 82 .5694 8 140 .9722 6 198 1.3T5 11 83 .5764 9 141 .9792 7 199 1.382 7. 84 .5834 10 142 .9861 8 200 1.389 1 85 .5903 11 143 .9931 9 201 1.396 2 86 .5972 12.0 144 1.000 10 202 1.403 3 87 .6042 1 145 1.007 11 203 1.41 4 88 .6111 2 146 1.014 17.0 204 1.417 . 5 89 .6181 3 147 1.021 1 205 1.424 6 90 .625 4 148 1.028 2 206 1.431 7 91 .6319 5 149 1.035 3 207 1.438 8 92 .6389 6 150 1.042 4 208 1.444 9 93 .6458 7 151 1.049 5 209 1.451 MATHEMATICAL TABLES. SQUARE: FEET IN Ft. and Ins. Long. Ins. Long. Square Feet. Ft. and Ins. Long. Ins. Long. Square Feet. Ft. and Ins. Long. Ins. Long. Square Feet. 17.6 210 1.458 22.5 269 1.868 27.4 328 2.278 211 1.465 6 270 1.875 5 329 2.285 8 21 2 1.472 7 271 1.882 6 330 2.292 9 213 1.479 8 272 1.889 7 331 2.299 10 214 1.486 9 273 1.896 8 332 2.306 11 215 1.493 10 274 1.903 9 3&3 2.313 18.0 216 1.5 11 275 1.91 10 334 2.319 1 217 1.507 38. 276 1.917 11 335 2.326 2 218 1.514 1 277 1.924 28.0 336 2.333 3 219 1.521 2 278 1.931 1 337 2.34 4 220 1.528 3 279 1.938 2 3:38 2.347 5 221 1.535 4 280 1.944 3 339 2.354 6 222 1.542 5 281 1.951 4 340 2.361 7 223 1.549 6 282 1.958 5 341 2.368 8 224 1.556 7 283 1.965 6 342 2.375 9 225 1.563 8 284 1.972 7 343 2.382 226 1.569 9 285 1.979 8 344 2.389 11 227 1.576 10 286 1.986 9 345 2.396 19.0 228 1.583 11 287 1.993 10 346 2.403 1 229 1.59 24.0 288 2 11 347 2.41 2 230 1.597 1 289 2.007 29. 348 2.417 3 231 1.604 2 290 2.014 1 349 2.424 4 232 1.611 3 291 2.021 2 350 2.431 5 233 1.618 4 292 2 028 3 351 2.438 6 234 1.6-25 5 293 2.035 4 352 2.444 7 235 1.632 6 294 2.042 5 353 2.451 8 236 1.639 7 295 2.049 6 354 2.458 9 237 1.645 8 296 2.056 7 355 2.465 10 238 1.653 9 297 2.0fi3 8 356 2.472 11 239 1.659 10 298 2.069 9 357 2.47d 20.0 240 1.667 11 299 2.076 10 358 2.486 241 1.674 25.0 300 2.083 11 359 2.493 2 242 1.681 1 301 2.09 30.0 360 2.5 3 243 1.688 2 302 2.097 1 361 2.507 4 244 1.694 3 303 2.104 2 362 2.514 5 245 1.701 4 304 2.111 3 363 2.521 6 246 1.708 5 305 2.118 4 364 2.528 7 247 1.715 6 306 2.125 5 365 2.535 8 248 1.722 7 307 2.132 6 366 2.542 9 249 1.729 8 308 2.139 7 367 2.549 10 250 1.736 9 309 2.146 8 368 2.556 11 251 1.743 10 310 2.153 9 369 2.563 21.0 252 1.75 11 311 2.16 10 370 2.569 1 253 1 .757 26.0 312 2.167 11 371 2.576 2 254 1.764 1 313 2.174 31.0 372 2.583 3 255 1.771 2 314 2.181 1 ' 373 2.59 4 256 1.778 3 315 2.188 2 374 2.597 5 257 1.785 4 316 2.194 3 375 2.604 6 258 1.792 5 317 2.201 4 376 2.611 7 259 1.799 6 318 2.208 5 377 2.618 8 260 1.806 7 319 2.215 6 378 2.625 9 261 1.813 8 3*0 2.222 7 379 2.632 10 262 1.819 9 321 2.229 8 380 2.639 11 263 1.826 10 322 2.236 9 381 2.646 22.0 264 1.833 11 323 2.243 10 382 2.653 1 265 1.84 27.0 324 2.25 11 383 2.66 2 266 1.847 1 325 2.257 32. 384 2.667 3 267 1.854 2 326 2.264 1 385 2.674 4 268 1.861 3 327 2.271 2 386 2.681 CAPACITY OF KECTAHGULAR TAHKS. 125 CAPACITIES OF RECTANGULAR TANKS IN U. S. GALLONS, FOR EACH FOOT IN DEPTH. 1 cubic foot = 7.4805 U. S. gallons. Width of Tank. Length of Tank. feet. 2 ft. in. 2 6 feet. 3 ft. in. 3 6 feet. 4 ft. in. 4 6 feet. 5 ft. in. 5 6 feet. 6 ft. in. 6 6 feet. 7 ft, in. 2 2 6 3 3 6 4 4 6 5 5 6 6 6 6 7 29.92 37.40 46.75 44.88 56.10 67.32 52.36 65.45 78.54 91.64 59.84 74.80 89.77 104.73 119.69 67.32 84.16 100.99 117.82 134.65 151.48 74.81 93.51 112.21 130.91 149.61 168.31 187.01 82.29 102.86 123.43 144.00 164.57 185.14 205.71 226.28 89.7" 112.21 134.6J 15701 179.5; 201.9' 224.4 1 246. 8( 269.3( * 97.25 121.56 > 145.87 ) 170.18 J 194.49 * 218.80 243.11 ) 267.43 ) 291.74 316.05 104.73 130.91 157.09 183.27 209.45 235.63 261.82 288.00 314.18 340.36 366.54 Width of Tank. Length of Tank. ft. in. 7 6 feet. 8 ft. in. 8 6 feet. 9 ft. in. 9 6 feet. 10 ft. in. 10 6 feet. 11 ft. in. 11 6 feet. 12 ft. in. 2 2 6 3 3 6 4 4 6 5 5 6 6 6 6 7 3 8 8 6 9 9 6 10 10 6 11 11 6 12 112.21 140.26 168.31 196.36 224.41 252.47 280.52 308.57 336.62 364.67 392.72 420.78 119.69 149.61 179.53 209.45 239.37 269.30 299.22 329.14 359.06 388.98 418.91 448.83 478.75 127.17 158.96 190.75 222.54 254.34 286.13 317.92 349.71 381.50 413.30 44509 476.88 508.67 540.46 134.65 168.31 202.97 235.63 269.30 302.96 336.62 370.28 403.94 437.60 471.27 504.93 538.59 572.25 605.92 14213 177.66 213.19 248.73 284.26 319.79 355.32 390.85 426.39 461.92 497.45 532.98 568.51 604.05 639.58 675.11 149.61 187.01 22441 261.82 299.22 336.62 374.03 411.43 448.83 486.23 523.64 561.04 598.44 635.84 673.25 710.65 748.05 157.09 196.36 235.63 274.90 314.18 353.45 392.72 432.00 471.27 510.54 549.81 589.08 628.36 66763 706.90 746.17 785.45 824.73 164.57 205.71 246.86 288.00 329.14 370.28 411.43 452.57 493.71 534.85 575.99 617.14 658.28 699.42 740.56 781.71 822.86 864.00 905.14 172.05 215.06 258.07 301.09 344.10 387.11 430.13 473.14 516.15 559.16 602.18 645.19 688.20 731.21 774.23 817.24 860.26 903.26 946.27 989.29 179.53 224.41 269.30 314.18 359.06 403.94 448.83 493.71 538.59 583.47 628.36 673.24 718.12 763.00 807.89 852.77 897.66 942.56 987.43 1032.3 1077.2 126 MATHEMATICAL TABLES. NUMBER OF BARRELS (31 1-2 GALLONS) IN CISTERNS AND TANKS. 1 Barrel = 31^ gallons = 31.5 X 231 = 4.21094 cubic fret. Reciprocal = .237477. Depth Diameter in Feet. in Feet. 5 6 7 8 9 10 11 12 13 14 1 4.663 6.714 9.139 11.93' ' 15.108 18.652 22.569 26.859 31.522 36.557 5 23.3 33.6 45.7 59.7 75.5 93.3 112.8 134.3 157.6 182.8 6 28.0 40.3 54.8 71.6 90.6 111.9 135.4 161.2 189.1 219 3 7 32.6 47.0 64.0 83.6 105.8 130.6 158.0 188.0 220.7 255.9 8 37.3 53.7 73.1 95.5 120.9 149.2 180.6 214.9 252.2 292.5 9 42.0 60.4 82.3 107.4 136.0 167.9 203.1 241.7 283.7 329.0 10 46.6 67.1 91.4 119.4 151.1 186.5 225.7 268.6 315.2 365.6 11 51.3 73.9 100.5 131.3 166.2 205.2 248 3 295.4 346.7 402.1 12 56.0 80.6 109.7 143.2 181.3 223.8 270.8 322.3 378.3 438.7 13 60.6 87.3 118.8 155.2 196.4 242.5 293.4 349.2 409.8 475 2 14 65.3 94.0 127.9 167.1 211.5 261.1 316.0 376.0 441.3 511.8 15 69.9 100.7 137.1 179.1 226.6 289.8 338.5 402.9 472.8 548.4 16 74.6 107.4 146.2 191.0 241.7 298.4 361.1 429.7 504.4 584.9 17 79.3 114.1 155.4 202.9 256.8 317.1 383.7 456.6 535.9 621 .5 18 83.9 120.9 164.5 214.9 271.9 335.7 406.2 483.5 567.4 658.0 19 88.6 127.6 173.6 226.8 287.1 354.4 428.8 510.3 598.9 694.6 20 93 3 134.3 182.8 238.7 302.2 373.0 451.4 537.2 630.4 731.1 Depth Diameter in Feet. in Feet. 15 16 17 18 19 20 21 22 1 iijwc 47.748 53.903 60.431 67.332 74.606 82.253 90.273 5 209.8 238.7 269.5 302.2 336.7 373.0 411.3 451.4 6 251.8 286.5 323.4 362 6 404.0 447.6 493.5 541.6 7 293.8 334.2 377.3 423 471.3 522.2 575.8 631.9 8 335.7 382.0 431.2 483.4 538.7 596.8 658.0 722.2 9 377.7 429.7 485.1 543.9 606.0 671.5 740.3 812.5 10 419.7 477.5 539.0 604.3 673.3 746.1 822.5 902.7 11 481.6 525.2 592.9 664.7 740.7 820.7 904.8 993.0 12 503.6 573.0 646.8 725.2 808.0 895.3 987.0 1083.3 13 545.6 620.7 700.7 785.6 875.3 969.9 1069.3 1173.5 14 587.5 668.5 754.6 846.0 942.6 1044.5 1151.5 1263.8 15 629.5 716.2 808.5 906.5 1010.0 1119.1 1233.8 1354.1 16 671.5 764.0 862.4 966.9 1077.3 1193.7 1316.0 1444.4 17 713.4 811.7 916.4 1027.3 1144.6 1268.3 1398.3 1534.5 18 755.4 859.5 970.3 1087.8 1212.0 1342.9 1480.6 1624.9 19 797.4 907.2 1024.2 1148.2 1279.3 1417.5 1562.8 1715.2 20 839.3 955.0 1078.1 1208.6 1346.6 1492.1 1645.1 1805.5 LOGARITHMS. 12? NUMBER OF BARRELS (31 1-2 GALLONS) IN CISTERNS AND TANKS. Continued. Depth in Feet. Diameter in Feet. 23 24 25 26 27 28 29 30 1 98.666 107.432 116.571 126.083 135.968 146.226 157.858 167. 86S 5 493.3 537.2 582.9 630.4 679.8 731.1 784.3 839.3 6 592.0 644.6 699.4 756.5 815.8 877.4 941.1 1007.2 7 690.7 752.0 816.0 882.6 951.8 1023.6 1098.0 1175 8 789.3 859.5 933.6 1008.7 1087.7 1169.8 1254 9 1342.9 9 888.0 966.9 1049.1 1134.7 1223.7 1316.0 1411.7 1510.8 10 986.7 1074.3 1165.7 1260.8 1359.7 1462.2 1568.6 1678.6 11 1085.3 1181.8 1282.3 1386.9 1495.6 1608.5 1725.4 1846.5 1'2 1184.0 1289.2 1398.8 1513.0 1631.6 1754.7 1882.3 2014.4 13 1282.7 1396.6 1515.4 1639.1 1767.6 1900.9 2039.2 2182.2 14 1381.3 1504 1632.0 1765.2 1903.6 2047.2 2196.0 2350.1 .15 1480.0 1611.5 1748.6 1891.2 2039.5 2193.4 2352.9 2517.9 16 1578.7 1718.9 1865.1 2017.3 2175.5 2339.6 2509.7 2685.8 17 1677.3 1826.3 1981.7 2143.4 2311.5 2485.8 2666.6 2853.7 18 1776.0 1933.8 2098.3 2269.5 2447.4 2632.0 2823.4 3021.5 19 1874.7 2041.2 2214.8 2395.6 2583.4 2778.3 2980.3 3189.4 20 1973.3 2148.6 2321.4 2521.7 2719.4 2924.5 3137.2 3357.3 LOGARITHMS. Logarithms (abbreviation log}. The log of a number is the exponent of the power to which it is necessary to raise a fixed number to produce the given number. The fixed number is called the base. Thus if the base is 10, the log of 1000 is 3, for 10 3 = 1000. There are two systems of logs in general use, the common, in which the base is 10, and the Naperian, or hyperbolic, in which the base is 2.718281828 .... The Naperian base is commonly de- noted by e, as in the equation e y = x, in which y is the Nap. log of x. In any system of logs, the log of 1 is 0; the log of the base, taken in that system, is 1. In any system the base of which is greater than 1, the logs of all numbers greater than 1 are positive and the logs of all numbers less than 1 are negative. The modulus of any system is equal to the reciprocal of the Naperian log of the base of that system. The modulus of the Naperian system is 1, that of the common system is .4342945. The log of a number in any system equals the modulus of that system X the Naperian log of the number. The hyperbolic or Naperian log of any number equals the common log X 2.3025851. Every log consists of two parts, an entire part called the characteristic, or index, and the decimal part, or mantissa. The mantissa only is given in the usual tables of common logs, with the decimal point omitted. The charac- teristic is found by a simple rule, viz., it is one less than the number of figures to the left of the decimal point in the number whose log is to be found. Thus the characteristic of numbers from 1 to 9.99 + is 0, from 10 to 99.99 + is 1, from 100 to 999 + is 2, from .1 to .99 -f is - 1, from .01 to .099 -** is - 2, etc. Thus log of 2000 is 3.30103; " " 200 " 2.30103; ' " 20 " 1.30103; * " 2 " 0.30103; log of .2 is - 1.30103; .02 " - 2.30103; .002 " - 3.30103; .0002 " - 4.30103. MATHEMATICAL TABLES. The minus sign is frequently written above the characteristic thus : log .002 = 3 .30103. The characteristic only is negative, the decimal part, or mantissa, being always positive. When a log consists of a negative index and a positive mantissa, it is usual to write the negative sign over the index, or else to add 10 to the index, and to indicate the subtraction of 10 from the resulting logarithm. Thus log .2 = Y- 3 9 10 3. and this may be written 9.30103 - 10. In tables of logarithmic sines, etc., the 10 is generally omitted, as being understood. Rules for use of the table of Logarithms. To find the log of any whole number. For 1 to 100 inclusive the log is given complete in the small table on page 129. For 100 to 999 inclusive the decimal part of the log is given opposite the given number in the column headed in the table (including the two figures to the left, making six figures). Prefix the characteristic, or index, 2. For 1000 to 9999 inclusive : The last four figures of the log are found opposite the first three figures of the given number and in the vertical column headed with the fourth figure of the given number ; prefix the two figures under column 0, and the index, which is 3. For numbers over 10,000 having five or more digits : Find the decimal part pf the log for the first four digits as above, multiply the difference figure in the last column by the remaining digit or digits, and divide by 10 if there be only one digit more, by 100 if there be two more, and so on ; add the quotient to the log of the first four digits and prefix the index, which is 4 if there are five digits, 5 if there are six digits, and so on. The table of pro- portional parts may be used, as shown below. To find the log of a decimal fraction or of a whole number and a decimal. First find the log of the quantity as if there were no decimal point, then prefix the index according to rule ; the index is one less than the number of figures to the left of the decimal point. Required log of 3.141593. log of 3.141 =0.497068. Diff. = 138 From proportional parts 5 = 690 09 = 1242 ** " * 003 = 041 log 3.141593 0.4971498 To find the number corresponding to a given log. Find in the table the log nearest to the decimal part of the given log and take the first four digits of the required number from the column N and the top or foot of the column containing the log which is the next less than the given log. To find the 5th and 6th digits subtract the log in the table from the given log, multiply the difference by 100, and divide by the figure in the Diff. column opposite the log ; annex the quotient to the four digits already found, and place the decimal point according to the rule ; the number *l figures to the left of the decimal point is one greater than the index. Find number corresponding to the log 0.497150 Next lowest log in table corresponds to 3141 497068 Diff. = 82 Tabular diff. = 138; 82 -* 138 = .59 -f The Index being 0, the number is therefore 3.14159 -f. To multiply two numbers by the use of logarithms,-- Add together the logs of the two numbers, and find the number whose log is the sum. To divide two numbers. Subtract the log of the divisor from the log of the dividend, and find the number whose log is the difference. To raise a number to any given power. Multiply the log of the number by the exponent of the power, and find the number whose log in the product. To find any root of a given number. Divide the log of the Dumber by the index of the root. The quotient is the log of the root. To find the reciprocal of a number. -Subtract the decimal part of the log of the number from 0, add 1 to the index and change the sign pf, the index. The result is the log of the reciprocal. LOGARITHMS. 129 Required the reciprocal of 3.141593. Log of 3.141593, as found above 0.4971498 Subtract decimal part from gives 0.5028502 Add 1 to the index, and changing sign of the index gives.. T.5028502 which is the log of 0.31831. To find the fourth term of a proportion by logarithms. Add the logarithms of tJ*e second and third terms, and from their sum subtract the logarithm of the first term. .. When one logarithm is to be subtracted from another, it may be more convenient to convert the subtraction into an addition, which may be done by first subtracting tLo given logarithm from 10, adding the difference to the other logarithm, and afterwards rejecting the 10. The difference between a given logarithm and 10 is called its arithmetical complement, or cologarithm. To subtract one logarithm from another is the same as to add its comple- ment and then reject 10 from the result. For a b = 10 b + a, 10. To work a proportion, then, by logarithms, add the complement of the logarithm of the first term to the logarithms of the second and third terms. The characteristic must afterwards be diminished by 10. Example In logarithms with a negative Index. Solve by 7686V" logarithms \101l7 , which means divide 526 by 1011 and raise the quotient to the 2.45 power. log 526 = log 1011 = 2.720986 3.004751 Jog of quotient = - 1.716235 Multiply by 2.45 - 2.581175 - 2.8 64940 - 1.43 2470 - 1.30 477575 = .20173, Ans. In multiplying - 1.7 by 5, we say: 5 x 7 - 35, 3 to carry; 5 x 1 = 5 less 4- 3 carried = 2. In adding -2-f-8-f3-fl carried from previous column, we say: 1 4- 3 + 8 = 12, minus 2 = 10, set down and carry 1; 1 -f 4 2 = 3. LOGARITHMS OF NUMBERS FROM 1 TO 100. N. Log. N. Log. N. Log. N. Log. N. Log. 1 0.000000 21 1.322219 41 1.612784 61 1.785330 81 1.908485 2 0.301030 22 1.342423 42 1.623249 62 1.792392 82 1.913814 3 0.477121 23 1.361728 43 1.633468 63 1.799341 83 1.919078 4 0.602060 24 1.380211 44 1.643453 64 1.806180 84 1.924279 5 0.698970 25 1.397940 45 1.653213 65 1.812913 85 1.929419 6 0.778151 26 1.414973 46 1.662758 66 1.819544 86 1.934498 7 0.845098 27 1.431364 47 1.672098 67 1.826075 87 1.939519 8 0.903090 28 1.447158 48 1.681241 68 1.832509 88 1.944483 9 0.954243 29 1.462398 49 1.690196 69 1.838849 89 1.949390 10 1.000000 30 1.477121 50 1.698970 70 1.845098 90 1.954243 11 1.041393 31 1.491362 51 1.707570 71 1.851258 91 1.959041 12 1.079181 32 1.505150 52 1.716003 72 1.857332 92 1.963788 13 1.113943 33 1.518514 53 1.724276 73 1.863323 93 1.968483 14 1.146128 34 1.531479 54 1.732394 74 1.869232 94 1.973128 15 1.176091 35 1.544068 55 1.740363 75 1.875061 95 1.977724 16 1.204120 36 1.556303 56 1.748188 76 1.880814 96 1.982271 17 1.230449 37 1.568202 57 1.755875 77 1.886491 97 1.986772 18 1.255273 38 1.579784 58 1.763428 78 1.892095 98 1.991226 19 1.278754 39 1.591065 59 1.770852 79 1.897627 99 1.995635 20 1.301030 40 1.602060 60 1.778151 80 1.903090 100 2.000000 LOGARITHMS OF LUMBERS. No. 100 L. 000.] [No. lu9 L. 040. *: 1 2 8 4 6 6 7 8 9 Diff. 100 000000 0434 0868 1301 1734 2166 2598 3029 3461 3891 432 i 4321 4751 5,181 5609 6038 6466 6894 7321 7748 8174 438 8600 9026 9451 9876 0300 0724 1147 1570 1993 2415 /KM 3 012837 3259 3680 4100 4521 4940 5360 5779 6197 6616 4K4 420 4 7033 7451 7868 8284 8700 9116 9532 9947 0361 0775 . , _ 5 021189 1603 2016 2428 2841 3252 3664 4075 4486 4896 412 6 5306 5715 6125 6533 6942 7350 7757 8164 8571 8978 408 7 9384 9789 0195 0600 1004 1408 1812 2216 261S 3021 4A1 8 033424 3826 4227 4628 5029 5430 5830 6230 6629 7028 400 9 7426 7825 8223 8620 9017 9414 9811 04 0207 0602 0998 397 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 434 43.4 86.8 130.2 173.6 217.0 260.4 3( )3.8 347.2 390.6 433 43.3 86.6 129.9 173.2 216.5 259 8 at )3.1 346.4 389.7 432 43.2 86.4 12 3.6 172.8 216.0 259 2 3( )2.4 345.6 388.8 431 43.1 86.2 129.3 172.4 215.5 258 6 301.7 344.8 387.9 430 43.0 86.0 129.0 172.0 215.0 258.0 301.0 344.0 387.0 429 42.9 85.8 12 8.7 171.6 214.5 257 4 3( K).3 343.2 386.1 428 42.8 85.6 128.4 171.2 214.0 256 8 2 ( . )9.6 342.4 385.2 427 42.7 85.4 128.1 170.8 213.5 256.2 21 )8.9 341.6 384.3 426 42.6 85.2 127.8 170.4 213.0 255.6 298.2 340.8 383.4 425 42.5 85.0 127.5 170.0 212.5 255 297.5 340.0 382.5 424 42.4 84.8 127 2 169.6 212.0 254 4 296.8 339.2 381.6 423 42.3 84.6 12 6.9 169.2 211.5 253 8 2J )6.1 338.4 380.7 422 42.2 84.4 126.6 168.8 211.0 253.2 295.4 337.6 379.8 421 42:1 84.2 126.3 168.4 210.5 252.6 294.7 336.8 378.9 420 42.0 84.0 12 6.0 168.0 210.0 252 % )4.0 336.0 378.0 419 43.9 83.8 125.7 167.6 209.5 251 4 20*. 3 335.2 377.1 418 41.8 83.6 12 5.4 167.2 209.0 250 8 2< )2.6 334.4 37'6.2 417 41.7 as. 4 125.1 166.8 208.5 250 8 291.9 333.6 375.3 416 41.6 83.2 124.8 166.4 208.0 249.6 291.2 332.8 374.4 415 41.5 83.0 124.5 166.0 207.5 249.0 290,5 332.0 373.5 414 41.4 82.8 124.2 165.6 207.0 248 4 289.8 331.2 372.6 413 41.3 82.6 12 3.9 165.2 206.5 247 8 2 39.1 330.4 371.7 412 41.2 82.4 12 3.6 164.8 206.0 247 2 2 38.4 329.6 370.8 411 41.1 82.2 123.3 164.4 205.5 246 6 287.7 328.8 309. 9 410 41.0 82.0 123.0 164.0 205.0 246.0 21 37.0 328.0 369.0 409 40.9 81.8 122.7 163.6 204.5 245 .4 286.3 327.2 368.1 408 40.8 81.6 12 2.4 163.2 204.0 244 .8 21 35.6 326.4 367.2 407 40.7 81.4 122.1 162.8 203.5 244 .2 284.9 325. 366.3 406 40.6 81.2 12 1.8 162.4 203.0 243 6 2! 34.2 324.8 365.4 405 40.5 81.0 121.5 162.0 202.5 243.0 2 33.5 324.0 364.5 404 40.4 80.8 121.2 161.6 202.0 242 .4 282.8 323.2 363.6 403 40.3 so.e 120.9 161.2 201.5 241 .8 282.1 322.4 362.7 40$ 40.2 80.4 I 1$ !0.6 160.8 201.0 241 2 21 31.4 321.6 361.8 401 40.1 80.2 120.3 160.4 200.5 240 .6 280.7 320.8 360.9 400 40.0 80-0 120.0 160.0 200.0 240 .0 280.0 320.0 360.0 39< 1 39.9 79 J 11 9.7 159.6 199.5 239 .4 2 1 79.3 319.2 359.1 39* \ 39.8 79!6 119.4 159.2 199.0 238.8 278.6 318.4 358.2 39 r r 39.7 79.^ ( 11 9.1 158.8 198.5 238 .2 2 77.9 317.6 357.3 396 39.6 79. $ > 118.8 158.4 198.0 237 .6 2 77.2 316.8 356.4 39 . 39.5 79.0 118.5 158.0 197.5 237.0 276.5 316.0 355.5 LOGARITHMS OF NUMBERS. No. 110 L. 041.] [No. 119 L. 078. N. 1 2 3 4 5 6 7 8 9 Diff. 110 041393 1787 2182 2576 2969 3362 3755 4148 4540 4932 393 1 5323 5714 6105 6495 6885 7275 7664 8053 8442 8830 390 2 9218 9606 9993 0380 0766 1153 1538 1924 2309 2694 OQC 3 053078 3463 3846 4230 4613 4996 5378 5760 6142 6524 oOD 383 4 6905 7286 7666 8046 8428 8805 9185 9563 9942 0320 379 5 060698 1075 1452 1829 2206 2582 2958 sass 3709 4083 376 6 4458 4832 5206 5580 5953 6326 6699 7071 7443 7815 373 7 8186 8557 8928 9298 9668 0038 0407 0776 1145 1514 370 8 071862 2250 2617 2985 3352 3718 4085 4451 4816 5182 366 9 5547 5912 6276 6640 7004 7368 7731 8094 8457 8819 363 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 395 394 39.5 39.4 79.0 78.8 118.5 118.2 158.0 157.6 197.5 197.0 237.0 236.4 276.5 275.8 316.0 815.2 355.5 354.6 393 39.3 78.6 11 7.9 157.2 196.5 235 .8 2 75.1 314.4 353.7 392 39.2 78.4 11 7.6 156.8 196.0 235.2 274.4 313.6 352.8 391 39.1 78.2 117.3 156.4 195.5 234 .0 273.7 312.8 351.9 390 39.0 78.0 11 7.0 156.0 195.0 234 .0 2 73.0 312.0 351.0 389 38.9 77.8 116.7 155.6 194.5 233 .4 272.3 311.2 350.1 388 38.8 77.6 11 6.4 155.2 194.0 232 .8 2 71.6 310.4 349.2 387 38.7 77.4 116.1 154.8 193.5 232.2 270.9 309.6 348.3 386 38.6 77.2 11 5.8 154.4 198.0 231 .e 2 70.2 308.8 347.4 385 38.5 77.0 115.5 154.0 192.5 231 .0 269.5 308.0 346.5 384 38.4 76.8 115.2 153.6 192.0 230.4 268.8 307.2 345.6 383 38.3 76.6 114.9 153.2 191.5 229.8 2 68.1 306.4 344.7 382 38.2 76.4 [ 11 4.6 152.8 191.0 228 .2 2 67.4 305.6 343.8 381 38.1 76.2 114.3 152.4 190.5 228.6 266.7 304.8 342.9 380 38.0 76.0 ) 11 4.0 152.0 190.0 228 .0 2 66.0 304.0 342.0 379 37.9 75. ! 11 3.7 151.6 189.5 227 .4 2 65.3 303.2 341.1 378 37.8 75.6 113.4 151.2 189.0 226.8 264.6 302.4 340.2 377 37.7 75.4 1 11 3.1 150.8 188.5 226 .2 2 63.9 301.6 339.3 376 37.6 75.2 112.8 150.4 188.0 225.6 263.2 300.8 338.4 375 37.5 75.0 112.5 150.0 187.5 225.0 I 62.5 300.0 337.5 374 37.4 74.8 112.2 149.6 187.0 224.4 261.8 299.2 336! 6 373 37.3 74. ( 1 11 1.9 149.2 186.5 22? .8 2 61.1 298.4 335.7 372 37.2 74.4 111.6 148.8 186.0 22c .2 260.4 297.6 334.8 371 37.1 74. $ J 11 1.3 148.4 185.5 22$ .6 2 59.7 296.8 333.9 370 37.0 74.0 111.0 148.0 185.0 22$ .0 259.0 296.0 333.0 369 36.9 73.* J 11 0.7 147-.6 184.5 221 .4 2 58.3 295.2 332.1 368 36.8 73.6 110.4 147.2 184.0 220.8 257.6 294.4 331.2 367 36.7 73.' 1 11 LO.l 146.8 183.5 22C ).2 i 56.9 293.6 830.3 366 36.6 73.2 109.8 146.4 183.0 219.6 256.2 292.8 329.4 365 36.5 73.0 109.5 146.0 182.5 219.0 255.7 292.0 328.5 364 36.4 72.8 109.2 145.6 182.0 218.4 254.8 291.2 327.6 363 36.3 72. ( 3 1( )8.9 145.2 181.5 217 .8 $ S54.1 290.4 326.7 362 36.2 72.4 108.6 144.8 181.0 .2 X 53.4 289.6 325.8 361 36.1 72.$ 2 1( )8.3 144.4 180.5 2ie .6 1 52.7 288.8 324.9 360 36.0 72. 3 1( )8.0 144.0 180.0 2ie .0 2 52.0 288.0 324.0 359 35.9 71.8 1( )7.7 143.6 179.5 215.4 251.3 287.2 323.1 358 35.8 71.6 107.4 143.2 179.0 214.8 250.6 286.4 322.2 357 35.7 71. I 1( )7.1 142.8 178.5 214 .2 2 49.9 285.6 321.3 356 35.6 71.2 106.8 142.4 178.0 213.6 249.2 284.8 320.4 LOGARITHMS OF NUMBERS. No. 120 L. 079.] [No. 134 L. 130. N. 1 2 3 4 5 6 7 8 9 Diff. 120 079181 9543 9904 0266 0626 II 0987 1347 1707 2067 2426 360 1 082785 3144 3503 3861 4219 4576 4934 5291 5647 6004 357 2 3 6360 9905 6716 7071 7426 7781 | 8136 8490 8845 9198 9552 355 0258 0611 0963 1315 1667 2018 2370 2721 3071 352 4 093422 3772 4122 4471 4820 5169 5518 5866 6215 6562 349 5 6910 7257 7604 7951 8298 8644 8990 9335 9681 0026 346 6 100371 0715 1059 1403 1747 2091 2434 2777 3119 3462 343 7 3804 4146 4487 4828 5169 5510 5851 6191 6531 6871 341 8 7210 7549 7888 8227 8565 8903 9241 9579 9916 0253 338 9 110590 0926 1263 1599 1934 2270 2605 2940 3275 3609 335 130 3943 4277 4611 4944 5278 5611 5943 6276 6608 6940 333 1 7271 7603 7934 8265 8595 8926 9356 9586 9915 0245 330 2 120574 0903 1231 1560 1888 2216 2544 2871 3198 3525 328 3 3852 4178 4504 4&30 5156 5481 5806 6131 6456 6781 325 4 7105 7429 7753 8076 8399 8722 9045 9368 9690 13 0012 323 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 355 a5.5 71.0 106.5 142.0 177.5 213 248.5 284.0 319.5 354 35.4 70.8 106.2 141.6 177.0 212.4 247.8 283.2 318.6 353 35.3 70.6 105.9 141.2 176.5 211.8 247.1 282.4 317.7 352 35.2 70.4 105.6 140.8 176.0 211.2 246.4 281.6 316.8 351 35.1 70.2 105.3 140.4 175.5 210.6 245.7 280.8 315.9 350 35.0 70.0 105.0 140.0 175.0 210.0 245.0 280.0 315,0 349 34.9 69.8 104.7 139.6 174.5 209.4 244.3 279.2 314.1 348 34.8 69.6 104.4 139.2 174.0 208.8 243.6 278.4 313.2 847 34.7 69.4 104.1 138.8 173.5 208.2 242.9 277.6 312.3 346 34.6 69.2 103.8 138.4 173.0 207.6 242.2 276.8 311.4 345 34.5 69.0 103.5 138.0 172.5 207.0 241.5 276.0 310.5 344 34.4 68.8 103.2 137.6 172.0 206.4 240.8 275.2 309.6 343 34.3 68.6 102.9 137.2 171.5 205.8 240.1 274.4 308.7 342 34.2 68.4 102.6 136.8 171.0 205.2 239.4 273.6 307.8 341 34.1 68.2 102.3 136.4 170.5 204.6 238.7 272.8 306.9 340 34.0 68.0 102.0 136.0 170.0 204.0 238.0 272.0 306.0 339 33.9 67.8 101.7 135.6 169-. 5 203.4 237.3 271.2 305 J 338 33.8 67.6 101.4 135.2 169.0 202.8 236.6 270.4 304.2 337 33.7 67.4 101.1 134.8 1G8.5 202.2 235.9 269.6 303.3 336 33.6 67.2 100.8 134.4 168.0 201.6 235.2 268.8 302.4 335 33.5 67.0 100.5 134.0 167.5 201.0 234.5 268.0 301.5 334 33.4 66.8 100.2 133.6 167.0 200.4 233.8 267.2 300.6 333 33.3 66.6 99.9 133.2 166.5 199.8 233.1 266.4 299.7 332 33.2 664 99.6 132.8 166.0 199.2 232.4 265.6 298.8 331 33.1 66,2 99.3 132.4 165.5 198.6 231.7 264.8 297.9 330 33.0 66.0 99.0 132.0 165.0 198.0 231.0 264.0 297.0 329 32.9 65.8 98.7 131.6 164.5 197.4 230.3 263.2 296.1 328 32.8 65.6 98.4 131.2 164.0 196.8 229.6 262.4 295.2 327 32.7 65.4 98.1 130.8 163.5 196.2 228.9 261.6 294.3 326 32.6 65.2 97.8 130.4 163.0 195.6 228.2 260.8 293.4 325 32.5 65.0 97.5 130.0 162.5 195.0 227.5 260.0 292.5 324 32.4 64.8 97.2 129.6 162.0 194.4 226.8 259.2 291.6 323 32.3 64.6 96.9 129.2 161.5 193.8 226.1 258.4 290.7 3S2 32.2 64.4 96.6 128.8 161.0 193.2 225.4 257.6 289.8 LOGARITHMS OE NUMBERS. No. 135 L. 130.] [No. 149 L. 175. N. 1 2 3 4 5 6 7 8 9 Diff. 135 130334 0655 0977 1298 1619 1939 2260 2580 2900 3219 321 6 3539 3858 4177 4496 4814 5133 5451 5769 6086 6403 318 7 g 6721 9879 7037 7354 7671 7987 8303 8618 8934 9249 9564 316 0194 0508 0822 1136 1450 1763 2076 2389 2702 314 9 143015 3327 3639 3951 4263 4574 4885 5196 5507 5818 311 140 j 6128 9219 6438 9527 6748 9835 7058 7367 7676 7985 8294 8603 8911 309 0142 0449 0756 1063 1370 1676 1982 307 2 152288 2594 2900 3205 3510 3815 4120 4424 4728 5032 305 3 5336 5640 5943 6246 6549 6852 7154 7457 7759 8061 303 4 8362 8664 8965 9266 9567 9868 0168 0469 0769 1068 301 5 161368 1667 1967 2266 25G4 2863 3161 3460 3758 4055 299 6 4353 4650 4947 5244 5541 5838 6134 6430 6726 7022 297 7 7317 7613 7908 8203 8497 8792 9086 9380 9674 9968 295 8 170262 0555 0848 1141 1434 1726 2019 2311 2603 2895 293 9 3186 3478 3769 4060 4351 4641 4932 5222 5512 5802 291 PROPORTIONAL PARTS. Diff. ' 1 2 3 4 5 6 7 8 9 321 32.1 64.2 96.3 128.4 160.5 192.6 2$ 4.7 256.8 288.9 320 32.0 64.0 96.0 128.0 160.0 192.0 224.0 256.0 288.0 319 31.9 63.8 95.7 127.6 159.5 191.4 2$ 53.3 255.2 287.1 318 31.8 63.6 95.4 127.2 159.0 190.8 222.6 254.4 286.2 317 31.7 63.4 95.1 126.8 158.5 190.2 & J1.9 253.6 285.3 316 31.6 63.2 94.8 126.4 158.0 189.6 221.2 252.8 284.4 315 31.5 63.0 94.5 126.0 157.5 189.0 2$ JO. 5 252.0 283.5 314 31.4 62.8 94.2 125.6 157.0 188.4 219.8 251.2 282.6 313 31.3 62.6 93.9 125.2 156.5 187.8 21 9.1 250.4 281.7 312 31.2 62.4 93.6 124.8 156.0 187.2 218.4 249.6 280.8 311 31.1 62.2 93.3 124.4 155.5 186.6 217.7 248.8 279.9 310 31.0 62.0 93.0 124.0 155.0 186.0 21 7.0 248.0 279.0 309 30.9 61.8 92.7 123.6 154.5 185.4 2] 6.3 247.2 278.1 308 30.8 61.6 92.4 123.2 154.0 184.3 215.6 246.4 277.2 307 30.7 61.4 92.1 122.8 153.5 184.2 214.9 245.6 276.3 306 30.6 61.2 91.8 122.4 153.0 183.6 21 4.2 244.8 275.4 305 30.5 61.0 91.5 122.0 152.5 183.0 21 3.5 244.0 274,5 304 30.4 60.8 91.2 121.6 152.0 182.4 212.8 243.2 273.6 303 30.3 60.6 90.9 121.2 151.5 181.8 2] L2.1 242.4 272.7 302 30.2 60.4 90.6 120.8 151.0 181.2 211.4 241.6 271.8 301 30.1 60.2 90.3 120.4 150.5 180.6 210.7 240.8 270.9 300 30.0 60.0 90.0 120.0 150.0 180.0 21 LO.O 240.0 270.0 299 29.9 59.8 89.7 119.6 149.5 179.4 209.3 239.2 269.1 298 29.8 59.6 89.4 119.2 149.0 178.8 2( )8.6 238.4 268.2 297 29.7 59.4 89.1 118.8 148.5 178.2 207.9 237.6 267.3 296 29.6 59.2 88.8 118.4 148.0 177.6 2( )7.2 236.8 266.4 295 29.5 59.0 88.5 118.0 147.5 177.0 206.5 '236.0 265:5 294 29.4 58.8 88.2 117.6 147.0 176.4 2( )5.8 235.2 264.6 293 29.3 58.6 87.9 117.2 146.5 175.8 205.1 234.4 263.7 292 29.2 58.4 87.6 116.8 146.0 175.2 204.4 233.6 262.8 291 29.1 58.2 87.3 116.4 145.5 174.6 203.7 232.8 261.9 290 29.0 58.0 87.0 116.0 145.0 174.0 2< )3.0 232.0 261.0 289 28.9 57.8 86.7 115.6 144.5 173.4 2( )2.3 231.2 260.1 288 28.8 57.6 86.4 115.2 144.0 172.8 2( )1.6 230.4 259.2 287 28.7 57.4 86.1 114.8 143.5 172.2 200.9 229.6 258.3 286 28.6 57.2 85.8 114.4 143.0 171.6 200.2 228.8 257.4 134 LOGARITHMS OF KUMBERS. No. 150 L,. 176.] [No. 169 L. 230. 1 N. 1 2 3 4 5 6 7 8 9 Diff. 150 176091 6381 6670 6959 7248 7536 7825 8113 8401 8689 289 j 8977 9264 9552 OH SO mo 0413 0699 0986 1272 1558 287 2 181844 2129 2415 2700 2985 3270 3555 3839 4123 4407 285 3 4691 4975 5259 5542 5825 6108 6391 6674 6956 7239 283 4 7521 7803 8084 8366 8647 8928 9209 9490 9771 0051 281 5 190332 0612 0892 1171 1451 1730 2010 2289 2567 2846 279 6 3125 3403 3681 3959 4237 4514 4792 5069 5346 5623 378 7 5900 6176 6453 6729 7005 7281 7556 7832 8107 8382 276 g 8657 8932 9206 9481 9755 0029 0303 0577 0850 1124 274 9 201397 1670 1943 2216 2488 2761 3033 3305 3577 3848 272 160 4120 4391 4663 4934 5204 5475 5746 6016 6286 6556 271 1 6826 7096 7365 7634 7904 8173 8441 8710 8979 9247 269 2 9515 9783 0051 0319 0586 0853 1121 1388 1654 1921 267 3 212188 2454 2720 2986 3252 3518 3783 4049 4314 4579 266 4 4844 5109 5373 5638 5902 6166 6430 6694 6957 7221 264 5 7484 7747- 8010 8273 8536 8798 9060 9323 9585 9846 262 6 220108 0370 0631 0892 1153 1414 1675 1936 2196 2456 261 7 2716 2976 3236 3496 3755 4015 4274 4533 4792 5051 259 8 5309 5568 5826 6084 6342 6600 6858 7115 7372 7630 258 9 7887 8144 8400 8657 8913 9170 9426 9682 9938 23 0193 256 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 285 28.5 57.0 85.5 114.0 142.5 171.0 199.5 228.0 256.5 284 28.4 56.8 85.2 113.6 142.0 170.4 198.8 227.2 255.6 283 28.3 56.6 84.9 113.2 141.5 169.8 198.1 226.4 254.7 282 28.2 56.4 84.6 112.8 141.0 169.2 197.4 225.6 253.8 281 28.1 56.2 84.3 112 4 140.5 168.6 196.7 224.8 252.9 280 28.0 56.0 84.0 112.0 140.0 168.0 196.0 224.0 252.0 279 27.9 55.8 83.7 111.6 139.5 167.4 195.3 223.2 251.1 278 27.8 55.6 83.4 111.2 139.0 166.8 194.6 222.4 250.2 277 27.7 55.4 83.1 110.8 138.5 166.2 193.9 221.6 249.3 276 27.6 55.2 82.8 110.4 138.0' 165.6 193.2 220.8 248.4 275 27.5 55.0 82.5 110.0 137.5 165.0 192.5 220.0 247.5 274 27.4 54.8 82.2 109.6 137.0 164.4 191.8 219.2 246.6 273 27.3 54.6 81.9 109.2 136.5 163.8 191.1 218.4 245.7 272 27.2 54.4 81.6 108.8 136.0 163.2 190.4 217.6 244.8 271 27.1 54.2 81.3 108.4 135.5 162.6 189.7 216.8 243.9 270 27.0 54.0 81.0 108.0 135.0 162,0 189.0 216.0 243.0 269 26.9 53.8 80.7 107.6 134.5 161.4 188.3 215.2 242.1 268 26.8 53.6 80.4 107.2 134.0 160.8 187.6 214.4 241.2 267 26.7 53.4 80.1 106.8 133.5 160.2 186.9 213 6 240.3 266 26.6 53.2 79.8 106.4 133.0 159.6 186.2 212.8 239.4 265 26.5 53.0 79.5 106.0 132.5 159.0 185.5 212.0 238.5 264 26.4 52.8 79.2 105.6 132.0 158.4 184.8 211.2 237.6 263 26.3 52.6 78.9 105.2 131.5 157.8 184.1 210.4 236.7 262 26.2 52.4 78.6 104.8 131.0 157.2 183.4 209.6 235.8 261 26.1 52.2 78.3 104.4 130.5 156.6 182.7 208.8 234.9 260 26.0 52.0 78.0 104.0 130.0 156.0 182.0 208.0 234.0 259 25.9 51.8 77.7 103.6 129.5 155.4 181.3 207.2 233.1 258 25.8 51.6 77.4 103.2 129.0 154.8 180.6 206.4 232.2 257 25.7 51.4 77.1 102.8 128.5 154.2 179.9 205.6 231.3 256 25.6 51.2 76.8 102.4 128.0 153.6 179.2 204.8 230.4 255 25.5 51.0 76.5 102.0 17.5 153.0 178.5 204.0 229.5 LOGARITHMS OF LUMBERS. No. 170 L. 230.] [No. 189 L. 278. N. 1 2 3 4 5 6 7 8 9 Diff. 170 230449 0704 0960 1215 1470 1724 1979 2234 2488 2742 255 1 2996 3250 3504 3757 4011 4264 4517 4770 5023 5276 253 2 5528 5781 6033 6285 6537 6789 7041 7292 7544 7795 252 8046 8297 8548 8799 9049 9299 9550 9800 0050 0300 250 4 240549 0799 1048 1297 1546 1795 2044 2293 2541 2790 249 5 3038 3286 3534 3782 4030 4277 4525 4772 5019 5266 248 6 5513 5759 6006 6252 6499 6745 6991 7237 7482 7728 246 7973 8219 8464 8709 8954 9198 9443 9687 9932 0176 245 8 250420 0664 0908 1151 1395 1638 1881 2125 2368 2610 243 9 2853 3096 3338 3580 3822 4064 4306 4548 4790 5031 242 180 5273 5514 5755 5996 6237 6477 6718 6958 7198 7439 241 1 7679 7918 8158 8398 8637 8877 9116 9355 9594 9833 239 2 260071 0310 0548 0787 1025 1263 1501 1739 1976 2214 238 3 2451 2688 2925 3162 3399 3636 3873 4109 4346 4582 237 4 4818 5054 5290 5525 5761 5996 6232 6467 6702 6937 235 5 7172 7406 7641 7875 8110 8344 8578 8812 9046 9279 234 9513 9746 9980 213 0446 0679 0912 1144 1377 1609 233 7 271842 2074 2306 2538 2770 3001 3233 3464 3696 3927 232 8 4158 4389 4620 4850 5081 5311 5542 5772 6002 6232 230 9 6462 6692 6921 7151 7380 7609 7838 8067 8296 8525 229 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 255 25,5 51,0 76,5 102.0 127,5 153,0 17S.5 204.0 229,5 254 25.4 50.8 76.2 101.6 127.0 152.4 177.8 203.2 228.6 253 25.3 50.6 75.9 101.2 126.5 151.8 177.1 202.4 227.7 252 25.2 50.4 75.6 100.8 126.0 151.2 176.4 201.6 226.8 251 25.1 50.2 75.3 100.4 125.5 150.6 175.7 200.8 225.9 250 25 50.0 75.0 100.0 125.0 150.0 175.0 200.0 225.0 249 24.9 49.8 74.7 99.6 124.5 149.4 174.3 199.2 224.1 248 24.8 49.6 74.4 99.2 124.0 148.8 173.6 198.4 223.2 247 24.7 49.4 74.1 98.8 123.5 148.2 172.9 197.6 222.3 246 24.6 49.2 73.8 98.4 123.0 147.6 172.2 196.8 221.4 245 24.5 49.0 73.5 98.0 122.5 147.0 171.5 196.0 220.5 244 24.4 48.8 73.2 97.6 122.0 146.4 170.8 195.2 219.6 243 24.3 48.6 72.9 97.2 121.5 145.8 170.1 194.4 218.7 242 24.2 48.4 72.6 96.8 121.0 145.2 169.4 193.6 217.8 241 24.1 48.2 72.3 96.4 120.5 144.6 168.7 192.8 216.9 240 24.0 48.0 72.0 96.0 120.0 144.0 168.0 192.0 216.0 239 23.9 47.8 71.7 95.6 119.5 143.4 167.3 191.2 215.1 238 23.8 47.6 71.4 95.2 119.0 142.8 166.6 190.4 214.2 237 23.7 47.4 71.1 94.8 118.5 142.2 165.9 189.6 213.3 236 23.6 47.2 70.8 94.4 118.0 141.6 165.2 188.8 212.4 235 23.5 47.0 70.5 94.0 < 117.5 141.0 164.5 188.0 211.5 234 23.4 46.8 70.2 93.6 117.0 140.4 163.8 187.2 210.6 233 23.3 46.6 69.9 93.2 116.5 139.8 163.1 186.4 209.7 232 23.2 46.4 69.6 92.8 116.0 139.2 162.4 185.6 208.8 .231 23.1 46.2 69.3 92.4 115.5 138.6 161.7 184.8 207.9 230 23.0 46.0 69.0 92.0 115.0 138.0 161.0 184.0 207.0 229 22.9 45.8 68.7 91.6 114.5 137.4 160.3 183.2 206.1 228 22.8 45.6 68.4 91.2 114.0 136.8 159.6 182.4 205.2 227 22.7 45.4 68.1 90.8 113.5 136.2 158.9 181.6 204.8 226 22.6 45.2 67.8 90.4 113.0 135.6 158 2 180.8 203.4 LOGARITHMS OF HUMBEK& No. 190 L. 278.] [No. 214 L. 332. N. 1 f 3 4 5 6 7 8 9 Diff. 190 278754 8982 9211 9439 9667 9895 0123 0351 0578 0806 228 1 281033 1261 1488 1715 1942 2169 2396 2022 2849 3075 227 2 3301 3527 3753 3979 4205 4431 4656 4882 5107 5332 226 3 5557 5782 6007 6232 6456 6681 6905 7130 7354 7578 225 4 7802- 8026 8249 8473 8096 8920 9143 9366 9589 9812 223 5 290035 0257 0480 0702 0925 1147 1369 1591 1813 2034 222 6 2256 2478 2699 2920 3141 3303 3584 3804 4025 4246 221 7 4466 4687 4907 5127 5347 5567 5787 6007 6226 6446 220 8 6665 68&4 7104 7323 7542 7761 7979 8198 &416 8635 219 9 8853 9071 9289 9507 9725 9943 flifii 0378 0595 0813 218 200 301030 1247 1464 1681 1898 2114 UJ.UJ. 2331 2547 2764 2980 217 1 3196 3412 3628 3844 4059 4275 4491 4706 4921 5136 216 2 5351 5566 5781 5996 6211 6425 6639 6854 7068 7282 215 3 7496 7710 7924 8137 8351 85G4 8778 8991 9204 9417 213 4 9630 9843 0056 0268 /VjQI flfiOQ flQftfi 1 1 18 iQQfj 1 *vd.9 919 5 311754 1966 2177 2389 Viol 2600 UOiAJ 2812 uyuo 3023 111O 3234 loou 3445 1O4X 3656 KOI 211 6 3867 4078 4289 4499 4710 4920 5130 5340 5551 5760 210 7 5970 6180 6390 6599 6809 7018 7227 7436 7646 7854 209 8 8063 8272 8481 8689 8898 9106 9314 9522 9730 9938 208 9 320146 0354 0562 0769 0977 1184 1391 1598 1805 2012 207 210 2219 2426 2633 2839 3046 3252 3458 3665 3871 4077 206 1 4282 4488 4694 4899 5105 5310 5516 5721 5926 6131 205 2 6336 6541 6745 6950 7155 7359 7563 7767 7972 8176 204 3 8380 8583 8787 8991 9194 9398 9601 9805 0008 0211 203 4 330414 0617 0819 1022 1225 1427 1630 1832 2034 2236 202 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 225 22.5 45.0 67.5 90.0 112.5 135.0 157.5 180.0 202.5 224 22.4 44.8 67.2 89.6 112.0 134.4 156.8 179.2 201.6 223 22.3 44.6 66.9 89.2 111.5 133.8 156.1 178.4 200.7 222 22.2 44.4 66.6 88.8 111.0 133.2 155.4 177.6 199.8 221 22.1 44.2 66.3 88.4 110.5 132.6 154.7 176.8 198.9 220 22.0 44.0 66.0 88.0 110.0 132.0 154.0 176.0 198.0 219 21.9 43.8 65.7 87.6 109.5 131.4 153.3 175.2 197.1 218 21.8 43.6 65.4 87.2 109.0 130.8 152.6 174.4 196.2 217 21.7 43.4 65.1 86.8 108.5 130.2 151.9 173.6 195.3 216 -21.6 43.2 64.8 86.4 108.0 129.6 151.2 172.8 194.4 215 21.5 43.0 64.5 86.0 107.5 129.0 150.5 172.0 193.5 214 21.4 42.8 64.2 85.6 107.0 128.4 149.8 171.2 192.6 213 21.3 42.6 63.9 85.2 106.5 127.8 149.1 170.4 191.7 212 21.2 42.4 63.6 84.8 106.0 127.2 148.4 169.6 190.8 211 21.1 42.2 63.3 84.4 105.5 126.6 147.7 168.8 189.9 210 21.0 42.0 63.0 84.0 105.0 126.0 147.0 168.0 189.0 209 20.9 41.8 62.7 83.6 104.5 125.4 146.3 167.2 188.1 208 20.8 41.6 62.4 83.2 104.0 124.8 145.6 1664 187.2 207 20.7 41.4 62.1 82.8 103.5 124.2 144.9 165.6 186.3 206 20.6 41.2 61.8 82.4 103.0 123.6 144.2 164.8 185.4 205 20.5 44.0 61.5 82.0 102.5 123.0 143.5 164.0 184.5 204 20.4 40.8 61.2 81.6 102.0 122.4 142.8 163.2 183.6 203 20.3 40.6 60.9 81.2 101.5 121.8 142.1 162.4 182.7 202 20.2 40.4 60.6 /0,8 101.0 121.2 141.4 161.6 181.8 LOGARITHMS OF NtTMBEHS. No. 215 L. 832.] [No. 239 L. 380. N. 1 2 3 4 5 6 7 8 9 Diff. 215 332438 2640 2842 3044 3246 3447 3649 3850 4051 4253 202 6 4454 4655 4856 5057 5257 5458 5658 5859 6059 6260 201 7 6460 6660 6860 7060 7260 7459 7659 7858 8058 8257 200 g 8456 8656 8855 9054 9253 9451 9650 9849 0047 0246 100 9 340444 0642 0841 1039 1237 1435 1632 1830 2028 2225 JUJJ 198 220 2423 2620 2817 3014 3212 3409 3606 3802 3999 4196 197 1 4392 4589 4785 4981 5178 5374 5570 5766 5062 6157 196 2 6353 6549 67'44 6939 7135 7330 7525 7720 7915 8110 195 3 8305 8500 8694 8889 9083 9278 9472 9666 9860 0054 194 4 350248 0442 ~0636~ 0829 1023 1216 1410 1603 1796 1989 193 5 2183 2375 2568 2761 2954 3147 3339 3532 3724 3916 193 6 4108 4301 4493 4685 4876 5068 5260 5452 5643 5834 192 7 6026 6217 6408 6599 6790 6981 7172 7363 7554 7744 191 8 Q 7935 9835 8125 8316 8506 8696 8886 9076 9266 9456 9646 190 0025 0215 0404 0593 0783 0972 1161 1350 1539 1ftQ 230 361728 1917 2105 2294 2482 2671 2859 3048 3236 3424 ioy 188 1 3612 3800 3988 4176 4363 4551 4739 4926 5113 5301 188 2 5488 5675 5862 6049 6236 6423 6610 6796 6983 7169 187 3 7356 7542 7729 7915 8101 8287 8473 8659 8845 9030 186 4 9216 9401 9587 9772 9958 O1 AQ nqoQ AK-jO OfiQft AQOQ 1QK 5 371068 1253 1437 1622 1806 U1<*O 1991 UcWo 2175 UO1O 2360 UOoO 2544 UOOO 2728 100 184 6 2912 3096 3280 3464 8647 3831 4015 4198 4382 4565 184 7 4748 4932. 5115 5298 5481 5664 5846 6029 6212 6394 183 8 6577 6759 6942 7124 7306 7488 7670 7852 8034 8216 182 9 8398 8580 8761 8943 9124 9306 9487 9668 9849 38 0030 181 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 202 201 20.2 20.1 40.4 40.2 60.6 60.3 80.8 80.4 101.0 100.5 121.2 120.6 141.4 140.7 161.6 160.8 181.8 180.9 200 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 180.0 199 19.9 39.8 59.7 79.6 99.5 119.4 139.3 159.2 179.1 198 19.8 39.6 59.4 79.2 99.0 118.8 138.6 158.4 178.2 197 19.7 89.4 59.1 78.8 98.5 118.2 137.9 157.6 177.3 196 19.6 39.2 58.8 78.4 98.0 117.6 137.2 156.8 176.4 195 19.5 39.0 58.5 78.0 97.5 117.0 136.5 156.0 175.5 194 19.4 38.8 58.2 77.G 97.0 116.4 135.8 156.2 174.6 193 19.3 38.6 57.9 77.2 96.5 115.8 135.1 154.4 173.7 192 19.2 38.4 57.6 76.8 96.0 115.2 134.4 153.6 172.8 191 19.1 38.2 57.3 76.4 95.5 114.6 133.7 152.8 171.9 190 19.0 38.0 57.0 76.0 95.0 114.0 133.0 152.0 171.0 189 18.9 37.8 56.7 75.6 94.5 113.4 132.3 151.2 170.1 188 18.8 37.6 56.4 75.2 94.0 112.8 131.6 150.4 169.2 187 18.7 374 56.1 74.8 93.5 112.2 130.9 149.6 168.3 186 18.6 37.3 55.8 74.4 93.0 111.6 130.2 148.8 167.4 185 18.5 37.0 55.5 74.0 92.5 111.0 129.5 148.0 166.5 184 18.4 36.8 55.2 73.6 92.0 110.4 128.8 147.2 165.6 183 18.3 36.6 54.9 73.2 91.5 109.8 128.1 146.4 164.7 182 18.2 36.4 54.6 72.8 91.0 109.2 127.4 145.6 163.8 181 18.1 36.2 54.3 72.4 90.5 108.6 126.7 144.8 162.9 380 18.0 36.0 54.0 72.0 90.0 108.0 126.0 144.0 162.0 179 17.9 35.8 53.7 71.6 89.5 107.4 125.3 143.3 161.1 138 LOGARITHMS OF NUMBERS. No. 240 L. 380.] [No. 269 L. 431. N. 1 | 3 4 5 6 7 8 9 Difl. 240 1 2 3 4 5 6 7 8 9 250 1 2 3 4 5 6 7 8 9 260 1 2 3 4 5 6 7 8 9 380211 2017 3815 5606 7390 9166 390935 2697 4452 6199 7940 9674 401401 3121 4834 6540 8240 9933 0392 2;97 3995 5785 756.8 9343 0573 2377 4174 5964 7746 9520 0754 2557 4353 6142 7924 9698 0934 2737 4533 6321 8101 9875 1115 2917 4712 6499 8279 1296 3097 4891 6677 8456 1476 3277 5070 6856 8634 1656 345G 5249 7034 8811 1837 3636 428 7212 8989 181 180 179 178 178 177 176 176 175 174 173 173 172 171 171 170 169 169 168 167 167 166 165 165 164 164 163 162 162 161 0051 1817 3575 5326 7071 8808 0228 1993 3751 5501 7245 8981 0405 2169 3926 5676 7419 9154 0582 2345 4101 5850 7593 9328 0759 2521 4277 6025 7766 9501 1112 2873 4627 6374 8114 9847 1288 3048 4802 6548 8287 1464 3224 4977 6722 8461 1641 3400 5152 6896 8634 0020 1745 3464 5176 6881 8579 0192 1917 3635 5346 7051 8749 0365 2089 .3807 5517 7221 8918 0538 2261 3978 5688 7391 9087 0711 2433 4149 5858 7561 9257 0883 2605 4320 6029 7731 9426 1056 2777 4492 6199 7901 9595 1228 2949 4663 6370 8070 9764 1573 3292 5005 6710 8410 0102 1788 3467 5140 6807 8467 0271 1956 3635 5307 6973 8633 0440 2124 3803 5474 7139 8798 0609 2293 3970 5641 7306 8964 0777 2461 4137 5808 7472 91^9 0946 2629 4305 5974 7638 9295 1114 2796 4472 6141 7804 9460 1283 2964 4639 6308 7970 9625 1451 3132 4806 6474 8135 9791 411620 3300 4973 6841 8301 9956 421604 3246 4882 6511 8135 9752 43 0121 1768 3410 5045 6674 8297 9914 0286 1933 3574 5208 6836 8459 0075 0451 2097 3737 5371 6999 8621 0236 0616 2261 3901 5534 7161 8783 i 0781 2426 4065 5697 7324 8944 0945 2590 4228 5860 7486 9106 1110 2754 4392 6023 7648 9268 '1275 2918 4555 6136 7811 9429 1439 3082 4718 6349 7973 9591 0398 0559 0720 0881 1042 1203 PROPORTIONAL PARTS. Diff. 1 , 2 3 4 5 6 106.8 106.2 105.6 105.0 104.4 103.8 103.2 102.6 102.0 101.4 100.8 100.2 99.6 99.0 98.4 97.8 97.2 96.6 7 8 9 178 17.8 177 17.7 176 17.6 175 17.5 174 17.4 173 17.3 172 17.2 171 17.1 170 17.0 169 16.9 168 16.8 167 16.7 166 16.6 165 16.5 164 16.4 163 16.3 162 16.2 161 16.1 35.6 53.4 35.4 53.1 35.2 52.8 35.0 52.5 34.8 52.2 34.6 51.9 34.4 51.6 34.2 51.3 34.0 51.0 33.8 50.7 33.6 50.4 33.4 50.1 33.2 49.8 33.0 49.5 32.8 49.2 32.6 48.9 32.4 48.5 32.2 48.3 71.2 70.8 70.4 70.0 69.6 69.2 68.8 68.4 68.0 67.6 67.2 66.8 66.4 66.0 65.6 65.2 64.8 64.4 89.0 88.5 88.0 87.5 87.0 86.5 86.0 85.5 85.0 84.5 84.0 83.5 83.0 82.5 82.0 81.5 81.0 80.5 124.6 123.9 123.2 122.5 121.8 121.1 120.4 119.7 119.0 118.3 117.6 116.9 116.2 115.5 114.8 114.1 113.4 112.7 142.4 141.6 140.8 140.0 139.2 138.4 137.6 136.8 136.0 135.2 134.4 133.6 132.8 132.0 131.2 130.4 129.6 128.8 160.2 159.3 158.4 157.5 156.6 155.7 154.8 153.9 153.0 152.1 151.2 150.3 149.4 148.5 147.6 146.7 145.8 144.9 LOGARITHMS OF NUMBERS. No. 270 L. 431.] [No. 299 L. 476. N. 1 2 3 4 5 7 8 9 Diff. 270 431364 1525 1685 1848 2007 2167 2328 2488 2649 2809 161 1 2969 3130 3290 3450 3610 3770 . 3930 4090 4249 4409 160 2 4569 4729 4888 5048 5207 5367 5526 5685 5844 6004 159 3 6163 6322 6481 6640 6799 6957 7116 7275 7433 7592 159 4 7751 7909 8067 8226 8384 8542 8701 8859 9017 9175 158 5 9333 9491 9648 ' Qftnft 9964 0122 0279 0437 0594 0752 158 6 440909 1066 1224 1381 isSf 1695 1852 2009 2166 2323 157 7 2480 2637 2793 2950 3106 3263 3419 3576 3732 3889 157 8 4045 4201 4357 4513 4669 4825 4981 5137 5293 5449 156 9 5604 5760 5915 6071 6226 6382 6537 6692 6848 7003 155 280 7158 7313 7468 7623 7778 7933 8088 8242 8397 8552 155 1 ,8706 8861 9015 9170 9324 9478 9633 9787 9941 0095 154 2 '450249 0403 0557 0711 0865 1018 117'2 1326 1479 1633 154 3 1786 1940 2093 2247 2400 2553 2706 2859 3012 3165 153 4 3318 3471 3624 3777 3930 4082 4235 4387 4540 4692 153 5 4845 4997 5150 5302 5454 5606 5758 5910 6062 6214 152 6 6366 6518 6670 6821 6973 7125 7276 7428 7579 7731 152 7 7882 8033 8184 8336 8487 8638 8789 8940 9091 9242 151 g 9392 9543 9694 9845 9995 0146 0296 0447 AKQ7 0748 JK-f 9 460898 1048 1198 1348 1499 1649 1799 1948 uoy< 2098 U(4O 2248 1O1 150 290 2398 2548 2697 2847 2997 3146 3296 3445 3594 3744 150 1 3893 4042 4191 4340 4490 4639 4788 4936 5085 5234 149 2 5383 5532 5680 5829 5977 6126 6274 6423 6571 6719 149 3 6868 7016 7164 7312 7460 7608 7756 7904 8052 8200 148 4 8347 8495 8643 8790 8938 9085 9233 9380 9527 9675 148 5 9822 9969 0116 0G3 0410 0557 0704 0851 0998 1145 147 6 471292 1438 1585 1732 1878 2025 2171 2318 2464 2610 itt 146 7 2756 2903 3049 3195 3341 3487 3633 3779 3925 4071 146 8 4216 4362 4508 4653 4799 4944 5090 5235 5381 5526 146 9 5671 5816 5962 6107 6252 6397 6542 6687 6832 6976 145 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 161 16.1 32.2 48.3 64.4 80.5 96.6 112.7 128.8 144.9 160 16.0 32.0 48.0 64.0 80.0 96.0 112.0 128.0 144.0 159 15.9 31.8 47.7 63.6 79.5 95.4 111.3 127.2 143.1 158 15.8 31.6 47.4 63.2 79.0 94.8 110.6 126.4 142.2 157 15.7 31.4 47.1 62.8 78.5 94.2 109.9 125.6 141.3 156 15.6 31.2 46.8 62.4 78.0 93.6 109.2 124.8 140.4 155 15.5 31.0 46.5 62.0 77.5 93.0 108.5 124.0 139.5 154 15.4 30.8 46.2 61.6 77.0 92.4 107.8 123.2 138.6 153 15.3 30.6 45.9 61.2 76.5 91.8 107.1 122.4 137.7 152 15.2 30.4 45.6 60.8 76.0 91.2 106.4 121.6 136.8 151 15.1 30.2 45.3 60.4 75.5 90.6 105.7 1208 135.9 150 15.0 30.0 45.0 60.0 75.0 90.0 105.0 120.0 135.0 149 14.9 29.8 44.7 59 6 74.5 89.4 104.3 119.2 134.1 148 14.8 29.6 44.4 59.2 74.0 88.8 103.6 118.4 133.2 147 14.7 29.4 44.1 58.8 73.5 88.2 102.9 117.6 132.3 146 14 6 29.2 43.8 58.4 73.0 87.6 102.2 116.8 131.4 145 14.5 29.0 43.5 58.0 72.5 87.0 101.5 116.0 130.5 144 14.4 28.8 43.2 57.6 72.0 86.4 100.8 115.2 129.6 143 14.3 28.6 42.9 57.2 71.5 85.8 100.1 114.4 128.7 142 14.2 28.4 42.6 56.8 71.0 85.2 99.4 113.6 127.8 141 14.1 28.2 42.3 56.4 70.5 84.6 98 '1 112.8 126.9 140 14.0 28.0 42.0 56.0 70.0 84.0 98.0 112.0 126.0 LOGARITHMS OP NUMBERS. No. 300 L, 4vr.] |No. 339 L. 531. N. 1 2 3 4 5 6 7 8 9 Diff. 300 1 2 3 4 5 6 r 8 9 310 1 2 3 4 5 6 8 9 320 1 2 3 4 5 6 7 8 9 330 1 2 3 4 5 6 7 8 9 477121 8566 7266 8711 7411 8855 7555 8999 7700 9143 7844 9287 7989 9431 8133 9575 8278 9719 8422 9863 145 144 144 143 143 142 142 141 141 140 140 139 139 139 138 138 137 137 136 13G 136 135 135 134 134 133 133 133 132 132 181 131 131 130 130 129 129 129 128 128 480007 1443 2874 4300 5721 7138 8551 9958 0151 1586 3016 4442 5863 7280 8692 0294 1729 3159 4585 6005 7421 8833 0438 1872 3302 4727 6147 7563 8974 0582 2016 3445 4869 6289 7704 9114 0725 2159 3587 5011 6430 7845 9255 0869 2302 3730 5153 657'2 7986 9396 1012 2445 3872 5295 6714 8127 9537 1156 2588 4015 5437 6855 8269 9677 1299 2731 4157 5579 6997 8410 9818 0099 1502 2900 4294 5683 7068 8448 9824 0239 1642 3040 4433 5822 7206 8586 9962 0380 1782 3179 4572 5960 7344 8724 0520 1922 3319 4711 6099 7483 8862 0661 2062 3458 4850 6238 7621 8999 0374 1744 3109 4471 5828 7181 8530 9874 0801 2201 3597 4989 6376 7759 9137 0941 2341 3737 5128 6515 5897 9275 1081 2481 S876 5267 6653 8035 9412 1)785~ 2154 3518 487'8 6234 7586 8934 1222 2621 4015 5406 6791 6173 9550 491362 2760 4155 5544 6930 8311 9687 0099 1470 2837 4199 5557 6911 8260 9606 0236 1607 297'3 4335 5693 7046 8395 9740 0511 1880 3246 4607 5964 7316 8G64 0648 2017 3382 4743 6099 7451 8799 0922 2291 8655 5014 6370 7721 9068 501059 2427 3791 5150 6505 7856 9203 1196 2564 3927 5286 6640 7991 9337 1333 2700 4063 5421 6776 8126 9471 0009 1349 9684 4016 5344 6668 79S7 9303 0143 1482 2818 4149 5476 6800 8119 9434 0745 2053 3356 4656 5951 7243 8531 9815 0277 1616 2951 4282 5609 6932 8251 9566 0411 1750 3084 4415 5741 7064 8S82 9697 510545 1883 3218 4548 5874 7196 8514 9828 521138 2444 3746 5045 G339 7630 8917 0679 2017 3351 4681 6006 7328 8646 9959 0813 2151 3484 4813 6139 7460 8777 0947 2284 3617 4946 6271 7592 8909 1081 2418 3750 5079 6403 7724 9040 1215 2551 3883 5211 6535 7855 9171 0090 1400 2705 4006 5304 6598 7888 9174 0221 1530 2835 4136 5434 6727 8016 9302 0353 1661 2966 4266 5563 6856 8145 9430 0484 1792 3096 4396 5693 6985 8274 9559 0615 1922 3226 4526 5822 7114 8402 9687 0876 2183 3486 4785 6081 7372 8060 9943 1007 2314 3616 4915 6210 7501 8788 1289 2575 3876 5174 6469 7759 9045 0072 1351 530200 0328 5 0456 0584 0712 0840 0968 1096 1223 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 139 13.9 138 13.8 137 13.7 136 13.6 135 13.5 134 13.4 133 13.3 132 13.2 131 131 130 130 129 12.9 128 12.8 127 12 7 27.8 41.7 27.6 41.4 27.4 41.1 27.2 40.8 27.0 40.5 26.8 40.2 26.6 39.9 26.4 39.6 26.2 89.3 26.0 89.0 25.8 38.7 25.6 38.4 25.4 38.1 55.6 55.2 54.8 54.4 54.0 53.6 53.2 52.8 52.4 52.0 51.6 51.2 50.8 69.5 69.0 68.5 68.0 67.5 67.0 66.5 66.0 65.5 65.0 64.5 64.0 63.5 83.4 82.8 82.2 81.6 81.0 80.4 79.8 79.2 78.6 78.0 77.4 76.8 76.2 97.3 96.6 95.9 95.2 94.5 93.8 93.1 92.4 91.7 91.0 90.3 i 89.6 88.9 111.2 110.4 109.6 108.8 108.0 107.2 106.4 105.3 104.8 104.0 103.2 102.4 101.6 125.1 124.2 123.3 122.4 121.5 120.6 119.7 118.8 117.9 117.0 116.1 115.2 114.3 kOGAKITHMS OP NUMBERS. No. 340 L. 531.] [No. 379 L.5,9. N. 1 2 3 4 5 6 7 8 9 Diff. 128 127 127 126 126 126 125 125 125 124 124 124 123 123 123 122 122 121 121 121 120 120 120 119 119 119 119 118 118 118 117 117 117 116 116 116 115 115 115 114 340 1 2 3 4 5 6 7 8 9 350 1 2 3 4 5 6 7 8 9 360 1 2 3 4 5 6 7 8 9 370 1 2 3 4 5 6 7 8 9 531479 2754 4026 5294 6558 7819 9076 1607 2882 4153 5421 6685 7945 9202 1734 3009 4280 5547 6811 8071 9327 1862 3136 4407 567'4 6937 8197 9452 1990 3264 4534 5800 7063 8322 9578 2117 3391 ! 4001 5927 7189 8448 9703 2245 3518 4787 6053 7315 8574 9829 2372 3645 4914 6180 7441 8699 9954 2500 3772 5041 6306 7567 8825 2627 3899 5167 6432 7693 8951 0079 1330 2576 3820 5060 6296 7529 8758 9984 0204 1454 2701 3944 5183 6419 7652 8881 540329 1579 2825 4068 5307 6543 7775 9003 0455 1704 2950 4192 5431 6666 7898 9126 0580 1829 3074 4316 5555 6789 8021 9249 0705 1953 3199 4440 5078 6913 8144 9371 0830 2078 3323 4564 5802 7036 8267 9494 0955 2203 3447 4688 5925 7159 8:389 9616 1080 2327 3571 4812 6049 7282 8512 9739 1205 2452 3096 4936 6172 7405 8635 9861 0106 1328 2547 3762 4973 6182 7387 a589 9787 550228 1450 2668 3883 5094 6303 7507 8709 9907 0351 1572 2790 4004 5215 6423 7627 8829 0473 1094 2911 4126 5336 6544 7?'48 8948 0595 1816 3033 4247 5457 6664 7868 90G8 0717 1938 3155 4368 5578 6785 7988 9188 0840 2000 3276 4489 5699 6905 8108 9308 0962 2181 3398 4610 5820 7026 8228 9428 1084 2303 3519 4731 5940 7146 8349 9548 1206 2425 3640 4852 6061 7267 8469 9007 0026 12^1 2412 3600 4784 5906 7144 8319 9491 0146 1340 2531 3718 4903 6084 7262 8436 9008 0265 1459 2050 3837 5021 6202 7379 8554 9725 0385 1578 2769 3955 5139 6320 7497 8671 9842 0504 1698 2887 4074 5257 6437 7614 8788 9959 0624 1817 3006 4192 5376 6555 7732 8905 0743 1936 3125 4311 5494 6673 7849 9023 0803 2055 3244 4429 5612 6791 7967 9140 0982 2174' 3362 4548 5730 6909 8084 9257 561101 2293 3481 4666 5848 7026 8202 9374 0076 1243 2407 3568 4726 5880 7032 8181 9326 0193 1359 2523 3684 4841 5996 7147 8295 9441 0309 1476 2639 3800 4957 6111 7262 8410 9555 0426 1592 2755 3915 5072 6226 7377 8525 9669 570543 1709 2872 4031 5188 6341 7492 8639 0660 1825 2988 4147 5303 6457 7607 8754 0776 1942 3104 42o3 5419 6572 7722 8868 0893 2058 3220 4379 5534 6687 7836 8983 1010 2174 3336 4494 5650 6802 7951 9097 1126 2291 3452 4610 5765 6917 8066 9212 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 128 12.8 127 12.7 126 12.6 125 12.5 124 12.4 123 12.3 122 12.2 121 12.1 120 12.0 119 11.9 25.6 25.4 25.2 25.0 24.8 24.6 24.4 24.2 24.0 23.8 38.4 38.1 . 37.8 37.5 37.2 36.9 36.6 36.3 86.0 35.7 51.2 50.8 50.4 50.0 49.6 49.2 48.8 48.4 48.0 47.6 64.0 63.5 63.0 62.5 62.0 61.5 61.0 60.5 60.0 59.5 76.8 76.2 75.6 75.0 74.4 73.8 73.2 72.6 72.0 71.4 89.6 88.9 88.2 87.5 86.8 86.1 85.4 84.7 84.0 83.3 102.4 101.6 100.8 100.0 99.2 98.4 97.6 96.8 96.0 95.2 115.2 114.3 113.4 112.5 111.6 110.7 109.8 108.9 108.0 107.1 LOGARITHMS OF LUMBERS. No. 380. I, 579.] [No. 414 L. 617. N. 380 1 2 3 4 5 6 7 8 9 390 1 2 3 4 5 G 7 8 9 400 1 2 3 4 5 6 7 8 9 410 1 2 3 4 1 2 3 4 5 6 7 8 9 Diff. 579784 9898 1 "0469" 1608 2745 3879 5009 6137 7262 8384 9503 0012 1153 2291 3426 4557 5686 6812 7935 9056 0126 1207 2404 3539 4670 5799 6925 8047 9167 0241 1381 2518 3652 4783 5912 7037 8100 9279 0355 1495 2031 3765 4896 6024 7149 8272 9391 0583 1722 2858 3992 5122 6250 7374 8496 9615 0697 1836 297'2 4105 5235 6362 7486 8608 9726 0811 1950 3085 4218 5348 6475 7599 8720 9838 0953 2066 3175 4282 5386 6487 7586 8681 9774 114 113 112 111 110 109 108 107 106 105 580925 2063 3199 4331 5461 6587 7711 8832 9950 1039 2177 3312 4444 5574 6700 7823 8944 0061 1176 2288 3397 4503 5606 6707 7805 8900 9992 ~1082~ 2169 3253 4334 5413 6489 7562 8633 9701 0173 1287 2339 3508 4614 5717 6817 7914 9009 0284 1399 2510 3018 47.24 5827 6927 8024 9119 0210 1299 2386 3469 4550 5628 6704 7777 8847 9914 0396 1510 2621 3729 4834 5937 7037 8134 9228 0507 1621 2732 3840 4945 6047 7146 8343 9337 0619 1732 2843 3950 5055 6157 7256 8353 9446 0730 1843 2954 4001 5165 6*67 7306 8462 9556 0842 1955 3004 4171 5270 6377 7476 8572 9665 0755 1843 2928 4010 5089 6166 7241 8312 9381 591065 2177 3286 4393 5496 6597 7695 8791 9883 600973 2060 3144 4226 5305 6381 7455 8526 9594 0101 1191 2277 3361 4442 5521 6596 7669 8740 9808 C319 1408 2494 3577 4G58 5736 6811 7884 8954 0428 1517 2603 3686 4706 5844 6919 7991 9001 0537 1625 2711 3794 4874 5951 7026 8098 9167 0646 1734 2819 3902 4982 6059 7133 8205 9274 0804 1951 3036 4118 5197 6274 7348 8419 9488 0021 1086 2148 3207 4264 5319 6370 7420 0128 1192 2254 3313 4370 5424 6476 7525 0234 1298 2360 3419 4475 5529 6581 7629 0341 1405 2466 3525 4581 5634 6686 7734 0447 1511 2572 3630 4686 5740 6790 7839 0554 1617 2678 8736 4792 5845 6895 7943 010660 1723 2784 3842 4897 5950 7000 0767 1829 2890 3947 5003 6055 7105 0873 1936 2996 4053 5108 6160 7210 0979 2042 3102 4159 5213 6265 7315 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 S 7 8 9 118 11.8 117 11.7 116 11.6 115 11.5 114 11.4 113 11.3 113 11.2 111 11.1 110 11.0 109 10.9 108 10.8 107 10.7 106 10.6 105 10.5 104 10.4 23.6 23.4 23.2 23.0 22.8 22.6 22.4 22.2 22.0 21.8 21.6 21.4 21.2 21.0 20.8 35.4 35.1 34.8 34.5 84.2 33.9 33.6 33.3 33.0 32.7 32.4 32.1 31.8 .81.5 31.2 47.2 46.8 46.4 46.0 45.6 45.2 44.8 44.4 44.0 43.6 43.2 42.8 42.4 42.0 41.6 59.0 58.5 58.0 57.5 57.0 56.5 56.0 55.5 55.0 54.5 54.0 53.5 53.0 52.5 52.0 70.8 70.2 69.6 69.0 68.4 67.8 67.2 66.6 66.0 65.4 64.8 64.2 63.6 63.0 62.4 82.6 81.9 81.2 80.5 79.8 79.1 78.4 77.7 77.0 76.3 75.6 74.9 74.2 73.5 72.8 94.4 93.6 92,8 92.0 91.2 90.4 89.6 88.8 88.0 87.2 86.4 85.6 84.8 84.0 83.2 100.2 105.3 104.4 103.5 102.6 101.7 100.8 99.9 99.0 98.1 97.2 96.3 95.4 94.5 93.0 LOGARITHMS OF NUMBERS. 143 No. 415 L. 618.] INo. 459 L. 662 N. 415 6 7 8 9 420 1 2 3 4 5 6 7 8 9 430 1 2 3 4 5 6 r 8 9 440 1 2 3 4 5 6 7 8 9 450 1 2 3 4 5 6 8 9 1 2 3 4 5 6 7 8 9 Diff. 618048 9093 8153 9198 8257 9302 8362 9406 8466 9511 8571 9615 8676 9719 0760 1799 2835 3869 4901 5929 6956 7980 9002 8780 9824 8884 9928 ~0968~ 2007 3042 4076 5107 6135 7161 8185 9206 8989 0032 107'2 2110 3146 4179 5210 6238 7263 8287 9308 105 104 103 102 101 100 99 98 97 96 95 620136 1176 2214 3249 4282 5312 6340 7366 8389 9410 0240 1280 2318 3353 4385 5415 6443 7468 8491 9512 0580 1545 2559 3569 4578 5584 6588 7590 8589 9586 0344 1384 2421 3456 4488 5518 6516 7571 8593 9613 0448 1488 2525 3559 4591 5621 6648 7673 8695 9715 0552 1592 2628 3663 4695 5724 6751 7775 8797 9817 0656 1695 2732 3766 4798 5827 6853 7878 8900 9919 0864 1903 2939 3973 5004 6032 7058 8082 9104 0021 1088 2052 3064 4074 5081 6087 7089 8090 9088 0123 1139 2153 3165 4175 5182 6187 7189 8190 9188 ~0183~ 1177 2168 3156 4143 5127 6110 7089 8067 9043 0224 1241 2255 3266 4276 5283 6287 7290 8290 9287 0283 1276 2267 3255 4342 5226 6208 7187 8165 9140 0326 1342 2356 3367 4376 5383 6388. 7'390 8389 9387 630428 1444 2457 3468 4477 5484 6488 7490 8489 9486 0631 1647 2660 3670 4679 5685 6688 7690 8689 9686 0733 1748 2761 3771 4779 5785 6789 7790 8789 9785 0835 1849 2862 3872 4880 5886 6889 7890 8888 9885 0936 1951 2963 3973 4981 5986 6989 7990 8988 9984 0084 1077 2069 3058 4044 5029 6011 6992 7969 8945 9919 0382 1375 2366 3354 4340 5324 6306 7285 8262 9237 640481 1474 2465 3453 4439 5422 6404 7383 8360 9335 0581 1573 2563 3551 4537 5521 6502 7481 8458 9432 0680 1672 2662 3650 4636 5619 6600 7579 8555 9530 0779 1771 2761 3749 4734 5717 6698 7676 8653 9627 0879 1871 2860 3847 4832 5815 6796 777'4 8750 9724, 0978 1970 2959 3946 4931 5913 6894 7872 8848 9821 0016 0987 1956 2923 3888 4850 5810 6769 7725 8679 9631 0581 1529 2475 0113 1084 2053 3019 3984 4946 5906 6864 7820 8774 9726 0676 1623 2569 0210 1181 2150 3116 4080 5042 6002 6960 7916 8870 9821 ~vm 171fi 2663 650308 1278 2246 3213 4177 5138 6098 7056 8011 8965 9916 0405 1375 2343 3309 4273 5235 6194 7152 8107 9060 0502 1472 2440 3405 4369 5331 6290 7247 8202 9155 0599 1569 2536 3502 4465 5427 6386 7343 8298 9250 0696 1666 2633 3598 4562 5523 6482 7438 8393 9346 0793 17'62 2730 3695 4658 5619 6577 7534 8488 9441 0890 1859 2826 3791 4754 5715 6673 7629 8584 9536 0011 0960 1907 0106 1055 2002 0201 1150 2096 0296 1245 2191 0391 1339 2286 0486 1434 2380 660865 1813 PROPORTIONAL PARTS. Diff. 1 234 5 6 7 8 9 105 10.5 104 10.4 103 10.3 102 10.2 101 10.1 100 10.0 99 9.9 21.0 31.5 42.0 20.8 31.2 41.6 20.6 30.9 41.2 20.4 30.6 40.8 20.2 30.3 40.4 20.0 30.0 40.0 19.8 29.7 39.6 52.5 52.0 51.5 51.0 50.5 50.0 49.5 63.0 73.5 62.4 72 8 61.8 721 61.2 714 60.6 70 7 60.0 70 59.4 69.3 84.0 83.2 82.4 81.6 80.8 80.0 79.2 94.5 93.6 92.7 91.8 90.9 90.0 89.1 LOGARITHMS OF LUMBERS. No. 4GO L. 662.] [No. 499 L. 698.' N, 1 2 8 4 5 C 7 8 9 Diff. 460 662758 2852 2947 3041 3135 3230 3324 3418 3512 3607 1 3701 3795 3889 398 3 4078 4172 4266 4 4454 4548 2 4642 4736 4830 4924 5018 5112 5206 5299 5393 5487 94 3 5581 5675 5769 586 2 5956 6050 6143 6$ 5:-i7 6331 6424 4 6518 6612 6705 6799 6892 6986 7079 7173 7266 7360 5 7453 7546 7640 773 3 7826 7920 8013 81 08 8199 8293 6 8386 8479 8572 866 5 8759 8852 8945 9038 9131 9224 9317 9410 9503 959 ft 9689 9782 9875 91 )t\7 0060 0153 93 8 670246 0339 0431 0524 0617 0710 0802 0895 0988 1080 9 1173 1265 1358 1451 1543 1636 1728 1821 1913 2005 470 2098 2190 2283 2375 2467 2560 2652 2744 2836 2929 1 3021 3113 3205 329 7 3390 3482 3574 3( 3758 3850 2 3942 4034 4126 4218 4310 4402 4494 4586 4677 4769 92 3 4861 4953 5045 5137 5228 5320 5412 5503 5595 5687 4 5778 5870 5962 605 3 6145 6236 6328 fr U9 6511 6602 5 6694 6785 6876 6968 7059 7151 7242 7333 7424 7516 6 7607 7698 7789 788 1 7972 8063 8154 nr> 8336 8427 7 8518 8609 8700 8791 8882 8973 9064 9155 9246 9337 91 g 9428 9519 9610 970 in 9791 9882 9973 0063 0154 0245 9 680336 0426 0517 0607 0698 0789 0879 0970 10GO 1151 480 1241 1332 1422 1513 1603 1693 1784 1874 1964 2055 1 2145 2235 2326 241 6 2506 2596 2686 2 77 2867 2957 2 3047 3137 3227 3317 3407 3497 3587 3677 37G7 3857 90 3 3947 4037 4127 4217 4307 4396 4486 4576 4666 4756 4 4845 4935 5025 511 4 5204 5294 5383 fr 173 5563 5652 5 5742 5831 5921 6010 6100 6189 6279 6 J68 6458 6547 6 6636 6726 6815 69C 4 6994 7083 7172 7, 861 7351 7440 7 7529 7618 7707 7796 7886 T975 8064 8153 8242 8331 89 8 8420 8509 8598 868 7 8776 8865 8953 9( m 9131 9220 9 9309 9398 9486 957 5 9664 9753 9841 Q )30 0019 0107 490 690196 0285 0373 0462 0550 0639 0728 0816 0905 0993 1 1081 1170 1258 134 1435 1524 1612 ji '00 1789 1877 2 1965 2053 2142 2:230 2318 2406 2494 2583 2671 2759 3 2847 2935 3023 311 1 3199 3287 3375 & 63 3551 3639 88 4 3727 3815 3903 3991 4078 4166 4254 4342 4430 4517 5 4605 4693 4781 48C 8 4956 5044 5131 5219 5307 5394 6 5482 5569 5657 574 4 5832 5919 6007 6( )94 6182 6269 7 6356 6444 6531 6618 6706 6793 6880 6968 7055 7142 8 7229 7317 7404 74[ 1 7578 7665 7752 7* 89 7926 8014 9 8100 8188 8275 8362 8449 8535 8622 8709 8796 8883 87 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 98 9.8 19.6 29.4 39.2 49.0 58.8 68.6 78.4 88.2 9? 9.7 19.4 29.1 38.8 48.5 58.2 67.9 77.6 87.8 96 9.6 19.2 28.8 38.4 48.0 57.6 67.2 76.8 86.4 95 9.5 19.0 28.5 38.0 47.5 57.0 66.5 76.0 85.5 94 9.4 18.8 28.2 37.6 47.0 56.4 65.8 75.2 84.6 93 9.3 18.6 27.9 37.2 46.5 55.8 65.1 74.4 83.7 92 9.2 18.4 27.6 36.8 46.0 55.2 64.4 73.6 82.8 91 9.1 18.2 27.3 36.4 45.5 54.6 63.7 72.8 81.9 90 9.0 18.0 27.0 36.0 45.0 54.0 63.0 72.0 81.0 89 8.9 17.8 26.7 35.6 44.5 53.4 62.3 71.2 80.1 88 8.8 17.6 26.4 35.2 44.0 52.8 61.6 70.4 79.2 87 8.7 17.4 26.1 34.8 43.5 52.2 60.9 69.6 78.3 86 8.6 17.2 25.8 34.4 43.0 51.6 60.2 68.8 77.4 LOGARITHMS OP NUMBERS. 145 [ No. 500 L. 698.] [No. 544 L. 736. N. 1 2 3 4 6 6 7 8 9 Diff. 500 698970 9057 9144 9231 9317 9404 9491 9578 9664 9751 1 9838 9924 0011 0098 0184 0271 0358 0444 0531 0617 2 700704 0790 0877 0963 1050 1136 1222 1309 1395 1482 3 1568 1654 1741 1827 1913 1999 2086 2172 2258 2344 4 2431 2517 2603 2689 2775 2861 2947 3033 3119 3205 5 3291 3377 3463 3549 3635 3721 3807 3893 3979 4065 86 6 4151 4236 4322 4408 4494 4579 4665 4751 4837 4922 7 5008 5094 5179 5265 5350 5436 5522 5607 5693 5778 8 5864 5949 6035 6120 62G6 6291 6376 6462 6547 6632 9 6718 6803 6888 6974 7059 7144 7229 7315 7400 7485 510 7570 7655 7740 7826 7911 7996 8081 8166 8251 8336 1 8421 8506 8591 8676 8761 8846 8931 9015 9100 9185 85 2 9270 9355 9440 9524 9609 9694 9779 9863 9948 0033 3 710117 0202 0287 0371 0456 0540 0625 0710 0794 0879 4 0963 1048 1132 1217 1301 1385 1470 1554 1639 1723 5 1807 1892 1976 2060 2144 2229 2313 2397 2481 2566 6 2650 2734 2818 2902 2986 3070 3154 3238 3323 3407 ft4 7 3491 3575 3659 3742 3826 3910 3994 4078 4162 4246 04 8 4330 4414 4497 4581 4665 4749 4833 4916 5000 5084 9 5167 5251 5335 5418 5502 5586 5669 5753 5836 5920 520 6003 6087 6170 6254 6337 6421 6504 6588 6671 6754 1 6838 6921 7004 7088 7171 7254 7338 7421 7504 7587 2 7671 7754 7837 7920 8003 8086 8169 8253 8336 8419 3 8502 8585 8668 8751 8834 8917 9000 9083 9165 9248 83 4 9331 9414 9497 9580 9663 9745 9828 9911 9994 0077 5 720159 0242 0325 0407 0490 0573 0655 0738 0821 0903 6 0986 1068 1151 1233 1316 1398 1481 1563 1646 1728 7 1811 1893 1975 2058 2140 2222 2305 2387 2469 2552 8 2634 2716 2798 2881 2963 3045 3127 3209 3291 3374 9 3456 3538 3620 3702 3784 3866 3948 4030 4112 4194 82 530 4276 4358 4440 4522 4604 4685 4767 4849 4931 5013 1 5095 5176 5258 5340 5422 5503 5585 5667 5748 5830 2 5912 '5993. 6075 6156 6238 6320 6401 6483 6564 6646 3 6727 6809 6890 697'2 7053 7134 7216 7297 7379 7460 4 7541 7623 7704 7785 7'866 7948 8029 8110 8191 8273 5 8354 8435 8516 8597 8678 8759 8841 8922 9003 9084 6 9165 9246 9327 9408 9489 9570 9651 9732 9813 9893 81 7 9974 0055 0136 0217 0298 0378 0459 0540 0621 0702 8 730782 0863 0944 1024 1105 1186 1266 1347 1428 1508 9 1589 1669 1750 1830 1911 1991 2072 2152 2233 2313 540 2394 2474 2555 2635 2715 2796 2876 2956 3037 3117 1 8197 3278 3358 3438 3518 3598 3679 3759 3839 3919 2 3999 4079 4160 4240 4320 4400 4480 4560 4640 4720 on 3 4800 4880 4960 5040 5120 5200 5279 5359 5439 5519 w 4 5599 5679 5759 5838 5918 5998 6078 6157 6237 6317 PROPORTIONAL PARTS. Diff. 1 234 5 678 9 87 8.7 17.4 26.1 34.8 43.5 52.2 60.9 69.6 78.3 86 8.6 17.2 25.8 34.4 43.0 51.6 60.2 68.8 77.4 85 8.5 17.0 25.5 34.0 42.5 51.0 59.5 68.0 76.5 84 8.4 16.8 25.2 33.6 42.0 50.4 58.8 67.2 75.6 LOGARITHMS OP CUMBERS. No. 545 L. 736.] [No. 584 L. 767. N. 1 2 8 4 5 6 7 8 9 Diff. 545 736397 6476 6556 6635 6715 1 6795 6874 6954 7034 7113 6 7193 7272 7352 7431 7511 7590 7670 7749 7829 7908 7 7987 8067 8146 8S& 25 8305 8384 8463 8543 862 2 8701 8 8781 8860 8939 9018 9097 9177 9256 9335 9414 9493 9 9572 9651 9731 98 10 9889 9968 0047 O19A H90 C\9RA 550 740363 0442 0521 0600 0678 0757 0836 U1/*D 0915 \J4\JO 0994 \)4& 1073 29 1 1152 1230 1309 13* S8 1467 1546 1624 1703 178 2 1860 2 1939 2018 2096 2175 2254 2332 2411 2489 2568 2647 3 2725 2804 2882 29t 51 3039 ! 3118 3196 3275 335 3 3431 4 3510 3588 3667 3745 3823 3902 3980 4058 4136 4215 5 4293 4371 4449 4528 4606 4684 4762 4840 4919 4997 6 5075 5153 5231 53( )9 5387 ; 5465 5543 5621 569 ) 5777 78 7 5855 5933 6011 601 S9 6167 6245 6323 6401 647 ) 6556 8 6634 6712 6790 68( 58 6945 7023 7101 7179 7256 7334 9 7412 7489 7567 7645 7722 7800 7878 7955 8oa J 8110 560 8188 8266 8343 8421 8498 1 8576 8653 8731 880 * 8885 1 8963 9040 9118 9195 9272 | 9350 9427 9504 9582 9659 2 9736 9814 9891 99( 58 0045 0123 0200 O977 AOR 4 fMOl 3 750508 0586 0663 0740 0817 0894 0971 \)/ii t 1048 UoO^ 1125 U4ol 1202 4 1279 1356 1433 1510 1587 1664 1741 1818 1895 1972 5 2048 2125 2202 2279 2356 2433 2509 2586 2663 2740 77 6 2816 2893 2970 3fr 17 3123 3200 3277 3a r >3 343 ) 3506 7 3583 3660 3736 3813 3889 3966 4042 4119 4195 4272 8 4348 4425 4501 45' '8 4654 4730 4807 4883 496 3 5036 9 5112 5189 5265 5341 5417 i 5494 5570 5646 5722 5799 5TO 5875 5951 6027 6103 6180 1 6256 6332 6408 6484 6560 1 6636 6712 6788 6864 6940 7016 7092 7168 7244 7320 76 2 7396 7472 7548 7ft 24 7700 i 7775 7851 7927 800 3 8079 3 8155 8230 8306 8382 8458 > 8533 8609 8685 87'61 8836 4 8912 8988 9063 91, 39 9214 i 9290 9366 9441 951 I 9592 5 9668 9743 9819 DM )4 997'0 0045 0121 0196 097 -> fiQ/17 6 760422 0498 0573 0649 0724 0799 0875 0950 \}4ii & 1025 Uo4< 1101 7 1176 1251 1326 1402 1477 1552 1627 1702 1778 1853 8 1928 2003 2078 21, 33 2228 i 2303 2378 2453 252 1 2604 9 2679 2754 2829 2904 29?'8 3053 3128 3203 3278 3353 73 580 3428 3503 3578 3653 3727 3802 3877 3952 4027 4101 1 4176 4251 4326 44( )0 4475 4550 4624 4699 477 i 4848 2 4923 4998 5072 5147 5221 5296 5370 5445 5520 5594 3 5669 5743 5818 58< K 5966 6041 6115 6190 62fr 1 6338 4 6413 6487 6562 6636 6710 6785 6859 6933 7007 7082 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 66.4 9 74.7 83 8.3 16.6 24.9 33.2 41.5 49.8 58.1 82 8.2 16.4 24.6 32.8 41.0 49.2 57.4 65.6 73.8 81 8.1 16.2 24.3 32.4 40.5 48.6 56.7 64.8 72.9 80 8.0 16.0 24.0 32.0 40.0 48.0 56.0 64.0 72.0 79 7.9 15.8 23.7 31.6 39.5 47.4 55.3 63.2 71.1 78 78 15.6 23.4 31.2 39.0 46.8 54.6 62.4 70.2 77 7.7 15.4 23.1 30.8 38.5 46.2 53.9 61.6 69.3 76 7.6 15.2 22.8 30.4 38.0 45.6 53.2 60.8 68.4 75 7.5 15.0 22.5 30.0 37.5 45.0 52.5 60.0 67.5 74 7.4 14.8 22.2 29.6 37.0 44.4 51.8 59.2 66.6 LOGARITHMS OF NUMBERS. 147 No. 585 L. 767.1 [No. 629 L. 799. N. 1 2 3 4 5 6 7 8 9 Diff. 585 767156 7230 7304 7379 7453 7527 7601 7675 7749 7823 6 7898 7972 8046 8120 8194 8268 8342 8416 8490 8564 74 7 8638 8712 8786 88t 8934 9008 9082 9 156 9230 9303 g 9377 9451 9525 95 9 9673 9746 9820 9 894 9968 0042 9 770115 0189 0263 0336 0410 0484 0557 0631 0705 0778 590 0852 0926 0999 1073 1146 1220 1293 1367 1440 1514 1 1587 1661 1734 180 8 1881 1955 2028 2 102 2175 2248 2 2322 2395 2468 2542 2615 2688 2762 2835 2908 2981 3 3055 3128 3201 327 4 3348 3421 3494 a 567 3640 3713 4 3786 3860 3933 4006 4079 4152 4225 4298 4371 4444 73 5 4517 4590 4663 473 6 4809 4882 4955 & 108 5100 5173 6 5246 5319 5392 5465 5538 5610 5683 5756 5829 5902 7 5974 6047 6120 619 3 6265 6338 6411 6- 188 6556 6629 8 6701 6774 6846 6919 6992 7064 7137 7209 7282 7354 9 7427 7499 7572 7644 7717 7789 7862 7934 8006 8079 600 8151 8224 8296 8368 8441 8513 8585 8 .58 8730 8802 1 8874 8947 9019 909 1 9163 9236 9308 9 M) 9452 9524 2 9596 9669 9741 981 3 9885 9957 0029 O mi 0173 0245 3 780317 0389 0461 0533 0605 0677 0749 0821 0893 0965 72 4 1037 1109 1181 1253 1324 1396 1468 1540 1612 1684 5 1755 1827 1899 197 1 2042 2114 2186 2 1 358 2329 2401 6 2473 2544 2616 268 8 2759 2831 2902 2 J?4 3046 3117 7 3189 3260 3332 3403 3475 3546 3618 8 >s<) 3761 3832 8 3904 3975 4046 4118 4189 4261 4332 4403 4475 4546 9 4617 4689 4760 483 1 4902 4974 5045 5116 5187 5259 610 5330 5401 5472 5543 5615 5686 5757 5 328 5899 5970 1 6041 6112 6183 625 4 6325 6396 6467 6 588 6609 6680 71 2 { 6751 6822 6893 6964 7035 7106 7177 7248 7319 7390 3 7460 7531 7602 767 a 7744 7815 7885 7 )56 8027 8098 4 8168 8239 8310 838 i 8451 8522 8593 8 >r>3 8734 8804 5 8875 8946 9016 908 7 9157 9228 9299 9 369 9440 9510 9581 9651 9722 979 9863 9933 0004 Ot "74 0144 0215 7 790285 0356 0426 0496 0567 0637 0707 0778 0848 0918 8 0988 1059 1129 119 9 1269 1340 1410 1 ISO 1550 1620 9 1691 1761 1831 1901 1971 2041 2111 2181 2252 2322 620 2392 2462 2532 2602 2672 2742 2812 2! m 2952 3022 70 1 3092 3162 3231 330 1 3371 3441 3511 a 581 3651 3721 2 3790 3860 3930 400 4070 4139 4209 4 279 4349 4418 3 4488 4558 4627 4697 4767 4836 4906 4976 5045 5115 4 5185 5254 5324 539 3 5463 5532 5602 5 >'~ 5741 5811 5 5880 5949 6019 6088 6158 6227 6297 6366 6436 6505 6 6574 6644 6713 678 8 6852 6921 6990 7( MJO 7129 7198 7 7268 7337 7406 7475 7545 7614 7683 7752 7821 7890 8 7960 8029 8008 816 7 8236 8305 8374 & i43 8513 8582 9 8651 8720 8789 8858 8927 8996 9065 9134 9203 927.8 69 PROPORTIONAL PARTS. DiflP. 1 2 3 4 5 6 7 8 9 75 7.5 15.0 22.5 30.0 37.5 45.0 52.5 60.0 67.5 74 7.4 14.8 22.2 29.6. 37.0 44.4 51.8 59.2 66.6 73 7.3 14.6 21.9 29.2 36.5 43.8 51.1 58.4 65.7 72 7.2 14.4 21.6 28.8 36.0 43.2 50.4 57.6 64.8 71 7.1 14.2 21.3 28.4 35.5 42.6 49.7 56.8 63.9 70 7.0 14.0 21.0 28.0 35.0 42.0 49.0 56.0 63.0 69 6.9 13.8 20.7 27.6 34.5 41.4 48.3 55.2 62.1 148 LOGARITHMS OP KUMBERS. No, 630 L. 799.] [No. 674 L. 829. N. 1 2 3 4 5 6 7 8 9 Diff. 630 799341 9409 9478 9547 9616 9685 9754 9823 9892 9961 1 800029 0098 0167 0236 0305 0373 0442 0511 0580 0640 2 0717 0786 0854 0923 0992 1061 1129 1198 1266 1335 3 1404 1472 1541 1609 1678 1747 1815 1884 1952 2021 4 2089 2158 2226 2295 2363 2432 2500 2568 2637 2705 5 2774 2842 2910 2979 3047 3116 3184 3252 3321 3389 6 3457 3525 3594 3662 3730 3798 3867 3935 4003 4071 7 4139 4208 4276 4344 4412 4480 4548 4616 4685 4753 8 4821 4889 4957 5025 5093 5161 5229 5297 5365 5433 68 9 5501 5569 5637 5705 5773 5841 5908 5976 6044 6112 640 806180 6248 6316 63&4 6451 6519 6587 6655 6723 6790 1 6858 6926 6994 7061 7129 7197 7264 7332 7400 7467 2 7535 7603 7670 7738 7806 7873 7941 8008 8076 8143 3 8211 8279 8346 8414 8481 8549 8616 8684 8751 8818 4 8886 8953 9021 9088 9156 9223 9290 9358 9425 9492 5 9560 9627 9694 9762 9829 9896 9964 0031 0098 0165 6 810233 0300 0367 0434 0501 0569 0636 0703 0770 0837 7 0904 0971 1039 1106 1173 1240 1307 1374 1441 1508 67 8 1575 1642 1709 1776 1843 1910 1977 2044 2111 2178 9 2245 2812 2379 2445 2512 2579 2646 2713 2780 2847 650 2913 2980 3047 3114 3181 3247 3314 3381 3448 3514 1 3581 3648 3714 3781 3848 3914 3981 4048 4114 4181 2 4248 431-4 4381 4447 45*4 4581 4647 4714 4780 4847 3 4913 4980 5046 5113 5179 5246 5312 5378 5445 5511 4 5578 5644 5711 5777 5843 5910 5976 6042 6109 6175 I 6241 6308 6374 6440 6506 6573 6639 6705 6771 6838 6 6904 6970 7036 7102 7169 7235 T301 7367 7433 7499 7 75G5 7631 7698 7764 7830 7896 7962 8028 8094 8160 8 8226 8292 8358 8424 8490 8556 8622 8688 8754 8820 9 8885 8951 9017 9083 9149 9215 9281 9346 9412 9478 Co 660 9544 9610 9676 9741 9807 9873 9969 0004 0070 0136 1 820201 0267 0333 0399 0464 0530 0595 0661 0727 0792 2 0858 0924 0989 1055 1120 1186 1251 1317 1382 1448 3 1514 1579 1645 1710 1775 1841 1906 1972 2037 2103 4 2168 2233 2299 2364 2430 2495 2560 2626 2691 2756 5 2822 2887 2952 3018 3083 3148 3213 3279 3344 3409 6 3474 3539 3605 3670 37'35 3800 3865 3930 3996 4061 7 4126 4191 4256 4321 4386 4451 4516 4581 4646 4711 fiK 8 4776 4841 4906 4971 5036 5101 5166 5231 5296 5361 DO 9 5426 5491 5556 5621 5686 5751 5815 5880 5945 6010 670 6075 6140 6204 6269 6334 6399 6464 6528 6593 6658 1 6723 6787 6852 6917 6981 7046 7111 7175 7240 7305 2 7369 7434 7499 7563 7628 7692 7757 7821 7886 7951 3 8015 8080 8144 8209 8273 8338 8402 8467 8531 8595 4 8660 8724 8789 8853 8918 8982 9046 9111 9175 9239 PROPORTIONAL PARTS. Diff 1 2 3 4 5 678 9 68 6.8 13.6 20.4 27.2 34.0 40.8 47.6 544 61.2 67 6.7 13.4 20.1 26.8 33.5 40.2 46.9 53,6 60.3 66 6.6 13.2 19.8 26.4 33.0 39.6 46.2 52.8 59.4 65 6.5 13.0 19.5 26.0 32.5 39.0 45.5 52.0 58.5 64 6.4 1.8 19.2 25. Q 32.0 38,4 44.8 51.2 57.6 LOGARITHMS OF NUMBERS. U9 No. 675 L. 829.? [No. 719 L. 857. N. 1 2 8 4 6 6 7 8 9 Diff. 675 829304 9368 9432 9497 9561 9625 9690 9754 9818 9882 g 9947 0011 0075 0139 0204 0268 0332 0396 0460 0525 7 830589 0653 0717 0781 0845 0909 0973 1037 1102 1166 8 1230 1294 1358 1422 1486 1550 1614 1678 1742 1806 64 9 1870 1984 1998 2062 2126 2189 2253 2317 2381 2445 680 2509 2573 2637 2700 2764 2828 2892 2956 3020 3083 1 3147 3211 3275 3338 3402 3466 3530 3593 3657 3721 2 3784 3848 3912 3975 4039 4103 4166 4230 4294 4357 3 4421 4484 4548 4611 4675 4739 4802 4866 4929 4993 4 5056 5120 5183 5247 5310 5373 5437 5500 5564 5627 5 5691 5754 5817 5881 5944 6007 6071 6134 6197 6261 6 6324 6387 6451 6514 6577 6641 6704 6767 6830 6894 7 6957 7020 7083 7146 7210 7273 7336 7399 7462 7525 8 7588 7652 7715 7778 7841 7904 7967 8030 8093 8156 9 8219 8282 8345 8408 8471 8534 8597 8660 8723 8786 63 690 8849 8912 8975 9038 9t01 9164 9227 9289 9&52 9415 1" 9478 9541 9604 9667 97'29 9792 9855 9918 9981 0043 2 840106 0169 0232 0294 0357 0420 0482 0545 0608 0671 3 0733 0796 0859 0921 0984 1046 1109 1172 1234 1297 4 1359 1422 1485 1547 1610 1672 1735 1797 1860 1922 '5 1985 2047 2110 2172 2235 2297 2360 2422 2484 2547 6 2609 2672 2734 2796 2859 2921 2983 3046 3108 3170 7 3233 3295 3357 3420 3482 3544 3606 3669 3731 3793 8 3855 3918 3980 4042 4104 4166 4229 4291 4353 4415 9 447.7 4539 4601 4664 4726 4788 4850 4912 4974 5036 700 5098 5160 5222 5284 5346 5408 5470 5532 5594 5656 62 1 5718 5780 5842 5904 5966 6028 6090 6151 6213 6275 2 6337 6399 6461 6523 6585 6646 6708 6770 6832 6894 3 6955 7017 7079 7141 7202 7264 7326 7388 7449 7511 4 7573 7634 7696 7758 7819 7881 7943 8004 8066 8128 5 8189 8251 8312 8374 8435 8497 8559 8620 8682 8743 6 8805 8866 8928 8989 9051 9112 9174 9235 9297 9358 7 9419 9481 95^ 9604 9665 9726 9788 9849 9911 9972 8 850033 0095 0156 0217 0279 0340 0401 0462 0524 0585 9 0646 0707 0769 0830 .0891 0952 1014 1075 1136 1197 710 1258 1320 1381 1442 1503 1564 1625 1686 1747 1809 1 1870 1931 1992 2053 2114 2175 2236 2297 2358 2419 2 2480 2541 2602 2663 2724 2785 2846 2907 2968 3029 61 3 3090 3150 3211 3272 3333 3394 3455 3516 3577 3637 4 3698 3759 3820 3881 3941 4002 4063 4124 4185 4245 5 4306 4367 4428 4488 4549 4610 4670 4731 4792 4852 6 4913 4974 5034 5095 5156 5216 5277 5337 5398 5459 7 5519 5580 5640 5701 5761 5822 5882 5943 6003 6064 8 6124 6185 6245 6306 6366 6427 6487 6548 6608 6668 9 6729 6789 6850 6910 6970 7031 7091 7152 7212 7272 PROPORTIONAL PARTS. Diff . 1 234 5 678 9 65 6.5 13.0 19.5 26.0 32.5 39.0 45.5 52.0 58.5 64 6.4 12.8 19.2 25.6 32.0 38.4 44.8 51.2 57.6 63 6.3 12.6 18.9 25.2 31.5 37.8 44.1 50.4 56.7 62 6.2 12.4 18.6 24.8 31.0 37.2 43.4 49.6 55 8 61 6.1 12.2 18.3 24.4 30.5 36.6 42.7 48,8 54.9 60 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 150 LOGARITHMS OF K UMBERS. No. 720 L. 857.] [No. 764 L. 883. . 9 Diff. 720 857332 7393 7453 7513 7574 7634 7694 7755 7815 7875 1 7935 7995 8056 8116 8176 8236 8297 8357 8417 8477 2 8537 8597 8657 8718 8778 8838 8808 8958 9018 9078 3 9138 9198 9258 9318 9879 9439 9499 9559 9619 9679 60 4 9739 9799 9859 9918 9978 0038 0098 0158 0218 0278 5 860338 0398 0458 0518 0578 0637 0697 0757 0817 0877 6 0937 0996 1056 1116 1176 1236 1295 1355 1415 1475 7 1534 1594 1654 1714 1773 1833 1893 1952 2012 2072 8 2131 2191 2251 2310 2370 2430 2489 2549' 2608 2668 9 2728 2787 2847 2906 2966 3025 3085 31 J4 3204 3263 730 3323 3382 3442 3501 3561 3620 3680 3739 3799 3858 1 3917 3977 4036 4096 4155 4314 4274 4333 4392 4452 2 4511 4570 4630 4689 4748 4808 4867 4926 4985 5045 3 v 5104 5163 5222 5282 5341 5400 5459 5519 5578 5637 4 5696 5755 5814 5874 5933 5992 6051 6110 6169 6228 5 6287 6346 6405 6465 6524 6583 G642 6701 6760 6819 6 6878 6937 6996 7055 7114 7173 7232 7291 7350 7409 59 7 7467 7526 7585 7644 7703 7762 7821 7880 7939 7998 8 8056 8115 8174 8233 8292 8350 8409 8468 8527 8586 9 8644 8703 8762 8821 8879 8938 8997 9056 9114 9173 740 9232 9290 9349 9408 9466 9525 9584 9642 9701 9760 1 9818 9877 9935 9994 0053 0111 0170 0228 O987 O'3AK 2 870404 0462 0521 0579 0638 0696 Ul t\J 0755 0813 Uo< 0872 Uo40 0930 3 0989 1047 1106 1164 1223 1281 1339 1398 1456 1515 4 1573 1631 1690 1748 1806 1865 1923 1981 2040 2008 5 2156 2215 2273 2331 2389 2448 2506 2564 2622 2681 6 2739 2797 2855 2913 2972 3030 3088 8146 3204 3262 7 3321 3379 3437 3495 3553 3611 3669 3727 3785 3844 8 3902 3960 4018 4076 4134 4192 4250 4308 4366 4424 58 9 4482 4540 4598 4656 4714 4772 4830 4888 4945 5003 750 5061 5119 5177 5235 5293 5351 5409 5466 5524 5582 1 5640 5698 5756 5813 5871 5929 5987 6045 6102 6160 2 6218 6276 6333 6391 6449 6507 6564 6622 6680 6737 3 6795 6853 6910 6968 7026 7083 7141 7199 7256 7314 4 7371 7429 7487 7544 7602 7659 7717 7774 7832 7889 5 7947 8004 8062 8119 8177 8234 8292 8349 8407 8464 6 8522 8579 8637 8694 8752 8809 8866 8924 8981 9039 7 9096 9153 9211 9268 9325 9383 9440 9497 9555 9612 g 9669 9726 9784 9841 9898 9956 0013 0070 0127 0185 9 880242 0299 0356 0413 0471 0528 0585 0642 0699 0756 760 0814 0871 0928 0985 1042 1099 1156 1213 1271 1328 1 1385 1442 1499 1556 1613 1670 1727 1784 1841 1898 2 1955 2012 2069 2126 2183 2240 2297 2354 2411 2468 57 3 2525 2581 2638 2695 2752 2809 2866 2923 2980 3037 4 3093 3150 3207 3264 3321 3377 3434 3491 3548 3605 PROPORTIONAL PARTS. Diff 1 2 3 4 5 678 9 59 5.9 11.8 17.7 23.6 29.5 35.4 41.3 47.2 53.1 58 5.8 11.6 17.4 23.2 29.0 S4.8 40.6 46.4 52.2 57 5.7 11.4 17.1 22.8 28.5 34.2 39.9 45.6 51.3 56 5.6 11.2 16.8 22.4 28.0 33.6 39.2 44.8 50.4 LOGARITHMS OF NUMBERS. 151 No. 765 L. 883.] [No. 809 L. 908. N. 1 2 3 4 5 6 7 8 9 Diff. 765 883661 3718 3775 3832 3888 3945 4002 4059 4115 4172 6 4229 4285 4342 4399 4455 4512 4569 4625 4682 4739 7 4795 4852 4909 4965 5022 5078 5135 5192 5248 5305 8 5361 5418 5474 5531 5587 5644 5700 5757 5813 5870 9 5926 5983 6039 6096 6152 6209 6265 6321 6378 6434 770 6491 6547 6604 6660 6716 6773 6829 6885 6942 6998 1 7054 7111 7167 7223 7280 7336 7392 7449 7505 7561 2 7617 7674 7730 7786 7842 7898 7955 8011 8067 8123 3 8179 8236 8292 8348 8404 8460 8516 8573 8629 8685 4 8741 8797 8853 8909 8965 9021 9077 9134 9190 9246 5 9302 9358 9414 9470 9526 9582 9638 9694 9750 9806 56 g 9862 9918 9974 0030 0086 0141 0197 0253 0309 0365 7 890421 0477 0533 0589 0645 0700 0756 0812 0868 0924 8 0980 1035 1091 1147 1203 1259 1314 1370 1426 1482 9 1537 1593 1649 1705 1760 1816 1872 1928 1983 2039 780 2095 2150 2206 2262 2317 2373 2429 2484 2540 2595 1 2651 2707 2762 2818 2873 2929 2985 3040 3096 3151 2 3207 3262 3318 3373 3429 3484 3540 3595 3651 3706 3 3762 3817 3873 3928 3984 4039 4094 4150 4205 4261 4 4316 4371 4427 4482 4538 4593 4648 4704 4759 4814 5 4870 4925 4980 5036 5091 5146 5201 5257 5312 5367 6 5423 5478 z:m 5588 5644 5699 5754 5809 5864 5920 7 5975 6030 6085 6140 6195 6251 6306 6361 6416 6471 8 6526 6581 6636 6692 6747 6802 6857 6912 6967 7022 9 7077 7132 7187 7242 7297 7352 7407 7462 7517 7572 790 7627 7682 7737 7792 7847 7902 7957 8012 8067 8122 55 1 8176 8231 8286 8341 8396 8451 8506 8561 8615 8670 2 8725 8780 sass 8890 8944 8999 9054 9109 9164 9218 3 4 9273 9821 9328 9875 9383 9G30 9437 9985 9492 9547 9G02 9656 9711 9766 0039 0094 0149 0203 0258 0312 5 900367 0422 0476 0531 0586 0640 0695 0749 0804 0859 6 0913 09G8 1022 1077 1131 1186 1240 1295 1349 1404 7 1458 1513 1567 1622 1676 1731 1785 1840 1894 1948 8 2003 2057 2112 2166 2221 2275 2329 2384 2438 2492 9 2547 2601 2655 2710 2764 2818 2873 2927 2981 3036 800 3090 3144 3199 3253 3307 3361 8416 3470 35^ 3578 1 3633 3687 3741 3795 3849 3904 3958 4012 4066 4120 2 4174 4229 4283 4337 4391 4445 4499 4553 4607 4661 3 4716 4770 4824 4878 4932 4986 5040 5094 5148 5202 54 4 5256 5310 5364 5418 5472 5526 5580 5634 5688 5742 5 5796 5850 5904 5958 6012 6066 6119 6173 6227 6281 6 6335 6389 6443 6497 6551 6604 6658 6712 6766 6820 7 6874 6927 6981 7035 7089 7143 7196 7250 7304 7358 8 7411 7465 7519 7573 7626 7680 7734 7787 7841 7895 9 7949 8002 8056 8110 8163 8217 8270 8324 8378 8431 PROPORTIONAL PARTS. Diff. 1 234 5 6 7 8 9 57 5.7 11.4 17.1 22.8 28.5 34.2 39.9 45.6 51.3 56 5.6 11.2 16.8 22.4 28.0 33.6 39.2 44.8 50.4 55 5.5 11.0 16.5 22.0 27.5 33.0 38.5 44.0 49.5 54 5.4 10.8 16.2 21.6 27.0 32.4 37.8 43.2 48.6 LOGARITHMS OF HUMBERS. No. 8K) L. 908.] [No. 854 L. 931. N. 1 2 3 4 6 6 7 8 9 Diff. 810 908485 8539 8592 8646 8699 8753 8807 8860 8914 8967 1 9021 9074 9128 9181 9235 9289 9342 9396 9449 9503 2 9556 9610 9663 9716 9770 9823 9877 9930 9984 0037 3 910091 0144 0197 0251 0304 0358 0411 0464 0518 0571 4 0624 0678 0731 0784 0838 0891 0944 0998 1051 1104 5 1158 1211 1264 1317 1371 1424 1477 1530 1584 1637 6 1690 1743 1797 1850 1903 1956 2009 2063 2116 2169 7 2222 2275 2328 2381 2435 2488 2541 2594 2647 2700 8 2753 2806 2859 2913 2966 3019 3072 3125 3178 3231 9 3284 3337 3390 3443 3496 3549 3602 3655 3708 3761 53 820 3814 3867 3920 3973 4026 4079 4132 4184 4237 4290 1 4343 4396 4449 4502 4555 4608 4660 4713 4766 4819 2 4872 4925 4977 5030 5083 5136 5189 5241 5294 5347 3 5400 5453 5505 5558 5611 5664 5716 5769 5822 5875 4 5927 5980 6033 6085 6138 6191 6243 6296 6349 6401 5 6454 6507 6559 6012 6664 6717 6770 6822 6875 6927 6 6980 7033 7085 7138 7190 7243 7295 7348 7400 7453 7 7506 7558 7611 7063 7716 7708 7820 7873 7925 7978 8 8030 8083 8135 8188 8240 8293 8345 8397 8450 8502 9 8555 8607 8659 8712 8764 8816 8869 8921 8973 9026 830 9078 9130 9183 9235 9287 9340 9392 9444 9496 9549 1 9601 9653 9706 9758 9810 9802 9914 9967 n/vm 0071 2 920123 0176 0228 0280 0,332 0384 0436 0489 0541 0598 3 0645 0697 0749 0801 0853 0906 0958 1010 1062 1114 4 1166 1218 1270 1322 1374 1426 1478 1530 1582 1634 52 5 1686 1738 1790 1842 1894 1946 1998 2050 2102 2154 6 2206 2258 2310 2362 2414 2466 2518 2570 2622 2674 7 2725 2777 2829 2881 2933 2985 3037 3089 3140 3192 8 3244 3296 3348 3399 3451 3503 3555 3607 3658 3710 9 3762 3814 3865 3917 3969 4021 4072 4124 4176 4228 840 4279 4331 4383 4434 4486 4538 4589 4641 4693 4744 1 4796 4848 4899 4951 5003 5054 5106 5157 5209 5261 2 5312 5364 6415 5467 5518 5570 5621 5673 5725 5776 3 5828 5879 6931 5982 6034 6085 6137 6188 6240 6291 4 6342 6394 6445 6497 6548 6600 6651 6702 6754 6805 5 6857 6908 6959 7011 7062 7114 7165 7216 7268 7319 6 7370 7422 7473 7524 7576 7627 7678 7730 7781 7832 7 7883 7935 7986 8037 8088 8140 8191 8242 8293 8345 8 8396 8447 8498 8549 8601 8652 8703 8754 8805 8857 9 8908 8959 9010 9061 9112 9163 9215 9266 9317 9368 850 9419 9470 9521 9572 9623 9674 9725 9776 9827 9879 1 9930 9981 51 0032 0083 0134 0185 0236 0287 0338 0389 2 930440 0491 0542 0592 0643 0694 0745 0796 0847 0898 3 0949 1000 1051 1102 1153 1204 1254 1305 1356 1407 4 1458 1509 1560 1610 1661 1712 1763 1814 1865 1915 PROPORTIONAL PARTS. Diff. 1 234 5 678 9 53 5.3 10.6 15.9 21.2 26.5 31.8 37.1 42.4 47.7 52 5.2 10.4 15.6 20.8 26.0 31.2 36.4 41.6 46.8 51 5.1 10.2 15.3 20.4 25.5 30.6 35.7 40.8 45.9 50 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 LOGARITHMS OF NUMBERS. 153 No. 855 L. 931.] [No. 899 L. 954. N. 1 2 3 4 6 6 7 8 9 Diff. 855 931966 2017 2068 2118 2169 2220 2271 2322 2372 2423 6 2474 2524 2575 2626 2677 2727 2778 2829 2879 2930 7 2981 3031 3082 3133 3183 3234 3285 3335 3386 3437 8 3487 3538 3589 3639 3690 3740 3791 3841 3892 3943 9 3993 4044 4094 4145 4195 4246 4296 4347 4397 4448 860 4498 4549 4599 4650 4700 4751 4801 4852 4902 4953 1 5003 5054 5104 5154 5205 5255 5306 5356 5406 5457 2 5507 5558 5608 5658 5709 5759 5809 5860 5910 5960 3 6011 6061 6111 6162 6212 6262 6313 6363 6413 6463 4 6514 6564 6614 6665 6715 6765 6815 6865 6916 6966 5 7016 7066 7116 7167 7217 7267 7317 7367 7418 74i8 6 7518 7568 7618 7668 7718 7769 7819 7869 7919 7969 7 8019 8069 8119 8169 8219 8269 8320 8370 8420 8470 50 8 8520 8570 8620 8670 8720 8770 8820 8870 8920 8970 9 9020 9070 9120 9170 9220 9270 9320 9369 9419 9469 870 9519 9569 9619 9669 9719 9769 9819 9869 9918 9968 1 940018 0068 0118 0168 0218 0267 0317 0367 0417 0467 2 0510 0566 0616 0666 0716 0765 0815 0865 0915 0964 3 1014 1064 1114 1163 1213 1263 1313 1362 1412 1462 4 1511 1561 1611 1GGO 1710 1760 1809 1859 1909 1958 5 2008 2058 2107 2157 2207 2256 2306 2355 2405 2455 6 2504 2554 2603 2653 2702 2752 2801 2851 2901 2950 7 3000 3049 3099 3148 3198 3247 3297 3346 3396 3445 8 3495 3544 3593 3643 3692 3742 3791 3841 3890 3939 9 3989 4038 4088 4137 4186 4236 4285 4335 4384 4433 880 4483 4532 4581 4631 4680 4729 4779 4828 4877 4927 1 4976 5025 5074 5124 5173 5222 5272 5321 5370 5419 2 5469 5518 5567 5616 5665 5715 5764 5813 5862 5912 3 5961 6010 6059 6108 6157 6207 6256 6305 6354 6403 4 6452 6501 6551 6600 6649 6698 6747 6796 6845 6894 5 6943 6992 7041 7090 7139 7189 7238 7287 7336 7385 6 7434 7483 7532 7581 7630 7679 7728 7777 7826 7875 49 7 7924 7973 8022 8070 8119 8168 8217 8266 8315 8364 8 8413 8462 8511 8560 8608 8657 8706 8755 8804 8853 9 8902 8951 8999 9048 9097 9146 9195 9244 9292 9341 890 9390 9439 9488 9536 9585 9634 9683 9731 9780 9829 1 9878 9926 9975 0024 0073 0191 0170 0219 O9fl7 nq-jft 2 950365 0414 0462 0511 0560 VIXl 0608 0657 0706 U/SJlM 6754 UolO 0803 3 0851 0900 0949 0997 1046 1095 1143 1192 1240 1289 4 1338 1386 1435 1483 1532 1580 1629 1677 1726 1775 5 1823 1872 1920 1969 2017 2066 2114 2163 2211 2260 G 2308 2356 2405 2453 2502 2550 2599 2647 2696 2744 7 2792 2841 2889 2938 2986 3034 3083 3131 3180 3228 8 3276 3325 3373 3421 3470 3518 3566 3615 3663 3711 9 3760 3808 3856 3905 3953 4001 4049 4098 4146 4194 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 51 50 49 48 5.1 5.0 4.9 4.8 10.2 10.0 9.8 9.6 15.3 20.4 15.0 20.0 14.7 19.6 14.4 19.2 25.5 25.0 24.5 24.0 30.6 30.0 29.4 28.8 35.7 40.8 35.0 40.0 34.3 39.2 33.6 38.4 45.9 45.0 44.1 43.2 LOGARITHMS OP KUMBERS. No 900 L. 954.1 [No. 944 L. 975. N. 1 2 3 4 5 6 7 8 9 Diff. 900 954243 4291 4339 4387 4435 4484 4532 4580 4628 4677 1 4725 4773 4821 4869 4918 4966 5014 5062 5110 5158 2 5207 5255 5303 5351 5399 5447 5495 5543 5592 5640 3 5688 5736 5784 5832 5880 5928 5976 6024 6072 6120 4 6168 6216 6265 6313 6361 6409 6457 6505 6553 6601 5 6649 6697 6745 6793 6840 6888 6936 6984 7032 7080 48 6 7128 7176 7224 7272 7320 7368 7416 7464 7512 7559 7 7607 7655 7703 7751 7799 7847 7894 7942 7990 8038 8 8086 8134 8181 8229 8277 8325 8373 8421 8468 8516 9 8564 8612 8659 8707 8755 8803 8850 8898 8946 8994 910 9041 9089 9137 9185 9232 9280 9328 9375 9423 9471 1 2 9518 9995 9566 9614 9661 9709 9757 9804 9852 9900 9947 0042 0090 0138 0185 0233 AOQA 0328 AQ7 ryoQ 3 960471 0518 0566 0613 0661 0709 UcoU 0756 0804 Uo<O 0851 U4/*O 0899 4 0946 0994 1041 1089 1136 1184 1231 1279 1326 1374 5 1421 1469 1516 1563 1611 1658 1706 1753 1801 1848 6 1895 1943 1990 2038 2085 2132 2180 2227 2275 2322 7 2369 2417 2464 2511 2559 2606 2653 2701 2748 2795 8 2843 2890 2937 2985 3032 3079 3126 3174 3221 3268 ! 9 3316 3363 3410 3457 3504 3552 3599 3646 3693 3741 [920 3788 3835 3882 3929 3977 4024 4071 4118 4165 4212 1 4260 4307 4354 4401 4448 4495 4542 4590 4637 4684 2 4731 4778 4825 4872 4919 4966 5013 5061 5108 5155 3 5202 5249 5296 5343 5390 5437 5484 5531 5578 5625 4 5672 5719 5766 5813 5860 5907 5954 6001 6048 6095 47 5 6142 6189 6236 6283 6329 6376 6423 6470 6517 6564 6. 6611 6658 6705 6752 6799 6845 6892 6939 6986 7033 7 7080 7127 7173 7220 7267 7314 7361 7408 7454 7501 8 7548 7595 7642 7688 7735 7782 7829 7875 7922 7969 9 8016 8062 8109 8156 8203 8249 8296 8343 8390 8436 930 8483 8530 8576 8623 8670 8716 8763 8810 8856 8903 1 8950 8996 9043 9090 9136 9183 9229 9276 9323 93G9 2 9416 9463 9509 9556 9602 9649 9695 9742 9789 9835 3 9882 9928 9975 0021 0068 0114 0161 0207 0254 0300 4 970347 0393 0440 0486 0533 0579 0626 0672 0719 0765 5 0812 0858 0904 0951 0997 1044 1090 1137 1183 1229 6 1276 1322 1369 1415 1461 1508 1554 1601 1647 1693 7 1740 1786 1832 1879 1925 1971 2018 20G4 2110 2157 8 2203 2249 2295 2342 2388 2434 2481 2527 2573 2619 9 2666 2712 2758 2804 2851 2897 2943 2989 3035 3082 940 3128 3174 3220 3266 3313 3359 3405 3451 3497 3543 1 3590 3636 3682 3728 3774 3820 3866 3913 3959 4005 2 4051 4097 4143 4189 4235 4281 4327 4374 4420 4466 3 4512 4558 4604 4650 4696 4742 4788 4834 4880 4926 4 4972 5018 5064 5110 5156 5202 5248 5294 5340 5386 46 PROPORTIONAL PARTS. Diff. 1 234 5 678 9 47 4.7 9.4 14.1 18.8 23.5 28.2 32.9 37.6 42.3 46 4.6 9.2 13.8 18.4 23.0 27.6 32.2 36.8 41.4 LOGARITHMS OF NUMBERS. 155 No. 945 L. 975.] [No. 989 L. 995. N. 1 2 3 4 5 6 7 8 9 Diff. 945 975432 5478 5524 5570 5616 5662 5707 5753 5799 5845 6 5891 5937 5983 6029 6075 6121 6167 6212 6258 6304 7 6350 6396 6442 6488 6533 6579 6625 6671 6717 6763 8 6808 6854 6900 6946 6992 7037 7083 7129 7175 7220 9 7266 7312 7358 7403 7449 7495 7541 7586 7632 7678 950 7724 7769 7815 7861 7906 7952 7998 8043 8089 8135 1 8181 8226 8272 8317 8363 8409 8454 8500 8546 8591 2 8637 8683 8728 8774 8819 8865 8911 8956 9002 9047 3 9093 9138 9184 9230 9275 9321 93G6 9412 9457 9503 4 9548 9594 9639 9685 9730 9776 9821 9867 9912 9958 5 980003 0049 0094 0140 0185 0231 0276 0322 0367 0412 G 0458 0503 0549 0594 0640 0685 0730 0776 0821 0867 7 0912 0957 1003 1048 1093 1139 1184 1229 1275 1320 8 1366 1411 1456 1501 1547 1592 1637 1683 1728 1773 9 1819 1864 1909 1954 2000 2045 2090 2135 2181 2226 960 2271 2316 2362 2407 2452 2497 2543 2588 2633 2678 1 2723 2769 2814 2859 2904 2949 2994 3040 3085 3130 2 3175 3220 3265 3310 3356 3401 3446 3491 3536 3581 3 3626 3671 3716 37'62 3807 3852 3897 3942 3987 4032 4 4077 4122 4167 4212 4257 4302 4347 4392 4437 4482 5 4527 4572 4617 4662 4707 4752 4797 4842 4887 4932 45 6 4977 5022 5067 5112 5157 5202 5247 5292 5337 5382 7 5426 5471 5516 5561 5606 5651 5696 5741 5786 5830 8 5875 5920 5965 6010 6055 6100 6144 6189 6234 6279 9 6324 6369 6413 6458 6503 6548 6593 6637 6682 6727 970 6772 6817 6861 6906 6951 6996 7040 7085 7130 7175 1 7219 7264 7309 7353 7398 7443 7488 7532 7577 7622 2 7666 7711 7756 7800 7845 7890 7934 7979 8024 8068 3 8113 8157 8202 8247 8291 8336 8381 8425 8470 8514 4 8559 8604 8648 8693 8737 8782 8826 8871 8916 8960 5 9005 9049 9094 9138 9183 9227 9272 9316 9361 9405 6 7 9450 9895 9494 9939 9539 9983 9583 9628 9672 9717 9761 9806 9850 0028 0072 0117 0161 0206 0250 0294 8 990339 0383 0428 0472 0516 0561 0605 0650 0694 0738 9 0783 0827 0871 0916 0960 1004 1049 1093 1137 1182 980 1226 1270 1315 1359 1403 1448 1492 1536 1580 1625 1 1669 1713 1758 1802 1846 1890 1935 1979 2023 2067 2 2111 2156 2200 2244 2288 2333 2377 2421 2465 2509 3 2554 2598 2642 2686 2730 2774 2819 2863 2907 2951 4 2995 3039 3083 3127 3173 3216 3260 3304 3348 3392 5 3436 3480 3524 3568 3613 3657 3701 3745 3789 3833 6 3877 3921 3965 4009 4053 4097 4141 4185 4229 4273 7 4317 4361 4405 4449 4493 4537 4581 4625 4669 4713 44 8 4757 4801 4845 4889 4933 4977 5021 5065 5108 5152 9 5196 5240 5284 5328 5373 5416 5460 55U4 5547 5591 PROPORTIONAL PARTS. Diff 1 234 5 678 9 46 4.6 9.2 13.8 18.4 23.0 27.6 32.2 36.8 41.4 45 4.5 9.0 33.5 18.0 22.5 27.0 31.5 86 40.5 44 4.4 8.8 13. 2 17.6 22.0 26.4 30.8 35.2 39.6 43 4.3 8.6 12.9 17.2 yi.5 25.8 30.1 34.4 38.7 156 No. 990 L. 995.] MATHEMATICAL TABLES. [No. 999 L. 999. N. 1 2 3 4 5 6 7 8 9 Diff. 990 995635 5679 5723 5767 5811 5854 5898 5942 5986 6030 1 6074 6117 6161 6205 6249' 6293 6337 6380 6424 6468 44 2 6512 6555 6599 6643 6687 6731 6774 6818 6862 6906 3 6949 6993 7037 7080 7124 7168 7212 7255 7299 7343 4 7386 7430 7474 7517 7561 7C05 7648 7692 7736 7779 5 7823 7867 7910 7954 7998 8041 8085 8129 8172 8216 6 8259 8303 8347 8390 8434 8477 8521 8564 8608 8652 7 8695 8739 8782 8826 8869 8913 8956 9000 9043 9087 8 9131 9174 9218 9261 9305 9348 9392 9435 9479 9522 9 9565 9609 9652 9696 9739 9783 9826 9870 9913 9957 43 HYPERBOLIC LOGARITHMS. No. Log. No. Log. No. Log. No. Log. No. Log. 1.01 .0099 1.45 .3716 1.89 .6366 2.33 .8458 2.77 1.0188 1.02 .0198 1,46 .3784 1.90 .6419 2.34 .8502 2.78 1.0225 1.03 .0296 1.47 .3853 1.91 .6471 2.35 .8544 2.79 .0260 1.04 .0392 1.48 .3920 1.92 .6523 2.36 .8587 2.80 .0296 1.05 .0488 1.49 .3988 1.93 .6575 2.37 .8629 2.81 .0332 1.06 .0583 1.50 .4055 1.94 .6627 2.38 .8671 2.82 .0367 1.07 .0677 1.51 .4121 1.95 .6678 2.39 .8713 2.83 .0403 1.08 .0770 1.52 .4187 1.96 .6729 2.40 .8755 2.84 .0438 1.09 .0862 1.53 .4253 1.97 .6780 2.41 .8796 2.85 .0473 1.10 .0953 1.54 .4318 1.98 .6831 2.42 .8838 2.86 .0508 1.11 .1044 1.55 .4383 1.99 .6881 2.43 .8879 2.87 .0543 1.12 .1133 1.56 .4447 2.00 .6931 2.44 .8920 2.88 .0578 1.13 .1222 1.57 .4511 2.01 .6981 2.45 .8961 2.89 .0613 1.14 .1310 1.58 .4574 2.02 .7031 2.46 .9002 2.90 .0647 1.15 .1398 1.59 .4637 2.03 .7080 2.47 .9042 o gi .0682 1.16 .1484 1.60 .4700 2.04 .7129 2.48 .9083 2^92 .0716 1.17 .1570 1.61 .4762 2.05 .7178 2.49 .9123 2.93 .0750 1.18 .1655 1.62 .4824 2.06 .7227 2.50 .9163 2.94 .0784 1.19 .1740 1.63 .4886 2.07 .7275 2.51 .9203 2.95 .Obis 1.20 .1823 1.64 .4947 2.08 .7324 2.52 .9243 2.96 .0852 1.21 .1906 1.65 ,5008 2.09 .7372 2.53 .9282 2.97 .0886 1.22 .1988 1.66 .5068 2.10 .7419 2.54 .9.322 2.98 .0919 1.23 .2070 1.67 .5128 2.11 .7467 2.55 .9361 2.99 .0953 1.24 .2151 1.68 .5188 2.12 .7514 2.56 .9400 3.00 .0986 1.25 .2231 1.69 .5247 2.13 .7561 2.57 .9439 3.01 .1019 1.26 .2311 1.70 .5306 2.14 .7608 2.58 .9478 3.02 .1053 1.27 .2390 1.71 .5365 2.15 .7655 2.59 .9517 3.03 .1086 1.28 .2469 .72 .5423 2.13 .7701 2.60 .9555 3.04 .1119 1.29 .2546 .73 .5481 2.17 .7747 2.61 .9594 3.05 .1151 1.30 .2624 .74 .5539 2.18 .7793 2.62 .9632 3.06 .1184 1.31 .2700 .75 .5596 2.19 .7839 2.63 .9670 3.07 .1217 1.32 .2776 .76 .5653 2.20 .7885 2.64 .9708 3.08 .1249 1.33 .2852 .77 .5710 2.21 .7930 2.65 .9746 3.09 .1282 1.34 .2927 .78 .5766 2.22 .7975 2.66 .9783 3.10 .1314 1.35 .3001 .79 .5822 2.23 .8020 2.67 .9821 3.11 .1346 1.36 .3075 .80 .5878 2.24 .8065 2.68 .9858 3.12 .1378 1.37 .3148 .81 .5933 2.25 .8109 2.69 .9895 3.13 .1410 1.38 .3221 .82 .5988 2.26 .8154 2.70 .9933 3.14 .1442 1.88 .3293 .83 .6043 2.27 .8198 2.71 .9969 3.15 .1474 1.40 .3365 .84 .6098 2.28 .8242 2.72 1.0006 3 16 .1506 1.41 .3436 1.85 .6152 2.29 .8286 2.73 1.0043 3.17 .1537 1.42 .3507 1.86 .6206 2.30 .8329 2.74 1.0080 3.18 .1569 1.43 .3577 1.87 .6259 2.31 .8372 2.75 1.0116 3.19 .1600 1.44 .3646 1.88 .6313 2.32 .8416 2.76 1.0152 3.20 .1632 HYPERBOLIC LOGARITHMS. 157 No, Log. No. Log. No. Log. No. Log. No. Log. 3.21 1.1663 3.87 1.3533 4.53 1.5107 5.19 1.6467 5.85 1.7664 3.22 1.1694 3.88 1.3558 4.54 1.5129 5.20 1.6487 5.86 1.7681 3.23 1.1725 3.89 1.3584 4.55 1.5151 5.21 1.6506 5.87 1.7699 3.24 1.1756 3.90 1.3610 4.56 1.5173 5.22 1.6525 5.88 1.7716 3.25 1.1787 3.91 1.3635 4.57 1.5195 5.23 1.6544 5.89 1.7733 3.26 1.1817 3.92 1.3661 4.58 1.5217 5.24 1.6563 5.90 1.7750 3.27 1.1848 3.93 1.3686 4.59 1.5239 5.25 1.6582 5.91 1.7766 3.28 1.1878 3.94 1.3712 4.60 1.5261 5.26 1.6601 5.92 1.7783 3.29 1.1909 3.95 1.3737 4.61 1.5282 5.27 1.6620 5.93 1.7800 3.30 1.1939 3.96 1.3762 4.62 1.5304 5.28 1.6639 5.94 1.7817 3.31 1.1969 3.97 1.3788 4.63 1.5326 5.29 1.6658 5.95 1.7834 3.32 1.1999 3.98 1.3813 4.64 1.5347 5.30 1.6677 5.96 1.7851 3.33 1.2030 3.99 1.3838 4.65 1.5369 5.31 1.6696 5.97 1.7867 3.34 1.2060 4.00 1.3863 4.66 1.5390 5.32 1.6715 5.98 1.7884 3.35 1.2090 4.01 1.3888 4.67 1.5412 5.33 1.6734 5.99 1.7901 3.36 1.2119 4.02 1.3913 4.68 1.5433 5.34 1.6752 6.00 1.7918 3.37 1.2149 4.03 1 .3938 4.69 1.5454 5.35 1.6771 6.01 1.7934 3.38 1.2179 4.04 1.3962 4.70 .5476 5.36 1.6790 6.02 1.7951 3.39 1.2208 4.05 1.3987 4.71 .5497 5.37 1.6808 6.03 1.7967 3.40 1.2238 4.06 1.4012 4.72 .5518 5.38 1.6827 6.04 1.7984 3.41 1.2267 4.07 1.4036 4.73 .5539 5.39 1.6845 6.05 1.8001 3.42 1.2296 4.08 1.4061 4.74 .5560 5.40 1.6864 6.06 1.8017 3.43 1.2326 4.09 1.4085 4.75 .5581 5.41 1.6882 6.07 1.8034 3.44 1.2355 4.10 1.4110 4.76 .5602 5.42 1.6901 6.08 1.8050 3.45 1.2384 4.11 1.4134 4.77 .5623 5.43 1.6919 6.09 1.8066 3.46 1.2413 4.12 1.4159 4.78 1.5644 5.44 1.6938 6.10 1.8083 3 47 1.2442 4.13 1.4183 4.79 1.5665 5.45 1.6956 6.11 1.8099 3.48 1.2470 4.14 1.4207 4.80 1.5686 5.46 1.6974 6.12 1.8116 8.49 1.2499 4.15 1.4231 4.81 1.5707 5.47 1.6993 6.13 1.8132 3.50 1.2528 4.16 1.4255 4.82 1.5728 5.48 1.7011 6.14 1.8148 3.51 1.2556 4.17 1.4279 4.83 1.5748 5.49 1.7029 6.15 1.8165 3.52 1.2585 4.18 1.4303 4.84 1.5769 5.50 1.7047 6.16 1.8181 3.53 1.2613 4.19 1.4327 4.85 1.5790 5.51 1.7066 6.17 1.8197 3.54 1.2641 4.20 1.4351 4.86 1.5810 5.52 1.7084 6.18 1-8213 3.55 1.3669 4.21 1.4375 4.87 1.5831 5.53 1.7102 6.19 1.8229 3.56 1.2698 4.22 1.4398 4.88 1.5851 5.54 1.7120 6.20 1.8245 3.57 1.2726 4.23 1.4422 4.89 .5872 5.55 1.7138 6.21 1.8262 3.58 1.2754 4.24 1.4446 4.90 .5892 5.56 1.7156 6.22 1.8278 3.59 1.2782 4.25 1.4469 4.91 .5913 5.57 1.7174 6.23 1.8294 3.60 1.2809 4.26 1.4493 4.92 .5933 5.58 1.7192 6.24 1.8310 3.61 1.2837 4.27 1.4516 4.93 .5953 5.59 1.7210 6.25 1.8326 3.62 1.2865 4.28 1.4540 4.94 .5974 5.60 1.7228 6.26 1.8342 3.63 1.2892 4.29 1.4563 4.95 .5994 5.61 1.7246 6.27 1.8358 3.64 1 .2920 4.30 1.4586 4.96 .6014 5.62 1.7263 6.28 1.8374 3.65 1.2947 4.31 1.4609 4.97 .6034 5.63 1.7281 6.29 1.8390 3.66 1.2975 4.32 1.4633 4.98 .6054 5.64 1.7299 6.30 1.8405 3.67 1.3002 4.33 .4656 4.99 .6074 5.65 1.7317 6.31 1.8421 3.68 1.3029 4.34 .4679 5.00 .6094 5.66 1.7334 6.32 1.8437 3.69 1.3056 4.35 .4702 5.01 .6114 5.67 1.7352 6.33 1.8453 3.70 1.3083 4.36 .4725 5.02 .6134 5.68 1.7370 6.34 1.8469 3.71 1.3110 4.37 .4748 5.03 .6154 5.69 1.7387 6.35 1.8485 3.72 1.3137 4.38 .4770 5.04 .6174 5.70 1.7405 6.36 1.8500 3.73 1.3164 4.39 .4793 5.05 .6194 5.71 1.7422 6.37 1.8516 3.74 1.3191 4.40 .4816 5.06 .6214 5.72 1.7440 6.38 1.8532 3.75 1.3218 4.41 .4839 5.07 .6233 5.73 1.7457 6.39 1.8547 3.76 1.3244 4.42 .4861 5.08 .6253 5.74 1.7475 6.40 1.8563 3.77 1.3271 4.43 .4884 5.09 .6273 5.75 1.7492 6.41 1,8579 3.78 1.3297 4.44 .4907 5.10 .6292 5.76 1.7509 6.42 1.8594 3.79 1.3324 4.45 .4929 5.11 .6312 5.77 1.7527 6.43 1.8610 3.80 1.3350 4.46 .4951 5.12 .6332 5.78 1.7544 6.44 1.8625 8.81 1.3376 4.47 .4974 5.13 .6351 5.79 1.7561 6.45 1.8641 3.82 1.3403 4.48 .4996 5.14 .6371 5.80 1.7579 6.46 1.8656 3.83 1.3429 4.49 .5019 5.15 .6390 5.81 1.7596 6.47 1.8672 3.84 1.3455 4.50 .5041 5.16 .6409 5.82 1.7613 6.48 1.8687 8.85 1.3481 4.51 .5063 5.17 .6429 5.83 1 .7630 6.49 1.8703 3.86 1.3507 4.52 .5085 5.18 1.6448 5.84 1.7647 6.50 1.8718 158 MATHEMATICAL TABLES. No. Log. No. Log. No. Log. No. Log. No. Log. 6.51 1.8733 7.15 .9671 7.79 2.0528 8.6(5 2.1587 9.94 2.2966 6.52 1.8749 7.16 .9685 7.80 2.0541 8.68 2.1610 9.96 2.2986 6.53 1.8764 7.17 .9699 7.81 2.0554 8.70 2.1633 9.98 2.3006 6.54 1.8779 7.18 .9713 7.82 2.0567 8.72 2.1656 10.00 2.3026 6.55 1.8795 7.19 .9727 7.83 2.0580 8.74 2.1679 10.25 2.3279 6.56 1.8810 7.20 .9741 7.84 2.0592 8.76 2.1702 10.50 2.3513 6.57 1.8825 7.21 .9754 7.85 2.0605 8.78 2.1725 10.75 2.3749 6.58 1.8840 7.22 .9769 7.86 2.0618 8.80 2.1748 11.00 2.3979 6.59 .8856 7.23 .9782 7.87 2.0631 8.82 2.1770 11.25 2.4201 6.60 .8871 7.24 .9796 7.88 2.0643 8.84 2.1793 11.50 2.4430 6.61 .8886 7.25 .9810 7.89 2.0656 8.86 2.1815 11.75 2.4636 6.62 .8901 7.26 1.9824 7.90 2.0669 8.88 2.1838 12.00 2.4849 6.63 .8916 7.27 1.9838 7.91 2.0681 8.90 2.1861 12.25 2.5052 6.64 .8931 7.28 1.9851 7.92 2.0694 8.92 2.1883 12.50 2.5262 6.65 .8946 7.29 1.9865 7.93 2.0707 8.94 2.1905 12.75 2.5455 6.66 .8961 7.30 1.9879 7.94 2.0719 8.96 2.1928 13.00 2.5649 6.67 .8976 7.31 1.9892 7.95 2.0732 8.98 2.1950 13.25 2.5840 6.68 .8991 7.32 1.9906 7.96 2.0744 9.00 2.1972 13.50 2.6027 6.69 .9006 7.33 1.9920 7.97 2.0757 9.02 2.1994 13.75 2.6211 6.70 .9021 7.34 1.9933 7.98 2.0769 9.04 2.2017 14.00 2.6391 6.71 .9086 7.35 1.9947 7.99 2.0782 9.06 2.2039 14.25 2.6567 6.72 .9051 7.36 1.9961 8-00 2.0794 9.08 2.2061 14.50 2.6740 6.73 .9066 7.37 1.9974 8.01 2.0807 9 10 2.2083 14.75 2.6913 6.74 .9081 7.38 1.9988 8.02 2.0819 9.12 2.2105 15.00 2.7081 6.75 .9095 7.39 2.0001 8.03 2.0832 9.14 2.2127 15.50 2.7408 6.76 .9110 7.40 2.0015 8.04 2.0844 9.1-6 2.2148 16.00 2.7726 6.77 .9125 7.41 2.0028 8.05 2.0857 9.18 2.2170 16.50 2.8034 6.78 1.9140 7.42 2.0041 8-06 2.0869 9.20 2.2192 17.00 2.8332 6.79 1.9155 7.43 2-0055 8.07 2.0882 9.22 2.2214 17.50 2.8621 6.80 1.9169 7.44 2-0069 8.08 2.0894 9.24 2.2235 18.00 2.8904 6.81 1.9184 7.45 2.0082 8-09 2.0906 9.26 2.2257 18.50 2.9178 6.82 1.9199 7.46 2.0096 8.10 2.0919 9.28 2.2279 19.00 2.9444 6.83 1.9213 7.47 2.0108 8.11 2.0931 9.30 2.2300 19.50 2.9703 6.84 1.9228 7.48 2.0122 8.12 2.0943 9.32 2.2322 20.00 2.9957 6.85 1.9242 7.49 2.0136 8.13 2.0958 9.34 2.2343 21 3.0445 6.86 1.9257 7.50 2.0149 8.14 2.0968 9.36 2.2364 22 3.0910 6.87 1.9272 7.51 2.0162 8.15 2.0980 9.38 2.2386 23 3.1355 6.88 1.9286 7.52 2.0176 8.16 2.0992 9.40 2.2407 24 3.1781 6.89 1.9301 7.53 2.0189 8-17 2.1005 9.42 2.2428 25 3.2189 6.90 1.9315 7.54 2.0202 8.18 2.1017 9.44 2.2450 26 3.2581 6.91 1.9330 7.55 2.0215 8-19 2.1029 9.46 2.2471 27 3.2958 6.92 1.9344 7.56 2.0229 8.20 2.1041 9.48 2.2492 28 3.3322 6.93 1.9359 7.57 2.0242 8.22 2.1066 9.50 2.2513 29 3.3673 6.94 1.9373 7.58 2.0255 8.24 2.1090 9.52 2.2534 30 3.4012 6.95 1.9387 7.59 2.0268 8.26 2.1114 9.54 2.2555 31 3.4340 6.96 1.9402 7.60 2.0281 8.28 2.1138 9.56 2.2576 32 3.4657 6.97 1.9416 7.61 2.0295 8.30 2.1163 9.58 2.2597 33 3.4965 6.98 .9430 7.62 2.0308 8.32 2.1187 9.60 2.2618 34 3.5263 6.99 .9445 7.63 2.0321 8.34 2.1211 9.62 2.2638 35 3.5553 7.00 .9459 7.64 2.0334 8.36 2.1235 9.64 2.ii659 36 3.5835 7.01 .9473 7.65 2.0347 8.38 2.1258 9.66 2.2680 37 3.6109 7.02 .9488 7.66 2.0360 8.40 2.1282 9.68 2.2701 38 3.6376 7.03 .9502 7.67 2.0373 8.42 2.1306 9.70 2.2721 39 3.6636 7.04 .9516 7.68 2.0386 8.44 2.1330 9.72 2.2742 40 3.6889 7.05 .9530 7.69 2.0399 8.46 2.1353 9.74 2.2762 41 3.7136 7.06 .9544 7.70 2.0412 8.48 2.1377 9.76 2.2783 42 3.7377 7.07 .9559 7.71 2.0425 8.50 2.1401 9.78 2.2803 43 3.7612 7.08 .9573 7.72 2.0438 8.52 2.1424 9.80 2.2824 44 3.7842 7.09 .9587 7.73 2.0451 8.54 2.1448 9.82 2.2844 45 3.8067 7.10 .9601 7.74 2.0464 8.56 2.1471 9.84 2.2865 46 3.8286 7.11 .9615 7.75 2.0477 8.58 2.1494 9.86 2.2885 47 3.8501 7.12 .9629 7.76 2.0490 8.60 2.1518 9.88 2.2905 48 3.8712 7.13 .9643 7.77 2.0503 8.62 2.1541 9.90 2.2925 49 3.8918 7.14 .9657 7.78 2.0516 8.64 2.1564 9.92 2.2946 50 3.9120 NATURAL TRIGONOMETRICAL FUNCTIONS. 159 NATURAL TRIGONOMETRICAL FUNCTIONS. M. Sine. Co-Vers. Cosec. Tang. Cotn. Secant. r er. Sin. CoHine. o 00000 .0000 nfinite 00000 Infinite 1.0000 .00000 1.0000 90 15 00436 .99564 229.18 00436 229.18 ! 1.0000 .00001 .99999 45 30 00873 .99127 14.59 00873 114.59 1.0000 .00004 .99996 30 45 01309 .98691 76.397 01309 76.390 1.0001 .00009 .99991 15 1 01745 .98255 57.299 01745 57.290 1.0001 .00015 .99985 89 15 02181 .97819 45.840 02182 45.829 1.0002 .00024 .99976 45 30 02618 .97382 38.202 02618 38.188 1.0003 .00034 .99966 30 45 03054 .96946 32.746 03055 32.730 1.0005 .00047 .99953 15 2 03490 .96510 28.654 03492 28.636 1.0006 .00061 .99939 88 15 03926 .96074 25.471 03929 25.452 1.0008 .00077 .99923 45 30 04362 .95638 22.926 04366 22.904 1.0009 .00095 .99905 30 45 04798 .95202 20.843 04803 20.819 1.0011 .00115 .99885 15 3 05234 .94766 19.107 05241 19.081 1.0014 .00137 .99863 87 15 05669 .94331 17.639 05678 17.611 1.0016 .00161 .99839 45 30 06105 .93895 16.380 06116 16.350 1.0019 .00187 .99813 30 45 06540 .93460 15.290 06554 15.257 1.0021 .00214 .99786 15 4 06976 .93024 14.336 06993 14.301 I 1.0024 .00244 .99756 86 15 07411 .92589 13.494 07431 13.457 1 1.0028 .00275 .99725 45 30 07846 .92154 12.745 07870 12.706 1.0031 .00308 .99692 30 45 08231 .91719 12.076 08309 12.035 1.0034 .00343 .99656 15 5 08716 .91284 11.474 08749 11.430 1.0038 .00381 .99619 85 15 09150 .90850 10.929 09189 10.883 1.0042 .00420 .99580 45 30 09585 .90415 10.433 09629 10.385 1.0046 .00460 .99540 30 45 10019 .89981 9.9812 10069 9.9310 1.0051 .00503 .99497 15 6 10453 .89547 9.5668 10510 9.5144 1.0055 .00548 .99452 84 15 10887 .89113 9.1855 10952 9.1309 1.0060 .00594 .99406 45 30 11320 .88680 8.8337 11393 8.7769 1.0065 .00643 .99357 30 45 11754 .88246 8.5079 11836 8.4490: 1.0070 .00693 .99307 15 7 12187 .87813 8.2055 12278 8.1443 1.0075 .00745 .99255 83 15 12620 .87380 7.9240 12722 7.8606 1.0081 .00800 .99200 45 30 13053 .86947 7.6613 13165 7.5958 1.0086 .00856 .99144 30 45 13485 .86515 7.4156 13609 7.3479 1.0092 .00913 .99086 15 8 13917 .86083 7.1853 14054 7.1154 1.0098 .00973 .99027 82 15 14349 .85651 6.9690 14499 6.8969; 1.0105 .01035 .98965 45 30 14781 .85219 6.7655 14945 6.6912 1.0111 .01098 .98902 30 45 15212 .84788 6.5736 15391 6.4971 1.0118 .01164 .98836 15 9 15643 .84357 6.3924 15838 6.3138 1.0325 .01231 .98769 81 15 16074 .83926 6.2211 16286 6.1402 1.0132 .01300 .98700 45 30 .16505 .83495 6.0589 16734 5.9758 1.0139 .01371 .98629 30 45 .16935 .83065 5.9049 .17183 5.8197 1.0147 .01444 .98556 15 10 .17365 .82635 5.7588 .17633 5.6713 1.0154 .01519 .98481 80 15 .17794 .82206 5.6198 .18083 5.5301 1.0162 .01596 .98404 45 30 .18224 .81776 5.4874 .18534 5.3955 1.0170 .01675 .98325 30 45 .18652 .81348 5.3612 .18986 5.2672 1.0179 .01755 .98245 15 11 .19081 .80919 5.2408 .19438 5.1446 1.0187 .01837 .98163 79 15 .19509 .80491 5.1258 .19891 5.0273 1.0196 .01921 .98079 45 30 .19937 .80063 5.0158 .20345 4.9152 1.0205 .02008 .97992 30 45 .20364 .79636 4.9106 .20800 4.8077 1.0214 .02095 .97905 15 12 .20791 .79209 4.8097 .21256 4.7046 1.0223 .02185 .97815 78 15 .21218 .78782 4.7130 .21712 4.6057 1.0233 .02277 .97723 45 30 .21644 .78356 4.6202 .22169 4.5107 1.0243 .02370 .97630 30 45 .22070 .77930 4.5311 .22628 4.4194 1.0253 .02466 .97534 15 13 .22495 .77505 4.4454 23087 4.3315 1.0263 .02563 .97437 77 15 .22920 .77080 4.3630 23547 4.2468 1.0273 .02662 .97338 45 30 .23345 .76655 4.2837 .24008 4.1653 1.0284 .02763 .97237 30 45 .23769 .76231 4.2072 .24470 4.0867 1.0295 .02866 .97134 15 14 .24192 .75808 4.1336 .24933 4.0108 1.0306 .02970 .97030 76 15 .24615 .75385 4.0625 .25397 3.9375 1.0317 .03077 .96923 45 30 .25038 .74962 3 993! .25862 3.8667 1.0329 .03185 .96815 30 45 .25460 .7454C 3*.92r .26328 3.7983 1.034 .03295 .96705 15 15 .25882 .74118 3.8637 .2679P 3.732C 1.0353 .03407 .96593 75 Cosine Ver. Sin Secant. Cotan. Tang. Cosec. Co-Vers Sine. M. From 75 to 90 read from bottom of table upwards. 1GO MATHEMATICAL TABLES. M. Sine. Co-Vers. Cosec. Tang. Cotan. Secant. Ver. Sin. Cosine. 15 .25882 .74118 3.8637 .26795 3.7320 1.0353 .03407 .96593 75 15 .26303 .73697 3.8018 .27263 3.6680 1.0365 .03521 .96479 45 30 .26724 .73276 3.7420 .27732 3.6059 1.0377 .03637 .96363 at 45 .27144 .72856 3.6840 .28203 3.5457 1.0390 .03754 .96246 15 16 .27564 .72436 3.6280 .286?4 3.4874 1.0403 .03874 .96126 74 15 .27983 .72017 3.5736 .29147 3.4308 1.0416 .03995 .96005 45 30 .28402 .71598 3.5209 .29621 3.3759 1.0429 .04118 .95882 30 45 .28820 .71180 3.4699 .30096 3.3226 1.0443 .04243 .95757 15 17 .29237 .70763 3.4203 .30573 3.2709 1.0457 .04370 .95630 7* 15 .29654 .70346 3.3722 .31051 3.2205 1.0471 .04498 .95502 45 30 .30070 .69929 3.3255 .31530 3.1716 1.0485 .04628 .95372 30 45 .30486 .69514 3.2801 .32010 3.1240 1.0500 .04760 .95240 15 18 .30902 .69098 3.2361 .32492 3.0777 1.0515 .04894 .95106 72 15 .31316 .68684 3.1932 .32975 3.0326 1.0530 .05030 .94970 45 30 .31730 .68270 3.1515 .33459 2.9887 1.0545 .05168 .94832 30 45 .32144 .67856 3.1110 .33945 2.9459 1.0560 .05307 .94693 15 19 .32557 .67443 3.0715 .34433 2.9042 1 .0576 .05448 .94552 71 15 .32969 .67031 3.0331 .34921 2.8636 1.0592 .05591 .94409 45 30 .33381 .66619 2.9957 .35412 2.8239 1.0608 .05736 .94264 30 45 .33792 .66208 2.9593 .35904 2.7852 1.0625 .05882 .94118 15 20 .3420:2 .65798 2.9238 .36397 2.7475 1 .0642 .06031 .93969 70 15 .34612 .65388 2.8892 .36892 2.7106 1.0659 .06181 .93819 45 30 .35021 .64979 2.8554 .37388 2.6746 1.0676 .06333 .93667 30 45 .35429 .64571 2.8225 .37887 2.6395 1.0694 .06486 .93514 15 21 .35837 .64163 2.7904 .38386 2.6051 1.0711 .06642 .93358 69 15 .36244 .63756 2.7591 .38888 2-5715 1.0729 .06799 .93201 45 30 .36650 .63350 2.7285 .39391 2.5386 1.0748 .06958 .93042 30 45 .37056 .62944 2.6986 .39896 2.5065 1.0766 .07119 .92881 15 22 .37461 .62539 2.6695 .40403 2-4751 1.0785 .07282 .92718 68 15 .37865 .62135 2.6410 .40911 2 4443 1.0804 .07446 .92554 45 30 .38268 .61732 2.6131 .41421 2-4142 1.0824 .07612 .92388 30 45 .38671 .61329 2.5859 .41933 2-3847 1.0844 .07780 .92220 15 23 .39073 .60927 2.5593 .42447 2-3559 1.0864 .07950 .92050 67 15 .39474 .60526 2.5333 .42963 2-3276 1.0884 .08121 .91879 45 30 .39875 60125 2.5078 .43481 2-2998 1.0904 .08294 .91706 30 45 .40275 .59725 2.4829 .44001 2 2727 1.0925 .08469 .91531 15 24 .40674 .59326 2.4586 .44523 2-2460 1.0946 .08645 .91355 66 15 .41072 .58928 2.4348 .45047 2.2199 1.0968 .08824 .91176 45 30 .41469 .58531 2.4114 .45573 2.1943 1.0989 .09004 .90996 30 45 41866 .58134 2.3886 .46101 2.1692 1.1011 .09186 .90814 15 25 .42262 .57738 2.3662 .46631 2.1445 1.1034 .09369 .90631 65 15 .42657 .57343 2.3443 .47163 2-1203 1.1056 .09554 .90446 45 30 .43051 .56949 2.3228 .47697 2-0965 1.1079 .09741 .90259 30 45 .43445 .56555 2.3018 .48234 2.0732 1.1102 .09930 .90070 15 26 .43837 .56163 2.2812 .48773 2-0503 1.1126 .10121 .89879 64 15 .44229 .55771 2.2610 .49314 2.0278 1.1150 .10313 .89687 45 30 .44620 .55380 2.2412 .49858 2.0057 1.1174 .10507 .89493 30 45 .45010 .54990 2.2217 .50404 .9840 1.1198 .107'02 .89298 15 27 .45399 .54601 2.2027 .50952 .9626 1.1223 .10899 .89101 63 15 .45787 .54213 2.1840 .51503 .9415 1.1248 .11098 .88902 45 30 .46175 .53825 2.1657 .52057 .9210 1.1274 .11299 .88701 30 45 .46561 .53439 2.1477 .52612 .9007 1.1300 .11501 .88499 15 2$ .46947 .53053 2.1300 .53171 .8807 1.1326 .11705 .88295 62 15 .47332 .52668 2.1127 .53732 .8611 1.1352 .11911 .88089 45 30 .47716 .52284 2.0957 .54295 .8418 1.1379 .12118 .87882 30 45 .48099 .51901 2.0790 .54862 .8228 1.1400 .12327 .87673 15 29 .48481 .51519 2.0627 .55431 .8040 1.1433 .12538 .87462 61 15 .48862 .51138 2.0466 .56003 .7856 1.1461 .12750 .87250 45 30 .49242 .50758 2.0308 .56577 .7675 1.1490 .12964 .87036 30 45 . 49622 .50378 2.0152 .57155 .7496 1.1518 .13180 .86820 15 30 .50000 .50000 2.0000 .57735 1.7320 1.1547 .13397 .86603 60 Cosine. Ver. Sin. Secant. Cotan. Tang. Cosec. Co-Vers. Sine. ' M. From 60 to 75 read from bottom of table upwards. NATURAL TRIGONOMETRICAL FUNCTIONS. 161 M. Sine. Co-Vers. Cosec. Tang. Cotan. Secant. Ver. Sin. Cosine. 80 .50000 .50000 2.0000 .57735 .7320 1.1547 .13397 .86603 60 15 .50377 .49623 .9850 .58318 .7147 1.1576 .13616 .86384 45 30 .50754 .49246 .9703 .58904 .6977 1.1606 .13837 .86163 30 45 .51129 .48871 .9558 .59494 .6808 1.1636 .14059 .85941 15 81 .51504 .48496 .9416 .60086 .6643 1.1666 .14283 .85717 59 15 .51877 .48123 .9276 .60681 .6479 .1697 .14509 .85491 45 30 .52250 .47750 .9139 .61280 .6319 .1728 .14736 .85264 30 45 .52621 .47379 .9004 .61882 1.6160 .1760 .14965 .85035 15 32 .52992 .47008 .8871 .62487 1.6003 .1792 .15195 .84805 58 15 .53361 .46639 .8740 .63095 1.5849 .1824 .15427 .84573 45 30 .53730 .46270 .8612 .63707 1.5697 .1857 .15661 .84339 30 45 .54097 .45903 .8485 .64322 1.5547 .1890 .15896 .84104 15 33 .54464 .45536 .8361 .64941 1.5399 1924 .16133 .83867 57 15 .54829 .45171 .8238 .65563 1.5253 .1958 .16371 .83629 45 30 .55194 .44806 .8118 .66188 1.5108 .1992 .16611 .83389 30 45 .55557 .44443 .7999 .66818 1.4966 .2027 .16853 .83147 15 34 .55919 .44081 .7883 .67451 1.4826 .2062 .17096 .82904 56 15 .56280 .43720 .7768 .68087 1.4687 .2098 .17341 .82659 45 30 ;56641 .43359 .7655 .68728 1.4550 .2134 .17587 .82413 30 45 .57000 .43000 .7544 . 69372 1.4415 .2171 .17835 .82165 15 85 .57358 .42642 .7434 .70021 1.4281 .2208 .18085 .S1915 55 15 .57715 .42285 .7327 .70673 1.4150 .2245 .18336 .81664 45 30 .58070 .41930 .72x>0 .71329 1.4019 .2283 .18588 .81412 30 45 .58425 .41575 .7116 .71990 1.3891 .2322 .18843 .81157 15 36 .58779 .41221 .7013 .72654 1.3764 .2361 .19098 .80902 54 15 .59131 .40869 .6912 .73323 1.3638 .2400 .19356 .80644 45 30 .59482 .40518 .6812 .73996 1.&514 .2440 .19614 .80386 30 45 .59832 .40168 .6713 .74673 1.3392 .2480 .19875 .80125 15 37 .60181 .39819 .6616 .75355 1.3270 .2521 .20136 .79864 53 15 .60529 .39471 .6521 .76042 1.3151 .2563 .20400 .79600 45 30 .60876 .39124 .6427 .76733 1.3032 .2605 .20665 .79335 30 45 .81222 .38778 .6334 . 77428 1.2915 .2647 .20931 .79069 15 38 .61566 .38434 .6243 .78129 1.2799 .2690 .21199 .78801 52 15 .61909 .38091 .6153 .78834 1.2685 1.2734 .21468 .78532 45 30 .62251 .37749 .6064 .79543 1.2572 1.2778 .21-739 .78261 3" 45 .62592- .37408 .5976 .80258 1.2460 1.2822 .22012 .77988 15 39 .62932 .37068 .5890 .80978 1.2349 1.2868 .22285 .77715 51 15 .63271 .36729 .5805 .81703 1.2239 1.2913 .22561 .77'439 45 30 .63608 .36392 .5721 .82434 1.2131 1.2960 .22838 .77162 30 45 .63944 .36056 .5639 .83169 1.2024 1.3007 .23116 .76884 15 40 .64279 .35721 .5557 .83910 1.1918 1.3054 . .23396 .76604 50 15 .64612 .35388 .5477 .84656 1.1812 1.3102 .23677 .76323 45 30 .64945 .35055 .5398 .85408 1.1708 1.3151 .23959 .76041 30 45 .65276 .34724 1.5320 .86165 1.1606 1.3200 .24244 .75756 15 41 .65606 .34394 1.5242 .86929 1.1504 1.3250 .24529 .75471 49 15 .65935 .34065 1.5166 .87698 1.1403 1.3301 .24816 .75184 45 30 .66262 .33738 1 5092 .88472 1.1303 1.3352 .25104 . 7489( 30 45 .66588 .33412 1.5018 .89253 1.1204 1.3404 .25394 .74606 15 42 .66913 .33087 1.4945 .90040 1.1106 1.3456 .25686 .74314 48 15 .67237 .32763 1.4873 .90834 1.1009 1.3509 .25978 .7402? 45 30 .67559 .32441 1.4802 .91633 1.0913 1.3563 .2627'2 .73728 30 45 .67830 .32120 1.4732 .92439 1.0818 1.3618 .26568 .73432 15 43 .68200 .31800 1.4663 .93251 1.0724 1.3673 .26865 .73135 47 15 .68518 .31482 1.4595 .94071 1.0630 1.3729 .27163 .72837 45 30 .68835 .31165 1.4527 .94896 1.0538 1 .3786 .27463 .72537 30 45 .69151* .30849 1.4461 .95729 1.044G 1.3843 .27764 .72236 15 44 .69466 .30534 1.43*96 .96569 1.0355 1.3902 .28066 .71934 46 15 .69779 .30221 1.4331 .97416 1.0265 1.3961 .28370 .71630 45 30 .70091 .29909 1.4267 .98270 1.0176 1.4020 .28675 .71325 30 45 .70401 .29599 1.4204 .99131 1.0088 1.4081 .28981 .71019 15 45 .70711 .29289 1.4142 1.0000' 1.0000 1.4142 .29289 .70711 45 Cosine. Ver. Sin. Secant. Cotan. Tang. Cosec. Co-Vers. Sine. o M. From 45 to 60 read from bottom of table upwards, 162 MATHEMATICAL TABLES. LOGARITHMIC SINES, ETC. Deg. Sine. Cosec. Versin. Tangent. Cotan. Covers. Secant. Cosine. Deg. In.Neg. Infinite. In.Neg. In.Ne?. Infinite. 10.00000 10.00000 10.00000 90 1 8.24186 11.75814 0.18271 8.24192 11.75808 9.99235 10.00007 9.99993 89 2 8.54282 11.45718 6.78474 8.54308 11.45692 9.98457 10.00026 9.99974 88 3 8.71880 11.28120 7.13687 8.71940 11.28060 9.97665 10.00060 9.99940 87 4 8.84358 11.15642 7.38667 8.84464 11.15536 9.96860 10.06106 9.99894 86 5 8.94030 11.05970 7.58039 8.94195 11.05805 9.96040 10.00166 9.99834 85 6 9.01923 10.98077 7.73863 9.02162 10.97838 9.95205 10.00239 9.99761 84 7 9.08589 10.91411 7.87238 9.08914 10.91086 9.94356 10.00325 9.99675 83 8 9.14356 10.85644 7.98820 9.14780 10.85220 9.93492 10.00425 9.99575 82 9 9.19433 10.80567 8.09032 9.19971 10.80029 9.92612 10.00538 9.99462 81 10 9.23967 10.76033 8.18162 9.24632 10.75368 9.91717 10.00665 9.99335 80 11 9.28060 10.71940 8.26418 9.28865 10.71135 9.90805 10.00805 9.99195 79 12 9.31788 10.68212 8.33950 9.32747 10.67253 9.89877 10.00960 9. 99f)40 78 13 9.35209 10.64791 8.40875 9.36336 10.636G4 9.88933 10.01128 9.98872 77 14 9.38368 10.61632 8.47282 9.39677 10.60323 9.87971 10.01310 9.98690 76 15 9.41300 10.58700 8.53243 9.42805 10.57195 9.86992 10.01506 9.98494 75 16 9.44034 10.55966 8.58814 9.45750 10.54250 9.85996 10.01716 9.98284 74 17 9.46594 10.53406 8.64043 9.48534 10.51460 9.84981 10.01940 9.98060 73 18 9.48998 10.51002 8.68969 9.51178 10.48822 9.83947 10.02179 9.97821 72 19 9.51264 10.48736 8.73625 9.53697 10.46303 9.82894 10.02433 9.97567 71 20 , 9. 53405 10.46595 8.78037 9.56107 10.43893 9.81821 10.02701 9.97299 70 21 9.55433 10.44567 8.82230 9.58418 10.41582 9.807*9 10.02985 9.97015 69 22 9.57358 10.42642 8.86223 9.60641 10.39359 9.79615 10.03283 9.96717 68 23 9.59188 10.40812 8.90034 9.62785 10.37215 9.78481 10.03597 9.96403 67 24 9.60931 10.39069 8.93679 9.64858 10.35142 9.77325 10.03927 9.96073 66 25 9.62595 10.37405 8.97170 9.66867 10.33133 9.7614C 10.04272 9.95728 65 26 9.64184 10.35816 9.00521 9.68818 10.31182 9.74945 10.04634 9.95366 64 27 9.65705 10.34295 9.03740 9.70717 10.29283 9.73720 10.05012 9.94988 63 28 9.67161 10.32839 9.06838 9.72567 10.27433 9.72471 10.05407 9.94593 62 29 9.68557 10.31443 9.09823 9.74375 10.25625 9.71197 10.05818 9.94182 61 30 9.69897 10.30103 9.12702 9.76144 10.23856 9.69897 10.06247 9.93753 60 31 9.71184 10.28816 9.15483 9.77877 10.22123 9.68571 10.06693 9.93307 59 32 9.72421 10.27579 9.18171 9.79579 10.20421 9.67217 10.07158 9.92842 58 33 9.73611 10.26389 9.20771 9.81252 10.18748 9.65836 10.07641 9.92359 57 34 9.74756 10.25244 9.23290 9.82899 10.17101 9.64425 10.08143 9.91857 56 35 9.75859 10.24141 9.25731 9.84523 10.15477 9.62984 10.08664 9.91336 55 36 9.76922 10.23078 9.28099 9.86126 10.13874 9.61512 10.09204 9.90796 54 SP- 9.77946 10.22054 9.30398 9.87711 10.12289 9.60008 10.09765 9.90235 53 SS 9.78934 10.21066 9.32631 9.89281 10.10719 9.58471 10.10347 9.89653 52 39 9.79887 10.20113 9.34802 9.90837 10.09163 9.56900 10.10950 9.89050 51 40 9.80807 10.19193 9.36913 9.92381 10.07619 9.55293 10.11575 9.88425 50 41 9.81694 10.18306 9.38968 9.93916 10.06084 9.53648 10.12222 9.87778 49 42 9.82551 10.17449 9.40969 9.95444 10.04556 9.51966 10.12893 9.87107 48 43 9.83378 10.16622 9.42918 9.96966 10.03034 9.50243 10.13587 9.86413 47 44 9.84177 10.15823 9.44818 9.98484 10.01516 9.48479 10.14307 9.85693 46 ^45 9.84949 10.15052 9.46671 10.00000 10.00000 9.46671 10.15052 9.84949 45 Cosine. Secant. Covers. Cotan. Tangent. Versin. Cosec. Sine. From 45 to 90 read from bottom of table upwards. SPECIFIC GRAVITY. MATERIALS. THE CHEMICAL ELEMENTS. Common Elements (42). 1o* 'i a-- 0^ "3-; If Name. si S'S It Name. s*S $% || Name. fl bJO '3 gco 4fje o^ <$ go> <: Al Aluminum 27.1 F Fluorine 19. Pd Palladium 106. Sb Antimony 120.4 Au Gold 197.2 P Phosphorus 31. As Arsenic 75.1 H Hydrogen 1.01 Pt Platinum 194.9 Ba Barium 137.4 I Iodine 126.8 K Potassium 39.1 Bi Bismuth 208.1 Ir I rid in m 193.1 Si Silicon 28.4 B Boron 10.9 Fe Iron 56. Ag Silver 107.9 Br Bromine 79.9 Pb Lead 206.9 Sodium 23. Cd Cadmium 111.9 Li Lithium 7.03 S? Strontium 87.6 Ca Calcium 40.1 Mg Magnesium 24.3 S Sulphur 32.1 C Carbon 12. Mn Manganese 55. Sn Tin 119. Cl Chlorine 35.4 Hg Mercury 200. Ti Titanium 48.1 Cr Chromium 52.1 Ni Nickel 58.7 W Tungsten 184.8 Co Cobalt 59. N Nitrogen 14. Va Vanadium 51.4 Cu Copper 63.6 Oxygen 16. Zn Zinc 65.4 j-tureu to \j = JD ttuu n = i.vuo. vvneii n. is ttt.i4.cii tt i, v^ = lu.o/y, uiiu 1110 other figures are diminished proportionately. (See Jour. Am. Chem. Soc., TV. o i.^V. -i OflC \ The Rare Elements (27). Beryllium, Be. Caesium, Cs. Cerium, Ce. Didymium, D. Erbium, E. Gallium, Ga. Germanium, Ge. Glucinum, G. Indium, In. Lanthanum, La. Molybdenum, Mo. Niobium, Nb. Osmium, Os. Rhodium, R. Rubidium, Rb. Ruthenium, Ru. Samarium, Sm. Scandium, Sc. Selenium, Se. Tantalum, Ta. Tellurium, Te. Thallium, Tl. Thorium, Th. Uranium, U. Ytterbium, Yr. Yttrium, Y. Zirconium, Zr. SPECIFIC GRAVITY. The specific gravity of a substance is its weight as compared with the weight of an equal bulk of pure water. To find the specific gravity of a substance. W = weight of body in air; w = weight of body submerged in water. W Specific gravity = w _ w - If the substance be lighter than the water, sink it by means of a heavier substance, and deduct the weight of the heavier substance. Specific-gravity determinations are usually referred to the standard of the weight of water at 62 F., 62.355 Ibs. per cubic foot. Some experimenters have used 60 F. as the standard, and others 32 and 39.1 F. There is no general agreement. Given sp. gr. referred to water at 39.1 F., to reduce it to the standard of 62 F. multiply it by 1.00112. Given sp. gr. referred to water at 62 F., to find weight per cubic foot mul- tiply by 62.355. Given weight per cubic foot, to find sp. gr. multiply by 0.016037. Given sp. gr., to find weight per cubic inch multiply by .036085. 164 MATERIALS. Weight and Specific Gravity of Metals. Specific Gravity. Range accord- ing to several Authorities. Specific Grav- ity. Approx. Mean Value, used in Calculation of Weight. Weight per Cubic Foot, Ibs. Weight per Cubic Inch, Ibs. 2.56 to 2.71 6.66 to 6.86 9.74 to 9.90 7.8 to 8.6 8.52 to 8.96 8.6 to 8.7 1.58 5.0 8.5 to 8.6 19.245 to 19.361 8.69 to 8.92 22.38 to 23. 6.85 to 7.48 7.4 to 7.9 11.07 to 11.44 7. to 8. 1.69 to 1.75 13.60 to 13.62 13.58 13.37 to 13.38 8.279 to 8.93 20.33 to 22.07 0.865 10.474 to 10.511 0.97 7.69* to 7.932t 7.291 to 7.409 5.3 17. to 17.6 6.86 to 7.20 2.67 6.76 9.82 rs.eo J8.40 1 8.36 [8.20 8.853 8.65 19.258 8.853 7.218 7.70 11.38 8. 1.75 13.62 13.58 13.38 8.8 21.5 10.505 7.854 7.350 7.00 166.5 421.6 612.4 536.3 523.8 521.3 511.4 552. 539. 1200.9 552. 1396. 450. 480. 709.7 499. 109. 849.3 846.8 834.4 548.7 1347.0 655.1 489.6 458.3 436.5 .0963 .2439 .3544 .3103 .3031 .3017 .2959 .3195 .3121 .6949 .3195 .8076 .2604 .2779 .4106 .2887 .0641 .4915 .4900 .4828 .3175 .7758 .3791 .2834 .2652 .2526 Antimony Bismuth . ... Brass: Copper 4- Zinc 1 80 20 I 70 30 >-.. 60 40 50 50 J Rron7p jCopper,95to80> onze lTin, 5 to 20 f Cadmium Calcium Cobalt Gold pure Copper Iron Cast " Wrought Lead Manganese Magnesium.. .. Mercury Nickel j 32 ....-< 60 (212 Platinum Potassium Silver Sodium Steel Tin Titanium Tungsten Zinc . * Hard and burned. t Very pure and soft. The sp. gr. decreases as the carbon is increased. In the first column of figures the lowest are usually those of cast metals, which are more or less porous; the highest are of metals finely rolled or drawn into wire. Specific Gravity of Liquids at 60 F. Acid, Muriatic 1.200 " Nitric 1.217 " Sulphuric 1.849 Alcohol, pure 794 " 95 per cent 816 " 50 " " 934 Ammonia, 27.9 per cent 891 Bromine 2.97 Carbon disulphide 1 .26 Ether, Sulphuric 72 Oil, Linseed 94 Compression of tlse following Fluids under a Pressure of 15 Ibs. per Square Inch. Water 00004663 I Ether 00006158 Alcohol 0000216 [Mercury 00000265 Oil, Olive 92 Palm 97 Petroleum 78 to .88 Rape 92 Turpentine 87 Whale 92 Tar 1. Vinegar 1.08 Water 1. " sea 1.026 tol.O SPECIFIC GRAVITY. 165 The Hydrometer, The hydrometer is an instrument for determining the density of liquids. It is usually made of glass, and consists of three parts: (1) the upper part, a graduated stem or fine tube of uniform diameter; (2) a bulb, or enlarge- ment of the tube, containing air ; and (3) a small bulb at the bottom, con- taining shot or mercury which causes the instrument to float in a vertical position. The graduations are figures representing either specific gravities, or the numbers of an arbitrary scale, as in Baume's, Twaddell's, Beck's, and other hydrometers. There is a tendency to discard all hydrometers with arbitrary scales and ;o use only those which read in terms of the specific gravity directly. Baume's Hydrometer and Specific Gravities Compared. Degrees Baume. Liquids Heavier than Water, sp. gr. Liquids Lighter than Water, sp. gr. Degrees Baume. Liquids Heavier than Water, sp. gr. Liquids Lighter than Water, sp. gr. Degrees Baum6. Liquids Heavier than Water, sp. gr. Liquids Lighter than Water, sp. gr. 1 1.000 1.007 19 20 1.143 1.152 .942 .936 38 8P 1.333 1.345 .839 .834 1 013 21 1 160 .930 40 1 357 .830 3 4 5 1.020 1.027 1.034 22 23 24 1.169 1.178 1.188 .924 .918 .913 41 42 44 1.369 1.382 1.407 .825 .820 .811 fi 1 041 25 1 197 .907 46 1.434 .802 7' 1.048 96 1.206 .901 48 1.462 .794 8 q 1.05G 1 063 27 98 1.216 1.226 .896 .890 50 53 1.490 1.520 .785 .777 10 11 12 13 14 15 1.070 1.078 1.086 1.094 1.101 1.109 1.000 .993 .986 .980 .973 .967 29 30 31 32 33 84 1.236 1.246 1.256 1.267 1.277 1.288 .885 .880 .874 .869 .864 .859 54 56 58 60 65 70 1.551 .583 .617 .652 .747 .854 .768 ,760 .753 .745 16 1 118 960 35 1 299 .854 75 1 974 17 1.126 .954 36 1 310 .849 76 2.000 18 1.134 .948 37 1.322 .844 Specific Gravity and Weight of Wood. Specific Gravity. Weight per Cubic Foot. Ibs. Specific Gravity. Weight per Cubic Foot, Ibs. Alder Avge. 0.56 to 0.80 .68 42 Hornbeam. . . Avge. .76 .76 47 Apple A.sh .73 to .79 .76 .60 to .84 .72 47 45 Juniper Larch .56 .56 .56 .56 35 35 Bamboo.. . . Beech . .31 to .40 .35 .62 to .85 .73 22 46 Lignum vitse Linden .65 to 1.33 1.00 .604 62 37 Birch .56 to .74 .65 41 Locust .728 46 Box, .91 to 1.33 1.12 70 Mahogany. .. .56 to 1.06 .81 51 Cedar ... . .49 to .75 .62 39 Maple .57 to .79 .68 42 Cherry Chestnut Cork .61 to .72 .66 .46 to .66 .56 .24 .24 41 35 15 Mulberry Oak, Live 44 White .56 to .90 .73 .96 to 1.26 1.11 .69 to .86 .77 46 69 48 Cypress.... Dogwood . . . Ebony Elm . . .41 to .66 .53 .76 .76 1.13 to 1.33 1.23 .55 to .78 .61 33 47 76 38 44 Red.... Pine, White. . 44 Yellow. Poplar .... .73 to .75 .74 .35 to .55 .45 .46 to .76 .61 .38 to .58 .48 46 28 -38 30 Fir Gum Hackmatack .48 to .70 .59 .84 to 1.00 .92 .59 .59 37 57 37 Spruce. Sycamore.... Teak .40 to .50 .45 .59 to .62 .60 .66 to .98 .82 28 37 51 Hemlock . . . Hickory .36 to .41 .38 69 to .94 .77 24 48 Walnut Willow. .50 to .67 .58 .49 to .59 .54 36 34 Holly....'!.!! .76 .76 47 166 MATE-KLAUS. Weight and Specific Oravlty of Stones, Brick, Cement, etc. Pounds per Cubic Foot. Specific Gravity. Asphaltum 87 Brick, Soft 100 " Common 112 41 Hard 125 " Pressed 135 " Fire 140 to 150 Brickwork in mortar 100 " cement 112 Cement, Rosendale, loose 60 *' Portland, " 78 Clay 120 to 150 Concrete 120 to 140 Earth, loose 72 to 80 rammed 90 to 110 Emery 250 Glass 156 to 172 " flint 180tol96 Gneiss I , A , ^ n Granite p ' 160 to 170 Gravel 100 to 120 Gypsum 130 to 150 Hornblende 200 to 220 Lime, quick, in bulk 50 to 55 Limestone 170 to 200 Magnesia, Carbonate 150 Marble 160 to 180 Masonry, dry rubble 140 to 160 " dressed 140 to 180 Mortar 90 to 100 Pitch 72 Plaster of Paris 74 to 80 Quartz 165 Sand 90 to 110 Sandstone 140 to 150 Slate 170tol80 Stone, various 135 to 200 Trap 170 to 200 Tile 110 to 120 Soapstone 166 to 175 1.39 1.6 1.79 2.0 2.16 2.24 to 2.4 1.6 1.79 .96 1.25 1.92 to 2.4 1.92 to 2. 24 1.15 to 1.28 1.44 to 1.76 4. 2.5 to 2.73 2. 88 to 3. 14 2.56 to 2.72 1.6 to 1.92 2. 08 to 2. 4 3.2 to 3. 52 > .8 to .88 2. 72 to 3. 2 2.4 2. 56 to 2. 88 2. 24 to 2. 56 2.24 to 2.88 1.44 to 1.6 1.15 1.18 to 1.28 2.64 1.44 to 1.76 2.24 to 2.4 2. 72 to 2. 88 2.16 to 3. 4 2. 72 to 3.4 1.76 to 1.92 2.65 to 2.8 Specific Gravity and Weight of Oases at Atmospheric Pressure and 32 F. (For other temperatures and pressures see pp. 459, 479.) Density, Air ='l. Air Oxygen, O Hydrogen, H Nitrogen, N Carbon monoxide, CO... Carbon dioxide, CO 2 Methane, marsh-ga s, CH 4 Eihylene, C 2 H 4 Acetylene, C 2 H 2 Ammonia, NH 3 Water vapor, H 2 O 1.0UOO 1.1052 0.0692 0.9701 0.9671 1.5197 0.5530. 0.9674 0.8982 0.5889 0.6218 Density, H = 1. 14.444 15.963 1.000 14.012 13.968 21 .950 7.987 13.973 12.973 8.506 8.981 per Litre. 1.2931 1.4291 0.0895 1.2544 1.2505 1.9650 0.7150 1.2510 1.1614 0.7615 0.8041 Lbs. per Cubic Ft. Cu. Ft, .080753 .08921 .00559 .07831 .07807 .12567 .04464 .07809 .07251 .04754 .05020 per Lb. 12.388 11.209 178.931 12.770 12.810 8.152 22.429 12.805 13.792 21.036 19.922 PROPERTIES OF THE USEFUL METAXS. 16? PROPERTIES OF THE USEFUL METALS. Aluminum, Al. Atomic weight 27.1. Specific gravity 2.6 to 2.7. The lightest of all the useful nietals except magnesium. A soft, ductile, malleable metal, of a white color, approaching silver, but with a bluish cast. Very non-corrosive. Tenacity about one third that of wrought-iron. For- merly a rare metal, but since 1890 its production and use have greatly in- creased on account of the discovery of cheap processes for reducing it from the ore. Melts at about 1160 F. For further description see Aluminum, under Strength of Materials. Antimony (Stibium), Sb. At. wt. 120.4. Sp. gr. 6.7 to 6.8. A brittle metal of a bluish-white color and highly crystalline or laminated structure. Melts at 842 F. Heated in the open air it burns with a bluish-white flame. Its chief use is for the manufacture of certain alloys, as type metal (anti- mony 1, lead 4), britannia (antimony 1, tin 9), and various anti-friction metals (see Alloys). Cubical expansion by heat from 92 to 212 F., 0.0070. Specific heat .050. Bismuth, Bi. At. wt. 208.1. Bismuth is of a peculiar light reddish color, highly crystalline, and so brittle that it can readily be pulverized. It melts at 510 F., and boils at about 2300 F. Sp. gr. 9.823 at 54 F., and 10.055 just above the melting-point:. Specific heat about .0301 at ordinary temperatures. Coefficient of cubical expansion from 32 to 212, 0.0040. Con- ductivity for heat about 1/56 and for electricity only about 1/80 of that of silver. Its tensile strength is about 6400 Ibs. per square inch. Bismuth ex- pands in cooling, and Tribe has shown that this expansion does not take place until after solidification. Bismuth is the most diamagrietic element known, a sphere of it being repelled by a magnet. Cadmium, Cd. At. wt. 112. Sp. gr. 8.6 to 8.7. A bluish-white metal, lustrous, with a fibrous fracture. Melts below 500 F. and volatilizes at about 680 F. It is used as an ingredient in some fusible alloys with lead, tin, and hismuth. Cubical expansion from 32 to 212 F., 0.0094. Copper, Cu. At. wt. 63.2. Sp. gr. 8.81 to 8.95. Fuses at about 1930 F. Distinguished from all other metals by its reddish color. Very ductile and malleable, and its tenacity is next to iron. Tensile strength 20,000 to 30,000 Ibs. per square inch. Heat conductivity 73. 6# of that of silver, and su- perior to that of other metals. Electric conductivity equal to that of gold and silver. Expansion by heat from 32 to 212 F., 0.0051 of its volume. Specific heat .093. (See Copper under Strength of Materials: also Alloys.) Gold (Aurum). Au At. wt. 197.2. Sp. gr., when pure and pressed in a die, 19.34. Melts at about 1915 F. The most malleable and ductile of all metals. One ounce Troy may be beaten so as to cover 160 sq. ft. of surface. The average thickness of gold-leaf is 1/282000 of an inch, or 100 sq. ft. per ounce. One grain may be drawn into a wire 500 ft. in length. The ductil- ity is destroyed by the presence of 1/2000 part of lead, bismuth, or antimony. Gold is hardened by the addition of silver or of copper. In U. S. gold coin there are 90 parts gold and 10 parts of alloy, which is chiefly copper with a little silver. By jewelers the fineness of gold is expressed in carats, pure gold being 24 carats, three fourths fine 18 carats, etc. Iridium. Indium is one of the rarer metals. It has a white lustre, re- sembling that of steel; its hardness is about equal to that of the ruby; in the cold it is quite brittle, but at a white heat it is somewhat malleable. It is one of the Heaviest of metals, having a specific gravity of *jy.3S. It is ex- tremely infusible and almost absolutely inoxiclizable. For uses of iridium, methods of manufacturing it, etc., see paper by W. D. Dudley on the "Iridium Industry," Trans. A. I. M. E. 1884. Iron (Ferrum), Fe. At. wt. 56. Sp. gr.: Cast, 6.85 to 7.48; Wrought, 7.4 to 7.9. Pure iron is extremely infusible, its melting point being above 3000 F , but its fusibility increases with the addition of carbon, cast iron fus* ing about 2500 F. Conductivity for heat 11.9, and for electricity 12 to 14.8, silver being 100. Expansion in bulk by heat: cast iron .0033, and wrought iron .0035, from 32 to 212 F. Specific heat: cast iron .1298, wrought iron .1138, steel .1165. Cast iron exposed to continued heat becomes permanently ex- panded 1^ to 3 per cent of its length. Grate-bars should therefore be allowed about 4 per cent play. (For other properties see Iron and Steel under Strength of Materials.) Lead (Plumbum), JPb. At. wt. 208.9. Sp. gr. 11.07 to 11.44 by different authorities. Melte at about 625 F., softens and becomes pasty at about 617 F. If broken by a sudden blow when just below the melting-point it ia quite brittle and the fracture appears crystalline. Lead is very malleable 168 'MATERIALS. and ductile, but its tenacity is such that it can be drawn into wire with great difficulty. Tensile strength, 1600 to 2400 Ibs. per square inch. Its elasticity is very low, and the metal flows under very slight strain. Lead dissolves to some extent in pure water, but water containing carbonates or sulphates forms over it a film of insoluble salt which prevents further action. Magnesium, Mg. At. wt. 24. Sp. gr. 1.69 to 1.75. Silver-white, brilliant, malleable, and ductile. It is one of the lightest of metals, weighing only about two thirds as much as aluminum. In the form of filings, wire, or thin ribbons it is highly combustible, burning with a light of dazzling brilliancy, useful for signal-lights and for flash-lights for photographers. It is nearly non-corrosive, a thin film of carbonate of magnesia forming on ex- posure to damp air, which protects it from further corrosion. It may be alloyed with aluminum, 5 per cent Mg added to Al giving about as much in- crease of strength and hardness as 10 per cent of copper. Cubical expansion by heat 0.0083, from 32 to 212 F. Melts at 1200 F. Specific heat .25. Manganese, Mn. At. wt. 55. Sp. gr. 7 to 8. The pure metal is not used iu tne arts, but alloys of manganese and iron, called spiegeleisen when containing below 25 per cent of manganese, and ferro-manganese when con- taining from 25 to 90 per cent, are used in the manuf ,cture of steel. Metallic manganese, when alloyed with iron, oxidizes rapidly in the air, and its func* tion in steel manufacture is to remove the oxygen from the bath of steel whether it exists as oxide of iron or as occluded gas. Mercury (Hydrargyrum), Hg. At. wt. 199.8. A silver-white metal, liquid at temperatures above 39 F., and boils at 680 F. Unchangeable as gold, silver, and platinum in the atmosphere at ordinary temperatures, but oxidizes to the red oxide when near its boiling-point. Sp.gr.: when liquid 13.58 to 13.59, when frozen 14.4 to 14.5. Easily tarnished by sulphur fumes, also by dust, from which it may be freed by straining through a cloth. No metal except iron or platinum should be allowed to touch mercury. The smallest portions of tni, lead, zinc, and even copper to a less extent, cause it to tarnish and lose its perfect liquidity. Coefficient of cubical expansion from 32 to 212 F. .0182; per deg. .000101. Nickel, Ni. At. wt. 58.3. Sp. gr. 8.27 to 8.93. A silvery- white metal with a strong lustre, not tarnishing on exposure to the air. Ductile, hard, and as tenacious as iron. It is attracted to the magnet and may be made magnetic like iron. Nickel is very difficult of fusion, melting at about 3000* F. Chiefly used in alloys with copper, as german-silver, nickel-silver, etc., and recently in the manufacture of steel to increase its hardness and strength, also for nickel-plating. Cubical expansion from 32 to 212 F., 0.0038. Specific heat .109. Platinum, Pt. At. wt. 195. A whitish steel-gray metal, malleable, very ductile, and as unalterable by ordinary agencies as gold. When fused and refined it is as soft as copper. Sp. gr. 21.15. It is fusible only by the pxyhydrogen blowpipe or in strong electric currents. When combined with indium it forms an alloy of great hardness, which has been used for gun- vents and for standard weights and measures. The most important uses of platinum in the arts are for vessels for chemical laboratories and manufac- tories, and for the connecting wires in incandescent electric lamps. Cubical expansion from 32 to 212 F., 0.0027, less than that of any other metal. ex- cept the rare metals, and almost the same as glass. Silver (Argentum), Ag. At. wt. 107.7. Sp. gr. 10.1 to 11.1, according to condition and purity. It is the whitest of the metals, very malleable and ductile, and in hardness intermediate between gold and copper. Melts at about 1750 F. Specific heat .056. Cubical expansion from 32 to 212 F., 0.0058. As a conductor of electricity it is equal to copper. As a conductor of heat it is superior to all other metals. Tin (Stannum) Sn. At. wt. 118. Sp. gr. 7.293. White, lustrous, soft : malleable, of little strength, tenacity about 3500 Ibs. per square inch. Fuses at 442 F. Not sensibly volatile when melted at ordinary heats. Heat con- ductivity 14.5, electric conductivity 12.4; silver being 100 in each case. Expansion of volume by heat .0069 from 32 to 212 F. Specific heat .055. Its chief uses are for coating of sheet-iron (called tin plate) and for making alloys with copper and other metals. Zinc, Zn. At. wt. 65. Sp. gr. 7.14. Melts at 780 F. Volatilizes and burns in the air when melted, with bluish-white fumes of zinc oxide. It is ductile and malleable, but to a much less extent than copper, and its tenacity, about 5000 to 6000 Ibs. per square inch, is about one tenth that of wrought iron. It is practically non-corrosive in the atmosphere, a thin film of car- bonate of zinc forming upon it. Cubical expansion between 32 and 212 F., MEASURES AKD WEIGHTS OF VARIOUS MATERIALS. 169 0.0088. Specific heat .096. Electric conductivity 29, heat conductivity 36, silver being 100. Its principal uses are for coating iron surfaces, called " galvanizing," and for making brass and other alloys. Table Showing the Order of Malleability. Ductility. Tenacity. Infusitoility. Gold Silver Aluminum Copper Tin Lead Zinc Platinum Iron Platinum Silver Iron Aluminum Zinc Tin Lead Iron Copper Aluminum Platinum Silver Zinc Gold Tin Lead Platinum Iron Copper Gold Silver Aluminum Zinc Lead Tin WEIGHT OF RODS, BARS, PLATES, TUBES, AND SPHERES OF DIFFERENT MATERIALS. Notation : b = breadth, t = thickness, s = side of square, d = external Diameter, d l = internal diameter, all in inches. Sectional areas : of square bars s 2 ; of flat bars = bt\ of round rods ,7854d a ; of tubes = .7854(d a - d, a ) = 3.1410(d* - f 2 ). Volume of 1 foot in length :" of square bars = 12s 2 ; of flat bars = 126 ; of round bars = 9.4248d a ; ot tubes = 9.4248(<i 2 - d-ft = 37.699(<i ? 2 ), in cu. in. Weight per foot length = volume X weight per cubic inch of the material. Weight of a sphere = diam. 3 X .5236 X weight per cubic inch. Material. Cast iron Wrought Iron Steel Copper & Bronze I (copper and tin) f Lead Aluminum Glass Pine Wood, dry . . . 8.855 11.38 2.G7 2.62 0.481 7.218450. 7.7 480. 7.854489.6 552. 8.393523.2 709.6 166.5 163.4 30.0 37.5 40. 40. Sit W 83 $&! .4s' 46. 43.63.633s 2 59.1 4. 93s 2 13.91.16s 2 13.61.13s* 2.50.21s 2 3.46* 3.8336* 3.6336* 4.936* 1.166* 1.136* 0.216* ,2604 ,27791. 28331.02 .31951.15 30291.09 ,41061.48 3470 09630. 09450.34 0174 1-16 15-16 2. 454d 2 2.618d 2 2.670d 2 3. Olid 2 2.854d 2 3.870d 2 0.164d 2 .0091d Weight per cylindrical in., 1 in. long, = coefficient of d 2 in ninth col. -v- 12. For tubes use the coefficient of d 2 in ninth column, as for rods, and multiply it into (d 2 c?r); or multiply it by 4(dt l 2 ). For hollow spheres use the coefficient of d 3 in the last column and multiply it into (d 3 rfj 3 ). For hexagons multiply the weight of square bars by 0.866 (short diam. of hexagon == side of square). For octagons multiply by 0.8284. MEASURES AND WEIGHTS OF VARIOUS MATERIALS (APPROXIMATE). Brickworlt. Brickwork is estimated by the thousand, and for various thicknesses of wall runs as follows: 8*4-in. wall, or 1 brick in tbickness v 14 bricks per superficial feet. 12% ** ** " 1U " * 21 '* " " 17 ' 2 21 28 35 An ordinary brick measures about $4X4X2 inches, which is equal to 66 cubic inches, or 26.2 bricks to a cubic foot. The average weight is % Ibs. 170 MATERIALS. Fuel. A bushel of bituminous coal weighs 76 pounds and contains 2688 cubic inches = 1.554 cubic feet. 29 .47 bushels = 1 gross ton. A bushel of coke weighs 40 Ibs. (35 to 42 Ibs.). One acre of bituminous coal contains 1600 tons of 2240 Ibs. per foot of thickness of coal worked. 15 to 25 per cent must be deducted for waste in mining. 41 to 45 cubic feet bituminous coal when broken down = 1 ton, 2240 Ibs. 34 to 41 " " anthracite, prepared for market = 1 ton, 2240 Ibs. 123 " ' of charcoal .= 1 ton, 2240 Ibs. 70.9 ** " "coke = 1 ton, 2240 Ibs. 1 cubic foot of anthracite coal (see also page 625) = 55 to 66 Ibs. 1 *' "bituminous 4 * , = 50 to 55 Ibs. 1 " '* Cumberland coal. = 53 Ibs. 1 " " Cannel coal = 50.3 Ibs. 1 " " charcoal (hardwood) = 18.5 Ibs. 1 " " " (pine) =181bs. A bushel of charcoal. In 1881 the American Charcoal-Iron Work- ers' Association adopted for use in its official publications for the standard bushel of charcoal 2748 cubic inches, or 20 pounds. A ton of charcoal is to be taken at 2000 pounds. This figure of 20 pounds to the bushel was taken as a fair average of different bushels used throughout the country, and it has since been established by law in some States. Ores, Earths, etc. 13 cubic feet of ordinary gold or silver ore, in mine = 1 ton = 2000 Ibs. 20 " " " broken quartz = 1 ton = 2000 Ibs. 18 feet of gravel in bank =1 ton. 27 cubic feet of gravel when dry = 1 ton. 25 " *' "sand = 1 ton. 18 ' " earth in bank = 1 ton. 27 ** " ** " when dry = 1 ton. 17 " clay =lton. Cement. English Portland, sp. gr. 1.25 to 1.51, per bbl 400 to 430 Ibs. Rosendale, U. S., a struck bushel 62 to 70 Ibs. liime. A struck bushel 72 to 75 Ibs. Grain. A struck bushel of wheat = 60 Ibs.; of corn = 56 Ibs. : of oats = 30 Ibs. Salt. A struck bushel of salt, coarse, Syracuse, N. Y. = 56 Ibs. ; Turk's Island = 76 to 80 Ibs. Weight of Earth Filling. (From Howe's " Retaining Walls.") Average weight in Ibs. per cubic foot. Earth, common loam, loose 72 to 80 " shaken 82 to 92 4 * rammed moderately 90 to 100 Gravel 90 to 106 Sand 90tol06 Soft flowing mud 104 to 120 Sand, perfectly wet 118 to 129 COMMERCIAL SIZES OF IRON BARS. Flats. Width. Thickness. Width. Thickness. Width. Thickness. 5* WEIGHTS OF WROUGHT IRON BARS. 171 Rounds : H to \% inches, advancing by 16ths, and \% to 5 inches by 8ths. Squares : 5/16 to 1J4 inches, advancing by 16ths, and 1J4 to 3 inches by 8ths. Half rounds: 7/16, %, %, 11/16, %, 1, % 1^, % 1%, 2 inches. Hexagons : % to 1^ inches, advancing by 8ths. Ovals : y% X y, % X 5/16, % x %, Vs X 7/16 inch. Half ovals : ^ X & % X 5/32, % X 3/16, % X 7/32, 1^ X H, 1% X %, 1% X % inch. Round-edge flats : 1^ X J4 1% X %, 1% X % inch. Rands : }4 to \y% inches, advancing by 8ths, 7 to 16 B. W. gauge. 1J4 to 5 inches, advancing by 4ths, 7 to 16 gauge up to 3* inches, 4 to 14 gauge, 3J4 to 5 inches. WEIGHTS OF SQUARE AND ROUND RARS OF WROUGHT IRON IN POUNDS PER LINEAL FOOT. Iron weighing 480 Ibs. per cubic foot. For steel add 2 per cent. Thickness or Diameter in Inches. Weight of Square Bar One Foot Long. Weight of Round Bar One Foot Long. Thickness or Diameter in Inches. Weight of Square Bar One Foot Long. Weight of Round Bar One Foot Long. Thickness or Diameter in Inches. Weight of Square Bar One Foot Long. Weight of Round Bar One Foot Long. 11/16 24.08 18.91 % 96.30 75.64 1/16 .013 .010 M 25.21 19.80 7/16 98.55 77.40 M .052 .041 13/16 26.37 20.71 H 100.8 79.19 3/16 .117 .092 27.55 21.64 9/16 103.1 81.00 M .208 .164 15/16 28.76 22.59 % 105.5 82.83 5/16 .326 .256 3 30.00 23.56 11/16 107.8 84.69 % .469 .368 1/16 31.26 24.55 H 110.2 86.56 7/16 .638 .501 \& 32.55 25.57 13/16 112.6 88.45 .833 .654 3/16 33.87 26.60 115.1 90.36 9/16 1.055 .828 \A 35.21 27.65 15/16 117.5 92.29 Ys 1.302 1.023 5/16 36.58 28.73 6 120.0 94.25 11/16 1.576 1.237 37.97 29.82 125.1 98.22 82 1.875 1.473 7/16 39.39 30.94 ^4 130.2 102.3 13/16 2.201 1.728 40.83 32.07 3X 135.5 106.4 % 2.552 2.004 9/16 42.30 33.23 L 140.8 110.6 15/16 . 2.930 2.301 % 43.80 34.40 To 146.3 114.9 3.333 2.618 11/16 45.33 35.60 % 151.9 119.3 1/16 3.763 2.955 46.88 36.82 VR 157.6 123.7 4.219 3.313 13/16 48.45 38.05 7 163.3 128.3 3/16 4.701 3.692 50.05 39.31 169.2 132.9 /4 5.208 4.091 15/16 51.68 40.59 x4 175.2 137.6 5/16 5.742 4.510 53.33 41.89 s2 181.3 142.4 6.302 4.950 1/16 55.01 43.21 i^ 187.5 147.3 7/16 6.888 5.410 56.72 44.55 % 193.8 152.2 7.500 5.890 3/16 58.45 45.91 % 200.2 157.2 9/16 8.138 6.392 60.21 47.29 y& 206.7 162.4 9& 8.802 6.913 5/16 61.99 48.69 8 213.3 167.6 11/16 9.492 7.455 'n. 63.80 50.11 226.9 178.2 M 10.21 8.018 7/16 65.64 51.55 vh 240.8 189.2 13/16 10.95 8.601 y 67.50 53.01 3^ 255.2 200.4 H 11.72 9.204 9/16 69.39 54.50 9 270.0 212.1 15/16 12.51 9.828 71.30 56.00 285.2 224.0 2 13.33 10.47 11/16 73.24 57.52 /^ 300.8 236.3 1/16 14.18 11.14 M 75.21 59.07 *M 316.9 248.9 ^0 15.05 11.82 18/16 77.20 60.63 10 333.3 261.8 3/16 15.95 12.53 79.22 62.22 350.2 275.1 U 16.88 13.25 15/16 81.26 63.82 V% 367.5 288.6 5/16 17.83 14.00 5 83.33 65.45 % 385.2 302.5 sk 18.80 14.77 1/16 85.43 67.10 11 403.3 316.8 7/16 19.80 15.55 87.55 68.76 421.9 331.3 20.83 16.36 3/16 89.70 70.45 ^ 440.8 346.2 9/16 21.89 17.19 k 91.88 72.16 34 460.2 361.4 22.97 18.04 5/16 94.08 73.89 12 480. 377. 172 MATERIALS. g: * i If ' ~***^^^z2zsz2s^z%%z%%$%% ^ ^ioot~t-ooo>ooWw^^oorcoo^Hg|g3^o N tt h * N -33 g ^3 g P 1^ W * 1 * g b ^ " ' "rH^T^(^WC^COCCCOTj<T}<Tj<JOlO*n)ODOOt>J>b-OOOOQOOSOSO5O pa O 9S?^ S^, SSS ' TH " r H T ^iHCi(NC^NooeococOT^r^^T^inooo)oi>i>{><>ocQO A<I.S S 9c4i WEIGHTS OF FLAT WROUGHT IKON. 173 ft 8888 X*> If at 1* 174 MATERIALS. WEIGHT OF IRON AND STEEL SHEETS. Weights per Square Foot. (For weights by Decimal Gauge, see page 32.) Thickness by Birmingham Gauge. Thickness by American (Brown and Sharpe's) Gauge. No. of Gauge. Thick- ness in Inches. Iron. Steel. No. of Gauge. Thick- ness in Inches. Iron. Steel. 0000 .454 18.16 18.52 0000 .46 18.40 18.77 000 .425 17.00 17.34 000 .4096 16.38 16.71 00 .38 15.20 15.50 00 .3648 14.59 14.88 .34 13.60 13.87 .3249 13.00 13.26 1 .3 12.00 12.24 1 .2893 11.57 11.80 2 .284 11.36 11.59 2 .2576 10.30 10.51 3 .259 10.36 10.57 3 .2294 9.18 9.36 4 .238 9.52 9.71 4 .2043 8.17 8.34 5 .22 8.80 8.98 5 .1819 7.28 7.42 6 .203 8.12 8.28 6 .1620 6.48 6.61 7 .18 7.20 7.34 7 .1443 5.77 5.89 8 .165 6.60 6.73 8 .1285 5.14 5.24 9 .148 5.92 6.04 9 .1144 4.58 4.67 10 .134 5.36 5.47 10 .1019 4.08 4.16 11 .12 4.80 4.90 11 .0907 3.63 3.70 12 .109 4.36 4.45 12 .0808 3.23 3.30 13 .095 3.80 3.88 13 .0720 2.88 2.94 14 .083 3.32 3.39 14 .0641 2.56 2.62 15 .072 2.88 2.94 15 .0571 2.28 2.33 16 .065 2.60 2.65 16 .0508 2.03 2.07 17 .058 2.32 2.37 17 .0453 1.81 .85 18 .049 .96 2.00 18 .0403 1.61 .64 19 .042 .68 1.71 19 .0359 1.44 .46 20 .035 .40 1.43 20 .0320 1.28 .31 21 .032 .28 1.31 21 .0,285 1.14 .16 22 .028 .12 1.14 22 .0253 1.01 .03 23 .025 .00 1.02 23 .0226 .904 .922 24 .022 .88 .898 24 .0201 .804 .820 25 .02 .80 .816 25 .0179 .716 .730 26 .018 .72 .734 26 .0159 .636 .649 27 .016 .64 .653 27 .0142 .568 .579 28 .014 .56 .571 28 .0126 .504 .514 29 .013 .52 .530 29 .0113 .452 .461 30 .012 .48 .490 30 .0100 .400 .408 31 .01 .40 .408 31 .0089 .356 .363 32 .009 .36 .367 32 .0080 .320 .326 33 .008 .32 .326 33 .0071 .284 .290 34 .007 .28 .286 34 .0063 .252 .257 35 .005 .20 .204 35 .0056 .224 .228 Specific gravity . I ron. Steel. .7 7.854 4SQ fi fnnt . 480 " g P " " inch 2778 .2833 As there are many gauges in use differing from each other, and even the thicknesses of a certain specified gauge, as the Birmingham, are not assumed the same by all manufacturers, orders for sheets and wires should always state the weight per square foot, or the thickness in thousandths of an inch. WEIGHT OF PLATE IRON. 175 8 iQ^S^SSS^SSgfiSSSSSSSSiSS^^SSSfiSSS THf-HT-H-IT-.l-H-frHT-.^T-11-lT-I^Hl-H^ ^ _ < ^oopT-.(?'roincot-aooT-iOJcoincoi>ocoinQOOecinQOOccnooocoin JL r o 09 p " OO Tf -<3< "tf ^ *?" : o =5 ? 05 ^' id co ^j 3; i^ p ? * o oo r ^ tr ??' os d o co" ?i op r? fa J, ^'ini^ooooc^rtiios<Nnt-ocoinooT^oc>?t^(>?QOcoosTf<oinoOTHt-.oi g 5 TOoaw^TjiTTTrininino^ooi-^i-i-ooaooosoo^^MTOcc^Tfinino 5"S 8inpoS8Spo8S8SpSpS8p888ooppooopppp I! KQ, ^2 oi-occ^ooiTJinQO^Tft'-o^inooi i-^fi coosinocooJoo^ocowcooo f 53 bM * 5_ gfe ^3 JL 2?S:Sffa?o r? co t Qt r c5^coinoQQ gl J . Djf g* * * 2 S ^ H ^ ^^ojotof^wwoiTOWOTWWccro^^^^^ooo^'oofic^^^Qddooo !l H* 1 few ll gl *8 2 H^ ^ ^SSS^SJSi 176 MATERIALS. WEIGHTS OF STK12L BLOOMS. Soft steel. 1 cub ; c inch = 0.284 Ib. 1 cubic foot = 490.75 Ibs. Sizes. Lengths. 1" 6" 12" 18" 24" 30" 36" 42" 48" 54" 60" 66" 12" x 4" 13.63 82 164 245 327 409 491 573 654 736 818 900 11 x 6 18.75 113 225 338 450 563 675 788 900 1013 1125 1238 x 5 15.62 94 188 281 375 469 562 656 750 843 937 1031 x 4 12.50 75 150 225 300 375 450 525 600 675 750 825 10 x 7 19.88 120 239 358 477 596 715 835 955 1074 1193 1312 x 6 17.04 102 204 307 409 511 613 716 818 920 1022 1125 x 5 14.20 85 170 256 341 426 511 596 682 767 852 937 x 4 11.36 68 136 205 273 341 409 477 546 614 682 750 x 3 8.52 51 102 153 204 255 306 358 409 460 511 562 9 x 7 17.89 107 215 322 430 537 644 751 859 966 1073 1181 x 6 15.34 92 184 276 368 460 552 644 736 828 920 1012 x 5 12.78 77 153 230 307 383 460 537 614 690 767 844 x 4 10.22 61 123 .184 245 307 368 429 490 552 613 674 8 x 8 18.18 109 218 327 436 545 655 764 873 982 1091 1200 x 7 15.9 95 191 286 382 477 572 668 763 859 954 1049 x 6 13.63 82 164 245 327 409 491 573 654 736 818 900 x 5 11.36 68 136 205 273 341 409 477 546 614 682 750 x 4 9.09 55 109 164 218 273 327 382 436 491 545 600 7 x 7 13.92 83 167 251 334 418 501 585 668 752 835 919 x 6 11.93 72 143 215 286 358 430 501 573 644 716 ,788 x 5 9.94 60 119 179 238 298 358 417 477 536 596 656 x 4 7.95 48 96 143 191 239 286 334 382 429 477 525 x 3 5.96 36 72 107 143 179 214 250 286 322 358 393 6^x W/ 2 12. 72 144 216 388 360 432 504 576 648 720 792 x 4 7.38 44 89 133 177 221 266 310 354 399 443 487 6 x 6 10.22 61 123 184 245 307 368 429 490 551 613 674 x 5 8.52 51 102 153 204 255 307 358 409 460 511 562 x 4 6.82 41 82 123 164 204 245 286 327 368 409 450 x 3 5.11 31 61 92 123 153 184 214 245 276 307 337 5^x sy 2 8.59 52 103 155 206 258 309 361 412 464 515 567 x 4 6.25 37 75 112 150 188 225 262 300 337 375 412 5 x 5 7.10 43 85 128 170 213 256 298 341 383 426 469 x 4 5.68 34 68 102 136 170 205 239 273 307 341 375 4^ x 4^ 5.75 35 69 104 138 173 207 242 276 311 345 380 x 4 5.11 31 61 92 123 153 184 215 246 276 307 338 4 x 4 4.54 27 55 80 109 136 164 191 218 246 272 300 x 3}4 3.97 24 48 72 96 119 143 167 181 215 238 262 x 3 3.40 20 41 61 82 102 122 143 163 184 204 224 3^ x*3J^ 3.48 21 42 63 84 104 125 146 167 188 209 230 x 3 2.98 18 36 54 72 89 107 125 14-3 161 179 197 3 x 3 2.56 15 31 46 61 77 92 108 123 138 154 169 SIZES AKD WEIGHTS OF STRUCTURAL SHAPES. 177 SIZES AND WEIGHTS OF STRUCTURAL SHAPES. Minimum, Maximum, and Intermediate Weights and Dimensions of Carnegie Steel I-Beams. Sec- tion Index Depth of Beam. Weight per Foot, Flange Width- Web Thick- ness. Sec- tion Index Depth of Beam. Weight pei- Foot. Flange Width. Web Thick- ness. ins. Ibs. ins. ins. ins. Ibs. ins. ins. Bl 24 100 7.25 0.75 B19 6 17.25 3.58 0.48 44 44 95 7.19 0.69 44 44 14.75 3.45 0.35 44 < 4 90 7.13 0.03 44 44 12.25 3.33 0.23 44 44 85 7.07 0.57 B21 5 14.75 3.29 0.50 44 44 8 9 7.00 0.50 44 44 12.25 3.15 0.36 B3 20 75 6.40 0.65 44 44 9.75 3.00 0.21 70 6.33 0.58 B23 4 1015 2.88 0.41 4k 44 65 6.25 0.50 44 44 9.5 2.81 0.34 B80 18 70 6.26 0.72 44 44 8:5 2.73 26 44 44 65 6.18 0.64 44 44 7.5 2.66 0.19 44 44 60 6.10 0.56 B77 3 7.5 2.52 0.36 . it 44 55 6.00 0.46 44 44 6.5 2.42 0.26 B7 15 55 5.75 0.06 44 44 5,5 2 33 0.17 44 50 5.65 0.56 B2 20 100 ^.28 0.88 44 44 45 5.55 0.46 44 44 95 *" 21 0.81 44 u 42 5.50 0.41 44 44 90 ".U 0.74 B9 12 35 5.09 0.44 44 44 85 *"' 06 0.66 44 31.5 5.00 0.35 44 44 80 ".00 0.60 Bll 10 40 5.10 0.75 B4 15 100 6.77 1.18 35 4.95 0.60 44 44 95 6.68 1.09 44 44 30 4.81 0.46 44 44 90 6.58 0.99 44 * 4 20 4.66 0.31 44 44 85 6.48 0.89 B13 9 35 4.77 0.73 44 44 80 6.40 0.81 44 44 30 4.61 0.57 B5 15 75 6.29 0.88 44 44 25 4.45 0.41 44 70 6.19 0.78 44 44 21 4.33 0.29 44 44 65 6.10 0.69 B15 8 25.5 4.27 0.54 4k 44 60 6.00 0.59 44 44 23 4.18 45 B8 12 55 5.61 0.82 44 44 20.5 4.09 0.36 44 44 50 5.49 0.70 44 44 18 4.00 27 44 44 45 5.37 0.58 B17 7 20 3.87 0.46 * 4 44 40 5.25 0.46 tt H 17.5 3.76 3.66 0.35 0.25 Sections B2, B4, B5, and B8 are " special ^ beams, the others are "standard." Sectional area = weight in Ibs. per ft. -*- 3.4, or X 0.2941. Weight in Ibs. per foot = sectional area X 3.4. Maximum and Minimum Weights and Dimensions of Carnegie Steel Deck Beams. Section Depth of Weight per Foot, Ibs. Flange Width. Web Thickness. Increase of Web and Flange per * , ' Ib. increase Min. Max. Min. Max. Min. Max. of Weight. B100 10 27.23 35.70 5.25 5.50 .38 .63 .029 B101 9 26.00 30.00 4.91 5.07 .44 57 .033 BIOS 8 20.15 24.48 5.00 5.16 31 .47 .037 B103 7 18.11 23.46 4.87 5.10 31 .54 .042 BIOS 6 15.30 18.36 4.38 4.53 .28 .43 .049 178 MATERIALS. Minimum, Maximum, and Intermediate Weights and Dimensions of Carnegie Standard Channels. 1 I ^"3 ! !s p t-H ^ ' froS _c ^ ! j a sd I s " s g l .SPo *& 0>S'J il 5 Web Th ness. Inchei Section dex. o.;R a & 4J O '-^ . ^^ A WebTh ness. Inche! 01 15 55 3.82 0.82 05 8 16.25 2.44 0.40 44 50 3.72 0.72 44 44 13.75 2.35 0.31 4 * 44 45 3.62 0.62 44 44 11.25 2.26 0.22 44 44 40 3.52 0.52 06 7 19.75 2^51 0.63 44 44 35 3.43 0.43 44 44 17.25 2.41 0.53 44 44 33 3.40 0.40 44 44 14.75 2.30 0.42 02 12 40 3.42 0.76 44 44 12.25 2.20 0.32 4 - 44 35 3.30 0.64 44 i4 9.75 2.09 0.21 44 11 30 3.17 0.51 07 6 15.50 2.28 0.56 4 ! 44 25 20.5 3.05 2.94 0.39 0.28 u ,4 13 10.50 2.16 2.04 0.44 0.32 03 10 35 3.18 0.82 44 8 1.92 0.20 44 44 30 3.04 0.68 08 5 11.50 2.04 0.48 44 44 25 2.89 0.53 44 ** 9 1.89 0.33 41 44 20 2.74 0.38 * 4 44 6.50 1.75 0.19 ** * 15 2.60 0.24 09 4 7 . 25 1.73 0.33 04 9 25 2.82 0.62 44 44 6^25 1.65 0.25 44 20 2.65 0.45 44 44 5.25 1.58 0.18 44 44 15 2.49 0.29 072 3 6 1.60 0.36 ! 11 13.25 2.43 0.23 * 44 5 1.50 0.26 05 8 21.25 2.62 0.58 M 44 4 1.41 0.17 ' " 18.75 2.53 0.49 Weight* and Dimensions of Carnegie Steel Z-Bars. Size. Size. 02 "3 a Section Index. $ -2 i! 2 o H Flanges. 43 4> il vft P Section Index. Thicknes of Meti Flanges. 1 Weight. Pounds Zl % 3 K 6 15.6 Z6 H 3 5/16 5 1/16 26.0 * 7{1Q 39/16 6 1/16 18.3 44 13/16 3 % 5 X 28.3 " & 3 % 6 Ys 21.0 Z7 Y4 3 1/16 4 8.2 Z2 9/16 3 i^ 6 22.7 44 5/16 3 y& 4 1/16 10.3 N T 9/16 6 1/16 25.4 44 % 3 3/16 4 K 12.4 * 11/16 3 % 6 Ys 28.0 Z8 7/16 3 1/16 4 13.8 Z3 13/16 3 fc 3 9/1 G 6 6 1/16 29.3 32.0 ;; $, 3 K 3 3/16 4 1/16 4 K 15.8 17.9 * l % 3 % 6 Ys 34.6 Z9 % 3 1/16 4 18.9 Z4 5/16 3 H 5 11.6 44 11/16 3 X 4 1/16 20.9 * % 3 5/16 5 1/16 13.9 14 % 3 3/16 4 J4 22.9 M 7/16 3 % 5 X 16.4 Z10 l /4 2 11/16 3 6.7 Z5 K 3 M 5 17.8 " 5/16 2 % 3 1/16 8.4 " 9/16 3 5/16 5 1/16 20.2 Zll % 2 11/18 3 9.7 " % 3 % 5 y s 22.6 4t 7/16 2 % 3 1/16 11.4 Z6 11/16 3 J4 5 23.7 Z12 Vk 2 11/16 a 12.5 ' 9/16 2 M 3 1/16 14.2 SIZES AND WEIGHTS OF STRUCTURAL SHAPES. 179 Pencoyd Steel Angles. EVEN LEGS. Size in Inches. Approximate Weight in Pounds per Foot for Various Thicknesses in Inches. .125 3/16 .1875 H .25 5/16 .3125 3 4 7/16 .4375 & 9/16 .5625 % .625 11/16 .6875 3 A .75 13/16 .8125 .75 15/16 .9375 1 1.00 8x8 26.4 29.8 33.2 36.6 39.0 42.4 45.8 49.3 52.8 6x6 14.8 17.3 19.7 22.0 24.4 26.5 28.8 31.0 33.4 35.9 5 x5 12.3 14.3 16.3 18.2 20.1 22.0 23.8 25.6 27.4 29.4 4 x4 8.2 9.8 11.3 12.8 14.5 15.8 17.2 18.6 31*2 x 3Jrfjj 7.1 8.5 9.8 11.1 12.4 13.7 3 x3 4.0 6.1 7.2 8.3 9.4 10.4 11.5 2% x 2f> 4.5 5.5 6.6 7 7 8.6 gi^j x 2^ 3.1 4.1 5.0 5.9 6.9 7.8 2^x2^4 2.7 3.6 4.5 5.4 2 x2 2.5 3.2 4.0 4.8 1% x 1% 2.1 2.8 3.5 4.1 1V6 x ll/ 1.2 1.8 2.4 2.9 3.5 1)4 x 1/4 1.0 1.5 2.0 1 x 1 0.8 1.2 1.5 UNEVEN LEGS. Size in Approximate Weight in Pounds per Foot for Various Thicknesses in Inches. Inches. H i/^ 5/16 % 7/16 i^ 9/16 % 11/16 $4 13/16 % 15/16 1 .186 .1875 .25 .3125 .375 .4375 .50 .5625 .625 .6875 .75 .8125 .875 .9375 1.00 8 x6 23.0 25.8 28.7 31.7 33.8 36.6 39.5 42.5 45.6 17.0 19.0 21.0 23.0 24.8 26.7 28.6 30.5 32.5 6J^x4 12.9 15.0 17.0 19.0 21.2 23.4 25.6 27.8 29.8 31.9 6 x4 12.2 14.3 16.3 18.1 20.1 22.0 23.8 25.6 27.4 29.4 6 x3J4 11.6 13.6 15.5 17.1 19.0 20.8 22.6 24.5 26.5 28.6 5L x 314 11.0 12.8 14.6 16.2 17.9 5 x4 11.0 12.8 14.6 16.2 17.9 19.6 21.3 5 x3^ 8.7 10.3 12.0 13.6 15.2 16.8 18.4 20.0 5x3" 8.2 9.7 11.2 12.8 14.2 15.7 17.2 18.7 4^x3 7.7 9.1 10.5 11.9 13.3 14.7 16.0 17.4 4 x3^ 7.7 9.1 10.5 11.9 13.3 14.7 16.0 17.4 4 x3 " 7.1 8.5 9.8 11.1 12.4 13.8 3^x3 6.6 7.8 9.1 10.3 11.6 12.9 4 9 6.1 7.2 8.3 9.4 gi/ x 2 I* 5.5 6.6 Q 21^C 4.' J 5.5 6.6 7.7 8.7 3 x2 4.1 5.0 5.9 6.9 7.9 2^x2 2.7 3. 4.5 5.4 6.2 7.0 2.3 3.7 4.4 2 xl}4 2.1 > 9 3.6 4.3 2 xl}4 1.9 2.'6 3.3 3.9 ANGLE-COVERS. Siz in Inches. 3/16 y* 5/16 % 7/16 X 9/16 % 3 x3 4.8 5.9 7.1 8.2 9.3 10.4 11.5 2^x2% 2^x2^ 2*4 x 2J4 3.0 2.6 4.4 4.0 3.5 5.5 5.0 4.4 6.6 6.0 5.3 7.7 7.0 8.8 8.1 2 x2 2.4 3.2 4.0 4.8 180 MATERIALS. SQUARE-ROOT ANGLES. Size in Inches. Approximate Weight in Pounds per Foot for Various Thicknesses in Inches. Size in Inches. Approximate Weight in Pounds per Foot for Various Thicknesses in Inches. I 5/16 .3125 .375 7/16 .4375 ^ .50 9/16 .5625 % .625 Ys .125 3/16 .1875 1 5/16 .3125 % .375 4 x4 3^x3^ 3 x3 2%x2% 2^x2^ 8*4 x2fc 4.9 4.5 4.1 3.6 7.1 6.1 5.6 5.1 4.5 9.8 8.5 7.2 6.7 6.1 5.4 11.4 9.9 8.3 7.8 7.1 13.0 11.4 9.4 8.9 8.2 14.6 16.2 2 x2 l%xl% l^xl^ iMxi^ 1 xl 0.82 1.80 1.53 1.16 3.3 2.9 2.4 2.04 1.53 4.1 3.6 3.0 2.55 4.9 4.4 Pencoyd Tees. Section Number. Size in Inches. Weight per Foot. Section Number. Size in Indies. Weight per Foot. EVEN TEES. UNEVEN TEES. 440T 441T 4 x4 4 x4 10.9 13.7 43T 4 x3 9.0 335T 3^3 x 3Jx> 7.0 44T 4 x3 10.2 336T 3J^x3>J 9.0 45T 4 x4^ 13.5 337T 3>Jx3^ 11.0 38T 31^x3 7.0 330T 3 x3 6.5 39T 3^x3 8.5 33 IT 3 x3 7.7 SOT 3 xlU 4.0 225T 2^x2^ 5.0 31T 3 x2J4 5.0 226T 5.8 32T 3 x2^ 6.0 227T 2L x ;;>ij? 6.6 33T 3 x2^J 7.0 222T 2^4 x 214 4.0 34T 3 x2U 8.0 223T 2J4 x 2*4 4.0 35T 3 x3i| 8.3 220T 2 x2 3.5 36T 3 X3U 9.5 117T l%xl% 2.4 28T 2%x 1% 6.6 115T l^xl^ 2.0 29T OHX x 2 7.2 112T 1*4*1*4 1.5 25T 2^x114 3.3 HOT 1 xl 1.0 ' 26T 2^x2% 5.7 27T 2^x3 6.0 24T 2J4x 9/16 2.2 UNEVEN TEES. 20T 2 9 T 2 x 9/16 2 xl 1/16 2.0 2.0 21T 2 xl 2.5 64T 65T 6x4 6x5J4 17.4 39.0 23T 17T 2 xl^ l%xl 1/16 3.0 1.9 53T 5x3^ 17.0 1ST l^xl^j 3.5 54T 5x4 15.3 15T \Y 2 x 15/16 1.4 42T 4x2 6.5 12T 1^4 x 15/16 1.2 Pencoyd Miscellaneous Shapes. Section Number. Section. Size in Inches. Weight per Foot in Pounds. 217M 210M 260M Heavy rails. Floor-bars. 3 1/16x4x3 6 i/i6x*4 to y 2 50.0 7.1 to 14.3 9.8 to 14.7 SIZES AND WEIGHTS OF KOOFIHG MATERIALS. 181 SIZES AND WEIGHTS OF HOOFING MATERIALS. Corrugated Iron. (The Cincinnati Corrugating Co.) SCHEDULE OF WEIGHTS. . aJ 02 to "1 Thickness in decimal parts of an inch. Flat. Weight per 100 sq. ft. Flat, Pain ted. Weight per 100 sq. ft. Corrugated and Painted. Weight per 100 sq. ft. Corrugated and Galvanized; Weight in oz. per sq. ft. Flat, Galvan- ized. No. 28 No. 26 No. 24 No. 22 No. 20 No. 18 No. 16 .015625 .01875 .025 .03125 .0375 .05 .0625 62^ Ib 75 100 125 150 200 250 s. 70 Ib 84 111 138 165 220 275 s. 86 Ib 99 127 154 182 286 291 s. 12^0 14J4 isU 22^3 26^ 342 42^ z. The above table is on the basis of sheets rolled according to the U. S. Standard Sheet-metal Gauge of 1893 (see page 31). It is also on the basis of ' " K ' in. corrugations. To estimate the weight per 100 sq. ft. on the roof when lapped one corru- gation at sides and 4 in. at ends, add approximately 12^$ to the weights per 100 sq. ft., respectively, given above. Corrugations 2^ in. wide by ^ or % in. deep are recognized generally as the standard size for both roofing and siding; sheets are manufactured usually in lengths 6, 7, 8, 9, and 10 ft., and have a width of 26^ or 26 in. out- side width ten corrugations, and will cover 2 ft. when lapped one corruga- tion at sides. Ordinary corrugated sheets should have a lap of 1^6 or 2 corrugations side- lap for roofing in order to secure water-tight side seams; if the roof is rather steep 1^ corrugations will answer. Some manufacturers make a special high-edge corrugation on sides of sheets (The Cincinnati Corrugating Co.), and thereby are enabled to secure a water-proof side-lap with one corrugation only, thus saving from 6$ to 12% of material to cover a given area. The usual width of flat sheets used for making the above corrugated material is 28J4 inches. No. 28 gauge corrugated iron is generally used for applying to wooden buildings; but for applying to iron framework No. 24 gauge or heavier should be adopted. Few manufacturers are prepared to corrugate heavier than No. 20 gauge, but some have facilities for corrugating as heavy as No. 12 gauge. Ten feet is the limit in length of corrugated sheets. Galvanizing sheet iron adds about 2% oz. to its weight per square foot. Corrugated Arches. For corrugated curved sheets for floor and ceiling construction in fire- proof buildings, No. 16, 18, or 20 gauge iron is commonly used, and sheets may be curved from 4 to 10 in. rise the higher the rise the stronger the arch. By a series of tests it has been demonstrated that corrugated arches give the most satisfactory results with a base length not exceeding 6 ft., and ff ft. or even less is preferable where great strength is required. These corrugated arches are usually made with 2^ X % i". corrugations, and in same width of sheet as above mentioned. Terra-Cotta. Porous terra-cotta roofing 3" thick weighs 16 Ibs, per square foot and 2" thick, 12 Ibs. per square foot. Ceiling made of the same material 2" thick weighs 11 Ibs. per square foot. Tiles. Flat tiles 6M" X 10J4" X % ff weigh from 1480 to 1850 Ibs. per square of roof (100 square feet), the lap being one-half the length of the tile. Tiles with grooves and fillets weigh from 740 to 925 Ibs. per square of roof. Pantiles 1%" X 10^ /x laid 10" to the weather weigh 850 Ibs. per square. 182 MATERIALS. Tin Plate Tinned Sheet Steel. The usual sizes for roofing tin are 14" X 20" and 20" X 28". Without allowance for lap or waste, tin roofing weighs from 50 to 62 Ibs. per square. Tin on the roof weighs from 62 to 75 Ibs. per square. Roofing plates or terne plates (steeJ plates coated with an alloy of tin and lead) are made only in 1C and IX thicknesses (29 and 27 Birmingham gauge). "Coke" and "charcoal' 1 tin plates, old names used when iron made with coke and charcoal was used for the tinned plate, are still used in the trade, although steel plates have been substituted for iron; a coke plate now commonly meaning one made of Bessemer steel, and a charcoal plate one of open-hearth steel. The thickness of the tin coating on the plates varies with different " brands. 1 ' For valuable information on Tin Roofing, see circulars of Merchant & Co., Philadelphia. The thickness and weight of tin plates were formerly designated in the trade, both in the United States and England, by letters, such as I.C., D.C., I.X., D.X., etc. A new system was introduced in the United States in 1898, known as the " American base-box system." The base-box is a package containing 32,000 square inches of plate. The actual boxes used in the trade contain 60, 120, or 240 sheets, according to the size. The number of square inches in any given box divided by 32,000 is known as the " box ratio." This ratio multiplied by the weight or price of the base-box gives the weight or price of the given box. Thus the ratio of a box of 120 sheets 14 X 20 in. is 33,600 -*- 32,000 = 1.05, and the price at $3.00 base is $3.00 X 1.05 = $3.15. The following tables are furnished by the American Tin Plate Co., Chicago, 111. Comparison of Gauges and Weights of Tin Plates. (Based on U. S. standard Sheet-metal Gauge.) ENGLISH BASE-BOX. (31,360 sq. in.) Gauge. Weight. No. 38. 00 54. 44 Ibs. 37.00 57.84 36.00 61.24 35.00 68.05 34.00... 74.85 33.24 ... 80.00 32.50 85/00 31.77 90.00 31.04 95.00 30.65 100.00 30.06 108.00 28.74 126.00 28.00 136.00 26.46 157.00 25.46 178.00 24.68 199.00 23.91 220.00 23.14 241.00 22.37 262.00 21.60 283.00 27.86 139.00 25.38 180.00 24.24 211.00 23.12 212.00 22.00 273.00 Weig 55 Ib 60 ' 65 * 70 ' 75 ' 80 4 85 ' 90 * 95 ' 100 ' 110 ' 130 ' 140 ' 160 ' 180 ' 200 ' 220 ' 240 l 260 l 280 ' 140 ' 180 ' 220 ' 240 ' 280 ' AMERICAN BASE-BO (32,000 sq. in.) ht. ( s M X. 3auge. o. 38.00 k 36.72 ' 35.64 1 34.92 ' 34.20 ' 33.48 ' 32 76 * 32.04 ' 31.32 ' 30.80 ' 30.08 ' 28.64 * 27.9^ 1 26.48 ' 25.5-) ' 24.85 1 24.0 8 ' 23.3 6 ' 22.64 ' 21.9a ' 27.9-2 * 25.52 ' 24.08 1 23.36 1 21.92 I.C.L. I.C. IX.L. IX. I.2X. I.3X. I.4X. I. 5X. I.6X. I.7X. I. 8X. D.C. D.X. D. 2X. D. 3X. D. 4X. American Packages Tin Plate. Inches Wide. Length. Sheets per Box Inches Wide. Length. Sheets per Box 9 to 16% 17 * 25% 26 ' 30 9 ' 10% 11 1 11% 12 ' 12% Square. Square. Square. All lengths. To 18 in. long, incl. 18J4 and longer. To 17 in. long, incl. 240 120 60 240 240 120 240 13 " 13% 13 to 13% 14 " 14% 14 " 14% 15 " 25% 26 " 30 17*4 and longer. To 16 in. long, incl. 16*4 and longer. To 15 in. long, incl. 15J4 and longer. All lengths. All lengths. 120 240 120 240 120 120 60 Small sizes of light base weights will be packed in double Tboxes. SIZES AND WEIGHTS OF ROOFING MATERIALS* 183 Slate. Number and superficial area of slate required for one square of roof. (1 square = 100 square feet.) Dimensions Number Superficial Dimensions Number Superficial in per Area in in per Area in Inches. Square. Sq. Ft. Inches. Square. Sq. Ft. 6x12 533 267 12x18 160 240 7x12 457 10x20 169 235 8x12 400 11 x20 !54 9x12 355 12x20 141 7x14 374 254 14x20 121 8x14 327 16x20 137 9x14 291 12x22 126 231 10x14 261 14x22 108 8x16 277 246 12x24 114 228 9x16 246 14x24 98 10x16 221 16 x 24 86 9x18 213 240 14 x 26 89 225 10x18 192 16x26 78 As slate is usually laid, the number of square feet of roof covered by one slate can be obtained from the following formula : width x (length 3 inches) e number of square feet of roof covered. s and thicknesses required for one square Weight of slate of various length of roof : Length in Inches. Weight in Pounds per Square for the Thickness. w M" V M % M % 12 14 16 18 20 22 24 26 483 460 445 434 425 418 412 407 724 688 667 650 637 626 617 610 967 920 890 869 851 836 825 815 1450 1379 1336 1303 1276 1254 1238 1222 1936 1842 1784 1740 1704 1675 1653 1631 2419 2301 2229 2174 2129 2093 2066 2039 2902 2760 2670 2607 2553 2508 2478 2445 3872 3683 3567 3480 3408 3350 3306 3263 The weights given above are based on the number of slate required for one square of roof, taking the weight of a cubic foot of slate at 175 pounds. Pine Shingles. Number and weight of pine shingles required to cover one square of roof : Number of Number of Weight in Inches Exposed to Weather. Shingles per Square of Roof. Pounds of Shingle on One-square Remarks. of Roofs. 4 900 216 The number of shingles per square is 4}x> 800 192 for common gable-roofs. For hip ly^ 720 655 -173 157 roofs add five per cent, to these figures. The weights per square are based on 6 600 144 the number per square. 184 MATERIALS. Skylight Glass. The weights of various sizes and thicknesses of fluted or rough plate-glass required for one square of roof. Dimensions in Inches. Thickness in Inches. Area in Square Feet. Weight in Lbs. per Square of Roof. 12x48 15x60 20x100 94x156 3.997 6.246 13.880 101.768 250 350 500 700 In the above table no allowance is made for lap. If ordinary window-glass is used, single thick glass (about 1-16") will weigli about 82 Ibs. per square, and double thick glass (about %") will weigh about 164 Ibs. per square, no allowance being made for lap. A box of ordinary window-glass contains as nearly 50 square feet as the size of the panes will admit of. Panes of any size are made to order by the manufacturers, but a great variety of sizes are usually kept in stock, ranging from 6x8 inches to 36 x 60 inches. APPROXIMATE WEIGHTS OF VARIOUS ROOF- COVERINGS. For preliminary estimates the weights of various roof coverings maybe taken as tabulated below (a square of roof = 10 ft. square = 100 sq. ft.); Name. Weight in Lbs. per Square of Roof. Cast-iron plates (%" thick) 1500 Copper 80-125 Felt and asphalt 100 Felt and gravel 800-1000 Iron, corrugated 100-375 Iron, galvanized, flat 100- 350 Lath and plaster 900-1000 Sheathing, pine, 1" thick yellow, northern .. 300 " southeru.. 400 Spruce, 1" thick 200 Sheathing, chestnut or maple, V thick 400 " ash, hickory, or oak, 1" thick.... 500 Sheet iron (1-16" thick) 300 " and laths 500 Shingles, pine 200 Slates W thick) 900 , Skylights (glass 3--16" to J" thick) . . .. 250- 700 Sheet lead 500- 800 Thatch ; 650 Tin 70-125 Tiles, flat 1500-2000 (grooves and fillets) 700-1000 pan 1000 " with mortar 2000-3000 Zinc ..... 100-200 Approximate Loads per Square Foot for Roofs of Span.* under 75 Feet, Including Weight of Truss. (Carnegie Steel Co.) Roof covered with corrugated sheets, unboarded 8 Ibs. Roof covered with corrugated sheets, on boards. 11 Roof covered with slate, on laths 13 Same, on boards, 1*4 in. thick 16 Roof covered with shingles, on laths : 10 Add to above if plastered below rafters 10 Snow, light, weighs per cubic foot . ... 5 to 12 For spans over 75 feet add 4 Ibs. to the above loads per square foot. It is customary to add 30 Ibs. per square foot to the above for gnow and when separate calculations are not made, WEIGHT OF CAST-IRON PIPES OR COLUMKS. 185 WEIGHT OF CAST-IRON PIPES OR COLUMNS. In L.bs. per Lineal Foot. Cast iron = 450 Ibs. per cubic foot. Bo*\i. Thick, of Metal. Weight per Foot. Bore. Thick, of Metal. Weight per Foot. Bore. Thick, of Metal. Weight per Foot. Ins. Ins. Lbs. Ins. Ins. Lbs. Ins. Ins. Lbs. 3 % 12.4 10 % 79.2 22 94 167.5 /^| 17.2 10}r<J i^ 54.0 % 196.5 % 22.2 ' % 68 2 23 94 174.9 3^2 % 14.3 M 82.8- % 205.1 v& 19.6 11 x^ 56.5 l 235.6 % 25.3 % 71.3 24 94 182.2 4 % 16.1 % 86.5 213.7 ^ 22.1 \\}/f> L<2 58.9 1 8 245.4 % 28.4 7& 74.4 25 94 189.6 4^3 % 17.9 g 90.2 % 222.3 1< 24.5 12 61.3 l 255.3 % 31.5 E^ 77.5 26 94 197.0 5 % 19.8 ^4 93.9 % 230.9 L 27.0 JO1Z /^ 63.8 l 265.1 2 34.4 % 80.5 27 204.3 5}^> ^ 21.6 % 97.6 v4 239.4 l2 29.4 13 /^ 66 3 1 274.9 76 37.6 % 83.6 28 94 211.7 6 % 23.5 94 101.2 % 248.1 1 " 31.8 14 /^ 71.2 l 284.7 6^ 40.7 % 89.7 29 94 219.1 gi^ 7& 25.3 108.6 256.6 LJJJ 34.4 15 % 95.9 i 8 294.5 Y8 43.7 4 116.0 30 265.2 7 % 27.1 % 136.4 i 8 304.3 1^3 36.8 16 % 102.0 343 7 % 46.8 4 123.3 31 % 273.8 71^ % 29.0 % 145.0 i 314.2 LJ 89.3 17 % 108.2 \\/. 354.8 % 49.9 94 130.7 32 % 282.4 8 % 30.8 % 153.6 l 324.0 Hi 41.7 18 % 114.3 i/"6 365.8 % 52.9 94 138.1 33 % 291.0 8J^ ^ 44.2 % 162.1 l 333.8 % 56.0 19 % 120.4 i/^ 376.9 94 68.1 94 145.4 34 % 299.6 9 M 46.6 % 170.7 i 343.7 % 59.1 20 % 126.6 ji^j 388.0 94 71.8 94 152.8 35 % 308.1 9^ ^ 49.1 % 179.3 l 353.4 % 62.1 21 % 132.7 \\ 399.0 M 75.5 94 160.1 36 % 316 6 10 /^ 51.5 % 187.9 1 363.1 % 65.2 22 % 138.8 ^ 410.0 The weight of the two flanges may be reckoned = weight of one foot- 186 MATERIALS. WEIGHTS OF CAST-IRON PIPE TO LAY 12 FEET LENGTH. Weights are Gross Weights, including Hull. (Calculated by F. H. Lewis.) Thickness. Inside Diameter. Inches. Equiv. Decimals. 4 // 6" 8" 10" 12" 14" 16" 18" 20" 1640 1810 1980 2152 2324 2498 2672 3024 saso 3739 ,&, 7-16 15-32 17-32 9-16 19-32 11--16 & % 15-16 1 1J| .375 .40625 .4375 .4687 .5 .53125 .5625 .59375 .625 .6875 .75 .8125 .875 .9375 1. 1.125 1.25 1.375 209 228 247 266 286 306 327 304 331 358 386 414 442 470 498 400 435 470 505 541 577 613 649 686 581 624 668 712 756 801 845 935 1026 692 744 795 846 899 951 1003 1110 1216 1324 1432 804 863 922 983 1043 1103 1163 1285 1408 1531 1656 1783 1909 1050 1118 1186 1254 1322 1460 1598 1738 1879 2021 2163 1177 1253 1329 1405 1481 1635 1789 1945 2101 2259 2418 2738 3062 3389 Thickness. Inside Diameter. Inches. Equiv. Decimals. 22" 24" 27" 30" 33" 36" 42" 48" 60" 9742 10740 11738 12744 13750 14763 15776 17821 19880 21956 11-16 H 13-16 J M I 1^4 f .625 .6875 '.8125 .875 .9375 '125 .25 .375 .5 .625 .75 .875 2. 2.25 2.5 2.75 1799 1985 2171 2359 2547 2737 2927 3310 3698 2160 2362 2565 2769 2975 3180 3598 4016 4439 2422 2648 2875 3103 3332 3562 4027 4492 4964 5439 2934 3186 3437 3690 3942 4456 4970 5491 6012 6539 3221 3496 3771 4048 4325 4886 5447 6015 6584 7159 7737 3507 3806 4105 4406 4708 5316 5924 6540 7158 7782 8405 4426 4773 5122 5472 6176 6880 7591 8303 9022 9742 10468 11197 5442 5839 6236 7034 7833 8640 9447 10260 11076 11898 12725 14385 CAST-IRON PIPE FITTINGS. 187 CAST-IRON PIPE FITTINGS. Approximate Weight. (Addyston Pipe and Steel Co., Cincinnati, Ohio.) Size in Inches. Weight in Lbs. Size in Inches. Weight in Lbs Size in Inches. Weight in Lbs. Size in Inches. Weight in Lbs. CROSSES. TEES. SLEEVES. REDUCERS. 2 3 3x2 4 4x3 4x2 6 6x4 6x3 8 8x6 8x4 8x3 10 10x8 10x6 10x4 10x3 12 12x10 12x8 12x6 - 12 x 4 12x3 14 x 10 14x8 14x6 16 16xl 4 16xl 2 16xl Q 16x8 16x6 16x4 18 20 20x12 20x10 20x8 20x6 20x4 24 24x20 24x6 30x20 30x12 30x8 40 110 90 120 114 90 200 160 160 325 280 265 225 575 415 450 390 350 740 650 620 540 525 495 750 635 570 1100 1070 1000 1010 825 700 650 1560 1790 1370 1225 1000 1000 1000 2400 2020 1340 2C35 2250 1995 8x4 8x3 10 10x8 10x6 10x4 10x3 12 12 x 10 12x8 12x6 12x4 14x12 14x10 14x8 14x6 14x4 14x3 16 16x14 16x12 16x10 16x8 16x6 16x4 18 20 20 x 16 20x12 20x10 20x8 20x6 20x4 20x10 24 24x12 24x8 24x6 30 30x24 30 x 20 30x12 30x10 30x6 36 36x30 36x12 250 220 390 330 370 350 310 600 555 515 550 525 650 650 575 545 525 490 790 850 850 850 755 680 655 1235 1475 1115 1025 1090 900 875 845 1465 2000 1425 1375 1450 3025 2640 2200 2035 2050 1825 5140 4200 4050 2 3 4 6 8 10 12 14 16 18 20 24 30 36 10 25 45 65 80 140 190 208 350 375 500 710 965 1200 8x3 10x8 10x6 10x4 * 12x10 12x8 12x6 12x4 14 x 12 14x 10 14x8 14x6 16 x 12 16x10 20x16 20x14 20x12 20x8 24x20 30x24 30x18 36x30 116 212 170 160 320 250 250 250 475 440 390 285 475 435 690 575 540 400 990 1305 1385 1730 90 ELBOWS. 3 4 6 8 10 12 14 16 18 20 24 30 14 34 55 120 150 260 370 450 660 850 900 1400 3000 ANGLE REDUC- ERS FOR GAS. 6x4 6x3 95 70 S PIPES. ^ or 45 BENDS. I 105 190 3 4 6 8 10 12 16 18 20 24 30 30 70 95 150 200 290 510 580 780 1425 2000 PLUGS. 2 3 4 6 8 10 12 14 16 18 20 24 30 3 10 10 15 30 46 66 90 100 130 150 185 370 1/16 or 2214 BENDS. 6 8 10 12 16 24 30 150 155 205 260 450 1280 2000 CAPS. TEES. 45 BRANCH PIPES. 3 4 6 8 10 12 20 25 60 75 100 120 2 3 3x2 4 4x3 4x2 6 6x4 6x3 6x2 8 8x6 2H 80 76 100 90 87 150 145 145 75 300 270 3 4 6 6x6x4 8 8x6 24 24 x 24 x 20 30 36 90 125 205 145 330 330 2765 2145 4170 10300 REDUCERS. 3x2 4x3 4x2 6x4 6x3 8x6 8x4 25 42 40 95 70 126 116 DRIP BOXES. 4 6 8 10 20 295 330 375 875 1420 188 MATERIALS. WEIGHTS OF CAST-IRON WATER- AND GAS-PIPE C (Addyston Pipe and Steel Co., Cincinnati, Ohio.) at Standard Water-pipe. *i Standard Gas -pipe. v N U m$ Per Foot. Thick- ness. Pei- Length. li MM Per Foot. Thick- ness. Pei- Length. 2 3 15 5/16 63 180 2 3 6 12K 5/16 48 150 3 17 ^ 204 4 22 L 264 4 17 % 204 6 33 ^ 396 6 30 7/16 360 8 42 /^ 504 8 40 7'/l 6 480 8 45 ^ 540 10 60 9/16 720 10 50 7/16 600 12 75 9/16 900 12 70 y 840 14 117 H 1400 14 84 9/16 1000 16 125 n 1500 16 100 9/16 1200 18 167 % 2000 18 134 11/16 1600 20 200 15/16 2400 20 150 11/16 1800 24 250 1 3000 24 184 % 2200 30 350 ji^ 4-,>00 30 250 % 3000 36 475 1% 5700 36 350 /& 4200 42 600 1% 7200 42 417 15/16 5000 48 775 1L 9300 48 542 \\ 6500 60 1330 2 15960 60 900 1% 10800 72 1835 2M 22020 72 1250 m 15000 THICKNESS OF CAST-IRON WATER-PIPES. P. H. Baermann, in a paper read before the Engineers' Club of Phila- delphia in 1882, gave twenty different formulas for determining the thick- ness of cast-iron pipes under pressure. The formulas are of three classes: 1. Depending upon the diameter only. 2. Those depending upon the diameter and head, and which add a con- stant. 3. Those depending upon the diameter and head, contain an additive >i subtractive term depending upon the diameter, and add a constant. The more modern formulas are of the third class, and are as follows: t= .OOOOS/id -f .01 d + .36 Shedd, No. 1. t = .00006/id -f .0133d -f .296 Warren Foundry, No. 2. t = .000058/id -f .0152d -f- .312 Francis, No. 3. t= .000048/i<2 + .013^4- .32 ...Dupuit, No. 4. t- .00004/td 4- .1 |/d-f.l5 Box, No. 5. t = .000135/id 4- .4 .OOlld Whitman, No. 6. t = .00006(/i 4- 230)d -f .333 - .0033d Fanning, No. 7. t = .00015/id 4- .25 - '.0052d Meggs, No. 8. In which t = thickness in inches, h = head in feet, d = diameter in inches. Rankine, "Civil Engineering," p. 721, says: "Cast-iron pipes should be made of a soft and tough quality of iron. Great attention should be paid to moulding them correctly, so that the thickness may be exactly uniform all round. Each pipe should be tested for Jr-bubMes and flav s by ringing it with a hammer, and for strength by exposing "t to *ou ie tlL intended greatest working pressure. " The rule for competing the Jiickness of a pipe to resist a given working pressure is t = -4-, where r is the radius in inches, p the pressure in pounds per square inch, and / the tenacity of the iron x>er square inch. When/ = 18000, and a factor of safety of 5 is used, the above expressed in terms of d and h becomes .5rf.4887t dh nnnnp *" "3600" = 16628 = 00006d/l "There are limitations, however, arising from difficulties in casting, and by the strain produced by shocks, which cause the thickness to be made greater than that given by the above formula." THICKHESS OF CAST-IROK PIPE. 189 Thickness of Metal and Weight per Length for Different Sizes of Cast-iron Pipes under Various Heads of Water. (Warren Foundry and Machine Co.) 50 Ft. Head. 100 Ft. Head. 150 Ft. Head. 200 Ft. Head. 250 Ft. Head. 300 Ft. Head. Size. Thickness of Metal. Weight i per Length. Thickness of Metal. Weight per Length. Thickness of Metal. Weight per Length. Thickness of Metal. Weight per Length. Thickness of Metal. Weight per Length. Thickness of Metal. Weight per Length. 3 .344 144 .353 149 .862 153 .371 157 .380 161 .390 166 4 .361 197 .373 204 .385 211 .397 218 .409 226 .421 235 5 .378 254 .393 265 .408 275 .423 286 .438 298 .453 309 6 .393 315 .411 330 .429 345 .447 361 .465 377 .483 393 8 .422 445 .450 475 .474 502 .498 529 .522 557 .546 584 10 .459 600 .489 641 .519 682 .549 723 .579 766 .609 808 12 .491 768 .527 826 .563 885 .599 944 .635 1004 .671 1064 14 .524 952 .566 1031 .608 1111 .650 1191 .692 1272 .734 1352 16 .557 1152 .604 1253 .652 1360 .700 1463 .748 1568 .796 1673 18 .589 1370 .643 1500 .697 1630 .751 1761 .805 1894 .859 2026 20 .622 1603 .682 1763 .742 1924 .802 2086 .862 2248 .922 2412 24 .687 2120 .759 2349 .831 2580 .903 2811 .975 3045 1.047 3279 30 .785 3020 .875 3376 .965 3735 1.055 4095 1.145 4458 1.235 4822 36 .882 4070 .990 4581 1.098 5096 1.206 5613 1.314 6133 1.422 6656 42 .980 5265 1.106 5958 1.232 6657 1.358 7360 1.484 8070 1.610 8804 48 1.078 6616 1.222 7521 1.366 8431 1.510 9340 1.654 10269 1.798 11195 All pipe cast vertically in dry sand; the 3 to 12 inch in lengths of 12 feet, all larger sizes in lengths of 12 feet 4 inches. Safe Pressures and Equivalent Heads of Water for Cast- iron Pipe of Different Sizes and Thicknesses, (Calculated by F. H. Lewis, from Fanning's Formula.) Size of Pipe. Thick- ness. k *s 10" 16" 18" 20" 112 140 168 116 141 166 190 MATERIALS. Safe Pressures, etc., for Cast-iron Pipe. (Continued.) Thick- ness. Size of Pipe. 22" 24" 27" 80" 33" 36" 42" 48" Pressure I in Pounds, j 1 Head in * i Feet. 1 1 Pressure in Pounds. Head in Feet. ! H C Head in Feet. Pressure in Pounds. Head in Feet. Pressure in Pounds. Head in Feet. Pressure In Pounds. ii |h Pressure in Pounds. Head in Feet. Pressure in Pounds. Head in Feet. Pressure in Pounds. Is ^ 11-16 3-4 13-16 7-8 15-16 1 1 1-8 1 1-4 1 3-8 1 1-2 1 5-8 1 3-4 1 7-8 2 2 1-8 2 1-4 21-2 23-4 40 60 80 101 121 142 182 224 92 138 184 233 279 327 419 516 30 49 68 86 105 124 161 199 237 69 113 157 198 242 286 371 458 546 19 36 52 69 85 102 135 169 202 236 64 83 120 159 196 S55 311 389 465 544 24 39 54 69 84 114 144 174 204 234 55 90 124 159 194 263 332 401 470 538 42 55 69 96 124 1^1 97 127 159 221 286 348 410 472 537 32 44 57 82 1(1? 132 157 182 207 74 101 131 189 247 304 362 419 477 38 59 81 103 124 145 167 188 210 88 136 187 237 286 334 385 433 484 24 43 62 81 99 118 136 155 174 193 212 55 99 143 187 228 272 313 357 401 445 488 Si 49 64 79 94 109 124 139 154 134 214 78 113 147 182 217 251 286 320 355 424 482 178 206 233 NOTE. The absolute safe static pressure which may be 2T S put upon pipe is given by the formula P = ~=r X -z-, in which formula P is the pressure per square inch.; T, the thickness of the shell; S, the ultimate strength per square inch of the metal in tension; and D, the inside diameter of the pipe. In the tables S is taken as 18000 pounds per square inch, with a working strain of one fifth this amount or 3600 pounds per square inch. The formula for the 7200 T absolute safe static pressure then is: P = . It is, however, usual to allow for "water-ram" by in- creasing tho thickness enough to provide for 100 pounds additional static pressure, ana, to insure sufficient metal for good casting and for wear and tear, a further increase equal to .333 (l JQQ)- The expression for the thickness then becomes: (P-flOO)D 883(l~^ 7200 M V 100/' and for safe working pressure The additional section provided as above represents an increased value under static pressure for the different sizes of pipe as follows (see table in margin). So that to test the pipes up to one fifth of the ultimate strength of the material, the pressures in the marginal table should be added to the pressure-values given in the table above. Size of Pipe. RIVETED HYDRAULIC PIPE. 191 RIVETED HYDRAULIC PIPE. (Pel ton Water Wheel Co.) Weight per foot with safe head for various sizes of double-riveted pipe. -S gj' 1 _ | _j- a 3| d += a o-g il 43 OJ t- ^j *o"o 11 +3 CO $_ , -S *" O T i- a qu-2 . t- a 031 1 i, of .^& "3-gJ . ^3 ^1 o u aT ^w 6 tj 111 S 1? S'l 5 s ^ "O 05 "3-3 C IS-S^ .Sf-S^ S3 U 5 . '~ Q T3 Pi"-* .s&s^ ft H T' w gHl. s *S H^O H Hl " l^cc ^,p. 3 18 .05 810 2)4 18 12 .109 295 2514 4 18 .05 607 3 18 11 .125 337 29 4 5 16 18 .062 .05 760 485 3% 3% 18 18 10 8 .14 .171 378 460 5* 5 16 .062 605 4^3 20 16 .062 151 16 5 14 .078 757 5% 20 14 .078 189 19% 6 18 .05 405 4J4 20 12 .109 265 6 16 .062 505 5J4 20 11 .125 304 3)i^ 6 14 .078 630 6^ 20 10 .14 340 35 7 18 .05 346 4% 20 8 .171 415 4514 7 16 .062 433 6 22 16 .062 138 17% 7 14 .078 540 7J><2 22 14 .078 172 22 8 16 .062 378 7 22 12 .109 240 30^3 8 14 .078 472 8% 22 11 .125 276 34(1 8 12 .109 660 12 22 10 .14 309 39 9 16 .062 336 7V*> 22 8 .171 376 50 9 14 .078 420 9J4 24 14 .078 158 23% 9 12 .109 587 12% 24 12 .109 220 32 10 16 .062 307 24 11 .125 253 37^ 10 14 .078 378 10)4 24 10 .14 283 42 10 12 .109 530 14J4 24 8 .171 346 50 10 11 .125 607 16J4 24 6 .20 405 59 10 10 .14 680 18J4 26 14 .078 145 25V* 11 16 .062 275 9 26 12 .109 203 351^ 11 14 .078 344 11 26 11 .125 233 39^ 11 12 .109 480 15/4 26 10 .14 261 44^ 11 11 .125 553 17V& 26 8 .171 319 54 11 10 .14 617 19V6 26 6 .20 373 64 12 16 .062 252 10 " 28 14 .078 135 27^ 12 14 .078 316 28 12 .109 188 38 12 12 .109 442 17 4 28 11 .125 216 12 11 .125 506 19^ 28 10 .14 242 47V^ 12 10 .14 567 21% 28 8 .171 295 58 13 16 .062 233 10i| 28 6 .20 346 69 13 14 .078 291 30 12 .109 176 39^ 13 12 .109 407 18 30 11 .125 202 45 13 11 .125 467 20L 30 10 .14 226 50^ 13 10 .14 522 23 30 8 .171 276 61% 14 16 .062 216 H/4 30 6 .20 323 73 14 14 .078 271 14 30 H .25 404 90 14 12 .109 378 36 11 .125 168 54 14 11 .125 433 22*4 36 10 .14 189 60^ 14 15 10 16 .14 .062 485 202 25 36 36 i .187 .25 252 337 81 109 15 14 .078 252 14% 36 .312 420 135 15 12 .109 352 20^4 40 10 .14 170 67^ 15 11 .125 405 23 J4 40 .187 226 90 15 10 .14 453 26 40 M .25 303 120 16 16 .062 190 13 40 j> .312 378 150 16 14 .078 237 16 40 % .375 455 180 16 12 .109 332 42 10 .14 162 71 16 11 .125 379 24/^1 42 A .187 216 94^ 16 10 .14 425 28^j 42 M .25 289 126 18 16 .062 168 14% 42 T B 5 .312 360 158 18 1 14 .078 210 18H 42 % .375 435 190 192 MATERIALS. STANDARD PIPE FLANGES. Adopted August, 1894, at a conference of committees of the American Society of Mechanical Engineers, and the Master Steam and Hot Water Fit- ters' Association, with representatives of leading manufacturers and users of pipe. Trans. A. S. M. E., xxi. 29. (The standard dimensions given have not yet, 1901, been adopted by some manufacturers on account of their un- willingness to make a change in their patterns.) The list is divided into two groups; for medium and high pressures, the first ranging up to 75 Ibs. per square inch, and the second up to 200 Ibs. a K 6 7 8 9. 10 12 14 15 16 18 20 22 24 26 28 30 36 42 j\ 6090 NOTES. Sizes up to 24 inches are designed for 200 Ibs. or less. Sizes from 24 to 48 inches are divided into two scales, one for 200 Ibs., the other for less. The sizes of bolts given are for high pressure. For medium pressures the diameters are % in. less for pipes 2 to 20 in. diameter inclusive, and % in. less for larger sizes, except 48-in. pipe, for which the size of bolt is 1% in. When two lines of figures occur under one heading, the single columns are for both medium and high pressures. Beginning with 24 inches, the left-hand columns tire for medium and the right-hand lines are for high pressures. The sudden increase in diameters at 16 inches is due to the possible inser- tion of wrought-iron pipe, making with a nearly constant width of gasket a greater diameter desirable. When wrought-iron pipe is used, if thinner flanges than those given are sufficient, it is proposed that bosses be used to bring the nuts up to the standard lengths. This avoids the use of a reinforcement around the pipe. Figures in the 3d, 4th, 5th, and last columns refer only to pipe for high pressure. In drilling valve flanges a vertical line parallel to the spindles should be midway between two holes on the upper side of the flanges. CAST-IRON" PIPE AKD PIPE FLANGES. 193 FLANGE DIMENSIONS, ETC., FOR EXTRA HEAVY PIPE FITTINGS (UP TO 250 LBS. PRESSURE). Adopted by a Conference of Manufacturers, June 28, 1901. Size of Pipe. Diam. of Flange. Thickness of Flange. Diameter of Bolt Circle. Number of Bolts. Size of Bolts. Inches. Inches. Inches. Inches. Inches. 2 6^ % 5 4 % % 7V 1 5^ 4 % 3 3^ SJ4 9 It* 8 8 1 4 4^ 10 ttM PU If 8 8 i 5 11 1% 9J4 8 8 6 12^ 1 7-16 10% 13 M 7 14 1^4 31% 12 y& ^8 15 1% 13 12 % 9 16 1M 14 12 7^ 10 12 g* 3* Wi im 16 16 1 14 15 22U 2^ 2 3-16 20 21 20 20 ft 16 25 214 22^ 20 18 27 2% 1V& 24 20 29^ 2Va 26^ 24 ^ 22 81JJ 2% 28% 28 /^ 24 84 234 31% 28 DIMENSIONS OF PIPE FLANGES AND CAST-IRON PIPES. (J. E. Codman, Engineers 1 Club of Philadelphia, 1889.) 4 5 6 8 10 12 14 16 18 20 22 24 32 34 40 42 44 46 48 .So 5 3 O Is 8 8 10 12 14 16 16 18 20 22 24 24 26 28 30 32 32 34 34 2 1-16 40 2^ Thickness of Pipe. Frac. Dec. 13-32 7-16 7-16 15-32 19-32 21-32 11-16 Has 27-32 31-32 1 1 1-16 1 l'5-32 1 3-16 1 5-16 111-32 1T-16 .373 .396 .420 .443 .466 .511 .557 .603 .649 .695 .741 .787 .833 .879 .925 .971 1.017 1.063 1.109 1.155 1.201 1.247 1.293 1.339 1.385 1.431 Cfe.-g o3 JUS 6.96 11.16 15.84 21.00 26.64 39.36 54.00 70.56 89.04 109.44 131.76 156.00 182.16 210.24 240.24 272.16 306.00 341.76 379.44 419.04 460.56 504.00 549.36 596.64 645.84 696.96 4.41 5.93 7.66 9.63 11.82 16.91 23.00 30.13 38.34 47.70 58.23 70.00 83.05 97.42 113.18 130.35 149.00 169.17 190.90 214.26 239.27 266.00 294 49 324.78 356.94 391.00 D = Diameter of pipe. All dimensions in inches. FORMUUE. Thickness of flange = 0.033D -f- 0.56 ; thickness of pipe 0'23D -4- 0.327* r*irrVif f\f i"kii-ck r-xjii. -F/-W/-W*- n f>A TY1 I O 7~ . ,, ^:_Ui xii _._ ' 194 MATERIALS. Df Perf ".2J2S9?2:*SeSJ52}2P<P Thread! No. of Threads aerlnch. . Tfioco^i-ieooDOiccot^i-toiocxtco^TEcoTttoo oo T~co T~ei ^ per Lin. Ft. S "~~^~'~^;S;^Sw33$SSo^oco of Pipe. ___^ r-,^-.t-i ^ Weight .TtCJOT*iNl~TtOOi-i^^OOTt<Oimr~GOOeoojer<-->^r-,<-->r-> - of Pipe per -ii-tS'totcoiot>osoW^GC2oc6coono:T Li ^ Ft i-^i^fr^co^Tr^ioo ^ __ .^ JyQ oyp'jpl i- c '"ssss-ssssnssigsg ^ i\u nt f?. %^ e ^ IPJ.llliillliSSSISIilgligigSlS ^ - 1 ^ ^ i-^ i-^ I-' o*' iy| eo I g _ ^ ^ CC ' ' "T^^H'^CJUJ'C o^ 02 " ^* T-I <ri y-t & a ai *& >+* Length j ^ OQOI>OOTt( ^ OQOO:)V ^ irao;)Tt ,j > j > ^ Heot , tOO ! Outside' fa os t- 10 -^wojcjoj i-^i-^d B& Surface.j ?rl?t ^Sg'g2Slgi?^^^^^^^^^^^^"^^^^ SSS ^ Inside fa -^ o' i> to *? oo ci oi i-i i-i TH ^ d ^ Surface. ^ ence. Internal .f3!S!iba Circum- S . '-J *. >. w f erence. l ~ l ^-< '-> c Thick- . GO oo as c A otual _^?^cocs'fjc^ : ^GOr : Hcbo : 3"55oTfoc$6cco^oSoioiocioioio ^sp ^.^1/UlO.i gg^J^^^QOQJlQ^^^CjJ^Q^^l^X^QS^^^^^^JQ^^J^^g^ ^\ Inside H ^ Diam. T-II-H i-< ri ^ 1-1 r-i 1-1 1-1 1-1 1-1 ?? g< g Dm e - ' ^"^^^^^^^sdSsasftftii* g- Nominal j Inside fl w Diam. ' WROUGHT-IROH PIPE. 195 For discussion of the Briggs Standard of Wrought-iron Pipe Dimensions, see Report of the Committee of the A. S. M. E. in " Standard Pipe and Pipe Threads," 1886. Trans., Vol. VIII, p. 29. The diameter of the bottom of the thread is derived from the formula J> (0.05D + 1.9) x , in which D = outside diameter of the tubes, and n the number of threads to the inch. The diameter of the top of the thread is derived from the formula 0.8 X 2 -f d, or 1.6 }- d, in which d is the diameter at the bottom of the thread at the end of the .pipe. The sizes for the diameters at the bottom and top of the thread at the end of the pipe are as follows: Diam. Diam. Diam. Diam. Diam. Diam. Diam. Diam. Diam. of Pipe, Nom- at Bot- tom of at Top of of Pipe, Nom- at Bot- tom of at Top of of Pipe, Nom- at Bot- tom of at Top of inai. Thread. Thread. inal. Thread. Thread. inal. Thread. Thread. in. in. in. In. in. in. in. in. in. .334 .393 **4 2.620 2.820 8 8.334 8.534 M .438 .522 3 3.241 3.441 9 9.327 9.527 % .568 .658 3^ 3.738 3.938 10 10.445 10.645 Ut .701 .815 4 4.234 4.434 11 11.439 11.639 % .911 1.025 4^ 4.731 4.931 12 12.433 12.633 I 1.144 1.283 5 5.290 5.490 13 13.675 13.875 w 1.488 1.627 6 6.346 6 546 14 14.669 14.869 i*i 1.727 1.866 7 7.340 7.540 15 15.663 15.863 8 2.223 2.339 Having the taper, length of full-threaded portion, and the sizes at bottom and top of thread at the end of the pipe, as given in the table, taps and dies can be made to secure these points correctly, the length of the imperfect threaded portions on the pipe, aud the length the tap is run into the fittings beyond the point at which the size is as given, or, in other words, beyond the end of the pipe, having no effect upon the standard. The angle of the thread is 60, and it is slightly rounded off at top and bottom, so that, instead of its depth being 0.866 its pitch, as is the case with a full V-thread, it is 4/5 the pitch, or equal to 0.8 -*- n, n being the number of threads per inch. Taper of conical tube ends, 1 in 32 to axis of tube = 2 inch to the foot total taper. L96 MATERIALS. WROUGHT-IRON WELDED TUBES, EXTRA STRONG. Standard Dimensions. Nominal Diameter. Actual Out- side Diameter. Thickness, Extra Strong. Thickness, Double Extra Strong. Actual Inside Diameter, Extra Strong. Actual Inside Diameter, Double Extra Strong. Inches. Inches. Inches. Inches. Inches. Inches. V6 0.405 0.100 0.205 \* 0.54 0.123 294 Z 0.675 0.127 0.421 H 0.84 0.149 0.298 0.542 0.244 fi 1.05 0.157 0.314 0.736 0.422 1 1.315 0.182 0.364 0.951 0.587 1/4 1.66 0.194 0.388 1.272 0.884 Ii2 1.9 0.203 0.406 1.494 1.088 2 2.375 0.221 0.442 1.933 1.491 2^ 2.875 0.280 0.560 2.315 1.755 3 3.5 0.304 0.608 2.892 2.284 3^ 4.0 0.321 0.642 3.358 2.716 4 4.5 0.341 0.682 3.818 3.136 STANDARD SIZES, ETC., OF LAP-WELDED CHAR- COAL-IRON BOILER-TUBES. (National Tube Works.) I A " , . IJ -o'Si oy * ^^M -,^D 5 s I O al Internal External E^.jj S^g 0^1 E Area. Area. 1^ I 1 ' | H pS I 3 11 fl* fll fill ftS 1 hj in. in. in. in. in. sq. in. sq.ft. sq. in. sq.ft. ft. ft. ft. Ibs. 1 .810 .095 2.545 3.142 .515 .0036 .785 .0055 4.479 3.820 4.149 .90 1 1-4 1.060 .095 3.330 3.927 .882 .0061 1.227 .0085 3.604 3.056 3.330 1.15 1 1-2 1.310 .095 4.115 4.712 1.348 .0094 1.767 .0123 2.916 2.547 2.732 1.40 13-4 1.560 .095 4.901 5.498 1.911 .0133 2.405 .0167 2.448 2.183 2.316 1.65 2 1.810 .095 5.686 6.283 2.573 .0179 3.142 .0218 2.110 1.910 2.010 1.91 2 1-4 2.060 .095 6.472 7.069 3.333 .0231 3.976 0276 1.854 1.698 1.776 2.16 2 1-2 2.282 .109 7.169 7.854 4.090 .0284 4.909 .0341 1.674 1.528 1.601 2.75 23-4 2.532 .109 7.955 8.639 6.035 .0350 5.940 .0412 1.508 1.389 1.449 3.04 3 2.782 .109 8.740 9.425 6.079 .0422 7.069 .0491 1.373 1.273 1.322 3.33 3 1-4 3.010 .120 9.456 10.210 7.116 .0494 8.296 .0576 1.269 1.175 1.222 3.96 31-2 3.260 .120 10.242 10,996 8.347 .0580 9.621 .0668 1.172 1.091 1.132 4.28 33-4 8.510 120 11.027 11.781 9.676 .0672 11.045 07G7 1.088 1.019 1.054 4.60 4 3.732 .134 11.724 J 2.566 10.939 .0760 12.566 0873 1.024 .955 .990 5.47 41-2 4.232 .134 13.295 14.137 14.066 .0977 15.904 .1104 .903 .849 .876 6.17 5 4.704 .148 14.778 15.708 17.379 .1207 19.635 .1364 .812 .764 .788 7.58 6 5.670 .165 17.813 18.850 25.250 .1750 28.274 .1963 .674 .637 .656 10.16 7 6.670 .165 20.954 21.991 34,942 .2427 38.485 .2673 .573 .546 .560 11.90 8 7.670 .165 24.096 25.133 46.204 .3209 50.266 .3491 .498 .477 .488 13.65 9 8.640 .180 27.143 28.274 58.630 .4072 63.617 .4418 .442 .424 .433 16.76 10 9.594 .203 30.141 31.416 72.292 .5020 78.540 -5454 .398 .382 .390 21.90 11 10.560 .220 33.175 34.558 87.583 .6082 95.033 .6600 .362 .347 .355 25.00 12 11.542 .229 36.260 37.699 104.629 .7266 113.098 .7854 .331 .318 .325 28.50 13 12.524 .233 39.345 40.841 123.190 .8555 132.733 .9217 .305 .294 .300 32.06 14 13.504 .248 42.424 43.982 143.224 .9946 153.938 1.0690 .283 .273 .278 36.00 15 14.482 .259 45.497 47.124 164.721 1.1439 176.715 1.2272 .264 .255 260 40.60 16 15.458 .271 48.563 50.266 187.671 1.3033 201.062 1.3963 .247 .239 .243 45.20 17 16.432 .284 51.623 53.407 212.066 1.4727 226.981 1.5763 .232 .225 .229 49.90 18 17.416 .292 54.714 56.549 238.225 1.6543 254.470 1.7671 .219 .212 .216 54.82 19 18.400 .300 57.805 59.690 265.905 1.8466 283.529 1.9690 .208 .201 .205 59.48 20 19.360 .320 60.821 62.832 294.375 2.0443 314.159 2.1817 .197 .191 .194 66.77 21 20.320 .340 63.837 65.974 324.294 2.2520 346.361 2.4053 .188 .182 .185 73.40 surface in bes) is to In estimating the effective steam-heating or boiler surface of tubes, the su contact with air or gases of combustion (whether internal or external to the tu be taken. For heating liquids by steam, superheating steam, or transferring heat from one liquid or gas to another, the mean surface of the tubes is to be taken. RIVETED TROK PIPE. 197 To find the square feet of surface, S, in a tube of a given length, L, in feet, and diameter, d, in inches, multiply the length in feet by the diameter in inches and by .2618. Or, 8 - - L -- - = .2618dL. For the diameters in the table below, multiply the length in feet by the figures given opposite the diameter. Inches, Diameter. Square Feet per Foot Length. Inches, Diameter. Square Feet per Foot Length. Inches, Diameter. Square Feet per Foot Length. 1 4 2 4 .0654 .1309 .1963 .2618 .3272 .3927 .4581 .5236 3 4 4 4 .5890 .6545 .7199 .7854 .8508 .9163 .9817 1.0472 5 6 7 8 9 10 11 12 1.3090 1.5708 1.8326 2.0944 2.3562 2.6180 2.8798 3.1416 RIVETED IRON PIPE. (Abendroth & Root Mfg. Co.) Sheets punched and rolled, ready for riveting, are packed in convenient form for shipment. The following table shows the iron and rivets required for punched and formed sheets. Number Square Feet of Iron required to make 100 Lineal Feet Punched and Formed I11H1 ^^^3c| Number Square Feet of Iron required to make 100 Lineal Feet Punched and Formed Sheets when put together. Sheets when put together. "el r S^"^ 3^ fijU^ & If jr | Diam- eter in Inches. Width of Lap in Inches. Square Feet. Diam-. eter in Inches. Width of Lap in Inches. Square Feet. oTS oS2fe oSCQ 3 1 90 1,600 14 JJX 397 2,800 4 1 116 1,700 15 l^i 423 2,900 5 150 1.800 16 ji^j 452 3,000 6 ji/ 178 1,900 18 l/^ 506 3,200 7 ]1Z 206 2,000 20 l/'ij 562 3,500 8 1^1 234 2,200 22 l/^ 617 3,700 9 JL 258 2,300 24 1^ 670 3,900 10 l^J 289 2,400 26 ^/^ 725 4,100 11 1^1 314 2,500 28 1/^j 779 4,400 12 ji/ 343 2,600 30 1^3 836 4,600 13 % 369 2,700 36 % 998 5,200 WEIGHT OF ONE SQUARE FOOT OF SHEET-IRON FOR RIVETED PIPE. Thickness by the Rirmiugliam Wire-Gauge. No. of Gauge. Thick- ness in Decimals of an Inch. Weight in Ibs., Black. Weight in Ibs., Galvan- ized. No. of Gauge. Thick- ness in Decimals of an Inch. Weight in Ibs., Black. Weight in Ibs., Galvan- ized. 26 24 22 20 .018 .022 .028 .035 .80 1.00 1.25 1.56 .91 1.16 1.40 1.67 18 16 14 12 .049 .065 .083 .109 1.82 2.50 J 3.12 4.37 2.16 2 67 3.34 4.73 198 MATERIALS. SPIRAL RIVETED PIPE. (Abendroth & Root Mfg. Co.) Thickness. Diam- eter, Inches. Approximate Weight in Ibs. per Foot in Length. Approximate Burst- ing Pressure in Ibs. per Square Inch. B. W. G. No. Inches. 26 24 22 20 18 16 14 12 .018 .022 .028 .035 .049 .065 .083 .109 3 to 6 1 3 to 12 3 to 14 3 to 24 3to2i 6 to 24 8 to 24 9 to 24 bs.= = ^ofd = .4 = .5 = .6 = .8 = 1.1 = 1.4 iam. in ns. 27001bs.-f-diam.inins. 3600 " H- *' 4800 " -T- " 6400 " -*- " 8000 " -*- " The above are black pipes. Galvanized weighs 10 to 30 % heavier. Double Galvanized Spiral Riveted Flanged Pressure Pipe, tested to 150 Ibs. hydraulic pressure. Inside diameters, inches.... Thickness, B. W. G Nominal wt. per foot, Ibs.. . 2020 71 8 O 1C 181818 91011 1816 811 13114151618202224 16 16 14 14 15' 20 22 24 29 34 4050 1212 DIMENSIONS OF SPIRAL PIPE FITTINGS. Diameter Inside Diameter. Outside Diameter Flanges. Number Bolt-holes. Diameter Bolt-holes.j Circles on which Bolt- holes are Sizes of Bolts. Drilled. ins. ins. ins. ins. ins. 3 6 * 4 % 4% 7/16 x 1% 4 7 8 i^ 5 15/16 7/16 x 1% 5 8 8 ^ 6 15/16 7/16x1% 6 8% 8 % 7% l^> x 1% 7 10 8 % 9 ^6 x 1% 8 11 8 5^ 10 1^x2 9 13 8 % tt4 Ux2 10 14 8 % 1214 >|x2 11 15 12 % 13% ^2*2 12 16 12 % i42 1^x2 13 17 12 % 15^4 1^x2 14 15 & 12 12 i 16*4 17 7/16 j|*2| 16 21 3/16 12 % 19^ L x 2J-J2 18 23^ 16 11/16 2ii| ^ x 2^ 20 25^ 16 11/16 23V ^ x 2V& 22 28^ 16 26 % X 2V^ 24 30 16 A 27% %*^ SEAMLESS BRASS TUBE. IRON-PIPE SIZES. (For actual dimensions see tables of Wrought-iron Pipe.) Nominal Size. Weight pr Foot. Nom. Size. Weight per Foot. Nom. Size. Weight per Foot. Nom. Size. Weight per Foot. ins. Ibs. .25 .43 .62 .90 ins. f i Ibs. 1.25 1.70 2.50 3. ins. 2 P 3^ Ibs. 4.0 5.75 8.30 10.90 ins. 4 9* 6 Ibs. 12.70 13.90 15.75 18.31 BRASS TUBING; COILED PIPES. 199 SEAMLESS DRAWN BRASS TUBING. (Randolph & Clowes, Waterbury, Conn.) Outside diameter 3/16 to 7% inches. Thickness of walls 8 to Gauge, length 12 feet. The following are the standard sizes: Outside Diam- eter. Length Feet. Stubbs' or Old Gauge. Outside Diam- eter. Length Feet. Stubbs' or Old Gauge. Outside Diam- eter. Length Feet. Stubbs 1 or Old Gauge. H 12 20 1% 12 14 2% 12 11 5-16 12 19 12 14 2M 12 11 % 12 19 1% 12 13 3 12 11 l/; 12 18 1% 12 13 3*4 12 11 % 12 18 1 13-16 12 13 31^1 12 11 % 12 17 m 12 12 4 10 to 12 11 13-16 12 17 1 15-16 12 12 5 10 to 12 11 % 12 17 2 12 12 5/4 10 to 12 11 15-16 12 17 % 12 12 51^3 10 to 12 11 1 12 16 2^4 12 12 5M 10 to 12 11 12 16 12 12 6 10 to 12 11 1J4 12 15 2J^ 12 11 BENT AND COILED PIPES. (National Pipe Bending Co., New Haven, Conn.) COILS AND BENDS OF IRON AND STEEL PIPE. Size of pipe Inches 8 6 1-4 $ 1 1M 1^ 2 2U 3 Least outside diameter of coil Inches 2 01^ fti< 4V fi 8 12 16 24 32 Size of pipe Inches 3U 4 41-6 5 Q 7 8 9 10 12 Least outside diameter of coil Inches 40 18 50 58 66 30 92 105 130 156 Lengths continuous welded up to 3-in. pipe or coupled as desired. COILS AND BENDS OF DRAWN BRASS AND COPPER TUBING. Size of tube, outside diameter Inches Least outside diameter of coil Inches 1* 2 * ^ 1 4 J JN Size of tube, outside diameter Inches m 1** W ^ 2M 23/ tf ^ 2 ^ Least outside diameter of coil Inches 8 9 10 12 14 16 18 20 Lengths continuous brazed, soldered, or coupled as desired. 90 BENDS. EXTRA-HEAVY WROUGHT-IRON PIPE. Diameter of pipe Inches Radius Inches Centre to end Inchei is 26 24 The radii given are for the centre of the pipe. *' Centre to end " means the perpendicular distance from the centre of one end of the bent pipe to a plane passing across the other end. Standard iron pipes of sizes 4 to 8 in. are bent to radii 8 in. larger than the radii in the above table; sizes 9 to 12 in. to radii 12 in. larger. Welded Solid I>rawn>steel Tubes, imported by P. S. Justice & Co., Philadelphia, are made in sizes from ^ to 4^ in. external diameter, varying by Vths, and with thickness of walls from 1/16 to 11/16 in. The maximum length is 15 feet. 200 MATERIALS. WEIGHT OF BRASS, COPPER, AND ZINC TUBING. Per Foot. Thickness by Brown & Sharpens Gauge. Brass, No. 17. Brass, No. 20. Copper, Lightning-rod Tube, No. 23. Inch. *\ & #> /& 2 4 $ Lbs. .107 .157 .185 .234 .266 .318 .333 .377 .462 .542 .675 .740 .915 .980 1.90 1.506 2.188 Inch. HI 3-16 5-16 $6 ! i m 8? Lbs. .032 .039 .063 .106 .126 .158 .189 .208 .220 .252 .284 .378 .500 .580 Inch. & A g Lbs. .162 .176 .186 .211 .229 Zinc, No. 20. r ijj .161 .185 .234 .272 311 .380 .452 LEAD PIPE IN LENGTHS OF 10 FEET. In. 3-8 Thick. 5-16 Thick. M Thick. 3-16 Thick. Ib. oz. Ib. oz. Ib. oz. Ib. oz. 2^ 17 14 11 8 3 20 16 12 9 3^3 23 18 15 9 S 4 25 21 16 12 8 4^ 18 14 5 31 20 LEAD WASTE-PIPE. in., 2 Ibs. per foot. "3 and 4 Ibs. per foot. " 3^6 and 5 Ibs. per foot. in., 4 Ibs. per foot. " 5, 6, and 8 Ibs. 6 and 8 Ibs. 5 in. 8, 10, and 12 Ibs. LEAD AND TIN TUBING. ^ inch. J4 inch. SHEET LEAD. Weight per square foot, 2^, 3, 3*4 4, Other weights rolled to order. 4 5, 6, 8, 9, 10 Ibs. and upwards. BLOCK-TIN PIPE. in., 4}4, 6}4, and 8 oz. per foot. " 6, 7^j, and 10 oz. " *' 8 and 10 oz. " 10 and 12 oz. " 1 in., 15, and 18 oz. per foot. 154 " 114 and lUlbs. " lj| " 2 and 2V Ibs. 2 4i 2^ and 3 Ibs. " LEAD PIPE. 201 LEAD AND TIN-LINED LEAD PIPE. (Tatham & Bros., New York.) .s . a 1 1 Weight per Foot and Rod. I 1- I Weight per Foot and Rod. S5 ^ 1 S H I Mil. E D 7 Ibs. per rod 10 oz. per foot 6 1 u in. E D \y% Ibs. per foot 2 " " 10 11 ** C 12 " " 8 41 C <>L " * 4 14 " B 1 Ib. 12 M B 31^ ** 17 ** A 1*4 " " 16 " A 4 || || 21 " AA 1*4 " " 19 '* AA 24 44 AAA 1% " 4 ' 27 4 ' AAA 6 " '* 30 7-16 in. 13 oz. " 1*4 in. E 2 " *' 10 " 1 Ib. " 4 D gi^ 4 * 12 ]/2 ' n E 9 Ibs. per rod 7 C 3 * ' 14 D % Ib. per foot 9 4 B 3% ' 16 44 C 1 44 4t 11 A 4^4 ' * 19 " B 1*4 " " 13 AA 54 ' ' 25 44 1*4 4t ** 1 AAA 6M ' ' M A \%L 4< 4< 16 IHJn. E 3 12 44 AA 2 " 19 D 3*4 * 14 23 C 17 * AAA 3 " 25 41 B 5 | | 19 P& in. E 12 " per rod 8 * A 23 44 D 1 ' per foot 9 44 AA 8 * ' 27 " C 1*4 44 4t 13 44 AAA 9 * 44 B 2 " " 16 1% in. C 4 13 4 A 2*^3 " 4 ' 20 " B 5 | | 17 ' AA 2% " " 22 44 A 21 4 AAA 3*4 " " 25 44 AA 8Va ' 4t 27 % in. E 1 " per foot 8 2 in. C 4M * * l 15 1 D 1*4 " 10 41 B 6 4 18 1 C jax *t 12 " A 7 ' 22 " B 0]X ' ** 16 44 AA 9 || ;; 27 44 A 3 ' " 20 * AAA 44 AA 3^3 " 4 ' 23 AAA 4% " 30 WEIGHT OF LEAD PIPE WHICH SHOULD BE USED FOR A OIVEN HEAD OF WATER. (Tatham & Bros., New York.) Head or Number Pressure Calibre and Weight per Foot. of Feet Fall. per sq. inch. Letter. %inch. *^inch. % inch. inch. 1 inch. lJ4in. 30 ft. 15 Ibs. D 10 oz. Ib, 1 Ib. 1*4 Ibs 2 Ibs. 2*4 Ibs. 50ft. 25 Ibs. C 12 oz. 1 Ib. 1*4 Ibs. 1% Ibs. 2*4 Ibs. 3 Ibs. 75ft. 38 Ibs. B 1 Ib. 1*4 Ibs. 2 Ibs. 2*4 Ibs. 3*4 Ibs. 3% Ibs. 100 ft. 50 Ibs. A 1*4 Ibs. 1M Ibs. 2*4 Ibs. 3 Ibs. 4 Ibs. 4% Ibs. 150 ft. 75 Ibs. AA 1*4 Ibs. 2 Ibs.i 2% Ibs. 3*4 Ibs. 4% Ibs. 6 Ibs. 200ft. 100 Ibs. AAA 1% Ibs. 3 Ibs. ! 3*| Ibs. i 4% Ibs. 6 Ibs. 6% Ibs. To find the thickness of lead pipe required when the head of water is given. (Chadwick Lead Works). RULE. Multiply the head in feet by size of pipe wanted, expressed deci- mally, and divide by 750; the quotient will give thickness required, in one- hundredths of an inch. EXAMPLE. Required thickness of half -inch pipe for a head of 25 feet. 25 X 0.50 -?- 750 = 0.16 inch. 202 MATERIALS. O O O g? OO *H O? O5 C? O 00 O O Tj< T- O rH O 00 OJ001>COCOOTjTj<COCOOOOJC<lOirH^THTH OJ 0> ^3 o5 ^ i- TH o* co co o r ^O* 8, $S8S88ii3$3SSSg *" I ^ tfj t^ O <> CO i - a SS f- | a^^, fiii52Siiilii|li > : , *J __ .~OOOOOOOOOOOOOOOOOOoS *r ri 02 M |j Sc a 35 " *i I 6C02 P ^OQO^COr-iOOJOOI>0150TTfCQCo"(?iNNOJT _Q M^r-.^r-.rHrHTH ^ fc^ 5,< ^ .fc^ CQ - $ O ?*9Pr^^95Otr?5Q^ ^W s& oi BOLT COPPER SHEET AND BAB BRASS. 203 WEIGHT OF ROUND BOLT COPPER, Per Foot. Inches. Pounds. Inches. Pounds. Inches. Pounds. 1 .425 .756 1.18 1.70 2.31 1 3.02 3.83 4.72 5.72 6.81 2 7.99 9.27 10.64 12.10 WEIGHT OF SHEET AND BAR BRASS. Thickness, Side or Diam. Sheets per sq. ft. Square Bars 1 ft. long. Round Bars 1 ft. long. Thickness, Side or Diam. Sheets per sq. ft. Square Bars 1 ft. long. Round Bars 1 ft. long. Inches. Inches. 1-16 2.72 .014 .011 1 1-16 46.32 4.10 3.22 K 5.45 .056 .045 Ug 49.05 4.59 3.61 3-16 8.17 .128 .100 1 3-16 51.77 5.12 4.02 1 A 10.90 .227 .178 154 54.50 5.67 4.45 5-16 13.62 .355 .278 1 5-16 57.22 6.26 4.91 % 16.35 .510 .401 1% 59.95 6.86 5.39 7-16 19.07 .695 .545 1 7-16 62.67 7.50 5.89 21.80 .907 .712 1*6 65.40 8.16 6.41 9-16 24.52 1.15 .902 1 9-16 68.12 8.86 6.95 % 27.25 1.42 1.11 m 70.85 9.59 7.53 11-16 29 97 1.72 1.35 1 11-16 73.57 10.34 8.12 H 13-16 3-2.70 35.43 2.04 2.40 1.60 1.88 1 fs-16 76.30 79.0-2 11.12 11.93 8.73 9.36 7 /8 38.15 2.78 2.18 1% 81.75 12.76 10.01 15-16 40.87 3.19 2.50 1 15-16 84.47 13.63 10.70 1 43.60 3.63 2.85 2 87.20 14.52 11.40 COMPOSITION OF VARIOUS GRADES OF ROLLED BRASS, ETC. Trade Name. Copper Zinc. Tin. Lead. Nickel. Common high brass 61.5 38 5 Yellow metal ... 60 40 Cartridge brass 66% 33^ Low brass 80 20* Clock brass 60 40 1L Drill rod 60 40 \y> to 2 Spring brass 33V<* 1V 18 per cent German silver. . 61J^ 20^ 18 The above table was furnished by the superintendent of a mill in Connec- ticut in 1894. He says: While each mill has its own proportions for various mixtures, depending upon the purposes for which the product is intended, the figures given are about the average standard. Thus, between cartridge brass with 33J per cent zinc and common high brass with 38U per cent zinc, there are any number of different mixtures known generally as " high brass," or specifically as "spinning brass," "drawing brass," etc., wherein the amount of zinc is dependent upon the amount of scrap used in the mix- ture, the degree of working to which the metal is to be subjected, etc. 204 MATERIALS. AMERICAN STANDARD SIZES OF DROP-SHOT. Diameter. No. of Shot to the oz. Diameter. No. of Shot to the oz. Diam- eter. No. of Shot to the oz. I Fine Dust. Dust... . No. 12. . . " 11. . " 10.. . " 10. . . " 9. . . " 9.. . 3-1 00" 4-100 5-100 6-100 Trap Shot 7-100" Trap Shot 8-100" 10784 4565 2326 1346 1056 848 688 568 No. 8 " 8 7 7 6 5 4 3 Trap Shot 9-100" Trap Shot 10-100" 11-100 12-100 13-100 14-100 472 399 338 291 218 168 132 106 No. 2... 1.. . B... BB. BBB T... TT.. F.. FF.. 15-100" 16-100 17-100 18-100 19-100 20-100 21-100 22-100 23-100 86 71 59 50 42 36 31 27 24 COMPRESSED BUCK-SHOT. Diameter. No. of Balls to the Ib. Diameter. No. of Balls to the Ib. No 3 . 25-100" 284 No. 00.... ... 34-100" 115 * 2 27-100 232 " 000 3-100 98 44 1 30 100 173 Balls 38-100 85 " 32 100 140 44-100 50 SCREW-THREADS, SELLERS OR U. S. STANDARD. In 1864 a committee of the Franklin Institute recommended the adoption of the system of screw-threads and bolts which was devised by Mr. William Sellers, of Philadelphia. This same system was subsequently adopted as the standard by both the Army and Navy Departments of the United States, and by the Master Mechanics 1 and Master Car Builders' Associations, so that it may now be regarded, and in fact is called, the United States Stand- ard. The rule given by Mr. Sellers for proportioning the thread is as follows : Divide the pitch, or, what is the same thiug, the side of the thread, into eight equal parts; take off one part from the top and fill in one part in the bottom of the thread; then the flat top and bottom will equal one eighth of the pitch, the wearing surface will be three quarters of the pitch, and the diameter of screw at bottom of the thread will be expressed by the for mula 1 299 diameter of bolt - er inch' For a sharp V thread with angle of 60 the formula is 1.733 diameter of bolt -- -=. -. -- = - - r no. of threads per inch The angle of the thread in the Sellers system is 60. In the Whitworth or English system it is 55, and the point and root of the thread are rounded. Screw-Threads, United States Standard. 5 B 3 ft a Q 5-16 11-16 20 18 16 14 13 12 11 11 13-16 &6 1 1-16 2 13-16 3 3 5-16 4 4 TT. S. OR SELLERS SYSTEM OF SCREW-THREADS. 205 Screw-Threads, Whltwortli (English) Standard. A g i ^ j J - f 4 LI ,d 3 u w ^ S 5 S O flH o S S K P S *4 20 & /8 11 8 194 5 3 3U l6 18 11-16 11 *6 7 1?6 4*6 3*4 3*4 *M6 16 14 13-16 10 10 n 7 6 2 4 S 3% 3^ I/ 12 9 ^ 6 2*6 4 4 3 9-16 12 15-16 9 % o 2% 3*6 U. S. OR SI^I.I.KKS S VSTK1TI OF SCREW-THREADS. BOLTS AND THREADS. HEX. NUTS AND HEADS. i Q 1 3* CM 0) S *o %& .s Root of ,d in Sq. s. 5*1 S:i *A 5 g 3 w 00 ? sf cc ll 1 S * 5 ~ EH P ,g 8 5 *% C60 25 ^ "eg o>J c? k& -< ll s -w S3 {->"-> ofe 02 f 81 e w I 1 |l Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. *4 20 .185 .0062 .049 .027 y 7-16 37-64 54 3-16 7-10 5^16 18 .2^0 .0074 .077 .045 19-32 17-32 11-16 5-16 *4 10-12 96 16 .294 .0078 .110 .068 11-16 % 51-64 % 5-16 63-64 7-16 14 .344 .0089 .150 .093 25-32 ^3-3"- 9-10 7-16 % 1 7-64 *6 13 .400 .0096 .196 .126 Vs 13-16 1 V> 7-16 1 15-64 9-16 12 .454 .0104 .249 .162 31-32 29-32 1*6 9-16 *6 1 23-64 % 11 .507 .0113 .307 .202 1 1-16 1 1 7-32 % 9-16 ]!/> M 10 .620 .0125 .442 .302 \y* 13-16 1 7-16 % 11-16 1 49-64 % 9 .731 .0138 .601 .420 1 7-16 1% 1 21-32 % 13-16 21-32 1 8 .837 .0156 .785 .550 !&/ 1 916 JT^ 1 15-16 219-64 1^ 7 .940 .0178 .994 .694 1 13-16 1% 2 3-32 1*6 1 1-16 29-16 IM 7 1.065 .0178 1 .227 .893 2 1 15-16 25-16 1/4 1 3-16 253-64 i^ 6 1.160 .0208 1.485 1.057 23-16 2*6 2 17-32 18 15-16 33-32 ji^j 6 1.284 .0208 1.767 1.295 2% 25-16 2M 1 7-16 3 23-64 1% 5*6 1.389 .0227 2.074 1.515 29-16 2*6 2 31-32 J 19-16 3% m 5 5 1.491 1.G16 .0250 .0250 2.405 2.761 1.746 2.051 2% 3 15-16 211-16 2% 33-16 313-32 1 Jo 1 11-16 1 13-16 3 57-64 45-32 2 4*6 1.712 .0277 3.142 2.302 3*6 3 1-16 3% 2 1 15-16 427-64 2)4 4*6 1.962 .0277 3.976 3.023 37-16 41-16 2*4 23-16 461-64 2J^j 4 2.176 .0312 4.909 3.719 gl 3 13-16 4*6 2U 2 7-16 5 31-64 2M 4 2.426 .0312 5.940 4.620 43-16 4 29-32 2% 211-16 6 3 3*6 2.629 .0357 7.069 5.428 m 49-16 5% |3 215-16 6 17-32 3)4 3^ 2.879 .0357 8.296 6.510 5 415-16 5 13-16 3U 33-16 7 1-16 354 3)4 3.100 .0384 9.621 7.548 5% 55-16 67-64 3*6 37-16 739 64 3M 3 3.317 .0413 11.045 8.641 5 11-16 6 21-32 3% 3 11-16 gi,^ 4 3 3.567 .0413 12.566 9.993 6*1 61-16 7 3-32 14 3 15-16 8 41-64 4)4 2% 3.798 .0435 14.186 11.329 67-16 7 9-li? 4*4 43-16 93-16 4*6 2% 4.028 .0454 15.904 12.743 (3 7^ 6 13-16 731-32,4*6 47-16 Q3/ 4M 2% 4.256 .0476 17.721 14.226 7**4 73-16 8 13-324% 411-16 10*4 5 2^ 4.480 .0500 19.635 15.763 <% 79-16 8 27-32 5 4 15-16 10 49-64 5)4 2*6 4.730 .0500 21.648 17.572 8 7 15-16 9 9-32 ! 5*4 53-16 11 23-64 5)^2 2% 4.953 .0526 23.758 19.267 8% 85-16 9 23-32 5*6 5 7-16 11% 5%i 2% 5.203 .0526 25.967 21.262 854 811-16 105-32 ! 5% 5 11-16 12% 6 5.423 .0555 28.274 23.098 9*6 9 1-16 10 19-32 6 515-16 12 15-16 LIMIT GAUGES FOR IRON FOR SCREW THREADS. In adopting the Sellers, or Franklin Institute, or United States Standard, as it is variously called, a difficulty arose from the fact that it is the habit of iron manufacturers to make iron over- size, and as there are no over-size 206 MATERIALS. screws in the Sellers system, if iron is too large it is necessary to cut it away with the dies. So great is this difficulty, that the practice of making taps and dies over-size has become very general. If the Sellers system is adopted it is essential that iron should be obtained of the correct size, or very nearly so. Of course no high degree of precision is possible in rolling iron, and when exact sizes were demanded, the question arose how much allowable variationjthere should be from the true size. It was proposed to make limit- gauges for inspecting iron with two openings, one larger and the other smaller than the standard size, and then specify that the iron should enter the large end and not enter the small one. The following table of dimen- sions for the limit-gauges was - commended by the Master Car-Builders' Association and adopted by letter ballot in 1883. Size of Size of Size of Size of Size of Large Small Differ- Size of Large Small Differ- Iron. End of End of ence. Iron. End of End of ence. Gauge. Gauge. Gauge. Gauge. Hin. 0.2550 0.2450 0.010 96 in. 0.6330 0.6170 0.016 5-16 0.3180 0.3070 0.011 0.7585 0.7415 0.017 % 0.3810 0.3690 0.012 so 0.8840 0.8660 0.018 7l?6 0.4440 0.4310 0.013 \ 1.0095 0.9905 0.019 ^ 0.5070 0.4930 0.014 i*i 1.1350 1.1150 0.020 9-16 0.5700 0.5550 0.015 m 1.2605 1.2395 0.021 Caliper gauges with the above dimensions, and standard reference gauges for testing them, are made by The Pratt & Whitney Co. THE MAXIMUM VARIATION IN SIZE OF ROUGH IRON FOR U. S. STANDARD BOLTS. Am. Mach., May 12, 1892. By the adoption of the Sellers or U. S. Standard thread taps and dies keep their size much longer in use when flatted in accordance with this system than when made sharp " V," though it has been found advisable in practice in most cases to make the taps of somewhat larger outside diameter than the nominal size, thus carrying the threads further towards the V -shape and giving corresponding clearance to *he tops of the threads when in the nuts or tapped holes. Makers of taps and dies often have calls for taps and dies, U. S. Standard, " for rough iron." An examination of rough iron will show that much of it is rolled out of round to an amount exceeding the limit of variation in size allowed. In view of this it may be desirable to know what the extreme variation in iron may be, consistent with the maintenance of U. S. Standard threads, i.e., threads which are standard when measured upon the angles, the only placo where it seems advisable to have them fit closely. Mr. Chas. A. Bauer, the general manager of the Warder, Bushnell & Glessner Co., at Springfield, Ohio, in 1884 adopted a plan which may be stated as follows : All bolts, whether cut from rough or finished stock, are standard size at the bottom and at the sides or angles cf the threads, the variation for fit of the nut and allowance for wear of taps being made in the machine taps. Nuts are punched with holes of such size as to give 85 per cent of a full thread, expe rience showing that the metal of wrought nuts will then crowd into the threads of the taps sufficiently to give practically a full thread, while if punched smaller some of the metal will be cut out by the tap at the bottom of the threads, which is of course undesirable. Machine taps are made enough larger than the nominal to bring the tops of the threads up sharp, plus the amount allowed for fit and wear of taps. This allows the iron to be enough above the nominal diameter to bring the threads up full (sharp) at top, while if i ia small the only effect is to give a flat at top of threads ; neither condition affecting the actual size of the thread at the point at which it is intended co bear. Limit gauges are furnished to the mills, by which the iron is rolled, the maximum size being shown in the third column of the table. The minimum diameter is not given, the tendency in rolling being nearly always to exceed the nominal diameter. In making the taps the threaded portion is turned to the size given in the eighth column of the table, which gives 6 to 7 thousandths of an inch allow- ance for fit and wear of tap. Just above the threaded portion of the tap a SIZES OF SCKEW-THEEADS FOE BOLTS AND TAPS. 207 place is turned to the size given in the ninth column, these sizes being the same as those of the regular U. S. Standard bolt, at the bottom of the thread, plus the amount allowed for fit and wear of tap ; or, in other words, d' = U. S. Standard d + (D f D). Gauges like the one in the cut, Fig. 72, are furnished for this sizing. In finishing the threads of the tap a tool FIG. 72. is used which has a removable cutter finished accurately to gauge by grind- ing, this tool being correct U. S. Standard as to angle, and flat at the point. It is fed in and the threads chased until the flat point just touches the por- tion of the tap which has been turned to size d'. Care having been taken with the form of the tool, with its grinding on the top face (a fixture being provided for this to insure its being ground properly), and also with the set- ting of the tool properly in the lathe, the result is that the threads of the tap are correctly sized without further attention. It is evident that one of the points of advantage of the Sellers-system is sacrificed, i.e., instead of the taps being flatted at the top of the} threads they are sharp, and are consequently not so durable as they otherwise would be ; but practically this disadvantage is not found to be serious, and is far overbalanced by the greater ease of getting iron within the prescribed limits ; while any rough bolt when reduced in size at the top of the threads, by filing or otherwise, will fit a hole tapped with the U. S. Standard hand taps, thus affording proof that the two kinds of bolts or screws made for the two different kinds of work are practically interchangeable. By this system \" iron can be .005" smaller or .0108" larger than the nominal diameter, or, in other words, it may have a total variation of .0158", while 1" iron can be .0105" smaller or .0309" larger than nominal a total variation of .0414" and within these limits it is found practicable to procure the iron. STANDARD SIZES OF SCREW-THREADS FOR BOLTS AND TAPS. (CHAS. A. BAUER.) 1 2 3 4 5 6 7 8 9 10 A n D d h / D' -D D' d' H Inches. Inches Inches. Inches. Inches. Inches. Inches. Inches H 20 .2G08 .1855 .0379 .0062 .006 .2668 .1915 .2024 5-16 18 .3245 .2403 .0421 .0070 .006 .3305 .2463 .2589 % 16 .3885 .2938 .0474 .0078 .006 .3945 .2998 .3139 7-16 14 .4530 .3447 .0541 .0089 .006 .4590 .3507 .3670 M 13 .5166 .4000 .0582 .0096 .006 .5223 .4060 .4236 9-16 12 .5805 .4543 .0631 .0104 .007 .5875 .4613 .4802 % 11 .6447 .5069 .0689 .0114 .007 .6517 .5139 .5346 34 10 .7717 .620! .0758 .0125 .007 .7787 .6271 .6499 % 9 .8991 .7307 .0842 .0139 .007 .9061 .7377 .7630 I 8 1.0271 .8376 .0947 .0156 .007 1.0341 .8446 .8731 V/B 7 1.1559 .9394 .1083 .0179 .007 1.1629 .9464 .9789 V/4, 7 1.2809 1.0644 .1083 .0179 .007 1.2879 1.0714 1.1039 A = nominal diameter of bolt. D = actual diameter of bolt. d = diameter of bolt at bottom of thread. n = number of threads per inch. / = flat of bottom of thread. h depth of thread. D f and d' diameters of tap. H = hole in nut before tapping. 208 MATERIALS. STANDARD SET-SCREWS AND CAP-SCREWS. American, Hartford, and Worcester Machine-Screw Companies. (Compiled by W. S. Dix.) (A) (B) (C) (D) (E) (F) (G) Diameter of Screw. . . . K 3-16 /4 5-16 % 7-16 7& Threads per Inch Size of Tap Drill* 40 No. 43 24 No. 30 No. 5 18 17-64 16 21-64 14 12 27-64 (H) (D (J) (K) (L) (M) (N) Diameter of Screw.. . . 9-16 Ys H % 1 1^6 1J4 Threads per Inch 12 11 10 9 8 7 7 Size of Tap Drill*.... 31-64 17-32 21-32 49-64 % 63-64 *M Set Screws. Hex. Head Cap-screws. Sq. Head Cap-screws. Short Diam. of Head (C) < (D) 5-16 $ & JM. (I) K (ft % (L) 1 (M) 1L (N) 1M Long Diam. of Head .44 .53 .62 .71 .80 .89 1.06 1.24 1.42 1.60 1.77 Lengths (under Head). Short Diam. of Head. 7-16 Long Diam, of Head. .51 .58 .65 .72 .87 .94 1.01 1.15 1.30 1.45 1.59 1.73 Lengths (under Head). Short Diam. of Head. to 3 7-16 9-16 11-16 Loug Diam. of Head. .53 .62 .71 1.06 1.24 1.60 1.77 1.95 2.13 Lengths (under Head). Round and Filister Head Cap-screws. Diam. of Head. 3-16 Lengths (under Head). Flat Head Cap-screws. Button-head Cap- screws. Diam. of Head. Lengths (including Head). Diam. of Head. 7-32 (.225) 5-16 7-16 9-16 13-16 15-16 1 Lengths (under Head). * For cast iron. For numbers of twist-drills see p. 29. Threads are U. S. Standard. Cap-screws are threaded % length up to and including I" diam. x 4" long, and &j length above. Lengths increase by J4" each regular size between the limits given. Lengths of heads, except flat and button, equal diam. of screws. The angle of the cone of the flat-head screw is 76, the sides making angles of 52 with the top. STANDARD MACHINE SCREWS. 209 STANDARD MACHINE SCREWS. No. Threads per Inch. Diam. of Body. Diam. of Flat Head. Diam. of Round Head. Diam. of Filister Head. Lengths. From To 2 56 .0842 .1631 .1544 .1332 3-16 2* 3 48 .0973 .1894 .1786 .1545 3-16 K 4 32, 36, 40 ,1105 .2158 .2028 .1747 3-16 % 5 32, 36, 40 .1236 .2421 .2270 .1985 3-16 % 6 30, 32 .1368 .2684 .2512 .2175 3-16 1 7 30,32 .1500 .2947 .2754 .2392 /4 l/*6 8 30, 32 .1631 .3210 .2936 .2610 /4 1/4 9 24, 30, 32 .1763 .3474 .3238 .2805 /4 I&2 10 24, 30, 32 .1894 .3737 .3480 .3035 /4 ji^j 12 20,24 .2158 .4263 .3922 .3445 % 1% 14 20, 24 .2421 .4790 .4364 .3885 % 2 16 16, 18, 20 .2684 .5316 .4866 .4300 % 2^4 18 16, 18 ,2947 .5842 .5248 .4710 i^j 2^J 20 16, 18 .3210 .6308 .5690 .5200 8 24 22 16,18 .3474 .6894 .6106 .5557 3 24 14, 16 .3737 .7420 .6522 .6005 /'is 3 26 14, 16 .4000 .7420 .6938 .6425 % 3 28 14, 16 .4263 .7946 .7354 .6920 % 3 30 14, 16 .4520 .8473 .7770 .7240 1 3 Lengths vary by 16ths from 3-16 to J^, by 8ths from ^ to 1J4 by 4ths from 1^ to 3. SIZES AND WEIGHTS OF SQUARE AND HEXAGONAL NUTS. United States Standard Sizes. Chamfered and trimmed. Punched to suit U. S. Standard Taps. s s y 5-16 7?16 9-?6 2 2^4 I I 2 15-16 13-64 1*9-64 11-32 25-64 29-64 33-64 39-64 47-64 53-64 59-64 1-16 5-32 9-32 1 13-32 1 23-32 1 15-16 2 3-16 2 7-16 2% S3' 11-16J 13-161 9-16 11-16 13-16 7-16 2' '1-16 11-16 2 5-161 1% 2 9-16 2 1-16 2 13-16 2 5-16 2 15-16 3 3-16 4 7-16 4 15-16 4 1-16 4 15-16 5 5-16 Square. Hexagon. 8 fee 8 gj .sj jb . C 03 8jg .s ~f i* 7270 4700 .0138 .0281 7615 5200 .0131 .0192 2350 .0426 3000 .0333 1630 .0613 2000 .050 1120 .0893 1430 .070 890 .1124 1100 .091 640 .156 740 .135 380 .263 450 .222 280 .357 309 .324 170 .588 216 .463 130 .769 148 .676 96 1.04 111 .901 70 1.43 85 1.18 58 1.72 68 1.47 44 2.27 56 1.79 34 2.94 40 2.50 30 3.33 37 2.70 23 4.35 29 3.45 19 5.26 21 4.76 12 8.33 15 6.67 9 11.11 11 9.09 IK 13.64 8^ 11.76 210 MATERIALS. tt 4 - R H C o o vj I OD . . J>. r-< lO O ^J 1 Oi CO O* T-t C ^ x .Q COTj< 1 i5'lOlOOCDI>-OOO rg * ^ cc >^ o 5 10 10 10 K ^O" COiOt^O5-> iCOiOC'-OS'-'COM -1 itnOiCOt>-' iCOOrfQCS^l-^^w C'COCOCOCO'^ 1 ^9''^ 1 ^T^3'lOiOlOCOCDCOi-t^-i."-aDQOOSOSOSOOO 2^-iT*iOOOOOOOOOOOOOOOO SOOOrH OiCOT QOOOOO ^aocooiio^-'J>cooco<?>co > *ocoGOT-iooT I ;i;oco;oi.~GcaoosooT-.i-,(?*cocoTfcoi--a SQOOOOCOCOCSWOOiOOlOpiqOOOO I _o t~ os rt" od o co* o io" oi oo i> I-H CD o c co" -<* ti TJ os ao co o co <M! o oo co' 10' -i< ^cocOrf-^iooeococDJ.--i-oOGOOSO5OT-('^cocOTf'Ocot.-aoc:osO'-iO? O CO CO OS OJ lO 00 i-; 10 0* OS t- 10 CO i-; O O O O O O "\ |SoO?if5J>OCOCOOST4rJ<t TH I SG^C* OJ WCOCOCOCO^TJiT >* TH O 10 O 10 O 10 O 10 O 10 O 10 O ' ??Tr^r?^90?oos^oOt-jcogi > CO OS 10 r-. t>G 50 M^ ^ ' i- i- 10 O 10 OT-K?JCOr>*lOt^GOOSC ^OiCO^C^OSCDCOOi.-^i-iGOlOC TRACK BOLTS. With United States Standard Hexagon Nuts, Rails used. Bolts. Nuts. No. in Keg, 200 Ibs. Kegs per Mile. \ %x4^ H 230 6.3 %x4 1? 240 6. 45to851bs.. J %x3% 8 254 260 5.7 5.5 1 % x 3^ /4 266 5.4 I %x3 M 283 5.1 r %x3^ 1-16 375 4. 30to401bs.. J %x3 1-16 1-16 410 435 3.7 3.3 1 ^ x2 H 1 1-16 465 3.1 [ 1^x3 % 715 2. 20to301bs..J ^xl^ n 760 800 2. 2. I 1^x2 ? 820 2. RIVETS TUIIKBUCKLES. CONK-HEAD BOILER RIVETS, WEIGHT PER 1OO. (Hoopes & Townsend.) Diam., in., Scant. 1/2 9/16 5/8 11/16 X 13/16 % 1 i* W Length. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. Ibs. %inch 8.75 18.7 16.20 H " 9.35 14.4 17.22 i ;' 10.00 15.2 18.25 21.70 26.55 10.70 16.0 19.28 23.10 28.00 /4 " 11.40 16.8 20.31 24.50 29.45 37.0 46 60 % " 12.10 17.6 21.34 25.90 30.90 38.6 48 63 95 L " 12.80 18.4 22.37 27.30 32.35 40.2 50 65 98 133 5X " 13.50 19.2 23.40 28.70 33.80 41.9 52 67 101 137 M " 14.20 20.0 24.43 30.10 35.25 43.5 54 69 104 141 y& " 14.90 20.8 25.46 31.50 36.70 45.2 56 71 107 145 2 " 15.60 21.6 26.49 32.90 38.15 47.0 58 74 110 149 jji^ ** 16.30 22.4 27.52 34.30 39.60 48.7 60 77 114 153 2J4 " 17.00 23.2 28.55 35.70 41.05 50.3 62 80 118 157 2% " 17.70 24.0 29.58 37.10 42.50 51.9 64 83 121 161 2Vji> " 18.40 24.8 30.61 38.50 43.95 53.5 66 86 124 165 2% " 19.10 25.6 31.64 39.90 45.40 55.1 68 89 127 169 m " 19.80 26.4 32.67 41.30 46.85 56.8 70 92 130 173 2% ' 20.50 27.2 33.70 42.70 48.30 58.4 72 95 133 177 3 21.20 28.0 34.73 44.10 49.75 60.0 74 98 137 181 3^4 ' 22.60 29.7 36.79 46.90 52.65 63.3 78 103 144 189 3Vi* ' 24.00 31 5 38.85 49.70 55.55 66.5 82 108 151 197 33 ' 25.40 33.3 40.91 52.50 58.45 69.8 86 113 158 205 4 * 26.80 35.2 42.97 55.30 61.35 73.0 90 118 165 213 4*4 ' 28.20 36.9 45.03 58.10 64.25 76.3 94 124 172 221 4^5 ' 29.60 38.6 47.09 60.90 67.15 79.5 98 130 179 229 4^4 ' 31.00 40.3 49.15 63.70 70.05 82.8 102 136 186 237 5 32.40 42.0 51.21 66.50 72.95 86.0 106 142 193 245 5/4 * 33.80 43.7 53.27 69.20 75.85 89.3 no 148 200 254 5^ ' 35.20 45.4 55.33 72.00 78.75 92.5 114 154 206 263 5M ' 36.60 47.1 57.39 74.80 81.65 95.7 118 160 212 272 6 38.00 48.8 59.45 77.60 84.55 99.0 122 166 218 281 6J4 ' 40.80 52.0 63.57 83.30 90.35 105.5 130 177 231 297 7 ' 43.60 55.2 67.69 88.90 96.15 112.0 138 188 245 314 Heads 5.50 8.40 11.50 13.20 18.00 23.0 29.0 38.0 56.0 77.5 * These two sizes are calculated for exact diameter. Rivets with button heads weigh approximately the same as cone-head rivets. T URN BUCK LES. (Cleveland City Forge and Iron Co.) Standard sizes made with right and left threads. D = outside diameter of screw. A = length in clear between heads = 6 ins. for all sizes. B = length of tapped heads = l^D nearly. C = 6 ins. + 3D nearly. 212 MATERIALS. SIZES OF WASHERS. Diameter in inches. Size of Hole, in inches. Thickness, Birmingham Wire-gauge. Bolt in inches. No. in 100 Ibs. a. 5-16 No. 16 M 29,300 a? H " 16 516 18,000 1 7-16 ** 14 % 7,600 9-16 " 11 14 3,300 jijj? % 44 11 9-16 2,180 1^3 11-16 ** 11 % 2,350 . 1% 13-16 * 11 % 1,680 2 31-32 * 10 % 1,140 2^ 1^ " 8 1 580 252 1J4 * 8 l/^ 470 3 1% tt 7 jix 360 3 ig * 6 ift 860 TRACK SPIKES* Rails used. Spikes. Number in Keg, 200 Ibs. Kegs per Mile, Ties 24 in. between Centres. 45 to 85 5^x9-16 880 30 40 " 52 5 x9-16 400 27 35 ** 40 5 xU 490 22 24 " 35 550 20 24 " 30 4J4 x 7-16 725 15 18 " 24 4 x7-16 820 13 16 " 20 8J4x% 1250 9 14 " 16 3 x % 1350 8 8 " 12 2^x% 1550 7 8 " 10 2^x5-16 2200 5 STREET RAILWAY SPIKES. Spikes. Number in Keg, 200 Ibs. Kegs per Mile, Ties 24 in. between Centres. 5^x9-16 5 x^ 4J^x7-16 400 575 800 30 19 13 BOAT SPIKES. Number in Keg of 200 Ibs. Length. H 5-16 H H 4 inch. 2375 5 " 2050 1230 940 6 ' 7 " 1825 1175 990 800 650 450 375 8 " 880 600 335 9 525 300 10 " 475 275 SPIKES; CUT KAILS. 213 WROUGHT SPIKES. Number of Nails in Keg of 15O Founds. Size. Min. 5-16 in. fcin. 7-16 in. Kin. 3 inches . 2250 3U " 1890 1208 f" .. 1650 1135 4U * 1464 1064 5^ " 1380 930 742 6 " 1292 868 570 7 " .. . 8 M 1161 662 635 482 455 445 384 306 256 9 573 424 300 240 10 * 391 270 222 11 " 249 203 IS * 236 180 WIRE SPIKES. Size. Approx. Size of Wire Nails. Ap. No. in 1 Ib. Size. Approx. Size of Wire Nails. Ap. No. in 1 Ib. lOd Spike.... 3 in. No. 7 50 60d Spike . . . 6 in. No. 1 10 16d " 3^ " " 6 35 6^ in. 44 .. . 6^ *' ** 1 9 20d * 4 "5 26 7 " " . . 7 7 30d 4^ " 4 20 8 " " .. . 8 " " 00 5 40d " 5 " "3 15 9 ** u 9 " 00 4 Hi 50d * % " ** 2 12 LENGTH AND NUJttRER OF CUT NAILS TO THE POUND. Size. ! Common. 1 '> i PR Finishing. | s Barrel. 1 03 1 Tobacco. Cut Spikes. M %in 800 7/| 500 2d 1 800 1100 1000 376 3d... WA 480 720 760 224 4d u& 288 523 368 180 398 5d m 200 410 130 6d 2 168 9^ 84 268 224 126 96 7d 124 74 61 188 98 82 8d .... 2V6 88 62 48 146 J28 75 68 9d 9g/ 70 *)S 8fi 130 110 65 lOd 3 58 46 30 102 91 55 28 )2d. 3V4 44 d 0^ 76 71 40 16d....i.. l& 34 ^S 62 54 27 99 20d f~ 23 88 16 54 40 14i SOd VA 18 33 12i/ 40d 5 14 27 9vl 50d 51^ 10 8 60d 6 /6 8 6 214 MATERIALS. S9ZJS fl : : o : <c o I'd'd'd 'etS'd'd'd'CJ'd' oj co eo <3<ir3?ooooo5O<? saqoui ' - SS3 :i :$2SS PITB qioouis PUB 'uisfcQ Suiqsmj.i paqa-e UOIUUIOQ sil^N noraraoo "^^ ,_ ,_ ^ ^ TH - 1 _ 4 ciwwwwcocoTf ^o o APPROXIMATE NUMBER OF WIRE NAILS PER POUND. 215 bT>!Ts co ^o ;.;;;;;;;;;; .M g ^rio i loVooo-* j I <Ot>OOOiT-iOOlOOO ^^Sc^S^o """"-"--""'^gg: : ; ; ; 5S^^^io?:SS2gg||||||| : : j 5 ' oot>^oS^^i^o^^"w^^^ x^f oc^^fooott-cOTHTt<' rfi?o(7't^JOr:oc^' 0\ rHr-ii-ir-iOi<MOCOTt<iOt-O!->iOOt>ir:O05CO ^o ... -S^i^S ~" T^ 1^1 TH 5^ CO ^ ^ j i j i : : j : : i i : i : Jill I- g w 2ib MATERIALS. SIZE, WEIGHT, LENGTH, AND STRENGTH OF IRON WIRE. (Trenton Iron Co.) Tensile Strength (Ap. No. by Wire Gauge. Diam. in Deci- mals of One Area of Section in Decimals of One Inch. Feet to the Pound. Weight of One Mile in pounds. proximate) of Charcoal Iron Wire in Pounds. Inch. Bright. Annealed. 00000 .450 .15904 1.863 2833.248 12598 9449 0000 .400 .12566 2.358 2238.878 9955 7466 000 .360 .10179 2.911 1813.574 8124 6091 00 .330 .08553 3.465 1523.861 6880 5160 .305 .07306 4.057 1301.678 5926 4445 1 .285 .06379 4.645 1136.678 5226 3920 2 .265 .05515 5.374 982 555 - 4570 3425 3 .245 .04714 6.286 839.942 3948 2960 4 .225 .03976 7.454 708.365 3374 2530 5 .205 .03301 8.976 588.139 2839 2130 6 .190 .02835 10.453 505.084 2476 1860 7 .175 .02405 12.322 428.472 2136 1600 8 .160 .02011 14.736 358.3008 1813 1360 9 .145 .01651 17.950 294.1488 1507 1130 10 .130 .01327 22.333 236.4384 1233 925 11 .1175 .01084 27.340 193.1424 1010 758 12 .105 .00866 34.219 154.2816 810 607 13 .0925 .00672 44.092 119.7504 631 473 14 .080 .00503 58.916 89.6016 474 356 15 .070 .00385 76.984 68.5872 372 280 16 .061 .00292 101.488 52.0080 292 220 17 .0525 .00216 137.174 38.4912 222 165 18 .045 .00159 186.335 28.3378 169 127 19 .040 .0012566 235.084 22.3872 137 103 20 .035 .0009621 308.079 17.1389 107 80 21 .031 .0007547 392.772 13.4429 22 .028 .0006157 481.234 10.9718 ' "22 2~ 23 .025 .000*909 603.863 8.7437 C?2iO <U~ ~ ~ 24 .0225 .0003976 745.710 7.0805 41 2*- 4 >a ^ 25 .020 .0003142 943.396 5.5968 5" i32S** ** 26 .018 .0002545 1164.689 4.5334 ^J'S | ! d 27 .017 .0002270 1305.670 4.0439 28 .016 .0002011 1476.869 3.5819 T3 ?J D-^ ol O * 2 29 .015 .0001767 1676.989 3.1485 43 S^'> -2''* tw fl 30 .014 .0001539 1925.321 2.7424 s S^li ^.s"*!^ ** 31 .013 .0001327 2232.653 2.3649 w -g SH .2S J^le o> ^ 32 .012 .0001131 2620.607 2.0148 2-2 ujS ^'j^^to O 33 .011 .0000950 3119.092 1.6928 id'~fi.h^ : ~~ M 34 35 .010 .0095 .00007854 .00007088 3773.584 4182.508 1.3992 1.2624 1 36 .009 .00006362 4657.728 1.1336 O-^-rt ^^ t >o 37 .0085 .00005675 5222.035 1.0111 * w'ejf! '"SW jf3 j| 88 .008 .00005027 5896.147 .89549 d>.Q E_S < o'!5 ; S*^ *" 39 .0075 .00004418 6724.291 .78672 40 .007 00003848 7698.253 .68587 TESTS OF 'TELEGRAPH WIRE. 217 GALVANIZED IRON WIRE FOR TELEGRAPH AND TELEPHONE LINES. (Trenton Iron Co.) WEIGHT PER MILE-OHM. This term is to be understood as distinguishing the resistance of material only, and means the weight of such material re- quired per mile to give the resistance of one ohm. To ascertain the mileage resistance of any wire, divide the " weight per mile-ohm " by the weight of the wire per mile. Thus in a grade of Extra Best Best, of which the weight per mile-ohm is 5000, the mileage resistance of No. 6 (weight per mile 525 Ibs.) would be about 9J^ ohms; and No. 14 steel wire, 6500 ibs. weight per mile-ohm (95 Ibs. weight per mile), would show about 69 ohms. Sizes of \Virc used in Telegraph and Telephone Lines. No. 4. Has not been much used until recently; is now used on important lines where the multiplex systems are applied. No. 5. Little used in the United States. No. 6. Used for important circuits between cities. No. 8. Medium size for circuits of 400 miles or less. No. 9. For similar locations to No. 8, but on somewhat shorter circuits ; until lately was the size most largely used in this country. Nos. 10, 11. For shorter circuits, railway telegraphs, private lines, police and fire-alarm lines, etc. No. 12. For telephone lines, police and fire-alarm lines, etc. Nos. 13, 14. For telephone lines and short private lines: steel wire is used most generally in these sizes. The coating of telegraph wire with zinc as a protection against oxidation is now generally admitted to be the most efficacious method. The grades of line wire are generally known to the trade as " Extra Best Best " (E. B. B.), " Best Best " (B. B.). and "Steel." " Extra Best Best " is made of the very best iron, as nearly pure as any commercial iron, soft, tough, uniform, and of very high conductivity, its weight per mile-ohm being about 5000 Ibs. The " Best Best 11 is of iron, showing in mechanical tests almost as good results as the E. B. B., but not quite as soft, and being somewhat lower in conductivity; weight per mile-ohm about 5700 Ibs. The Trenton " Steel " wire is well suited for telephone or short telegraph lines, and the weight per mile-ohm is about 6500 Ibs. The following are (approximately) the weights per mile of various sizes of galvanized telegraph wire, drawn by Trenton Iron Co.'s gauge: No. 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. Lbs. 720, 610, 525, 450, 375, 310, 250, 200, 160, 125, 95. TESTS OF TELEGRAPH WIRE. The following data are taken from a table given by Mr. Prescott relating to tests of E. B. B. galvanized wire furnished the Western Union Telegraph Co.: Size of Wire. Diam. Parts of One Inch. Weight. Length. Feet per pound. Resistance. Temp. 75.8 Fahr. Ratio of Breaking Weight to Weight per mile. Grains, per foot. Pounds per mile. Feet per ohm. Ohms per mile. 4 .238 1043.2 886.6 6.00 958 5.51 5 .220 891.3 673.0 7.85 727 7.26 6 .203 758.9 572.2 9.20 618 8.54 3.05 7 .180 596.7 449.9 11.70 578 10.86 3.40 8 .165 501.4 378.1 14.00 409 12.92 3.07 9 .148 403.4 304.2 17.4 328 16.10 3.38 10 .134 330.7 249.4 21.2 269 19.60 3.37 11 .120 265.2 200.0 26.4 216 24.42 2.97 12 .109 218.8 165.0 32.0 179 29.60 3.43 14 .083 126.9 95.7 55.2 104 51.00 3.05 JOINTS IN TELEGRAPH WIRES. The fewer the joints in a line the better. All joints should be carefully made and well soldered over, for a bad joint may cause as much resistance to the electric current as several miles of wire. 218 MATERIALS. ooi-ie< - i T-i -i eo IH T- >H U O O5 -* -^ >H o> m Ill 15" 11 H^SSc5<N DIMENSIONS, WEIGHT, RESISTANCE OF OOPPEB WIRE. #19 Op. & A o 5 s s oo 320 MATERIALS. ^t-^ liil 3^^0*5 IS <NOJo5^g Hr-lSnSlNOJCO-^lOOOOOC^ SOOOOOOOOOOrti- 3SSoS83 M 3Sll2llSl8ll ?SooIo^5t2t2?2ooaco2H jJTH^oiinNoJt-lto^sceooii-tiHrHO *iSg?2||i|ssis|IsgigS| rf OHJI>. to 10 eo ***< o> ooooco5)5>SS HARD-DRAWK COPPER WIRE; INSULATED WIRE. 221 HARD-DRAWN COPPER TELEGRAPH WIRE. (J. A. Roebling's Sons Co.) Furnished in half-mile coils, either bare or insulated. Size, B. & S. Gauge. Resistance in Ohms per Mile. Breaking . Strength. Weight per Mile. Approximate Size of E. B.B. Iron Wire equal to Copper. 9 4.30 625 209 2 t? 10 5.40 525 166 3 I 11 6.90 420 131 4 I 12 8.70 330 104 6 | 13 10.90 270 83 6^3 14 13.70 213 66 8 15 17.40 170 52 9 16 22.10 130 41 10 <g CD In handling this wire the greatest care should be observed to avoid kinks, bends, scratches, or cuts. Joints should be made only with Mclntire Con- nectors. On account of its conductivity being about five times that of Ex. B. B. Iron Wire, and its breaking strength over three times its weight per mile, copper maybe used of which the section is smaller and the weight less than an equivale&t iron wire, allowing a greater number of wires to be strung on the poles. Besides this advantage, the reduction of section materially decreases the electrostatic capacity, while its non-magnetic character lessens the self-in- duction of the line, both of which features tend to increase the possible speed of signalling in telegraphing, and to give greater clearness of enunci- ation over telephone lines, especially those of great length. INSULATED COPPER WIRE, WEATHERPROOF INSULATION. Double Braid. Triple Braid. Approximate Num- Weights, bers, Outside Weights, Outside Weights, Pounds. B. &S. Diame- Pounds. Diame- Pounds. Gauge. ters in ters in 32ds Inch. 1000 Feet. Mile. Inch. 1000 Feet. Mile. Reel. Coil. 0000 20 716 3781 24 775 4092 2000 250 000 18 575 3036 22 630 3326 2000 250 00 17 465 2455 18 490 2587 500 250 15 375 1980 17 400 2112 500 250 1 15 285 1505 16 306 1616 500 250 2 14 245 1294 15 268 1415 500 250 I 13 190 1003 14 210 1109 500 250 11 152 803 12 164 866 250 125 5 10 120 G34 11 145 766 260 130 6 9 98 518 10 112 691 275 140 8 8 66 349 9 78 412 200 100 10 7 45 238 8 55 290 200 100 12 6 30 158 7 35 185 .... 25 14 5 20 106 6 26 137 25 16 4 14 74 5 20 106 .... 25 18 3 10 53 4 16 .... 25 MATERIALS, Power Cables. Lead Incased, Jute or Paper Insulated* (John A. Roebling's Sons Co.) Nos,. B.&S.G. Circular Mils. Outside Diam. Inches. Weights, 1000 feet. Pounds. Nos., B.&S.G. Circular Mils. Outside Diam. Inches. Weights, 1000 feet. Pounds. 1000000 900000 800000 750000 700000 650000 600000 550000 500000 450000 400000 350000 1 13/16 1 23/32 21/32 19/32 9/16 17/32 1/ie V 5/16 6685 6228 5773 5543 5315 5088 4857 4630 4278 3923 3619 3416 300000 250000 211600 168100 133225 105625 83521 66564 52441 41616 26244 1M 1 3/16 1 3/32 1 1/16 15/16 29/32 % 25/32 % 11/16 3060 2732 2533 2300 2021 1772 1633 1482 1360 1251 1046 0000 " ' 000 00 1 2 3 4 6 Stranded Weather-proof Feed Wire. Circular Mils. 1000000 900000 800000 750000 700000 650000 600000 Outside Diam. Inches. Weights. Pounds. Approximate length on reels. Feet. Circular Mils. Outside Diam. Inches. Weights. Pounds. Approximate 1 length on reels. Feet. 1000 feet. Mile. 1000 feet. Mile. 1 13/32 1 11/32 1 5/16 1 9/32 H 1 7/32 3550 3215 2880 2713 2545 2378 2210 18744 16975 15206 14325 13438 12556 11668 800 800 850 850 900 900 1000 550000 500000 450000 400000 350000 300000 250000 3/16 X 3/32 1/16 15/16 29/32 2043 1875 1703 1530 1358 1185 1012 10787 9900 8992 8078 7170 6257 5343 1200 1320 1400 1450 1500 1600 1600 The table is calculated for concentric strands. Rope-laid strands are larger. Approximate Rules for the Resistance of Copper Wire. The resistance of any copper wire at 20 C. or 66 F., according to Mat: thiessen's standard, is E = ^p in which E is the resistance in inter- national ohms, I the length of the wire in feet, and d its diameter in mils. (1 mil = 1/1000 inch.) A No. 10 Wire, A.W.G., .1019 in. diameter (practically 0.1 in.), 1000 ft. in length, has a resistance of 1 ohm at 68 F. and weighs 31.4 Ibs. If a wire of a given length and size by the American or Brown & Sharpc gauge has a certain resistance, a wire of the same length and three numbers higher has twice the resistance, six numbers higher four times the resist- ance, etc. Wire gauge, A.W.G. No 000 1 4 7 10 13 16 19 22 Relative resistance 16 8 4 2 11/2 1/4 1/8 1/16 section or weight.. 1/16 1/8 1/4 1/212 4 8 16 Approximate rules for resistance at any temperature : v f : R = resistance at 0, R f = resistance at the temperature t C., I - in feet, d - diameter iii c mils. (See Copper Wire Table, p. 1034.) length STEEL WIRE CABLES. 223 GALVANIZED STEEL-WIRE STRAND. For Smokestack Guys, Signal Strand, etc. (J. A. Roebling's Sons Co.) This strand is composed of 7 wires, twisted together into a single strand. L 4>.J br.~ LI S^" dg 5 ^ h eg'd ^3? -S^bJC a> -2*2 "Ss qj ^S I .1 "So ss HI a> | 5?o |ll (0 I || III p ^ 1 O 1 , 3 ^ W in. Ibs. Ibs. in. Ibs. Ibs. in. Ibs. Ibs. \$ 51 8,320 9/32 18 2,600 5/32 4J^ 700 15/32 48 7,500 17/64 15 2,250 9/64 3Vi* 525 7/16 37 30 6,000 4,700 7/32 \\V 2 1,750 1,300 3/32 2^4 2 375 320 5^16 21 3,300 3/16 *y* 1,000 For special purposes these strands can be made of 50 to 100 per cent greater tensile strength. When used to run over sheaves or pulleys the use of soft-iron stock is advisable. FLEXIBLE STEEL-WIRE CABLES FOR VESSELS. (Trenton Iron Co., 1886.) With numerous disadvantages, the system of working ships' anchors with chain cables is still in vogue. A heavy chain cable contributes to the hold- ing-power of the anchor, and the facility of increasing that resistance by paying out the cable is prized as an advantage. The requisite holding- power is 'obtained, however, by the combined action of a comparatively light anchor and a corresponding^ 7 great mass of chain of little service in proportion to its weight or to the weight of the anchor. If the weight and size of the anchor were increased so as to give the greatest holding-power required, and it were attached by means of a light wire cable, the combined weight of the cable and anchor would be much less than the total weight of the chain and anchor, and the facility of handling would be much greater. English shipbuilders have taken the initiative in this direction, and many of the largest and most serviceable vessels afloat are fitted with steel- wire cables. They have given complete satisfaction. The Trenton Iron Co/s cables are made of crucible cast-steel wire, and guaranteed to fulfil Lloyd's requirements. They are composed of 72 wires subdivided into six strands of twelve wires each. In order to obtain great flexibility, hempen centres are introduced in the strands as well as in the completed cable. FLEXIBLE STEEL-WIRE HAWSERS. These hawsers are extensively used, They are made with six strands of twelve wires each, hemp centres being inserted in the individual strands as well as in the completed rope. The material employed is crucible cast steel, galvanized, and guaranteed to fulfil Lloyd's requirements. They are only one third the weight of hempen hawsers; and are sufficiently pliable to work round any bitts to which hempen rope of equivalent strength can be applied. 13-inch tarred Russian hemp hawser weighs about 39 Ibs. per fathom. 10-inch white manila hawser weighs about 20 Ibs. per fathom. 1^-inch stud chain weighs about (58 Ibs. per fathom. 4-inch galvanized steel hawser weighs about 12 /6s. per fathom. Each of the above named has about the same tensile strength. 224 MATERIALS. SPECIFICATIONS FOR GALVANIZED IRON WIRE. Issued by the British. Postal Telegraph Authorities. Weight per Mile. Diameter. Tests for Strength and Ductility. sk* &* * * s * o -2 g V) oS^lau Sg T3 t^ ^9 a fl .2 rti 5 C 8 -ed Standar Allowed. d Standard Allowed. Breakir Weigh d ing Weight ss than d ~ :ing Weight ss than 0* ft Resistance of the S Size at 6 t, being Sta t x Resista 'o* g s i 1 | 1 5 | 8 3 3 | & S a i I S 1 M i M a I g^ i 1 1 M | | o "3 o I N 1 O Ibs. Ibs. Ibs. mils. mils. mils. Ibs. Ibs. Ibs. ohms. 800 767 833 242 237 247 2480 15 2550 14 2620 13 6.75 5400 600 571 629 209 204 214 1860 17 1910 16 1960 15 9.00 5400 450 424 477 181 176 186 1390 19 1425 18 1460 17 12.00 5400 400 377 424 171 166 176 1240 21 1270 20 1300 19 13.50 5400 200 190 213 121 118 125 620 30 638 28 655 26 27.00 5400 STRENGTH OF PIANO-WIRE. The average strength of English piano- wire is given as follows by Web ster, Horsfals & Lean: Numbers Equivalents Ultimate Numbers Equivalents Ultimate. in Music- in Fractions Tensile in Music- in Fractions Tensile wire Gauge. of Inches in Diameters. Strength in Pounds. wire Gauge. of inches in Diameters. Strength in Pounds. 12 .029 225 18 .041 395 13 .031 250 19 .043 425 14 .033 285 20 .045 500 15 .035 305 21 .047 540 16 .037 340 22 .052 650 17 .039 360 ' These strengths range from 300,000 to 340,000 Ibs. per sq. in. The compo- sition of this wire is as follows: Carbon, 0.570; silicon, 0.090; sulphur, C Oil; phosphorus, 0.018; manganese, 0.425. " PLOUGH "-STEEI, 1IRE. The term "plough," given in England to steel wire of high quality, was derived from the fact that such wire is used for the construction of ropes used for ploughing purposes. It is to be hoped that the term will not be Tised in this country, as it tends to confusion of terms. Plough-steel is known here in some steel- works as the quality of plate steel used for the mould-boards of ploughs, for which a very ordinary grade is good enough. Experiments by Dr. Percy on the English plough-steel (so-called) gave the following results: Specific gravity, 7.814; carbon, 0.828 per cent; manga- nese, 0.587 per cent; silicon, 0.143 per cent; sulphur, 0.009 percent; phos- phorus, nil; copper, 0.030 per cent. No traces of chromium, titanium, or tungsten were found. The breaking strains of the wire were as follows: Diameter, inch 093 .132 .159 .191 Pounds per sq. inch 344,960 257,600 224,000 201,600 The elongation was only from 0.75 to 1.1 per cent. SPECIFICATIONS FOR HARD-DRAWH COPPER WIRE. 225 WIRES OF DIFFERENT METALS AND ALLOYS. (J. Bucknall Smith's Treatise oil Wire.) Brass "Wire is commonly composed of an alloy of 1 3/4 to 2 parts of copper to 1 part of zinc. The tensile strength ranges from 20 to 40 tons per square inch, increasing with the percentage of zinc in the alloy. German or Nickel Silver, an alloy of copper, zinc, and nickel, ia practically brass whitened by the addition of nickel. It has been drawn into -wire as fine as .002" diam. Platinum wire may be drawn into the finest sizes. On account of its high price its use is practically confined to special scientific instruments and electrical appliances in which resistances to high temperature, oxygen, and acids are essential. It expands less than other metals when heated, which property permits its being- sealed in glass without fear of cracking. It is therefore used in incandescent electric lamps. Phosphor-bronze Wire contains from 2 to 6 per cent of tin and from 1/20 to 1/8 per cent of phosphorus. The presence of phosphorus is detrimental to electric conductivity. " Delta-metal " wire is made from an alloy of copper, iron, and zinc. Its strength ranges from 45 to 62 tons per square inch. It is used for some Mnds of wire rope, also for wire gauze. It is not subject to deposits of ver- digris. It has great toughness, even when its tensile strength is over 80 tons per square inch. Aluminum Wire. Specific gravity .268. Tensile strength only about 10 tons per square inch. It has been drawn as fine as 11,400 yards to the ounce, or .042 grains per yard, Aluminum Bronze, 90 copper, 10 aluminum, has high strength and ductility; is inoxidizable, sonorous. Its electric conductivity is 12.6 percent. Silicon Bronze, patented in 1882 by L. Weiler of Paris, is made as follows : Fluosilicate or potash, pounded glass, chloride of sodium and cal- cium, carbonate of soda and lime, are heated in a plumbago crucible, and after the reaction takes place the contents are thrown into the molten bronze to be treated. Silicon-bronze wire has a conductivity of from 40 to 98 per cent of that of copper wire and four times more than that of iron, while its tensile strength is nearly that of steel, or 28 to 55 tons per square inch of section. The conductivity decreases as the tensile strength in- creases. Wire whose conductivity equals 95 per cent of that of pure copper gives a tensile strength of 28 tons per square inch, but when its conductivity is 34 per cent of pure copper, its strength is 50 tons per square inch. It is being largely used for telegraph wires. It has great resistance to oxidation. Ordinary Drawn and Annealed Copper Wire has a strength of from 15 to 20 tons per square inch, SPECIFICATIONS FOR HARD-DRAWN COPPER WIRE. The British Post Office authorities require that hard-drawn copper wire supplied to them shall be of the lengths, sizes, weights, strengths, and con- ductivities as set forth in the annexed table. Weight per Statute Mile. Approximate Equiva- lent Diameter. ! *l ill 43 - wl a 1^ gs gCO I.S G CQ :imum Re ice per Mi ire (when 60 Fahr. 5 8 g J> ij.H ^E ill 11 3.3 cr 1 fl a a I 1 1 I a 1 | | I| a i 1 m a OQ & S % d a 1 3 ti& 08 rt p* Ibs. Ibs. Ibs. mils. mils. mils. Ibs. ohms. Ibs. 100 97^ 10^^> 79 78 80 330 30 9.10 50 150 1461^ 153M 97 95^ 98 490 25 6.05 50 200 195 205 112 110^ 113J4 650 20 4.53 50 400 390 410 158 155X8 160^ 1300 10 2.27 50 226 , MATERIALS. WIRE ROPES. List adopted by manufacturers in 1892. See pamphlets of John A. Roebling's Sons Co., Trenton Iron Co., and other makers, Pliable Hoisting Rope, With 6 strands of 19 wires each. IRON. Trade Number. q Circumference in inched Weight per foot in pounds. Rope with Hemp Cen- tre. Breaking Strain, tons of 2000 Ibs. Proper Working Load in tons of 2000 Ibs. Circumference of new Manila Rope of equal Strength. Min. Size of Drum [ or Sheave in- feet.j 1 SM 6% 8.00 74 15 14 13 2 2 6 6.30 65 13 13 12 3 1% 5^ 5.25 54 11 12 10 4 1% 5 4.10 44 9 11 8J4 5 Jl^ 4M 8.65 39 8 10 5^ 1% 4% 3.00 33 6*4 7 6 1/4 4 2.50 27 5^ 8*2 6^ 7 i/ij 3^ 2.00 20 4 7/^ 6 8 l 1.58 16 8 6^ 5M 9 2M 1.20 11.50 2^3 5V<2 XIX 10 'M 2!4 0.88 8.64 m 494 4 10M % 2 0.60 5.13 m lOJ'i 9-16 1^ 0.48 4.27 % v% 25^ 10% H ]i,^ 0.39 3.48 H 3 2M 10a 7-16 3% 0.29 3.00 n 2% 2 10% % 1J4 0.23 2.50 * 2^ 1& CAST STEEL. 1 2J4 6M 8.00 155 31 8U 2 2 6 6.30 125 25 8 3 5V 5.25 106 21 714 4 ^1 5^ 4.10 86 17 15 5 /^3 4% 3.65 77 15 14 5% 5^ % 4% 3.00 63 12 13 5^ 6 /4 4 2.50 52 10 12 5 7 1 3^ 2.00 42 8 11 4^ 8 3^/ 1.58 33 6 9^ 4 9 % 2% 1.20 25 5 w* 3^ 10 % 2*4 0.88 18 3^ 7 3 10)4 % 2 0.60 12 gi^j 5^ 2H 1014 9-16 1% 0.48 9 1M 5 1% 10% ^ 1^ 0.39 7 ip 4^j \\ 10a 7-16 ^% 0.29 5^ 3% j^4 10% % 1M 0.23 4H % 1 Cable-Traction Ropes. According to English practice, cable-traction ropes, of about 3^ in. in circumference, are commonly constructed with six strands of seven or fif- teen wires, the lays in the strands varying from, say. 3 in. to 3^ in., and the lays in the ropes from, say, 7^ in. to 9 in. In the United States, however, strands of nineteen wires are generally preferred, as being more flexible; but, on the other hand, the smaller external wires wear out more rapidly. The Market-street Street Railway Company, San Francisco, has used ropes 1J4 in. in diameter, composed of six strands of nineteen steel wires, weighing 2^ Ibs. per foot, the longest continuous length being 24,125 ft. The Chicago City Railroad Company has employed cables of identical construction, the longest length being 27,700 ft. On the New York and Brooklyn Bridge cable- railway steel ropes of 11,500 ft.* long, containing 114 wires, have been used. WIRE ROPES. 227 Transmission and Standing Rope. With 6 strands of 7 wires each. IRON. .2 Q.S a O 11 i i ^O 0*0 S ^8 11 s 1 ll d ^.s O 1) fc u -M <t> M a jfl bJC^ 1! l* II cc,a 1 s 1 fill M w* 3 |ll Jill ^ i 11 Lj 4% 3.37 36 9 10 13 12 13 ft 4% 4 8.77 2.28 30 25 pi 9 12 14 /^2 31^ 1.82 20 5 71^2 9^ 15 31^ 1.50 16 4 6J^ 8^ 16 % 2% 1.12 12.3 3 5% 7^ 17 18 11^16 ^ 0.92 0.70 8.8 7.6 1^ Si 6 4 19 % 2 0.57 5.8 JL^ 4 5^4 20 9-16 Ja^ 0.41 4.1 1 21 Ji J1Z 0.31 2.83 2% 4 22 7-16 1% 0.23 2.13 Hi 31^ 3^4 23 8^ 1 V<< 0.21 1.65 O1/J 03/f 24 5-16 1 0.16 1.38 2 2V^ 25 9-32 0.125 1.03 1% 2V4 CAST STEEL. 11 1^ m 3.37 62 13 13 8^ 12 1% 4% 2.77 52 10 12 8 13 1^4 4 2.28 44 9 11 14 31^ 1.82 36 7L 10 6/4 15 1 31^ 1.50 30 6 9 5% 16 % 2-M 1.12 22 4/^ 8 5 17 18 ! 2^ 0.92 0.70 17 14 3 2 7 6 19 % 2 8 0.57 11 2*4 3V 20 9-16 1% 0.41 8 1% 4^4 3 21 i^ JL/ 0.31 6 ji^ 4 22 7-16 1% 0.23 4^ 1/4 31^ 2^j 23 1^4 0.21 4 1 3/>4 2 /V 24 5-16 j 0.16 3 ^ 2M 1^< 25 9-32 % 0.12 2 *! 2^ 3i Plough-Steel Rope. Wire ropes of very high tensile strength, which are ordinarily called "Plough-steel Ropes," are made of a high grade of crucible steel, which, when put in the form of wire, will bear a strain of from 100 to 150 tons per square inch. Where it is necessary to use very long or very heavy ropes, a reduction of the dead weight of ropes becomes a matter of serious consideration. It is advisable to reduce all bends to a minimum, and to use somewhat larger drums or sheaves than are suitable for an ordinary crucible rope hav ing a strength of 60 to 80 tons per square inch. Before using Plough-stee Ropes it is best to have advice on the subject of adaptability. 228 MATERIALS. Plough-Steel Rope. With 6 strands of 19 wires each. Trade Number. Diameter in inches. Weight pel- foot in pounds. Breaking Strain in tons of 2000 Ibs. Proper Work- ing Load. Min. Size of Drum or Sheave in feet. 1 & 8.00 240 46 9 2 2 \ 6.30 189 37 8 3 l^v 5.25 157 31 7J4 4 % 4.10 123 25 6 5 ^ 3.65 110 22 51^8 5^ % 3.00 90 18 5J4 8 J4 2.50 75 15 5 7 iHj 2.00 60 12 4^ 8 1 1.58 47 9 4J4 9 % 1.20 37 7 3-M 10 % 0.88 27 5 SH 10M 7 0.60 18 m 3 10^ 9-16 0.44 13 2U 2^ 10% M 0.39 10 2^ 2 With 7 Wires to the Strand. 15 1 1.50 45 9 % 16 7^ 1.12 33 6^ 5 17 18 11-16 0.92 0.70 25 21 5 4 4 3^ 19 % 0.57 16 3% 3 20 9-16 0.41 12 2M 21 K 0.31 9 1% 2^2 22 7-16 0.23 5 IV 2 23 % 0.21 4 1 1^ Galvanized Iron Wire Rope. For Ships' Rigging and Guys for Derricks. CHARCOAL ROPE. Circum- ference in inches. Weight per Fath- om in pounds. Cir. of new Manila Rope of equal Strength. Break- ing Strain in tons of 2000 pounds 43 40 35 33 30 26 23 20 16 14 12 10 Circum- ference in inches Weight per Fathom Cir. of new Manila Rope of pounds. j&$QL Break- ing Strain in tons of 2000 pounds WIRE ROPES. 229 Galvanized Cast-steel Yacht Rigging. Circum- ference in inches. Weight per Fath- om in pounds. Cir. of new Manilla Rope of equal Strength. Break- ing Strain in tons of 2000 pounds Circum- ference in inches Weight per Fathom in pounds. Cir. of new Manilla Rope of equal Strength. Break- ing Strain in tons of 2000 pounds fa %& O1/ 2*4 M T 4$ 13 11 y^ 8H 8 7 66 43 32 27 22 18 2 VA i 2 2 1% % 3 4 14 10 8 Steel Hawsers. For Mooring, Sea, and Lake Towing. Size of Size of Circumfer- Breaking Manilla Haw- Circumfer- Breaking Manilla Haw- ence. Strength. ser of equal Strength. ence. Strength. ser of equal Strength. Inches. Tons. Inches. Inches. Tons. Inches. 2^ 15 6^ 3^ 29 9 2% 18 7 4 35 10 3 22 8^ Steel Flat Ropes. (J. A. Roebling's Sons Co.) Steel-wire Flat Ropes are composed of a number of strands, alternately twisted to the right and left, laid alongside of each other, and sewed together with soft iron wires, These ropes are use'd at times in place of round ropes in the shafts of mines. They wind upon themselves on a narrow winding- drum, which takes up less room than one necessary for a round rope. The Soft-iron sewing-wires wear out sooner than the steel strands, and then it becomes necessary to sew the rope with new iron wires. Width and Thickness in inches. Weight per foot in pounds. Strength in pounds. Width and Thickness in inches. Weight per foot in pounds. Strength in pounds. %x2 1.19 35,700 1^x3 2.38 71,400 %x2}4 1.86 55,800 ^x3VS 2.97 89,000 %x3 2.00 60,000 ^x4 3.30 99,000 %x3J^ 2.50 75,000 ^x4^ 4.00 120,000 %x4 2.86 85,800 }^x 5 4.27 128,000 %x4J4 3.12 93,600 ^x5^ 4.82 144,600 %x5 3.40 100,000 1^x6 5.10 153,000 %x5^ 3.90 110,000 1^x7 5.90 177,000 For safe working load allow from one fifth to one seventh of the breaking stress. " Lang I*ay Rope. In wire rope, as ordinarily made, the component strands are laid up into rope in a direction opposite to that in which the wires are laid into strands; that is, if the wires in the strands are laid from right to left, the strands are laid into rope from left to right. In the " Lang Lay," sometimes known as 01 Universal Lay," the wires are laid into strands and the strands into rope in the same direction ; that is, if the wire is laid in the strands from right to left, the strands are also laid into rope from right to left. Its use has been found desirable under certain conditions and for certain purposes, mostly for haulage plants, inclined planes, and street railway cables, although it has also been used for vertical hoists in mines, etc. Its advantages are that 230 MATERIALS. GALVANIZED STEEL CABLES* For Suspension Bridges. (Roebling's.) 220 200 180 13 11.3 10 2 m t II 3 v il 155 110 100 8.64 6.5 5.8 il 95 75 65 5.6 4 35 3.7 COMPARATIVE STRENGTHS OF FLEXIBLE GAL- VANIZED STEEL-WIRE HAWSERS, With Chain Cable, Tarred Russian Hemp, and White Manila Ropes. Patent Flexible Steel-wire Hawsers and Cables. Chain Cable. Tarred Rus- sian Hemp Rope. White Manilla Ropes. 5K 11 7 9 12 15 IS 22 20 33 39 64 74 88 102 116 130 150 9-16 10-16 11-16 12-16 13-16 15-16 1 17-32 166 1' 15-16 204 2 1-16 2 3-16 256 5-16 280 14 21 30 101 35 4815' 54 68 112 1434 23! 1 02 "o 2 PH I 107 1-10 12014 134^ 51 35^62 42 -" 22% NOTE. This is an old table, and its authority is uncertain. The figures in the fourth column are probably much too small for durability. WIRE ROPES. 231 it is somewhat more flexible than rope of the same diameter and composed of the same number of wi^es laid up in the ordinary manner; and (especi- ally) that owing to the fact that the wires are laid more axially in the rope, longer surfaces of the wire are exposed to wear, and the endurance of the rope is thereby increased. (Trenton Iron Co.) Notes on the Use of Wire Rope. (J. A. Koebling's Sons Co.) Several kinds of wire rope are manufactured. The most pliable variety contains nineteen wires in the strand, and is generally used for hoisting and running rope. The ropes with twelve wires and seven wires in the strand are stiffer, and are better adapted for standing rope, guys, and rigging. Or- ders should state the use of the rope, and advice will be given. Ropes are made up to three inches in diameter, upon application. For safe working load, allow one fifth to one seventh of the ultimate strength, according to speed, so as to get good wear from the rope. When substituting wire rope for hemp rope, it is good economy to allow for the former the same weight per foot which experience has approved for the latter. Wire rope is as pliable as new hemp rope of the same strength; the for- mer will therefore run over the same-sized sheaves and pulleys as the latter. But the greater the diameter of the sheaves, pulleys, or drums, the longer wire rope will last. The minimum size of drum is given in the table. Experience has demonstrated that the wear increases with the speed. It is, therefore, better to increase the load than the speed. Wire rope is manufactured either wilh a wire or a hemp centre. The lat- ter is more pliable than the former, and will wear better where there is short bending. Orders should specify what kind of centre is wanted. Wire rope must not be coiled or uncoiled like hemp rope. When mounted on a reel, the latter should be mounted on a spindle or flat turn-table to pay off the rope. When forwarded in a small coil, without reel, roll it over the ground like a wheel, and run off the rope in that way. All untwisting or kinking must be avoided. To preserve wire rope, apply raw linseed-oil with a piece of sheepskin, wool inside; or mix the oil with equal parts of Spanish brown or lamp-black. To preserve wire rope under water or under ground, take mineral or vege- table tar, and add one bushel of fresh-slacked lime to one barrel 9f tar, which will neutralize the acid. Boil it well, and saturate the rope with the hot tar. To give the mixture body, add some sawdust. The grooves of cast-iron pulleys and sheaves should be filled with well- seasoned blocks of hard wood, set on end, to be renewed when worn out. This end-wood will save wear and increase adhesion. The smaller pulleys or rollers which support the ropes on inclined planes should be constructed on the same plan. When large sheaves run with very great velocity, the grooves should be lined with leather, set on end, or with India rubber. This is done in the case of sheaves used in the transmission of power between distant points by means of rope, which frequently runs at the rate of 4000 feet per minute. Steel ropes are taking the place of iron ropes, where it is a special object to combine lightness with strength. But in substituting a steel rope for an iron running rope, the object in view should be to gain an increased wear from the rope rather than to reduce the size. Locked \Virc Rope. Fig. 74 shows what is known as the Patent Locked Wire Rope, made by the Trenton Iron Co. It is claimed to wear two to three times as long as an FIG. 74. ordinary wire rope of equal diameter and of like material. Sizes made are Irom y% to ly^ inches diameter. 232 MATERIALS. CRANE .CHAINS. (Bradlee & Co., Philadelphia.) 11 D. B. G." Special Crane. Crane. i c 0) be "S to "3 "5s l Li 1 ^ Is a fl tJ " M 4J - a 1 ^ of >$ W-S |1 feij ! <D M g PQft 1^1 w 3 3 -3 tj a5 d^'O cc-gg O % P..FH ^O o "3 rt O c 5 t>i 53 o Jsr^ O hS o cs'S jr A 2 c a & r I l O ^ g r P-. 8 ' 1 CC qj .So O y 25-32 Ys y 1932 3864 1288 1680 3300 1120 5-16 27-32 1 1 1-16 2898 5796 1932 2520 5040 1680 % 31-32 17-10 1H 4186 8372 2790 3640 7280 2427 7-16 1 5-32 2 1% 5796 11592 3864 5040 10080 3360 Lj 1 11-32 31^ 1 11-16 7728 15456 5182 6720 13440 4480 9-16 1 15-32 3 -,'-10 1% 9660 19320 6440 8400 16800 5600 5 1 23-32 21-16 11914 23828 7942 10360 20720 6907 11-16 1 27-32 5 8 2^4 14490 28980 9660 12600 25200 8400 H 1 31-32 5% 2^ 17388 34776 11592 15120 30240 10080 13-16 23-32 67-10 2 fl-16 20286 40572 13524 17640 35280 11760 % 27-32 8 2% 22484 44968 14989 20440 40880 13627 15-16 215-32 9 31-16 25872 51744 17248 23520 47040- 15680 1 2 19-32 10 7-10 3J4 29568 59136 19712 26880 53760 17920 1 1-16 2 23-32 11 2-10 35-16 33264 66538 22176 30240 60480 20160 u| 2 27-32 3-M 37576 75152 25050 34160 68320 22773 1 3-16 35-32 13 7-10 3% ' 41888 83776 27925 38080 76160 25387 1^4 37-32 16 46200 92400 30800 42000 84000 28000 1' 5-16 3 15-32 4% 50512 101024 33674 45920 91840 30613 j^ 3^Ha 18 4 2 -10 49-16 55748 111496 37165 50680 101360 33787 1 7-16 3 25-32 19 7-10 4% 60368 120736 40245 54880 109760 36587 JJ 3 31-32 21 7-10 5 66528 133056 41352 60480 120960 40320 The distance from centre of one link to centre of next is equal to the in- side length of link, but in practice 1/32 inch is allowed for weld. This is ap- proximate, and where exactness is required, chain should be made so. FOR CHAIN SHEAVES. The diameter, if possible, should be not less than twenty times the diameter of chain used. EXAMPLE. For 1-inch chain use 20-inch sheaves. WEIGHTS OF LOOS, LUMBER, ETC. Weight of Green Logs to Scale 1,000 Feet, Board Measure. Yellow pine (Southern). . 8,000 to 10,000 Ibs, Norway pine (Michigan). 7,000 to 8,000 " Whit* ninp nVTiVhi>fln J off of Stump 6,000 to 7,000 " White pine (Michigan) ^ out Qf water _ 7,000 to 8,000 " White pine (Pennsylvania), bark off 5,000 to 6,000 " Hemlock (Pennsylvania), bark off 6,000 to 7,000 a Four acres of water are required to store 1,000,000 feet of logs. Weight of 1,OOO Feet of Lumber, Board Measure. Yellow or Norway pine Dry, 3,000 Ibs. Green, 5.000 Ibs. White pine 2,500 " 4,000 " Weight of 1 Cord of Seasoned Wood, 128 CuMc Feet per Cord. Hickory or sugar maple 4,500 Jbs White oak .- 3,850 " Beech, red oak or black oak 3,250 " Poplar, chestnut or elm 2,350 " Pine (white or Norway) 2,000 " Hemlock bark, dry 2,200 " SIZES OF FIKE-BRICK. 233 _ \ ^ \ , thick x 4^ to 4 inches 3 iam. SIZES OF FIRE-BRICK, 9-inch straight 9 x 4^ x 2^ inches. Soap 9 x 2J^ x 2J*j Checker 9x3 x3 " 2-inch 9x4^x2 ** Split... 9x4 Jamb 9 x 4 No. Ikey 9x2; wide. 113 bricks to circle 12 feet inside diam. No.2key ... 9x2^ thick x 4^ to 3 inches wide. 63 bricks to circle 6 ft. inside diam. No. 3 key 9x2^ thick x 4^ to inches wide. 38 bricks to circle 3 ft. inside diam. No. 4 key / 9x2^ thick x 4^ to 2*4 inches wide. 25 bricks to circle 1^ ft. inside diam. No. 1 wedge (or bullhead). 9x4^ wide x 2*4 to 2 in. thick, tapering lengthwise. 98 bricks to circle 5 ft. inside diam. No. 2 wedge 9 x 4*4 x 2^ to 1^ in. thick. 60 bricks to circle 2J4 ft. inside diam. No. larch.., 9x4^x2^ to 2 in. thick, tapering breadthwise. 72 bricks to circle 4 ft. inside diam. No.2arch 9x4^x ;_ 42 bricks to circle 2 ft. inside No. 1 skew 9 to 7 x Bevel on one end. No. 2 skew 9x2^x4^ Equal bevel on both edges. No. 3skew 9x2^x4^ to Taper on one edge. 24 inch circle 8*4 to 5J4 x 4V x 2>. Edges curved, 9 bricks line a 24-inch circle. 36-inch circle 8% to 6^ x 4J4 x 2^. 13 bricks line a 36-inch circle. 48-inch circle 8% to 7J4 x 4^ x 2J4 17 bricks line a 48-inch circle. inch straight 13^ x 2^ x 6. inch key No. 1 13^ x 2^ x 6 to 5 inch. 90 bricks turn a 12-ft. circle. 13i^-inch key No. 2 13^ x 2^ x 6 to 4% inch. 52 bricks turn a 6-ft. circle. Bridge wall, No. 1 13x6^x6. Bridge wall, No. 2 13x6^x3. Mill tile 18,20,or24x6x3. Stock-hole tiles 18, 20, or 24 x 9 x 4. 18-inch block 18x9x6. Flat back 9x6x2^. Flat back arch 9 x 6 x 314 to 2^. 22-inch radius, 56 bricks to circle. Locomotive tile 32 x 10 x 3. 34 x 10 x 3. 34x 8x3. 36 x 8x3. 40x10x3. Tiles, slabs, and blocks, various sizes 12 to 30 inches long, 8 to 30 inches wide, 2 to 6 inches thick. , , Cupola brick, 4 and 6 inches high, 4 and 6 inches radial width, to line shells 23 to 66 in diameter. A 9-inch straight brick weighs 7 Ibs. and contains 100 cubic inches. (=120 Ibs. per cubic foot. Specific gravity 1.93.) One cubic foot of wall requires 17 9-inch bricks, one cubic yard requires 460. Where keys, wedges, and other " shapes " are used, add 10 per cent in estimating the number required. 234 MATERIALS. One ton of fire-clay should be sufficient to lay 3000 ordinary bricks. To secure the best results, fire-bricks should be laid in the same clay from which they are manufactured. It should be used as a thin paste, and not as mor- tar. The thinner the joint the better the furnace wall. In ordering bricks the service for which they are required should be stated. NUMBER OF FIRE-BRICK REQUIRED FOR VARIOUS CIRCLES. g log ft KEY BRICKS. ARCH BRICKS. WEDGE BRICKS. ^ d & w | OJ 6 fc 6 ft . I <?* 6 ft d fc OS a o E-t si 6 & 0* O5 1 ft. in. 1 6 2 2 6 3 3 6 4 4 6 5 5 6 6 6 6 7 7 6 8 8 6 9 9 6 10 10 6 11 11 6 12 12 6 25 17 9 25 30 34 38 42 46 51 55 59 63 67 71 76 80 84 88 92 97 101 105 109 113 117 '42 31 21 10 13 25 38 32 25 19 13 6 10 21 32 42 53 63 58 52 47 42 37 31 26 21 16 11 5 9 19 29 38 47 57 66 76 85 94 104 113 113 'is' 36 54 73 42 49 57 64 72 60 48 36 24 '26' 40 59 6fl 68 76 83 91 98 106 113 121 128 136 144 151 159 166 174 181 189 196 304 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 8 15 23 30 38 45 53 60 68 75 83 90 98 105 113 121 80 87 95 102 110 117 125 132 140 147 155 162 170 177 185 193 12 79 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 "*8 15 23 30 38 46 53 61 68 76 83 91 98 106 .... .... For larger circles than 12 feet use 113 No. 1 Key, and as many 9-inch brick as may be needed in addition. ANALYSES OF MET. SAVAGE FIRE-CLAY. (1) 1871 Mass. Institute of Technology. -, 50.457 35.904 0.133 0.018 trace 12.744 100.760 (2) 1877. Report on (8) 1878. 56.80 *sey Silica Survey of Pennsylvania. 44.395 80.08 ,. 33.558 1.15 Titanic acid , i 530 1 12 Peroxide iron 1 080 Lime Magnesia 0.108 0.80 Potash (alkalies). 0.247 10.50 Water and inorg. matter. 14.575 (4) 100.450 100.493 56.15 33.295 *6".59" 0.17 0.115 '9! ',68 100.000 MAGNESIA BRICKS. 230 MAGNESIA BRICKS. ** Foreign Abstracts " of the Institution of Civil Engineers, 1893, gives a paper by C. Bischof on the production of magnesia bricks. The material most in favor at present is the magnesite of Styria, which, although less pure considered as a source of magnesia than the Greek, has the property of fritting at a high temperature without melting. The composition of the two substances, in the natural and burnt states, is as follows: Magnesite. Styrian. Greek. Carbonate of magnesia 90.0 to 96.0# 94.46# " lime 0.5 to 2.0 4.49 " " iron 3.0 to 6.0 FeO 0.08 Silica 1.0 0.52 Manganous oxide 0.5 Water 0.54 Burnt Magnesite. Magnesia 77.6 82.46-95.36 Lime 7.3 0.8310.92 Alumina and ferric oxide 13.0 0.56 3.54 Silica 1.2 0.737.98 At a red heat magnesium carbonate is decomposed into carbonic acid and caustic magnesia, which resembles lime in becoming hydrated and recar- bonated when exposed to the air, and possesses a certain plasticity, so that it can be moulded when subjected to a heavy pressure. By long-continued or stronger heating the material becomes dead-burnt, giving a form of mag- nesia of high density, sp. gr. 3.8, as compared with 3.0 in the plastic form, which is unalterable in the air but devoid of plasticity. A mixture of two volumes of dead-burnt with one of plastic magnesia can be moulded into bricks which contract but little in firing. Other binding materials that have been used are: clay up to 10 or 15 per cent; gas -tar, perfectly freed from water, soda, silica, vinegar as a solution of magnesium acetate which is readily decomposed by heat, and carbolates of alkalies or lime. Among magnesium compounds a weak solution of magnesium chloride may also be used. For setting the bricks lightly burnt, caustic magnesia, with a small proportion of silica to render it less refractory, is recommended. The strength of the bricks may be increased by adding iron, either as oxide or silicate. If a porous product is required, sawdust or starch may be added to the mixture. When dead-burnt magnesia is used alone, soda is said to be the best binding material. See also papers by A. E. Hunt, Trans. A. I. M. E., xvi, 720, and by T. Egles- ton, Trans. A. I. M. E., xiv, 458. Asbestos. J. T. Donald, Eng. and M. Jour., June 27, 1891. ANALYSIS. Canadian. Italian. Broughton. Templeton. Silica 40.30# 40.57 40.52 Magnesia 43.37 41.50 42.05 Ferrous oxide 87 2.81 1.97 Alumina 2.27 .90 2.10 Water 13.72 13.55 13.46 100.53 99.33 100.10 Chemical analysis throws light upon an important point in connection with asbestos, i.e., the cause of the harshness of the fibre of some varieties. Asbestos is principally a hydrous silicate of magnesia, i.e., silicate of mag- nesia combined with water. When harsh fibre is analyzed it is found to contain less water than the soft fibre. In fibre of very fine quality from Black Lake analysis showed 14.38$ of water, while a harsh-fibred sample gave only 11.70#. If soft fibre be heated to a temperature that will drive off a portion of the combined water, there results a substance so brittle that it may be crumbled between thumb and finger. There is evidently some con- nection between the consistency of the fibre and the amount of water in its composition. 236 STRENGTH OF MATERIALS. STRENGTH OP MATERIALS. Stress and Strain. There is much confusion among writers on strength of materials as to the definition of these terms. An external force applied to a body, so as to pull it apart, is resisted by an internal force, or resistance, and the action of these forces causes a displacement of the mole- cules, or deformation. By some writers the external force is called a stress, and the internal force a strain; others call the external force a strain, and the internal force a stress: this confusion of terms is not of importance, as the words stress and strain are quite commonly used synonymously, but the use of the word strain to mean molecular displacement, deformation, or dis- tortion, as is the custom of some, is a corruption of the language. See En- gineering News, June 23, 1892. Definitions by leading authorities are given below. Stress. A stress is a force which acts in the interior of a body, and re- sists the external forces which tend to change its shape. A deformation is the amount of change of shape of a body caused by the stress. The word strain is often used as synonymous with stress and sometimes it is also used to designate the deformation. (Merriman.) The force by which the molecules of a body resist a strain at any point is called the stress at that point. The summation of the displacements of the molecules of a body for a given point is called the distortion or strain at the point considered. (Burr). Stresses are the forces which are applied to bodies to bring into action their elastic and cohesive properties. These forces cause alterations of the forms of the bodies upon which they act. Strain is a name given to the kind of alteration produced by the stresses. The distinction between stress and strain is not always observed, one being used for the other. (Wood.) Stresses are of different kinds, viz. : tensile, compressive, transverse, tor- sional, and shearing stresses. A tensile stress, or pull, is a force tending to elongate a piece. A com- pressive stress, or push, is a force tending to shorten it. A transverse stress tends to bend it. A torsional stress tends to twist it. A shearing stress tends to force one part of it to slide over the adjacent part. Tensile, compressive, and shearing stresses are called simple stresses. Transverse stress is compounded of tensile and compressive stresses, and torsional of tensile and shearing stresses. To these five varieties of stresses might be added tearing stress, which is either tensile or shearing, but in which the resistance of different portions of the material are brought into play in detail, or one after the other, in- stead of simultaneously, as in the simple stresses. Effects of Stresses. The following general laws for cases of simple tension or compression have been established by experiment. (Merriman): 1. When a small stress is applied to a body, a small deformation is pro- duced, and on the removal of the stress the body springs back to its original form. For small stresses, then, materials may be regarded as perfectly elastic. 2. Under small stresses the deformations are approximately proportional to the forces or stresses which produce them, and also approximately pro- portional to the length of the bar or body. 3. When the stress is great enough a deformation is produced which is partly permanent, that is, the body does not spring back entirely to its original form on removal of the stress. This permanent part is termed a set. In such cases the deformations are not proportional to the stress. 4. When the stress is greater still the deformation rapidly increases and the body finally ruptures. 5. A sudden stress, or shock, is more injurious than a steady stress or than a stress gradually applied. Elastic Limit. The elastic limit is defined as that point at which the deformations cease to be proportional to the stresses, or, the point at which the rate of stretch (or other deformation) begins to increase. It is also defined as the point at which the first permanent set becomes visible. The last definition is not considered as good as the first, as it is found that with some materials a set occurs with any load, no matter how small, and that with others a set which might be called permanent vanishes with lapse of time, and as it is impossible to get the point of first set without removing STRESS AKD STRAIN. 237 the whole load after each increase of load, which is frequently inconvenient. The elastic limit, defined, however, as the point at which the extensions be- gin to increase at a higher ratio than the applied stresses, usually corresponds very nearly with the point of first measurable permanent set. Apparent Elastic lamit. Prof. J. B. Johnson (Materials of Con- struction, p. 19) defines the " apparent elastic limit " as "the point on the stress diagram [a plotted diagram in which the ordinates represent loads and the abscissas the corresponding elongations] at which the rate of de- formation is 50$ greater than it is at the origin," [the minimum rate]. An equivalent definition, proposed by the author, is that point at which the modulus of extension (length X increment of load per unit of section H- in- crement of elongation) is two thirds of the maximum. For steel, with a modulus of elasticity of 30,000,000, this is equivalent to that point at which the increase of elongation in an 8-inch specimen for 1000 Ibs. per sq. in. increase of load is 0.0004 in. Yield-point. The term yield-point has recently been introduced into the literature of the strength of materials. It is defined as that point at which the rate of stretch suddenly increases rapidly. The difference be- tween the elastic limit, strictly defined as the point at which the rate of stretch begins to increase, and the yield-point, at which the rate increases suddenly, may in some cases be considerable. This difference, however, will not be discovered in short test-pieces unless the readings of elongations are made by an exceedingly fine instrument, as a micrometer reading to of an inch. In using a coarser instrument, such as calipers reading to 1/100 of an inch, the elastic limit and the yield-point will appear to be simultane- ous. Unfortunately for precision of language, the term yield-point was not introduced until long after the term elastic limit had been almost univer- sally adopted to signify the same physical fact which is now defined by the term yield-point, that is, not the point at which the first change in rate, observable tnly by a microscope, occurs, but that later point (more or less indefinite as to its precise position) at which the increase is great enough to be seen by the naked eye. A most convenient method of determining the point at which a sudden increase of rate of stretch occurs in short speci- mens, when a testing-machine in which the pulling is done by screws is used, is to note the weight on the beam at the instant that the beam " drops.' 1 During the earlier portion of the test, as the extension is steadily increased by the uniform but slow rotation of the screws, the poise is moved steadily along the beam to keep it in equipoise; suddenly a point is reached at which the beam drops, and will not rise until the elongation has been considerably increased by the further rotation of the screws, the advancing of the poise meanwhile being suspended. This point corresponds practically to the point at which the rate of elongation suddenly increases, and to the point at which an appreciable permanent set is first found. It is also the point which has hitherto been called in practice and in text-books the elastic limit, and it will probably continue to be so called, although the use of the newer term "yield-point" for it, and the restriction of the term elastic limit to mean the earlier point at which the rate of stretch begins to increase, as determin- able only by micrometric measurements, is more precise and scientific. In tables of strength of materials hereafter given, the term elastic limit is used in its customary meaning, tue point at which the rate of stress has be- gun to increase, as observable by ordinary instruments or by the drop of the beam. With this definition it is practically synonymous with yield- point. Coefficient (or Modulus) of Elasticity. This is a term express- ing the relation between the amount of extension or compression of a mate- rial and the load producing that extension or compression. It is defined as the load per unit of section divided by the extension per uuit of length. Let P be the applied load, fc the sectional area of the piece, I the length of the part extended, A the amount of the extension, and E the coefficient of elasticity. Then P -f- fc = the load on a unit of section ; A -*- 1 = the elonga- tion of a unit of length. The coefficient of elasticity is sometimes defined as the figure expressing the load which would be necessary to elongate a piece of one square inch section to double its original length, provided the piece would not break, and the ratio of extension to the force producing it remained constant. This definition follows from the formula above given, thus: If fcssoiie square inch, I and t- each = one inch, then E = P. Within the elastic limit, when the deformations are proportional to the 238 STRENGTH OF MATERIALS. stresses, the coefficient of elasticity is constant, but beyond the el&stic limit it decreases rapidly. In cast iron there is generally no apparent limit of elasticity, the deforma- tions increasing at a faster rate than the stresses, and a permanent set being produced by small loads. The coefficient of elasticity therefore is not con- stant during any portion of a test, but grows smaller as the load increases. The same is true in the case of timber. In wrought iron and steel, however, there is a well-defined elastic limit, and the coefficient of elasticity within that limit is nearly constant. Resilience, or Work of Resistance of a Material. Within the elastic limit, the resistance increasing uniformly from zero stress to the stress at the elastic limit, the work done by a load applied gradually is equal to one half the product of the final stress by the extension or other deforma- tion. Beyond the elastic limit, the extensions increasing more rapidly than the loads, and the strain diagram approximating a parabolic form, the work is approximately equal to two thirds the product of the maximum stress by the extension. The amount of work required to break a bar, measured usually in inch- pounds, is called its resilience; the work required to strain it to the elastic limit is called its elastic resilience. (See page 270.) Under a load applied suddenly the momentary elastic distortion is equal to twice that caused by the same load applied gradually. When a solid material is exposed to percussive stress, as when a weight falls upon a beam transversely, the work of resistance is measured by the product of the weight into the total fall. Elevation of Ultimate Resistance and Elastic Limit. It was first observed by Prof. R. H. Thurstqn, and Commander L. A. Beards lee, U. S. N., independently, in 1873, that if wrought iron be subjected to a stress beyond its elastic limit, but not beyond its ultimate resistance, and then allowed to "rest" for a definite interval of time, a considerable in- crease of elastic limit and ultimate resistance may be experienced. In other words, the application of stress and subsequent '* rest " increases the resist- ance of wrought iron. This " rest " may be an entire release from stress or a simple holding the test-piece at a given intensity of stress. Commander Beardslee prepared twelve specimens and subjected them to an intensity of stress equal to the ultimate resistance of the material, with- out breaking the specimens. These were then allowed to rest, entirely free from stress, from 24 to 30 hours, after which period they were again stressed until broken. The gain in ultimate resistance by the rest was found to vary from 4.4 to 17 per cent. This elevation of elastic and ultimate resistance appears to be peculiar to iron and steel: it has not been found in other metals. Relation of tlie Elastic Limit to Endurance under Re- peated Stresses (condensed from Engineering, August 7, 1891). When engineers first began to test materials, it was soon recognized that if a specimen was loaded beyond a certain point it did not recover its origi- nal dimensions on removing the load, but took a permanent set; this point was called the elastic limit. Since below this point a bar appeared to recover completely its original form and dimensions on removing the load, it ap peared obvious that it had not been injured by the load, and hence the work- ing load might be deduced from the elastic limit by using a small factor of safety. Experience showed, however, that in many cases a bar would not carry safely a stress anywhere near the elastic limit of the material as determined by these experiments, and the whole theory of any connection between the elastic limit of a bar and its working load became almost discredited, and engineers employed the ultimate strength only in deducing the safe working load to which their structures might be subjected. Still, as experience accu- mulated it was observed that a higher factor of safety was required for a live load than for a dead one. In 1871 Wohler published the results of a number of experiments on bars of iron and steel subjected to live loads. In these experiments the stresses were put on and removed from the specimens without impact, but it was, nevertheless, found that the breaking stress of the materials was in every case much below the statical breaking load. Thus, a bar of Krupp's axle steel having a tenacity of 49 tons per square inch broke with a stress of 28.6 tons per square inch, when the load was completely removed and replaced without impact 170,000 times. These experiments were made on a large STRESS AXD STKAIS7 239 number Of different brands of iron and steel, and the results were concor- dant in showing that a bar would break with an alternating stress of only, say, one third the statical breaking strength of the material, if the repetitions of stress were sufficiently numerous. At the same time, however, it ap- peared from the general trend of the experiments that a bar would stand an indefinite number of alternations of stress, provided the stress was kept below the limit. Prof. Bauschinger defines the elastic limit as the point at which stress ceases to be sensibly proportional to strain, the latter being measured with a mirror apparatus reading to r^th of a millimetre, or about </vinnA in. OUUU lUUUUU This limit is always below the yield-point, and may on occasion be zero. On loading a bar above the yield-point, this point rises with the stress, and the rise continues for weeks, months, and possibly for years if the bar is left at resl under its load. On the other hand, when a bar is loaded beyond its true elastic limit, but below its 3 ; ield-point, this limit rises, but reaches a maxi- mum as the yield-point, is approached, and then falls rapidly^ reaching even to zero. On leaving the bar at rest under a stress exceeding that of its primitive breaking-down point the elastic limit begins to rise again, and may, if left a sufficient time, rise to a point much exceeding its previous value. This property of the elastic limit of changing with the history of a bar has done more to discredit it than anything else, nevertheless it now seems as if it, owing to this very property, were once more to take its former place in the estimation of engineers, and this time with fixity of tenure. It had long been known that the limit of elasticity might be raised, as we have said, to almost any point within the breaking load of a bar. Thus, in some experi- ments by Professor Styffe, the elastic limit of a puddled-steel bar was raised 16,000 Ibs. by subjecting the bar to a load exceeding its primitive elastic limit. A bar has two limits of elasticity, one for tension and one for compression. Bauschinger loaded a number of bars in tension until stress ceased to be sensibly proportional to strain. The load was then removed and the bar tested in compression until the elastic limit in this direction had been ex- ceeded. This process raises the elastic limit in compression, as would be found on testing the bar in compression a second time. In place of this, however, it was now again tested in tension, when it was found that the artificial raising of the limit in compression had lowered that in tension be- low its previous value. By repeating the process of alternately testing in tension and compression, the two limits took up points at equal distances from the line of no load, both in tension and compression. These limits Bauschinger calls natural elastic limits of the bar, which for wrought iron correspond to a stress of about 8^ tons per square inch, but this is practically the limiting load to which a bar of the same material can be strained alter- nately in tension and compression, without breaking when the loading is repeated sufficiently often, as determined by Wohler's method. As received from the rolls the elastic limit of the bar in tension is above the natural elastic limit of the bar as defined by Bauschinger, having been artificially raised by the deformations to which it has been subjected in the process of manufacture. Hence, when subjected to alternating stresses, the limit in tension is immediately lowered, while that in compression is raised until they both correspond to equal loads. Hence, in Wohler's ex- periments, in which the bars broke at loads nominally below the elastic limits of the material, there is every reason for concluding that the loads ' were really greater than true elastic limits of the material. This is con- firmed by tests on the connecting-rods of engines, which of course work under alternating stresses of equal intensity. Careful experiments on old rods show that the elastic limit in compression is the same as that in ten- sion, and that both are far below the tension elastic limit of the material as received from the rolls. The common opinion that straining a metal beyond its elastic limit injures it appears to be untrue. It is not the mere straining of a metal beyond one elastic limit that injures it, but the straining, many times repeated, beyond its two elastic limits. Sir Benjamin Baker has shown that in bending a shell plate for a boiler the metal is of necessity strained bej^ond its elastic limit, so that stresses of as much as 7 tons to 15 tons per square inch may obtain in it as it comes from the rolls, and unless the plate is annealed, these stresses will still exist after it has been built into the boiler. In such a case, however, when exposed to the additional stress due to the pressure inside 240 STRENGTH OF MATERIALS. the boiler, the overstrained portions of the plate will relieve themselves by stretching and taking a permanent set, so that probably after a year's work- ing very little difference could be detected in the stresses in a plate built in- to the boiler as it came from the bending rolls, and in one which had been annealed, before riveting into place, and the first, in spite of its having been strained beyond its elastic limits, and not subsequently annealed, would be as strong as the other. Resistance of Metals to Repeated Shocks, More than twelve years were spent by Wohler at the instance of the Prus- sian Government in experimenting upon the resistance of iron and steel to repeated stresses. The results of his experiments are expressed in what is known as Wohler's law, which is given in the following words in Dubois'S translation of Weyrauch: " Rupture may be caused not only by a steady load which exceeds the carrying strength, but also by repeated applications of stresses, none of which are equal to the carrying strength. The differences of these stresses are measures of the disturbance of continuity, in so far as by their increase the minimum stress which is still necessary for rupture diminishes." A practical illustration of the meaning of the first portion of this law may be given thus: If 50,000 pounds once applied will just break a bar of iron or steel, a stress very much less than 50,000 pounds will break it if repeated sufficiently often. This is fully confirmed by the experiments of Fairbairn and Spangenberg, as well as those of Wohler; and, as is remarked by Weyrauch, it may be considered as a long-known result of common experience. It partially ac- counts for what Mr. Holley has called the tl intrinsically ridiculous factor of safety of six." Another "long-known result of experience " is the fact that rupture may be caused by a succession of shocks or impacts, none of which alone would be sufficient to cause it. Iron axles, the piston-rods of steam hammers, and other pieces of metal subject to continuously repeated shocks, invariably break after a certain length of service. They have a "life " which is lim- ited. Several years ago Fairbairn wrote: * We know that in some cases wrought iron subjected to continuous vibration assumes a crystalline structure, and that the cohesive powers are much deteriorated, but we are ignorant of the causes of this change." We are still ignorant, not only of the causes of this change, but of the conditions under which it takes place. Who knows whether wrought iron subjected to very slight continuous vibration will en- dure forever? or whether to insure final rupture each of the- continuous small shocks must amount at least to a certain percentage of single heavy shock (both measured in foot-pounds), which would cause rupture with one applica- tion ? Wohler found in testing iron by repeated stresses (not impacts) that in one case 400,000 applications or a stress of 500 centners to the square inch caused rupture, while a similar bar remained sound after 48,000,000 applica- tions of a stress of 300 centners to the square inch (1 centner = 110.2 Ibs.). Who knows whether or not a similar law holds true in regard to repeated shocks ? Suppose that a bar of iron would break under a single impact of 1000 foot-pounds, how many times would it be likely to bear the repetition of 100 foot-pounds, or would it be safe to allow it to remain for fifty years subjected to a continual succession of blows of even 10 foot-pounds each ? Mr. William Metcalf published in the Metallurgical Rev ieiv, Dec. 1877, the results of some tests of the life of steel of different percentages of carbon under impact. Some small steel pitmans were made, the specifications for . which required that the unloaded machine should run 4J4 hours at the rate of 1200 revolutions per minute before breaking. The steel was all of uniform quality, except as to carbon. Here are the results; The .30 C. ran 1 h. 21 m. Heated and bent before breaking. .490. ". Ih. 28m., " " " " * .43 C. " 4 h. 57 m. Broke without heating. .65 C. " 3 h. 50 m. Broke at weld where imperfect. .80 C. " 5h. 40m. .84 C. " 18 h. .87 C. Broke in weld near the end. .96 C. Ran 4.55 m., and the machine broke down. Some other experiments by Mr. Metcalf confirmed his conclusion, viz. STRESS AKB STRAIH. 241 that high-carbon steel was better adapted to resist repeated shocks and vi- brations than low-carbon steel. These results, however, would scarcely be sufficient to induce any en- gineer to use .84 carbon steel in a car-axle or a bridge-rod. Further experi- ments are needed to confirm or overthrow them. (See description of proposed apparatus for such an investigation in the author's paper in Trans. A. I. M. E M vol. viii., p. 76, from which the above extract is taken.) Stresses Produced by Suddenly Applied Forces and Shocks. (Mansfield Merriman, R. R. & Eng. Jour., Dec. 1889.) Let P be the weight which is dropped from a height h upon the end of a bar, and let y be the maximum elongation which is produced. The work performed by the falling weight, then, is TF=P(fc + y), and this must equal the internal work of the resisting molecular stresses. The stress in the bar, which is at first 0, increases up to a certain limit Q t which is greater than P; and if the elastic limit be not exceeded the elonga- tion increases uniformly with the stress, so that the internal work is equaJ to the mean stress 1/2Q multiplied by the total elongation y, or W=l/2Qy. Whence, neglecting the work that may be dissipated in heat, l/2Qy=Ph + Py. If e be the elongation due to the static load P, within the elastic limit y p e\ whence ........ (1) which gives the momentary maximum stress. Substituting this value of Q, there results which is the value of the momentary maximum elongation. A shock results when the force P, before its action on the bar, is moving with velocity, as is the case when a weight P falls from a height h. The above formulas show that this height h may be small if e is a small quan- tity, and yet very great stresses and deformations be produced. For in- stance, let h = 4e, then Q = 4P and y = 4e ; also let h = 12e, then Q = 6P and y = 6 e. Or take a wrought-iron bar 1 in. square and 5 ft. long: under a steady load of 5000 Ibs. this will be compressed about 0.012 in., supposing that no lateral flexure occurs; but if a weight of 5000 Ibs. drops upon its end from the small height of 0.048 in. there will be produced the stress of 20,000 Ibs. A suddenly applied force is one which acts with the uniform intensity P upon the end of the bar, but which has no velocity before acting upon it. This corresponds to the case of h in the above formulas, and gives Q =s 2P and y = 2e for the maximum stress and maximum deformation. Profoi I ably the action of a rapidly-moving train upon a bridge produces stressed of this character. Increasing the Tensile Strength of Iron Bars by Twist- ing them. Ernest L. Ransome of San Francisco has obtained an English Patent, No. 16221 of 1888, for an " improvement in strengthening and testing wrought metal and steel rods or bars, consisting in twisting the same in a cold state. . . . Any defect in the lamination of the metal which would otherwise be concealed is revealed by twisting, and imperfections are shown at once. The treatment may be applied to bolts, suspension-rods or bars subjected to tensile strength of any description." Results of tests of this process were reported by Lieutenant F. P. Gilmore, U. S. N., in a paper read before the Technical Society of the Pacific Coast, published in the Transactions of the Society for the month of December, 1888. The experiments include inajs wuu unrty-nine bars, twenty-nine of which were variously twisted, from three-eighths of one turn to six turns per foot. The test-pieces were cut from one and the same bar, and accurately 242 STRENGTH OF MATERIALS. measured and numbered. From each lot two pieces without twist were tested for tensile strength and ductility. One group of each set was twister, until the pieces broke, as a guide for the amount of twist to be given those to be tested for tensile strain. The following is the result of one set of Lieut. Gilmore's tests, on iron bars 8 in. long, .719 in. diameter. No. of Bars. Conditions. Twists in Turns. Twists per ft. Tensile Strength. Tensile per sq. in. Gain per cent. 2 Not twisted. 22,000 54,180 2 Twisted cold. ^ y 23,900 59,020 9 2 " " 1 iJ2 25,800 63,500 17 2 2 3 26,300 64,750 19 1 5 3M 26,400 65,000 20 Tests that corroborated these results were made by the University of California in 1889 and by the Low Moor Iron Works, England, in 1890. TENSILE STRENGTH. The following data are usually obtained in testing by tension in a testing-- machine a sample of a material of construction : The load and the amount of extension at the elastic limit. The maximum load applied before rupture. The elongation of the piece, measured between gauge-marks placed a stated distance apart before the test; and the reduction of area at the point of fracture. The load at the elastic limit and the maximum load are recorded in pounds per square inch of the original area. The elongation is recorded as a per- centage of the stated length between the gauge-marks, and the reduction area as a percentage of the original area. The coefficient of elasticity is cal- culated from the ratio the extension within the elastic limit per inch of length bears to the load per square inch producing that extension. On account of the difficulty of making accurate measurements of the frac- tured area of a test-piece, and of the fact that elongation is more valuable than reduction of area as a measure of ductility and of resilience or work of resistance before rupture, modern experimenters are abandoning the custom of reporting reduction of area. The "strength per square inch of fractured section " formerly frequently used in reporting tests is now almost entirely abandoned. The data now calculated from the results of a tensile test for commercial purposes are: 1. Tensile strength in pounds per square inch of original area. 2. Elongation per cent of a stated length between gauge-marks, usually 8 inches. 3. Elastic limit in pounds per square inch of original area. The short or grooved test specimen gives with most metals, especially with wrought iron and steel, an appaient tensile strength much higher than the real strength. This form of test-piece is now almost entirely aban- doned. The following results of the tests of six specimens from the same 1*4" steel bar illustrate the apparent elevation of elastic limit and the changes in other properties due to change in length of stems which were turned down in each specimen to .798" diameter. (Jas. E. Howard, Eng. Congress 1893 Section G.) Description of Stem. Elastic Limit, Lbs. per Sq. In. Tensile Strength, Lbs. per Sq. In. Contraction of Area, per cent. 1.00" long... 64,900 94,400 49.0 .50 "..... .... 65 320 97,800 43 4 68,000 102,420 39.6 Semicircular groove, A" radius. ... ... 75 000 116,380 31.6 Semicircular groove, %" radius 80,000, about 134,960 23.0 V-shaped groove 90,000, about 117,000 Indeterminate. TENSILE STRENGTH. 243 Tests plate made by the author in 1879 of straight and grooved test-pieces Of boiler-plate steel cut from the same gave the following results : 5 straight pieces, 56,605 to 59,012 Ibs. T. S. Aver. 57,566 Ibs. 4 grooved " 64,341 to 67,400 " " ** 65,452 " Excess of the short or grooved specimen, 21 per cent, or 12,114 Ibs. Measurement of Elongation. In order to be able to compare records of elongation, it is necessary not only to have a uniform length of section between gauge-marks (say 8 inches), but to adopt a uniform method of measuring the elongation to compensate for the difference between the apparent elongation when the piece breaks near one of the gauge-marks, and when it breaks midway between them. The following method is rec- ommended (Trans. A. S. M. E., vol. xi., p. 622): Mark on the specimen divisions of 1/2 inch each. After fracture measure from the point of fracture the length of 8 of the marked spaces on each fractured portion (or 7 -}- on one side and 8 -f- on the other if the fracture is not at one of the marks). The sum of these measurements, less 8 inches, is the elongation of 8 inches of the original length. If the fracture is so near one end of the specimen that 7 + spaces are not left on the shorter portion, then take the measurement of as many spaces (with the fractional part next to the fracture) as are left, and for the spaces lacking add the measurement of as many corresponding spaces of the longer portion as are necessary to make the 7 + spaces. Shapes of Specimens for Tensile Tests. The shapes shown in Fig. 75 were recommended by the author in 1882 when he was connected No. 1. Square or flat bar, as rolled. No. 2. Round bar, as rolled. No. 3. Standard shape for flats or squares. Fillets % inch radius. No. 4. Standard shape for rounds. Fillets J^ in. radius. No. 5. Government, shape for marine boiler-plates of iron. Not recommended for other tests, as results are generally in error. r* 16 Vso" FIG. 75. with the Pittsburgh Testing Laboratory. They are now in most general use, the earlier forms, with 5 inches or less in length between shoulders, being almost entirely abandoned. Precautions Required in making Tensile Tests, The testing-machine itself should be tested, to determine whether its weighing apparatus is accurate, and whether it is so made and adjusted that in the test of a properly made specimen the line of strain of the testing-machine is absolutely in line with the axis of the specimen. The specimen should be so shaped that it will not give an incorrect record of strength. It should be of uniform minimum section for not less than five inches of its length. Regard must be had to the i/ira* occupied in making tests of certain mate- rials. Wrought iron and soft steel can be made to show a higher than their actual apparent strength by keeping them under strain for a great length of time. Tn testing soft alloys, copper, tin, zinc, and the like, which flow under con- stant strain their highest apparent strength is obtained by testing them rapidly. In recording tests or such materials the length of time occupied in the test should be stated. 244 STRENGTH OF MATERIALS. For very accurate measurements of elongation, corresponding to incre- ments of load during the tests, the electric contact micrometer, described in Trans. A. S. M. E., vol. vi., p. 479, will be found convenient. When read- ings of elongation are then taken during the test, a strain diagram may be plotted from the reading, which is useful in comparing the qualities of dif- ferent specimens. Such strain diagrams are made automatically by the new Olsen testing-machine, described in Jour. Frank. Inst. 1891. The coefficient of elasticity should be deduced from measurement ob~ served between fixed increments of load per unit section, say between 2000 and 12,000 pounds per square inch or between 1000 and 11,000 pounds instead of between and 10,000 pounds. COMPRESSIVE: STRENGTH. What is meant by the term "compressive strength " has not yet been settled by the authorities, and there exists more confusion in regard to this term than in regard to any other used by writers on strength of materials. The reason of this may be easily explained. The effect of a compressive stress upon a material varies with the nature of the material, and with the shape and size of the specimen tested. While the effect of a tensile stress is to produce rupture or separation of particles in the direction of the line of strain, the effect of a compressive stress on apiece of material may be either to cause it to fly into splinters, to separate into two or more wedge-shaped pieces and fly apart, to bulge, buckle, or bend, or to flatten out and utterly re- sist rupture or separation of particles. A piece of speculum metal under compressive stress will exhibit no change of appearance until rupture takes place, and then it will fly to pieces as suddenly as if blown apart by gun- powder. A piece of cast iron or of stone will generally split into wedge- shaped fragments. A piece of wrought iron will buckle or bend. A piece of wood or zinc may bulge, but its action will depend upon fts shape and si:;e. A piece of lead will flatten out and resist compression till the last degree; that is, the more it is compressed the greater becomes its resistance. Air and other gaseous bodies are compressible to any extent as long as they retain the gaseous condition. Water not confined in a vessel is com- ressed by its own weight to the thickness of a mere film, while when con- ned in a vessel it is almost incompressible. It is probable, although it has not been determined experimentally, that solid bodies when confined are at least as incompressible as water. When they are not confined, the effect of a compressive stress is not only to shorten them, but also to increase their lateral dimensions or bulge them. Lateral strains are therefore induced by compressive stresses. The weight per square inch of original section required to produce any given amount or percentage of shortening of any material is not a constant quantity, but varies with both the length and the sectional area, with the shape of this sectional area, and with the relation of the area to the length. The " compressive strength' 1 of a material, if this term be supposed to mean the weight in pounds per square inch necessary to cause rupture, may vary with every size and shape of specimen experimented upon. Still more diffi- cult would it be to state what is the 4t compressive strength " of a material which does not rupture at all, but flattens out. Suppose we are testing a cylinder of a soft metal like lead, two inches in length and one inch in diam- eter, a certain weight will shorten it one per cent, another weight ten per cent, another fifty per cent, but no weight that we can place upon it will rupture it, for it will flatten out to a thin sheet. What, then, is its compres- sive strength ? Again, a similar cylinder of soft wrought iron would prob- ably compress a few per cent, bulging evenly all around ; it would then com- mence to bend, but at first the bend would be imperceptible to the eye and too small to be measured. Soon this bend would be great enough to be noticed, and finally the piece might be bent nearly double, or otherwise dis- torted. What is the "compressive strength 1 ' of this piece of iron ? Is it the weight per square inch which compresses the piece one per cent or five per cent, that which causes the first bending (impossible to be discovered), or that which causes a perceptible bend ? As showing the confusion concerning the definitions of compressive strength, the following statements from different authorities on the strength of wrought iron are of interest. Wood's Resistance of Materials states, " comparatively few experiments have been made to determine how much wrought iron will sustain at the point of crushing. Hodgkinson gives 65,000, Rondulet 70,800, Weisbach 72,000 COMPKESSIVE STKEKGTH. 245 Rankine 30,000 to 40,000. It is generally assumed that wrought iron will resist about two thirds as much crushing as to tension, but the experiments fail to give a" very definite ratio." Mr. Whipple, in his treatise on bridge-building, states that a bar of good wrought iron will sustain a tensile strain of about 60,000 pounds per square inch, and a compressive strain, in pieces of a length not exceeding twice the least diameter, of about 90,000 pounds. The following values, said to be deduced from the experiments of Major Wade, Hodgkinson, and Capt. Meigs, are given by Haswell : American wrought iron 127,720 Ibs. " (mean) 85,500 " TTnHih " J 65 > 200 " knglisn -j 40j00 o . Stoney states that the strength of short pillars of any given material, all having the same diameter, does not vary much, provided the length of the piece is not less than one and does not exceed four or five diameters, and that the weight which will just crush a short prism whose base equals one square Inch, and whose height is not less than 1 to 1J^ and does not exceed 4 or 5 diameters, is called the crushing strength of the material. It would be well if experimenters would all agree upon some such definition of the term " crushing strength, 1 ' and insist that all experiments which are made for the purpose of testing the relative values of different materials in com- pression be made on specimens of exactly the same shape and size. An arbitrary size and shape should be assumed and agreed upon for this pur- pose. The size mentioned by Stoney is definite as regards area of section, viz., one square inch, but is indefinite as regards length, viz., from one to five diameters. In some metals a specimen five diameters long would bend, and give a much lower apparent strength than a specimen having a length of one diameter. The words " will just crush " are also indefinite for ductile materials, in which the resistance increases without limit If the piece tested does not bend. In such cases the weight which causes a certain percentage of compression, as five, ten, or fifty per cent, should be assumed as the crushing strength. For future experiments On crushing strength three things are desirable : First, an arbitrary standard shape and size of test specimen for comparison of all materials. Secondly, a standard limit of compression for ductile materials, which shall be considered equivalent to fracture in brittle mate- rials. Thirdly, an accurate knowledge of the relation of the crushing strength of a specimen of standard shape and size to the crushing strength of specimens of all other shapes and sizes. The latter can only be secured by a very extensive and accurate series of experiments upon all kinds of materials, and on specimens of a great number of different shapes and sizes. The author proposes, as a standard shape and size, for a compressive test specimen for all metals, a cylinder one inch in length, and one half square inch in sectional area, or 0.798 inch diameter; and for the limit of compres- sion equivalent to fracture, ten per cent of the original length. The term "compressive strength," or "compressive strength of standard specimen," would then mean the weight per square inch required to fracture by com- pressive stress a cylinder one inch long and 0.798 inch diameter, or to reduce its length to 0.9 inch if fracture does not take place before that reduc- tion in length is reached. If such a standard, or any standard size whatever, had been used by the earlier authorities on the strength of materials, we never would have had such discrepancies in their statements in regard to the compressive strength of wrought iron as those given above. The reasons why this particular size is recommended are : that the sectional area, one-half square inch, is as large as can be taken in the ordinary test- ing-machines of 100,000 pounds capacity, to include all the ordinary metals of construction, cast and wrought iron, and the softer steels; and that the length, one inch, is convenient for calculation of percentage of compression. If the length were made two inches, many materials would bend in testing, and give incorrect results. Even in cast iron Hodgkinson found as the mean of several experiments on various grades, tested in specimens % inch in height, a compressive strength per square inch of 94,730 pounds, while the mean of the same number of specimens of the same irons tested in pieces 1J inches in height was only 88,800 pounds. The best size and shape of standard specimen should, however, be settled upon only after consultation and agreement among several authorities. 246 STllEKGTH OF MATERIALS. The Committee on Standard Tests 01 the American Society of Mechanical Engineers say (vol. xi., p. 624) : " Although compression tests have heretofore been made on diminutive sample pieces, it is highly desirable that tests be also made on long pieces from 10 to 20 diameters in length, corresponding more nearly with actual practice, in order that elastic strain and change of shape may be determined by using proper measuring apparatus. The elastic limit, modulus or coefficient of elasticity, maximum and ulti- mate resistances, should be determined, as well as the increase of section at various points, viz., at bearing surfaces and at crippling point. The use of long compression-test pieces is recommended, because the in- vestigation of short cubes or cylinders has led to no direct application of the constants obtained by their use in computation of actual structures, which have always been and are now designed according to empirical for- mulae obtained from a few tests of long columns." COLUMNS, PILLARS, OR STRUTS. Hodgkinson's Formula for Columns. P = crushing weight in pounds; d = exterior diameter in inches; d l = in- terior diameter in inches; L = length in feet. Kind of Column. Both ends rounded, the length of the column exceeding 15 times its diameter. P = 33,380 - ' = 95,850 p- Both ends flat, the length of the column exceeding 30 times its diameter. (3.56 >= 98,920-^ P = 99, ^- Solid cylindrical col- ) umns of cast iron ) Hollow cylindrical col- ) umns of cast iron ) Solid cylindrical col- ) umns of wrought iron. > Solid square pillar of \ Dantzic oak (dry) ) Solid square pillar of ) red deal (dry) . . f The above formulae apply only in cases in which the length is so great that the column breaks by bending and not by simple crushing. If the column be shorter than that given in the table, and more than four or five times its diameter, the strength is found by the following formula : PCK P = 299,600- P = 24,540^ in which P= the value given by the preceding formulae, K= the transverse section of the column in square inches, C = the ultimate compressive resis- tance of the material, and W = the crushing strength of the column. Hodgkinson's experiments were made upon comparatively short columns, the greatest length of cast-iron columns being 60^ inches, of wrought iron 90% inches. The following are some of his conclusions: 1 In all long pillars of the same dimensions, when the force is applied m the direction of the axis, the strength of one which has flat ends is about three times as great as one with roun L ed ends. 2 The strength of a pillar with ^ne nd rounded and the other flat is an arithmetical mean between the two given in the preceding case of the same dimensions. 3. The strength of a pillar having both ends firmly fixed is the same as one of half the length with both ends rounded. 4. The strength of a pillar is not increased more than one seventh by en- larging it at the middle. MOMENT OF INEKTIA AND RADIUS OF GYRATION. 247 Gordon's formulae deduced from Hodgkinson's experiments are more generally used than Hodgkinson's own. They are: Columns with both ends fixed or flat, P = - ; fs Columns with one end flat, the other end round, P = ' f sf Columns with both ends round, or hinged, P = - -; ~ 8 = area of cross-section in inches; P ultimate resistance of column, in pounds; / = crushing strength of the material in Ibs. per square inch; . , _ Moment of inertia r least radius of gyration, in inches, ?- 2 = area of section ' I length of column in inches; a a coefficient depending upon the material; /and a are usually taken as constants; they are really empirical variables, dependent upon the dimensions and character of the column as well as upon the material. (Burr.) For solid wrought-iron columns, values commonly taken are: / = 36,000 to 40,000; a = 1/36,000 to 1/40,000. For solid cast-iron columns, / = 80,000, a = 1/6400. 80 non For hollow cast-iron columns, fixed ends, p - - -- , I length and ficients derived from Hodgkinson's experiments, for cast-iron columns is to he deprecated. See Strength of Cast-iron Columns, pp. 250, 251. Sir Benjamin Baker gives, For mild steel, / = 67,000 Ibs., a = 1/22,400. For strong steel, /= 114,000 Ibs., a = VH400 Prof. Burr considers these only loose approximations for the ultimate resistances. See his formulae on p. 259. For dry timber Rankine gives/ = 7200 Ibs., a = 1/3000. MOMENT OF INERTIA AND RADIUS OF GYRATION. The moment of inertia of a section is the sum of the products of each elementary area of the section into the square of its distance from an assumed axis of rotation, as the neutral axis. The radius of gyration of the section equals the square root of the quotient of the moment of inertia divided by the area of the section. If E = radius of gyration, 1= moment of inertia and A area, The moments of inertia of various sections are as follows; d = diameter, or outside diameter; d } = inside diameter; 6 = breadth; h = depth; 6,, &, inside breadth and diameter; Solid rectangle I = l/126/i3; Hollow rectangle I = l/12(67i - Mi 3 ); Solid square 7= 1/126*; Hollow square 7= 1/12(6* - 6,*); Solid cylinder I- l/647rd 4 ; Hollow cylinder I- l/647r(d 4 - c^ 4 ). Moments of Inertia and Radius of Gyration for Various Sections, and their Use in the Formulas for Strength of Girders and Columns, The strength of sections to resist strains, either as girders or as columns, depends not only on the area but also on the form of the section, and the property of the section which forms the basis of the constants used in the formulas for strength of girders and columns to express the effect of the form, is its moment of inertia about its neutral axis. The modulus of resistance of any section to transverse bending is its 248 STRENGTH OF MATERIALS. moment of inertia divided by the distance from the neutral axis to the fibres farthest removed from that axis; or Moment of inertia I = Distance of extreme fibre from axis* " y' Moment of resistance = section modulus X unit stress on extreme fibre. IHoment of Inertia of Compound Shapes. (Pencoyd Iron Works.) The moment of inertia of any section about any axis is equal to the I about a parallel axis passing through its centre of gravity -f (the area of the section X the square of the distance between the axes). By this rule, the moments of inertia or radii of gyration of any single sec- tions being known, corresponding values may be obtained for any combina- tion of these sections. Radius of Gyration of Compound Shapes, In the case of a pair of any shape without a web the value of R can always be found with- out considering the moment of inertia. The radius of gyration for any section around an axis parallel to another axis passing through its centre of gravity is found as follows: Let r = radius of gyration around axis through centre of gravity; R =t radius of gyration around another axis parallel to above; d = distance be- tween axes: R = Vd'* -f- r*. When r is small, R may be taken as equal to d without material error. Graphical Method for Finding Radius of Gyration. Ben j. F. La Hue, Eng. Neius, Feb. 2, 1893, gives a short graphical method for finding the radius of gyration of hollow, cylindrical, and rectangular col- umns, as follows: For cylindrical columns: Lay off to a scale of 4 (or 40) a right-angled triangle, in which the base equals the outer diameter, and the altitude equals the inner diameter of the column, or vice versa. The hypothenuse, measured to a scale of unity (or 10), will be the radius of gyration sought. This depends upon the formula 'Mom, oflnertia _ ^D* + d 2 Area 4 in which A = area and D = diameter of outer circle, a area and d = dia- meter of inner circle, and G = radius of gyration. ^D' 2 -f d 2 is the expres- sion for the hypothenuse of a right-angled triangle, in which D and d are the base and altitude. The sectional area of a hollow round column is .7854(D 2 d 2 ). By con- structing a right-angled triangle in which D equals the hypothenuse and d equals the altitude, the base will equal 4/D 2 d 2 . Calling the value of this expression for the base J5, the area will equal .78541? 2 . Value of G for square columns: Lay off as before, but using a scale of 10, a right-angled triangle of whicfc the base equals D or the side of the outer square, and the altitude equals d, the side of the inner square. With a scale of 3 measure the hypotheuuse, which will be, approximately, the radius of gyration. This process for square columns gives an excess of slightly more than 4#. By deducting 4% from the result, a close approximation will be obtained. A very close result is also obtained by measuring the hypothenuse with the same scale by which the base and altitude were laid off, and multiplying by the decimal 0.29; more exactly, the decimal is 0.28867. The formula is This may also be applied to any rectangular column by using the lesser diameters of an unsupported column, and the greater diameters if the col- umn is supported in the direction of its least dimensions. ELEMENTS OF USUAI, SECTIONS. Moments refer to horizontal axis through centre of gravity. This table is intended for convenient application where extreme accuracy is not impor- tant. Some of the terms are only approximate; those marked * are correct. Values for radius of gyration in flanged beams apply to standard minimum sections only; A = area of section; b ^ bjeadth; h = depth; D = diameter. ELEMENTS OF USUAL SECTIONS. 249 Shape of Section. Moment of Inertia. Section Modulus. Square of Least Radius of Gyration. Least Radius of Gyration. .._.... Solid Rect- angle. bh* * 12 ~6~ (Least side)2* Least side * 12 3.46 * Hollow Rect- angle. 6W-Mi * bV-bfa** / t a _f Ttja * MtM JJgli Vrb+ 12 6/1 12 4.89 T Solid Circle. AD* * 16 AD* 8 D* * 16 "4 (*- D 1 Hollow Circle. A, area of large section ; a, area of small section. AD*-ad* AD* -ad* D2+ef2* 16 D + rf 16 SD 5.64 s Solid Triangle. bh* 36 bh* 24 The least of of the two: 18 F 24 The least of the two: h b 4.24 r 4.9 Even Angle. Ah* 10.2 Ah 7.2 b* 25 6 5 JE Uneven Angle. Ah* Ah 6.5 0*" hb 9.5 13(/i 2 + 62) 2.6(71 -f 6) H8 Even Cross. Ah* 19 Ah 9.5 M 22.5 & 4.74 i Even Tee. Ah* 11.1 Ah 8 62 22.5 6 4.74 ^ I Beam. Ah* 6.66 Ah 3.2 b* 21 6 4.58 lirt^x] Channel. Ah* 7.34 Ah 3.67 12T5 6 3.54 111 Deck Beam. Ah* 6.9 Ah 4 62 36.5 b 6 Distance of base from centre of gravity, solid triangle, ^; even angle, -^-- 6 6.6 uneven angle, $-=; ven tee, ^-5? deck beam, - ; all other shapes given in o.o o.o 4.6 the table, ~ or . 250 STRENGTH OF MATERIALS. The Strength of Cast-iron Columns. Hodgkinson's experiments (first published in Phil. Trans. Royal Socy., 1840, and condensed in Tredgold on Cast Iron, 4th ed., 1846), and Gordon's formula, based upon them, are still used (1898) in designing cast-iron col- umns. That they are entirely inadequate as a basis of a practical formula suitable to the present methods of casting columns will be evident from what follows. Hodgkinson's experiments were made on nine " long " pillars, about 7^ ft. long, whose external diameters ranged from 1.74 to 2.23 in., and average thickness from 0.29 to 0.35 in., the thickness of each column also varying, and on 18 "short " pillars, 0.733 ft. to 2.251 ft. long, with external diameters from 1.08 to 1.26 in., all of them less than J4 in. thick. The iron used was Low Moor, Yorkshire, No. 3, said to be a good iron, not very hard, earlier experiments on which had given a tensile strength of 14,535 and a crushing strength of 109,801 Ibs. per sq. in. The results of the experiments on the " long " pillars were reduced to the equivalent breaking weight of a solid pillar 1 in. diameter and of the same length, 714 ft., which ranged from 2969 to 3587 Ibs. per sq. in., a range of over 12 per cent, although the pillars were made from the same iron and of nearly uniform dimensions. From the 13 experiments on " short " pillars a formula was derived, and from it were obtained the " calculated " breaking weights, the actual breaking weights ranging from about 8 per cent above to about 8 per cent below the calcu- lated weights, a total range of about 16 per cent. Modern cast-iron columns, such as are used in the construction of buildings, are very different in size, S:oportions, and quality of iron from the slender " long" pillars used in odgkinson's experiments. There is usually no check, by actual tests or by disinterested inspection, upon the quality of the material. The tensile, com- pressive, and transverse strength of cast iron varies through a great range (the tensile strength ranging from less than 10,000 to over 40,000 Ibs. per sq. in.), with variations in the chemical composition of the iron, according to laws which are as yet very imperfectly understood, and with variations in the method of melting and of casting. There is also a wide variation in the strength of iron of the same melt when cast into bars of different thick- nesses. It is therefore impossible to predict even approximately, from the data given by Hodgkinson of the strength of columns of Low Moor iron in pillars 7% ft. long, 2 in. diam., and % in. thick, what will be the strength of a column made of American cast iron, of a quality not stated, in a column 16 ft. long, 12 or 15 in. diam., and from % in. to 1^ in. thick. Another difficulty in obtaining a practical formula for the strength of cast- iron columns is due to the uncertainty of the quality of the casting, and the danger of hidden defects, such as internal stresses due to unequal cooling, cinder or dirt, blow-holes, u cold-shuts, 1 ' and cracks on the inner surface, which cannot be discovered by external inspection. Variation in thick- ness, due to rising of the core during casting, is also a common defect. In addition to^the above theoretical or a priori objections to the use of Gordon's formula, based on Hodgkinson's experiments, for cast-iron columns, we have the data of recent experiments on full-sized columns, made by the Building Department of New York City (Eng'g News, Jan. 13 and 20, 1898). Ten columns in all were tested, six 15-inch, 190J inches long, two 8-inch, 160 inches long, and two 6-inch, 120 inches long. The tests were made on the large hydraulic machine of the Phoenix Bridge Co., of 2,000,000 pounds capacity, which was calibrated for frictiorml error by the repeated testing within the elastic limit of a large Phoenix column, and the compari- son of these tests with others made on the government machine at the Watertown Arsenal. The average frictional error was calculated to be 15.4 per cent, but Engineering Neivs, revising the data, makes it 17.1 per cent, with a variation of 3 per cent either way from the average with differ- ent loads. The results of the tests of the volumes are given on the opposite page. Column No. 6 was not broken at the highest load of the testing machine. Columns Nos. 3 and 4 were taken from the Ireland Building, which col- lapsed on August 8, 1895; the other four 15-inch columns were made from drawings prepared by the Building Department, as nearly as possible duplicates of Nos. 3 and 4. Nos. 1 and 2 were made by a foundry in New York with no knowledge of their ultimate use. Nos. 5 and 6 were made by a foundry in Brooklyn with the knowledge that they were to be tested. Nos. 7 to 10 were made from drawings furnished by the Department. THE STRENGTH OF CAST-IROK COLUMNS. 251 TESTS OF CAST-IRON COLUMNS. Thickness. Breaking Load. Niimhpi* Diam. Inches. Max. Min. Average. Pounds. Pounds per sq. in. 1 15 1 1 1 1,356,000 80,830 2 15 1 5/16 1 *6 1,330,000 27,700 3 15 1*4 1 *6 1,198,000 24.900 4 15J^ 1 7/32 1 *6 1,246,000 25,200 5 15 1 11/16 1 11/64 1,632,000 32,100 6 15 1*4 1*6 3/16 2,082,000 + 40,400 -f 7 7% to 8M 1*4 % 651,000 31,900 8 8 1 3/32 1 3/61 612,800 26,800 9 61/16 1 5/32 1*6 9/64 400,000 22,700 10 6 3/32 1*6 1 1/16 7/64 455,200 26,300 lying Gordon's formula, as used by the Building Department, ^ ^, to these columns gives for the breaking strength per square inch of the 15-inch columns 57,143 pounds, for the 8-inch columns 40,000 pounds, and for the 6-inch columns 40,000. The strength of columns Nos. 3 and 4 as calculated is 128 per cent more than their actual strength; their actual strength is less than 44 per cent of their calculated strength; and the factor of safety, supposed to be 5 in the Building Law, is only 2.2 for central loading, no account being taken of the likelihood of eccentric loading. Prof. Lanza, in Jhis Applied Mechanics, p. 372, quotes the records of 14 tests of cast-iron mill columns, made on the Watertown testing-machine in 1887-88, the breaking strength per square inch ranging from 25,100 to 63,310 pounds, and showing no relation between the breaking strength per square inch and the dimensions of the columns. Only 3 of the 14 columns had a strength exceeding 33,500 pounds per square inch. The average strength of the other 11 was 29,600 pounds per square inch. Prof. Lanza says that it is evident that in the case of such columns we cannot rely upon a crushing strength of greater than 25,000 or 30,000 pounds per square inch of area of section. He recommends a factor of safety of 5 or 6 with these figures for crush- ing strength, or 5000 pounds per square inch of area of section as the highest allowable safe load, and in addition makes the conditions that the length of the column shall not be greatly in excess of 20 times the diameter, that the thickness of the metal shall be such as to insure a good strong casting, and that the sectional area should be increased if necessary to insure that the extreme fibre stress due to probable eccentric loading shall not be greater than 5000 pounds per square inch. Prof. W. H. Burr (Eng'g News, June 30, 1898) gives a formula derived from plotting, the results of the Watertown and Phoenixville tests, above described, which represents the average strength of the columns in pounds per square inch. It isp = 30,500 - IQOl/d. It is to be noted that this is an average value, and that the actual strength of many of the columns was much lower. Prof. Burr says: " If cast-iron columns are designed with anything like a reasonable and real margin of safety, the amount of metal required dissipates any supposed economy over columns of mild steel." Transverse Strength of Cast-iron Water-pipe. (Technology Quarterly, Sept. 1897.) Tests of 31 cast-iron pipes by transverse stress gave a maximum outside fibre stress, calculated from maximum load, assuming each half of pipe as a beam fixed at the ends, ranging from 12,800 Ibs. to 26,300 Ibs. per sq. in. Bars 2 in. wide cut from the pipes gave moduli of rupture ranging from 28,400 to 51,400 Ibs. per sq. in. Four of the tests, bars and pipes: Moduli of rupture of bar 28,400 34,400 40,000 51 ,400 Fibre stress of pipe ... 18,300 12,800 14,500 26,300 These figures show a great variation in the strength of both bars and pipes, and also that the strength of the bar does not bear any definite rela- tion to the strength of the pipe. 252 STRENGTH OP MATERIALS. Safe Load, in Tons of 200O I/bs., for Round Cast-iron Columns, with Turned Capitals and Bases, Loads being not eccentric, and length of column not exceeding 20 times the diameter. Based on ultimate crushing strength of 25,000 Ibs. per sq. in. and a factor of safety of 5. (For eccentric loads see page 254.) Thick- ness, /nches. Diameter, inches. 6 7 8 9 10 54.5 62.7 70.7 78.4 85.9 93.1 11 12 13 14 15 16 18 1 IVii IK 2 26.4 30.9 35.2 39.2 31.3 36.8 42.1 47.1 42.7 48.9 55.0 60.8 48.6 55.8 62.8 69.6 76.1 69.6 78.5 87.2 95.7 103.9 76.5 86.4 96.1 105.5 114.7 123.7 94.2 104.9 115.3 125.5 135.5 102.1 113.8 125.2 136. 3 147.8 168.4 110.0 122.6 135.0 147.1 159.0 182.1 204.2 131.4 144.8 157.9 170.8 195.8 219.9 164.' 179. ( 194. < 223.! 251.; .... For lengths greater than 20 diameters the allowable loads should be decreased. How much they should be decreased is uncertain, since suf- ficient data of experiments on full-sized very long columns, from which a formula for the strength of such columns might be derived, are as yet lacking. There is, however, rarely, if ever, any need of proportioning cast* iron columns with a length exceeding 20 diameters. Safe Loads in Tons of 2000 Pounds for Cast-iron Columns^ (By the Building Laws of New York City, Boston, and Chicago, 1897.) New York. Boston. Chicago. 8a 5a 5a Square columns Round columns. ., 1 -f j: Sa ! + i 5a 1 + I* 400<2 a 1 + 800d a a = sectional area in square inches; I = unsupported length of column m inches; d = side of square column or thickness of round column in inches. The safe load of a 15-inch round column !$ inches diameter, 16 feet long, according to the laws of these cities would be, in New York, 361 tons; in Boston, 264 tons; in Chicago, 250 tons. The allowable stress per square inch of area of such a column would be, in New York, 11,350 pounds; in Boston, 8300 pounds; in Chicago, 7850 pounds. A safe stress of 5000 pounds per square inch would give for the safe load on the column 159 tons. Strengtn of Brackets on Cast-iron Columns, The columns tested by the New York Building Department referred to above had brackets cast upon them, each bracket consisting of a rectangular shelf supported by one or two triangular ribs. These were tested after the columns had been broken in the principal tests. In 17 out of 22 cases the brackets broke by tearing a hole in the body of the column, instead of by shearing or transverse breaking of the bracket itself. The results were surprisingly low and very irregular. Reducing them to strength per square inch of the total vertical section through the shelf and rib or ribs, they ranged from 2450 to 5600 Ibs., averaging 4200 Ibs., for a load concentrated at the end of the shelf, and 4100 to 10,900 Ibs., averaging 8000 Ibs., for a dis- tributed load. (Eng'g News, Jan. 20, 1898.) SAFE LOAD OF CAST-IROK COLUMKS. 253 Safe Loads, in Tons, for Round Cast Columns. In accordance with the Building Laws of Chicago.*) Diame ter in Inches Thick- ness in Inches. Unsupported Length in Feet. 6 8 10 12 14 16 18 20 22 24 26 28 30 /> ( % 50 43 37 32 27 KT, 6 1 H 57 5( 42 36 31 Formula: u ; ~ ' Za ' aJ % 62 56 49 43 38 33 1 -f- - 7 i YB 71 64 57 49 43 38 w = safe load in tons of i f 8( 97 (59 79 89 71 81 56 64 72 50 57 63 44 50 56 39 44 50 2000 pounds; a = cross-section of col- umn; \ 7 A 101 94 86 78 70 63 57 I = unsupported length 9 i 1 113 126 105 117 97 107 88 97 79 88 71 79 fr 7] in inches; d = diameter in inches. r % 116 109 101 93 85 78 71 64 i 1 130 122 114 105 96 88 80 72 1 l^ 145 136 126 ir 107 97 8 S 80 i 158 149 139 128 117 107 88 f 1 147 139 131 122 113 104 96 88 8 11 j l^ 163 155 146 136 126 116 100 ,97 8 1 1/4 179 170 160 149 138 127 119 jo; 9 l 1% 195 185 174 162 150 138 12 r 10 r 1^ 181 174 165 155 145 135 125 115 10 98 19 j 1M 199 191 181 170 159 148 13* 11 108 A 1 1% 217 207 197 1ST 173 161 149 jl 12 117 1 1% 234 224 212 200 187 173 161 13 12b f 1/^j 200 192 184 174 164 154 144 134 125 116 107 13 l^ 219 211 202 191 180 169 158 U7 127 117 1% 239 230 220 208 196 184 172 160 14 138 128 1 l^fj 258 248 237 225 212 199 186 173 16 149 138 r 1J4 232 223 213 202 191 180 168 157 147 137 128 1% 253 243 232 220 207 195 183 17 160 149 139 l^j 27 263 251 238 224 211 198 185 173 161 150 I 1% 293 282 269 255 241! 22? 212 198 185 173 161 r 1% 266 255 243 231 219 206 194 182 171 160 150 ,J 1^3 287 276 263 250 236 223 210 197 185 173 162 1 1% 309 296 283 268 254 239 225 211 198 186 174 I 1M 329 316 301 286 271 255 240 225 211 198 185 I 1/^ 301 288 275 262 248 235 222 209 197 185 16-\ j^ 323 310 296 282 267 253 239 225 212 199 ( 1% 345 331 316 300 285 270 254 239 225 212 l% 366 351 337 322 307 293 279 264 251 18-] 1% 391 375 360 344 328 313 298 282 268 1 1% 415 399 383 366 349 333 317 300 285 r 1% 435 420 404 389 373 357 341 326 on J 1% 463 447 431 414 397 380 363 347 ^ V ^ 2 ' 490 473 456 43S 420 402 384 367 I 2^j 517 499 481 462 443 425 406 387 f 1% 480 464 448 432 416 400 384 1 l/'O 511 494 478 461 443 426 409 1 541 524 506 488 470 452 434 1 %}& 581 562 543 524 504 485 465 f 2^ 626 608 589 570 550 531 oj 2*4 668 639 620 600 579 559 ** 1 2% 691 671 650 629 608 587 i *" 724 703 681 659 637 614 From tables published by The Expanded Metal Co., Chicago, 1897.) 254 STRENGTH OF MATERIALS. ECCENTRIC LOADING OF COLUMNS. In a given rectangular cross-section, such as a masonry joint under press- ure, the stress will be distributed uniformly over the section only when the resultant passes through the centre of the section ; any deviation from such a central position will bring a maximum unit pressure to one edge and a minimum to the other; when the distance of the resultant from one edge is one third of the entire width of the joint, the pressure at the nearer edge is twice the mean pressure, while that at the farther edge is zero, and that when the resultant approaches still nearer to the edge the pressure at the farther edge becomes less than zero; in fact, becomes a tension, if the material (mortar, etc., there is capable of resisting tension. Or, if, as usual in masonry joints, the material is practically incapable of resisting tension, the pressure at the nearer edge, when the resultant approaches it nearer than one third of the width, increases very rapidly and dangerously, becom- ing theoretically infinite when the resultant reaches the edge. With a given position of the resultant relatively to one edge of the joint or section, a similar redistribution of the pressures throughout the section may be brought about by simply adding to .or diminishing the width of the section. Let P = the total pressure on any section of a bar of uniform thickness. w = the width of that section area of the section, when thickness = 1. p = P/w the mean unit pressure on the section. M the maximum unit pressure on the section. m = the minimum unit pressure on the section. d = the eccentricity of the resultant = its distance from the centre of the section. ThenM = p (l+~ ) and m = p (l - ^). When d = - w then M = %p and m = 0. When d is greater than l/6w, the resultant in that case being less than one third of the width from one edge, p becomes negative. (J. C. Traut- wine, Jr., Engineering News, Nov. 23, 1893.) Eccentric Loading of Cast-iron Columns. Prof. Lanza writes the author as follows: The table on page 252 applies when the resultant of the loads upon the column acts along its central axis, i.e., passes through the centre of gravity of every section. In buildings and other construc- tions, however, cases frequently occur when the resultant load does not pass through the centre of gravity of the section ; and then the pressure is not evenly distributed over the section, but is greatest on the side where the resultant acts. (Examples occur when the loads on the floors are not uniformly distributed.) In these cases the outside fibre stresses of the column should be computed as follows, viz.: Let P = total pressure on the section; d = eccentricity of resultant = its distance from the centre of gravity of the section; A = area of the section, and Jt its moment of inertia about an axis in its plane, passing through its centre of gravity, and perpendicular to d (see page 26?) ; Cj = distance of most compressed and c a = that of least compressed fibre from above stated axis; *j = maximum and s a = minimum pressure per unit of area. Then *-+2& and H-Z Having assumed a certain trial section for the column to be designed, s l should be computed, and, if it exceed the proper safe value, a different section should be used for which Sj does not exceed this value. The proper safe value, in the case of cast-iron columns whose ratio of length to diameter does not greatly exceed 20, is 5000 pounds per square inch when the eccentricity used in the computation of s l is liable to occur fre- quently in the ordinary uses of the structure; but when it is one which can only occur in rare cases the value 8000 pounds per square inch may be used. A long cap on a column is more conducive to the production of eccen- tricity of loading than a short one, hence a long cap is a source of weakness in a column. ULTIMATE STRENGTH OF WROUGHT-IROK COLUMNS. 255 ULTIMATE STRENGTH OF WROUGHT-IRON COL.UMNS. (Pottsville Iron and Steel Co.) Computed by Gordon's formula, p = 14-0 p = ultimate strength in Ibs. per square inch; I = length of column in inches; r least radius of gyration in inches; /= 40,000; C = 1/40,000 for square end-bearings; 1/30,000 for one pin and one square bearing; 1/20,000 for two pin-bearings. For safe working load on these columns use a factor of 4 when used in buildings, or when subjected to dead load only; but when used in bridges the factor should be 5. WROUGHT-IRON COLUMNS. Ultimate Strength in Ibs. per square inch. Safe Strength in Ibs. per square inch Factor of 5. I I r r Square Ends. Pin and Square End. Pin Ends. Square Ends. Pin and Square End. Pin Ends. 10 39944 39866 39800 10 7989 7973 7960 15 39776 39702 39554 15 7955 7940 7911 20 39604 39472 39214 20 7921 7894 7843 25 39384 39182 38788 25 7877 7836 7758 30 39118 38834 38278 30 7821 7767 7656 35 38810 38430 37690 35 7762 7686 7538 40 38460 37974 37036 40 7692 7595 7407 45 38072 37470 36322 45 7614 7494 7264 50 37646 36928 35525 50 7529 7386 7105 55 37186 36336 34744 55 7437 7267 6949 60 36697 35714 33898 60 7339 7143 6780 65 36182 34478 33024 65 7236 6896 6605 TO 35634 34384 32128 70 7127 6877 6426 75 35076 33682 31218 75 7015 6736 6244 80 34482 32966 30288 80 6896 6593 6058 85 33883 32236 29384 85 6777 6447 5877 90 33264 31496 28470 90 6653 6299 5694 95 32636 30750 27562 95 6527 6150 5512 100 32000 30000 26666 100 6400 6000 5333 105 31357 29250 25786 105 6271 5850 5-157 Maximum Permissible Stresses in columns used in buildings. (Building Ordinances of City of Chicago, 1893.) For riveted or other forms of wrought-iron columns: # _ 12000a I = length of column in inches; , Z 2 r = least radius of gyration in inches; of column in f square inches. a = are; For riveted or other steel columns, if more than 60r in length: = 17,000 - . S = 13,500a. a = area of post in square inches ; ~*ijleast side of rectangular post in inches; I =Tength v of post in inches; I 600 for" *w hite*x> r Jfl_r waw pine ; c = < 800 for oak ; ?~ \ ( 900 for long-leaf yellow pine. 256 STRENGTH OF MATERIALS. BUILT COLUMNS. From experiments by T. D. Lovett, discussed by Burr, the values of / and a in several cases are determined, giving empirical forms of Gordon's for- mula as follows: p = pounds crushing strength per square inch of section, I = length of column in inches, r = radius of gyration in inches. Keystone Keystone Columns. 39,500 1-f ~1 Z 2 18,300 r 2 (D 36,000 i(2) Flat Ends. Square Columns. 39,000 (4) Phoenix Columns. 42,000 1 Za Am. Dr. Co. American Bridge Co. Columns. (6) 35,000 r 2 50,000 r 2 Flat Ends, Swelled* 36,000 1 Z 2 (91 46,000 ; 36,000 - * * ( r 15,000 r Pin Ends. 39,000 (5) 42,000 :(7) i . - 1 _ 1.1- _ ^ 17,000 r 2 ^ 22,700 r 2 Pin Ends, Swelled* Round Ends. 42,000 1 12,500 r 36,000 1 J_ 21,500ra (10) 36,000 1-f- 1 Z 2 (11) 11,500 r 2 With great variations of stress a factor of safety of as high as 6 or 8 may be used, or it may be as low as 3 or 4, if the condition of stress is uniform or essentially so. Burr gives the following general principles which govern the resistance of built columns : The material should be disposed as far as possible from the neutral axis of the cross-section, thereby increasing ?; There should be no initial' internal stress; The individual portions of the column should be mutually supporting; The individual portions of the column should be so firmly secured to each ofher that no relative motion can take place, in order that the column may fail as a whole, thus maintaining the original value of r. Stoney says: **When the length of a rectangular wrought- iron tubular column does not exceed 30 times its least breadth, it fails by the bulging or buckling of a short portion of the plates, not by the flexure of th6 pillar as a whole." In Trans. A. S. C. E., Oct. 1880, are given the following formulae for the ultimate resistance of wrought-iron columns designed by C. Shaler Smith : BUILT COLUMNS. 257 Flat Ends. - (! 1 + 5820 d Phoenix Column. 42,500 1 Z 2 1 j *_ j_ ^4500 d 8 American Bridge Co. Column. (15) 36,500 Common Column. 36,500 (1> -- 3750 d* 2700 d One Pin End. 38,500 14 i- -^ ^3000 d a (13) 40,000 - 14- i- - ^2250 d a (16) 36,500 (19) 36,500 (22) 2250 Two Fin Ends. ootn ^75 *"T 1500 36,600 36,500 1750 d 36,500 14- - ^1200 d (23) The "common " column consists of two channels, opposite, with flanges outward, with a plate on one side and a lattice on the other. The formula for " square " columns may be used without much error for the common-chord section composed of two channel-bars and plates, with the axis of the pin passing through the centre of gravity of the cross- section. (Burr). Compression members composed of two channels connected by zigzag bracing may be treated by formulae 4 and 5, using / = 36,000 instead of 89,000. Experiments on full-sized Phoenix columns in 1873 showed a close agree- ment of the results with formulae 6-8. Experiments on full-sized Phoenix columns on the Watertown testing-machine in 1881 showed considerable dis- crepancies when the value of I -*- r became comparatively small. The fol- lowing modified form of Gordon's formula gave tolerable results through the whole range of experiments : Phoenix columns, flat end, p 40,000 ( 14-T-J 14-50,000 r a (24) Plotting results of three series of experiments on Phcenix columns, a more simple formula than Gordon's is reached as follows : Phcenix columns, flat ends, p = 39,640 46-, when I -*- r is from 30 to 140; p = 64,700 - 4600 \/- when I -*- r is less than 30. Dimensions of Phoenix Columns* (Phoenix Iron Co.) The dimensions are subject to slight variations, which are unavoidable In rolling iron shapes. The weights of columns given are those of the 4, 6, or 8 segments of which they are composed. The rivet heads add from 2g to 5# to the weights given. Rivets are spaced 3, 4, or 6 in. apart from centre to centre, and somewhat more closely at the ends than towards the centre of the column. G columns have 8 segments, E columns 6 segments, C, 2 , B l , and A have 4 segments. Least radius of gyration = D X .3636. The safe loads given are computed as being one-fourth of the breaking load, and as producing a maximum stress, in an axial direction, on a square- end column of not more than 14,000 Ibs, per sq. in, for lengths of 90 radii and under, 258 STRENGTH OF MATERIALS. Dimensions of Phoenix Steel Columns. (Least radius of gyration equals D x .3G?6.) One Segment. Diameters in Inches. One Column. S3 jS . gj . S'a ^? *M 1-173 9 tt * 202 ^TJ ijtj "~ i-> 03 03 Q) Q) .S 3 1 3 > c? <*-, S CO &2 "S o JS S * ^3 ^H q p O J3$ o-2| SP 5 * -v <t ~ > *> f II or^ 03 O o Si2 1 o5 J'S.S sjjj t^ < P 02 3/16 9.7 4 6 1/16 3.8 12.9 1.45 18.2 ^4 12.2 A 4/^ 6 3/16 4.8 16.3 1.50 23.9 5/16 14.8 3% 4J4 6 5/16 5.8 19.7 1.55 30.0 % 17.3 4% 6 7/16 6.8 23.1 1.59 35.9 5/1 6 16.3 19.9 f /B si/16 6.4 7.8 21.8 26.5 1.95 2.00 36.4 45.1 % 23.5 5% 8 5/16 9.2 31.3 2.04 54.4 7/16 27.0 B.I 5% 8 7/16 10.6 36.0 2.09 63.9 /^ 30.6 4/ 5% gl^f. 12.0 40.8 2.13 73.3 9/16 34.2 6 8 9/16 13.4 45.6 2.18 83.2 % 37.7 % 8 11/16 14.8 50.3 2.23 93.1 24 18.9 6 9/16 91^ 7.4 25.2 2.39 48.3 5/16 22.9 6 11/16 9% 9.0 30.6 2.43 59.5 % 27.0 6 13/16 9 7/16 10.6 36.0 2.48 70 7 7/16 31.1 35.2 B.2 6 1/16 6 15/16 7 1/16 12.2 13.8 41.5 46.9 2.52 2.57 82.3 93.9 9/16 39.3 7 3/16 ^M 15.4 52.4 2.61 105.8 % 43.3 7 5/16 9 13/16 17.0 57.8 2.66 111.9 M 25^ 7 13/16 11 11/16 10.0 34.0 2.84 70.0 5/16 31 7 15/16 HM 12.1 41.3 2.88 85.1 % 36 8 1/16 11 13/16 14.1 48.0 2.93 98.8 a 6 41 8 3/16 11% 16.0 54.6 2.97 112.5 46 8 5/16 11 15/16 18.0 61 3 3.01 126.3 9/16 51 8 7/16 12 19.9 68.0 3.06 140.0 % 56 C 8 9/16 12 1/16 21.9 74.6 3.11 153.7 11/16 62 7% 8 11/16 12 3/16 24.3 82.6 3.16 170.2 M 68 8 13/16 12 5/16 26.6 90.6 3.20 186.7 13/16 73 8 15/16 12 7/16 28.6 97.3 3.24 200.3 % 78 9 1/16 12^ 30.6 104.0 3.29 214.2 1 89 9 5/16 12% 34.8 118.6 3.34 244.3 1% 99 9 9/16 12 13/10 38.8 132.0 3.48 271.7 1M 109 9 13/16 13 42.7 145.3 3.57 299.2 J4 28 11 9/16 151/ ' 16.5 56.0 4.20 115.3 5/16 32^ 11 11/16 15% 19.1 65.0 4.25 133.8 % 37 11 13/16 15% 21.7 74.0 4.29 152.4 7/16 42 11 15/16 15% 24.7 84.0 4.34 173.0 47 12 1/16 15 15/16 27.6 94.0 4.38 193 6 9/16 52 12 3/16 16 1/16 30.6 104.0 4.43 214.1 % 57 E 12 5/16 16 3/16 33.5 114.0 4.48 234.7 11/16 62 11 1/16 12 7/16 16 5/16 36.4 124.0 4.52 255.3 % 68 12 9/16 16 7/16 40.0 136.0 4.56 280.0 13/16 73 12 11/16 16 9/16 43.0 146.0 4.61 300 6 % 78 12 13/16 16 11/16 45.9 156.0 4.66 321.2 1 88 13 1/16 16 13/16 51.7 176.0 4.73 36',>.4 1% 98 13 5/16 17 1/16 57.6 196.0 4.84 403.6 JJ4 108 13 9/16 17 5/16 63.5 216.0 4.93 444.7 5/16 31 15H P* 24.2 82.6 5.54 170.2 % 36 G 15% 28.1 96.0 5.59 197.7 7/16 41 14% 1514 32.0 109.3 5.64 225.1 ^ 46 i!5% 1/16 36.0 122.6 5.68 252.6 FORMULAE FOR IROK AtfD STEEL STRUTS. 259 One Segment. Diameters in Inches. One Column. II a $ - V* K a C!A Thickness i Inches. Weight in L per Yard, d Inside. D Outside . 2 bo 5> h Area of Cro Section, S( Inches. Weight per in Pounds. ' Least Radii of Gyratio in Inches. Safe Load i Tons for 11 Lengths. 9/16 51 15% ]9 39.9 136.0 5.73 280.0 ?8 56 15% 19% 43.8 149.3 5.77 307.4 ]1/16 61 16 20 47.7 162.6 5.82 334.9 M 66 20% 51.7 176.0 5.88 362.4 13/16 71 G 16^4 55.6 189.3 5.91 389.8 7 76 16% 20% 59.6 202.6 5.95 417.3 1 86 16% 20% 67.4 229.3 6.04 472.1 1% 96 16% 20% 75.3 256.0 6.13 527.3 ig 106 17% 21 83.1 282.6 6.27 582.0 15i 116 17% 21*4 90.9 309.3 6.32 636.9 Working Formula; for Wr ought-iron and Steel Struts of various Forms. Burr gives the following practical formulae, which he believes to possess advantages over Gordon's: Pi = Working Strength = 1/5 Ultimate, Ibs. per sq. in. of Section. p = Ultimate Strength, Ibs. per sq. in. of Section. Kind of Strut. Flat and flxed end iron angles and tees 44000 - 140 (1) 8800-28 ~ (2) Hinged-end iron angles and tees 46000-175 r 9200-35 r I Flat-end iron channels and I beams.... 40000- 110 (5) 8000-22 (6) Flat-end mild-steel angles 52000-180 (7) 10400-36 (8) I Flat-end high-steel angles 76000- 290 (9) Pin-end solid wroughMron columns.. . .32000- 80 " 1(11) 15200-58- (10) 6400-16-1 32000-277 - [ 6400-55 | d) dJ Equations (1) to (4) are to be used only between = 40 and = 200 (5) and (6) " "" " " " = 20 (7) to (10) ' " " " = 40 (11) and (12)" " = ^o =200 =200 =200 or - = 6 and -- = 65 d a rro, s ' P r operly made, of steel ranging in specimens from 65,000 to 73,000 Ibs. per square inch should give a resistance 25 to 33 per cent in ex- cess of that of wrought-iron columns with the same value of I H- r, provided that ratio does not exceed 140. The ^ u on^ pport . ed Y idth of a P late in a compression member should not exceed 30 tunes its thickness. In built columns the transverse distance between centre lines of rivets securing plates to angles or channels, etc., should not exceed 35 times the elate thickness. If this width is exceeded, longitudinal buckling of the ^60 STRENGTH OF MATERIALS. plate takes place, and the column ceases to fail as a whole, but yields in detail. The same tests show that the thickness of the leg of an angle to which latticing is riveted should not be less than 1/9 of the length of that leg or side if the column is purely and wholly a compression member. The above limit may be passed somewhat in stiff ties and compression members de- signed to carry transverse loads. The panel points of latticing should not be separated by a greater distance than 60 times the thickness of the angle-leg to which the latticing is riveted, if the column is wholly a compression member. The rivet pitch should never exceed 16 times the thickness of the thinnest metal pierced by the rivet, and if the plates are very thick it should never nearly equal that value. Merrimaii's Rational Formula for Columns (Eng. News, July 19, 1894). (2) B = unit-load on the column = total load P-*-area of cross-section A\ C = maximum compressive unit-stress on the concave side of the column: I length of the column; r = least radius of gyration of the cross-section E = coefficient of elasticity of the material ; n = 1 for both ends round n = 4/9 for one end round and one fixed; n -* y\ for both ends fixed. Thift formula is for use with strains within the eristic limit only: it does not hold good when the strain C exceeds the elasUc limit. Prof. Merriman takes the mean value otEfot timber = 1,500,000, for cast iron = 15,000,000, for wrought-iron = 25,000,000, nud for steel = 30,000,000, and 7T 2 = 10 as a close enough approximation. With these values he com- putes the following tables from formula (1): I. Wrought-iron Columns wiftb fttonnd Ends. Unit- load. Maximum Compressive Unit-stress C. p 1 = 20 - = 40 1 = 60 1 = 80 1 = 100 i~l -1=140 1=160 A T ' r r r r r 9* r r 5,000 6,000 5,040 6,055 5,170 6,240 5,390 6,560 5,730 7,090 6,250 7,890 6,980 9,0v>0 8-2PO 11,330 10,250 15,56(1 7,000 7,080 7,330 7,780 8,530 9,720 11,610 15,510 24,720 8,000 8,100 8,430 9,040 10,060 11,660 14,640 21,460 9000 9 130 9550 10340 11 690 14,060 18,380 10,000 10,160 10,680 11,680 13,440 16,670 23,090 11 000 11 200 11 750 13070 15 310 19 640 12000 12 240 13000 14 500 17320 23080 13,000 13,280 14,180 15,990 19,480 ' STRENGTH OF WROUGHT IROK AKD STEEL COLUMNS. 261 II. Wrought-iron Column* with Fixed Ends* Unit- load. Maximum Compressive Unit-stress C. ~orB. A i = 2 1 = 40 1 = 60 1 = 80 1 = 100 l-o ~ = 140 1=160 6,000 7,000 8,000 9,000 10,000 11,000 12,000 13,000 14,000 6,010 7,020 8,025 9,030 10,040 11,050 12,060 13,070 14,080 6,060 7,080 8,100 9,130 10,160 11,200 12,240 13,280 14,320 6,130 7,180 8,240 9,300 10,370 11,450 12,540 13,640 14,740 6,240 7,330 8,430 9,550 10,710 11,830 13,000 14,210 15,380 6,380 7,530 8,700 9,890 11,110 12,360 13,640 14,940 16,280 6,570 7,780 9,040 10,340 11,680 13,070 14.510 15,990 17,530 6,800 8,- 110 9,490 10,930 12,440 14,020 15,690 17,440 19,290 7,090 8,530 10,060 11,690 13,440 15,310 17,320 19,480 21,820 III. Steel Columns with Round Ends. Unit- load. Maximum Compressive Unit-stress O. 5 rjB< i = 20 7 = o 1 = 60 r 1 = 80 r 1 = 100 r 1 =120 r 1 = 140 7 = 160 6,000 7,000 8,000 9,000 10,000 11,000 12,000 13,000 14,000 6,050 7,070 8,090 9,110 10,130 11,160 12.200 13,330 14,250 6,200 7,270 8,380 9,450 10,560 11,690 12,820 13,970 15,130 6,470 7,650 8,770 10,090 11,360 12,670 14,020 15,400 16,830 6,880 8,230 9,650 11,140 12,710 14,370 16,130 18,000 19,960 7,500 9,130 10,870 12,850 15,000 17,370 20,000 22,940 26,250 8,430 10,540 12,990 15,850 19,230 23,300 28,300 9,870 12,900 16,760 20,930 28,850 12,300 17,400 24,590 IV. Steel Columns with Fixed Ends. Unit- load. Maximum Compressive Unit-stress 01 ^or. 1 = 20 1 = 40 1 = 60 7,150 8,200 9,250 10,310 11,380 12,450 13,530 14,610 15,710 7 = 80 1 = 100 r 1=1*0 1=140 1=160 7,000 8,000 9,000 10,000 11,000 12,000 13,000 14,000 15,000 7,020 8,020 9,030 10,030 11,040 12,050 13,060 14,070 15,080 7,070 8,090 9,110 10,130 11,160 12,200 13,230 14,250 15,310 7,270 8,380 9,450 10,560 11,690 12,820 13,970 15,130 16,310 7,430 8,570 9,730 10,910 12,110 13,330 14,580 15,850 17,140 7,650 8,770 10,090 11,360 12,670 14,020 15,400 16,830 18,290 7,900 9,200 10,550 11,810 13,410 14,930 16,500 18,150 19,870 8,230 9,650 11,140 12,710 14,370 16,130 17,990 19,960 22,060 The design of the cross-section of a column to carry a given load with maximum unit-stress C may be made by assuming dimensions, and then STRENGTH OF MATERIALS. computing C by formula (1). If the agreement between the specified and computed values is not sufficiently close, new dimensions must be chosen, and the computation be repeated. By the use of the above tables the work will be shortened. The formula (1) may be put in another form which in some cases will ab- breviate the numerical work. For B substitute its value P-^t4, and for Ar* write /, the least moment of inertia of the cross-section; then Jn which I and r 2 are to be determined. For example, let it be required to find the size of a square oak column with fixed ends when loaded with 24 000 Ibs. and 16 ft. long, so that the maximum compressive stress C shall be 1000 Ibs. per square inch. Here 7 = 24,000, C = 1000, n = M, * 2 = 10, E = 1,500,000, I = 16 X 12, and (3) be- comes I - 24r = 14.75. Now let x be the side of the square; then so that the equation reduces to x* 24# 2 = 177, from which x* is found to be 29.92 sq. in., and the side x = 5.47 in. Thus the unit-load B is about 802 Ibs. per square inch. WORKING STRAINS ALLOWED IN BRIDGE MEMBERS. Theodore Cooper gives the following in his Bridge Specifications : Compression members shall be so proportioned that the maximum load shall in no case cause a greater strain than that determined by the follow- ing formula : 8000 P = - for square-end compression members ; P me - for compression members with one pin and one square end; 1 ~*~ 30,000r 8000 P= for compression members^with pin-bearings; 1 ~*~20,000r (These values may be increased in bridges over 150 ft. span. See Cooper's Specifications.) P = the allowed compression per square inch of cross-section; I = the length of compression member, in inches; r = the least radius of gyration < . f gyration of the section in inches. No compression member, however, shall have a length exceeding 45 times its least width. Tension Members. All parts of the structure shall be so proportioned that the maximum loads shall in no case cause a greater tension than the following (except in spans exceeding 150 feet) : Pounds per sq. in. On lateral bracing 15,000 On solid rolled beams, used as cross floor-beams and stringers. 9,000 On bottom chords and main diagonals (forged eye-bars) 10,000 On bottom chords and main diagonals (plates or shapes), net section 8,000 On counter rods anri long verticals (forged eye-bars) 8,000 On counter and long verticals (plates or shapes), net section.. 6,500 On bottom flange of riveted cross-girders, net section 8,000 On bottom flange of riveted longitudinal plate girders over 20ft. long, net section 8,000 WORKING STRAINS ALLOWED IN BRIDGE MEMBERS. 263 On bottom flange of riveted longitudinal plate girders under 20 ft. long, net section ..'.'.. 7,000 On floor-beam hangers, and other similar members liable to sudden loading (bar iron with forged ends) 6,000 On floor-beam hangers, and other similar members liable to sudden loading (plates or shapes), net section 5,000 Members subject to alternate strains of tension and compression shall be proportioned to resist each kind of strain. Both of the strains shall, how- ever, be considered as increased by an amount equal to 8/10 of the least of the two strains, for determining the sectional area by the above allowed strains. The Phoenix Bridge Co. (Standard Specifications, 1895) gives the follow- ing : The greatest working stresses in pounds per square inch shall be as fol- lows : Tension. Steel. Iron. P = OOP f 1 I Min ' Stress 1 Forbars P= 75o o ri i MiD - stress 1 L Max. stressj forged ends. |_ Max. stress J P - 8 500 fl -I Min ' Stress 1 PIatesor p _ 700o r i , Mia, stress"] uu L T Max. stressj shapes net. J ~ r |/ T Max. stressj 8,500 pounds. Floor-beam hangers, forged ends 7,000 pounds. 7,500 Floor-beam hangers, plates or shapes, net section 6,000 " 10,000 " Lower flanges of rolled beams. 8,000 " 50,000 ** Outside fibres of pins 15,000 " 30,000 " Pins for wind-bracing 22,500 " 20,000 " Lateral bracing 15,000 " Shearing. 9,000 pounds. Pins and rivets 7,500 pounds. Hand-driven rivets 20# less unit stresses. For bracing increase unit stresses 50%. 6,000 pounds. Webs of plate girders 5,000 pounds. Bearing. 16,000 pounds. Projection semi-intrados pins and rivets.. . . 12,000 pounds. Hand-driven rivets 20# less unit stresses. For bracing increase unit stresses 50#. Compression. Lengths less than forty times the least radius of gyration, P previously found. See Tension. Lengths more than forty times the least radius of gyration, P reduced by following formulae: For both ends fixed, b = For one end hinged, For both ends hinged, 18,000 r P = permissible stress previously found (see Tension) ; b = allowable working stress per square inch; I = length of member in inches; r = least radius of gyration of section in inches. No compression member, how- ever, shall have a length exceeding 45 times its least width. io,ooo(i 264 STRENGTH OF MATEEIAL8. Pounds per sq. in. In counter web members 10,500 In long verticals 10,000 In all main-web and lower-chord eye-bars 13,200 In plate hangers (net section) 9,000 In tension members of lateral and transverse bracing 19,000 In steel-angle lateral ties (net section) 15,000 For spans over 200 feet in length the greatest allowed working stresses per square inch, in lower-chord and end main-web eye-bars, shall be taken at min. total stress \ max. total stress J whenever this quantity exceeds 13,200. The greatest allowable stress in the main-web eye-bars nearest the centre of such spans shall be taken at 13,200 pounds per square inch ; and those for the intermediate eye-bars shall be found by direct interpolation between the preceding values. The greatest allowable working stresses in steel plate and lattice girders and rolled beams shall be taken as follows : Pounds per sq. in. Upper flange of plate girders (gross section) 10,000 Lower flange of plate girders (net section) 10,000 In counters and long verticals of lattice girders (net section) . . 9,000 In lower chords and main diagonals of lattice girders (net section) " 10,000 In bottom flanges of rolled beams 10,000 In top flanges of rolled beams 10,000 RESISTANCE OF HOLLOW CYLINDERS TO COLLAPSE. Fairbairn's empirical formula (Phil. Trans. 1858) is .i p = 9,675,600 '-rv-, . . , (1) Id where p = pressure in Ibs. per square inch, t = thickness of cylinder, d = diameter, and I = length, all in inches ; or, p = 806,300 ~^, if L is in feet (2) He recommends the simpler formula p = 9,675,600^ (3) as sufficiently accurate for practical purposes, for tubes of considerable diameter and length. The diameters of Fairbairn's experimental tubes were 4", 6", 8", 10", and 12", and their lengths ; between the cast-iron ends, ranged between 19 inches and 60 inches. His formula (3) has been generally accepted as the basis of rules for ascertaining the strength of boiler-flues. In some cases, however, limits are fixed to its application by a supplementary formula. Lloyd's Register contains the following formula for the strength of circular boiler-flues, viz., 89,600^ LA () The English Board of Trade prescribes the following formula for circular flues, when the longitudinal joints are welded, or made with riveted butt- straps, viz., - M.000< For lap-joints and for inferior workmanship the numerical factor may be reduced as low as 60,000. RESISTANCE OF HOLLOW CYLINDERS TO COLLAPSE. 265 The rules of Lloyd's Register, as well as those of the Board of Trade, pre- scribe further, that in no case the value of P must exceed the amount given by the following equation, viz., In formulae (4), (5), (6) P is the highest working pressure in pounds per square inch, t and d are the thickness and diameter in inches, L is the length of the flue in feet measured between the strengthening rings, in case it is fitted with such. Formula (4) is the same as formula (3), with a factor of safety of 9. In formula (5) the length L is increased by 1 ; the influence which this addition has on the value of P is, of course, greater for short tubes than for long ones. Nystrom has deduced from Fairbairn's experiments the following formula for the collapsing strength of flues : ............ where p, , and d have the same meaning as in formula (1), L is the length in feet, and Tis the tensile strength of the metal in pounds per square inch. If we assign to T the value 50,000, and express the length of the flue in inches, equation (7) assumes the following form, viz., p = 692,800 -. .......... (8) d yl Nystrom considers a factor of safety of 4 sufficient in applying his formula. (See "A New Treatise on Steam Engineering," by J. W. Nystrom, p. 106.) Formula (1), (4), and (8) have the common defect thai they make the collapsing pressure decrease indefinitely with increase of length, and vice versa. M. Love has deduced from Fairbairn's experiments an equation of a different form, which, reduced to English measures, is as follows, viz., p= 5,358,150 ^ + 41,906^+ 1323 j, ...... (9) where the notation is the same as in formula (1) . D. K. Clark, in his " Manual of Rules," etc., p. 696, gives the dimensions of six flues, selected from the reports of the Manchester Steam-Users Associa- tion, 1862-69, which collapsed while in actual use in boilers. These flues varied from 24 to 60 inches in diameter, and from 8-16 to % inch in thickness. They consisted of rings of plates riveted together, with one or two longitud- inal seams, but all of them unfortified by intermediate flanges or strength- ening rings. At the collapsing pressures the flues experienced compressions ranging from 1.53 to 2.17 tons, or a mean compression of 1.82 tons per square inch of section. From these data Clark deduced the following formula "for the average resisting force of common boiler-flues," viz., where p is the collapsing pressure in pounds per square inch, and d and t are the diameter and thickness expressed in inches. C. R. Roelker, in Tan Nostrand's Magazine, March, 1881, discussing f'e above and other formulae, shows that experimental data are as yet insuffi- cient to determine the value of any of the formulae. He says that Nystrom 's formula, (8), gives a closer agreement of the calculated with the actual col- lapsing pressures in experiments on flues of every description than any of the other formulae. Collapsing Pressure of Plain Iron Tubes or Flues. (Clark, S. E., vol. i. p. 643.) The resistance to collapse of plain-riveted flues is directly as the square of the thickness of the plate, and inversely as the square of the diameter. The support of the two ends of the flue does not practically extend over a length of tube greater than twice or three times the diameter. The collapsing pressure of long tubes is therefore practically independent of the length. 266 STRENGTH OF MATERIALS. Instances of collapsed flues of Cornish and Lancashire boilers collated by Clark, showed that the resistance to collapse of flues of %-iuch plates, 18 to 43 feet long, and 30 to 50 inches diameter, varied as the 1 75 power of the diameter. Thus, for diameters of ....................... 30 35 40 45 50 inches, the collapsing pressures were ......... 76 58 45 37 30 Ibs. per sq. in; for 7-16-inch plates the collapsing pressures were ........... ............ . . . 60 49 42 For collapsing pressures of plain iron flue-tubes of Cornish and Lanca shirs steam-boilers, Clark gives: _ 200,000*2 P = collapsing pressure, in pounds per square inch; t = thickness of the plates of the furnace tube, in inches. d = internal diameter of the furnace tube, in inches. For short lengths the longitudinal tensile resistance may be effective in augmenting the resistance to collapse. Flues efficiently fortified by flange= joints or hoops at intervals of 3 feet may be enabled to resist from 50 Ibs. to 60 Ibs. or 70 Ibs, pressure per square inch more than plain tubes, accord. ing to the thickness of the plates. Strength of Small Tubes. The collapsing resistance of solid- drawn tubes of small diameter, and from .134 inch to .109 inch in thickness, Has been tested experimentally by Messrs. J. Russell & Sons. The results lor wrought-iron tubes varied from 14.33 to 20.07 tons per square-inch sec- tion of the metal, averaging 18.20 tons, as against 17.57 to 24.28 tons, averag- ing 22.40 tons, for the bursting pressure. (For strength of Segmental Crowns of Furnaces and Cylinders see Clark, S. E., vol. i, pp. 649-651 and pp. 627, 628.) Formula for Corrugated Furnaces (Bng'g* July 24, 1891. p. 102). As the result of a series of experiments on the resistance to collapse of Fox's corrugated furnaces, the Board of Trade and Lloyd's Registry altered their formulae for these furnaces in 1891 as follows: Board of Trade formula is altered from T = thickness in inches; D = mean diameter of furnace; WP = working pressure in pounds per square inch. Lloyd's formula is altered from 1000 X (T - '!} = wp to 1S84XCT.-2) = WR T = thickness in sixteenths of an inch; D = greatest diameter of furnace; WP = working pressure in pounds per square inch. TRANSVERSE STRENGTH. In transverse tests the strength of bars of rectangular section is found to rary directly as the breadth of the specimen tested, as the square of its depth, and inversely as its length. The deflection under any load varies as the cube of the length, and inversely as the breadth and as the cube of the depth. Represented algebraically, if S = the strength and D the deflection, 1 the length, 6 the breadth, and d the depth, 7,,?3 19 8 varies as -r- and D varies as ^. For the purpose of reducing the strength of pieces of various sizes to a common standard, the term modulus of rupture (represented by K) is used. Its value is obtained by experiment on a bar of rectangular section TRANSVERSE STRENGTH. 267 supported at the ends and loaded in the middle and substituting numerical values in the following formula : to which P= the breaking load in pounds, I = the length in inches, b the breadth, and d the depth. The modulus of rupture is sometimes defined as the strain at the instant of rupture upon a unit of the section which is most remote from the neutral axis on the side which first ruptures. This definition, however, is based upon a theory which is yet in dispute among authorities, and it is better to define it as a numerical value, or experimental constant, found by the ap- plication of the formula above given. From the above formula, making I 12 inches, and b and d each 1 inch, it follows that the modulus of rupture is 18 times the load required to break a bar one inch square, supported at two points one foot apart, the load being applied in the middle. .. span in feet X load at middle in Ibs. Coefficient of transverse strength = ^^ in inches x (dep[h in ^^^ = th of the modulus of rupture. lo Fundamental Formulae for Flexure of Beams (Merriman). Resisting shear = vertical shear; Resisting moment =? bending moment; Sum of tensile stresses = sum of compressive stresses; Resisting shear = algebraic sum of all the vertical components of the in- ternal stresses at any section of the beam. Tf A be the area of the section and Ss the shearing unit stress, then resist- ing shear = ASs; and if the vertical shear = V, then V ASs. The vertical shear is the algebraic sum of all the external vertical forces on one side of the section considered. It is equal to the reaction of one sup- port, considered as a force acting upward, minus the sum of all the vertical downward forces acting between the support and the section. The resisting moment algebraic sum of all the moments of the inter- nal horizontal stresses at any section with reference to a point in that sec- or tion, = , in which 8 = the horizontal unit stress, tensile or compressive c as the case may be, upon the fibre most remote from the neutral axis, c = the shortest distance from that fibre to said axis, and / = the moment of inertia of the cross-section with reference to that axis. The bending moment M is the algebraic sum of the moment of the ex- ternal forces on one side of the section with reference to a point in that sec- tion moment of the reaction of one support minus sum of moments of loads between the support and the section considered. he bending moment is a compound quantity = product of a force by the (Distance of its point of application from the section considered, the distance being measured on a line drawn from the section perpendicular to the direction of the action of the force. Concerning the above formula, Prof. Merriman, Eng. News, July 21, 1894, says: The formula just quoted is true when the unit-stress <S on the part of the beam farthest from the neutral axis is within the elastic limit of the material. It is not true when this limit is exceeded, because then the neutral axis does not pass through the centre of gravity of the cross-section, and because also the different longitudinal stresses are not proportional to their distances from that axis, these two requirements being involved in the de- duction of the formula. But in all cases of design the permissible unit- stresses should not exceed the elastic limit, and hence the formula applies rationally, without regarding the ultimate strength of the material or any of the circumstances regarding rupture. Indeed so great reliance is placed upon this formula that the practice of testing beams by rupture has been almost entirely abandoned, and the allowable unit-stresses are mainly de- rived from tensile and compressive tests. 268 STRENGTH OF MATERIALS. + ft, 18 '1*1 MBS g 1,5 SjB II II ~t ~l8S i !! fe -h ftj J I I i : : ^ > <D : ' ii 3 ^ *--s tJ > 1 1 a s a "3 a * S 3 ii 5 5 1 | g n. el APPROXIMATE SAFE LOADS IK LBS. OK STEEL BEAMS. Formulae for Transverse Strength of Beams* Referring to table on preceding page, P = load at middle; W= total load, distributed uniformly; I = length, 6 = breadth, d = depth, in inches; E =s modulus of elasticity; R = modulus of rupture, or stress per square inch of extreme fibre; / =r moment of inertia; c = distance between neutral axis and extreme fibre. For breaking load of circular section, replace 5d 2 by 0.59d 9 . For good wrought iron the value of R is about 80,000, for steel about 120,000, the percentage of carbon apparently having no influence. (Thurston, Iron and Steel, p. 491), For cast iron the value of R varies greatly according to quality. Thurston found 45,740 and 67,980 in No. 2 and No. 4 cast iron, respectively. For beams fixed at both ends and loaded in the middle, Barlow, by experi- ment, found the maximum moment of stress = 1/6PI instead of 1&PI, the result given by theory. Prof. Wood (Resist. Matls. p. 155) says of this case: The phenomena are of too complex a character to admit of a thorough and exact analysis, and it is probably safer to accept the results of Mr. Barlow in practice than to depend upon theoretical results. APPROXIMATE: GREATEST SAFE LOADS IN LBS. ON STEEL BEAMS. (Pencoyd Iron Works.) Based on fibre strains of 16,000 Ibs. for steel. (For iron the loads should be one-eighth less, corresponding to a fibre strain of 14,000 Ibs. per square inch.) L = length in feet between supports; a = interior area in square A = sectional area of beam in square inches; inches; d = interior depth in inches. D = depth of beam in inches. w = working load in net tons. Shape of ' Section. Greatest Safe Load in Pounds. Deflection in Inches. Load in Middle. Load Distributed. Load in Middle. Load Distributed. Solid Rect- angle. 890.4D 1 780.4 D wL* S2AD* tc3 L L 52AD* HollowRect- angle. 890UD-orf) 1780C4D-ad) wL* wL* L L 32UZ)-ad 2 ) 52UZ)2-ad a ) Solid Cylin- der. M7AD 13334Z) wLs 24AD* wl? 3SAD* L L Hollow Cylinder. 667UD-ad) 1333(AD-ad) wL* wL* L L 24(AD*-ad*) 38(AD*-ad?) Even-legged Angle or Tee. S85AD 1710AD wL* wL* L L 32^Z> 52AD* Channel or Zbar. 1525AD 3Q5QAD wL? wL* 85AD* L L 53^D 2 Deck Beam. 1380 AD 2760 AD wL* 504D* wL* L L SOAD* I Beam. I 1695^1) mQAD mL* wL* L L 5&AD* 934Z>' II III IV V 270 STRENGTH OF MATERIALS. The above formulae for the strength and stiffness of rolled beams of va- rious sections are intended for convenient application in cases where strict accuracy is not required. The rules for rectangular and circular sections are correct, while those for the flanged sections are approximate, and limited in their application to the standard shapes as given in the Pencoyd tables. When the section of any beam is increased above the standard minimum dimensions, the flanges re- maining unaltered, and the web alone being thickened, the tendency will be for the load as found by the rules to be in excess of the actual; but within the limits that it is possible to vary any section in the rolling, the rules will apply without any serious inaccuracy. The calculated safe loads will be approximately one half of loads that would injure the elasticity of the materials. The rules for deflection apply to any load below the elastic limit, or less than double the greatest safe load by the rules. If the beams are long without lateral support, reduce the loads for the ratios of width to span as follows : Length of Beam. 20 times flange width. 30 " " " 40 " ' ** 50 ** * * 60 " 70 " " " These rules apply to beams supported at each end. For beams supported otherwise, alter the coefficients of the table as described below, referring to the respective columns indicated by number. Proportion of Calculated Load forming Greatest Safe Load. Whole calculated load. 9-10 " " 8-10 " " 7-10 6-10 " 5-10 " " Changes of Coefficients for Special Forms of Beams. Kind of Beam. Coefficient for Safe Load. Coefficient for Deflec- tion. Fixed at one end, loaded at the other. One fourth of the coeffi- cient, col. II. One sixteenth of the co- efficient of col. IV. Fixed at one end, load evenly distributed. One fourth of the coeffi- cient of col. III. Five forty-eighths of the coefficient of col. V. Both ends rigidly fixed, or a continuous beam, with a load in middle. Twice the coefficient of col. II. Four times the coeffi- cient of col. IV. Both ends rigidly fixed, or a continuous beam, with load evenly dis- tributed. One and one-half times the coefficient of col. III. Five times the coefficient of col. V. ELASTIC RESILIENCE. In a rectangular beam tested by transverse stress, supported at the ends and loaded in the middle, 2 Rbd* p -3-~T~ ; 1 PJ3 ~lEbd* ' in which, if P is the load in pounds at the elastic limit, R = the modulus of transverse strength, or the strain on the extreme fibre, at the elastic limit, E= modulus of elasticity, A = deflection, I, 6, and d= length, breadth, and depth in inches. Substituting for P in (2) its value in (1), we have 1 Rl* 6 JEtT BEAMS OF UNIFORM STRENGTH THROUGHOUT LENGTH. 271 The elastic resilience = half the product of the load and deflection and the elastic resilience per cubic inch _1 PA "~ 2 Ibd ' Substituting the values of P and A, this reduces to elastic resilience per cubic inch = jg^ which is independent of the dimensions; and therefore fhe elastic resilience per cubic inch for transverse strain may be used as a modulus expressing one valuable quality of a material. Similarly for tension: Let P = tensile stress in pounds per square inch at the elastic limit; e = elongation per unit of length at the elastic limit; E = modulus of elasticity = P -*- e\ whence e P-*- E. Then elastic resilience per cubic inch = y%Pe . 2 E BEAMS OF UNIFORM STRENGTH THROUGHOUT THEIR LENGTH. The section is supposed in all cases to be rectangular throughout. The beams shown in plan are of uniform depth throughout. Those shown in elevation are of uniform breadth throughout. B iireadth of beam. D = depth of beam. Fixed at one end, loaded at the other; curve parabola, vertex at loaded end; BD Z proportional to distance from loaded end. The beam may be reversed, so that the up- per edge is parabolic, or both edges may be parabolic. Fixed at one end, loaded at the other; triangle, apex at loaded end; BD* propor- tional to the distance from the loaded end. Fixed at one end; load distributed; tri- angle, apex at unsupported end; BD'* pro- portional to square of distance from unsup- ported end. Fixed at one end; load distribut