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Consulting Engineer
Sv <&/*, t r\^
MM. 3T3.,
,
Consulting Engineer
The Publishers and the Author will be grateful to
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their attention to any errors of omission or of commis
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rarely happens that the early editions of works of any
size are free from errors; but it is the endeavor of the
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WORKS OF WILLIAM KENT
PUBLISHED BY
JOHN WILEY & SONS.
The Mechanical Engineers' PocketBook.
A Reference Book of Rules, Tables, Data, and
Formulae, for the Use of Engineers, Mechanics,
and Students, xxxiif noo pages, i6mo, morocco,
$5.00.
SteamBoiler Economy.
A Treatise on the Theory and Practice of Fuel
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xiv + 458 P a S es > T 3 6 figures, 8vo, cloth, $4.00.
THE
MECHANICAL ENGINEER'S
POCKETBOOK
A REFERENCEBOOK 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 FORTYFIVE 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 pocketbook, as he proceeds in study and
practice, to suit his particular business." The manuscript
pocketbook 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 Pocketbook 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 "longfelt 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 Safetyvalves and Crankpins.
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, refrigeratingma*
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 Castiron 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 Wirerope 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 crossreferences
which should be made to the items in the Appendix. Users of the
book may find it advisable to write in the margin such crossrefer
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, Steamengines, 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
Twistdrill and Steelwire 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 Antifriction 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 Rightangled Triangles C
Solution of Obliqueangled 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
" * Terracotta 181
" * Tiles 181
" " Tin Plates ...181
" Slates 183
" " PineShingles 183
' " Skylight Glass . 184
Weights of Various Ropfcoverings 184
Castiron Pipes or Columns 185
" " " 12 ft. lengths 186
* " Pipefittings 187
" " Water and Gaspipe 188
and thickness of Castiron Pipes 189
Safe Pressures on Cast Iron Pipe 189
CONTENTS. IX
PAGE
Sheetiron Hydraulic Pipe.. 191
Standard Pipe Flanges 192
Pipe Flanges and Castiron Pipe 193
Standard Sizes of W rough tiron Pipe 194
Wroughtiron 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 Tinlined 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
Screwthread, U. S. Standard 204
Limitgauges for Screwthreads 205
Size of Iron for Standard Bolts 206
Sizes of Screwthreads 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 Boltheads 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 Steelwire Strand 223
Steelwire Cables for Vessels 223
Specifications for Galvanized Iron Wire 224
Strength of Piano Wire 224
Ploughsteel Wire 224
Wires of different metals 225
Specifications for Copper Wire 225
Cabletraction Ropes 226
Wire Ropes 226, 227
Ploughsteel 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 Castiron Columns 250
Transverse Strength of Cast iron Waterpipe 251
Safe Load on Castiron Columns 252
Strength of Brackets on Castiron Columns 252
Eccentric Loading of Columns 254
Wroughtiron 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 Castiron 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 Wroughtiron Bolts.: 293
CONTENTS. xi
PAGE
Initial Strain on Bolts 292
Stand Pipes and their Design , 292
Riveted Steel Waterpipes 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, Wirerope , 301
Ropes, Hemp, and Cotton 301
Belting, Canvas 302
Stones, Brick, Cement * 302
Tensile Strength of Wire 303
Watertown Testingmachine Tests 303
Riveted Joints 303
Wroughtiron Bars, Compression Tests 304
Steel Eyebars ^ 304
Wroughtiron Columns ... 305
Cold Drawn Steel 305
American Woods 306
Shearing Strength of Iron and Steel 306
Holding Power of Boilertubes 307
Chains, Weight, Proof Test, etc 307
Wroughtiron 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 Indiarubber 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
Coppertinzinc Alloys.
Liquation or Separation of Metals.
Alloys used in Brs
Jrass, Foundries 325
Coppernickel Alloys 326
Copperzinciron 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
Tungstenaluminum Alloys 331
Aluminumtin 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
Whitemetal Alloys 336
Typemetal 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
Phosphorbronze 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
Staybolt 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
Nonoxidizing Process of Annealing 387
Manganese Plating of Iron 389
Steel.
Relation between Chemical and Physical Properties 389
Variation in Strength 391
Openhearth 392
Bessemer 392
Hardening Soft Steel 393
Effect of Cold Rolling 393
Comparison of Fullsized 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, Splicebars, Structural Steel 398
Boilerplate 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, Footpound 428
Power, Horsepower 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
Animalpower, Manpower 433
Work of aHorse . ... 434
Manwheel 434
Horsegin 434
Resistance of Vehicles 435
Elements of Machines,
The Lever 435
The Bent Lever 436
The Moving Strut 436
The Togglejoint 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
Toothedwheel 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
Copperball Pyrometer 451
Thermoelectric Pyrometer 451
Temperatures in Furnaces 451
Wiborgh Air Pyrometer , 453
Seeger's Fireclay Pyrometer , 453
Mesur6 and Nouel's Pyrometer 453
Uehling and Steinbart's Pyrometer 453
Airthermometer 454
High Temperatures judged by Color.... 454
Boilingpoints of Substances 455
Meltingpoints 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
Steampipe Coverings . . 470
Transmission through Plates 471
in Condenser Tubes 473
* ** Castiron 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
Airmanometer 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
Horsepower Required for Compression 501
Table for Adiabatic Compression 502
Mean Effective Pressures 502
Mean and Terminal Pressures 503
Aircompressors. 503
Practical Results 505
Efficiency of Compressedair Engines 506
Requirements of Rockdrills , 506
Popp Compressedair System 507
Small Compressedair Motors 507
Efficiency of Airheating Stoves 507
Efficiency of Compressedair Transmission 508
Shops Operated by Compressed Air 509
Pneumatic Postal Transmission t 509
Mekarski Compressedair 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 Steamcoils for the Blower System of Heating 519
Sturtevant Steel Pressureblower 519
Diameter of Blastpipes 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
Steamjet 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
Mineventilation 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 Steampipes 537
Indirect Heatingsurface 537
Boiler Heatingsurface Required 538
Proportion of Gratesurface to Radiatorsurface 538
Steamconsumption in Carheating 538
Diameters of Steam Supply Mains 539
Registers and Coldair Ducts 539
Physical Properties of Steam and Condensed Water 540
Size of Steampipes 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 Hotwater 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
Boilingpoint 550
Freezingpoint 550
Seawater 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 Boilerscale 552
Hardness of Water 553
Purifying Feedwater 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 20inch Pipe 566
Velocities in Smooth Castiron Waterpipes 567
Table of Flow of Water in Circular Pipes 568573
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 Gradeline 578
Flow of Water in Houseservice Pipes 578
Airbound 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 Fireengines 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
Waterpower*
Power of a Fall of Water .................................................. 588
Horsepower of a Running Stream ..... ., > ............................... 589
Current Motors. ... .................... ...... . ....................... 589
Horsepower of Water Flowing in a Tube... 7 . .......................... 589
Maximum Efficiency of a Long Conduit .................. ............... 589
Millpower ............. . ............................................. ,.,.. 689
Value of Waterpower ........................... , ..................... , . 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 Waterwheel .................................................. 597
Pumps.
Theoretical capacity of a pump ....... . ................................. 601
Depth of Suction .......... ............................................... 602
Amount oi Water raised by a Singleacting Liftpump ................... 602
Proportioning the Steamcylinder of a Directacting Pump .............. 602
Speed of Water through Pipes and Pump passages .................... 602
Sizes of Directacting 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
Boilerfeed 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
Airlift 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
Downwarddraught 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 Byproducts in Coke manufacture 638
Making Hard Coke 638
Generation of Steam from the Waste Heat and Gases from Cokeovens. 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
Dustfuel Dust Explosions 642
Peat or Turf 643
Sawdust as Fuel 643
Horsemanure as Fuel 643
Wet Tanbark 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&
Producergas from One Ton of Coal 649
Natural Gas in Ohio and Indiana 649
Combustion of Producergas 650
Use of Steam in Producers 650
Gas Fuel for Small Furnaces *. 651
Illuminating Gas,
Coalgas 651
Watergas 652
Analyses of Watergas and Coal gas 653
Calorific Equivalents of Constituents 654
Efficiency of a Watergas Plant 654
Space Required for a Watergas Plant 656
Ruelvalue of Illuminatinggas 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 Steampipes for Stationary Engines 673
Sizes of Steampipes for Marine Engines 674
Steam Pipes.
Burstingtests of Copper Steampipes 674
Thickness of Copper Steampipes.. , 675
Reinforcing Steampipes , 675
Wirewound Steam pipes 675
Riveted Steel Steampipes.. 675
Valves in Steampipes 675
Failure of a Copper Steampipe 676
The Steam Looj> 676
Loss from an Uncovered Steampipe , 676
THE STEAM BOILER.
The Horsepower of a Steam boiler .... 677
Measures for Comparing the Duty of Boilers 678
Steamboiler Proportions 678
Heatingsurface 678
Horsepower, Builders' Rating 679
Gratesurface 680
AreasofFlues 680
Airpassages Through Gratebars 681
Performance of Boilers , ,...., 681
Conditions which Secure Economy , 682
Efficiency of a Boiler .. ...683
Tests of Steamboilers 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 Steamboilers.
Rules for Construction... 700
Shellplate 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
Safeworking Pressures 707
Rules Governing Inspection of Boilers in Philadelphia 708
Flues and Tubes for Steam Boilers 709
Flatstayed Surfaces 4. 709
Diameter of Staybolts. 710
Strength of Stays 710
Staybolts in Curved Surfaces . 710
Boiler Attachments, Furnaces, etc*
Fusible Plugs 710
Steam Domes 711
Height of Furnace. 711
Mechanical Stokers 711
The Hawley Downdraught Furnace 712
Underfeed Stokers 712
Smoke Prevention 712
Gasfired Steamboilers 714
Forced Combustion .....i 714
Fuel Economizers. 715
Incrustation and Scale 716
Boilerscale 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 Safetyvalves ..... 721
Springloaded Safetyvalves 724
The Injector.
Equation of the In jector 725
Performance of Injectors ............. 726
Boilerfeeding Pumps 726
Feedwater Heaters.
Strains Caused by Cold Feedwater 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
Sheetiron Chimneys 741
THE STEAM ENGINE*
Expansion of Steam , 742
Mean and Terminal Absolute Pressures 743
CONTENTS.
Condensers, Airpumps, Circulatingpumps, etc.
PAGE
The Jet Condenser 839
Ejector Condensers 840
The Surface Condenser 840
Condenser Tubes 840
Tubeplates 841
Spacing of Tubes 841
Quantity of Cooling Water 841
Airpump 841
Area through Valveseats 842
Circulatingpump. . . . 843
Feedpumps 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 1IOTAIU ENGINES.
Gasengines 847
Efficiency of the Gasengine 848
Tests of the Simplex Gasengine 848
A 320H.P. Gasengine 848
Test of an Otto Gasengine 849
Temperatures and Pressures Developed 849
Test of the Clerk Gasengine 849
Combustion of the Gas in the Otto Engine 849
Use of Carburetted Air in Gasengines 849
The Otto Gasolineengine . 850
The Priestman Petroleumengine 850
Test of a 5H.P. Priestman Petroleumengine 850
Naphthaengines 851
Hotair or Caloric Engines 851
Test of a Hotair 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 Freesteaming Locomotive 855
Wootten's Locomotive 855
Grate surf ace, Smokestacks, and Exhaustnozzles for Locomotives .... 856
Exhaust Nozzles 856
Firebrick 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 Highspeed Locomotive 859a
Locomotive Linkmotion 859a
Dimensions of Some American Locomotives 859862
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
Narrowgauge Railways 865
Petroleum burning Locomotives 865
Fireless Locomotives 866
SHAFTING.
Diameters to Resist Torsional Strain 867
Deflection of Shafting 868
Horsepower 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
Horsepower of a Belt one mch wide 878
A. F. Nagle's Formula : 878
Width of Belt for Given Horsepower 879
Taylor's Rules for Belting 880
Notes on Belting 882
Lacing of Belts 883
Setting a Belt on Quartertwist 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, Pitchcircle, etc 887
Diametral and Circular Pitch 888
Chordal Pitch 889
Diameter of Pitchline of Wheels from 10 to 100 Teeth 889
Proportions of Teeth 889
Proportion of Gearwheels 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 Bevelwheels. 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 Machinecut Spurgear 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 Chainblocks 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 Windingengines 909
Cranes.
Classification of Cranes .". 911
Position of the Inclined Brace in a Jib Crane 912
A Large Travellingcrane ; 912
A 150ton Pillar Crane 912
Compressedair Travelling Cranes .' 912
Coalhandling Machinery.
Weight of Overhead Bins 912a
Supplypipes from Bins 912a
Types of Coal Elevators 912a
Combined Elevators and Conveyors ; 912a
Coal Conveyors 912a
Weight of Chain 9126
Weight of Flights 912c
Horsepower of Conveyors 912c
Bucket Conveyors 912c
Screw Conveyors 912d
Belt Conveyors Ql2d
Capacity of Belt Conveyors 9l2d
Wirerope Haulage.
Selfacting Inclined Plane 913
Simple Engine Plane 913
Tailrope System 913
Endless Rope System 914
Wirerope Tramways ^ 914
Suspension Cableways and Cable Hoists 915
Stress in Hoistingropes 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
WIREROPE TRANSMISSION.
Elastic Limit of Wire Ropes 917
Bending Stresses of Wire Ropes  918
Horsepower Transmitted 919
Diameters of Minimum Sheaves 919
Deflections of the Rope 920
Longdistance Transmission 921
ROPE DRIVING.
Formulae for Rope Driving 922
Horsepower 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 Ropedriving 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 Antifriction Metals 932
Castiron for Bearings 933
Friction of Metal Under Steampressure 933
Morin's Laws of Friction . . . 933
Laws of Friction of welllubricated Journals 934
Allowable Pressures on Bearingsurface 935
Oilpressure in a Bearing 937
Friction of Carjournal 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 Pivotbearing 939
Mercurybath Pivot 940
Ball Bearings 940
Friction Rollers. . . 940
Bearings for Very High Rotative Speed 941
Friction of Steamengines 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, Fibregraphite, 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
Changegears for Lathes 956
Metric Screwthreads 956
Setting the Taper in a Lathe 956
Speed of Drilling Holes 956
Speed of Twistdrills 957
Milling Cutters 957
Speed of Cutters 958
Results with Millingmachines 959
Milling with or Against Feed 960
Millingmachine vs. Planer 960
Power Required for Machine Tools 960
Heavy Work on a Planer 960
Horsepower 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 Horsepower 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
Copperwire Table 1034, 1035
Electric Transmission, Direct Currents.
Section of Wire Required for a Given Current 1033
Weight of Copper for a Given Power 1036
Shortcircuiting 1036
Economy of Electric Transmission 1036
Wire Table for 110, 220, 500, 1000, and 2000 volt Circuits 1037
Efficiency of Longdistance Transmission 1038
Table of Electrical Horsepowers 1039
Cost of Copper for Longdistance Transmission 1040
Systems of Electrical Distribution 1041
Electric Lighting.
Arc Lights 1042
Incandescent Lamps 1042
Variation in Candlepower and Life 1042
Specifications for Lamps 1043
Special Lamps 1043
Nernst Lamp 1043
Electric Welding 1044
Electric Heaters 1044
Electric Accumulators or Storagebatteries.
Description of Storagebatteries 1045
Sizes and Weights of Storagebatteries 1048
General Rules for Storagecells 1048
Electrolysis 1048
Electrochemical Equivalents 1049
Efficiency of a Storagecell 1048
Electromagnets.
Units of Electromagnetic 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 Electromagnets 1054
Determining the Direction of a Current 1054
Dynamoelectric Machines.
Kinds of Dynamoelectric Machines as regards Manner of Winding. . . 1055
Moving Force of a Dynamoelectric Machine 1055
Torque of an Armature : . 1056
Electromotive 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
Selfinductance of Lines and Circuits 1066
Capacity of Conductors 1067
Singlephase and Polyphase Currents 1068
Measurement of Power in Polyphase Circuits 1069
Alternatingcurrent Generators 1070
Transformers, Converters, etc 1070
Synchronous Motors 1071
Induction Motors 1072
Calculation of Alternatingcurrent Circuits 1072
Weight of Copper Required in Different Systems. .' 1074
Electrical Machinery.
Directcurrent Generators and Motors 10741076
Alternatingcurrent Generators 1077
Induction Motors 1077
Symbols Used in Electrical Diagrams 1078
APPENDIX.
Strength of Timber.
Safe Load on Whiteoak 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 Hotblast or Plenum Heating with Fans and Blowers. . 1081
Waterwheels.
Waterpower Plants Operating under High Pressure 1G81
Formulae for Power of Jet Waterwheels 1082
Gas Fuel.
Composition Energy, etc., of Various Gases 1082
Steamboilers.
Rules for Steamboiler Construction 1083
Boiler Feeding 1083
Feedwater Heaters 1083
The Steamengine.
Current Practice in Engine Proportions 1084
Work of Steamturbines 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 TEXTBOOKS 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 Steamengine.
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 Dynamoelectric 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., horsepower.
I.H.P., indicated horsepower.
B.H.P., brake horsepower,
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. 1516 =
.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* = ,a2 = 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_76_ 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.
164
.015625
1764
.265625
3364
.515625
4964
.765625
132
.03125
932
.28125
1732
.53125
2532
.78125
364
.046875
1964
.296875
3584
.546875
5164
.796875
116
.0625
516
.3125
916
.5625
1316
.8125
564
.078125
2164
.328125
3764
.578125
5364
.828125
332
.09375
1132
.34375
1932
.59375
2732
.84375
764
.109375
2364
.359375
3964
.609375
5564
.859375
18
.125
38
.375
58
.625
78
.875
964
.140625
2564
.390625
4164
.640625
5764
.890625
532
.15625
1332
.40625
2132
.65625
2932
.90625
1164
316
.171875
.1875
2764
716
.421875
.4375
4364
1116
.671875
.6875
5964
1516
.921875
.9375
1364
.203125
2964
.453125
4564
.703125
6164
.953125
732
.21875
1532
.46875
2332
.71875
3132
.96875
1564
.234375
3164
.484375
4764
.734375
6364
.984375
14
.25
12
.50
34
.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 ii
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 nonrecurring 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 fivetenths power, two to the one and
fivetenths 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 lefthand period, and place it as the first
figure in the quotient. Subtract its square from the lefthand 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 righthand 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 lefthand period ; write its root as the first figure
in the required root, Subtract the cube from the lefthand 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 =
453872049896345.1f
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) _ (rl)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
= nl nl. '
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. . . . 76,295,220 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
id+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 TI CO <M GC 1C <?* O O 1O i~> 00 CO T
00 T. CO GO* 0OOSt>t C040*OTrt COCOCOCOCi TITH
^ CO <7^ > I T< T I T1
T^
!
OCOOC005 T^C<
iSS $g
OcOCOrt* i>COOO
o
FH
ce
^
o
5
5
3
^TM iOTicoooi." TIO;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 ostTjNu
OtOOWO TtJOriG
?^c?^ g^^S
o
*
tO T I TI W CO W Th CO QC t
ostcocjnj oio^^oo
ftGOCOlC Ot^O1<?CO
31TCO OSC7OOO
s
^
I
5co wrt c
ft O CO TH C
5 ^^^^g glTSS^S
3R
OTfTfOOCO TQOGOO5Tti C^GOCOOSC?
lO OS I ri iO iO TJ< <* C? rt ift O tO t CO
^' ^ ^J ,_; i>I co o' r' o' Ti co j  TI co w
?t^i2? COOtOOSO
O5CO WTI

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 sinkingfunds 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 fiveyear 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 ora 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 halfminute 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
measuringbox, and the plank 5 inches high above the aperture, thus mak
ing a 6inch 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 495 (the cubic feet allowed for a ton).
The result will be the tonnage. For a doubledecker 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 platinumiridium bar deposited at the International Bureau.
The International Standard Kilogramme is a mass of platinumiridium
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. (freezingpoint) ............................ 62.418 Ibs.
" 39.1 F. (maximum density) .......... . ............ 62.425 "
" 62 F. (standard temperature) ....... ............... 62.355 "
u 212 F. (boilingpoint, 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 footpound, i.e., a pressure of one pound exerted
through a space of one foot.
Horsepower. 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
Horsepower 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 footpounds.
28
ARITHMETIC.
WIRE AND SHEETMETAL, 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 centralstation 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 milfoot, 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 flS fl
5 73 r2 3
lf.N
eusia
^SQ 03
%$ *
28 
fls1
ggM
gl 1
<3 ^
111
I{
sill.
iili
ft*!
}l
III
S3*
i
^.s
'~S !'
0000000
12
0.5
12.7
320
20.
9.072
97.65
215.28
000000
1532
0.46875
11.90625
300
18.75
8.505
91.55
201.82
00000
716
0.4375
11.1125
280
17.50
7.938
85.44
188.37
0000
1332
0.40625
10.31875
260
16.25
7.371
79.33
174.91
000
38
0.375
9.525
240
15.
6.804
73.24
161.46
00.
1132
0.31375
8.73125
220
13.75
6.237
67.13
148.00
516
0.3125
7.9375
200
12.50
5.67
61.03
134.55
1
932
0.28125
7.14375
180
11.25
5.103
54.93
121.09
2
1764
0.265625
6.746875
170
10.625
4.819
51.88
114.37
3
14
0.25
6.35
160
10.
4.536
48.82
107.64
4
1564
0.234375
5.953125
150
9.375
4.252
45.77
100.91
5
732
0.21875
5.55625
140
8.75
3.969
42.72
94.18
6
1364
0.203125
5.159375
130
8.125
3.685
39.67
87.45
7
316
0.1875
4.7625
120
7.5
3.402
36.62
80.72
8
1164
0.171875
4.365625
110
6.875
3.118
33.57
74.00
9
532
0.15625
3.9S875
100
6.25
2.835
30.52
67.27
10
964
0.140625
3.571875
90
5.625
2.552
27.46
60.55
11
18
0.125
3.175
80
5.
2.268
24.41
53.82
1<2
764
0.109375
2.778125
70
4.375
1.984
21.36
47.09
13
332
0.09375
2.38125
60
3.75
1.701
18.31
40.36
14
564
0.078125
1.984375
50
3.125
1.417
15.26
33.64
15
9128
0.0703125
1.7859375
45
2.8125
1.276
13.73
30.27
16
116
0.0625
1.5875
40
2.5
1.134
12.21
26.91
17
9160
0.05625
1 .42875
36
2.25
1.021
10.99
24.22
18
120
0.05
1.27
32
2.
0.9072
9.765
21.53
19
7160
0.04375
1.11125
28
1.75
0.7938
8.544
18.84
20
380
0.0375
0.9525
24
1.50
0.6804
7.324
16.15
21
11320
0.034375
0.873125
22
1.375
0.6237
6.713
14.80
22
132
0.03125
0.793750
20
1.25
0.567
6.103
13 46
23
9320
0.028125
0.714375
18
1.125
0.5103
5.493
12.11
24
140
0.025
0.635
16
1.
0.4536
4.882
10.76
25
7320
0.021875
0.555625
14
0.875
0.3969
4.272
9.42
26
3160
0.01875
0.47625
12
0.75
0.3402
3.662
8.07
27
11640
0.0171875
0.4365625
11
0.6875
0.3119
3.357
7.40
28
164
0.015625
0.396875
10
0.625
0.2835
3.052
6.73
29
9640
0.0140625
0.3571875
9
0.5625
0.2551
2.746
6.05
30
180
0.0125
0.3175
8
0.5
0.2268
2.441
5.38
81
7640
0.0109375
0.2778125
7
0.4375
0.1984
2.136
4.71
32
131280
0.01015625
0.25796875
gi^
0.40625
0.1843
1.9R3
4.37
33
3320
0.009375
0.238125
6
0.375
0.1701
1.831
4.04
34
111280
0.00859375
0.21828125
5^
0.34375
0.1559
1.678
3 70
35
5640
0.0078125
0.1984375
5
0.3125
0.1417
1.526
3.36
36
91280
0.00703125
0.17859375
41^
0.28125
0.1276
1.373
3.03
37
172560
0.006640625
0.168671875
4/4
0.265625
0.1205
1.297
2.87
38
1160
0.00625
0.15875
4
0.25
0.1134
1.221
2.69
MATHEMATICS.
Tlie Decimal Gauge. The legalization of the standard sheetmetaj
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

DBp
* '
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 byaf b.
Ans. a 2 6 + a6 2 .
Multiply a f 6 by a \b. Ans. (af 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:
(af6) 2 = a 2 +2a6f 6 2 ; (a  6) 2 =a 2  2a6f 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 f6" 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;
(afU) = a*la r .J4,  a (a+l) + %. (4^) 2 =4 2 4 44^^=
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 ^(af 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?bxraby =  = ; = = 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 ri
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 scrollwork.
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 (7Fdiaw 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 60degree 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 semicircumference
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 dotlining.
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 setscrews. 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 semiaxes, 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 semiminor 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 semimajor 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 semiaxes,
for the third radius (= p), and employ these radii for the eightcentred 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 antifriction 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 semicircumference 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 abovementioned
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 rightangled 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 rightangled.
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 foursided 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 threesided plane figure.
Varieties. Rightangled, having one right angle; obtuseangled, 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 rightangled triangle = 90.
Hypothenuse of a rightangled 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 rightangled 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 rightangled 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 47
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'
147311
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
tj Length. 4
FIG. 69.
In a figure of very irregular outline, as an indicatordiagram from a high
ipeed steamengine, 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 crosssection paper. Count the number
of squares that are entirely included within the boundary; then estimate
ruling o the crosssection 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; ;=2Vr; = 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 = GQD27V 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 1.2732 = 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 onehalf 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.
Tofind 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 semiaxes 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 = ry. 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
generatingpoint 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.GCOJ540 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 rightangled 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 rightangled triangle, the right angle
being equal to 90, each of the acute angles is the complement of the other.
In all rightangled 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 righthand 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)
cosCf 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 (AB) =
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 Rightangled 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 rightangled 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 Obliqueangled 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 et e C = ten C.
Or, in other words, solve B <?, A e and B e C as rightangled 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 rightangled 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 rightangled 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 semitransverse 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 semitransverse axis to the semiconjugate.
Any ordinate of a circle inscribed in an ellipse is to the corresponding ordi
nate of the ellipse as the semiconjugate axis to the semitransverse.
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 subnormal, 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 semitransverse axis and B the semiconjugate 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*dydz.
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 mf 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  (af 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 rightangle 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;
= versin ~" 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 = versin 2; f C;
=: tan f C;
In "Wf'Cj
/,
:r= = sin ~~ * f O;
/a 2  w 2
/~ dtC __ = cos 1 4C;
/a 2  w* <*
/ U = versin ~ 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 versin 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 textbooks 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
12
.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
.00221215
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
.00187266
9
.00166945
340
.00294118
5
.00246914
470
.00212760
5
.00186916
600
.00166667
1
.00293255
6
.00216305
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
.00210961
480
.00208333
.00183486
610
.00163934
]
.00284900
16
.00240385
1
.00207900
.00183150
11
.0016361)6
.00784091
17
.00239808
f
.00207469
.00182815
12
.001 64399
f
.00283286
18
.00J39234
3
.00207039
.00182482
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
.00204918
.00180832
18
.00161812
9
.00278551
4
.00235849
9
.00204499
.00180505
19
.00161551
360
.00277778
5
.00235294
490
.C0204082
.00180180
620
.00161290
1
.00277008
6
.00234742
1
.00203666
.CO 179856
1
.00161031
2
.00276243
7
.0023419!?
.00203252
.00179533
f
.00160772
.00275482
8
.00283645
c
.00202840
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
.00201207
2
.00177936
1
.00159490
8
.00271739
3
.00230947
8
.00200803
3
.00177620
8
.00159236
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
.00158479
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
.00155521
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
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.00150602
9
.00137174
L
.00125945
9
.00116414
4
.00108225
t
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730
.00136986
f
.00125786
860
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6
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.00149925
2
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1
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2
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8
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8
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3
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8
.00107759
]
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i
.00136240
9
.00125156
t
.00115741
c
.00107643
670
.00149254
5
.00136054
800
.00125000
5
.00115607
930
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.00149031
1
.00135870
.00124844
6
.00115473
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2
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7
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68
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8li
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.00114286
941
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.00134048
11
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.00114155
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12
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13
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14
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75
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16
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5
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t
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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
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i
.000948767
19
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4
.000844595
9
.000800640
990
.00101010
5
.000947867
1120
.000892857
5
.000843882
1250
.000800000
.00100908
6
.000946970
3
.000892061
6
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1
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\
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7
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2
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8
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j
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S
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3
.000798085
c
.00100604
c
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<.
.000889680
9
.000841043
4
.000797448
t
.00100502
1060
.000943396
5
.000888889
1190
.000840336
5
.000796813
(
.00100J02
.000942507
6
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1
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6
.000796178
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.00100301
.000941620
\
.000887311
2
.000838926
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8
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{
.000940734
8
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3
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8
.000794913
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t
.000939850
9
.000885740
4
.000837521
9
.000794281
1000
.00100000
5
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1130
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5
.000836820
1260
.000793651
.000999001
<
.000938086
.000884173
6
.000836120
1
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2
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7
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7
.000835422
*
.000792393
j
.000997009
8
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8
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3
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t
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i
.000881834
9
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4
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5
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1070
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5
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1200
000833333
5
.000790514
(
.000994036
.000933707
6
.000880282
1
.000832639
6
.000789889
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2
.000932836
r
.000879508
2
.000831947
7
.000789266
I
.000992063
j
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8
.000878735
3
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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
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16
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6
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11
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6
.000783699
1"
.000983284
j
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\
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12
.000825082
.000783085
18
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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
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<
.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
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2
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j
.000777001
8
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<
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8
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3
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8
.000776397
9
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i
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(
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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
.000732601
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
.000676138
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
970299
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
4574296
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
13225
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
5832000
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
34225
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
2924207
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
3176523
12.1244
5.2776
202
40804
8242408
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
22801
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
225
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
2464J171
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
15813J51
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
89915393
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
23389
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
192100033
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 16815933. 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
320
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
11.4853
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
961 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
64
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
78426.72
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
621
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
682215.69
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
1316
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
1516
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
x4
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>\NKl>\aSWS .. _ . _
>i iOOiOOJOOTii lCOiOCOaOOrirlCO^
h
i e* o o* c* o QO i 5P gp ri 3" t^ o co o o eo CD OB g} *o GO it o oo e 3! fc: QS? 5r ff ^
T.TiTi(?4^0<^5COCOrfT}iriOiOOiO35COJ.J.J>QOOOQ00500JOO
M JO & 30 O O T O
rT ' 1 "
r^coio?oacooTicoiooaDO' ' 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
1.9722221
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
.022
.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.
132
.00307
.00002
3 M
33.183
17.974
9 Vs
306.36
504.21
116
.01227
.00013
516
34.472
19.031
10.
314.16
523.60
332
.02761
.00043
35.784
20.129
322.06
543.48
t*
.04909
.07670
.00102
.00200
716
y*
37.122
38.484
21.268
22.449
?!
330.06
338.16
563.86
584.74
316
.11045
.00345
916
39.872
23.674
L
346.36
606.13
732
.15033
.00548
%
41.283
24.942
%
354.66
628.04
.19635
.00818
1116
42.719
26.254
4
363.05
650.46
932
.24851
.01165
M
44.179
27.611
7
371.54
673.42
516
.30680
.01598
1316
45.664
29.016
11.
380.13
696.91
1132
.37123
.02127
Vs
47.173
30.466
388.83
720.95
.44179
.02761
1516
48.708
31.965
M
397.61
745.51
1332
.51848
.03511
4.
50.265
33.510
%
406.49
770.64
716
.60132
.04385
53.456
36.751
x^2
415.48
796.33
1532
.69028
.05393
/4
56.745
40.195
%
424.50
822.58
L
.78540
.06545
%
60.133
43.847
54
433.73
849.40
916
.99403
.09319
i^
63.617
47.713
Xo
443.01
876.79
%
1.2272
.12783
%
67.201
51.801
12.
452.39
904.78
1116
1.4849
.17014
%:
70.883
56.116
^
471.44
962.52
1.7671
.22089
%
74.663
60.663
i^
490.87
1022.7
1316
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
1516
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
116
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
316
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
516
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
716
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
916
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
1116
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
1316
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
1516
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
116
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
316
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
516
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
716
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
916
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
1116
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
1316
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
1516
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
116
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
316 .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
518
.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
916
.0017
.0129
8
.3491
2.611
2H/2
2.521
18.86
%
.0021
.0159
8J4
.3712
2.777
22
8.640
19.75
1116
.0020
.0193
gL
.3941
2'. 9 48
221^
2.761
20.UO
H
.0031
.0230
8%
.4176
3.125
23
2.885
21.58
1316
.0030
.0269
9
.4418
3.305
23^
3.012
22.53
%
.0042
.0312
9J4
.4667
3.491
24
3.142
23.50
1516
.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
1.625
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 12 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 12 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 2f8f3fl 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
800
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
AKjO
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
JKf
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
4314
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
nqjft
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
18213
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
800
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
9.16
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
806
2.0869
9.20
2.2192
17.00
2.8332
6.79
1.9155
7.43
20055
8.07
2.0882
9.22
2.2214
17.50
2.8621
6.80
1.9169
7.44
20069
8.08
2.0894
9.24
2.2235
18.00
2.8904
6.81
1.9184
7.45
2.0082
809
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
817
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
819
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.
CoVers.
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.
CoVers
Sine.
M.
From 75 to 90 read from bottom of table upwards.
1GO
MATHEMATICAL TABLES.
M.
Sine.
CoVers.
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
25715
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
24751
1.0785
.07282
.92718
68
15
.37865
.62135
2.6410
.40911
2 4443
1.0804
.07446
.92554
45
30
.38268
.61732
2.6131
.41421
24142
1.0824
.07612
.92388
30
45
.38671
.61329
2.5859
.41933
23847
1.0844
.07780
.92220
15
23
.39073
.60927
2.5593
.42447
23559
1.0864
.07950
.92050
67
15
.39474
.60526
2.5333
.42963
23276
1.0884
.08121
.91879
45
30
.39875
60125
2.5078
.43481
22998
1.0904
.08294
.91706
30
45
.40275
.59725
2.4829
.44001
2 2727
1.0925
.08469
.91531
15
24
.40674
.59326
2.4586
.44523
22460
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
21203
1.1056
.09554
.90446
45
30
.43051
.56949
2.3228
.47697
20965
1.1079
.09741
.90259
30
45
.43445
.56555
2.3018
.48234
2.0732
1.1102
.09930
.90070
15
26
.43837
.56163
2.2812
.48773
20503
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.
CoVers.
Sine.
'
M.
From 60 to 75 read from bottom of table upwards.
NATURAL TRIGONOMETRICAL FUNCTIONS.
161
M.
Sine.
CoVers.
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
.21739
.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.
CoVers.
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
jtureu 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.
Specificgravity 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
MATEKLAUS.
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, marshga 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 noncorrosive. Tenacity about one third that of wroughtiron. 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 bluishwhite color and highly crystalline or laminated structure.
Melts at 842 F. Heated in the open air it burns with a bluishwhite 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 antifriction
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 meltingpoint:. 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 bluishwhite 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 goldleaf 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. Gratebars 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 meltingpoint 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. Silverwhite,
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 signallights and for flashlights for photographers. It
is nearly noncorrosive, 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 ferromanganese 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 silverwhite 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 boilingpoint. 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 germansilver, nickelsilver,
etc., and recently in the manufacture of steel to increase its hardness and
strength, also for nickelplating. Cubical expansion from 32 to 212 F.,
0.0038. Specific heat .109.
Platinum, Pt. At. wt. 195. A whitish steelgray 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 sheetiron (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 bluishwhite 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 noncorrosive 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  dft = 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 116
1516 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*4in. 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 CharcoalIron 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.
Roundedge 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~tooo>ooWw^^oorcoo^Hgg3^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>bOOOOQOOSOSO5O
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
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H*
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few
ll
gl
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2
H^
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^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
143
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 IBeams.
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
tH
^ '
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 ZBars.
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
ANGLECOVERS.
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.
SQUAREROOT 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.
Floorbars.
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 Sheetmetal 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 watertight side seams; if the roof is
rather steep 1^ corrugations will answer.
Some manufacturers make a special highedge corrugation on sides of
sheets (The Cincinnati Corrugating Co.), and thereby are enabled to secure
a waterproof sidelap 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.
TerraCotta.
Porous terracotta 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 onehalf 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 openhearth 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 basebox system." The basebox 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 basebox 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 Sheetmetal Gauge.)
ENGLISH BASEBOX.
(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 BASEBO
(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
' 27.92
* 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
Onesquare
Remarks.
of Roofs.
4
900
216
The number of shingles per square is
4}x>
800
192
for common gableroofs. 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 plateglass
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 windowglass is used, single thick glass (about 116") 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
windowglass 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.
Castiron plates (%" thick) 1500
Copper 80125
Felt and asphalt 100
Felt and gravel 8001000
Iron, corrugated 100375
Iron, galvanized, flat 100 350
Lath and plaster 9001000
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 (116" thick) 300
" and laths 500
Shingles, pine 200
Slates W thick) 900
, Skylights (glass 316" to J" thick) . . .. 250 700
Sheet lead 500 800
Thatch ; 650
Tin 70125
Tiles, flat 15002000
(grooves and fillets) 7001000
pan 1000
" with mortar 20003000
Zinc ..... 100200
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 CASTIRON PIPES OR COLUMKS. 185
WEIGHT OF CASTIRON 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 CASTIRON 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
,&,
716
1532
1732
916
1932
1116
&
%
1516
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
1116
H
1316
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
CASTIRON PIPE FITTINGS.
187
CASTIRON 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 CASTIRON WATER AND GASPIPE C
(Addyston Pipe and Steel Co., Cincinnati, Ohio.)
at
Standard Waterpipe.
*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 CASTIRON WATERPIPES.
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 castiron 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 /df.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: "Castiron 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 JrbubMes 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 CASTIROK PIPE.
189
Thickness of Metal and Weight per Length for Different
Sizes of Castiron 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 Castiron 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
^
1116
34
1316
78
1516
1
1 18
1 14
1 38
1 12
1 58
1 34
1 78
2
2 18
2 14
212
234
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 "waterram" 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:
(PflOO)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 pressurevalues 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 doubleriveted pipe.
S
gj'
1
_  _j
a
3
d
+= a
og
il
43
OJ
t ^j
*o"o
11
+3 CO
$_ , S
*" O T
i a
qu2 .
t a
031 1
i, of
.^&
"3gJ
. ^3
^1 o
u aT
^w 6
tj
111
S 1?
S'l 5
s ^
"O 05
"33 C
ISS^
.SfS^
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 48in. 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 lefthand
columns tire for medium and the righthand lines are for high pressures.
The sudden increase in diameters at 16 inches is due to the possible inser
tion of wroughtiron pipe, making with a nearly constant width of gasket a
greater diameter desirable.
When wroughtiron 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.
CASTIRON" 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^
1 716
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^
2 316
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 CASTIRON
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
2 116
40 2^
Thickness
of Pipe.
Frac. Dec.
1332
716
716
1532
1932
2132
1116
Has
2732
3132
1
1 116
1
l'532
1 316
1 516
11132
1T16
.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"kiick rxjii. 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^iieooDOiccot^itoiocxtco^TEcoTttoo oo T~co T~ei ^
per
Lin. Ft. S "~~^~'~^;S;^Sw33$SSo^oco
of Pipe. ___^ r,^.ti ^
Weight .TtCJOT*iNl~TtOOii^^OOTt<Oimr~GOOeoojer<>^r,<>r> 
of Pipe
per iitS'totcoiot>osoW^GC2oc6coono:T
Li ^ Ft i^i^fr^co^Tr^ioo
^ __ .^ JyQ
oyp'jpl i c '"ssssssssnssigsg ^
i\u nt f?. %^ e
^ IPJ.llliillliSSSISIilgligigSlS ^
 1 ^ ^ i^ i^ I' o*' iy eo
I
g
_ ^ ^
CC ' ' "T^^H'^CJUJ'C
o^
02 " ^* TI <ri yt & 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 ii ii 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. TIIH i< ri ^ 11 ri 11 11 11 11 ?? g< g
Dm e  ' ^"^^^^^^^sdSsasftftii* g
Nominal j
Inside fl w
Diam. '
WROUGHTIROH PIPE.
195
For discussion of the Briggs Standard of Wroughtiron 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 fullthreaded 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 Vthread, 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.
WROUGHTIRON 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 LAPWELDED CHAR
COALIRON BOILERTUBES.
(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 14
1.060
.095
3.330
3.927
.882
.0061
1.227
.0085
3.604
3.056
3.330
1.15
1 12
1.310
.095
4.115
4.712
1.348
.0094
1.767
.0123
2.916
2.547
2.732
1.40
134
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 14
2.060
.095
6.472
7.069
3.333
.0231
3.976
0276
1.854
1.698
1.776
2.16
2 12
2.282
.109
7.169
7.854
4.090
.0284
4.909
.0341
1.674
1.528
1.601
2.75
234
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 14
3.010
.120
9.456
10.210
7.116
.0494
8.296
.0576
1.269
1.175
1.222
3.96
312
3.260
.120
10.242
10,996
8.347
.0580
9.621
.0668
1.172
1.091
1.132
4.28
334
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
412
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 steamheating 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 SHEETIRON
FOR RIVETED PIPE.
Thickness by the Rirmiugliam WireGauge.
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.fdiam.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
Boltholes.
Diameter
Boltholes.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 2JJ2
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. IRONPIPE SIZES.
(For actual dimensions see tables of Wroughtiron 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
516
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 1316
12
13
31^1
12
11
%
12
17
m
12
12
4
10 to 12
11
1316
12
17
1 1516
12
12
5
10 to 12
11
%
12
17
2
12
12
5/4
10 to 12
11
1516
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
14
$
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
416
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 3in. 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. EXTRAHEAVY WROUGHTIRON 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,
Lightningrod 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
316
516
$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.
38 Thick.
516 Thick.
M Thick.
316
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 WASTEPIPE.
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.
BLOCKTIN 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 TINLINED 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
716 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^^,
fiii52Siiiliili > : , *J
__ .~OOOOOOOOOOOOOOOOOOoS *r ri
02 M
j
Sc
a
35 "
*i I
6C02 P ^OQO^COriOOJOOI>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.
116
2.72
.014
.011
1 116
46.32
4.10
3.22
K
5.45
.056
.045
Ug
49.05
4.59
3.61
316
8.17
.128
.100
1 316
51.77
5.12
4.02
1 A
10.90
.227
.178
154
54.50
5.67
4.45
516
13.62
.355
.278
1 516
57.22
6.26
4.91
%
16.35
.510
.401
1%
59.95
6.86
5.39
716
19.07
.695
.545
1 716
62.67
7.50
5.89
21.80
.907
.712
1*6
65.40
8.16
6.41
916
24.52
1.15
.902
1 916
68.12
8.86
6.95
%
27.25
1.42
1.11
m
70.85
9.59
7.53
1116
29 97
1.72
1.35
1 1116
73.57
10.34
8.12
H
1316
32.70
35.43
2.04
2.40
1.60
1.88
1 fs16
76.30
79.02
11.12
11.93
8.73
9.36
7 /8
38.15
2.78
2.18
1%
81.75
12.76
10.01
1516
40.87
3.19
2.50
1 1516
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 DROPSHOT.
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.. .
31 00"
4100
5100
6100
Trap Shot
7100"
Trap Shot
8100"
10784
4565
2326
1346
1056
848
688
568
No. 8
" 8
7
7
6
5
4
3
Trap Shot
9100"
Trap Shot
10100"
11100
12100
13100
14100
472
399
338
291
218
168
132
106
No. 2...
1.. .
B...
BB.
BBB
T...
TT..
F..
FF..
15100"
16100
17100
18100
19100
20100
21100
22100
23100
86
71
59
50
42
36
31
27
24
COMPRESSED BUCKSHOT.
Diameter.
No. of Balls
to the Ib.
Diameter.
No. of Balls
to the Ib.
No 3 .
25100"
284
No. 00.... ...
34100"
115
* 2
27100
232
" 000
3100
98
44 1
30 100
173
Balls
38100
85
"
32 100
140
44100
50
SCREWTHREADS, SELLERS OR U. S. STANDARD.
In 1864 a committee of the Franklin Institute recommended the adoption
of the system of screwthreads 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.
ScrewThreads, United States Standard.
5
B
3
ft
a
Q
516
1116
20
18
16
14
13
12
11
11
1316
&6
1 116
2 1316
3
3 516
4 4
TT. S. OR SELLERS SYSTEM OF SCREWTHREADS. 205
ScrewThreads, 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
1116
11
*6
7
1?6
4*6
3*4
3*4
*M6
16
14
1316
10
10
n
7
6
2
4 S
3%
3^
I/
12
9
^
6
2*6
4
4
3
916
12
1516
9
%
o
2%
3*6
U. S. OR SI^I.I.KKS S VSTK1TI OF SCREWTHREADS.
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
716
3764
54
316
710
5^16
18
.2^0
.0074
.077
.045
1932
1732
1116
516
*4
1012
96
16
.294
.0078
.110
.068
1116
%
5164
%
516
6364
716
14
.344
.0089
.150
.093
2532
^33"
910
716
%
1 764
*6
13
.400
.0096
.196
.126
Vs
1316
1
V>
716
1 1564
916
12
.454
.0104
.249
.162
3132
2932
1*6
916
*6
1 2364
%
11
.507
.0113
.307
.202
1 116
1
1 732
%
916
]!/>
M
10
.620
.0125
.442
.302
\y*
1316
1 716
%
1116
1 4964
%
9
.731
.0138
.601
.420
1 716
1%
1 2132
%
1316
2132
1
8
.837
.0156
.785
.550
!&/
1 916
JT^
1
1516
21964
1^
7
.940
.0178
.994
.694
1 1316
1%
2 332
1*6
1 116
2916
IM
7
1.065
.0178
1 .227
.893
2
1 1516
2516
1/4
1 316
25364
i^
6
1.160
.0208
1.485
1.057
2316
2*6
2 1732
18
1516
3332
ji^j
6
1.284
.0208
1.767
1.295
2%
2516
2M
1 716
3 2364
1%
5*6
1.389
.0227
2.074
1.515
2916
2*6
2 3132
J
1916
3%
m
5
5
1.491
1.G16
.0250
.0250
2.405
2.761
1.746
2.051
2%
3 1516
21116
2%
3316
31332
1 Jo
1 1116
1 1316
3 5764
4532
2
4*6
1.712
.0277
3.142
2.302
3*6
3 116
3%
2
1 1516
42764
2)4
4*6
1.962
.0277
3.976
3.023
3716
4116
2*4
2316
46164
2J^j
4
2.176
.0312
4.909
3.719
gl
3 1316
4*6
2U
2 716
5 3164
2M
4
2.426
.0312
5.940
4.620
4316
4 2932 2%
21116
6
3
3*6
2.629
.0357
7.069
5.428
m
4916
5% 3
21516
6 1732
3)4
3^
2.879
.0357
8.296
6.510
5
41516
5 1316 3U
3316
7 116
354
3)4
3.100
.0384
9.621
7.548
5%
5516
6764
3*6
3716
739 64
3M
3
3.317
.0413
11.045
8.641
5 1116
6 2132 3%
3 1116
gi,^
4
3
3.567
.0413
12.566
9.993
6*1
6116
7 332 14
3 1516
8 4164
4)4
2%
3.798
.0435
14.186
11.329
6716
7 9li? 4*4
4316
9316
4*6
2%
4.028
.0454
15.904
12.743
(3 7^
6 1316
73132,4*6
4716
Q3/
4M
2%
4.256
.0476
17.721
14.226
7**4
7316
8 13324%
41116
10*4
5
2^
4.480
.0500
19.635
15.763
<%
7916
8 2732 5
4 1516
10 4964
5)4
2*6
4.730
.0500
21.648
17.572
8
7 1516
9 932 ! 5*4
5316
11 2364
5)^2
2%
4.953
.0526
23.758
19.267
8%
8516
9 2332 5*6
5 716
11%
5%i
2%
5.203
.0526
25.967
21.262
854
81116
10532 ! 5%
5 1116
12%
6
5.423
.0555
28.274
23.098
9*6
9 116
10 1932 6
51516
12 1516
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 oversize
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 oversize 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 limitgauges was  commended by the Master CarBuilders'
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
516
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
916
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 SCKEWTHEEADS 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 Sellerssystem 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 SCREWTHREADS 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
516
18
.3245
.2403
.0421
.0070
.006
.3305
.2463
.2589
%
16
.3885
.2938
.0474
.0078
.006
.3945
.2998
.3139
716
14
.4530
.3447
.0541
.0089
.006
.4590
.3507
.3670
M
13
.5166
.4000
.0582
.0096
.006
.5223
.4060
.4236
916
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 SETSCREWS AND CAPSCREWS.
American, Hartford, and Worcester MachineScrew Companies.
(Compiled by W. S. Dix.)
(A)
(B)
(C)
(D)
(E)
(F)
(G)
Diameter of Screw. . . .
K
316
/4
516
%
716
7&
Threads per Inch
Size of Tap Drill*
40
No. 43
24
No. 30
No. 5
18
1764
16
2164
14
12
2764
(H)
(D
(J)
(K)
(L)
(M)
(N)
Diameter of Screw.. . .
916
Ys
H
%
1
1^6
1J4
Threads per Inch
12
11
10
9
8
7
7
Size of Tap Drill*....
3164
1732
2132
4964
%
6364
*M
Set Screws.
Hex. Head Capscrews.
Sq. Head Capscrews.
Short
Diam.
of Head
(C) <
(D) 516
$ &
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.
716
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
716
916
1116
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
Capscrews.
Diam. of
Head.
316
Lengths
(under
Head).
Flat Head Capscrews.
Buttonhead Cap
screws.
Diam. of
Head.
Lengths
(including
Head).
Diam. of
Head.
732 (.225)
516
716
916
1316
1516
1
Lengths
(under
Head).
* For cast iron. For numbers of twistdrills see p. 29.
Threads are U. S. Standard. Capscrews 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 flathead 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
316
2*
3
48
.0973
.1894
.1786
.1545
316
K
4
32, 36, 40
,1105
.2158
.2028
.1747
316
%
5
32, 36, 40
.1236
.2421
.2270
.1985
316
%
6
30, 32
.1368
.2684
.2512
.2175
316
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 316 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
516
7?16
9?6
2
2^4
I
I
2 1516
1364
1*964
1132
2564
2964
3364
3964
4764
5364
5964
116
532
932
1 1332
1 2332
1 1516
2 316
2 716
2%
S3'
1116J
13161
916
1116
1316
716
2' '116 1116
2 5161 1%
2 916 2 116
2 1316 2 516
2 1516
3 316
4 716
4 1516
4 116
4 1516
5 516
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* Tt C
^ x .Q COTj< 1 i5'lOlOOCDI>OOO
rg
*
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>^ o
5 10 10 10 K
^O" COiOt^O5> iCOiOC'OS''COM
1 itnOiCOt>' iCOOrfQCS^l^^w
C'COCOCOCO'^ 1 ^9''^ 1 ^T^3'lOiOlOCOCDCOit^i."aDQOOSOSOSOOO
2^iT*iOOOOOOOOOOOOOOOO
SOOOrH OiCOT
QOOOOO
^aocooiio^'J>cooco<?>co > *ocoGOTiooT
I ;i;oco;oi.~GcaoosooT.i,(?*cocoTfcoia
SQOOOOCOCOCSWOOiOOlOpiqOOOO
I _o t~ os rt" od o co* o io" oi oo i> IH CD o c co" <* ti TJ os ao co o co <M! o oo co' 10' i<
^cocOrf^iooeococDJ.ioOGOOSO5OT('^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
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> CO OS 10 r. t>G
50 M^ ^
' i i 10 O 10
OTK?JCOr>*lOt^GOOSC
^OiCO^C^OSCDCOOi.^iiGOlOC
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^
116
375
4.
30to401bs.. J
%x3
116
116
410
435
3.7
3.3
1
^ x2 H
1 116
465
3.1
[
1^x3
%
715
2.
20to301bs..J
^xl^
n
760
800
2.
2.
I
1^x2
?
820
2.
RIVETS TUIIKBUCKLES.
CONKHEAD 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 conehead
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
Wiregauge.
Bolt in
inches.
No. in 100 Ibs.
a.
516
No. 16
M
29,300
a?
H
" 16
516
18,000
1
716
** 14
%
7,600
916
" 11
14
3,300
jijj?
%
44 11
916
2,180
1^3
1116
** 11
%
2,350 .
1%
1316
* 11
%
1,680
2
3132
* 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^x916
880
30
40 " 52
5 x916
400
27
35 ** 40
5 xU
490
22
24 " 35
550
20
24 " 30
4J4 x 716
725
15
18 " 24
4 x716
820
13
16 " 20
8J4x%
1250
9
14 " 16
3 x %
1350
8
8 " 12
2^x%
1550
7
8 " 10
2^x516
2200
5
STREET RAILWAY SPIKES.
Spikes.
Number in Keg, 200 Ibs.
Kegs per Mile, Ties 24 in.
between Centres.
5^x916
5 x^
4J^x716
400
575
800
30
19
13
BOAT SPIKES.
Number in Keg of 200 Ibs.
Length.
H
516
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.
516 in.
fcin.
716 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 paqae
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>OOOiTiOOlOOO
^^Sc^S^o
"""""""'^gg: : ; ; ;
5S^^^io?:SS2gg : : j
5 '
oot>^oS^^i^o^^"w^^^
x^f oc^^fooottcOTHTt<' rfi?o(7't^JOr:oc^'
0\ rHriiiriOi<MOCOTt<iOtO!>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
22 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 MILEOHM. 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 mileohm " by the weight of
the wire per mile. Thus in a grade of Extra Best Best, of which the weight
per mileohm 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
mileohm (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 firealarm lines, etc.
No. 12. For telephone lines, police and firealarm 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 mileohm 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 mileohm about 5700 Ibs.
The Trenton " Steel " wire is well suited for telephone or short telegraph
lines, and the weight per mileohm 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.
ooiie<
 i Ti
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
HrlSnSlNOJCO^lOOOOOC^
SOOOOOOOOOOrti
3SSoS83 M 3Sll2llSl8ll
?SooIo^5t2t2?2ooaco2H
jJTH^oiinNoJtlto^sceooiitiHrHO
*iSg?2issisIsgigS
rf
OHJI>. to 10 eo ***< o>
ooooco5)5>SS
HARDDRAWK COPPER WIRE; INSULATED WIRE. 221
HARDDRAWN COPPER TELEGRAPH WIRE.
(J. A. Roebling's Sons Co.)
Furnished in halfmile 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 nonmagnetic character lessens the selfin
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 Weatherproof 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. Ropelaid 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 STEELWIRE 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 softiron stock is advisable.
FLEXIBLE STEELWIRE 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
ingpower 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 holdingpower
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 caststeel 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 STEELWIRE 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.
13inch tarred Russian hemp hawser weighs about 39 Ibs. per fathom.
10inch white manila hawser weighs about 20 Ibs. per fathom.
1^inch stud chain weighs about (58 Ibs. per fathom.
4inch 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 PIANOWIRE.
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. Ploughsteel is
known here in some steel works as the quality of plate steel used for the
mouldboards of ploughs, for which a very ordinary grade is good enough.
Experiments by Dr. Percy on the English ploughsteel (socalled) 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 HARDDRAWH 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.
Phosphorbronze 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.
" Deltametal " 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. Siliconbronze 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 HARDDRAWN COPPER
WIRE.
The British Post Office authorities require that harddrawn 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
916
1^
0.48
4.27
%
v%
25^
10%
H
]i,^
0.39
3.48
H
3
2M
10a
716
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
916
1%
0.48
9
1M
5
1%
10%
^
1^
0.39
7
ip
4^j
\\
10a
716
^%
0.29
5^
3%
j^4
10%
%
1M
0.23
4H
%
1
CableTraction Ropes.
According to English practice, cabletraction 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 Marketstreet 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
916
Ja^
0.41
4.1
1
21
Ji
J1Z
0.31
2.83
2%
4
22
716
1%
0.23
2.13
Hi
31^
3^4
23
8^
1 V<<
0.21
1.65
O1/J
03/f
24
516
1
0.16
1.38
2
2V^
25
932
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
%
2M
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
916
1%
0.41
8
1%
4^4
3
21
i^
JL/
0.31
6
ji^
4
22
716
1%
0.23
4^
1/4
31^
2^j
23
1^4
0.21
4
1
3/>4
2 /V
24
516
j
0.16
3
^
2M
1^<
25
932
%
0.12
2
*!
2^
3i
PloughSteel Rope.
Wire ropes of very high tensile strength, which are ordinarily called
"Ploughsteel 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 Ploughstee
Ropes it is best to have advice on the subject of adaptability.
228
MATERIALS.
PloughSteel 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
3M
10
%
0.88
27
5
SH
10M
7
0.60
18
m
3
10^
916
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
1116
0.92
0.70
25
21
5
4
4
3^
19
%
0.57
16
3%
3
20
916
0.41
12
2M
21
K
0.31
9
1%
2^2
22
716
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&$QL
Break
ing
Strain
in tons
of 2000
pounds
WIRE ROPES.
229
Galvanized Caststeel Yacht Rigging.
Circum
ference
in inches.
Weight
per Fath
om in
pounds.
Cir. of
new
Manilla
Rope of
equal
Strength.
Break
ing
Strain
in tons
of 2000
pounds
Circum
ference
in inches
Weight
per
Fathom
in
pounds.
Cir. of
new
Manilla
Rope of
equal
Strength.
Break
ing
Strain
in tons
of 2000
pounds
fa
%&
O1/
2*4
M
T
4$
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.)
Steelwire 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
Softiron sewingwires 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 STEELWIRE HAWSERS,
With Chain Cable, Tarred Russian Hemp, and White
Manila Ropes.
Patent Flexible
Steelwire 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
916
1016
1116
1216
1316
1516
1 1732
166
1' 1516 204
2 116
2
316 256
516 280
14
21
30 101
35
4815'
54
68
112
1434
23!
1
02
"o
2
PH
I
107 110
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 samesized 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
turntable 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 linseedoil with a piece of sheepskin,
wool inside; or mix the oil with equal parts of Spanish brown or lampblack.
To preserve wire rope under water or under ground, take mineral or vege
table tar, and add one bushel of freshslacked 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 castiron pulleys and sheaves should be filled with well
seasoned blocks of hard wood, set on end, to be renewed when worn out.
This endwood 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
>$
WS
1
feij
!
<D M
g
PQft
1^1
w 3
3 3
tj a5
d^'O
ccgg
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
2532
Ys
y
1932
3864
1288
1680
3300
1120
516
2732
1
1 116
2898
5796
1932
2520
5040
1680
%
3132
1710
1H
4186
8372
2790
3640
7280
2427
716
1 532
2
1%
5796
11592
3864
5040
10080
3360
Lj
1 1132
31^
1 1116
7728
15456
5182
6720
13440
4480
916
1 1532
3 ,'10
1%
9660
19320
6440
8400
16800
5600
5
1 2332
2116
11914
23828
7942
10360
20720
6907
1116
1 2732
5 8
2^4
14490
28980
9660
12600
25200
8400
H
1 3132
5%
2^
17388
34776
11592
15120
30240
10080
1316
2332
6710
2 fl16
20286
40572
13524
17640
35280
11760
%
2732
8
2%
22484
44968
14989
20440
40880
13627
1516
21532
9
3116
25872
51744
17248
23520
47040
15680
1
2 1932
10 710
3J4
29568
59136
19712
26880
53760
17920
1 116
2 2332
11 210
3516
33264
66538
22176
30240
60480
20160
u
2 2732
3M
37576
75152
25050
34160
68320
22773
1 316
3532
13 710
3% '
41888
83776
27925
38080
76160
25387
1^4
3732
16
46200
92400
30800
42000
84000
28000
1' 516
3 1532
4%
50512
101024
33674
45920
91840
30613
j^
3^Ha
18 4 2 10
4916
55748
111496
37165
50680
101360
33787
1 716
3 2532
19 710
4%
60368
120736
40245
54880
109760
36587
JJ
3 3132
21 710
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 1inch chain use 20inch 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 FIKEBRICK.
233
_
\
^
\
,
thick x 4^ to 4 inches
3
iam.
SIZES OF FIREBRICK,
9inch straight 9 x 4^ x 2^ inches.
Soap 9 x 2J^ x 2J*j
Checker 9x3 x3 "
2inch 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 24inch circle.
36inch circle 8% to 6^ x 4J4 x 2^.
13 bricks line a 36inch circle.
48inch circle 8% to 7J4 x 4^ x 2J4
17 bricks line a 48inch circle.
inch straight 13^ x 2^ x 6.
inch key No. 1 13^ x 2^ x 6 to 5 inch.
90 bricks turn a 12ft. circle.
13i^inch key No. 2 13^ x 2^ x 6 to 4% inch.
52 bricks turn a 6ft. circle.
Bridge wall, No. 1 13x6^x6.
Bridge wall, No. 2 13x6^x3.
Mill tile 18,20,or24x6x3.
Stockhole tiles 18, 20, or 24 x 9 x 4.
18inch block 18x9x6.
Flat back 9x6x2^.
Flat back arch 9 x 6 x 314 to 2^.
22inch 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 9inch 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 9inch 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 fireclay should be sufficient to lay 3000 ordinary bricks. To
secure the best results, firebricks 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 FIREBRICK 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
Et
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 9inch brick
as may be needed in addition.
ANALYSES OF MET. SAVAGE FIRECLAY.
(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 82.4695.36
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 longcontinued
or stronger heating the material becomes deadburnt, 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 deadburnt 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 deadburnt 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 harshfibred 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 8inch specimen for 1000 Ibs. per sq. in.
increase of load is 0.0004 in.
Yieldpoint. The term yieldpoint 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 yieldpoint, at which the rate increases
suddenly, may in some cases be considerable. This difference, however, will
not be discovered in short testpieces 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 yieldpoint will appear to be simultane
ous. Unfortunately for precision of language, the term yieldpoint 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 yieldpoint, 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 testingmachine 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 textbooks the elastic limit, and
it will probably continue to be so called, although the use of the newer term
"yieldpoint" 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 welldefined 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
testpiece 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 yieldpoint, and may on occasion be zero. On
loading a bar above the yieldpoint, 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 ; ieldpoint, this limit rises, but reaches a maxi
mum as the yieldpoint, is approached, and then falls rapidly^ reaching even
to zero. On leaving the bar at rest under a stress exceeding that of its
primitive breakingdown 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 puddledsteel 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 connectingrods 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 longknown result of common experience. It partially ac
counts for what Mr. Holley has called the tl intrinsically ridiculous factor of
safety of six."
Another "longknown 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 pistonrods 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 footpounds), 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 footpounds, how many times would it be likely to bear the repetition
of 100 footpounds, or would it be safe to allow it to remain for fifty years
subjected to a continual succession of blows of even 10 footpounds 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 highcarbon steel was better adapted to resist repeated shocks and vi
brations than lowcarbon steel.
These results, however, would scarcely be sufficient to induce any en
gineer to use .84 carbon steel in a caraxle or a bridgerod. 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 wroughtiron 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 rapidlymoving 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, suspensionrods 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 unrtynine bars, twentynine of
which were variously twisted, from threeeighths of one turn to six turns per
foot. The testpieces 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 gaugemarks 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 gaugemarks, 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 testpiece, 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
gaugemarks, 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 testpiece 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
Vshaped groove
90,000, about
117,000
Indeterminate.
TENSILE STRENGTH.
243
Tests plate made by the author in 1879 of straight and grooved testpieces
Of boilerplate 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 gaugemarks (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 gaugemarks,
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 boilerplates 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
testingmachine 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 testingmachine
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 testingmachine, 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 wedgeshaped
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 bridgebuilding, 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, onehalf square inch, is as large as can be taken in the ordinary test
ingmachines 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 compressiontest 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 castiron 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 crosssection 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 wroughtiron columns, values commonly taken are: / = 36,000 to
40,000; a = 1/36,000 to 1/40,000.
For solid castiron columns, / = 80,000, a = 1/6400.
80 non
For hollow castiron columns, fixed ends, p    , I length and
ficients derived from Hodgkinson's experiments, for castiron columns is to
he deprecated. See Strength of Castiron 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 rightangled 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 rightangled 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 rightangled 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 rightangled 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.
6WMi *
bVbfa**
/ 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 Castiron 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 castiron 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 castiron 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, blowholes, u coldshuts, 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 castiron
columns, we have the data of recent experiments on fullsized 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 15inch, 190J inches long,
two 8inch, 160 inches long, and two 6inch, 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 15inch 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 CASTIROK COLUMNS.
251
TESTS OF CASTIRON 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 15inch columns 57,143 pounds, for the 8inch columns 40,000
pounds, and for the 6inch 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 castiron mill columns, made on the Watertown testingmachine in
188788, 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 castiron 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 Castiron Waterpipe. (Technology
Quarterly, Sept. 1897.) Tests of 31 castiron 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 Castiron
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 fullsized 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 Castiron 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 15inch 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 Castiron 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 CASTIROK 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 = crosssection 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 crosssection, 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 Castiron 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 HZ
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 castiron 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 WROUGHTIROK COLUMNS. 255
ULTIMATE STRENGTH OF WROUGHTIRON
COL.UMNS.
(Pottsville Iron and Steel Co.)
Computed by Gordon's formula, p =
140
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 endbearings; 1/30,000 for one pin and one square
bearing; 1/20,000 for two pinbearings.
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.
WROUGHTIRON 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
5157
Maximum Permissible Stresses in columns used in buildings.
(Building Ordinances of City of Chicago, 1893.)
For riveted or other forms of wroughtiron 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 longleaf 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
1f
~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
1f
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 crosssection, 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 wroughtiron 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 commonchord section composed of two channelbars 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 fullsized Phoenix columns in 1873 showed a close agree
ment of the results with formulae 68. Experiments on fullsized Phoenix
columns on the Watertown testingmachine 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 ( 14TJ
1450,000 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 onefourth 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
1173
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$
o2
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 oughtiron 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) 880028 ~ (2)
Hingedend iron angles and tees 46000175
r
920035
r
I
Flatend iron channels and I beams.... 40000 110 (5) 800022 (6)
Flatend mildsteel angles 52000180 (7) 1040036 (8)
I
Flatend highsteel angles 76000 290 (9)
Pinend solid wroughMron columns.. . .32000 80
" 1(11)
1520058 (10)
6400161
32000277  [ 640055 
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 wroughtiron 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 angleleg 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 = unitload on the column = total load P*area of crosssection A\
C = maximum compressive unitstress on the concave side of the column:
I length of the column; r = least radius of gyration of the crosssection
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 wroughtiron = 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. Wroughtiron Columns wiftb fttonnd Ends.
Unit
load.
Maximum Compressive Unitstress 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
82PO
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. Wroughtiron Column* with Fixed Ends*
Unit
load.
Maximum Compressive Unitstress C.
~orB.
A
i = 2
1 = 40
1 = 60
1 = 80
1 = 100
lo
~ = 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 Unitstress 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 Unitstress 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 crosssection of a column to carry a given load with
maximum unitstress 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 crosssection; 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 unitload 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 squareend compression members ;
P me  for compression members with one pin and one square end;
1 ~*~ 30,000r
8000
P= for compression members^with pinbearings;
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 crosssection;
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 floorbeams and stringers. 9,000
On bottom chords and main diagonals (forged eyebars) 10,000
On bottom chords and main diagonals (plates or shapes), net
section 8,000
On counter rods anri long verticals (forged eyebars) 8,000
On counter and long verticals (plates or shapes), net section.. 6,500
On bottom flange of riveted crossgirders, 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 floorbeam hangers, and other similar members liable to
sudden loading (bar iron with forged ends) 6,000
On floorbeam 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. Floorbeam hangers, forged ends 7,000 pounds.
7,500 Floorbeam 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 windbracing 22,500 "
20,000 " Lateral bracing 15,000 "
Shearing.
9,000 pounds. Pins and rivets 7,500 pounds.
Handdriven 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 semiintrados pins and rivets.. . . 12,000 pounds.
Handdriven 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 mainweb and lowerchord eyebars 13,200
In plate hangers (net section) 9,000
In tension members of lateral and transverse bracing 19,000
In steelangle lateral ties (net section) 15,000
For spans over 200 feet in length the greatest allowed working stresses
per square inch, in lowerchord and end mainweb eyebars, shall be taken at
min. total stress \
max. total stress J
whenever this quantity exceeds 13,200.
The greatest allowable stress in the mainweb eyebars nearest the centre
of such spans shall be taken at 13,200 pounds per square inch ; and those
for the intermediate eyebars 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 castiron 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 boilerflues. 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
boilerflues, 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 lapjoints 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 SteamUsers Associa
tion, 186269, which collapsed while in actual use in boilers. These flues
varied from 24 to 60 inches in diameter, and from 816 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 boilerflues," 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 plainriveted 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 716inch plates the collapsing
pressures were ........... ............ . . . 60 49 42
For collapsing pressures of plain iron fluetubes of Cornish and Lanca
shirs steamboilers, 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 wroughtiron tubes varied from 14.33 to 20.07 tons per squareinch 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. 649651 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 crosssection 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 unitstress <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 crosssection, 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 unitstresses 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
oneeighth 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.
890UDorf)
1780C4Dad)
wL*
wL*
L
L
32UZ)ad 2 )
52UZ)2ad a )
Solid Cylin
der.
M7AD
13334Z)
wLs
24AD*
wl?
3SAD*
L
L
Hollow
Cylinder.
667UDad)
1333(ADad)
wL*
wL*
L
L
24(AD*ad*)
38(AD*ad?)
Evenlegged
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.
910 " "
810 " "
710
610 "
510 " "
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 fortyeighths 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 onehalf 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 distributed ; curves
two parabolas, vertices touching each other
at unsupported end; BD* proportional to
distance from unsupported end.
Supported at both ends; load at any one
point; two parabolas, vertices at the points
of support, bases at point loaded ; BD* pro
portional to distance from nearest point oi
support. The upper edge or both edges
may also be parabolic.
Supported at both ends; load at any one
point; two triangles, apices at points of sup
port, bases at point loaded; BD* propor
tional to distance from the nearest point of
support.
Supported at both ends; load distributed;
curves two parabolas, vertices at the middle
of the beam; bases centre line of beam; BD*
proportional to product of distances from
points of support.
Supported at both ends; load distributed;
curve semiellipse; BD* proportional to the
product of the distances from the points of
support.
272 STKENGTH OF MATERIALS.
PROPERTIES OF ROLLED STRUCTURAL STEEL.
Explanation of Tables of the Properties of I Reams,
Channels, Angles, Deck Beams, Bulb Angles, Z Bars,
Tees, Trough and Corrugated Plates.
(Tne Carnegie Steel Co., Limited.)
The tables for I beams and channels are calculated for all standard
weights to which each pattern is rolled. The tables for deckbeams and
angles are calculated for the minimum and maximum weights of the
various shapes, while the properties of Z bars are given for thicknesses
differing by 1/16 inch.
For tees, each shape can be rolled to one weight only.
Column 12 in the tables for I beams and channels, and column 9 for
deckbeams, give coefficients by the help of which the safe, uniformly
distributed load may be readily determined. To do this, divide the coeffi
cient given by the span or distance between supports in feet. If the weight
of the deckbeams is intermediate between the minimum and maximum
weights given, add to the coefficient for the minimum weight the value given
for one pound increase of weight multiplied by the number of pounds
the section is heavier than the minimum.
If a section is to be selected (as will usually be the case), intended to carry
a certain load for a length of span already determined on, ascertain the
coefficient which this load and span will require, and refer to the table for a
section having a coefficient of this value. The coefficient is obtained by mul
tiplying the load, in pounds uniformly distributed, by the span length in feet.
In case the load is not uniformly 'distributed, but is concentrated at the
middle of the span, multiply the load by 2, and then consider it as uniformly
distributed. The deflection will be 8/10 of the deflection for the latter load.
For other cases of loading obtain the bending moment in ft.lbs.; this
multiplied by 8 will give the coefficient required.
If the loads are quiescent, the coefficients for a fibre stress of 16,000 Ibs.
per square inch for steel may be used ; but if moving loads are to be pro
vided for, a coefficient of 12,500 Ibs. should be taken. Inasmuch as the effects
of impact may be very considerable (the stresses produced in an unyielding
inelastic material by a load suddenly applied being double those produced
by the same load in a quiescent state), it will sometimes be advisable to use
still smaller fibre stresses than those given in the tables. In such cases the
coefficients may be determined by proportion. Thus, for a fibre stress of
8,000 Ibs. per square inch the coefficient will equal the coefficient for 16,000
Ibs. fibre stress, from the table, divided by 2.
The section moduli, column 11, are used to determine the fibre stress per
square inch in a beam, or other shape, subjected to bending or transverse
stresses, by simply dividing the bending moment expressed in inchpounds
by the section modulus.
In the case of T shapes with the neutral axis parallel to the flange, there
will be two section moduli, and the smaller is given. The fibre stress cal
culated from it will, therefore, give the larger of the two stresses in the
extreme fibres, since these stresses are equal to the bending moment divided
by the section modulus of the section.
For Z bars the coefficients (C) may be applied for cases where the bars are
subjected to transverse loading, as in the case of roofpurlins.
For angles, there will be two section moduli for each position of the neutral
axis, since the distance between the neutral axis and the extreme fibres has
a different value on one side of the axis from what it has on the other. The
section modulus given in the table is the smaller of these two values.
Column 12 in the table of the properties of standard channels, giving the
distance of the center of gravity of channel from the outside of web, is used
to obtain the radius of gyration for columns or struts consisting of two
channels latticed, for the case of the neutral axis passing through the centre
of the crosssection parallel to the webs of the channels. This radius of
gyration is equal to the distance between the centre of gravity of the chan
nel and the centre of the section, i.e., neglecting the moments of inertia of
the channels around their own axes, thereby introducing a slight error on
the side of safety.
(For much other important information concerning rolled structural
shapes, see the "Pocket Companion " of The Carnegie Steel Co., Limited,
Pittsburg, Pa., price $2.)
PROPERTIES OF ROLLED STRUCTURAL SHAPES. 273
Properties of Carnegie Standard I Beams Steel.
1
2
3
4
5
6
7
8
9
10
11
12
i
rfog
"
 i o
.sS
g
fo.2
J3
P^
13s
ss^*^
131
& c
action Index.
epth of Beam.
r eight per Foot,
a
.2
1
o
hickness of We
7idth of Flange
foment of In<
Neutral Axis
pendicular to
at Centre.
oment of Im
Neutral Axis
cident with C
Line of Web.
adius of Gyre
Neutral Axis
pendicular to
at Centre.
5 v
ffcll
ection Modulus,
tral Axis Perpe
ular to Web al
tre.
oefficient of Str
for Fibre Stre
1 6,000 Ibs. per s
02
3
&
<J
^
f
5
s
5_
in
Ibs.
sq. in.
in
in.
I
//
r
r /
8
~C
Bl
24
100
29.41
0.75
7.25
2380.3
48.56
9.00
.28
198.4
2115800
95
27.94
0.69
7.19
2309.6
47.10
9.09
.30
192.5
2052900
k
"
90
26.47
0.63
7.13
2239.1
45.70
9.20
.31
186.6
1990300
a
4k
85
25.000.577.07
2168.6
44.35
9.31
.33
180.7
1927600
"
11
80
23.320.50
7.00
2087.9
42.86
9.46
.36
174.0
1855900
B3
20
75
22.06
0.65
6.40
1268.9
30.25
7.58
.17
126.9
1353500
M
70
20.590.57
6.32
1219.9
29.04
7.70
.19
122.0
1301200
44
"
65
19.080.506.25
1169.6
27.86
7.83
.21
117.0
1247600
B80
18
70
20.59
0.72
6.26
921.3
24.62
6.69
.09
102.4
1091900
65
19.12,0.646.18
881.5
23.47
6.79
.11
97.9
1044800
44
41
60
17.65
0.55
6.09
841.8
22.38
6.91
.13
93.5
997700
44
11
55
15.93,0.46 6.00
795.6
21.19
7.07
.15
88.4
943000
B7
15
55
16.18
0.66
5.75
511.0
17.06
5.23
.95
68 1
726800
'
50
14.71;0.56'5.65
483.4
16.04
5.73
.04
64.5
687500
M
u
45
13.240.46!5.55
455.8
15.09
5.87
.07
60.8
648200
M
'*
42
12.48
0.41
5.50
441.7
14.62
5.95
.08
58.9
628300
B9
12
35
10.290.44!5.09
228.3
10.07
4.71
0.99
38.0
405800
'
4i
31.5
9.26
0.35
5.00
215.8
9.50
4.83
1.01
36.0
383700
Bll
10
40
11.78 0.7615.10
158.7
9.50
3.67
0.90
31.7
338500
41
35
10.290.60,4.95
146.4
8.52
3 77
0.91
29.3
312400
44
"
30
8.82
0.45
4.80
134.2
7.65
3.90
0.93
26.8
286300
"
"
25
7.370.31
4.66
122.1
6.89
4.07
0.97
24.4
260500
B13
9
35
10.29
0.73
4.77
111.8
7.31
3.29
0.84
24.8
265000
41
30
8.820.57
4.61
101.9
6.42
3.40
0.85
22.6
241500
4 '
25
7.35
0.41
4.45
91.9
5.65
3.54
0.88
20.4
217900
K
21
6.310.29'4.33
84.9
5.16
3.67
0.90
18.9
201300
B15
8
25.5
7.500.544.27
68.4
4.75
3.02
0.80
17.1
182500
44
"
23
6.76
0.45
4.18
64.5
4.39
3.09
0.81
16 1
172000
44
41
20.5
6.030.364.09
60.6
4.07
3 17
0.82
15.1
161600
14
"
18
5.33
0.27
4.00
56.9
3.78
3.27
0.84
14.2
151700
B17
7
20
5.880.463.87
42.2
3.24
2.68
0.74
12.1
128600
44
tk
17.5
5.150.353.76
39.2
2.94
2.76
0.76
11.2
119400
44
44
15
4.43
0.25
3.66
36.2
2.67
2.86
0.78
10.4
110400
B19
6
17M
5.070.483.58
26.2
2.36
2.27
0.68
8.7
93100
44
14
14% 4.34
0.35
3.45
24.0
2.09
2.35
0.69
8.0
85300
"
44
12J4 3.610.233.33
21.8
1.85
2.46
0.72
7.3
77500
B21
5
14%
4.34
0.50
3.29
15.2
1.70
1.87
0.63
6.1
64600
44
44
12J4
3.600.363.15
13.6
1.45
1.94
0.63
5.4
581 CO
44
44
m
2.870.21 3.00
12.1
1.23
2.05
0.65
4.8
51600
B23
4
10.5
3.09
0.41
2.88
7.1
1.01
1.52
0.57
3.6
38100
44
44
9.5
2.790.34
2.80
6.7
0.93
1.55
0.58
3.4
36000
44
"
8.5
2.50
0.26
2.73
6.4
0.85
1.59
0.58
3.2
33900
"
4t
7.5
2.21
0.192.66
6.0
0.77
1.64
0.59
3.0
31800
B77
3
7.5
2.21
0.36
2.52
2.9
0.60
1.15
0.52
1.9
20700
'
6.5
1.91
0.26 2.42
2.7
0.53
1.19
0.52
1.8
19100
vj
"
5.5 1.630.17
2.33
2.5
0.46
1.23
0.53
1.7
17600
L = safe loads in Ibs., uniformly distributed; I = span in feet;
M = moment of forces in ft.lbs. ; C = coefficient given above.
12 '
/ = fibre stress.
274
STRENGTH OF MATERIALS.
Properties of Special I Beams Steel.
1
2
3
4
5
6
7
8
9
10
11
12
.'8
.5 S
gfe
= cS
i, a
~3*oT
"o
V
oJ
be
P*^
 _cc O
J w
1?
5S
o
^'a* 3
'i
M
1
!
.2
c
rt
^5"^
^Mfi
6''*^
Ox
3^^
^!
'O
Q
a
O>
s
05
fa
O ^2 4J
11 S ^
o j'^p:
M ^
^w^^
O co ^
W 00
'+
o
P<
CO
05
0>
S
*
^.'g ii'o
? ^"c
^ S^*o
^ X o
c
o
s
o
a
rW
^
s^^a
"p
'OQ
s"s a '
0^1 1!
a,
D
60
'5
.2
g
S^ 5
0^1 &S
0^1^
iaats
ll?5
ilfl
1^^'
<i)
02
5
H
?
S
s
PH
P5
03^ '
o
n.
Ibs.
sq. in.
in.
in.
I
/'
r
r'
5
C
B2
20
100
29.41
0.88
7.28
1655.8
52.65
7.50
.34
165 6
1766100
k >
95
27.94
0.81
7.21
1606.8
50.78
7.58
.35
160.7
1713900
"
'I
90
26.47
0,74
7.14
1557.8
48.98
7.67
.36
155.8
1661600
'
"
85
25.00
0.66
7.06
1508.7
47.25
7.77
.37
150.9
1609300
"
"
80
23.73
0.60
7.00
1466.5
45.81
7.86
.39
146.7
1564300
B4
15
100
29.41
1.18
5.77
900.5
50.98
5.53
.31
120.1
1280700
95
27.94
1.08
6.67
872.9
48.37
5.59
.32
116.4
1241500
*
90
26.47
0.99
6.58
845.4
45.91
5.65
.32
112.7
1202300
M
85
25.00
0.89
6.48
817.8
43.57
5.72
.32
109.0
1163000
"
80
23.81
0.81
6.40
795.5
41.76
5.78
.32
106.1
1131300
B5
15
75
22.06
0.88
6.29
691.2
30.68
5.60
.18
92.2
983000
"
70
20.59
0.78
6.19
663.6
29.00
5.68
.19
88.5
943800
'*
14
65
19.12
0.69
6.10
636.0
27.42
5 77
.20
84.8
904600
*
M
60
17.67
0.59
6.00
609.0
25.96
5.87
.21
81.2
866100
B8
12
55
16.18
0.82
5.61
3^1.0
17.46
4.45
.04
53.5
570600
50
14.71
0.70
5.49
303.3
16.12
4.54
.05
50.6
539200
"
M
45
13.24
0.58
5.37
285.7
14.89
4.65
.06
47.6
507900
"
"
40
11.84
0.46
5.25
268.9
13.81
4.77
.08
44.8
478100
Properties of Carnegie Trough Plates Steel.
Section
Index.
Size,
in
Inches.
Weight
per
Foot.
Area
of Sec
tion.
Thick
ness in
Inches.
Moment of
Inertia,
Neutral
Axis
Parallel to
Length.
Section
Modulus,
Axis as
before.
Radius
of Gyra
tion,
Axis as
before.
MIO
Mil
M12
M!3
M14
9^x3%
9^ x 3%
9^ x 3%
Ibs.
16.32
18.02
19.72
21.42
23.15
sq. in.
4.8
5.3
5.8
6.3
6.8
H
9/16
%
11/16
H
I
3.68
4.13
4.57
5.02
5.46
S
1.38
1.57
1.77
1.96
2.15
0.91
0.91
0.90
0.90
0.90
Properties of Carnegie Corrugated Plates Steel.
Moment of
Section
Index.
Size,
m
Inches.
Weight
per
Foot.
Area
of Sec
tion.
Thick
ness in
Inches.
Inertia,
Neutral
Axis
Parallel to
Length.
Section
Modulus,
Axis as
before.
Radius
of Gyra
tion,
Axis as
before.
Ibs.
sq. in.
/
8
r
M30
gs/ x ji/
8.06
2.4
y*
0.64
0.80
0.52
M31
8M x \\4>
10.10
3.0
5/16
0.95
1.13
0.57
M32
33? x lVj>
12.04
3.5
%
1.26
1.42
0.62
M33
12 3/16x2%
17.75
5.2
%
4.79
3.33
0.96
M34
M35
12 3/16 x 2%
12 3/16x294
20.71
23.67
6.1
7.0
7/16
g
5.81
6.82
3.90
4.46
0.98
0.99
PROPERTIES OF ROLLED STRUCTURAL STEEL. 275
ri IH OO OO O OOOOOOOO
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STRENGTH OF MATERIALS.
M
10
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CO
o
t rf co w w
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10 *
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PROPERTIES OF ROLLED STRUCTURAL STEEL. 27?
Properties of Standard Channels Steel.
2
3
4
5
6
7
8
9
10
11
12
ja
p

HI
111
lie
c K r.
* W cc
V
<D
0)
1
CM
be
c
~Jl
ll
3

fc<S
* 1 h <
^P,
Oo,o
&
"8
O
S
'"o j5
'o<jO
w 4 *
M<!O
O oo c3
CD aj
"o*" 1 ^
P,
2

o
^ijfl
^ g5g
IJfl
Si!'
'^
oJsC
or o
I
d
1
g
o
ll*
i^
3 Is ^>
S'Sr
o
g
'33
4)
2
'O
o^i ftts
o^ o
Irt^H P,C3
aS^r 1 O
w * 8
OB CD 'S
^
^
H
^
>i
(B
g
H
QQ
O
5
Ibs.
sq. in.
in.
in.
i
I'
r
r ,
s
C
X
55.
J6.18
0.823.82
430.2
12.19
5.16
.868
57.4
611900
.823
50.
14.71
0.723.72
402.7
11.22
5.23
.873
53.7
572700
.803
45.
13.24
0.623.62
375.1
10.29
5.32
.882
50.0
533500
.788
40.
11.76
0.523.52
347.5
9.39
5.43
.893
46.3
494200
.783
35.
10.29
0.43
3.43
320.0
8.48
5.58
.908
42.7
455000
.789
33.
9.90
0.403.40
312.6
8.23
5.62
.912
41.7
444500
794
40.
11.76
0.763.42
197.0
6.63
4.09
.751
32.8
350200
.722
35.
10.29
0.643.30
179.3
5.90
4.17
.757
29.9
318800
.694
30.
8.82
0.51 3.17
161.7
5.21
4.28
.768
26.9
287400
.677
25.
7.35
0.393.05
144.0
4.53
4.43
.785
24.0
256100
.678
20.5
6 03
0.282.94
128.1
3.91
4.61
,805
21.4
227800
.704
35.
10.29
0.823.18
115.5
4.66
3.35
.672
23.1
246400
.695
30.
8.82
0.683.04
103.2
3.99
3.42
.672
20.6
220300
.651
25.
7.35
0.532.89
91.0
3.40
3.52
.680
18.2
194100
.620
20.
5.88
0.382.74
78.7
2.85
3.66
.696
15.7
168000
.609
15.
4.46
0.24:2.60
66.9
2.30
3.87
.718
13.4
142700
.639
25.
7.35
0.612.81
70.7
2.98
3.10
.637
15.7
167600
.615
20.
5.88
0.45 2.65
60.8
2.45
3.21
.646
13.5
144100
.585
15.
4.41
0.292.49
50.9
1.95
3.40
665
11.3
120500
.590
13V4
3.89
0.232.43
47.3
1.77
3.49
.674
10.5
112200
.607
2114
6.25
0.58 2.62
47.8
2.25
2.77
.600
11.9
127400
.587
5.51
0.49 2.53
43.8
2.01
2.82
.603
11.0
116900
.567
16V4
4.78
0.402.44
39.9
1.78
2.89
.610
10.0
106400
.556
13%
4.04
0.31]2.35
36.0
1.55
2.98
.619
9.0
96000
.557
11J4
3.35
0.222.26
32.3
1.33
3.11
.630
8.1
86100
.576
19%
5.81
0.632.51
33.2
1.85
2.39
.565
9.5
101100
.583
17/4
5.07
0.532.41
30.2
1.62
2.44
.564
8.6
92000
.555
14%
4.34
0.422.30
27.2
1.40
2.50
.568
7.8
82800
.535
12J4
3.60
0.322.20
24.2
1.19
2.59
.575
6.9
73700
.528
9%
2.85
0.21
2.09
21.1
0.98
2.72
.586
6.0
66800
.546
15.5
4.56
0.562.28
19.5
1.28
2.07
.529
6.5
69500
.546
13.
3.82
0.442.16
17.3
1.07
2.13
.529
5.8
61600
,517
10.5
3.09
0.322.04
15.1
0.88
2.21
.534
5.0
53800
.503
8.
2.38
0.201.92
13.0
0.70
2.34
.542
43
46200
.517
11.5
3.38
0.482.04
10.4
0.82
1.75
.493
4.2
44400
.508
9.
2.65
0.33
1.89
8.9
0.64
1.83
.493
3.5
37900
.481
6.5
1.95
0.19
1.75
7.4
0.48
1.95
.498
3.0
31600
489
714
2.13
0.32
1.72
4.6
0.44,
1.46
.455
2.3
24400
.463
6^4
.84
0.25
1.65
4.2
0.38
1.51
.454
2.1
22300
.458
.55
0.18
1.58
3.8
0.32
1.56
.453
1.9
20200
.464
6. 4
.76
0.36
1.60
2.1
0.31
1.08
.421
1.4
14700
.459
5.
.47
0.26
1.50
1.8
0.25
1.12
.415
1.2
13100
.443
4.
.19
0.17
1.41
1.6
0.20
1.17
.409
1.1
11600
.443
L = safe load in Ibs., uniformly distributed; I = span in feet;
M moment of forces in ft.lbs.; C coefficient given above.
f=^; C=Ll = 8M =
12 *'
/ = fibre stress.
278 PROPERTIES OF ROLLED STRUCTURAL STEEL.
Carnegie Peckbeams,
1
2
3
4'
5
G
7
8
9
10 1 11
ft
1
,
1
OJ
i
5
Eftjjo
A
rfl
3*3
II
'Z"* o
a* 3,u
II
Ka
(Li
l^o
Depth of Beai
Weight per F
Area of Secti
Thickness of
Width of Fla
Moment of In
Neutral Axi
pendicular
Web.
Section Modu
Neutral Axi
pendicular
Web.
Radius of Gyr
Neutral Axi
pendicular
Web.
Coefficient of
Strengthfor
Stress of 16,0
per sq. in.
Moment of In
Neutral Axi
incident wi1
Centre Line
Web.
Radius of Gyr
Neutral Axi
incident wit
Centre Line
Wet.
in.
Ihs.
sq.in.
in
in.
I
s
r
C
F
?'
10
35.70
10.5
.63
5.50
139.9
25.7
3.64
274100
7.41
0.84
10
27.23
8.0
.38
5.25
118.4
21.2
3.83
226100
6.12
0.87
9
30.00
8.8
.57
5.07
93.2
19.6
3.25
208500
5.18
0.75
9
26.00
7.6
.44
4.94
85.2
17.7
3.35
189100
4.61
0.76
8
24.48
7.2
.47
5.16
62.8
14.1
2.97
150100
4.45
0.79
8
20.15
5.9
.31
5.00
55.6
12.2
3.08
129800
3.90
0.82
7
23.46
6.9
.54
5.10
45.5
11.7
2.57
124600
4.30
0.79
7
18.11
5.3
.31
4.87
38.8
9.7
2 70
103000
3.55
0.82
6
18.36
5.4
.43
4.53
26.8
8.2
2.25
87700
2.73
0.72
6
15. 3C
4.5
.28
4.38
24.0
7.3
2.33
77400
2.38
0.73
Add to coefficient C for every Ib. increase in weight of beam, for 10in.
beams, 4900 Ibs.; 9in., 4500 Ibs.; 8in., 4000 Ibs.; 7in., 3400 Ibs., 6in., 3000 Ibs.
Carnegie Bulb Angles,
10
26.50
7.80
48
3 5
104.2
19.9
3.66
211700
9
21.80
6.41
44
3,5
69.3
14.5
3.33
154200
8
19.23
5 6fi
41
3.5
48.8
11.7
2.95
124800
7
18.25
5.37
44
3
34.9
9,6
2.56
102300
6
17 ?0
5 06
50
3
23.9
7.6
2.16
80500
6
13 75
4.04
.38
3
20.1
6.6
2.21
70400
6
12.30
3.62
31
3
18.6
5.7
2.28
60400
5
10.00
2.94
.31
2.5
10.2
4.1
1.86
43300
Carnegie T Shapes.
1
2
3
4 I 5
6
7
8
9
10
11
S
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o
^
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^
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a
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5
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n^'
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^ x 5
N
1
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P
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Q
1
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02
PH
6
in.
Ibs.
sq.in.
in.
I
S
r
I'
S't
P
c
5 X3
13.6
3.99
0.75
2.6
1.18
0.82
5.6
2 22
'1.19
9410
5 X2^
11.0
2.24
0.65
1.6
0.86
0.71
4.3
K70
1.16
6900
4^X3^
4^X3
15.8
8.5
4.65
2.55
1.11
0.73
5.1
1.8
2.13
0.81
r.04
0.87
3.7
2.6
1.65
1.16
0.90
1.03
17020
6490
4^X3
10.0
3.00
0.75
2.1
0.94
0.86
3.1
1.38
1.04
7540
4 Vo X 2^
8.0
2.40
0.58
1.1
0.56
0.69
2.6
1.16
1.07
4520
4^X2Vi>
9.3
2.79
0.60
1.2
0.65
0.68
3.1
1.38
1.08
5230
4 X5
15.6
4.56
1.56
10.7
3.10
1.54
2.8
1.41
0.79
24800
4 X5
12.0
3.54
1.51
8.5
2.43
1.56
2.1
1.06
0.78
19410
14.6
4.29
1.37
8.0
2.55
1.37
2.8
1.41
0.81
20400
4 X4Vjj
11.4
3.36
1.31
6.3
1.98
1.38
2.1
1.06
0.80
15840
4 X4
13.7
4.02
1.18
5.7
2.02
1.20
2.8
1.40
0.84
16190
4 X4
10.9
3.21
1.15
4.7
1.64
1.23
2.2
1.09
0.84
13100
4 X3
9.3
2.73
0.78
2.0
0.88
0.86
2.1
1.05
0.88
7070
4 X2J<
8.6
2.52
0.63
1.2
0.62
0.69
2.1
1.05
0.92
4980
STRENGTH OP MATERIALS.
279
Carnegie T Shapes (Continued).
1
2
3
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5
6
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7.3
2.16
0.60
1.0
0.55
0.70
1.8
0.88
0.91
4380
4 X2)4
5.8
1.71
0.56
0.81
0.42
0.71
1.4
0.71
0.94
3350
4 X2
7.9
2.31
0.48
0.60
0.40
0.52
2.1
1.05
0.96
3180
4 X2
6.6
1.95
0.51
0.54
0.34
0.51
1.8
0.88
0.95
2700
3)4X4
12.8
3.75
1.25
5.5
1.98
.21
1.89
1.08
0.72
15870
3)4X4
9.9
2.91
1.19
4.3
1.55
.22
1.42
0.81
0.70
12380
3)4X3)4
11.7
3.45
1.06
3.7
1.52
.04
1.89
1.08
0.74
12000
3)4x3)^
9.2
2.70
1.01
3.0
1.19
.05
1.42
0.81
0.73
9530
3^>X3^*
6.8
2.04
0.98
2.3
0.93
.09
1.07
0.61
0.73
7450
3)4X3
11.73
3.45
1.01
2.9
1.43
0.92
1.74
1.00
0.72
11470
3)4X3
10.9
3.21
0.88
2.4
1.13
0.87
1.88
1.08
0.77
9050
3)4X3
8.5
2.49
0.83
1.9
0.88
0.88
1.41
0.81
0.75
7040
7.8
2.28
0.78
1.6
0.72
0.89
1.18
0.68
0.76
5790
3 3 X4
11.8
3.48
1.32
5.2
1.94
.23
1.21
0.81
0.59
15480
3 X4
10.6
3.12
1.32
4.8
1.78
.25
1.09
0.72
0.60
14270
3 X4
9.3
2.7J
1.29
4.3
1.57
.26
0.93
0.62
0.59
12540
3 X3)4
10.9
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1.12
3.5
1.49
.06
1.20
0.80
0.62
11910
3 X3)4
9.8
2!88
1.1
3.3
1.37
.08
1.31
0.88
0.68
10990
3 X3)4
8.5
2.49
1.09
2.9
1.21
.09
0.93
0.62
0.61
9680
3 X3
10.0
2.94
0.9
2.3
1.10
0.88
1.20
0.80
0.64
8780
8 X3
9.1
2.67
0.92
2.1
1.01
0.90
1.08
0.72
0.64
8110
3 X3
7.8
2.28
0.88
1.8
0.86
0.90
0.90
0.60
0.63
6900
3 X3
6.6
1.95
0.86
1.6
0.74
0.90
0.75
0.50
0.62
5900
3 X2)4
7.2
2.10
0.7
1.1
0.60
0.72
0.89
0.60
0.66
4800
3 X2)4
6.1
1.80
0.6
0.94
0.52
0.73
0.75
0.50
0.65
4100
2%X2
7.4
2.16
0.53
1.1
0.75
0.71
0.62
0.45
0.54
6000
2%Xl^
6.6
1.95
0.64
0.56
0.50
0.53
0.61
0.44
0.56
4000
2^X3
7.2
2.10
0.9
1.8
0.87
0.92
0.54
0.43
0.51
6960
2)4X3
6.1
1.80
0.9
1.6
0.76
0.94
0.44
0.35
0.51
6110
2)4X2%
6.7
1.98
0.8
1.4
0.73
0.84
0.66
0.53
0.58
5860
2)4X2^
5.8
1.71
0.83
1.2
0.60
0.83
0.44
0.35
0.51
4830
2)4X2)^
6.4
1.89
0.76
1.0
0.59
0.74
0.52
0.42
0.53
4700
5.5
1.620.74
87
0.50
0.74
0.44
0.35
0.52
4000
2)xi^
2.9
0.84
0.29