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

GIFT OF 




Consulting Engineer 

Sv <&/*, t r\^ 



MM. 3T3., 



, 



Consulting Engineer 




The Publishers and the Author will be grateful to 
any of the readers of this volume who will kindly call 
their attention to any errors of omission or of commis- 
sion that they may find therein. It is intended to make 
our publications standard works of study and reference, 
and, to that end, the greatest accuracy is sought. It 
rarely happens that the early editions of works of any 
size are free from errors; but it is the endeavor of the 
Publishers to have them removed immediately upon being 
discovered, and it is therefore desired that the Author 
may be aided in his task of revision, from time to time, 
by the kindly criticism of his readers. 

JOHN WILEY & 
13 & 45 EAST NINETEENTH STREET. 




gnilli 



WORKS OF WILLIAM KENT 



PUBLISHED BY 



JOHN WILEY & SONS. 



The Mechanical Engineers' Pocket-Book. 

A Reference Book of Rules, Tables, Data, and 
Formulae, for the Use of Engineers, Mechanics, 
and Students, xxxii-f- noo pages, i6mo, morocco, 
$5.00. 

Steam-Boiler Economy. 

A Treatise on the Theory and Practice of Fuel 
Economy in the Operation of Steam-Boilers. 
xiv + 458 P a S es > T 3 6 figures, 8vo, cloth, $4.00. 



THE 

MECHANICAL ENGINEER'S 
POCKET-BOOK 



A REFERENCE-BOOK OF RULES, TABLES, DATA, 

AND FORMULAE, FOR THE USE OF 

ENGINEERS, MECHANICS, 

AND STUDENTS. 



BY 

WILLIAM KENT, A.M., M.E., 

Dean and Professor of Mechanical Engineering in the L. C. Smith 

College of Applied Science, Syracuse University, 
Member Amer. Soc'y Mechl Engrs. and Amer. Inst. Mining Engrs. 



SEVENTH EDITION, REVISED AND ENLARGED 

TEXTH THOUSAND. 
TOTAL ISSTO FORTY-FIVE THOUSAND. 



NEW YORK : 

JOHN WILEY & SONS. 

LONDON: CHAPMAN & HALL, LIMITED. 

1906. 



COPYRIGHT, 1895, 1902, 

BY 

WILLIAM KENT. 



PRESS OF 

BRAUNWORTH & CO. 

BOOKBINDERS AND PRINTERS 

BROOKLYN. N. Y. 



PREFACE. 

MORE than twenty years ago the author began to follow 
the advice given by Nystrom : " Every engineeer should 
make his own pocket-book, as he proceeds in study and 
practice, to suit his particular business." The manuscript 
pocket-book thus begun, however, soon gave place to more 
modern means for disposing of the accumulation of engi- 
neering facts and figures, viz., the index rerum, the scrap- 
book, the collection of indexed envelopes, portfolios and 
boxes, the card catalogue, etc. Four years ago, at the re- 
quest of the publishers, the labor was begun of selecting 
from this accumulated mass such matter as pertained to 
mechanical engineering, and of condensing, digesting, and 
arranging it in form for publication. In addition to "this, a 
careful examination was made of the transactions of engi- 
neering societies, and of the most important recent works 
on mechanical engineering, in order to fill gaps that might 
be left in the original collection, and insure that no impor- 
tant facts had been overlooked. 

Some ideas have been kept in mind during the prepara- 
tion of the Pocket-book that will, it is believed, cause it to 
differ from other works of its class. In the first place it 
was considered that the field of mechanical engineering was 
so great, and the literature of the subject so vast, that as 
little space as possible should be given to subjects which 
especially belong to civil engineering. While the mechan- 
ical engineer must continually deal with problems which 
belong properly to civil engineering, this latter branch is 
so well covered by Trautwine's " Civil Engineer's Pocket- 
book " that any attempt to treat it exhaustively would not 
only fill no "long-felt want," but would occupy space 
which should be given to mechanical engineering. 

Another idea prominently kept in view by the author has 
been that he would not assume the position of an "au- 
thority " in giving rules and formulae for designing, but 
only that of compiler, giving not only the name of the 
originator of the rule, where it was known, but also the 
volume and page from which it was taken, so that its 

lii 



288901 



17 PREFACE. 

derivation may be traced when desired. When different 
formulae for the same problem have been found they have 
been given in contrast, and in many cases examples have 
been calculated by each to show the difference between 
them. In some cases these differences are quite remark- 
able, as will be seen under Safety-valves and Crank-pins. 
Occasionally the study of these differences has led to the 
author's devising a new ^formula, in which case the deriva 
tion of the formula is given. 

Much attention has been paid to the abstracting' of data 
of experiments from recent periodical literature, and numer- 
ous references to other data are given. In this respect 
the present work will be found to differ from other Pocket- 
books. 

The author desires to express his obligation to the many 
persons who have assisted him in the preparation of the 
work, to manufacturers who have furnished their cata- 
logues and given permission for the use of their tables, 
and to many engineers who have contributed original data 
and tables. The names of these persons are mentioned in 
their proper places in the text, and in all cases it has been 
endeavored to give credit to whom credit is due. The 
thanks of the author are also due to the following gentle- 
men who have given assistance in revising manuscript or 
proofs of the sections named : Prof. De Volson Wood, 
mechanics and turbines ; Mr. Frank Richards, compressed 
air ; Mr. Alfred R. Wolff, windmills ; Mr. Alex. C. 
Humphreys, illuminating gas ; Mr. Albert E. Mitchell, 
locomotives ; Prof. James E. Denton, refrigerating-ma* 
chinery ; Messrs. Joseph Wetzler and Thomas W. Varley, 
electrical engineering ; and Mr. Walter S. Dix, for valuable 
contributions on several subjects, and suggestions as to their 
treatment. WILLIAM KENT. 

PASSAIC, N. J M April^ 1895. 

FIFTH EDITION, MARCH, 1900. 

Some typographical and other errors discovered in the fourth 
edition have been corrected. New tables and some additions 
have been made under the head of Compressed Air. The new 
(1899) code of the Boiler Test Committee of the American 
Society of Mechanical Engineers has been substituted for the 
old (1885) code. W. K. 



PREFACE TO FOURTH EDITION. 

IN this edition many extensive alterations have been made. 
Much obsolete matter has been cut out and fresh matter substi- 
tuted. In the first 170 pages but few changes have been found 
necessary, but a few typographical and other minor errors have 
been corrected. The tables of sizes, weight, and strength of 
materials (pages 172 to 282) have been thoroughly revised, many 
entirely new tables, kindly furnished by manufacturers, having 
been substituted. Especial attention is called to the new matter 
on Cast-iron Columns (pages 250 to 253). In the remainder of 
the book changes of importance have been made in more than 100 
pages, and all typographical errors reported to date have been 
corrected. Manufacturers' tables have been revised by reference 
to their latest catalogues or from tables furnished by the manufac- 
turers especially for this work. Much new matter is inserted 
under the heads of Fans and Blowers, Flow of Air in Pipes, and 
Compressed Air. The chapter on Wire-rope Transmission (pages 
917 to 922) has been entirely rewritten. The chapter on Electrical 
Engineering has been improved by the omission of some matter 
that has become out of date and the insertion of some new matter. 

It has been found necessary to place much of the new matter of 
this edition in an Appendix, as space could not conveniently be 
made for it in the body of the book. It has not been found possi- 
ble to make in the body of the book many of the cross-references 
which should be made to the items in the Appendix. Users of the 
book may find it advisable to write in the margin such cross-refer- 
ences as they may desire. 

The Index has been thoroughly revised and greatly enlarged. 

The author is under continued obligation to many manufacturers 
who have furnished new tables and data, and to many individual 
engineers who have furnished new matter, pointed out errors in 
the earlier editions, and offered helpful suggestions. He will be 
glad to receive similar aid, which will assist in the further 
improvement of the book in future editions. 

WILLIAM KENT. 

PASSAIC, N. J., September^ 1898. 

SIXTH EDITION. DECEMBER, 1902. 

THE chapter on Electrical Engineering has been thoroughly 
revised, much of the old matter cut out and new matter sub- 
stituted. Fourteen new pages have been devoted to the sub- 
ject of Alternating Currents. The chapter on Locomotives has 
been revised. Some new matter has been added under Cast 
Iron, Specifications for Steel, Springs, Steam-engines, and 
Friction and Lubrication. Slight changes and corrections in 
the text have been made in nearly a hundred pages. 



IV& PBEFACE. 

SEVENTH EDITION, OCTOBER 1904. 

AN entirely new index has been made, with about twice as 
many titles as the former index. The electrical engineering 
chapter has been further revised and some new matter added. 
Four pages on Coal Handling Machinery have been inserted 
at page 911, .and numerous minor changes have been made. 

W. K. 

SYRACUSE, N. Y. 



CONTENTS. 



(For Alphabetical Index see page 1093.) 

MATHEMATICS. 

Arithmetic. 

PAGE 

Arithmetical and Algebraical Signs 1 

Greatest Common Divisor 2 

Least Common Multiple 2 

Fractions 2 

Decimals 3 

Table. Decimal Equivalents of Fractions of One Inch 3 

Table. Products of Fractions expressed in Decimals 4 

Compound or Denominate Numbers 5 

Reduction Descending and Ascending 5 

Ratio and Proportion 5 

Involution, or Powers of Numbers 6 

Table. First Nine Powers of the First Nine Numbers 7 

Table. First Forty Powers of 2 7 

Evolution. Square Root 7 

CubeRoot 8 

Alligation. 10 

Permutation 10 

Combination < 10 

Arithmetical Progression 11 

Geometrical Progression 11 

Interest 13 

Discount 13 

Compound Interest 14 

Compound Interest Table, 3, 4, 5, and 6 per cent 14 

Equation of Payments 14 

Partial Payments 15 

Annuities 16 

Tables of Amount, Present Values, etc., of Annuities 16 

Weights and Measures. 

Long Measure 17 

Old Land Measure 17 

Nautical Measure 17 

Square Measure 18 

Solid or Cubic Measure 18 

Liquid Measure 18 

The Miners' Inch 18 

Apothecaries' Fluid Measure 18 

Dry Measure * 18 

Shipping Measure 19 

Avoirdupois Weight 19 

Troy Weight 19 

Apothecaries' Weight 19 

To Weigh Correctly on an Incorrect Balance 19 

Circular Measure 20 

Measure of time , 20 

V 



y: CONTENTS. 

Board and Timber Measure ] 20 

Table. Contents in Feet of Joists, Scantlings, and Timber 20 

French or Metric Measures 21 

British and French Equivalents 21 

Metric Conversion Tables 23 

Compound Units. 

of Pressure and Weight 27 

of Water, Weight, and Bulk f 27 

of Work, Power, and Duty P 27 

of Velocity 27 

of Pressure per unit area 27 

Wire and Sheet Metal Gauges , 28 

Twist-drill and Steel-wire Gauges 28 

Music- wire Gauge 29 

Circular- mil Wire Gauge 30 

NewU. S. Standard Wire and Sheet Gauge, 1893 30 

Decimal Gauge 32 

Algebra. 

Addition, Multiplication, etc 33 

Powers of Numbers 33 

Parentheses, Division 34 

Simple Equations and Problems 34 

Equations containing two or more Unknown Quantities 35 

Elimination 35 

Quadratic Equations 35 

Theory of Exponents.. 36 

Binomial Theorem.... 36 

Geometrical Problems of Construction 37 

of Straight Lines 37 

of Angles 38 

of Circles , 39 

of Triangles 41 

of Squares and Polygons 42 

oftheEllipse 45 

of the Parabola , 48 

of the Hyperbola 49 

of the Cycloid... 49 

of the Tractrix or Schiele Anti-friction Curve 50 

oftheSpiral 50 

of the Catenary . 51 

of the Involute 52 

Geometrical Propositions 53 

' Mensuration, Plane Surfaces. 

Quadrilateral, Parallelogram, etc 54 

Trapezium and Trapezoid c . . .. 54 

Triangles 54 

Polygons. Table of Polygons. . 55 

Irregular Figures 55 

Properties of the Circle 57 

Values of ir and its Multiples, etc 57 

Relations of arc, chord, etc 58 

Relations of circle to inscribed square, etc 58 

Sectors and Segments 59 

Circular Ring 59 

The Ellipse 59 

The Helix 60 

TheSpiral 60 

Mensuration, Solid Bodies. 

Prism ... 60 

Pyramid 60 

Wedge , 61 

The Prismoidal Formula > 

Rectangular Prismoid. 61 

Cylinder 61 

Cone * > > > *1 



CONTENTS. Vll 

PAGE 

Sphere 61 

Spherical Triangle J 

Spherical Polygon j 

Spherical Zone "2 

Spherical Segment 62 

Spheroid or Ellipsoid 

Polyedron 62 

Cylindrical Ring 62 

Solids of Revolution 62 

Spindles J 

Frustrum of a Spheroid " 

Parabolic Conoid t 

Volume of a Cask 64 

Irregular Solids 64 

Plane Trigonometry. 

Solution of Plane Triangles 65 

Sine, Tangent, Secant, etc t 

Signs of the Trigonometric Functions t 

Trigonometrical Formulae ,. C 

Solution of Plane Right-angled Triangles C 

Solution of Oblique-angled Triangles 68 

Analytical Geometry. 

Ordinates and Abscissas 69 

Equations of a Straight Line, In tersections, etc C 

Equations of the Circle 70 

Equations of the Ellipse . 70 

Equations of the Parabola 70 

Equations of the Hyperbola 70 

Logarithmic Curves . 71 

Differential Calculus. 

Definitions 72 

Differentials of Algebraic Functions 72 

Formulae for Differentiating 73 

Partial Differentials 73 

Integrals.. . . 73 

Formulae for Integration 74 

Integration between Limits 74 

Quadrature of a Plane Surface 74 

Quadrature of Surfaces of Revolution 75 

Cubature of Volumes of Revolution 75 

Second, Third, etc., Differentials , 75 

Maclaurin's and Taylor's Theorems 76 

Maxima and Minima.. 76 

Differential of an Exponential Function 77 

Logarithms.. 77 

Differential Forms which have Known Integrals 78 

Exponential Functions 78 

Circular Functions 78 

The Cycloid 79 

Integral Calculus 79 

Mathematical Tables. 

Reciprocals of Numbers 1 to 2000 80 

Squares, Cubes, Square Roots, and Cube Roots from 0.1 to 1600 86 

Squares and Cubes of Decimals 101 

Fifth Roots and Fifth Powers 102 

Circumferences and Areas of Circles, Diameters 1 to 1000 103 

Circumferences and Areas of Circles, Advancing by Eighths from ^ to 

100 108 

Decimals of a Foot Equivalent to Inches and Fractions of an Inch 112 

Circumferences of Circles in Feet and Inches, from 1 inch to 32 feet 11 

inches in diameter 113 

Lengths of Circular Arcs, Degrees Given 114 

Lengths of Circular Arcs, Height of Arc Given . 115 

Areas of the Segments of a Circle 116 



viii CONTENTS. 

PAGE 

Spheres 118 

Contents of Pipes and Cylinders, Cubic Feet and Gallons 120 

Cylindrical Vessels, Tanks, Cisterns, etc 121 

Gallons in a Number of Cubic Feet 122 

Cubic Feet in a Number of Gallons 122 

Square Feet in Plates 3 to 32 feet long and 1 inch wide 123 

Capacities of Rectangular Tanks in Gallons 125 

Number of Barrels in Cylindrical Cisterns and Tanks 126 

Logarithms 127 

Table of Logarithms 129 

Hyperbolic Logarithms 156 

Natural Trigonometrical Functions 159 

logarithmic Trigonometrical Functions 162 

MATEKIAL.S. 

Chemical Elements 165 

Specific Gravity and Weight of Materials 163 

Metals, Properties of 164 

The Hydrometer 165 

Aluminum 166 

Antimony 166 

Bismuth 166 

Cadmium 167 

Copper 167 

Gold .- 167 

Iridium 167 

Iron 167 

Lead 167 

Magnesium 168 

Manganese 168 

Mercury 1 68 

Nickel 168 

Platinum 168 

Silver 168 

Tin 168 

Zinc 168 

Miscellaneous Materials. 

Order of Malleability, etc., of Metals 169 

Formulae and Table for Calculating Weight of Rods, Plates, etc 169 

Measures and Weights of Various Materials 169 

Commercial Sizes of Iron Bars 170 

Weights of Iron Bars 171 

of Flat Rolled Iron 172 

of Iron and Steel Sheets 174 

of Plate Iron 175 

of Steel Blooms 176 

of Structural Shapes 177 

Sizes and Weights of Carnegie Deck Beams 177 

" Steel Channels 178 

" " ZBars 178 

" " Pencoyd Steel Angles 179 

* " " Tees... 180 

' Channels 10 

" Roofing Materials 181 

" * Terra-cotta 181 

" * Tiles 181 

" " Tin Plates ...181 

" Slates 183 

" " PineShingles 183 

' " Sky-light Glass . 184 

Weights of Various Ropf-coverings 184 

Cast-iron Pipes or Columns 185 

" " " 12- ft. lengths 186 

* " Pipe-fittings 187 

" " Water and Gas-pipe 188 

and thickness of Cast-iron Pipes 189 

Safe Pressures on Cast Iron Pipe 189 



CONTENTS. IX 

PAGE 

Sheet-iron Hydraulic Pipe.. 191 

Standard Pipe Flanges 192 

Pipe Flanges and Cast-iron Pipe 193 

Standard Sizes of W rough t-iron Pipe 194 

Wrought-iron Welded Tubes ... 196 

Riveted Iron Pipes 197 

Weight of Iron for Riveted Pipe , 197 

Spiral Riveted Pipe 198 

Seamless Brass Tubing 198, 199 

Coiled Pipes 199 

Brass, Copper, and Zinc Tubing 200 

Lead and Tin-lined Lead Pipe , 201 

Weight of Copper and Brass Wire and Plates 202 

Round Bolt Copper 203 

44 Sheet and Bar Brass 203 

Composition of Rolled Brass 203 

Sizesof Shot 204 

Screw-thread, U. S. Standard 204 

Limit-gauges for Screw-threads 205 

Size of Iron for Standard Bolts 206 

Sizes of Screw-threads for Bolts and Taps 207 

Set Screws and Tap Screws 208 

Standard Machine Screws 209 

Sizes and Weights of Nuts 209 

Weight of Bolts with Heads 210 

Track Bolts 210 

Weights of Nuts and Bolt-heads 211 

Rivets 211 

Sizes of Turnbuckles 211 

Washers 212 

Track Spikes 212 

Railway Spikes 212 

Boat Spikes 212 

Wrought Spikes 213 

Wire Spikes ... 213 

Cut Nails 213 

Wire Nails , 214, 215 

Iron Wire, Size, Strength, etc 216 

Galvanized Iron Telegraph Wire 217 

Tests of Telegraph Wire 217 

Copper Wire Table, B. W. Gauge 218 

" Edison or Circular Mil Gauge.... 219 

" 4t B.&S.Gauge 220 

Insulated Wire 221 

Copper Telegraph Wire 221 

Electric Cables 221,222 

Galvanized Steel-wire Strand 223 

Steel-wire Cables for Vessels 223 

Specifications for Galvanized Iron Wire 224 

Strength of Piano Wire 224 

Plough-steel Wire 224 

Wires of different metals 225 

Specifications for Copper Wire 225 

Cable-traction Ropes 226 

Wire Ropes 226, 227 

Plough-steel Ropes 227, 228 

Galvanized Iron Wire Rope 228 

Steel Hawsers 223, 229 

Flat Wire Ropes 2*9 

Galvanized Steel Cables ... 230 

Strength of Chains and Ropes 230 

Notes on use of Wire Rope 231 

' Locked Wire Rope 231 

Crane Chains 232 

Weights of Logs, Lumber, etc 232 

Sizes of Fire Brick 233 

Fire Clay, Analysis 234 

Magnesia Bricks 235 

Asbestos 235 



X CONTENTS. 

Strength of Materials. 

_ , . _. PAQK 

Stress and Strain 236 

ElasticLimit . 236 

Yield Point 237 

Modulus of Elasticity 237 

Resilience 238 

Elastic Limit and Ultimate Stress 238 

Repeated Stresses 238 

Repeated Shocks 240 

Stresses due to Sudden Shocks 241 

Increasing Tensile Strength of Bars by Twisting 241 

Tensile Strength 242 

Measurement of Elongation 243 

Shapes of Test Specimens 243 

Coinpressive Strength 244 

Columns, Pillars, or Struts 246 

Hodgkinson's Formula 246 

Gordon's Formula. 247 

Moment of Inertia 247 

Radius of Gyration 247 

Elements of Usual Sections 248 

Strength of Cast-iron Columns 250 

Transverse Strength of Cast iron Water-pipe 251 

Safe Load on Cast-iron Columns 252 

Strength of Brackets on Cast-iron Columns 252 

Eccentric Loading of Columns 254 

Wrought-iron Columns 255 

Built Columns . . . . 256 

Phoenix Columns 257 

Working Formulae for Struts 259 

Merriman's Formula for Columns 2GC 

Working Strains in Bridge Members 263 

Working Stresses for Steel 263 

Resistance of Hollow Cylinders to Collapse 264 

Collapsing Pressure of Tubes or Flues 265 

Formula for Corrugated Furnaces 266 

Transverse Strength 266 

Formulae for Flexure of Beams 267 

Safe Loads on Steel Beams 269 

Elastic Resilience 270 

Beams of Uniform Strength 271 

Properties of Rolled Structural Shapes 272 

" SteellBeams 273 

Spacing of Steel I Beams 276 

Properties of Steel Channels 277 

" T Shapes 278 

** ' Angles 279a 

" Zbars 280 

Size of Beams for Floors 280 

Flooring Material.. 281 

Tie Rods for Brick Arches 281 

Torsional Strength 281 

Elastic Resistance to Torsion 282 

Combined Stresses , 282 

Stress due to Temperature 283 

Strength of Flat Plates 283 

Strength of Unstayed Flat Surfaces 284 

Unbraced Heads of Boilers 285 

Thickness of Flat Cast-iron Plates ... 286 

Strength of Stayed Surfaces 286 

Spherical Shells and Domed Heads 286 

Stresses in Steel Plating under Water Pressure 287 

Thick Hollow Cylinders under Tension 287 

Thin Cylinders under Tension 289 

Hollow Copper Balls 289 

Holding Power of Nails, Spikes, Bolts, and Screws 289 

Cut versus Wire Nails 29(1 

Strength of Wrought-iron Bolts.: 293 



CONTENTS. xi 

PAGE 

Initial Strain on Bolts 292 

Stand Pipes and their Design , 292 

Riveted Steel Water-pipes 295 

Mannesmann Tubes 296 

Kirkaldy's Tests of Materials 296 

Cast Iron 296 

Iron Castings 297 

Iron Bars, Forgings, etc 297 

Steel Rails and Tires 298 

Steel Axles, Shafts, Spring Steel 299 

Riveted Joints 299 

Welds 300 

Copper, Brass, Bronze, etc 300 

Wire, Wire-rope , 301 

Ropes, Hemp, and Cotton 301 

Belting, Canvas 302 

Stones, Brick, Cement * 302 

Tensile Strength of Wire 303 

Watertown Testing-machine Tests 303 

Riveted Joints 303 

Wrought-iron Bars, Compression Tests 304 

Steel Eye-bars ^ 304 

Wrought-iron Columns ... 305 

Cold Drawn Steel 305 

American Woods 306 

Shearing Strength of Iron and Steel 306 

Holding Power of Boiler-tubes 307 

Chains, Weight, Proof Test, etc 307 

Wrought-iron Chain Cables , 308 

Strength of Glass 308 

Copper at High Temperatures 309 

Strength of Timber 309 

Expansion of Timber 311 

Shearing Strength of Woods 312 

Strength of Brick, Stone, etc . ... 312 

" Flagging 313 

" ** Lime and Cement Mortar 313 

Moduli of Elasticity of Various Materials 314 

Factors of Safety 314 

Properties of Cork 316 

Vulcanized India-rubber 316 

XylolithorWoodstone 316 

Aluminum, Properties and Uses 317 

Alloys. 

Alloys of Copper and Tin, Bronze 319 

Copper and Zinc, Brass 321 

Variation in Strength of Bronze 321 

Copper-tin-zinc Alloys. 



Liquation or Separation of Metals. 
Alloys used in Brs 



Jrass, Foundries 325 

Copper-nickel Alloys 326 

Copper-zinc-iron Alloys 326 

Tobin Bronze 326 

Phosphor Bronze 327 

Aluminum Bronze 328 

Aluminum Brass 329 

Caution as to Strength of Alloys 329 

Aluminum hardened 330 

Alloys of Aluminum, Silicon, andiron 330 

Tungsten-aluminum Alloys 331 

Aluminum-tin Alloys 331 

Manganese Alloys 331 

Manganese Bronze 331 

German Silver , 332 

Alloys of Bismuth 332 

Fusible Alloys , 333 

bearing Metal Alloys , , ,,...,.., 333 



Xll CONTENTS. 

PAGE 

Alloys containing Antimony 03 336 

White-metal Alloys 336 

Type-metal 336 

Babbitt metals 336 

Solders 338 

Ropes and Chains. 

Strength of Hemp, Iron, and Steel Ropes , 333 

FlatRopes t 339 

Working Load of Ropes and Chains ... 339 

Strength of Ropes and Chain Cables 340 

Rope for Hoisting or Transmission 340 

Cordage, Technical terms of 341 

Splicing of Ropes 341 

Coal Hoisting 343 

Manila Cordage, Weight, etc 344 

Knots, how to make ., 344 

Splicing Wire Ropes 346 

Springs. 

Laminated Steel Springs 847 

Helical Steel Springs- 347 

Carrying Capacity of Springs 349 

Elliptical Springs .. 352 

Phosphor-bronze Springs 352 

Springs to Resist Torsional Force 352 

Helical Springs for Cars, etc .1 353 

Riveted Joints. 

Fairbairn's Experiments 354 

Loss of Strength by Punching .'. 354 

Strength of Perforated Plates 354 

Hand vs. Hydraulic Riveting 355 

Formulae for Pitch of Rivets 357 

Proportions of Joints 358 

Efficiencies of Join ts 359 

Diameter of Rivets 360 

Strength of Riveted Joints 361 

Riveting Pressures 362 

Shearing Resistance of Rivet Iron 363 

Iron and Steel. 

Classification of Iron and Steel 364 

Grading of Pig Iron 365 

Influence of Silicon Sulphur, Phos. and Mn on Cast Iron 365 

Tests of Cast Iron 369 

Chemistry of Foundry Iron 370 

Analyses of Castings , 373 

Strength of Cast Iron 374 

Specifications for Cast Iron 374 

Mixture of Cast Iron with Steel 375 

Bessemerized Cast Iron 375 

Bad Cast Iron , 375 

Malleable Cast Iron 375 

Wrought Iron ; 377 

Chemistry of Wrought Iron 377 

Influence of Rolling on Wrought Iron 377 

Specifications for Wrought Iron 378 

Stay-bolt Iron 378 

Formulae for Unit Strains in Structures 379 

Permissible Stresses in Structures 381 

Proportioning Materials in Memphis Bridge 382 

Tenacity of Iron at High Temperatures 382 

Effect of Cold on Strength of Iron 383 

Expansion of Iron by Heat 385 

Durability of Cast Iron 885 

Corrosion of Iron and Steel 386 

Preservative Coatings ; Paints, etc 387 



CONTENTS Xlll 

PAGE 

Non-oxidizing Process of Annealing 387 

Manganese Plating of Iron 389 

Steel. 

Relation between Chemical and Physical Properties 389 

Variation in Strength 391 

Open-hearth 392 

Bessemer 392 

Hardening Soft Steel 393 

Effect of Cold Rolling 393 

Comparison of Full-sized and Small Pieces 393 

Treatment of Structural Steel 394 

Influence of Annealing upon Magnetic Capacity 396 

Specifications for Steel 397 

Chemical Requirements 397 

Kinds of Steel used for Different Purposes 397 

Castings, Axles, Forgings 397 

Tires, Rails, Splice-bars, Structural Steel 398 

Boiler-plate and Rivet Steel 399 

May Carbon be Burned out of Steel ? 402 

Recalescerice of Steel 402 

Effect of Nicking r. Bar 402 

Electric Conductivity 403 

Specific Gravity 403 

Occasional Failures 403 

Segregation in Ingots 404 

Earliest Uses for Structures 405 

Steel Castings 405 

Manganese Steel 407 

Nickel Steel 407 

Aluminum Steel 409 

Chrome Steel 409 

Tungsten Steel 409 

Compressed Steel 410 

Crucible Steel 410 

Effect of Heat on Grain 412 

1 ' Hammering, etc 412 

Heating and Forging 412 

Tempering Steel 413 

MECHANICS. 

Force, Unit of Force 415 

Inertia 415 

Newton's Laws of Motion 415 

Resolution of Forces 415 

Parallelogram of Forces 416 

Moment of a Force 416 

Statical Moment, Stability 417 

Stability of a Dam 417 

Parallel Forces 417 

Couples 418 

Equilibrium of Forces 418 

Centre of Gravity 418 

Moment of Inertia 419 

Centre of Gyration 420 

Radius of Gyration 420 

Centre of Oscillation 421 

Centre of Percussion 422 

The Pendulum 422 

Conical Pendulum 423 

Centrifugal Force 423 

Acceleration 423 

Falling Bodies 424 

Value of g 424 

Angular Velocity 425 

Height due to Velocity 425 

Parallelogram of Velocities 426 

Mass 427 



XIV CONTENTS. 

PAGE 

Force of Acceleration, 427 

Motion on Inclined Planes : 428 

Momentum 428 

Vis Viva 428 

Work, Foot-pound 428 

Power, Horse-power 429 

Energy 429 

Work of Acceleration ....... 430 

Force of a Blow . . , , 430 

Impact of Bodies . . . . . . .... 431 

Energy of Recoil of Guns 431 

Conservation of Energy 432 

Perpetual Motion 432 

Efficiency of a Machine 432 

Animal-power, Man-power 433 

Work of aHorse . ... 434 

Man-wheel 434 

Horse-gin 434 

Resistance of Vehicles 435 

Elements of Machines, 

The Lever 435 

The Bent Lever 436 

The Moving Strut 436 

The Toggle-joint 436 

The Inclined Plane 437 

The Wedge 437 

The Screw 437 

The Cam 438 

The Pulley 438 

Differential Pulley 439 

Differential Windlass 439 

Differential Screw 439 

Wheel and Axle 439 

Toothed-wheel Gearing 439 

Endless Screw , 440 

Stresses in Framed Structures. 

Cranes and Derricks 440 

Shear Poles and Guys 442 

King Post Truss or Bridge 442 

Queen Post Truss 442 

Burr Truss 443 

Pratt or Whip pie Truss 443 

Howe Truss 445 

Warren Girder . 445 

Roof Truss 446 

HEAT. 

Thermometers and Pyrometers 448 

Centigrade and Fahrenheit degrees compared 449 

Copper-ball Pyrometer 451 

Thermo-electric Pyrometer 451 

Temperatures in Furnaces 451 

Wiborgh Air Pyrometer , 453 

Seeger's Fire-clay Pyrometer , 453 

Mesur6 and Nouel's Pyrometer 453 

Uehling and Steinbart's Pyrometer 453 

Air-thermometer 454 

High Temperatures judged by Color.... 454 

Boiling-points of Substances 455 

Melting-points 455 

Unit of Heat 455 

Mechanical Equivalent of Heat 456 

Heat of Combustion 456 

Specific Heat 457 

Latent Heat of Fusion 459, 461 

Expansion by Heat 460 

Absolute Temperature 461 

Absolute Zero ; 461 



CONTENTS. XY 

PAGE 

Latent Heat 461 

Latent Heat of Evaporation 462 

Total Heat of Evaporation 462 

Evaporation and Drying 462 

Evaporation from Reservoirs 463 

Evaporation by the Multiple System 463 

Resistance to Boiling 463 

Manufacture of Salt 464 

Solubility of Salt and Sulphate of Lime 464 

Salt Contents of Brines 464 

Concentration of Sugar Solutions 465 

Evaporating by Exhaust Steam 465 

Drying in Vacuum 466 

Radiation of Heat 467 

Conduction and Convection of Heat 468 

Rate of External Conduction 469 

Steam-pipe Coverings . . 470 

Transmission through Plates 471 

in Condenser Tubes 473 

* ** Cast-iron Plates 474 

** from Air or Gases to Water 474 

' from Steam or Hot Water to Air 475 

' through Walls of Buildings 478 

Thermodynamics 478 

PHYSICAL PROPERTIES OF GASES. 

Expansion of Gases 479 

Boyle and Marriotte's Law.... 479 

Law of Charles, Avogadro's Law 479 

Saturation Point of Vapors 480 

Law of Gaseous Pressure , 480 

Flow of Gases 480 

Absorption by Liquids 480 

AIR. 

Properties of Air 481 

Air-manometer 481 

Pressure at Different Altitudes 481 

Barometric Pressures .... 482 

Levelling by the Barometer and by Boiling Water 482 

To find Difference in Altitude 483 

Moisture in Atmosphere 483 

Weight of Air and Mixtures of Air and Vapor 484 

Specific Heat of Air 484 

Flow of Air. 

Flow of Air through Orifices 484 

Flow of Air in Pipes 485 

Effect of Bends in Pipe 488 

Flow of Compressed Air 488 

Tables of Flow of Air 489 

Anemometer Measurements 491 

Equalization of Pipes 491 

Loss of Pressure in Pipes 493 

Wind. 

Force of the Wind 493 

Wind Pressure in Storms 495 

Windmills 495 

Capacity of Windmills 497 

Economy of Windmills 498 

Electric Power from Windmills 499 

Compressed Air. 

Heating of Air by Compression 499 

Loss of Energy in Compressed Air 499 

Volumes and Pressures...., , 500 



CONTENTS. 



Loss due to Excess of Pressure *.... 501 

Horse-power Required for Compression 501 

Table for Adiabatic Compression 502 

Mean Effective Pressures 502 

Mean and Terminal Pressures 503 

Air-compressors. 503 

Practical Results 505 

Efficiency of Compressed-air Engines 506 

Requirements of Rock-drills , 506 

Popp Compressed-air System 507 

Small Compressed-air Motors 507 

Efficiency of Air-heating Stoves 507 

Efficiency of Compressed-air Transmission 508 

Shops Operated by Compressed Air 509 

Pneumatic Postal Transmission t 509 

Mekarski Compressed-air Tramways 510 

Compressed Air Working Pumps in Mines 511 

Fans and Blowers. 

Centrifugal Fans 511 

Best Proportions of Fans 512 

Pressure due to Velocity 513 

Experiments with Blowers 514 

Quantity of Air Delivered 514 

Efficiency of Fans and Positive Blowers 516 

Capacity of Fans and Blowers 517 

Table of Centrifugal Fans 518 

Engines, Fans, and Steam-coils for the Blower System of Heating 519 

Sturtevant Steel Pressure-blower 519 

Diameter of Blast-pipes 519 

Efficiency of Fans 520 

Centrifugal Ventilators for Mines 521 

Experiments on Mine Ventilators 522 

Disk Fans 524 

Air Removed by Exhaust Wheel , 525 

Efficiency of Disk Fans 525 

Positive Rotary Blowers 526 

Blowing Engines . . . . 526 

Steam-jet Blowers <> 527 

Steam -jet for Ventilation 527 

HEATING AND VENTILATION. 

Ventilation 528 

Quantity of Air Discharged through a Ventilating Duct 530 

Artificial Cooling of Air , 531 

Mine-ventilation 531 

Friction of Air in Underground Passages 531 

Equivalent Orifices 533 

Relative Efficiency of Fans and Heated Chimneys 533 

Heating and Ventilating of Large Buildings 534 

Rules for Computing Radiating Surfaces *. 536 

Overhead Steam-pipes 537 

Indirect Heating-surface 537 

Boiler Heating-surface Required 538 

Proportion of Grate-surface to Radiator-surface 538 

Steam-consumption in Car-heating 538 

Diameters of Steam Supply Mains 539 

Registers and Cold-air Ducts 539 

Physical Properties of Steam and Condensed Water 540 

Size of Steam-pipes for Heating 540 

Heating a Greenhouse by Steam 541 

Heating a Greenhouse by Hot Water 542 

Hot- water Heating , 542 

Law of Velocity of Flow 542 

Proportions of Radiating Surfaces to Cubic Capacities 543 

Diameter of Main and Branch Pipes 543 

Rules for Hot-water Heating w 544 

Arrangements of Mains ,..,..,.... tt .,.,,..,.,...... 544 



CONTENTS. XVll 

PAGE 

Blower System of Heating and Ventilating 6 ;u 545 

Experiments with Radiators . . , 545 

Heating a Building to 70 F. ., 545 

Heating by Electricity 546 

WATER. 

Expansion of Water .... 547 

Weight of Water at different temperatures 547 

Pressure of Water due to its Weight 549 

Head Corresponding to Pressures , 549 

Buoyancy 550 

Boiling-point 550 

Freezing-point 550 

Sea-water 549,550 

Ice and Snow 550 

Specific Heat of Water 550 

Compressibility of Water 551 

Impurities of Water 551 

Causes of Incrustation 551 

Means for Preventing Incrustation 552 

Analyses of Boiler-scale 552 

Hardness of Water 553 

Purifying Feed-water 554 

Softening Hard Water 555 

Hydraulics. Flow of Water. 

Fomulae for Discharge through Orifices 555 

Flow of Water from Orifices 555 

Flow in Open and Closed Channels 557 

General Formulae for Flow , . 557 

Table Fall ofFeet per mile, etc 558 

Values of Vr for Circular Pipes 559 

Kutter's Formula 559 

Molesworth's Formula 562 

Bazin's Formula ... 563 

IV Arcy's Formula 563 

Older Formulae 564 

Velocity of Water in Open Channels 564 

Mean, Surface and Bottom Velocities 564 

Safe Bottom and Mean Velocities 565 

Resistance of Soil to Erosion 565 

Abrading and Transporting Power of Water 565 

Grade of Sewers 566 

Relations of Diameter of Pipe to Quantity discharged 566 

Flow of Water in a 20-inch Pipe 566 

Velocities in Smooth Cast-iron Water-pipes 567 

Table of Flow of Water in Circular Pipes 568-573 

Loss of Head . . 573 

Flow of Water in Riveted Pipes 574 

Fractional Heads at given rates of discharge 577 

Effect of Bend and Curves 578 

Hydraulic Grade-line 578 

Flow of Water in House-service Pipes 578 

Air-bound Pipes 579 

VerticalJets 579 

Water Delivered through Meters 579 

Fire Streams 579 

Friction Losses in Hose 580 

Head and Pressure Losses by Friction , 580 

Loss of Pressure in smooth 2^-inch Hose 580 

Rated capacity of Steam Fire-engines 580 

Pressures required to throw water through Nozzles 581 

The Siphon 581 

Measurement of Flowing Water 582 

Piezometer 582 

Pitot Tube Gauge ... . 583 

The Venturi Meter... 583 

Measurement of Discharge by means of Nozzles 584 



XV111 CONTENTS. 

PAGE 

Flow through Rectangular Orifices ....................... . ...... t ........ 584 

Measurement of an Open Stream ......................................... 584 

Miners' Inch Measurements ..... ............ ............................ 585 

Flow of Water over Weirs .............................................. 586 

Francis's Formula for Weirs ............................................. 586 

Weir Table ............................................................... 587 

Bazin's Experiments .................... .. ............................... 587 

Water-power* 
Power of a Fall of Water .................................................. 588 

Horse-power of a Running Stream ..... ., > ............................... 589 

Current Motors. ... .................... ...... . ....................... 589 

Horse-power of Water Flowing in a Tube... 7 . .......................... 589 

Maximum Efficiency of a Long Conduit .................. ............... 589 

Mill-power ............. . ............................................. ,.,.. 689 

Value of Water-power ........................... , ..................... , . 590 

The Power of Ocean Waves ................. ...... ....................... 599 

Utilization of Tidal Power ............................................... 600 

Turbine Wheels. 
Proportions of Turbines ..... ........................................ ..... 591 

Tests of Turbines ............... .......................................... 596 

Dimensions of Turbines ......... ....................................... 597 

The Pelton Water-wheel .................................................. 597 

Pumps. 
Theoretical capacity of a pump ....... . ................................. 601 

Depth of Suction .......... ............................................... 602 

Amount oi Water raised by a Single-acting Lift-pump ................... 602 

Proportioning the Steam-cylinder of a Direct-acting Pump .............. 602 

Speed of Water through Pipes and Pump -passages .................... 602 

Sizes of Direct-acting Pumps ................... ................... ----- 603 

The Deane Pump .............................. .......................... 603 

Efficiency of Small Pumps .............................. ....... .. ..... 603 

The Worthington Duplex Pump .......................................... 604 

Speed of Piston ............... ,. ......................................... 605 

Speed of Water through Valves. ........... . .......................... .. 605 

Boiler-feed Pumps ............... . ....................................... 605 

Pump Valves ........... ................................................. 606 

Centrifugal Pumps ............ ....................... .................. 606 

Lawrence Centrifugal Pumps .......................................... 607 

Efficiency of Centrifugal and Reciprocating Pumps ...................... 608 

Vanes of Centrifugal Pumps ............................................. 609 

The Centrifugal Pump used as a Suction Dredge ........................ 609 

Duty Trials of Pumping Engines... ............................... 609 

Leakage Tests of Pumps ............................................... 611 

Vacuum Pnmps .... ............ . ....................................... 612 

The Pulsometer... ...................................... ................ 612 

TheJetPump ........................ .................................. 614 

The Injector ................................................ . .............. 614 

Air-lift Pump ................................................ .............. 6i4 

The Hydraulic Ram ............................ .......................... 614 

Quantity of Water Delivered by the Hydraulic Ram . ............ , ........ 615 

Hydraulic Pressure Transmission. 
Energy of Water under Pressure .......................... . ............ 616 

Efficiency of Apparatus .................................................. 616 

Hydraulic Presses -------- ................... ........................... 617 

Hydraulic Power in London ............. ..... .......................... 617 

Hydraulic Riveting Machines ............................................. 618 

Hydraulic Forging ................. . . . ............ ....................... 618 

The Aiken Intensifler ......... , .......................... ................ 619 

Hydraulic Engine ..................................... ................... 61S 



FUEL. 



Theory of Combustion 
Total Heat of Combustion 



CONTENTS. XIX 

PACK 

Analyses of Gases of Combustion 622 

Temperature of the Fire 622 

Classification of Solid Fuel 623 

Classification of Coals 624 

Analyses of Coals 624 

Western Lignites 631 

Analyses of Foreign Coals 631 

Nixon's Navigation Coal 632 

Sampling Coal for Analyses 632 

.Relative Value of Fine Sizes 632 

Pressed Fuel... 632 

Relative Value of Steam Coals , ... 633 

Approximate Heating Value of Coals 634 

Kind of Furnace Adapted for Different Coals 635 

Downward-draught Furnaces 635 

Calorimetric Tests of American Coals 636 

E vaporati ve Power of Bituminous Coals 636 

Weathering of Coal , 637 

Coke 637 

Experiments in Coking 637 

Coal Washing 633 

Recovery of By-products in Coke manufacture 638 

Making Hard Coke 638 

Generation of Steam from the Waste Heat and Gases from Coke-ovens. 638 

Products of the Distillation of Coal 639 

Wood as Fuel 639 

Heating Value of Wood 639 

Composition of Wood 640 

Charcoal 640 

Yield of Charcoal from a Cord of Wood 641 

Consumption of Charcoal in Blast Furnaces 641 

Absorption of Water and of Gases by Charcoal 641 

Composition of Charcoals 642 

Miscellaneous Solid Fuels 642 

Dust-fuel Dust Explosions 642 

Peat or Turf 643 

Sawdust as Fuel 643 

Horse-manure as Fuel 643 

Wet Tan-bark as Fuel.... 643 

Straw as Fuel 643 

Bagasse as Fuel in Sugar Manufacture 643 

Petroleum. 

Products of Distillation 645 

Lima Petroleum 645 

Value of Petroleum as Fuel 645 

Oil vs. Coal as Fuel 646 

Fuel Gas* 

Carbon Gas 646 

Anthracite Gas , 647 

Bituminous Gas 647 

Water Gas 64& 

Producer-gas from One Ton of Coal 649 

Natural Gas in Ohio and Indiana 649 

Combustion of Producer-gas 650 

Use of Steam in Producers 650 

Gas Fuel for Small Furnaces *. 651 

Illuminating Gas, 

Coal-gas 651 

Water-gas 652 

Analyses of Water-gas and Coal gas 653 

Calorific Equivalents of Constituents 654 

Efficiency of a Water-gas Plant 654 

Space Required for a Water-gas Plant 656 

Ruel-value of Illuminating-gas 666 



XX CONTENTS. 

PAGE 

Flow of Gas in Pipes 657 

Service for Lamps 658 

STEAM. 

Temperature and Pressure > 659 

Total Heat 659 

Latent Heat of Steam 659 

Latent Heatof Volume 660 

Specific Heat of Saturated Steam 660 

Density andVolume 660 

Superheated Steam 661 

Regnault's Experiments 661 

Table of the Properties of Steam .- 662 

Flow of Steam. 

Napier's Approximate Rule .... 669 

Flow of Steam in Pipes , 669 

Loss of Pressure Due to Radiation 671 

Resistance to Flow by Bends 672 

Sizes of Steam-pipes for Stationary Engines 673 

Sizes of Steam-pipes for Marine Engines 674 

Steam Pipes. 

Bursting-tests of Copper Steam-pipes 674 

Thickness of Copper Steam-pipes.. , 675 

Reinforcing Steam-pipes , 675 

Wire-wound Steam- pipes 675 

Riveted Steel Steam-pipes.. 675 

Valves in Steam-pipes 675 

Failure of a Copper Steam-pipe 676 

The Steam Looj> 676 

Loss from an Uncovered Steam-pipe , 676 

THE STEAM BOILER. 

The Horse-power of a Steam -boiler .... 677 

Measures for Comparing the Duty of Boilers 678 

Steam-boiler Proportions 678 

Heating-surface 678 

Horse-power, Builders' Rating 679 

Grate-surface 680 

AreasofFlues 680 

Air-passages Through Grate-bars 681 

Performance of Boilers , ,...., 681 

Conditions which Secure Economy , 682 

Efficiency of a Boiler .. ...683 

Tests of Steam-boilers 685 

Boilers at the Centennial Exhibition 685 

Tests of Tubulous Boilers 686 

High Rates of Evaporation , 687 

Economy Effected by Heating the Air..., 687 

Results of Tests with Different Coals 688 

Maximum Boiler Efficiency with Cumberland Coal . , 689 

Boilers Using Waste Gases 689 

Boilers for Blast Furnaces 689 

Rules for Conducting Boiler Tests , 690 

Table of Factors of Evaporation 695 

Strength of Steam-boilers. 

Rules for Construction... 700 

Shell-plate Formulae 701 

Rules for Flat Plates 701 

Furnace Formulae 702 

Material for Stays 703 

Loads allowed on Stays 703 

Girders 703 

Rules for Construction of Boilers in Merchant Vessels iu U. 8 705 



CONTENTS. Xxi 

PAGB 

U. S. Rule for Allowable Pressures 706 

Safe-working Pressures 707 

Rules Governing Inspection of Boilers in Philadelphia 708 

Flues and Tubes for Steam Boilers 709 

Flat-stayed Surfaces 4. 709 

Diameter of Stay-bolts. 710 

Strength of Stays 710 

Stay-bolts in Curved Surfaces . 710 

Boiler Attachments, Furnaces, etc* 

Fusible Plugs 710 

Steam Domes 711 

Height of Furnace. 711 

Mechanical Stokers 711 

The Hawley Down-draught Furnace 712 

Under-feed Stokers 712 

Smoke Prevention 712 

Gas-fired Steam-boilers 714 

Forced Combustion .....i 714 

Fuel Economizers. 715 

Incrustation and Scale 716 

Boiler-scale Compounds.. 717 

Removal of Hard Scale 718 

Corrosion in Marine Boilers 719 

TJseofZinc 720 

Effect of Deposit on Flues 720 

Dangerous Boilers 720 

Safety Valves. 

Rules for Area of Safety-valves ..... 721 

Spring-loaded Safety-valves 724 

The Injector. 

Equation of the In jector 725 

Performance of Injectors ............. 726 

Boiler-feeding Pumps 726 

Feed-water Heaters. 

Strains Caused by Cold Feed-water 727 

Steam Separators* 

Efficiency of Steam Separators 728 

Determination of Moisture in Steam* 

Coil Calorimeter. 729 

Throttling Calorimeters 729 

Separating Calorimeters 730 

Identification of Dry Steam 730 

Usual Amount of Moisture in Steam 731 

Chimneys* 

Chimney Draught Theory 731 

Force or Intensity of Draught. . 732 

Rate of Combustion Due to Height of Chimney 733 

High Chimneys not Necessary 734 

Heights of Chimneys Required for Different Fuels 734 

Table of Size of Chimneys 734 

Protection of Chimney from Lightning 736 

Some Tall Brick Chimneys 737 

Stability of Chimneys . 738 

Weak Chimneys 739 

Steel Chimneys 740 

Sheet-iron Chimneys 741 

THE STEAM ENGINE* 

Expansion of Steam , 742 

Mean and Terminal Absolute Pressures 743 



CONTENTS. 
Condensers, Air-pumps, Circulating-pumps, etc. 

PAGE 

The Jet Condenser 839 

Ejector Condensers 840 

The Surface Condenser 840 

Condenser Tubes 840 

Tube-plates 841 

Spacing of Tubes 841 

Quantity of Cooling Water 841 

Air-pump 841 

Area through Valve-seats 842 

Circulating-pump. . . . 843 

Feed-pumps for Marine Engines 843 

An Evaporative Surface Condenser 844 

Continuous Use of Condensing Water 844 

Increase of Power by Condensers 846 

Evaporators and Distillers 817 

GAS, PETROLEUM, AND 1IOT-AIU ENGINES. 

Gas-engines 847 

Efficiency of the Gas-engine 848 

Tests of the Simplex Gas-engine 848 

A 320-H.P. Gas-engine 848 

Test of an Otto Gas-engine 849 

Temperatures and Pressures Developed 849 

Test of the Clerk Gas-engine 849 

Combustion of the Gas in the Otto Engine 849 

Use of Carburetted Air in Gas-engines 849 

The Otto Gasoline-engine . 850 

The Priestman Petroleum-engine 850 

Test of a 5-H.P. Priestman Petroleum-engine 850 

Naphtha-engines 851 

Hot-air or Caloric Engines 851 

Test of a Hot-air Engine 851 

LOCOMOTIVES. 

Resistance of Trains 851 

Inertia and Resistance at Increasing Speeds 853 

Efficiency of the Mechanism of a Locomotive 854 

Size of Locomotive Cylinders 855 

Size of Locomotive Boilers 855 

Qualities Essential for a Free-steaming Locomotive 855 

Wootten's Locomotive 855 

Grate -surf ace, Smokestacks, and Exhaust-nozzles for Locomotives .... 856 

Exhaust Nozzles 856 

Fire-brick Arches 857 

Size. Weight, Tractive Power, etc 857 

Leading American Types 858 

Steam Distribution for High Speed 858 

Speed of Railway Trains 859 

Formulae for Curves 859a 

Performance of a High-speed Locomotive 859a 

Locomotive Link-motion 859a 

Dimensions of Some American Locomotives 859-862 

Indicated Water Consumption 862 

Locomotive Testing Apparatus 863 

Waste of Fuel in Locomotives 863 

Advantages of Compounding 863 

Counterbalancing Locomotives 864 

Maximum Safe Load on Steel Rails 865 

Narrow-gauge Railways 865 

Petroleum -burning Locomotives 865 

Fireless Locomotives 866 

SHAFTING. 

Diameters to Resist Torsional Strain 867 

Deflection of Shafting 868 

Horse-power Transmitted by Shafting 869 

Table for Laying Out Shafting 871 



CONTEXTS. XXV 

PULLETS. 

PAGE 

Proportions of Pulleys 873 

Convexity of Pulleys 874 

Cone or Step Pulleys 874 

BELTING. 

Theory of Belts and Bands 876 

Centrifugal Tension 876 

Belting Practice, Formulae for Belting 877 

Horse-power of a Belt one mch wide 878 

A. F. Nagle's Formula : 878 

Width of Belt for Given Horse-power 879 

Taylor's Rules for Belting 880 

Notes on Belting 882 

Lacing of Belts 883 

Setting a Belt on Quarter-twist 883 

To Find the Length of Belt 884 

To Find the Angle of the Arc of Contact 884 

To Find the Length of Belt when Closely Rolled 884 

To Find the Approximate Weight of Belts 884 

Relations of the Size and Speeds of Driving and Driven Pulleys 884 

Evils of Tight Belts 885 

Sag of Belts 885 

Arrangements of Belts and Pulleys 885 

Care of Belts 886 

Strength of Belting 886 

Adhesion, Independent of Diameter 886 

Endless Belts 886 

Belt Data 886 

Belt Dressing 887 

Cement for Cloth or Leather 887 

Rubber Belting 887 

GEARING. 

Pitch, Pitch-circle, etc 887 

Diametral and Circular Pitch 888 

Chordal Pitch 889 

Diameter of Pitch-line of Wheels from 10 to 100 Teeth 889 

Proportions of Teeth 889 

Proportion of Gear-wheels 891 

Width of Teeth 891 

Rules for Calculating the Speed of Gears and Pulleys 891 

Milling Cutters for Interchangeable Gears 892 

Forms of the Teeth. 

The Cycloidal Tooth 892 

The Involute Tooth 894 

Approximation by Circular Arcs 896 

Stepped Gears 897 

Twisted Teeth 897 

Spiral Gears 897 

Worm Gearing 897 

Teeth of Bevel-wheels. 898 

Annular and Differential Gearing 898 

Efficiency of Gearing 899 

Strength of Gear Teeth. 

Various Formulas for Strength 900 

Comparison of Formulae 903 

Maximum Speed of Gearing 905 

A Heavy Machine-cut Spur-gear 905 

Fractional Gearing 905 

Frictional Grooved Gearing 906 

HOISTING. 

Weight and Strength of Cordage 906 

Working Strength of Blocks 906 



XXVI CONTENTS. 



PAGE 

Efficiency of Chain-blocks 907 

Proportions of Hooks 907 

Power of Hoisting Engines. 908 

Effect of Slack Rope on Strain in Hoisting 908 

Limit of Depth for Hoisting ' 908 

Large Hoisting Records 908 

Pneumatic Hoisting 909 

Counterbalancing of Winding-engines 909 

Cranes. 

Classification of Cranes ."-. 911 

Position of the Inclined Brace in a Jib Crane 912 

A Large Travelling-crane ; 912 

A 150-ton Pillar Crane 912 

Compressed-air Travelling Cranes .' 912 

Coal-handling Machinery. 

Weight of Overhead Bins 912a 

Supply-pipes from Bins 912a 

Types of Coal Elevators 912a 

Combined Elevators and Conveyors ; 912a 

Coal Conveyors 912a 

Weight of Chain 9126 

Weight of Flights 912c 

Horse-power of Conveyors 912c 

Bucket Conveyors 912c 

Screw Conveyors 912d 

Belt Conveyors Ql2d 

Capacity of Belt Conveyors 9l2d 

Wire-rope Haulage. 

Self-acting Inclined Plane 913 

Simple Engine Plane 913 

Tail-rope System 913 

Endless Rope System 914 

Wire-rope Tramways ^ 914 

Suspension Cableways and Cable Hoists 915 

Stress in Hoisting-ropes on Inclined Planes 915 

Tension Required to Prevent Wire Slipping on Drums 916 

Taper Ropes of Uniform Tensile Strength 916 

Effect of Various Sized Drums on the Life of Wire Ropes 917 

WIRE-ROPE TRANSMISSION. 

Elastic Limit of Wire Ropes 917 

Bending Stresses of Wire Ropes - 918 

Horse-power Transmitted 919 

Diameters of Minimum Sheaves 919 

Deflections of the Rope 920 

Long-distance Transmission 921 

ROPE DRIVING. 

Formulae for Rope Driving 922 

Horse-power of Transmission at Various Speeds 924 

Sag of the Rope Between Pulleys 925 

Tension on the Slack Part of the Rope 925 

Miscellaneous Notes on Rope-driving 926 

FRICTION AND LUBRICATION. 

Coefficient of Friction 928 

Rolling Friction 928 

Friction of Solids 928 

Friction of Rest . . 928 

Laws of Unlubricated Friction 928 

Friction of Sliding Steel Tires 928 

Coefficient of Rolling Friction 929 

Laws of Fluid Friction 929 

Angles of Repose 929 



CONTENTS. XXV11 



PAGE 

Friction of Motion 929 

Coefficient of Friction of Journal 930 

Experiments on Friction of a Journal 931 

Coefficients of Friction of Journal with Oil Bath 932 

Coefficients of Friction of Motion and of Rest 932 

Value of Anti-friction Metals 932 

Cast-iron for Bearings 933 

Friction of Metal Under Steam-pressure 933 

Morin's Laws of Friction . . . 933 

Laws of Friction of well-lubricated Journals 934 

Allowable Pressures on Bearing-surface 935 

Oil-pressure in a Bearing 937 

Friction of Car-journal Brasses 937 

Experiments on Overheating of Bearings 938 

Moment of Friction and Work of Friction 938 

Pivot Bearings 939 

The Schiele Curve. ^ 939 

Friction of a Flat Pivot-bearing 939 

Mercury-bath Pivot 940 

Ball Bearings 940 

Friction Rollers. . . 940 

Bearings for Very High Rotative Speed 941 

Friction of Steam-engines 941 

Distribution of the Friction of Engines 941 

Lubrication. 

Durability of Lubricants. 942 

Qualifications of Lubricants 943 

Amount of Oil to run an Engine 943 

Examination of Oils 943 

Penna. R. R. Specifications 944 

Soda Mixture for Machine Tools 945 

Solid Lubricants , 945 

Graphite, Soapstone, Fibre-graphite, Metaline 945 

THE FOUNDRY. 

Cupola Practice 946 

Charging a Cupola > 948 

Charges in Stove Foundries 949 

Results of Increased Driving 949 

Pressure Blowers 950 

Loss of Iron in Melting 950 

Use of Softeners. . 950 

Shrinkage of Castings 951 

Weight of Castings from Weight of Pattern 952 

Moulding Sand 952 

Foundry Ladles 952 

THE MACHINE SHOP. 

Speed of Cutting Tools 953 

Table of Cutting Speeds 954 

Speed of Turret Lathes 954 

Forms of Cutting Tools 955 

Rule for Gearing Lathes -. 955 

Change-gears for Lathes 956 

Metric Screw-threads 956 

Setting the Taper in a Lathe 956 

Speed of Drilling Holes 956 

Speed of Twist-drills 957 

Milling Cutters 957 

Speed of Cutters 958 

Results with Milling-machines 959 

Milling with or Against Feed 960 

Milling-machine vs. Planer 960 

Power Required for Machine Tools 960 

Heavy Work on a Planer 960 

Horse-power to run Lathes 961 



XXX CONTENTS. 

Electrical Resistance. 

Laws of Electrical Resistance 1027 

Electrical Conductivity of Different Metals and Alloys 1028 

Conductors and Insulators 1028 

Resistance Varies with Temperature 1028 

Annealing 1029 

Standard of Resistance of Copper Wire 1029 

Direct Electric Currents. 

Ohm's Law 1029 

Series and Parallel or Multiple Circuits 1030 

Resistance of Conductors in Series and Parallel 1030 

Internal Resistance 1031 

Electrical, Indicated, and Brake Horse-power 1031 

Power of the Circuit 1031 

Heat Generated by a Current 1031 

Heating of Conductors 1032 

Fusion of Wires 1032 

Heating of Coils 1032 

Allowable Carrying Capacity of Copper Wires 1033 

Underwriters' Insulation 1033 

Copper-wire Table 1034, 1035 

Electric Transmission, Direct Currents. 

Section of Wire Required for a Given Current 1033 

Weight of Copper for a Given Power 1036 

Short-circuiting 1036 

Economy of Electric Transmission 1036 

Wire Table for 110, 220, 500, 1000, and 2000 volt Circuits 1037 

Efficiency of Long-distance Transmission 1038 

Table of Electrical Horse-powers 1039 

Cost of Copper for Long-distance Transmission 1040 

Systems of Electrical Distribution 1041 

Electric Lighting. 

Arc Lights 1042 

Incandescent Lamps 1042 

Variation in Candle-power and Life 1042 

Specifications for Lamps 1043 

Special Lamps 1043 

Nernst Lamp 1043 

Electric Welding 1044 

Electric Heaters 1044 

Electric Accumulators or Storage-batteries. 

Description of Storage-batteries 1045 

Sizes and Weights of Storage-batteries 1048 

General Rules for Storage-cells 1048 

Electrolysis 1048 

Electro-chemical Equivalents 1049 

Efficiency of a Storage-cell 1048 

Electro-magnets. 

Units of Electro-magnetic Measurements 1050 

Lines of Loops of Force 1050 

The magnetic Circuit 1051 

Permeability 1052 

Tractive or Lifting Force of a Magnet 1053 

Magnet Windings 1053 

Determining the Polarity of Electro-magnets 1054 

Determining the Direction of a Current 1054 

Dynamo-electric Machines. 

Kinds of Dynamo-electric Machines as regards Manner of Winding. . . 1055 

Moving Force of a Dynamo-electric Machine 1055 

Torque of an Armature : . 1056 

Electro-motive Force of the Armature Circuit 1056 

Strength of the Magnetic Field 1057 

Dynamo Design 1058 



COKTEOTS. 
Alternating Currents. 

PAGE 

Maximum, Average, and Effective Values 1061 

Frequency 1061 

Inductance, Capacity, Power Factor 1062 

Reactance, Impedance, Admittance 1063 

Skin Effect Factors 1063 

Ohm's Law Applied to Alternating Currents 1064 

Impedance Polygons 1066 

Capacity of Conductors 1066 

Self-inductance of Lines and Circuits 1066 

Capacity of Conductors 1067 

Single-phase and Polyphase Currents 1068 

Measurement of Power in Polyphase Circuits 1069 

Alternating-current Generators 1070 

Transformers, Converters, etc 1070 

Synchronous Motors 1071 

Induction Motors 1072 

Calculation of Alternating-current Circuits 1072 

Weight of Copper Required in Different Systems. .' 1074 

Electrical Machinery. 

Direct-current Generators and Motors 1074-1076 

Alternating-current Generators 1077 

Induction Motors 1077 

Symbols Used in Electrical Diagrams 1078 

APPENDIX. 

Strength of Timber. 

Safe Load on White-oak Beams 1079 

Mathematics. 

Formula for Interpolation 1080 

Maxima and Minima without the Calculus 1080 

Riveted Joints. 

Pressure Required to Drive Hot Rivets 1080 

Heating and Ventilation. 

Capacities for Hot-blast or Plenum Heating with Fans and Blowers. . 1081 
Water-wheels. 

Water-power Plants Operating under High Pressure 1G81 

Formulae for Power of Jet Water-wheels 1082 

Gas Fuel. 

Composition Energy, etc., of Various Gases 1082 

Steam-boilers. 

Rules for Steam-boiler Construction 1083 

Boiler Feeding 1083 

Feed-water Heaters 1083 

The Steam-engine. 

Current Practice in Engine Proportions 1084 

Work of Steam-turbines 1085 

Relative Cost of Different Sizes of Engines 1085 

Gearing:. 

Efficiency of Worm Gearing 1086 

Hydraulic Formulae. 
Flow of Water from Orifices, etc 1087 

Tin and Terne Plate. 

Penna. R. R. Co.'s Specifications 1088 

LIST OF AUTHORITIES 1089 



NAMES AND ABBREVIATIONS OF PERIODICALS 
AND TEXT-BOOKS FREQUENTLY REFERRED TO 
IN THIS WORK. 



Am. Mach. American Machinist. 

App. Cyl. Mech. Appleton's Cyclopaedia of Mechanics, Vols. I and n. 

Bull. I. & S. A. Bulletin of the American Iron and Steel Association 
(Philadelphia). 

Burr's Elasticity and Resistance of Materials. 

Clark, E. T. D. D. K. Clark's Rules, Tables, and Data for Mechanical En- 
gineers. 

Clark, S. E. D. K. Clark's Treatise on the Steam-engine. 

Col. Coll. Qly. Columbia College Quarterly. 

Eugg. Engineering (London). 

Eng. News. Engineering News. 

Engr. The Engineer (London). 

Fairbairn's Useful Information for Engineers. 

Flynn's Irrigation Canals and Flow of Water. 

Jour. A. C. I. W. Journal of American Charcoal Iron Workers' Association. 

Jour. F. I. Journal of the Franklin Institute. 

Kapp's Electric Transmission of Energy. 

Lanza's Applied Mechanics. 

Merriman's Strength of Materials. 

Modern Mechanism. Supplementary volume of Appleton's Cyclopaedia of 
Mechanics. 

Proc. Inst. C. E. Proceedings Institution of Civil Engineers (London). 

Proc. Inst. M. E. Proceedings Institution of Mechanical Engineers (Lon- 
don). 

Peabody's Thermodynamics. 

Proceedings Engineers' Club of Philadelphia. 

Rankine, S. E. Rankine's The Steam Engine and other Prime Movers. 

Rankine's Machinery and Millwork. 

Rankine, R. T. D. Rankine's Rules, Tables, and Data. 

Reports of U. S. Test Board. 

Reports of U. S. Testing Machine at Watertown, Massachusetts. 

Rontgen's Thermodynamics, 

Seaton's Manual of Marine Engineering. 

Hamilton Smith, Jr.'s Hydraulics. 

The Stevens Indicator. 

Thompson's Dynamo-electric Machinery. 

Thurston's Manual of the Steam Engine. 

Thurstou's Materials of Engineering. 

Trans. A. I. E. E. Transactions American Institute of Electrical Engineers. 

Trans. A. I. M. E. Transactions American Institute of Mining Engineers. 

Trans. A. S. C. E. Transactions American Society of Civil Engineers. 

Trans. A. S. M. E. Transactions American Soc'ty of Mechanical Engineers 

Trautwine's Civil Engineer's Pocket Book. 

The Locomotive (Hartford, Connecticut). 

Unwin's Elements of Machine Design. 

Weisbach's Mechanics of Engineering. 

Wood's Resistance of Materials. 

Wood's Thermodynamics. 

xxzii 



MATHEMATICS. 



a Alpha 
/3 Beta 
y Gamma 
6 Delta 

e Epsilon 
Zeta 


H 

I 
K 
A 
M 



Eta 
Theta 
Iota 
Kappa 
Lambda 
Mu 


N v 
H f 
O -o 

n TT 
p p 

2 <r * 


Nu 
Xi 
Omicron 
Pi 
Rho 
Sigma 


T 
Y 
* 
X 

* 
n 


T 
V 

X 
w 


Tau 
Upsilon 
P& 
Chi 
Psi 
Omega 



Greek Letters. 

B 

r 
A 

E 
Z 

Arithmetical and Algebraical Signs and Abbreviation*. 

angle. 

L right angle. 

perpendicular to. 
sin., sine, 
cos., cosine, 
tang., or tan., tangent, 
sec., secant, 
versin., versed sine, 
cot., cotangent. 
cosec., cosecant, 
covers., co- versed sine. 

In Algebra, the first letters of the 
alphabet, a, 6, c, d, etc., are gener- 
ally used to denote known quantities, 
and the last letters, w, x, y, z, etc., 
imknown quantities. 

Abbreviations and Symbols com- 
monly used. 

d, differential (in calculus). 
/, integral (in calculus). 

J *, integral between limits a and 6. 

A, delta, difference. 

2. sigma, sign of summation. 

IT, pi, ratio of circumference of circle 

to diameter = 3.14159. 
g, acceleration due to gravity = 32.16 

ft. per sec. per sec. 

Abbreviations frequently used in 

this Book. 

L., 1., length in feet and inches. 
B., b., breadth in feet and inches. 
D., d., depth or diameter. 
H., h., height, feet and inches. 
T., t., thickness or temperature. 
V.,v., velocity. 
F., force, or factor of safety. 
f., coefficient of friction. 
E M coefficient of elasticity. 
R., r., radius. 
W., w., weight. 
P., p., pressure or load. 
H.P., horse-power. 
I.H.P., indicated horse-power. 
B.H.P., brake horse-power, 
h. p., hif 1 - 



4- plus (addition). 

4- positive. 

- minus (subtraction). 

negative. 

i: plus or minus. 

=F minus or plus. 

z= equals. 

x multiplied by. 

ab or a.b = a x b. 

^- divided by 

/ divided by. 

- = a/6 = a -f- b. 15-16 = 

.2 = , .002 =^. 

V* square root. 

V cube root. 

V 4th root. 

: is to, :: so is, : to (proportion). 

2 : 4 x 3 : 6, as 2 is to 4 so is 3 to 6. 

: ratio; divided by. 

2 : 4, ratio of 2 to 4 = 2/4. 
/. therefore. 
> greater than. 
< less than, 
n square. 
O round. 

degrees, arc or thermometer. 

' minutes or feet. 

11 seconds or inches. 
' " '" accents to distinguish letters, as 

a', a", a'". 

i 2< 3< ;,' <V r ad a sub 1, a sub 6, 
etc. 

( ) C ] { } vincula, denoting 

that the numbers enclosed are 
to be taken together ; as, 
(a -f 6)c = 4 + 3 x 5 = 35. 

a 8 , a 8 , a squared, a cubed. 

a n , a raised to the_nth power. 

a3 = |/2 ? a f = |/ a 3. 

a-* = -,a-2 = -L 
a a a 

10 = 10 to the 9th power = 1,000 000 - 

000. 

sin. a = the sine of a. 
sin. J a= the arc whose sine is a. 

sin. a-* = -; 

sin. a. 

log. = logarithm. 

log. or hyp. log. = hyperbolic loga- 
rithm. 



. 

. p., high pressure. 
. p., intermediate 



pressure. 

1. p., low pressure. 
A.W. G., American Wire Gauge 

(Brown & Sharpe). 
B.W.G., Birmingham Wire Gauge. 
r. p. m., or revs, permin., revolutions 
per minute. 



MATHEMATICS. 



ARITHMETIC. 

The user of this book is supposed to have had a training in arithmetic as 
well as in elementary algebra. Only those rules are given here which are 
apt to be easily forgotten. 

GREATEST COMMON MEASURE. OR GREATEST 
COMMON DIVISOR OF TWO NUMBERS. 

Rule. Divide the greater number by the less ; then divide the divisor 
by the remainder, and so on, dividing always the last divisor by the last 
remainder, until there is no remainder, and the last divisor is the greatest 
common measure required. 

LEAST COMMON MULTIPLE OF TWO OR MORE 
NUMBERS. 

Rule. Divide the given numbers by any number that will divide the 
greatest number of them without a remainder, and set the quotients with 
the undivided numbers in a line beneath. 

Divide the second line as before, and so on, until there are no two numbers 
that can be divided ; then the continued product of the divisors and last 
quotients will give the multiple required. 

FRACTIONS. 

To reduce a common fraction to its lowest terms. Divide 
both terms by their greatest common divisor: 39/52 = 3/4. 
To change an improper fraction to a mixed number. 

Divide the numerator by the denominator; the quotient is the whole number, 
and the remainder placed over the denominator is the fraction: 39/4 = 9%. 

To change a mixed number to an improper fraction. 
Multiply the whole number by the denominator of the fraction; to the prod- 
uct add the numerator; place the sum over the denominator: 1% = 15/8. 

To express a whole number in the form of a fraction 
with a given denominator. Multiply the whole number by the 
given denominator, and place the product over that denominator: 13 = 39/3. 

To reduce, a compound to a simple fraction, also to 
multiply fractions. Multiply the numerators together for a new 
numerator and the denominators together for a new denominator: 



To reduce a complex to a simple fraction. The numerator 
and denominator must each first be given the form of a simple fraction; 
then multiply the numerator of the upper fraction by the denominator of 
the lower for the new numerator, and the denominator of the upper by the 
numerator of the lower for the new denominator: 



To divide fractions. Reduce both to the form of simple fractions, 
invert the divisor, and proceed as in multication: 

3 3 5 3 v 4 12 3 

_^ 1M= _ 4 ._ = _ X _ = _ == _. 

Cancellation of fractions. In compound or multiplied fractions, 
divide any numerator and any denominator by any number which will 
divide them both without remainder, striking out the numbers thus divided 
and setting down the quotients in their stead. 

To reduce fractions to a common denominator. Reduce 
each fraction to the form of a simple fraction ; then multiply each numera- 



DECIMALS. 



tor by all the denominators except its own for the new numerators, and all 
the denominators together for the common denominator: 

1 1 3_21 14 18 
2' 3' 7 " 42' 42' 42* 

To add fractions. Reduce them to a common denominator, then 
add the numerators and place their sum over the common denominator: 



_ 

2 ,3 ' 7 



__ 

42 ~ 42 ~~ 



To subtract fractions. Reduce them to a common denominator, 
subtract the numerators and place the difference over the common denomi- 
nator: 

1 3_7-6_ 1 

2 7~ 14 ~14* 

DECIMALS. 

To add decimals. Set down the figures so that the decimal points 
are one above the other, then proceed as in simple addition: 18.75+ .012 = 
18.762. 

To subtract decimals. Set down the figures so that the decimal 
points are one above the other, then proceed as in simple subtraction: 18.75 
- .012 = 18.738. 

To multiply decimals. Multiply as in multiplication of whole 
numbers, then point off as many decimal places as there are in multiplier 
and multiplicand taken together: 1.5 X .02 = .030 = .03. 

To divide decimals. Divide as in whole numbers, and point off in 
the quotient as many decimal places as those in the dividend exceed those 
in the divisor. Ciphers must be added to the dividend to make its decimal 
places at least equal those in the divisor, and as many more as it is desired 
to have in the quotient: 1.5 -*- .25 = 6. 0.1 -f- 0.3 = 0.10000 *- 0.3 = 0.3333 -f- 

Decimal Equivalents of Fractions of One Incb. 



1-64 


.015625 


17-64 


.265625 


33-64 


.515625 


49-64 


.765625 


1-32 


.03125 


9-32 


.28125 


17-32 


.53125 


25-32 


.78125 


3-64 


.046875 


19-64 


.296875 


35-84 


.546875 


51-64 


.796875 


1-16 


.0625 


5-16 


.3125 


9-16 


.5625 


13-16 


.8125 


5-64 


.078125 


21-64 


.328125 


37-64 


.578125 


53-64 


.828125 


3-32 


.09375 


11-32 


.34375 


19-32 


.59375 


27-32 


.84375 


7-64 


.109375 


23-64 


.359375 


39-64 


.609375 


55-64 


.859375 


1-8 


.125 


3-8 


.375 


5-8 


.625 


7-8 


.875 


9-64 


.140625 


25-64 


.390625 


41-64 


.640625 


57-64 


.890625 


5-32 


.15625 


13-32 


.40625 


21-32 


.65625 


29-32 


.90625 


11-64 
3-16 


.171875 
.1875 


27-64 
7-16 


.421875 
.4375 


43-64 
11-16 


.671875 
.6875 


59-64 
15-16 


.921875 
.9375 


13-64 


.203125 


29-64 


.453125 


45-64 


.703125 


61-64 


.953125 


7-32 


.21875 


15-32 


.46875 


23-32 


.71875 


31-32 


.96875 


15-64 


.234375 


31-64 


.484375 


47-64 


.734375 


63-64 


.984375 


1-4 


.25 


1-2 


.50 


3-4 


.75 


1 


1. 



To convert a common fraction into a decimal. Divide the 

numerator by the denominator, adding to che numerator as many ciphers 
prefixed by a decimal point as are necessary to give the number of' decimal 
places desired in the result: % = 1.0000 -=-3 = 0.3333 -f. 

To convert a decimal into a common fraction. Set down 
the decimal as a numerator, and place as the denominator 1 with as many 
ciphers annexed as there are decimal places in the numerator; erase the 



0*0 



fe 



joo 



ARITHMETIC. 



TH O T- 

1> ~ GO 



g 



CO t- i-i 



s s 



C CO CO 

to 10 rf< 



CO CO 

CO -^1 



8 g g 

O O O O 



CO 00 



COMPOUKD NUMBERS. 5 

decimal point in the numerator, and reduce the fraction thus formed to It* 
lowest terms: 



To reduce a recurring decimal to a common fraction. 

Subtract the decimal figures that do not recur from the whole decimal in- 
cluding one set of recurring figures; set down the remainder as the numer- 
ator of the fraction, and as many nines as there are recurring figures, fol- 
lowed by as many ciphers as there are non-recurring figures, in the denom- 
inator. Thus: 

.79054054, the recurring figures being 054. 
Subtract 79 

J17 

= (reduced to its lowest terms) ^ 

COMPOUND OR DENOMINATE NUMBERS. 

Reduction descending. To reduce a compound number to a lower 
denomination. Multiply the number by as many units of the lower denomi- 
nation as makes one of the higher. 

3 yards to inches: 3 X 36 = 108 inches. 
.04 square feet to square inches: .04 X 144 = 5.76 sq. in. 

If the given number is in more than one denomination proceed in steps 
from the highest denomination to the next lower, and so on to the lowest, 
adding in the units of each denomination as the operation proceeds. 

3 yds. 1 ft. 7 in. to inches: 3x3 = 9, -f 1 = 10, 10 X 12 = 120, -f 7 = 127 in. 

Reduction ascending* To express a number of a lower denomi- 
nation in terms of a higher, divide the number by the numb r of units of 
the lower denomination contained in one of the next higher; the quotient is 
in the higher denomination, and the remainder, if any, in the lower. 

127 inches to higher denomination. 

127 -*- 12 = 10 feet + 7 inches ; 10 feet -*- 3 = 3 yards + 1 foot. 

Ans. 3 yds. 1 ft. 7 in. 

To express the result in decimals of the higher denomination, divide the 
given number by the number of units of the given denomination contained 
in one of the required denomination, carrying the result to as many places 
of decimals as may be desired. 

127 inches to yards: 127 -*- 36 = 3f = 3.5277 -f yards. 
RATIO AND PROPORTION. 

Ratio is the relation of one number to another, as obtained by dividing 
one by the other. 

Ratio of 2 to 4, or 2 : 4 = 2/4 = 1/2. 
Ratio of 4 to 2, or 4 : 2 = 2. 

Proportion is the equality of two ratios. Ratio of 2 to 4 equals ratio 
of 3 to 6, 2/4 = 3/6; expressed thus, 2 : 4 : : 3 : 6; read, 2 is to 4 as 3 is to 6. 

The first and fourth terms are called the extremes or outer terms, the 
second and third the means or inner terms. 

The product of the means equals the product of the extremes: 

2 : 4 : : 3 : 6; 2 X 6 = 12; 3 X 4 = 12. 

Hence, given the first three terms to find the fourth, multiply the second 
and third terms together and divide by the first. 

4 v 3 
2 : 4 : : 3 : what number ? Ans, = 6, 



6 ARITHMETIC. 

Algebraic expression of proportion. a : b : : c : d; =- = %;ad 

be . be . ad ad 

= be; from which a ; d ; & = ; c = -= . 
d a c b 

Mean proportional between two given numbers, 1st and 2d, is such 
a number that the ratio which the first bears to it equals the ratio which it 
bears to the second. Thus, 2 : 4 : : 4 : 8; 4 is a mean proportional between 
2 and 8. To find the mean proportional between two numbers, extract the 
square root of their product. 

Mean proportional of 2 and 8 = V% x 8 = 4. 

Single Rule of Three ; or, finding the fourth term of a proportion 
when three terms are given. Rule, as above, when the terms are stated in 
their proper order, multiply the second by the third and divide by the first. 
The difficulty is to state the terms in their proper order. The term which is 
of the same kind as the required or fourth term is made the third; the first 
and second must be like each other in kind and denomination. To deter- 
mine which is to be made second and which first requires a little reasoning. 
If an inspection of the problem shows that the answer should be greater 
than the third term, then the greater of the other two given terms should 
be made the second term otherwise the first. Thus, 3 men remove 54 cubic 
feet of rock in a day; how many men will remove in the same time 10 cubic 
yards ? The answer is to be men make men third term; the answer is to 
be more than three men, therefore make the greater quantity, 10 cubic 
yards, the second term ; but as it is not the same denomination as the other 
term it must be reduced, = 270 cubic feet. The proportion is then stated: 

3 X 270 
54 : 270 : : 3 : x (the required number) ; x = = 15 men. 

The problem is more complicated if we increase the number of given 
terms. Thus, in the above question, substitute for the words " in the same 
time " the words " in 3 days." First solve it as above, as if the work were 
to be done in the same time; then make another proportion, stating it thus: 
If 15 men do it in the same time, it will take fewer men to do it in 3 days; 
make 1 day the 2d term and 3 days the first term 3:1 : : 15 men : 5 men. 

Compound Proportion, or Double Rule of Three. By this 
rule are solved questions like the one just given, in which two or more stat- 
ings are required by the single rule of three. In it as in the single rule, 
there is one third term, which is of the same kind and denomination as the 
fourth or required term, but there may be two or more first and second 
terms. Set down the third term, take each pair of terms of the same kind 
separately, and arrange them as first and second by the same reasoning as 
is adopted in the single rule of three, making the greater of the pair the 
second if this pair considered alone should require the answer to be 
greater. 

Set down all the first terms one under the other, and likewise all the 
second terms. Multiply all the first terms together and all the second terms 
together. Multiply the product of all the second terms by the third term . and 
divide this product by the product of all the first terms. Example: If 3 men 
remove 4 cubic yards in one day, working 12 hours a day, how many men 
working 10 hours a day will remove 20 cubic yards in 3 days ? 



Yards 4 
Days 3 

Hours 10 



Products 120 



20 

1 : : 3 men. 
12 
240 : : 3 : 6 men. Ans. 



To abbreviate by cancellation, any one of the first terms may cancel 
either the third or any of the second terms; thus, 3 in first cancels 3 in third, 
making it 1, 10 cancels into 20 making the latter 2, which into 4 makes it 2, 
which into 12 makes it 6, and the figures remaining are only 1 : 6 : : 1 : 6. 

INVOLUTION, OR POWERS OF NUMBERS. 

Involution is the continued multiplication of a number by itself a 
given number of times. The number is called the root, or first power, and 
the products are called powers. The second power is called the square and 



POWERS OF HUMBERS. 



the third power the cube. The operation may be indicated without being 
performed by writing a small figure called the index or exponent to the 
right of and a, little above the root; thus, 3 3 = cube of 3, = 27. 

To multiply two or more powers of the same number, add their exponents; 
thus, 22 x 2 3 ' = 2 5 , or 4 X 8 = 32 = 2 5 . 

To divide two powers of the same number, subtract their exponents; thus, 

2 3 -r- 2 2 = 2 1 = 2; 2 2 -f- 2 4 = 2~ 2 = = -. The exponent may thus be nega- 
tive 2 3 -t- 2 3 = 2 = 1, whence the zero power of any number = 1. The 
first power of a number is the number itself. The exponent may be frac- 
tional, as 2*, 23, which means that the root is to be raised to a power whose 
exponent is the numerator of the fraction, and the root whose sign is the 
denominator is to be extracted (see Evolution). The exponent may be a 
deeimal, as 2' 5 , 2 1 * 6 ; read, two to the five-tenths power, two to the one and 
five-tenths power. These powers are solved by means of Logarithms (which 
see). 

First Nine Powers of the First Nine Numbers. 



1st 


3d 


3d 


4th 


5th 


6th 


7th 


8th 


9th 


Pow'r 


Pow'r 


Power. 


Power. 


Power. 


Power. 


Power. 


Power. 


Power. 


1 


1 


1 


j 


1 


1 


1 


1 


1 


2 


4 


8 


16 


32 


64 


128 


256 


512 


3 


9 


27 


81 


243 


729 


2187 


6561 


19683. 


4 


16 


64 


256 


1024 


4096 


16384 


65536 


262144 


5 


25 


125 


625 


3125 


15625 


78125 


390625 


1953125 


6 


36 


216 


1296 


7776 


46656 


279936 


1679616 


10077696 


7 


49 


343 


2401 


16807 


117649 


823543 


5764801 


40353607 


8 


64 


512 


4096 


32768 


262144 


2097152 


16777216 


134217728 


9 


81 


729 


6561 


59049 


531441 


4782969 


43046721 


387420489 



The First Forty Powers of 2. 






h 


o 


L* 


c 


L 





L 


oJ 


"3 


1 

o 


"3 


i 


1 


o 


1 


o 


1 


> 


fi 




p. 


> 


PH 


> 


PH 




l 


9 


512 


18 


262144 


27 


134217728 


36 


68719476736 


2 


10 


1024 


19 


524288 


28 


268435456 


37 


137438953472 


4 


11 


2048 


20 


1048576 


29 


536870912 


38 


274877906944 


8 


12 


4096 


21 


2097152 


30 


1073741824 


39 


549755813888 


16 


13 


8192 


22 


4194304 


31 


2147483048 


40 


1099511627776 


32 


14 


16384 


23 


8388608 


32 


4294967296 






64 


15 


32768 


24 


16777216 


33 


8589934592 






128 


16 


65536 


25 


33554432 


34 


17179869184 






256 


17 


131072 


26 


67108864 


35 


34350738368 







EVOLUTION. 

Evolution is the finding of the root (or extracting the root) of any 
number the power of which is given. 



I/ V 



* the 



The sign tf indicates that the square root is to be extracted : 
cube root, 4th root, ?ith root. 

A fractional exponent with 1 for the numerator of the fraction is also 
used to indicate that the operation of extracting the root is to be performed; 

thus, 2*, 2* = V2, Vs. 

When the power of a number is indicated, the involution not being per- 
formed, the extraction of any root of that power may also be indicated by 



8 ARITHMETIC. 

dividing the index of the power by the index of the root, indicating the 
division by a fraction. Thus, extract the square root of the 6th power of 2: 

|/2 = 2$ _ 2 f _ g 3 _ 8< 
The 6th power of 2, as in the table above, is 64 ; |/64 ss 8. 

Difficult problems in evolution are performed by logarithms, but the 
square root and the cube root may be extracted directly according to the 
rules given below. The 4th root is the square root of the square root. The 
6th root is the cube root of the square root, or the square root of the cube 
root ; the 9th root is the cube root of the cube root etc. 

To Extract tlie Square Root. Point off the given number into 
periods of two places each, beginning with units. If there are decimals, 
point these off likewise, beginning at the decimal point, and supplying 
as many ciphers as may be needed. Find the greatest number whose 
square is less than the first left-hand period, and place it as the first 
figure in the quotient. Subtract its square from the left-hand period, 
and to the remainder annex the two figures of the second period for 
a dividend. Double the first figure of the quotient for a partial divisor ; 
find how many times the latter is contained in the dividend exclusive 
of the right-hand figure, and set the figure representing that number of 
times as the second figure in the quotient, and annex it to the right of 
the partial divisor, forming the complete divisor. Multiply this divisor by 
the second figure in the quotient and subtract the product from the divi- 
dend. To the remainder bring down the next period and proceed as before, 
in each case doubling the figures in the root already found to obtain the 
trial divisor. Should the product of the second figure in the root by the 
completed divisor be greater than the dividend, erase the second figure both 
from the quotient and from the divisor, and substitute the next smaller 
figure, or one small enough to make the product of the second figure by the 
divisor less than or equal to the dividend. 



3.141 5926536 1 [1.77245 -f 

27T274 



34712515 
1 2489 
3542 8692 
7084 

35444 160865 
1141776 



354485 1908936 
1772425 



To extract the square root of a fraction, extract the root of numerator 

/4 2 

and denominator separately. \/ - -, or first convert the fraction into a 
p 9 3 

decimal, j/|= 4/.4444 + = .6666 + . 
T 9 

To Kxtract the Cube Root. Point off the number into periods of 
3 figures each, beginning at the right hand, or unit's place. Point off deci- 
mals in periods of 3 figures from the decimal point. Find the greatest cube 
that does not exceed the left-hand period ; write its root as the first figure 
in the required root, Subtract the cube from the left-hand period, and to 
the remainder bring down the next period for a dividend. 

Square the first figure of the root; multiply by 300, and divide the product 
into the dividend for a trial divisor ; write the quotient after the first figure 
of the root as a trial second figure. 

Complete the divisor by adding to 3CO times the square of the first figure, 
30 times the product of the first by the second figure, and the square of the 
second figure. Multiply this divisor by the second figure; subtract the 
product from the remainder. (Should the product be greater than the 
remainder, the last figure of the root and the complete divisor are too large ; 



CUBE ROOT. 



substitute for the last figure the next smaller number, and correct the trial 
divisor accordingly.) 

To the remainder bring down the next period, and proceed as before to 
find the third figure of the root that is, square the two figures of the root 
already found; multiply by 300 for a trial divisor, etc. 

If at any time the trial divisor is greater than the dividend, bring down an- 
other period of 3 figures, and place in the root and proceed. 

The cube root of a number will contain as many figures as there are 
periods of 3 in the number. 

Shorter Methods of Extracting the Cube Root,- 1, From 
Went worth's Algebra: 



300 x 
30x 



x 2 = 



1,881, 365,963,625 1 12345 
1 

300 881 



728 







64 153365 


300 x 
30 x 


122 = 
12 x 3 = 


43200 
1080 


300 x 
30 x 


123 x 4 = 
4,2 = 


442891 132867 


1089 J 20498963 


4538700 1 
14760* 
16| 



4553476 ) 18213904 

_ I 4 ! 7 !! 25J85059625 
300 x 12342 = 456826800 
30 x 1234 x 5 = 185100 
52= 25 

457011925 2285059625 



After the first two figures of the root are found the next trial divisor is 
found by bringing down the sum of the 60 and 4 obtained in completing the 
preceding divisor; then adding the three lines connected by the brace, and 
annexing two ciphers. This method shortens the work in long examples, as 
is seen in the case of the last two trial divisors, saving the labor of squaring 
123 and 1234. A further shortening of the work is made by obtaining the 
last two figures of the root by division, the divisor employed being three 
times the square of the part of the root already found ; thus, after finding 
the first three figures: 



3 x 123 = 



45387|20498963|45.1-f 
181548 ~ 
234416 
226935 



74813 

The error due to the remainder is not sufficient to change the fifth figure of 
the root. 
2. By Prof. H. A. Wood (Stevens Indicator, July, 1890): 

I. Having separated the number into periods of three figures each, count- 
ing from the right, divide by the square of the nearest root of the first 
period, or first two periods ; the nearest root is the trial root. 

II. To the quotient obtained add twice the trial root, and divide by 3. 
This gives the root, or first approximation. 

III. By using the first approximate root as a new trial root, and proceed- 
ing as before, a nearer approximation is obtained, which process may be 
repeated until the root has been extracted, or the approximation carried as 
far as desired. 



10 ARITHMETIC. 

EXAMPLE. Required the cube root of 20. The nearest cube to 20 is 3*. 
32 = 9)20.0 



2.2 

6_ 

3)871 

2.7 IstT. R. 
7.29)20.000 



3)8.143 

2.714, 1st ap. cube root, 
2.714 2 = 7.365796)20.0000000 

2.7152534 
5.428 

3)8.1432534 
2.7144178 2d ap. cube root. 

REMARK. In the example it will be observed that the second term, or 
first two figures of the root, were obtained by using for trial root the root of 
the first period. Using, in like manner, these two terms for trial root, we 
obtained four terms of the root ; and these four terms for trial root gave 
seven figures of the root correct. In that example the last figure should be 
7. Should we take these eight figures for trial root we should obtain at least 
fifteen figures of the root correct. 

To Extract a Higher Root than the Cube, The fourth root is 
the square root of the square root ; the sixth root is the cube root of the 
square root or the square root of the cube root. Other roots are most con- 
veniently found by the use of logarithms. 

ALLIGATION 

shows the value of a mixture of different ingredients when the quantity 
and value of each is known. 

Let the ingredients be a, 6, c, d, etc., and their respective values per unit 
w>> x, y, z, etc. 

A = the sum of the quantities = a-\-b-\-c-}-d, etc. 
P mean value or price per unit of A. 
AP = aw -f bx -f- cy + dz, etc. 
_ aw -\-bx-\-cy-\-dz 
A 
PERMUTATION 

shows in how many positions any number of things may be arranged in a 
vrow; thus, the letters a, b, c may be arranged in six positions, viz. abc, acb, 
'cab, cba, bac, bca. 

Rule. Multiply together all the numbers used in counting the things; thus, 
permutations of 1, 2, and 3 = 1X2X3 = 6. In how many positions can 9 
things in a row be placed ? 

1X2X3X4X5X6X7X8X9 = 362880. 
COMBINATION 

shows how many arrangements of a few things may be made out of a 
greater number. Rule : Set down that figure which indicates the greater 
number, and after it a series of figures diminishing by 1, until as many are 
set down as the number of the few things to be taken in each combination. 
Then beginning under the last one set down said number of few things ; 
then going backward set down a series diminishing by 1 until arriving under 
the first of the upper numbers. Multiply together all the upper numbers to 
form one product, and all the lower numbers to form another; divide the 
upper product by the lower one. 



GEOMETRICAL PROGRESSION. 11 

How many combinations of 9 things can be made, taking 3 in each com- 
bination ? 

9X8X7 _ 504 _ 84 
1X2X3" 6 

ARITHMETICAL PROGRESSION, 

in a series of numbers, is a progressive increase or decrease in each succes- 
sive number by the addition or subtraction of the same amount at each step, 
as 1, 2, 3, 4, 5, etc., or 15, 12, 9, 6, etc. The numbers are called terms, and the 
equal increase or decrease the difference. Examples in arithmetical pro- 
gression may be solved by the following formulae : 

Let a = first term, I = last term, d = common difference, n = number of 
terms, s = sum of the terms: 

I = a -f (u l)d, 

_2s _ 

~~ n ~ ' 



= d + a) , 




- d) + Sds 



OE01TIKTRICAI, PROGRESSION, 

in a series of numbers, is a progressive increase or decrease in each sue. 
cessive number by the same multiplier or divisor at each step, as 1, 2, 4, 8, 
16. etc., or 243, 81, 27, 9, etc. The common multiplier is called the ratio. 

Let a = first term, I = last term, r ratio or constant multiplier, n =: 
number of terms, m = any term, as 1st, 2d, etc., s = sum of the terms: 

' a -r- (r - I) _ (r-l)sr~ - l 

l = ar*-l> -p- -7nr~ 

iog Z = log a + (n - 1) log r, f( - l) n ~ x - a( - a) n - J = 0. 

m = af"* - a * log w = log a -f- (m - 1) log r. 

n - 1 / n 1/~~^ 
yj n - y a n 

= n-l- n-l.- ' 



ARITHMETIC. 




=: o. 



log I - log a 1 
logr " 
log I log a 
'' log (s - a) - log (* - " 



log a = log I - (n 1) log r. 

log I log a 
logr = n __! 



. lg [ft + (r l)s] log a 

log r 

log Z - log [?/ - (r - l)s] 
log r 



Population of the United States. 

(A problem in geometrical progression.^ 



Tear. 

1860 
1870 
1880 
1890 
1900 
1905 
1910 



Population. 
81,443,821 
39,818,449* 
50,155,783 
62,622,250 
76,295,220 
Est. 83,577,000 
" 91554,000 



Increase in 10 Annual Increase, 
Years, per cent. per cent. 



26.63 
25.96 
24.86 
21.834 



2.39 
2.33 
2.25 

1.994 

Est. 1.840 
44 1.840 



Est. 20.0 

Estimated Population in Each Year from 1870 to 1909. 
(Based on the above rates of increase, in even thousands.) 



1870. . . . 


39,818 


1880.. . 


50,156 


1890.. 


62,622 


1900. .. 


76,295 


1871 .... 


40,748 


1881.. . 


51,281 


1891. 


63,871 


1901. .. 


77,699 


1872. .. 


41,699 


1882 . . 


52,433 


1892. . 


65,145 


1902. .. 


79,129 


1873.... 


42,673 


1883.. . 


53,610 


1893. . 


66,444 


1903. .. 


80,585 


1874.... 


43,670 


1884.. . 


54,813 


1894. 


67,770 


1904. .. 


82,067 


1875... 


44,690 


1885.. 


56,043 


1895. 


69,122 


1905. .. 


83,577 


1876.... 


45,373 


1886.. . 


57,301 


1896. 


70,500 


1906. .. 


85,115 


1877.... 


46,800 


1887.. . 


58,588 


1897. 


71,906 


1907. .. 


86,681 


1878 .. 


47,893 


1888.. . 


59,903 


1898. 


73,341 


1908. .. 


88,276 


1879.... 


49,011 


1889.. . 


61,247 


1899. 


74,803 


1909. .. 


89,900 



The above table has been calculated by logarithms as follows : 
log r = log I - log a -*- (n - 1), log m = log a -f- (w* 1) lo S r 

Pop. 1900. . . . 76,295,2-20 log = 7.8824988 = log I 

" 1890 . . . 62,022,250 log = 7.7967285 = log a 

cliff. = .0857703 

n = 11, n - 1 = 10; diff. * 10 = .00857703 = log r, 
add log for 1890 7.7967285 = log a 

log for 1891 = 7.80530553 No. = 63,871 . . . 
add again .00857703 

log for 1892 7.81388256 No. = 65,145 . . . 

Compound interest is a form of geometrical progression ; the ratio be- 
ing 1 plus the percentage. 

* Corrected by addition of 1,260,078, estimated error of the census of 1870, 
Census Bulletin No, 16, Dec, 12, 1890, 



DISCOUNT. 13 

INTEREST AND DISCOUNT. 

Interest is money paid for the use of money for a given time; the fao 
tors are : 

p, the sum loaned, or the principal: 

t, the time in years; 

r, the rate of interest; 

t, the amount of interest for the given rate and time; 

a = p + * = the amount of the principal with interest 

at the end of the time. 
Formulae : 

i = interest = principal X time X rate per cent = i = ^~; 

a = amount = principal -f- interest = p -{- ^55; 
_ 100* 
- pt> 



, ,. 100* 

t = time = . 

pr 

If the rate is expressed decimally as a per cent, thus, 6 per cent = .06, 
the formulae become 



pt 1 pr 7 tr 

Rules for finding Interest. Multiply the principal by ;the rate 
per annum divided by 100, and by the time in yc *i ; and fractions of a year. 

If the time is given in days, interest = Principal X rate X no. of days 

oo5 X 100 

In banks interest is sometimes calculated on the basis of 360 days to a 
year, or 12 months of 30 days each. 

Short rules for interest at 6 per cent, when 360 days are taken as 1 year: 

Multiply the principal by number of days.and divide by 6000. 

Multiply the principal by number of months and divide by 200. 

The interest of 1 dollar for one month is ^ cent. 

Interest of 100 Dollars for Different Times and Rates. 

Time. 2# 3# 4# 6# 6 8 10* 

lyear $2.00 $3.00 $4.00 $5.00 $6.00 $8.00 $10.00 

1 month .16f .25 .33$ .41f .50 .66| .83$ 

1 day = 3 $s year .0055f .0083$ .0111$ .0138| .0166 .0222* .0277$ 

1 day = 3$ 5 year .005479 .008219 .010959 .013699 .016438 .0219178 .0273973 

Discount is interest deducted for payment of money before it is due. 

True discount is the difference between the amount of a debt pay- 
able at a future date without interest and its present worth. The present 
worth is that sum which put at interest at the legal rate will amount to the 
debt when it is due. 

To find the present worth of an amount due at future date, divide the 
amount by the amount of $1 placed at interest for the given time. The dis- 
count equals the amount minus the present worth. 

What discount should be allowed on $103 paid six months before it is due, 
interest being 6 per cent per annum ? 

103 
T = $100 present worth, discount = 8.00. 

1 + 1 X .06 X | 

Bank discount is the amount deducted by a bank as interest on 
money loaned on promissory notes. It is interest calculated not on the act- 
ual sum loaned, but on the gross amount of the note, from which the dis- 
count is deducted in advance. It is also calculated on the basis of 360 days 
in the year, and for 3 (in some banks 4) days more than the time specified "in 
the note. These are called days of grace, and the note is not payable till 
tfce last of these days. In some States days of grace have been abolished. 



14 



ARITHMETIC. 



What discount will be deducted by a bank in discounting a note for $108 
payable 6 months hence ? Six months = 182 days, add 3 days grace = 185 



days 



,103 X 185 
6000 



= $3.176. 



Compound Interest. In compound interest the interest is added to 
the principal at the end of each year, (or shorter period if agreed upon). 

Let p = the principal, r = the rate expressed decimal^, n = no of years, 
and a the amount : 

a = amount = p (1 + r) n ; r = rate = 



p = principal = 



no . ot years _ n = 



Compound Interest Table. 

(Talue of one dollar at compound interest, compounded yearly, at 
3, 4, 5, and 6 per cent, from 1 to 50 years.) 






3# 


4* 


6* 


w 


05 

1 

F 


3* 


W 


5* 


w 


i 


1.03 


3.04 


1.05 


1.06 


16 


1.6047 


1.8730 


2.1829 


2.5403 


2 


1.0609 


1.0816 


1.1025 


1.1236 


17 


1.6528 


1.9479 


2.2920 


2.6928 


3 


1.0927 


1.1249 


1.1576 


1.1910 


18 


1.7024 


2.0258 


2.4066 


2.8543 


4 


.1255 


1.1699 


1.2155 


1.2625 


19 


1.7535 


2.1068 


2.5269 


3.0256 


5 


.1593 


1.2166 


1.2763 


1.3382 


20 


1.8061 


2.1911 


2.6533 


3.2071 


6 


.1941 


1.2653 


1.3401 


1.4185 


21 


1.8603 


2.2787 


2.7859 


3.3995 


7 


.2299 


1.3159 


.4071 


1.5036 


22 


1.9161 


2.3699 


2.9252 


3.6035 


8 


.2668 


1.3686 


.4774 


1.5938 


23 


1.9736 


24647 


3.0715 


3.8197 


9 


.3048 


.4233 


.5513 


1.6895 


24 


2.0328 


2.5633 


3.2251 


40487 


10 


.3439 


.4802 


.6289 


1.7908 


25 


2.0937 


2.6658 


3.3863 


4.2919 


11 


1.3842 


.5394 


.7103 


1.8983 


30 


2.4272 


3.2433 


4.3219 


5.7435 


12 


1.4258 


.6010 


.7958 


2.0122 


35 


2.8138 


3.9460 


5.5159 


7.6862 


13 


1.4685 


.6651 


.8856 


2.1329 


40 


3.2620 


4.8009 


7.0398 


10.2858 


14 


1.5126 


.7317 


.9799 


2.2609 


45 


3.7815 


5.8410 


8.9847 


13.7648 


15 


1.5580 


1.8009 


2.0789 


2.3965 


50 


4.3838 


7.1064 


11.4670 


18.4204 



At compound interest at 3 per cent money will double itself in 23J^ years, 
at 4 per cent in 17% years, at 5 per cent in 14.2 years, and at 6 per cent in 
11. 9 years. 

EQUATION OF PAYMENTS. 

By equation of payments we find the equivalent or average time in which 
one payment should be made to cancel a number of obligations due at dif- 
ferent dates ; also the number of days upon which to calculate interest or 
discount upon a gross sum which is composed of several smaller sums pay- 
able at different dates. 

Rule. Multiply each item by the time of its maturity in days from a 
fixed date, taken as a standard, and divide the sum of the products by the 
sum of the items: the result is the average time in days from the standard 
date. 

A owes B $100 due in 30 days, $200 due in 60 days, and $300 due in 90 days. 
In how many days may the whole be paid in one sum of $600 ? 

100 x 30 -f- 200 x 60 -f 300 x 90 = 42,000 ; 42,000 -f- 600 = 70 days, ana. 

A owes B $100, $200, and $300, which amounts are overdue respectively 30, 
60, and 90 days. If he now pays the whole amount, $600, how many days' 
interest should he pay on that sum ? Ans. 70 days. 



ANNUITIES. 



15 



PARTIAL PAYMENTS. 

To compute Interest on notes and bonds when partial payments have been 
made: 

United States Rule. Find the amount of the principal to the time 
of the first payment, and, subtracting the payment from it, find the amount 
of the remainder as a new principal to the time of the next payment. 

If the payment is less than the interest, find the amount of the principal 
to the time when the sum of the payments equals or exceeds the interest 
due, and subtract the sum of the payments from this amount. 

Proceed in this manner till the time of settlement. 

Note. The principles upon which the preceding rule is founded are: 

1st. That payments must be applied first to discharge accrued interest, 
and then the remainder, if any, toward the discharge of the principal. 

26. That only unpaid principal can draw interest. 

Mercantile Method. When partial payments are made on short 
notes or interest accounts, business men commonly employ the following 
method : 

Find the amount of the whole debt to the time of settlement ; also find 
the amount of each payment from the time it was made to the time of set- 
tlement. Subtract the amount of payments from the amount of the debt; 
the remainder will be the balance due. 

ANNUITIES. 

An Annuity is a fixed sum of money paid yearly, or at other equal times 
agreed upon. The values of annuities are calculated by the principles of 
compound interest. 

1. Let i denote interest on $1 for a year, then at the end of a year the 
amount will be 1 + i. At the end of n years it will be (1 + i). 

2. The sum which in n years will amount to 1 is n or (l + i)~ w , or the 

present value of 1 due in n years. 

(1 I i) n 1 

3. The amount of an annuity of 1 in any number of years n is v ~ . - . 

4. The present value of an annuity of 1 for any number of years n is 



5. The annuity which 1 will purchase for any number of years n is 



6. The annuity which would amount to 1 in n years is - 



Amounts, Present Values, etc., at 5% Interest. 



Years 


(1) 


(2) 


(3) 


(4) 


(5) 


(6) 




(l+i) n 


(l_f i)-n 


(1 4- i)n - 1 


l-(l+9- 


i 


i 








i 


i 


i-d+9-" 


(1 + 9" -1 


1 


1.05 


.952381 


1. 


.952381 


1.05 


1. 


2 


1.1025 


.907029 


2.05 


1.859410 


.537805 


.487805 


3 


1.157625 


.863838 


3.1525 


2.723248 


.367209 


.317209 


4 


1.215506 


.822702 


4.310125 


3.545951 


.282012 


.232012 


5 


1.276282 


.783526 


5.525631 


4.329477 


.230975 


.180975 


6... 


.340096 


.746215 


6.801913 


5.075692 


.197017 


.147018 


7 


.407100 


.710681 


8.142008 


5.786373 


.172820 


.122820 


8 


.477455 


.676839 


9.549109 


6.463213 


.154722 


.104722 


9 


1. 5513">8 


.644609 


11.026564 


7.107822 


.140690 


.090690 


10 


.628895 


.613913 


12.577893 


7.721735 


.129505 


.078505 



16 



ARITHMETIC. 



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WEIGHTS AND MEASUBES. 



TABLES FOR CALCULATING SIN It ING -FUNDS AND 
PRESENT VALUES. 

Engineers and others connected with municipal work an d*indus trial enter- 
prises often find it necessary to calculate payments to sinking-funds which 
will provide a sum of money sufficient to pay off a bond issue or other debt 
at the end of a given period, or to determine the present value of certain 
annual charges. The accompanying tables were computed by Mr. John W. 
Hill, of Cincinnati, Eng'g News, Jan. 25, 1894. 

Table I (opposite page) shows the annual sum at various rates of interest 
required to net $1000 in from 2 to 50 years, and Table II shows the present 
value at various rates of interest of an annual charge of $1000 for from 5 to 
50 years, at five-year intervals and for 100 years. 

Table II. Capitalization of Annuity of $1000 for 
from 5 to 1OO Years. 



5 
10 
15 
20 
25 


Rate of Interest, per cent. 


w 


3 


SM 


4 


* 


5 


6* 


6 


4,645 88 
8,752.17 
12,381.41 
15,589.215 
18,424.67 


4,579.60 
8,530.13 
11,937.80 
14,877.27 
17,413.01 


4,514.92 
8,316.45 
11,517.23 
14,212.12 
16,481.28 


4,451.68 
8,110.74 
11,118.06 
13,590.21 
15,621.93 


4,389.91 
7,912.67 
10,739.42 
13,007.88 
14,828.12 


4,329.45 
7,721.73 
10,379.53 
12,462.13 
14,093.86 


4,268.09 
7,537.54 
10,037.48 
11,950.26 
13,413.82 


4,212.40 
7,860.19 
9,712.30 
11,469.96 
12,783.38 


30 
35 
40 
45 
50 
100 


20,930.59 
23,145.31 
25,103.53 
26,833.15 
28,362.48 
36,614.21 


19,600.21 
21,487.04 
23,114.36 
24,518.49 
25,729.58 
31,598.81 


18,391.8517,291.86 
20,000.43 18,664.37 
21,354.8319,792.65 
22,495.2320,719.89 
23,455.21i21,482.08 
27,655.3624,504.96 


16,288.77 
17,460.89 
18,401.49 
19,156.24 
19,761.93 
21,949.21 


15,372.36 
16,374.36 
17,159.01 
17,773.99 
18,255.86 
19,847.90 


14,533.63 
15,390.48 
16,044.92 
16,547.65 
16,931.97 
18,095.83 


13,764.85 
14,488.65 
15,046.31 
15,455.85 
15,761.87 
16,612.64 



WEIGHTS AND MEASURES. 

Long Measure. Measures of Length. 

12 inches = 1 foot. 

3 feet = 1 yard. 

1760 yards, or 5280 feet = 1 mile. 

Additional measures of length in occasional use : 1000 mils = 1 inch ; 
4 inches = 1 hand; 9 inches = 1 span; 2*4 feet = 1 military pace; 2 yards = 
1 fathom; 5V yards, or 16^> feet = 1 rod (formerly also called pole or perch). 
Old Land Measure. 7.92 inches = 1 link; 100 links, or 66 feet, or 4 
rods =i 1 chain; 10 chains, or 220 yards = 1 furlong; 8 furlongs = 1 mile; 
10 square chains = 1 acre. 

Nautieal Measure* 

6 8 ut 6 e f mlles r 1>15156 Stat " \ = 1 nautical mile, or knot.* 
3 nautical miles = 1 league. 

60 Ta'tuTe milll' P 69 ' 168 [ = 1 degree (at the ec l uator )- 
360 degrees = circumference of the earth at the equator. 

*The British Admiralty takes the round figure of 6080 ft. which is the 
length of the *' measured mile 1 ' used in trials of vessels. The value varies 
from 6080.26 to 6088.44 ft. according to different measures of the earth's di- 
ameter. There is a difference of opinion among writers as to the use of the 
word " knot" to mean length or-a distance some holding that it should be 
used only to denote a rate of speed. The length between knots on the log 
line is 1/120 of a nautical mile, or 50.7 ft., when a half-minute glass is used; 
so that a speed of 10 knots is equal to 10 nautical miles per hour. 



18 ARITHMETIC. 

Square Measure. Measures of Surface. 

144 square incites, or 183.35 circular \ _ ^ square f oot . 

9 square feet = 1 square yard . 
30]4 square yards, or 272> square feet = 1 square rod. 

10 sq. chains, or 160 sq. rods, or 4840 sq. | * 

yards, or 43560 sq. feet, f - - 1 e< 

640 acres ' = 1 square mile. 

An acre equals a square whose side is 208.71 feet. 

Circular Inch; Circular Mil. A circular inch is the area of a 
circle 1 inch in diameter = 0.7854 square inch. 

I square inch = 1.2732 circular inches. 

A circular mil is the area of a circle 1 mil, or .001 inch in diameter. 
1000 2 or 1,000,000 circular mils = 1 circular inch. 

1 square inch = 1,273,239 circular mils. 

The mil and circular mil are used in electrical calculations involving 
the diameter and area of wires. 

Solid or Cubic Measure. Measures of Volume. 

1728 cubic inches = 1 cubic foot. 
27 cubic feet = 1 cubic yard. 

1 cord of wood = a pile, 4x4x8 feet = 128 cubic feet. 
1 perch of masonry = 16^X 1MX1 foot =24^ cubic feet. 

Liquid Measure. 

4 gills = 1 pint. 

2 pints = 1 quart. 

j U. S. 231 cubic inches. 
4 quarts = 1 gallon J Eng 2?7 274 cubic inchegi 

31 K gallons = 1 barrel. 

42 gallons = 1 tierce. 

2 barrels, or 63 gallons = 1 hogshead. 
84 gallons, or 2 tierces = 1 puncheon. 

2 hogsheads, or 126 gallons = 1 pipe or butt. 
2 pipes, or 3 puncheons = 1 tun. 
A gallon of water at 62 F. weighs 8.3356 Ibs. 

The U. S. gallon contains 281 cubic inches; 7.4805 gallons = 1 cubic foot. 
A cylinder 7 in. diam. and 6 in. high contains 1 gallon, very nearly, or 230.9 
cubic inches. The British Imperial gallon contains 277.274 cubic inches 
= 1.20032 U. S. gallon, or 10 Ibs. of water at 62 F. 

Tlie Miner's Inch. (Western U. S. for measuring flow of a stream 
of water). 

The term Miner's Inch is more or less indefinite, for the reason that Cali- 
f ornip, water companies do not all use the same head above the centre of 
the aperture, and the inch varies from 1.36 to 1.73 cubic feet per minute 
each; but the most common measurement is through an aperture 2 inches 
high and whatever length is required, and through a plank 1 inches thick. 
The lower edge of the aperture should be 2 inches above the bottom of the 
measuring-box, and the plank 5 inches high above the aperture, thus mak- 
ing a 6-inch head above the centre of the stream. Each square inch of this 
opening represents a miners inch, which is equal to a flow of H cubic feet 
per minute. 

Apothecaries' Fluid Measure. 

60 minims = 1 fluid drachm. 8 drachms = 1 fluid ounce. 

In the U. S. a fluid ounce is the 128th part of a U. S. gallon, or 1 805 cu. ins. 
It contains 456.3 grains of water at 39 F, In Great Britain the fluid ounce 
is 1.732 cu. ins. and contains i ounce avoirdupois, or 437.5 grains of water at 
62 F. 

Dry Measure, U. S. 

2 pints = 1 quart. 8 quarts = 1 peck. 4 pecks = 1 bushel. 

The standard U. S. bushel is the Winchester bushel, which is in cylinder 



WEIGHTS AND MEASURES. 19 

form, 18>3 inches diameter and 8 inches deep, and contains 3150.42 cubic 
inches. 

A struck bushel contains 2150.42 cubic inches = 1.2445 cu. ft.; 1 cubic foot 
- 0.80356 struck bushel. A heaped bushel is a cylinder 18^ inches diam- 
eter and 8 inches deep, with a heaped cone not less than 6 inches high. 
It is equal to 1^ struck bushels. 

The British Imperial bushel is based on the Imperial gallon, and contains 
8 such gallons, or 2218.192 cubic inches = 1.2837 cubic feet. The English 
quarter = 8 Imperial bushels. 

Capacity of a cylinder in U. S. gallons = square of diameter, in inches X 
height in inches X .0034. (Accurate within 1 part in 100,000.) 

Capacity of a cylinder in U. S. bushels = square of diameter in inches X 
height in inches X .0003652. 

Shipping Measure* 

Register Ton. For register tonnage or for measurement of the entire 
internal capacity of a vessel : 

100 cubic feet = 1 register ton. 

This number is arbitrarily assumed to facilitate computation. 
Shipping Ton. For the measurement of cargo : 

1 U. S. shipping ton. 



40 cubic feet = 



42 cubic feet = 



31. 16 Imp. bushels. 
32.143 U. S. " 
1 British shipping ton. 
32.719 Imp. bushels. 



33.75 U. S. 

Carpenter's Rule. Weight a vessel will carry = length of keel X breadth 
at main beam X depth of hold in feet -4-95 (the cubic feet allowed for a ton). 
The result will be the tonnage. For a double-decker instead of the depth 
of the hold take half the breadth of the beam. 

Measures of Weight. Avoirdupois, or Commercial 
Weight. 

16 drachms, or 437.5 grains = 1 ounce, oz. 
16 ounces, or 7000 grains = 1 pound, Ib. 
28 pounds =1 quarter, qr. 

4 quarters = 1 hundredweight, cwt. = 112 Ibs. 

20 hundred weight = 1 ton of 2240 pounds, or long ton. 

2000 pounds = 1 net, or short ton. 

2204.6 pounds = 1 metric ton. 

1 stone = 14 pounds ; 1 quintal = 100 pounds. 

The drachm, quarter, hundredweight, stone, and quintal are now seldom 
used in the United States. 

Tr*y Weight. 

24 grains = 1 pennyweight, dwt. 

20 pennyweights = 1 ounce, oz. = 480 grains. 

12 ounces = 1 pound, Ib. = 5760 grains. 

Troy weight is used for weighing gold and silver. The grain is the same 
in Avoirdupois, Troy, and Apothecaries' weights. A carat, used in weighing 
diamonds = 3.168 grains = .205 gramme. 

Apothecaries 9 Weight. 

20 grains = 1 scruple, 3 
3 scruples = 1 drachm, 3 = 60 grains. 
8 drachms = 1 ounce, = 480 grains. 

12 ounces = 1 pound, Ib. = 5760 grains. 

To determine whether a balance has unequal arms. 

After weighing an article and obtaining equilibrium, transpose the article 
and the weights. If the balance is true, it will remain in equilibrium ; if 
untrue, the pan suspended from the longer arm will descend. 

To weigh correctly on an incorrect balance. First, by 
substitution. Put the article to be weighed in one pan of the balance and 



20 ARITHMETIC. 

counterpoise it by any convenient heavy articles placed on the other pan. 
Remove the article to be weighed and substitute for it standard weights 
until equipoise is again established. The amount of these weights is the 
weight of the article. 

Second, by transposition. Determine the apparent weight of the article 
as usual, then its apparent weight after transposing the article and the 
weights. If the difference is small, add half the difference to the smaller 
of the apparent weights to obtain the true weight. If the difference is 2 
per cent the error of this method is 1 part in 10,000. For larger differences, 
or to obtain a perfectly accurate result, multiply the two apparent weights 
together and extract the square root of the product. 

Circular Measure* 

60 seconds, " = 1 minute, '. 
60 minutes, ' = 1 degree, . 
90 degrees = 1 quadrant. 
360 " = circumference. 

Time. 

60 seconds = 1 minute. 
60 minutes = 1 hour. 
24 hours = 1 day. 
7 days = 1 week. 
365 days, 5 hours, 48 minutes, 48 seconds = 1 year. 

By the Gregorian Calendar every year whose number is divisible by 4 is a 
leap year, and contains 366 days, the other years containing 365 days, ex- 
cept that the centesimal years are leap years only when the number of the 
year is divisible by 400. 

The comparative values of mean solar and sidereal time are shown by the 
following relations according to Bessel : 

365.24222 mean solar days = 366.24222 sidereal days, whence 
1 mean solar day = 1.00273791 sidereal days; 

1 sidereal day = 99726957 mean solar day; 
24 hours mean solar time = 24 h 3 m 56 8 .555 sidereal time; 
24 hours sidereal time = 23 h 56 m 4 8 .091 mean solar time, 

whence 1 mean solar day is 3 m 55 8 .91 longer than a sidereal day, reckoned in 
mean solar time. 

BOARD AND TIMBER MEASURE. 

Board Measure. 

In board measure boards are assumed to be one inch in thickness. To 
obtain the number of feet board measure (B. M.) of a board or stick of 
square timber, multiply together the length in feet, the breadth in feet, and 
the thickness in inches. 

To compute the measure or surface in square feet. When 
all dimensions are in feet, multiply the length by the breadth, and the pro- 
duct will give the surface required. 

When either of the dimensions are in inches, multiply as above and divide 
the product by 12. 

When all dimensions are in inches, multiply as before and divide product 
by 144. 

Timber Measure. 

To compute the vol ume of round timber. When all dimen- 
sions are in feet, multiply the length by one quarter of the product of the 
mean girth and diameter, and the product will give the measurement in 
cubic feet. When length is given in feet and girth and diameter in inches, 
divide the product by 144 ; when all the dimensions are in inches, divide by 
1728. 

To compute the volume of square timber. When all dimen- 
sions are in feet, multiply together the length, breadth, and depth; the 
product will be the volume in cubic feet. When one dimension is given in 
inches, divide by 12; when two dimensions are in inches, divide by 144; when 
all three dimensions are iu inches, divide by 1728. 



WEIGHTS AKD MEASURES. 



Contents in Feet of Joists, Scantling, and Timber. 

Length in Feet. 



Size. 


12 


14 


16 


18 


20 


22 


24 


26 


28 


30 



Feet Board Measure. 



2X 4 


8 


9 


11 


12 


13 


15 


16 


17 


19 


20 


2X6 


12 


14 


16 


18 


20 


22 


24 


26 


28 


30 


2X8 


16 


19 


21 


24 


27 


29 


32 


35 


37 


40 


2 X 10 


20 


23 


27 


30 


33 


37 


40 


43 


47 


50 


2 X 12 


24 


28 


32 


36 


40 


44 


48 


52 


56 


60 


2 X 14 


28 


33 


37 


42 


47 


51 


56 


61 


65 


70 


3X8 


24 


28 


32 


36 


40 


44 


48 


52 


56 


60 


3 X 10 


30 


35 


40 


45 


50 


55 


60 


65 


70 


75 


3 X 12 


36 


42 


48 


54 


60 


66 


72 


78 


84 


90 


3X14 


42 


49 


56 


63 


70 


77 


84 


91 


98 


105 


4X4 


16 


19 


21 


24 


27 


29 


32 


35 


37 


40 


4X6 


24 


28 


32 


36 


40 


44 


48 


52 


56 


60 


4X8 


32 


37 


43 


48 


53 


59 


64 


69 


75 


80 


4 X 10 


40 


47 


53 


60 


67 


73 


80 


87 


93 


100 


4 X 12 


48 


56 


64 


72 


80 


88 


96 


104 


112 


120 


4 X 14 


56 


65 


75 


84 


93 


103 


112 


121 


131 


140 


6X6 


36 


42 


48 


54 


60 


66 


72 


78 


84 


90 


6X8 


48 


56 


64 


72 


80 


88 


96 


104 


112 


120 


6 X 10 


60 


70 


80 


90 


100 


110 


120 


130 


140 


150 


6X 12 


72 


84 


96 


108 


120 


132 


144 


156 


168 


180 


6X 14 


84 


98 


112 


126 


140 


154 


168 


182 


196 


210 


8X* 8 


64 


75 


85 


96 


107 


117 


128 


139 


149 


160 


8 X 10 


80 


93 


107 


120 


133 


147 


160 


173 


187 


200 


8 X 12 


96 


112 


128 


144 


160 


176 


192 


208 


224 


240 


8 X 14 


112 


131 


149 


168 


187 


205 


224 


243 


261 


280 


10 X 10 


100 


117 


133 


150 


167 


183 


200 


217 


233 


250 


10 X 12 


120 


140 


160 


180 


200 


2^0 


240 


260 


280 


300 


10 X 14 


140 


J63 


187 


210 


233 


257 


280 


303 


327 


350 


12 X 12 


144 


168 


192 


216 


240 


264 


288 


312 


336 


360 


12 X 14 


168 


196 


224 


252 


280 


308 


336 


364 


392 


420 


14 X 14 


196 


229 


261 


294 


327 


359 


392 


425 


457 


490 



FRENCH OR METRIC MEASURES. 

The metric unit of length is the metre = 39.37 inches. 
The metric unit of weight is the gram = 15.432 grains. 
The following prefixes are used for subdivisions and multiples; Milli = T0 ^ 
Centi = T fo, Deci = j^, Deca = 10, Hecto = 100, Kilo = 1000, Myria = 10,000. 

FRENCH AND BRITISH (AND AMERICAN) 
EQUIVALENT MEASURES. 

Measures of Length. 

FRENCH. BRITISH and U. S. 

1 metre = 39.37 inches, or 3.28083 feet, or 1.09361 yards. 

.8048 metre = 1 foot. 

1 centimetre = .3937 inch. 
54 centimetres = 1 inch. 

1 milimetre = .03937 inch, or 1/25 inch, nearly. 
25.4 millimetres = 1 inch. 

1 kilometre = 1093.61 yards, or 0.62137 mile. 



22 ARITHMETIC. 

Measures of Surface. 

FRENCH. BRITISH and U. S. 

isquaremetre = j '; 

.836 square metre = 1 square yard. 

.0929 square metre = 1 square foot. 

1 square centimetre = .155 square inch. 
6.452 square centimetres = 1 square inch. 

1 square millimetre = .00155 sq. in. = 1973.5 circ. mils. 
645.2 square millimetres = 1 square inch. 
1 centiare = 1 sq. metre = 10.764 square feet. 

1 are = 1 sq. decametre = 1076.41 " 4 * 

1 hectare = 100 ares = 107641 " " = 2.4711 acres. 

1 sq. kilometre = .386109 sq. miles = 247.11 " 

1 sq. myriametre =38.6109" " 

Of Volume. 

FRENCH. BRITISH and U. S. 

1 onbip metre - 4 35 ' 314 cubic feet 

- 1 1.308 cubic yards. 
.7645 cubic metre = 1 cubic yard. 

.02832 cubic metre = 1 cubic foot. 

1 cubic decimetre = {;<, ^ic inches, 

28.32 cubic decimetres = 1 cubic foot. 

1 cubic centimetre = .061 cubic inch. 
16.387 cubic centimetres = 1 cubic inch. 
1 cubic centimetre = 1 millilitre = .061 cubic inch. 
1 centilitre = = .610 " " 

1 decilitre = = 6.102 " 

1 litre = 1 cubic decimetre = 61.023 " " = 1.05671 quarts, U. S. 
1 hectolitre or decistere = 3.5314 cubic feet = 2.8375 bushels, " 

1 stere, kilolitre, or cubic metre = 1.308 cubic yards = 28.37 bushels, " 

Of Capacity. 

FRENCH. BRITISH and U. S. 

f 61. 023 cubic inches, 

( 1 niihin rlppimpfr^ - J - 08531 Cubic foot, 

1 .2642 gallon (American), 
[2.202 pounds of water at 62 F. 
28.317 litres = 1 cubic foot. 

4.543 litres = 1 gallon (British). 

3.785 litres = 1 gallon (American). 

Of Weight. 

FRENCH. BRITISH and U. S. 

1 gramme = 15.432 grains. 

.0648 gramme = 1 grain. 

28.35 gramme = 1 ounce avoirdupois. 

1 kilogramme = 2.2046 pounds. 

.4536 kilogramme = 1 pound. 

1 tonne or metric ton = j - 
1000 kilogrammes = 1 



Mr. O. H. Titmann, in Bulletin No. 9 of the U. S. Coast and Geodetic Sur- 
vey, discusses the work of various authorities who have compared the yard 
and the metre, and by referring all the observations to a common standard 
has succeeded in reconciling the discrepancies within very narrow limits. 
The following are his results for the number of inches in a metre according 
to the eomparisc/zis of the authorities named: 

1817. Hassler ......................... 39.36994 inches. 

1818. Kater ........................... 39.36990 " 

1835. Baily ........................... 39.36973 " 

1866. Clarke .......................... 39.36970 " 

1885. Comstock ...................... 39.36984 " 

The mean of these is ......... ... 39.36982 " 



METKIC WEIGHTS AND MEASURES. 23 

METRIC CONVERSION TABLES. 

The following tables, with the subjoined memoranda, were published in 
1890 by the United States Coast and Geodetic Survey, office of standard 
weights and measures, T. C. Mendenhall, Superintendent. 

Tables for Converting U. S. Weights and Pleasures 
Customary to Metric. 

LINEAR. 





Inches to Milli- 
metres. 


Feet to Metres. 


Yards to Metres. 


Miles to Kilo- 
metres. 


J _ 


25.4001 


0.304801 


0.914402 


1.60935 


2 = 


50.8001 


0.609601 


1.828804 


3.21869 


O 


76.2002 


0.914402 


2.743205 


4.82804 


4 = 


101.6002 


1.219202 


3.657607 


6.43739 


5 = 


127.0003 


1.524003 


4.572009 


8.04674 


6 = 


152.4003 


1.828804 


5.486411 


9.65608 


7 = 


177.8004 


2.133604 


6.400813 


11.26543 


8 = 


203.2004 


2.438405 


7.315215 


12.87478 


9 = 


228.6005 


2.743205 


8.229616 


14.48412 



SQUARE. 





Square Inches to 
Square Centi- 
metres. 


Square Feet to 
Square Deci- 
metres. 


Square Yards to 
Square Metres. 


Acres to 
Hectares. 


1 


6.452 


9 290 


0.836 


0.4047 


2 = 


12.903 


1&581 


1.672 


0.8094 


3 = 


19.355 


27.871 


2.508 


1.2141 


4 = 


25.807 


37.161 


3.344 


1.6187 


5 = 


32.258 


46.452 


4.181 


2.0234 


6 = 


38.710 


55.742 


5.017 


2.4281 


7 = 


45.161 


65.032 


5.853 


2.8328 


8 = 


51.613 


74.323 


6.689 


3 2375 


9 = 


58.065 


83.613 


7.525 


3.6422 



CUBIC. 





Cubic Inches to 
Cubic Centi- 
metres. 


Cubic Feet to 
Cubic Metres. 


Cubic Yards to 
Cubic Metres. 


Bushels to 
Hectolitres. 


1 = 


16.387 


0.02832 


0.765 


0.35242 


2 = 


32.774 


0.05663 


1.529 


0.70485 


o 


49.161 


0.08495 


2.294 


1.05727 


4 = 


65.549 


0.11327 


3.058 


1.40969 


5 = 


81.936 


0.14158 


3.823 


1.76211 


6 = 


98.323 


0.16990 


4.587 


2.11454 


7 = 


114.710 


0.19822 


5.352 


2.46696 


8 = 


131 097 


0.22654 


6.116 


2.81938 


9 = 


147.484 


0.25485 


6.881 


3.17181 



ARITHMETIC. 
CAPACITY. 





Fluid Drachms 










to Millilitres or 


Fluid Ounces to 


Quarts to Litres. 


Gallons to Litres. 




Cubic Centi- 


Millilitres. 








metres. 








1 = 


3.70 


29.57 


0.94636 


3.78544 


2 = 


7.39 


59.15 


1.89272 


7.57088 


3 = 


11.09 


88.72 


2.83908 


11.35632 


4 = 


14.79 


118.30 


3.78544 


15.14176 


5 = 


18.48 


147.87 


4.73180 


18.92720 


6 = 


22.18 


177.44 


5.67816 


22.71264 


7 = 


25.88 


207.02 


6.62452 


26.49808 


8 = 


29.57 


236.59 


7.57088 


30.28352 


9 = 


33.38 


266.16 


8.51724 


34.06896 



WEIGHT. 













Grains to Milli- 
grammes. 


Avoirdupois 
Ounces to 
Grammes. 


Avoirdupois 
Pounds to Kilo- 
grammes. 


Troy Ounces to 
Grammes. 


1 = 


64.7989 


28.3495 


0.45359 


31.10348 


2 = 


129.5978 


56.6991 


0.90719 


62.20696 


3 = 


194.3968 


85.0486 


1.36078 


93.31044 


4 = 


259.1957 


113.3981 


1.81*37 


124.41392 


5 = 


323.9946 


141.7476 


2.26796 


155.51740 


6 = 


388.7935 


170.0972 


2.72156 


186.62089 


7 = 


453.5924 


198.4467 


3.17515 


217.72437 





518.3914 


226.7962 


3.62874 


248.82785 


9 = 


583.1903 


255.1457 


4.08233 


279.93133 



1 chain = 20.1169 metres. 

1 square mile 259 hectares. 
1 fathom = 1.829 metres. 

1 nautical mile = 1853.27 metres. 
1 foot = 0.304801 metre. 

1 avoir, pound = 453.5924277 gram. 
15432.35639 grains = 1 kilogramme. 



Tables for Converting 17. S. Weights and Measures 
Metric to Customary. 

LINEAR. 



1 = 

2 = 
3 = 
4 = 
5 = 


Metres to 
Inches. 


Metres to 
Feet. 


Metres to 
Yards. 


Kilometres to 
Miles. 


39.3700 
78.7400 
118.1100 
157.4800 
196.8500 


3.28083 
6.56167 
9.84250 
13.12333 
16.40417 


1.093611 

2.187222 
3.280833 
4.374444 
5.468056 


0.62137 
1.24274 
1.86411 

2.48548 
3.10685 


II II II II 
0i>ooca 1 


236.2200 
275.5900 
314.9600 
354.3300 


19.68500 
22.96583 
26.24667 
29.52750 


6.561667 
7.655278 
8.748889 
9.842500 


3.72822 
4.34959 
4.97096 
5.59233 



METRIC CONVERSION TABLES. 
SQUARE. 





Square Centi- 
metres to 
Square Inches. 


Square Metres 
to Square Feet. 


Square Metres 
to Square Yards. 


Hectares to 
Acres. 


1 = 


0.1550 


10.764 


1.196 


2.471 


2 


0.3100 


21.528 


2.392 


4.942 


3 = 


0.4650 


32.292 


3.588 


7.413 


4 = 


0.6200 


43.055 


4.784 


9.884 


5 = 


0.7750 


53.819 


5.980 


12.355 


6 = 


0.9300 


64.583 


7.176 


14.826 


7 = 


1.0850 


75.347 


8.372 


17.297 


8 = 


1.2400 


86.111 


9.568 


19.768 


9 = 


1.3950 


96.874 


10.764 


22.239 



CUBIC. 





Cubic Centi- 
metres to Cubic 
Inches. 


Cubic Deci- 
metres to Cubic 
Inches. 


Cubic Metres to 
Cubic Feet. 


Cubic Metres to 
Cubic Yards. 


1== 


0.0610 


61.023 


35.314 


1.308 


2 = 


0.12^0 


122.047 


70.629 


2.616 


3 = 


0.1831 


183.070 


105.943 


3.924 


4 = 


0.2441 


244.093 


141.258 


5.232 


5 = 


0.3051 


305.117 


176.572 


6.540 


6 = 


0.3661 


366.140 


211.887 


7.848 


7 = 


. 0.4272 


427.163 


247.201 


9.156 


8 = 


0.4882 


488.187 


282.516 


10.464 


9 = 


0.5492 


549.210 


317.830 


11.771 



CAPACITY. 





Millilitres or 
Cubic Centi- 
litres to Fluid 
Drachms. 


Centilitres 
to Fluid 
Ounces. 


Litres to 
Quarts. 


Dekalitres 
to 
Gallons. 


Hektolitres 
to 
Bushels. 


1 = 


0.27 


0.338 


1.0567 


2.6417 


2.8375 


2 


0.54 


0.676 


2.1134 


5.2834 


5.6750 


3 = 


0.81 


1.014 


3.1700 


7.9251 


8.5125 


4 = 


1.08 


1.352 


4.2267 


10.5668 


11.3500 


5 = 


1.35 


1.691 


5.2834 


13.2085 


14.1875 


6 = 


1.62 


2.029 


6.3401 


15.8502 


17.0250 


7 


1.89 


2.368 


7.3968 


18.4919 


i 19.8625 


8 = 


*.16 


2.706 


8.4534 


21.1336 


! 22.7000 


9 = 


2.43 


3.043 


9.5101 


23.7753 


25.5375 



26 



ARITHMETIC. 
WEIGHT. 



1 = 
2 = 
3 = 
4 = 

5 = 


Milligrammes 
to Grains. 


Kilogrammes 
to Grains. 


Hectogrammes 
(100 grammes) 
to Ounces Av. 


Kilogrammes 
to Pounds 
Avoirdupois. 


0.01543 
0.03086 
0.04630 
0.06173 
0.07716 


15432.36 
30864.71 
46297.07 
61729.43 
77161.78 


3.5274 
7.0548 
10.5822 
14.1096 
17.6370 


2.20462 
4.40924 
6.61386 
8.81849 
11.02311 


6 = 

8 = 
9 = 


0.09259 
0.10803 
0.12346 
0.13889 


92594.14 
108026.49 
123458.85 
138891 .21 


21.1644 
24.6918 
28.2192 
31.7466 


13.22773 
15.43235 
17.63697 
19.84159 



WEIGHT (Continued). 





Quintals to 
Pounds Av. 


Milliers or Tonnes to 
Pounds Av. 


Grammes to Ounces, 
Troy. 


W^CdO*-* 

II II II II II 


220.46 
440.92 
661.38 
881.84 
1102.30 


2204 6 
4409.2 
6613.8 
8818.4 
11023.0 


0.03215 
0.06430 
0.09645 
0.12860 
0.16075 


6 = 

7 = 

8 = 
9 = 


1322.76 
1543.22 
1763.68 
1984.14 


13227.6 
15432.2 
17636.8 
19841.4 


0.19290 
0.22505 
0.25721 
0.28936 

i 



The only authorized material standard of customary length is the 
Troughton scale belonging to this office, whose length at 59.62 Fahr. con- 
forms to the British standard. The yard in use in the United States is there- 
fore equal to the British yard. 

The only authorized material standard of customary weight is the Troy 
pound of the mint. It is of brass of unknown density, and therefore not 
suitable for a standard of mass. It was derived from the British standard 
Troy pound of 1758 by direct comparison. The British Avoirdupois pound 
was also derived from the latter, and contains 7000 grains Troy. 

The grain Troy is therefore the same as the grain Avoirdupois, and the 
pound Avoirdupois in use in the United States is equal to the British pound 
Avoirdupois. 

The metric system was legalized in the United States in 1866. 

By the concurrent action of the principal governments of the world an 
International Bureau of Weights and Measures has been established near 

The International Standard Metre is derived from the Metre des Archives, 
and its length is defined by the distance between two lines at Centigrade, 
on a platinum-iridium bar deposited at the International Bureau. 

The International Standard Kilogramme is a mass of platinum-iridium 
deposited at the same place, and its weight in vacua is the same as that of 
the Kilogramme des Archives. 

Copies of these international standards are deposited in the office of 
standard weights and measures of the U. S. Coast and Geodetic Survey. 

The litre is equal to a cubic decimetre of water, and it is measured by the 
quantity of distilled water which, at its maximum density, will counterpoise 
the standard kilogramme in a vacuum; the volume of such a quantity of 
water being, as nearly as has been ascertained, equal to a cubic decimetre. 



WEIGHTS AND MEASURES COMPOUND UNITS. 27 

COMPOUND UNITS. 

Measures of Pressure and Weight. 

f 144 Ibs. per square foot. 

2.0355 ins. of mercury at 32* F. 
1 Ib. per square inch. = i 2.0416 " " " " 62 F. 

2.309 ft. of water at 62 F. 
[ 27.71 ins. " " " 62 F. 



1 ounce per sq. In. = { 



f 2116.3 Ibs. per square foot. 
I 33.947 ft. of water at 62 F. 
1 atmosphere (14.7 Ibs. per sq. in.) = -j 30 ins. of mercury at 62 F. 



- 

29.922 ins. of mercury at 32 F. 
1.76 



. . . 

.760 millimetres of mercury at 32 F. 

.03609 Ib. or .5774 oz. per sq. in. 
1 inch of water at 62 F. =< 5.196 Ibs. per square foot. 

.0736 in. of mercury at 62 F. 



1 
< 
( 

j 
1 foot of water at F. = \ 



1 inch of water at 38 F. = 5 ;gg}^.I? r 



( .491 Ib. or 7.86 oz. per sq. in. 

1 inch of mercury at 62 F. t=4 1.132ft. of water at 62 F. 

f 13. 58 ins. " " * 62 F. 

Weight of One Cubic Foot of Pure Water. 

At 32 F. (freezing-point) ............................ 62.418 Ibs. 

" 39.1 F. (maximum density) .......... . ............ 62.425 " 

" 62 F. (standard temperature) ....... ............... 62.355 " 

u 212 F. (boiling-point, under 1 atmosphere) ........ 59.76 * 

American gallon = 231 cubic ins. of water at 62 F. = 8.3356 Ibs. 

British = 277.274 " " " " " " =10 Ibs. 

Measures of Work, Power, and Duty. 

Work. /The sustained exertion of pressure through space. 

Unit of work. One foot-pound, i.e., a pressure of one pound exerted 
through a space of one foot. 

Horse-power. The rate of work. Unit of horse- power = 33,000 ft.- 
Ibs. per minute, or 550 ft. -Ibs. per second = 1,980,000 ft. -Ibs. per hour. 

Heat unit = heat required to raise 1 Ib. of water 1 F. (from 39 to 40). 

33000 

Horse-power expressed in heat units = ~^g~ = 42.416 heat units per min- 

ute = .707 heat unit per second = 2545 heat units per hour. 
1 Ib. of f ue! per H. P. per hour= 



1,000,000 ft.-lbs. per Ib. of fuel = 1.98 Ibs. of fuel per H. P. per hour. 

5280 22 
Velocity. Feet per second = ^^ = 15 x miles per hour. 

Gross tons per mile = ^ = Ibs. per yard (single rail.) 

French and British Equivalents of Compound Units. 

FRENCH. BRITISH. 

ramme per square millimetre = 1.422 Jbs. per square inch. 

*' 



1 g 
1 ki 



. 

ilogramme per square *' = 1422.32 

1 " J* centimetre = 14.223 " * " j 

1.0335 kg. per sq. cm. 1 atmosphere = 14.7 

0.070308 kilogramme per square centimetre = 1 Ib. per square inch. 

1 gramme per litre = 0.062428 Ib. per cubic foot. 

1 kilogrammetre = 7.2330 foot-pounds. 



28 



ARITHMETIC. 



WIRE AND SHEET-METAL, GAUGES COMPARED. 



Number of 
Gauge. 


Birmingham 
(or Stubs' Iron) 
Wire Gauge. 


American or 
Brown and 
Sharpe Gauge. 


Roebling's and 
Washburn 
& Moen's 
Gauge. 


2| 

4> <M 

fit! 

QQ &0 <S 
CQ 


British Imperial 
Standard 
Wire Gauge. 
(Legal Standard 
in Great Britain 
since 
March 1, 1884.) 


U. S. Standard 
Gauge for 
Sheet and Plate 
Iron and Steel. 
(Legal Standard 
since July 1, 1893.) 


Number of 
Gauge. 




inch. 


inch. 


inch. 


inch. 


inch. 


millim. 


inch. 




0000000 






.49 




.500 


12.7 


.5 


7/0 


000000 






.46 




.464 


11.78 


.469 


6/0 


00000 






.43 




.432 


10.97 


.438 


5/0 


0000 


.454 


.46 


.393 




.4 


10.16 


.406 


4/0 


000 


.425 


.40964 


.362 




.372 


9.45 


.375 


3/0 


00 


.38 


.3648 


.331 




.348 


8.84 


.344 


2/0 





.34 


.32486 


.307 




.324 


8.23 


.313 





1 


.3 


.2893 


.283 


.227 


.3 


7.62 


.281 


1 


2 


.284 


.25763 


.263 


.219 


.276 


7.01 


.266 


2 


3 


.259 


.22942 


.244 


.212 


.252 


6.4 


.25 


3 


4 


.238 


.20431 


.225 


.207 


.232 


5.89 


.234 


4 


5 


.22 


.18194 


.207 


.204 


.212 


5.38 


.219 


5 


6 


.203 


.16202 


.192 


.201 


.192 


4.88 


.203 


6 


7 


.18 


.14428 


.177 


.199 


.176 


4.47 


.188 


7 


8 


.165 


.12849 


.162 


.197 


.16 


4.06 


.172 


8 


9 


.148 


.11443 


.148 


.194 


.144 


3.66 


.156 


9 


10 


.134 


.10189 


.135 


.191 


.128 


3.25 


.141 


10 


11 


.12 


.09074 


.12 


.188 


.116 


2.95 


.125 


11 


12 


.109 


.08081 


.105 


.185 


.104 


2.64 


.109 


12 


13 


.095 


.07196 


.092 


.182 


.092 


2.34 


.094 


13 


14 


.083 


.06408 


.08 


.180 


.08 


2.03 


.078 


14 


15 


.072 


.05707 


.072 


.178 


.072 


1.83 


.07 


15 


16 


.065 


.05082 


.063 


.175 


.064 


1.63 


.0625 


16 


17 


.058 


.04526 


.054 


.172 


.056 


1.42 


.0563 


17 


18 


.049 


.0403 


.047 


.168 


.048 


1 . 22 


.05 


18 


19 


.042 


.03589 


.041 


.164 


.04 


1,02 


.0438 


19 


20 


.035 


.03196 


.035 


.161 


.036 


.91 


.0375 


20 


21 


.032 


.02846 


.032 


.157 


.032 


.81 


.0344 


21 


22 


.028 


.02535 


.028 


.155 


.028 


.71 


.0313 


22 


23 


.025 


.02257 


.025 


.153 


.024 


.61 


.0281 


23 


24 


.022 


.0201 


.023 


.151 


.022 


.56 


.025 


24 


25 


.02 


.0179 


.02 


.148 


.02 


.51 


.0219 


25 


26 


.018 


.01594 


.018 


.146 


.018 


.46 


.0188 


26 


27 


.016 


.01419 


.017 


.143 


.0164 


.42 


.0172 


27 


28 


.014 


.01264 


.016 


.139 


.0148 


.38 


.0156 


28 


29 


.013 


.01126 


.015 


.134 


.0136 


.35 


.0141 


29 


30 


.012 


.01002 


.014 


.127 


.0124 


.31 


.0125 


30 


31 


.01 


.00893 


.0135 


.120 


.0116 


.29 


.0109 


31 


32 


.009 


.00795 


.013 


.115 


.0108 


.27 


.0101 


32 


33 


.008 


.00708 


.011 


.112 


.01 


.25 


.0094 


33 


34 


.007 


.0063 


.01 


.110 


.0092 


.23 


.0086 


34 


35 


.005 


.00561 


.0095 


.108 


.0084 


.21 


.0078 


35 


36 


004 


.005 


.009 


.106 


.0076 


.19 


.007 


36 


37 




.00445 


.0085 


.103 


.0068 


.17 


.0066 


37 


38 




.00390 


.008 


.101 


.006 


.15 


.0063 


38 


39 




.00353 


.0075 


.099 


.0052 


.13 




39 


40 




.00314 


.007 


.097 


.0048 


.12 




40 


41 








.095 


.0044 


.11 




41 


42 








.092 


.004 


.10 




42 


43 








.088 


.0036 


.09 




43 


44 








.085 


.0032 


.08 




44 


45 








.081 


.0028 


.07 




45 


46 








.079 


.0024 


.06 




46 


47 








.077 


.002 


.05 




47 


48 








.075 


.0016 


.04 




48 


49 








.072 


.0012 


.03 




49 


50 






.069 


.001 


.025 




50 



WIRE GAUGE TABLES. 



EDISON, OR CIRCULAR ftllL GAUGE, FOR ELEC- 
TRICAL WIRES. 



Gauge 
Num- 
ber. 


Circular 

Mils. 


Diam- 
eter 
in Mils. 


Gauge 
Num- 
ber. 


Circular 
Mils. 


Diam- 
eter 
in Mils. 


Gauge 
Num- 
ber. 


Circular 
Mils. 


Diam- 
eter 
in Mils. 


3 


3,000 


54.78 


70 


70,000 


264.58 


190 


190,000 


435.89 


5 


5,000 


70.72 


75 


75,000 


273.87 


200 


200.000 


447.22 


8 


8,000 


89.45 


80 


80,000 


282.85 


220 


220,000 


469.05 


12 


12,000 


109.55 


85 


85,000 


291.55 


240 


240,000 


489.90 


15 


15,000 


122.48 


90 


90,000 


300.00 


260 


260,000 


509.91 


20 


20,000 


141.43 


95 


95,000 


308.23 


280 


280,000 


529.16 


25 


25,000 


158.1? 


100 


100,000 


316.23 


300 


300,000 


547.73 


30 


30,000 


173.21 


110 


110,000 


331.67 


320 


320,000 


565.69 


35 


35,000 


187.09 


120 


120,000 


346.42 


340 


340,000 


583.10 


40 


40,000 


200.00 


130 


130,000 


360.56 


360 


360,000 


600.00 


45 


45,000 


212.14 


140 


140,000 


374.17 








50 


50,000 


223.61 


150 


150,000 


387.30 








55 


55,000 


234.53 


160 


160,000 


400.00 








60 


60,000 


244.95 


17'0 


170,000 


412.32 








65 


65,000 


254.96 


180- 


180,000 


424.27 









TWIST DRILL AND STEEL WIRE GAUGE. 

(Morse Twist Drill aud Machine Co.) 



No. 


Size. 


No. 


Size. 


No. 


Size. 


No. 


Size. 


No. 


Size. 


No. 


Size. 




inch. 




inch 




inch. 




inch. 




inch. 




inch. 


1 


.2280 


11 


.1910 


21 


.1590 


31 


.1200 


41 


.0960 


51 


.0670 


y 


.2210 


12 


.1890 


22 


.1570 


32 


.1160 


42 


.0935 


52 


.0635 


3 


.2130 


13 


.1850 


23 


.1540 


33 


.1130 


43 


.0890 


53 


.0595 


4 


.2090 


14 


.1620 


24 


.1520 


34 


.1110 


44 


.0860 


54 


.0550 


5 


.2055 


15 


.1800 


25 


.1495 


35 


.1100 


45 


.0820 


55 


.0520 


6 


.2040 


16 


.1770 


26 


.1470 


36 


.1065 


46 


.0810 


56 


.0465 


7 


.2010 


17 


.1730 


27 


.1440 


37 


.1040 


47 


.0785 


57 


.0430 


8 


.199J 


18 


.1695 


28 


.1405 


38 


.1015 


48 


.0760 


58 


.0420 


9 


.1960 


19 


.1660 


29 


.1360 


39 


.0995 


49 


.0730 


59 


.0410 


10 


.1935 


20 


.1610 


30 


.1285 


40 


.0980 


50 


.0700 


60 


.0400 



STUBS' STEEL WIRE GAUGE. 
(For Nos. 1 to 50 see table on page 28.) 



No. 


Size. 


No. 


Size. 


No. 


Size. 


No. 


Size. 


No. 


Size. 


No. 


Size. 




inch. 




inch. 




inch. 




inch. 




inch. 




inch. 


Z 


.413 


P 


.323 


F 


.257 


51 


.066 


61 


.038 


71 


.026 


Y 


.404 


O 


.316 


E 


.250 


52 


.063 


62 


.037 


72 


.024 


X 


.397 


N 


.302 


D 


.246 


53 


.058 


63 


.036 


73 


.023 


W 


.386 


M 


295 


C 


.242 


54 


.055 


64 


.035 


74 


.022 


V 


.377 


L 


!&o 


B 


.238 


55 


.050 


65 


.033 


75 


.020 


U 


.368 


K 


.281 


A 


.234 


56 


.045 


66 


.032 


76 


.018 


T 


.358 


J 


.277 


1 


(See 


57 


.042 


67 


.031 


77 


.016 


S 


.348 


I 


.272 


to 


1 page 


58 


.041 


68 


.030 


78 


.015 


K 


.339 


H 


.266 


50 


( 28 


59 


.040 


69 


.029 


79 


*.014 


Q 


.332 


G 


.261 






60 


.039 


70 


.027 


80 


.013 



The Stubs' Steel Wire Gauge is used in measuring drawn steel wire or 
drill rods of Stubs' make, and is also used by many makers of American 
drill rods, 



30 AKITHMETIC. 

THE: EDISON OR CIRCULAR MIL, WIRE GAUGE. 

(For table of copper wires by this'gauge, giving weights, electrical resist 
ances, etc., see Copper Wire.) 

Mr. C. J. Field (Stevens Indicator, July, 1887) thus describes the origin of 
the Edison gauge: 

The Edison company experienced inconvenience and loss by not having a 
wide ^nough range nor sufficient number of sizes in the existing gauges. 
This was felt more particularly in the central-station work in making 
electrical determinations for the street system. They were compelled to 
make use of two of the existing gauges at least, thereby introducing a 
complication that was liable to lead to mistakes by the contractors and 
linemen. 

In the incandescent system an even distribution throughout the entire 
system and a uniform pressure at the point of delivery are obtained by cal- 
culating for a given maximum percentage of loss from the potential as 
delivered from the dynamo. In carrying this out, on account of lack of 
regular sizes, it was often necessary to use larger sizes than the occasion 
demanded, and even to assume new sizes for large underground conductors. 
It was also found that nearly all manufacturers based their calculation for 
the conductivity of their wire on a variety of units, and that not one used 
the latest unit as adopted by the British Association and determined from 
Dr. Matthiesseif s experiments ; and as this was the unit employed in the 
manufacture of the Edison lamps, there was a further reason for construct- 
ing a new gauge. The engineering department of the Edison company, 
knowing the requirements, have designed a gauge that has the widest 
range obtainable and a large number of sizes which increase in a regular 
and uniform manner. The basis of the graduation is the sectional area, and 
the number of the wire corresponds. A wire of 100,000 circular mils area ts 
No. 100 ; a wire of one half the size will be No. 50 ; twice the size No. .200. 

In the older gauges, as the number increased the size decreased. With 
this gauge, however, the number increases with the wire, and the number 
multiplied by 1000 will ^ive the circular mils. 

The w r eight per mil-foot, 0.00000302705 pounds, agrees with a specific 
gravity of 8.889, which is the latest figure given for copper. The ampere 
capacity which is given was deduced from experiments made in the com- 
pany's laboratory, and is based on a rise of temperature of 50 F. in the wire. 

In 1893 Mr. Field writes, concerning gauges in use by electrical engineers: 

The B. and S. gauge seems to be in general use for the smaller sizes, up 
to 100,000 c. m., and in some cases a little larger. From between one and 
two hundred thousand circular mils upwards, the Edison gauge or its 
equivalent is practically in use, and there is a general tendency to designate 
all sizes above this in circular mils, specifying a wire as 200,000, 400,000, 500,- 
000, or 1,000,000 c. m. 

In the electrical business there is a large use of copper wire and rod and 
other materials of these large sizes, and in ordering them, speaking of them, 
specifying, and in every other use, the general method is to simply specify 
the circular milage. I think it is going to be the only system in the future 
for the designation of wires, and the attaining of it means practically the 
adoption of the Edison gauge or the method and basis of this gauge as the 
correct one for wire sizes. 

THE U. S. STANAR GAUGE FOR SHEET AND 
PL. ATE IRON AN STEEL., 1893. 

There is in this country no uniform or standard gauge, and the same 
numbers in different gauges represent different thicknesses of sheets or 
plates. This has given rise to niHch misunderstanding and friction between 
employers and workmen and mistakes and fraud between dealers and con- 
sumers. 

An Act of Congress in 1893 established the Standard Gauge for sheet iron 
and^teel which is given on the next page. It is based on the fact that a 
cubic foot of iron weighs 480 pounds. 

A sheet of iron 1 foot square and 1 inch thick weighs 40 pounds, or 640 
ounces, and 1 ounce in weight should be 1/640 inch thick. The scale has 
been arranged so that each descriptive number represents a certain number 
of ounces in weight and an equal number of 640ths of an inch in thickness. 
The law enacts that on and after July 1, 1893, the new gauge shall be used 
in determining duties and taxes levied on sheet and plate iron and steel; and 
that in its application a variation of 2^4 per cent either way may be allowed. 



GAUGE FOE SHEET AND PLATE IKON AND STEEL. 31 



U. S. STANDARD GAUGE FOR SHEET AND PliATE 

IRON AND STEEL., 1893. 





fliM 


.2 a 


S 


SH- 5 


"o 03 


Si "ft 2 


- fe s 




Number oi 
Gauge. 


'i fl-S fl 


5 73 r2 3 

lf.N 

eusia 

^SQ 03 

%$ * 


28 | 

fls-1 

g-g-M 

gl 1 

<3 ^ 


-111 
||I{ 


sill. 

iili 
ft*! 


}l| 


III 

S3* 

i 

^.s 


'~S !'- 


0000000 


1-2 


0.5 


12.7 


320 


20. 


9.072 


97.65 


215.28 


000000 


15-32 


0.46875 


11.90625 


300 


18.75 


8.505 


91.55 


201.82 


00000 


7-16 


0.4375 


11.1125 


280 


17.50 


7.938 


85.44 


188.37 


0000 


13-32 


0.40625 


10.31875 


260 


16.25 


7.371 


79.33 


174.91 


000 


3-8 


0.375 


9.525 


240 


15. 


6.804 


73.24 


161.46 


00. 


11-32 


0.31375 


8.73125 


220 


13.75 


6.237 


67.13 


148.00 





5-16 


0.3125 


7.9375 


200 


12.50 


5.67 


61.03 


134.55 


1 


9-32 


0.28125 


7.14375 


180 


11.25 


5.103 


54.93 


121.09 


2 


17-64 


0.265625 


6.746875 


170 


10.625 


4.819 


51.88 


114.37 


3 


1-4 


0.25 


6.35 


160 


10. 


4.536 


48.82 


107.64 


4 


15-64 


0.234375 


5.953125 


150 


9.375 


4.252 


45.77 


100.91 


5 


7-32 


0.21875 


5.55625 


140 


8.75 


3.969 


42.72 


94.18 


6 


13-64 


0.203125 


5.159375 


130 


8.125 


3.685 


39.67 


87.45 


7 


3-16 


0.1875 


4.7625 


120 


7.5 


3.402 


36.62 


80.72 


8 


11-64 


0.171875 


4.365625 


110 


6.875 


3.118 


33.57 


74.00 


9 


5-32 


0.15625 


3.9S875 


100 


6.25 


2.835 


30.52 


67.27 


10 


9-64 


0.140625 


3.571875 


90 


5.625 


2.552 


27.46 


60.55 


11 


1-8 


0.125 


3.175 


80 


5. 


2.268 


24.41 


53.82 


1<2 


7-64 


0.109375 


2.778125 


70 


4.375 


1.984 


21.36 


47.09 


13 


3-32 


0.09375 


2.38125 


60 


3.75 


1.701 


18.31 


40.36 


14 


5-64 


0.078125 


1.984375 


50 


3.125 


1.417 


15.26 


33.64 


15 


9-128 


0.0703125 


1.7859375 


45 


2.8125 


1.276 


13.73 


30.27 


16 


1-16 


0.0625 


1.5875 


40 


2.5 


1.134 


12.21 


26.91 


17 


9-160 


0.05625 


1 .42875 


36 


2.25 


1.021 


10.99 


24.22 


18 


1-20 


0.05 


1.27 


32 


2. 


0.9072 


9.765 


21.53 


19 


7-160 


0.04375 


1.11125 


28 


1.75 


0.7938 


8.544 


18.84 


20 


3-80 


0.0375 


0.9525 


24 


1.50 


0.6804 


7.324 


16.15 


21 


11-320 


0.034375 


0.873125 


22 


1.375 


0.6237 


6.713 


14.80 


22 


1-32 


0.03125 


0.793750 


20 


1.25 


0.567 


6.103 


13 46 


23 


9-320 


0.028125 


0.714375 


18 


1.125 


0.5103 


5.493 


12.11 


24 


1-40 


0.025 


0.635 


16 


1. 


0.4536 


4.882 


10.76 


25 


7-320 


0.021875 


0.555625 


14 


0.875 


0.3969 


4.272 


9.42 


26 


3-160 


0.01875 


0.47625 


12 


0.75 


0.3402 


3.662 


8.07 


27 


11-640 


0.0171875 


0.4365625 


11 


0.6875 


0.3119 


3.357 


7.40 


28 


1-64 


0.015625 


0.396875 


10 


0.625 


0.2835 


3.052 


6.73 


29 


9-640 


0.0140625 


0.3571875 


9 


0.5625 


0.2551 


2.746 


6.05 


30 


1-80 


0.0125 


0.3175 


8 


0.5 


0.2268 


2.441 


5.38 


81 


7-640 


0.0109375 


0.2778125 


7 


0.4375 


0.1984 


2.136 


4.71 


32 


13-1280 


0.01015625 


0.25796875 


gi^ 


0.40625 


0.1843 


1.9R3 


4.37 


33 


3-320 


0.009375 


0.238125 


6 


0.375 


0.1701 


1.831 


4.04 


34 


11-1280 


0.00859375 


0.21828125 


5^ 


0.34375 


0.1559 


1.678 


3 70 


35 


5-640 


0.0078125 


0.1984375 


5 


0.3125 


0.1417 


1.526 


3.36 


36 


9-1280 


0.00703125 


0.17859375 


41^ 


0.28125 


0.1276 


1.373 


3.03 


37 


17-2560 


0.006640625 


0.168671875 


4/4 


0.265625 


0.1205 


1.297 


2.87 


38 


1-160 


0.00625 


0.15875 


4 


0.25 


0.1134 


1.221 


2.69 



MATHEMATICS. 



Tlie Decimal Gauge. The legalization of the standard sheet-metaj 
gauge of 1893 and its adoption by some manufacturers of sheet iron have 
only added to the existing confusion of gauges. A joint committee of the 
American Society of Mechanical Engineers and the American Railway 
Master Mechanics' Association in 1895 agreed to recommend the use of the 
decimal gauge, that is, a gauge whose number for each thickness 3s the 
number of thousandths of an inch in that thickness, and also to recommend 
*'the abandonment and disuse of the various other gauges now in use, as 
tending to confusion and error. 1 " A notched gauge of oval form, shown in 
the cut below, has come into use as a standard form of the decimal gauge. 

In 1904 The Westinghouse Electric & Mfg. Co. abandoned the use of gaug 
numbers in referring to wire, sheet metal, etc. 

Weight of Sheet Iron and Steel. Thickness by Decimal 
Oauge 



uge 





00 

c 


1 


Weight per 
Square Foot 




03 
E 


to 

I 


Weight per 
Square Foot 




o 


1 


in Pounds. 


. 


o 


s 


in Pounds. 


fi 

be 

2 


!. 


a 




1 


JL 


| 




DB-p 




* ' 




I 


| 





5 < 


So> 


3 

O 


*! 





gfe 


^ 


1 


|a 


2 


|o 


|;> 


"3 


gg 


H 

2 


M 


^ 






ft 










ft 


c ^ 




1 


& o 


& 


ft 


i3 O 


02 


ft o 






i/ (-1 


Q 


3 







02 


H 


< 


5 


1 


02 


0.002 


1/500 


0.05 


0.08 


0.082 


0.060 


1/16 - 


1.52 


2.40 


2.448 


0.004 


1/250 


0.10 


0.16 


0.163 


0.065 


13/200 


1.65 


2.60 


2.652 


0.006 


3/500 


0.15 


0.24 


0.245 


0.070 


7/100 


1.78 


2.80 


2.856 


0.008 


1/125 


0.20 


0.32 


0.326 


0.075 


8/40 


1.90 


3.00 


3.060 


0.010 


1/100 


0.25 


0.40 


0.408 


0.080 


2/25 


2.03 


3.20 


3.264 


0.012 


3/250 


0.30 


0.48 


0.490 


0.085 


17/200 


2.16 


8.40 


3.468 


0.014 


7/500 


C.36 


0.56 


0.571 


0.090 


9/100 


2.28 


3.60 


3.672 


0.016 


1/64 -f 


41 


0.64 


0.653 


0.095 


19/200 


2.41 


3.80 


3.876 


0.018 


9/500 


0.46 


0.72 


0.734 


0.100 


1/10 


2.54 


4.00 


4.080 


0.020 


1/50 


0.51 


0.80 


0.816 


0.110 


11/100 


2.79 


4.40 


4.488 


0.022 


11/500 


0.56 


0.88 


0.898 


0.125 


1/8 


3.18 


5.00 


5.100 


0.025 


1/40 


0.64 


1.00 


1.020 


0.135 


27/200 


3.43 


5.40 


5.508 


0.028 


7/250 


0.71 


1.12 


1.142 


0.150 


3/20 


3.81 


6.00 


6.120 


0.032 


1/32 + 


0.81 


1.28 


1.306 


0.165 


33/200 


4.19 


6.60 


6.732 


0.036 


9/250 


0.91 


1.44 


1.469 


0.180 


9/50 


4.57 


7.20 


7.344 


0.040 


1/25 


1.02 


1.60 


1.632 


0.200 


1/5 


5.08 


8.00 


8.160 


0.045 


9/200 


1.14 


1.80 


1.836 


0.220 


11/50 


5.59 


8.80 


8.976 


0.050 


1/20 


1.27 


2 00 


2.040 


0.240 


8/25 


6.10 


9 60 


9.792 


0.055 


11/200 


1.40 


2.20 


2.244 


0.250 


1/4 


6.35 


10.00 


10.200 




ALGEBRA. 33 



ALGEBRA. 

Addition. Add a and ft. Ans. a-\-b. Add a, 6, and -c. Ans. a-\-b c. 
Ad'l 2a and 3. Ans. a. Add 2ab t Sab, c, 3c. Ans. - ab 4c. 
Subtraction. Subtract a from 6. Ans. b a. Subtract a from 6. 

Aus. b -f a. 

Subtract b -f c from a. Ans. a b c. Subtract 3a 2 6 9c from 4a 2 6 -f- c. 
Ans. 2 6 4- lOc. RULE: Change the signs of the subtrahend and proceed as 
in addition. 

Multiplication. Multiply a by 6. Ans. ab. Multiply ab bya-f b. 
Ans. a 2 6 + a6 2 . 

Multiply a -f 6 by a -\-b. Ans. (a-f 6)(a + 6) = a 2 -f 2a6 + 6 2 . 

Multiply a by b. Ans. a6. Multiply a by 6. Ans. ab. Like 
signs give plus, unlike signs minus. 

Powers of numbers. The product of two or more powers of any 
number is the number \\ith an exponent equal to the sum of the powers: 
a 2 x a 3 = o 5 ; a* IP x ab = a 3 6 3 ; - 7ab x 2ac = - 14 a 2 6c. 

To multiply a polynomial by a monomial, multiply each term of the poly- 
nomial by the monomial and add the partial products: (6a 36) x 3c = 18ac 
- 96c. 

To multiply two polynomials, multiply each term of one factor by each 
term of the other and add the partial products: (5a 66) x (3a 46) = 
15a 2 - 38a6 + 246 2 . 

The square of the sum of two numbers sum of their squares -f twice 
their product. \ 

The square of the difference of two numbers the sum of their squares 
twice their product. 

The product of the sum and difference of two numbers = the difference 
of their squares: 

(a-f6) 2 = a 2 +2a6-f 6 2 ; (a - 6) 2 =a 2 - 2a6-f 6 2 ; 
(a + 6) x (a- 6) = a 2 -6 2 . 

The square of half the sums of two quantities is equal to their product pi us 
the square of half their difference: ( a "^ J = ab -f- ( a ~ J 

The square of the sum of two quantities is equal to four times their prod- 
ucts, plus the square of their difference: (a -f 6; 2 = 4a6 + (a 6) 2 

The sum of the squares of two quantities equals twice their product, plus 
the square of their difference: a 2 -j- 6 2 = 2ab -{- (a 6) 2 . 

The square of a trinomial = the square of each term -f twice the product 
of each term by each of the terms that follow it: (a +6 + c) 2 = a 2 -f-6" -f- 
c 2 -4- 2ab -f 2ac +26f; (a - 6 - c) 2 = 2 + 6 2 + c 2 - 2ab - Sac -f 2bc. 

The square of (any number -f ^) = square of the number -4- the number 
-f- 14; =*the number X (the number -f 1) -f- J4; 
(a-f-U) = a*-l-a- r .J4, - a (a+l) + %. (4^) 2 =4 2 -4- 4-4^^= 



Tiie product of any number -f y> by any other number + *4 = product of 
the numbers + half their sum -f J4- ( a + ^) X 6 + ^) = a6 -f ^(a-f- 6)-f- J4. 
4> X 6^ = 4 X G -f- 1^(4 + 6) + H = 24 + 5 + H = 29^. 

Square, cube, 4tli poiver, etc., of a binomial a 4 6. 

(a + 6) 2 = a 2 -f- 26 -f 6 2 ; (a + 6) 3 = a* + 3a 2 6 -f 3a6 2 + 6 3 ; 

4a6 3 + 6 4 . 



In each case the number of terms is one greater than the exponent of 
the power to which the binomial is raised. 

2. In the first term the exponent of a is the same as the exponent of the 
power to which the binomial is raised, and it decreases by 1 in each succeed- 
ing term. 

3. 6 appears in the second term with the exponent 1, and its exponent 
increases by 1 in each succeeding term. 

4. The coefficient of the first term is 1. 

5. The coefficient of the second term is the exponent of the power to 
which the binomial is raised. 

6. The coefficient of each succeeding term is found from the next pre- 
ceding term }yy multiplying its coefficient by the exponent of a, and divid- 
ing the product by a number greater by 1 than the exponent of 6. (See 
Binomial Theorem, below.) 



34 ALGEBRA." 

Parentheses* When a parenthesis is preceded by a plus sign it may be 
removed wuhout changing the value of the expression: a -f b -f (a -f ft) 
2a -f- 2b. When a parenthesis is preceded by a minus sign it may be removed 
if we change the signs of all ihe terms within the parenthesis: 1 (a b 
c) \ a -{- ft + c. When a parenthesis is within a parenthesis remove 

the inner one first: a |ft ( c (d e) i a I ft \c d -f e ! | 

= a [6 c -f- "d e] = a b -\- c d -f- e. 

A multiplication sign, X, has the effect of a parenthesis, in that the oper- 
ation indicated by it must be performed before the operations of addition 
or subtraction, a -f- b X a + b a -f- ab -f- ft; while (a -f- b) X (a + b) = 
a 2 -f 2ab -f ft 2 , and (a -f b) X a + b = a 2 -f- aft + ft. 

Division. The quotient is positive when the dividend and divisor 
have like signs, and negative when they have unlike signs: abc -*- b = ac; 
abc -. b = ac. 

To divide a monomial by a monomial, write the dividend over the divisor 
with a line between them. If the expressions have common factors, remove 
the common factors: 

a 2 bx ax a 4 a 3 

a?bx-r-aby = - = ; = = CK 

aby y a 3 a 5 -- 

To divide a polynomial by a monomial, divide each term of the polynomial 
by the monomial: (Sab 12ac) -* 4a = 2b 3c. 

To divide a polynomial by a polynomial, arrange both dividend and divi- 
sor in the order of the ascending or descending powers of some common 
letter, and keep this arrangement throughout the operation. 

Divide the first term of the dividend by the first term of the divisor, and 
write the result as the first term of the quotient. 

Multiply all the terms of the divisor by the first term of the quotient and 
subtract the product from the dividend. If there be a remainder, consider 
it as a new dividend and proceed as before: (a 2 ft 2 ) -*- (a + b). 
a 2 - ft 2 | a + b. 
a* -\-fib | a"- b. 
-ab- ft 2 . 
- ab- ft 2 . 



The difference of two equal odd powers of any two numbers is divisible 
by their difference and also by their sum: 

(a 3 - ft 3 ) -t- (a - b) = a 2 -f ab + ft 2 ; (a 3 - ft 3 ) -*- (a -f- ft) = a 2 - ab + ft 2 . 

The difference of two equal even powers of two numbers is divisible by 
their difference and also by their ^um: (a 2 ft 2 ) -*- (a ft) = a + b. 

The sum of two equal even powers of two numbers is not divisible by 
either the difference or the sum of the numbers; but when the exponent 
of each of the two equal powers is composed of an odd and an even factor, 
the sum of the given power is divisible by the sum of the powers expressed 
by the even factor. Thus x* -f- y* is not divisible by x -f y or by x y^ but is 
divisible by x* + ?/ 2 . 

Simple equations. An equation is a statement of equality between 
two expressions; as, a -f- ft = c -f- d. 

A simple equation, or equation of the first degree, is one which contains 
only the first power of the unknown quantity. If equal changes be made 
(by addition, subtraction, multiplication, or division) in both sides of an 
equation, the results will be equal. 

Any term may be changed from one side of an equation to another, pro- 
vided its sign be changed: a -\- b = c -}- d; a = c -\- d ft. To solve an 
equation having one unknown quantity, transpose all the terms involving 
the unknown quantity to one side of the equation, and all the other terms 
to the other side; combine like terms, and divide both sides by the coefficient 
of the unknown quantity. 

Solve 8x - 29 = 26 - 3x. 8x -f 3x = 29 + 26; llx = 55; x = 5, ans. 

Simple algebraic problems containing one unknown quantity are solved 
by making x = the unknown quantity, and stating the conditions of the 
problem in the form of an algebraic equation, and then solving the equa- 
tion. What two numbers are those whose sum is 48 and difference 14 ? Let 
x = the smaller number, x + 14 the greater, x + x -f- 14 = 48. 2x = 34, x 
= 17; a; + 14 = 31, ans. 

Find a number whose treble exceeds 50 as much as its double falls short 
of 40. Let x = the number. 3x - 50 = 40 - 2x; 5x = 90; x - 18, ans. Prov- 
ing, 54 - 50 = 40 - 36. 



ALGEBRA, 35 

Equations containing tfwo unknown quantities. It one 

equation contains two unknown quantities, x and ?/, an indefinite number of 
pairs of values of x and y may be found that will satisfy the equation, but if 
a second equation be given only one pair of values can be found that will 
satisfy both equations. Simultaneous equations, or those that may be satis- 
fied by the same values of the unknown quantities, are solved by combining 
the equations so as to obtain a single equation containing only one unknown 
quantity. This process is called elimination. 

Elimination by addition or subtraction. Multiply the equation by 
such numbers as will make the coefficients of one of the unknown quanti- 
ties equal in the resulting equation. Add or subtract the resulting equa- 
tions according as they have unlike or like signs. 

Solve J 2 * + 3 2/ = 7. Multiply by 2: 4x + Vy = U 

7 } 4 X _ ty - 3. Subtract: 4x - 5y = 3 \\y = 11; y = 1. 

Substituting value of ?/ in first equation, 2x -f- 3 = 7; x = 2. 

Elimination by substitiLtion. From one of the equations obtain the 
value of one of the unknown quantities in terms of the other. Substitu- 
tute for this unknown quantity its value in the other equation and reduce 
the resulting equations. 



j2o; + 3/ = 8. (1). From (1) we find x = -. 

TQ l3x+7y = 7. (2). 

Substitute this value in (2): s( ~ ) + 7y = 7; = 24 - 9y -f 14y = 14, 

whence y = - 2. Substitute this value in (1): 2x - 6 = 8; x = 7. 

Elimination by comparison. From each equation obtain the value of 
one of the unknown quantities in terms of the other. Form an equation 
from these equal values, and reduce this equation. 



-9y=il. (1). From (1) we find x = 11 

Solve-! 

I 3x - 4y = 7. (2). From (2) we find x- l 
(. <* 

Equating these values of x, ~\ } ' ' * - - ?/ ; IQy = - 19; y = - 1. 

Substitute this value of ?/ in (1):~2# -J- 9 = 11; x = 1. 

If three simultaneous equations are given containing three unknown 
quantities, one of the unknown quantities must be eliminated between two 
pairs of the equations; then a second between the two resulting equations. 

Quadratic equations. A quadratic equation contains the square 
of the unknown quantity, but no higher power. A pure quadratic contains 
the square only; an affected quadratic both the square and the first power. 

To solve a pure quadratic, collect the unknown quantities on one side, 
and the known quantities on the other; divide by the coefficient of the un- 
known quantity and extract the square root of each side of the resulting 
equation. 

Solve 3# 2 - 15_= 0. 3tf 2 = 15; a; 2 = 5; x = |/5 

A root like ^5, which is indicated, but which can be found only approxi- 
mately, is called a surd. 

Solve 3o; 2 + 15 = 0. 3x* = - 15; x* = - 5; x = V- 5. 

The square root of 5 cannot be found even approximately, for the square 
of any number positive or negative is positive; therefore a root which is in- 
dicated, but cannot be found even approximately, is called imaginary. 

To solve an affected quadratic.]. Convert the equation into the form 
a 2 ^ 2 2abx = c, multiplying or dividing the equation if necessary, so as 
to make the coefficient of x' 2 a square number. 

2. Complete the square of the first member of the equation, so as to con- 
vert it to the form of a% 2 2abx + fc 2 , which is the square of the binomial 
ax b, as follows: add to each side of the equation the square of the quo- 
tient obtained by dividing the second term by twice the square root of the 
first term. 

3. Extract the square root of each side of the resulting equation. 

Solve 3x 2 - 4x = 32. To make the coefficient of x"* a square number, 
multiply by 3: 9a* 2 - 12# = 96; 12x H- (2 x 3x) = 2; 2 2 = 4. 
Complete the square: 9# 2 - 12x -j- 4 = 100 Extract the root: 3x 2 = 



36 ALGEBRA. 

10, whence x 4 or 2 2/3. The square root of 100 is either -f 10 or - 10, 
since the square of - 10 as well as -f 10 2 = 100. 

Problems involving quadratic equations have Apparently two solutions, as 
a quadratic has two roots. Sometimes botli will be true solutions, but gen- 
erally one only will be a solution and the other be inconsistent with the 
conditions of the problem. 

The sum of the squares of two consecutive positive numbers is 481. Find 
the numbers. 

Let x = one number, a; -f 1 the other, x* -f (x + I) 2 = 481. 2x"* -f 2x -f 1 
= 481. 

re 2 -f- x = 240. Completing the square, # 2 -f x + 0.25 = 240.25. Extracting 
the root we obtain x -f- 0.5 = !5.5; x = 15 or 16. 

The positive root gives for the numbers 15 and 16. The negative root 
16 is inconsistent with the conditions of the problem. 

Quadratic equations containing two unknown quantities require different 
meihods for their solution, according to the form of the equations. For 
these methods reference must be made to works on algebra. 
n - 

Theory of exponents. \a when n is a positive integer is one of n 

n - 

equal factors of a. \ a m means a is to be raised to the with power and the 
u th root extracted. 

(y ~a; means that the nth root of a is to be taken and the result 
raised to the with power. 

. -- / , -- \7 

y a m = Vy<* / = a n . When the exponent is a fraction, the numera- 
tor indicates a power, and the denominator a root. l = T* = a 3 ; i = 

VVr.3 = a 1 ' 5 . 

To extract the root of a quantity raised to an indicated power, divide 
the exponent by the index of the required root; as, 

n, . 3,- 6 

\a m =z a ' r a 6 = a 3 = a 2 . 

Subtracting 1 from the exponent of a is equivalent to dividing by a : 

a 2 -i =a = a; a 1 - 1 ^ a = - =1; a - 1 = a - 1 = - ; a - 1 - = a - a = 

A number with a negative exponent denotes the reciprocal of the number 
with the corresponding positive exponent. 

A factor under the radical sign whose root can be taken may, by having 
the root taken, be removed from under the radical sign: 

|/o2/7 = |/o2 x |/b = a tyb. 

A factor outside the radical sign may be raised to the corresponding 
power and placed under it: 



Binomial Theorem. To obtain any power, as the nth, of an ex- 
pression of the form x -{- a 

(a + * = i- f W- i + '""-r"-^ + 

etc. 

The following laws hold for any term in the expansion of (a -f x) n . 
The exponent of x is less by one than the number of terms. 
The exponent of a is n minus the exponent of x. 

The last factor of the numerator is greater by one than the exponent of a, 
The last factor of the denominator is the same as the exponent of x. 
In the rth term the exponent of x will be r - 1. 
The exponent of a will be n (r 1), or n r + 1. 
The last factor of the numerator will be n r + 2. 
The last factor of the denominator will be = r 1. 
Hence the rth term = L- 2> - -- ' + ^ - , + i x r-i 



GEOMETRICAL PROBLEMS. 



37 



GEOMETRICAL PROBLEMS. 



f E 



1. To bisect a straight line, 
or an arc of a circle (Fig. l}. 

From the ends A, B, as centres, de- 
scribe arcs intersecting at C and Z>, 
and draw a line through C and D 
which will bisect the line at E or the 
arc at F. 






2. To draw a perpendicular 
to a straight line, or a radial 
line to a circular arc. Same as 
in Problem 1. C D is perpendicular to 
the line A B, and also radial to the arc. 

3. To draw a perpendicular 
to a straight line from a given 
point in that line (Fig. 2). With 
any radius, from the given point A in 
the line B C, cut the line at B and C. 
With a longer radius describe arcs 
from B and (7, cutting each other at 
Z), arid draw the perpendicular D A. 

4. From the end A. of a given 
line A D to erect a perpendic- 
ular A E (Fig. 3). From any centre 
j<', above A D, describe a circle passing 
through the given point A, and cut- 
ting the given line at D. Draw D F 
and produce it to cut the circle at E, 
and draw the perpendicular A E. 

Second Method (Fig. 4). From the 
given point A set off a distance A E 
equal to three parts, by any scale ; 
and on the centres A and E, with radii 
of four and five parts respectively, 
describe arcs intersecting at C. Draw 
the perpendicular A C. 

NOTE. This method is most useful 
on very large scales, where straight 
edges are inapplicable. Any multiples 
of the numbers 3, 4, 5 may be taken 
with the same effect as 6, 8, 10, or 9, 
12, 15. 



5. To draw a perpendlcula 
o a, straight line from an 



lar 

to a straight line' from any 
point -without it (Fig. 5.) From 
the point A, with a sufficient radius 
cut the given line at F and G, and 
from these points describe arcs cut- 
ting at E. Draw the perpendicular 
AE. 



A B 

FIG. 6. 



6. To draw a straight line 
parallel to a given line, at a 
given distance apart (Fig. 6). 
From the centres A, B, in the given 
line, with the given distance as radius, 
describe arcs C, D, and draw the par- 
allel lines C D touching the arcs. 



38 



GEOMETRICAL PROBLEMS. 
G 




7. 'jfo divide a straight line 
into a number of equal parts 

(Fig. 7). To divide the line A B into, 
say, five parts, draw the line A C at 
an angle from ^4; set off five equal 
parts; draw B 5 and draw parallels to 
it from the other points of division in 
A C. These parallels divide A B as 
required. 

NOTE. By a similar process a line 
may be divided into a number of un- 
equal parts; setting off divisions on 
A O, proportional by a scale to the re- 
quired divisions, and drawing parallel 
cutting A B. The triangles All, A22, 
A83, etc., are 'similar triangles. 



FIG. 8. 





8. Upon a straight line to 
draw an angle equal to a 
given angle (Fig. 8). Let A be the 
given angle and F G the line. From 
the point A with any radius describe 
the arc D E. From F with the same 
radius describe I H. Set off the arc 
/ H equal to D K, and draw F H. The 
angle F is equal to A, as required. 



9. To draw angles of 60 

and 30 (Fig. 9). From f\ with 
any radius FL describe an arc IH ; 
and from 7, with the same radius, cut 
the arc at H and draw F H to form 
the required angle I F H. Draw the 
perpendicular H K to the base line to 
form the angle of 30- F H K. 



1O. To draw an angle of 45 

(Fig. 10). Set off the distance FT, 
draw the perpendicular I H equal to 
IF, and .loin HFto form the angle at 
F. The angle at H is also 45. 



11. To bisect an angle (Fig. 

11). Let A C B be the angle; with G 
as a centre draw an arc cutting the 
sides at A, B. From A and B as 
centres, describe arcs cutting each 
other at D. Draw C D, dividing the 
angle into two equal "parts. 



FIG. 1 



12. Through two given 
points to describe an arc of 
a circle with a given radius 

(Fig. 12). From the points A and B 
as centres, with the given radius, de< 
scribe arcs cutting at C , and from 
Cwith the same radius describe an 
arc A B. 



GEOMETRICAL PROBLEMS. 



39 




FIG. 13. 




FIG. 14. 



13. To find I lie centre of a 
circle or of an arc of a circle 

(Fig. 13). Select three points, A, B, 
(7, in the circumference, well apart; 
with the same radius describe arcs 
from these three points, cutting each 
other, and draw the two lines, D E, 
F G, through their intersections. The 
point O, where they cut, is the centre 
of the circle or arc. 

To describe a circle passing 
through three given points. 
Let A, B, G be the given points, and 
proceed as in last problem to find the 
centre O, from which the circle may 
be described. 

14. To 'describe an arc of 
a circle passing through 
three given points when 
the centre is not available 
(Fig. 14). From the extreme points 
A, B, as centres, describe arcs A H< 
B G. Through the third point O 
draw A E, B F, cutting the arcs. 
Divide A F and B E into any num- 
ber of equal parts, and set off a 
series of equal parts of the same 
length on the upper portions of the 
arcs beyond the points E F. Draw 
straight lines, B L, B M, etc., to 
the divisions in A F, and A I, A K, 
etc., to the divisions in E G. The 
successive intersections N, O, etc., 
of these lines are points in the 
circle required between the given 
points A and C. which may be 
drawn in ; similarly the remaining 
part of the curve B C may be 
described. (See also Problem 54.) 

15. To draw a tangent to 
a circle from a given point 
in the circumference (Fig. In). 
Through the given point A, draw the 
radial line A (7, and a perpendicular 
to it, F #, which is the tangent re- 
quired. 



16. To draw tangents to a 
circle from a point without 

it (Fig. 16). From A. with the radius 
A C, describe an arc B C Z>, and from 
C. with a radius equal to the diameter 
of the circle, cut the arc at B D. Join 
B <7, C D, cutting the circle at E F, 
and draw A E, A F, the tangents. 

NOTE. When a tangent is already 
drawn, the exact point of contact may 
be found by drawing a perpendicular 
to it from the centre. 



17. Between two inclined lines to draw a series of cir- 
cles touching these lines and touching each other (Fig. 1?). 
Bisect the inclination of the given lines A B, CD, by the line NO. From 
a point P in this line draw the perpendicular P B to the line A B, and 




FIG. 15. 




40 




GEOMETRICAL PROBLEMS. 
A 



on P describe the circle B D, touching 
the lines and cutting the centre line 
at E. From E d raw E F perpendicular 
to the centre line, cutting A B at F, 
and from F describe an arc E G, cut- 
ting A B at G. Draw G H parallel to 
B P, giving H, the centre of the next 
circle, to be described with the radius 
C JJ E, and so on for the next circle IN. 
Inversely, the largest circle may be 
described first, and the smaller ones 
in succession. This problem is of fre- 
quent use in scroll-work. 

18. Between two inclined 
lines to draw a circular seg- 
ment tangent to tlie lines and 
passing through a point !' 
on tlie line /' C which bisects 
the angle of the lines (Fig. 18). 
Through .Fclraw D A at right angles 
to F C ; bisect the angles A and D, as 
in Problem 11, by lines cutting at C, 
and from C with radius (7-Fdiaw the 
arc H F G required. 

19. To draw a circular arc 
that will he tangent to two 
given lines A Jl and C 1) in- 
clined to one another, one 
tangential point E being 
given (Fig. 19). Draw the centre 
line G F. From ldraw E Fat right 
to angles A B ; then F is the centre 
of the circle required. 

20. To describe a circular 
arc joining two circles, and 
touching one of them at a 
given point (Fig. 20). To join the 
circles .4 B, F G, by an arc touching 
one of them at F, draw the radius E f\ 
and produce it both ways. Set off F H 
equal to the radius A C of the other 
circle; join C H and bisect it with the 
perpendicular LI, cutting E F at L 
On the centre /, with radius IF, de- 
scribe the arc F A as required. 

21. To draw a circle with a 
given radius It that will be 
tangent to two given circles 

A. and J* (Fig. 21) From centre 
of circled with radius equal R plus 
radius of A, and from centre of B with 
radius equal to R + radius of B, draw 
two arcs cutting each other in (7, which 
will be the centre of the circle re- 
quired. 

22. To construct an equi- 
lateral triangle, the sides 
heing given (Fig. aa). On the ends 
of one side, A, B, with A B as radius, 
describe arcs cutting at C, and draw 
AC, CB. 



GEOMETRICAL PROBLEMS. 




B- 
C- 



FIG. 23. 





23. To construct a triangle 
of unequal sides (Fig. 28). On 
either end of the base A Z>, with the 
side B as radius, describe an arc; 
and with the side C as radius, on the 
other end of the base as a centre, cut 
the arc at E. Join A E, D E. 



24. To construct ft square 
on a given straight line A JK 

(Fig. 24). With A B as radius and A 
and B as centres, draw arcs A D and B 
C, intersecting at E. Bisect EB at F. 
With E as centre and E F as radius, 
cut the arcs A D and B C in D and C. 
Join A C, C Z>, and D J5 to form the 
square. 



25. To construct a rect- 
angle witli given base ./<; /' 
and height -E Jf (Fig. 25). On the 
base E Fdraw the perpendiculars EH, 
F G equal to the height, and join Q H. 



26. To describe a circle 
about a triangle (Fig. 26). 
Bisect two sides A B, A C of the tri- 
angle at E F, and from these points 
draw perpendiculars cutting at K. On 
the centre K, with the radius K A, 
draw the circle A B C. 



27. To inscribe a circle in 
a triangle (Fig. 27). Bisect two of 
the angles A, <7, of the triangle by lines 
cutting at D ; from D draw a per- 
pendicular D Eto any side, and with 
D E as radius describe a circle. 

When the triangle is equilateral, 
draw a perpendicular from one of the 
angles to the opposite side, and from 
the side set off one third of the per- 
pendicular. 

28. To describe a circle 
about a square, and to in- 
scribe a square in a circle (Fig. 
28). To describe the circle, draw the 
diagonals A B, C D of the square, cut- 
ting at E. On the centre E. with the 
radius A E, describe the circle. 

To inscribe the square. 
Draw the two diameters, A 5, CD, at 
right angles, and join the points A, B, 
C /), to form the square. 

NOTE. In the same way a circle may 
be described about a rectangle. 



GEOMBTEICAL PROBLEMS. 







29. To inscribe a circle in a 
square (Fig. 29). To inscribe the 
Circle, draw the diagonals A B, CD 
Of the square, cutting at E\ draw the 
perpendicular E F to one side, and 
with the radius E F describe the 
circle. 



30. To describe a square 
about a circle (Fig. 30). Draw two 
diameters A B, CD at right angles. 
With the radius of the circle and A, B, 
C and D as centres, draw the four 
half circles which cross one another 
in the corners of the square. 

9 

31. To inscribe a pentagon 
in a circle (Fig. 31). Draw diam- 
eters AC, B D at right angles, cutting 
at o. Bisect A o at E, and from E, 
with radius E B, cut A C at F ; from 
B, with radius B F, cut the circumfer- 
ence at Gr, H, and with the same radius 
step round the circle to /and K; join 
the points so found to form the penta 
gon. 



32. To construct a penta- 
gon on a given line A B (Fig. 

82). Frotti B erect a perpendicular 
B C half the length of A B\ join A C 
and prolong it to D, making CD = B C. 
Then B D is the radius of the circle 
circumscribing the pentagon. From 
A and B as centres, with B Das radius, 
draw arcs cutting each other in O, 
which is the centre of the circle. 

33. To construct a hexagon 
upon a given straight line 

(Fig. 33). From A and B, the ends of 
the given line, with radius A B, de- 
scribe arcs cutting at g ; from g, with 
the radius g A, describe a circle ; with 
the same radius set off the arcs A G, 
G F, and B D, D E. Join the points so 
found to form the hexagon. The side 
of a hexagon = radius of its circum- 
scribed circle, 

34. To inscribe a hexagon 
in a circle (Fig. 34). Draw a diam- 
eter A CB. From A audl? as centres, 
with the radius of the circle A (7, cut 
the circumference at D, E, F, G, and 
drawyl D, D E, etc., to form the hexa- 
gon. The radius of the circle is equal 
to the side of the hexagon ; therefore 
the points D, E, etc., may also be 
found by stepping the radius six 
times round the circle. The angle 
between the diameter and the sides of 
a hexagon and also the exterior angle 
between a side and an adjacent side 

Erolonged is 60 degrees; therefore a 
exagon may conveniently be drawn 
by the use of a 60-degree triangle. 



GEOMETKICAL PROBLEMS. 



43 






m l 



35. To describe a hexagon 
about a circle (Fig. 35). Draw a 
diameter AD B, and with the radius 
A D, on the centre A, cut the circum- 
ference at C ; join A C, and bisect it 
with the radius D E ; through E draw 
FG, parallel to A O, cutting the diam- 
eter at F, and with the radius D F de- 
scribe the circumscribing circle F H. 
Within this circle describe a hexagon 
by the preceding problem.. A more 
convenient method is by use of a 60- 
degree triangle. Four of the sides 
make angles of 60 degrees with the 
diameter, and the other two are par- 
allel to the diameter. 

36. To describe an octagon 
on a given straight line {Fig. 
36). Produce the given line A B both 
ways, and draw perpendiculars A E, 
B F', bisect the external angles A and 
B by the lines A H, B C, which make 
equal to A B. Draw C D and H G par- 

allel to A E, and equal to A B ; from 
the centres G, D, with the radius A B, 
cut the perpendiculars at E, F, and 
draw E F to complete the octagon. 

37. To convert a square 
into an octagon (Fig. 37). Draw 
the diagonals of the square cutting at 
e ; from the corners A, B, C, D, with 
A e as radius, describe arcs cutting 
the sides at gn, /fc, /im, and ol, and 
join the points so found to form the 
octagon. Adjacent sides of an octa- 
gon make an angle of 135 degrees. 





38. To inscribe an octagon 
in a circle (Fig. 38). Draw two 
diameters, A C, B D at right angles; 
bisect the arcs A B t B (7, etc., at ef\ 
etc., and join A e, e B, etc., to form 
the octagon. 



39. To describe an octagon 
about a circle (Fig. 39). Desci itr 
a square about the given circle A B , 
draw perpendiculars h k, etc. . to the 
diagonals, touchiDg the circle to form 
the octagon. 



4O. To describe a polygon of any number of sides upon 
** given straight line (Fig. 40). Produce the given line A B, and on A, 



44 



GEOMETRICAL PROBLEMS. 







with the radius A B, describe a semi- 
circle; divide the semi-circumference 
into as many equal parts as there are 
to be sides in the polygon say, in this 
example, five sides. Draw lines from 
A through the divisional points D, 6, 
and c, omitting one point a ; and on 
the centres J5, D, with the radius A B, 
cut A b at E and A c at F. Draw D E, 
E F, F B to complete the polygon. 

41. To Inscribe a circle 
within a polygon (Figs. 41, 42). 
When the polygon has an even number 
of sides (Fig. 41), bisect two opposite 
sides at A and B; draw A B, and bisect 
it at C by a diagonal D E, and with 
the radius C A describe the circle. 

When the number of sides is odd 
(Fig. 42), bisect two of the sides at A 
and B : and draw lines A E, B D to the 
opposite angles, intersecting at (7; 
from C, with the radius C A, describe 
the circle. 



42. To describe a circle 
without a polygon (Figs. 41, 42). 
Find the centre (J as before, and with 
the radius C D describe the circle. 



43. To inscribe a polygon 
of any number of sides with* 
in a circle (Fig. 43). Draw the 
diameter A B and through the centre 
E draw the perpendicular EC, cutting 
the circle at F. Divide E F into four 
equal parts, and set off three parts 
equal to those from F to C. Divide 
the diameter A B into as many equal 
parts as the polygon is to have sides ; 
and from C draw CD, through the 
second point of division, cutting the 
circle at D. Then A D is equal to one 
side of the polygon, and by stepping 
round the circumference with the 
length A D the polygon may be com- 
pleted. 



TABLE OF POLYGONAL ANGLES. 



Number 
of Sides. 


Angle 
at Centre. 


1 Number 
of Sides. 


Angle 
at Centre. 


Number 
of Sides. 


Angle 
at Centre. 


No. 


Degrees. 


No. 


Degrees. 


No. 


Degrees. 


3 


120 


9 


40 


15 


24 


4 


90 


10 


36 


16 


22 


5 


72 


11 


32 T " T 


17 


21 T 3 7 


6 


60 


12 


80 


18 


20 


7 
8 


S 


13 
14 


If 


19 
20 


19 

18 



GEOMETRICAL PROBLEMS. 



45 



C 




In this table the angle at the centre is found by dividing 360 degrees, the 
number of degrees in a circle, by the number of sides in the polygon; and 
by setting off round the centre of the circle a succession of angles by means 
of the protractor, equal to the angle in the table due to a given number of 
sides, the radii so drawn will divide the circumference into the same number 
of parts. 

44. To describe an ellipse 
when the length and breadth 
are given (Fig. 44). A B, transverse 
axis; C D, conjugate axis; F G, foci. 
The sum of the distances from C to 
.Fand G, also the sum of the distances 
from F and G to any other point in 
the curve, is equal to the transverse 
axis. From the centre C, with A E as 
radius, cut the axis AB at .Fand G, 
the foci ; fix a couple of pins into the 
axis at F and G, and loop on a thread 
or cord upon them equal in length to 
the axis A B, so as when stretched to 
reach to the extremity C of the con- 
jugate axis, as shown in dot-lining. 
Place a pencil inside the cord as at H, 
and guiding the pencil in this way, 
keeping the cord equally in tension, 
carry the pencil round the pins .F, G, 
and so describe the ellipse. 

NOTE. This method is employed in 
setting off elliptical garden - plots, 
walks, etc. 

2d Method (Fig. 45). Along the 
straight edge of a slip of stiff paper 
mark off a distance a c equal to A C, 
half the trans verse axis; and from the 
same point a distance a b equal to 
C D, half the conjugate axis. Place 
the slip so as to bring the point 6 on 
the line A B of the transverse axis, 
and the point c on the line D E ; and 
set off on the drawing the position of 
the point a. Shifting the slip so that 
the point b travels on the transverse 
axis, and the point c on the conjugate 
axis, any number of points in the 
curve may be found, through which 
the curve'may be traced. 

3d Method (Fig. 46). The action of 
the preceding method may be em- 
bodied so as to afford the means of 
describing a large curve continuously 
by means of a bar m fc, with steel 
points m, 7, fc, riveted into brass slides 
adjusted to the length of the semi- 
axis and fixed with set-screws. A 
rectangular cross E G, with guiding- 
slots is placed, coinciding with the 
two axes of the ellipse A C and B H. 
By sliding the points k, I in the slots, 
and carrying round the point m, the 
curve may be continuously described. 
A pen or pencil may be fixed at m. 

4th Method (Fig. 47). Bisect the 
transverse axis at C, and through C 
draw the perpendicular D E, making 
C D and C E each equal to half the 
conjugate axis. From D or E, with 
the radius A C, cut the transverse 
axis at F, F', for the foci. Divide 
A C into a number of parts at the 




FIG. 45. 




46 



GEOMETRICAL PROBLEMS. 




FIG. 48. 



P 2 ' & et v c * ^ th fche radhls ^ 7 on F and *" as centres, describe 
aics, and with the radius B I on the same centres cut these arcs as shown. 

Repeat the operation for the other 
divisions of the transverse axis. The 
series of intersections thus made are 
points in the curve, through which the 
curve may be traced. 

5th Method (Fig. 48). On the two 
axes A B, D E &s diameters, on centre 

C, describe circles; from a number of 
points a, 6, etc., in the circumference 
AFB, draw radii cutting the inner 
circle at a', b', etc. From a, b, etc., 
draw perpendiculars to AB; and from 
a', b', etc., draw parallels to A B, cut- 
ting the respective perpendiculars at 
n, o, etc. The intersections are points 
in the curve, through which the curve 
may be traced. 

6th Method (Fig. 49). When the 
transverse and conjugate diameters 
are given, A B, C D, draw the tangent 
EF parallel to A B. Produce CD, 
and on the centre G with the radius 
of half A B, describe a semicircle 
HDK; from the centre G draw any 
number of straight lines to the points 
E, r, etc., in the line E F, cutting the 
circumference at I, m, n, etc. ; from 
the centre O of the ellipse draw 
straight lines to the points E, r, etc. ; 
and from the points I, m, n, etc., draw 
parallels to G C, cutting the lines O E, 
Or, etc., at Z,, M, N, etc. These are 
points in the circumference of the 
ellipse, and the curve may be traced 
through them. Points in the other 
half of the ellipse are formed by ex- 
tending the intersecting lines as indi- 
cated in the figure. 

45. To describe an ellipse 
approximately by means of 
circular arcs. First. With arcs 
of two radii (Fig. 50j. Find the differ- 
ence of the semi-axes, and set it off 
from the centre O to a and c on O A 
and OC; draw ac, and set off half 
a c to d ; draw d i parallel to a c; set 
off O e equal to O d; join e i, and draw 
the parallels e m, d m. From m, with 
radius m C, describe an arc through 
C ; and from i describe an arc through 
Z); from d and e describe arcs through 
A and B. The four arcs form the 
ellipse approximately. 

NOTE. This method does not apply 
satisfactorily when the conjugate axis 
is less than two thirds of the trans- 
verse axis. 

2d Method (by Carl G. Earth, 
Fig. 51). -In Fig. 51 a & is the major 
and c d the minor axis of the ellipse 
to be approximated. Lay off b e equal 
to the semi-minor axis c 0, and use a e 
as radius for the arc at each extremity 
of the minor axis. Bisect e o at / and 
lay off e g equal to e /, and use g b as 
radius for the arc at each extremity 
of the major axis. 





FIG. 51. 



GEOMETRICAL PROBLEMS. 



47 




The method is not considered applicable for cases in which the minor 
axis is less than two thirds of the major. 

3d Method : With arcs of three radii 
^~- T-^^ (Fig. 52). On the transverse axis A B 

V ^- -^.-^-^Cl 7^... _f draw the rectangle B G on the height 

l\ / B^~ ^ OC; to the diagonal A C draw the 

perpendicular G H D\ set off OK 
equal to O C, and describe a semi- 
circle on A K, and produce O Cto I/; 
set off M equal to C L, and from D 
describe an arc with radius D M ; from 
A, with radius O -L, cut A B at JV; from 
H, with radius HJV, cut arc a 6 at a. 
Thus the five centres D, a, 6, H, H' 
are found, from which the arcs are 
described to form the ellipse. 

This process works well for neaily 
all proportions of ellipses. It is used 
in striking out vaults and stone bridges. 
4th Method (by F. R,. Honey, Figs. 53 and 54). Three radii are employed. 
With the shortest radius describe the two arcs which pass through the ver- 
tices of the major axis, with the longest the two arcs which pass through 
the vertices of the minor axis, and with the third radius the four arcs which 
connect the former. 

A. simple method of determining the radii of curvature is illustrated in 

Fig. 53. Draw the straight 
lines a f and a c, forming any 
angle at a. With a as a Centre, 
and with radii a b and a c, re- 
spectively, equal to the semi- 
minor and semi-major axes, 
draw the arcs b e and c d. Join 
ed, and through b and c re- 
spectively draw b g and c / 
parallel to e d, intersecting a c 
at g, and af at/; af is the 
radius of curvature at the ver- 
tex of the minor axis; and a g 




vertex of the major axis. 



the radius of curvature at the 



Lay off d h (Fig. 53) equal to one eighth of b d. Join e h, and draw c k and 
6 I parallel to e h. Take a k for the longest radius (= R), a I for the shortest 
radius (= rl and the arithmetical mean, or one half the sum of the semi-axes, 
for the third radius (= p), and employ these radii for the eight-centred oval 
as follows: 

Let a b and c d (Fig. 54) 
be the major and minor 
axes. Lay off a e equal 
to r, and af equal to p: 
also lay off c g equal to R, 
and c h equal to p. With 
g as a centre and g h as a 
radius, draw the arc h 7c; 
with the centre e and 
radius e f draw the arc / fc, 
intersecting hk at k. Draw 
the line g k and produce it, 
making g I equal to R. 
Draw ke and produce it, 
making k m equal to jp. 
With the centre g and 
radius g c ( R) draw the 
arc c I ; with the centre k 
and radius k I (= p) draw 
the arc I m, and with the 
centre e and radius e m 
(=r) draw the arc m a- 

The remainder of the 
work is symmetrical with 
respect to the axes. 




48 



GEOMETRICAL PROBLEMS. 



E 




A 


G 


2 


rxj 


F 


\ 


J 


O 


\ 


~\rc 


J 


o 


\ 


Y 


/ 


o 




\ 


D B 

b 

FIG 


. 55. 



' 46. The Parabola. A parabola 
(D A C, Fig. 55) is a curve such that 
every point in the curve is equally 
distant from the directrix KL&ud the 
focus F. The focus lies in the axis 
A B drawn from the vertex or head of 
the curve A, so as to divide the figure 
into two equal parts. The vertex A 
is equidistant from the directrix and 
the focus, or A e = A F. Any line 
parallel to the axis is a diameter. A 
straight line, as EG or DC, drawn 
across the figure at right angles to the 
axis is a double ordinate, and either 
half of it is an ordinate. The ordinate 
to the axis E F G, drawn through the 
focus, is called the parameter of the 
axis. A segment of the axis, reckoned 
from the vertex, is an abscissa of the 
axis, and it is an abscissa of the ordi- 
nate drawn, from the base of the ab- 
scissa. Thus, A B is an abscissa of 
the ordinate B C. 

Abscissae of a parabola are as the squares of their ordinates. 
To describe a parabola when an abscissa and its ordi- 
nate are given (Fig. 55). Bisect the given ordinate B Cat a, draw A a, 
and then a b perpendicular to it, meeting the axis at b. Set off A e, A F, 
each equal to B b; and draw KeL perpendicular to the axis. Then K L is 
the directrix and F is the focus. Through F and any number of points, o, o, 
etc., in the axis, draw double ordinates, n o n, etc , and from the centre F r 
with the radii Fe, o e, etc., cut the respective ordinates at E } G, n, n, etc. 
The curve may be traced through these points as shown. 
-iL 



2d Method : By means of a square 
and a cord (Fig. 56). Place a straight- 
edge to the directrix EN, and apply 
to it a square LEG. Fasten to the 
end G one end of a thread or cord 
equal in length to the edge E G, and 
attach the other end to the focus F', 
slide the square along the straight- 
edge, holding the cord taut against the 
e(ige of the square by a pencil D, by 
which the curve is described. 



3d Method: When the height and 
the base are given (Fig. 5?). Let A B 
be the given axis, and C D & double 
ordinate or base; to describe a para- 
bola of which the vertex is at A. 
Through A draw E F parallel to CD, 
and through C and D draw C E and 
D F parallel to the axis. Divide B C 
and B D into any number of equal 
parts, say five, at 'a, b, etc., and divide 
C E and DF into the same number of 
parts. Through the points a, ft, c, d in 
the base C D on each side of the axis 
draw perpendiculars, and through 
a, 6, c, d in C E and D F draw lines to 
the vertex A, cutting the perpendicu- 
lars at e. /, g, h. These are points in 
the parabola, and the curve C A D may 
be traced as shown, passing through 
then;. 




FIG. 56. 



A 




__ 


V 


t- 


e 


^x 


i^i 




' 


^ 


3 






./ 


\^ 


. : 


^/ 


t/ 










g 


x> 

















1 



C d cbaBabcd 
FIG. 57. 



GEOMETRICAL PROBLEMS. 



49 




FIG. 58. 




FIG. 59. 



47. The Hyperbola (Fig. 58). A hyperbola is a plane curve, such 
that the difference of the distances from any point of it to two fixed points 

is equal to a given distance. The fixed 
points are called the foci. 

To construct a hyperbola. 
Let F' and F be the foci, and F' F 
the distance between them. Take a 
ruler longer than the distance F r F, 
and fasten one of its extremities at the 
focus F'. At the other extremity, H, 
attach a thread of such a length that 
the length of the ruler shall exceed 
the length of the thread by a given 
distance A B. Attach the other ex- 
tremity of the thread at the focus F. 

Press a pencil, P, against the ruler, 
and keep the thread constantly tense, 
while the ruler is turned around F' as 
a centre. The point of the pencil will 
describe one branch of the curve. 

2d Method: By points (Fig. 59). 
From the focus F' lay off a distance 
F' N equal to the transverse axis, or 
distance between the two branches of 
the curve, and take any other distance, 
as F'H, greater than F'N. 

With F' as a centre and F'H as a 
radius describe the arc of a circle. 
Then with Fa,s a centre and N H as a 
radius describe an arc intersecting 
the arc before described at p and q. 

These will be points of the hyperbola, for F' q Fq is equal to the trans- 
verse axis A B. 

If, with F as a centre and F' H as a radius, an arc be described, and a 
second arc be described with F' as a centre and NH as a radius, two points 
in the other branch of the curve will be determined. Hence, by changing 
the centres, each pair of radii will determine two points in each branch. 

The Equilateral Hyperbola, The transverse axis of a hyperbola 
is the distance, on a line joining the foci, between the two branches of the 
curve. The conjugate axis is a line perpendicular to the transverse axis, 
drawn from its centre, and of such a length that the diagonal of the rect- 
angle of the transverse and conjugate axes is equal to the distance between 
the foci. The diagonals of this rectangle, indefinite!} 7 prolonged, are the 
asymptotes of the hyperbola, lines which the curve continually approaches, 
but touches only at an infinite distance. If these asymptotes are perpen- 
dicular to each other, the hyperbola is called a rectangular or equilateral 
hyperbola. It is a property of this hyperbola that if the asymptotes are 
taken as axes of a rectangular system of coordinates (see Analytical Geom- 
etry), the product of the abscissa and ordinate of any point in the curve is 
equal to the product of the abscissa and ordinate of any other point ; or, if 
p is the ordinate of any point and v its abscissa, and p^ and v t are the ordi- 
iiate and abscissa of any other point, pv=p* v\ ; or pv = a constant. 

48. The Cycloid 

K J5_ / (Fig. 60). If a circle Ad 

be rolled along a straight 
line 46, any point of the 
circumference as A will 
describe a curve, which is 
called a cycloid . The circle 
is called the generating 
circle, and A the generat- 
ing point. 

To draw a cycloid. 
Divide the circumference 
of the generating circle into an even number of equal parts, as A 1, 12, etc., 
and set off these distances on the base. Through the points 1, 2, 3, etc., on 
the circle draw horizontal lines, and on them set off distances la = A\, 
26 = A2, 'ic = A3, etc. The points A, a, 6, c, etc., will be points in the cycloid, 
through which draw the curve. 




50 



GEQMETKICAL PROBLEMS. 




49. The Epicycloid (Fig. 61) is 
generated by a point D in one circle 
D C rolling upon the circumference of 
another circle A C B, instead of on a 
flat surface or line; the former being 
the generating circle, and the latter 
the fundamental circle. The generat- 
ing circle is shown in four positions, in 
which the generating point is succes- 
sively marked D, D', D", D"'. A D'" B 
is the epicycloid. 



50. The Hypocycloid (Fig. 62) 

is generated by a point in the gener- 
ating circle rolling on the inside of the 
fundamental circle. 

When the generating circle radius 
of the other circle, the hypocycloid 
becomes a straight line. 



51. Tlie Traetrix or 
Schiele's anti-friction curve 

(Fig. 63). R is the radius of the shaft, 
C, 1,2, etc.. the axis. From O set off 
on R a small distance, o a; with radius 
R and centre a cut the axis at 1, join 
a 1, and set off a like small distance 
a 6; from b with radius R cut axis at 
2, join 6 2, and so on, thus finding 
points o, a, 6, c, d, etc., through which 
the curve is to be drawn. 
FIG. 63. 
52. The Spiral. The spiral is a curve described by a point which 

moves along a straight line according to any given law, the line at the same 

time having a uniform angular motion. The line is called the radius vector. 

If the radius vector increases directly 
as the measuring angle, the spires, 
or parts described in each revolution, 
thus gradually increasing their dis- 
tance from each other, the curve is 
known as the spiral of Archimedes 
(Fig. 64). 

This curve is commonly used for 
cams. To describe it draw the radius 
vector in several different directions 
around the centre, with equal angles 

between them; set off the distances 1, 2, 3, 4, etc., corresponding to the scale 

upon which the curve is drawn, as shown in Fig. 64. 
In the common spiral (Fig. 64) the pitch is uniform; that is, the spires are 

equidistant. Such a spiral is made by rolling up a belt of uniform thickness. 



To construct a spiral with 
four centres (Fig. 65). Given the 
pitch of the spiral, construct a square 
abont the centre, with the sum of the 
four sides equal to the pitch. Prolong 
the sides in one direction as shown; 
the corners are the centres for each 
arc of the external angles, forming a 
quadrant of a spire. 





Fig. 65. 



GEOMETRICAL PROBLEMS. 



51 




FIG. 



53. To find the diameter of a circle into which a certain 
number of rings will fit on its inside (Fig. 66). For instance, 
what is the diameter of a circle into which twelve J^-inch rings will fit, as 
per sketch ? Assume Uiat we have found the diameter of the required 

circle, and have drawn the rings inside 
of it. Join the centres of the rings 
by straight lines, as shown : we then 
obtain a regular polygon with 12 
sides, each side being equal to the di- 
ameter of a given ring. We have now 
to find the diameter of a circle cir- 
cumscribed about this polygon, and 
add the diameter of one ring to it; the 
sum will be the diameter of the circle 
into which the rings will fit. Through 
the centres A and D of two adjacent 
rings draw the radii CA and CD; 
since the polygon has twelve sides the 
angle A C D = 30 and A C B = 15. 
One half of the side A D is equal to 
A B. We now give the following pro- 
portion : The sine of the angle A C B 
is to A B as 1 is to the required ra- 
dius. From this we get the following 
rure : Divide A B by the sine of the angle A CB ; the quotient will be the 
radius of the circumscribed circle ; add to the corresponding diameter the 
diameter of one ring ; the sum will be the required diameter F G. 

54. To describe an arc of a circle which is too large to 
be drawn by a beam compass, by means of points in the 
arc, radius being given. Suppose the radius is 20 feet and it is 
desired to obtain five points in an arc whose half chord is 4 feet. Draw a 
line equal to the half chord, full size, or on a smaller scale if more con- 
venient, and erect a perpendicular at one end, thus making rectangular 
axes of coordinates. Erect perpendiculars at points 1, 2, 3, and 4 feet from 
the first perpendicular. Find values of y in the formula of the circle. 
#2 -f 2/2 = ^2 D y substituting for x the values 0, 1, 2, 3, and 4, etc.. and fov_R* 
the_squajre of _the radius, or 400. The values will be y = V R* a 2 = ^400, 
^399, ^396, ^391, ^384; = 20, 19.975, 19.90, 19.774, 19.596. 
Subtract the smallest, 

or 19.596, leaving 0.404, 0.379, 0.304, 0.178, feet. 

Lay off these distances on the five perpendiculars, as ordinates from the 
fcuUf chord, and the positions of five points on the arc will be found. 

Through these the curve may be 
drawn. (See also Problem 14.) 

55. The Catenary is the curve 
assumed by a perfectly flexible cord 
when its ends are fastened at two 
points, the weight of a unit length 
being constant. 
The equation of the catenary is 




- 

~ 



iV \ 
|, in 



which e is the 

base of the Naperian system of log- 
arithms. 
To plot the catenary. Let o 

(Fig. 67) be the origin of coordinates. 
Assigning to a any value as 3, the 
equation becomes 



FIG. 67. 



To find the lowest point of the curve. 

'0 -o 1 



GEOMETRICAL PROBLEMS. 



Thenput* = 1; .-. y = l\e* + e 3 ) = | (1.396 -f 0.717) = 8.17. 

= ? (1.948 -I- 0.513) = 3.69. 

Fut x = 3, 4, 5, etc., etc., and find the corresponding values of y. For 
each value of y we obtain two symmetrical points, as for example p 'and p l . 

In this way, by making a successively equal to 2, 3, 4, 5, 6, 7, and 8, the 
curves of Fig. 67 were plotted. 

In each case the distance from the origin to the lowest point of the curve 
is equal to a ; for putting x o, the general equation reduces to y a. 

For values of a = 6, 7, and 8 the catenary closely approaches the parabola. 
For derivation of the equation of the catenary see Bowser's Analytic 
Mechanics. For comparison of the catenary with the parabola, see article 
by F. R. Honey, Amer. Machinist, Feb. 1, 1894. 

56. The Involute is a name given to the curve which is formed by 

the end of a string which is unwound 
from a cylinder and kept taut ; con- 
sequently the string as it is unwound 
will always lie in the direction of a 
tangent to the cylinder. To describe 
the involute of any given circle, Fig. 
68, take any point A on its circum- 
ference, draw a diameter A B, and 
f rom B draw B b perpendicular to AB. 
Make Bb equal in length to half the 
circumference of the circle. Divide 
Bb and the semi-circumference into 
the same number of equal parts, 
say six. From each point of division 
1, 2, 3, etc., on the circumference draw 
lines to the centre C of the circle. 
Then draw 1 a perpendicular to C 1 ; 
2a 2 perpendicular to C2; and so on. 

F IG 68. Make la equal to b b, ; 2a 2 equal 

to b 6 2 ; 3 a 3 equal to b b 2 ; and so on. 

Join the points A, a^, a 2 , a 3 , etc., by a curve; this curve will be the 

required involute. 

57. Method of plotting angles without using a protrac- 
tor. The radius of a circle whose circumference is 360 is 57.3 (more ac- 
curately 57.296). Striking a semicircle with a radius 57.3 by any scale, 
spacers set to 10 by the same scale will divide the arc into 18 spaces of 10 
each, and intermediates can be measured indirectly at the rate of 1 by scale 
for each 1, or interpolated by eye according to the degree of accuracy 
required. The following table shows the chords to the above-mentioned 
radius, for every 10 degrees from up to 1JO. By means of one of these, 




Angle. Chord. 

1 0.999 

10 9.988 

20 19.899 

30 29.658 

40 39.192 

50.. .. 48.429 



Angle. Chord. 

60. 57.296 

70 65.727 

80 73.658 

90 81.029 

100 87.782 

110 93.869 



a 10 point is fixed upon the paper next less than the required angle, and 
the remainder is laid off at the rate of 1 by scale for each degree. 



GEOMETRICAL PROPOSITIONS. 53 



GEOMETRICAL PROPOSITIONS. 

In a right-angled triangle the square on the hypothenuse is equal to the 
sum of the squares on the other two sides. 

If a triangle is equilateral, it is equiangular, and vice versa. 

If a straight line from the vertex of an isosceles triangle bisects the base, 
it bisects the vertical angle and is perpendicular to the base. 

If one side of a triangle is produced, the exterior angle is equal to the sum 
of the two interior and opposite angles. 

If two triangles are mutually equiangular, they are similar and their cor- 
responding sides are proportional. 

If the sides of a polygon are produced in the same order, the sum of the 
exterior angles equals four right angles. (Not true if the polygon has re- 
entering angles ) 

In a quadrilateral, the sum of the interior angles equals four right angles. 

In a parallelogram, the opposite sides are equal ; the opposite angles are 
equal; it is bisected by its diagonal, and its diagonals bisect each other. 

If three points are not in the same straight line, a circle may be passed 
through them. 

If two arcs are intercepted on the same circle, they are proportional to 
the corresponding angles at the centre. 

If two arcs are similar, they are proportional to their radii. 

The areas of two circles are proportional to the squares of their radii. 

If a radius is perpendicular to a chord, tt bisects the chord and it bisects 
the arc subtended by the chord. 

A straight line tangent to a circle meets it in only one point, and it is 
perpendicular to the radius drawn to that point. 

If from a point without a circle tangents are drawn to touch the circle, 
there are but two; they are equal, and they make equal angles with the 
chord joining the tangent points. 

If two lines are parallel chords or a tangent and parallel chord, they 
Intercept equal arcs of a circle. 

If an angle at the circumference of a circle, between two chords, fa sub- 
tended by the same arc as an angle at the centre, between two ntcJii, the 
angle at the circumference is equal to half the angle at the centre. 

If a triangle is inscribed in a semicircle, it is right-angled. 

If two chords intersect each other in a circle, the rectangle of the seg- 
ments of the one equals the rectangle of the segments of the other. 

And if one chord is a diameter and the other perpendicular to it, the 
rectangle of the segments of the diameter is equal to the square on half the 
other chord, and the half chord is a mean proportional between the seg- 
ments of the diameter. 

If an angle is formed by a tangent and chord, it is measured by one half 
of the arc intercepted by the chord; that is, it is equal to half the angle at 
the centre subtended by the chord. 

Degree of a Railway Curve. This last proposition is useful in staking out 
railway curves. A curve is designated as one of so many degrees, and the 
degree is the angle at the centre subtended by a chord of 100 ft. To lay out 
a curve of n degrees the transit is set at its beginning or " point of curve, 1 ' 
pointed in the direction of the tangent, and turned through Y%n degrees; a 
point 100 ft. distant in the line of sight will be a point in the curve. The 
transit is then swung y z n degrees further and a 100 ft. chord is measured 
from the point already found to a point in the new line of sight, which is a 
second point or " station " in the curve. 

The radius of a 1 curve is 5729.05 ft., and the radius of a curve of any 
degree is 5729.05 ft. divided by the number of degrees. 



54 MENSURATION. 



MENSURATION. 

PLANE SURFACES. 

Quadrilateral. A four-sided figure. 

Parallelogram. A quadrilateral with opposite sides parallel. 

Varieties. Square : four sides equal, all angles right angles. Rectangle: 
opposite .sides equal, all angles right angles. Rhombus: four sides equal, 
opposite angles equal, angles not right angles. Rhomboid: opposite sides 
equal, opposite angles equal, angles not right angles. 

Trapezium. A quadrilateral with unequal sides. 

Trapezoid. A quadrilateral with only one pair of opposite sides 
parallel. 

Diagonal of a square = 4/2 x side 2 = 1.4142 X side. 



IMag. of a rectangle = |/sum of squares of two adjacent sides. 

Area of any parallelogram = base X altitude. 

Area of rhombus or rhomboid = product of two adjacent sides 
X sine of angle included between them. 

Area of a trapezium = half the product of the diagonal by the sum 
of the perpendiculars let fall on it from opposite angles. 

Area of a trapezoid = product of half the sum of the two parallel 
sides by the perpendicular distance between them. 

To find the area of any quadrilateral figure. Divide the 
quadrilateral into two triangles; the sum of the areas of the triangles is the 
area. 

Or, multiply half the product of the two diagonals by the sine of the angle 
at their intersection. 

To find the area of a quadrilateral inscribed in a circle. 
From half the sum of the four sides subtract each side severally; multi- 
ply the four remainders together; the square root of the product is the area. 

Triangle. A three-sided plane figure. 

Varieties. Right-angled, having one right angle; obtuse-angled, having 
one obtuse angle ; isosceles, having two equal angles and two equal sides? 
equilateral, having three equal sides and equal angles. 

The sum of the three angles of every triangle = 180. 

The sum of the two acute angles of a right-angled triangle = 90. 

Hypothenuse of a right-angled triangle, the side opposite the right angle, 
= |/sum of the squares of the other two sides. If a and 6 are the two sides 
and c the hypothenuse, c 2 = a 2 + b 2 ; a = f'c 2 - 6- = \/(c -f b)(c b). 

To find the area of a triangle : 

RULE 1. Multiplj 7 the base by half the altitude. 

RULE 2. Multiply half the product of two sides by the sine of the included 
angle. 

RULES. From half the sum of the three sides subtract each side severally; 
multiply together the half sum and the three remainders, and extract the 
square root of the product. 

The area of an equilateral triangle is equal to one fourth the square of one 

of its sides multiplied by the square root of 3, = , a being the side; or 

4 
a 2 X .433013. 

Hypothenuse and one side of right-angled triangle given, to find other side, 
Required side = ^hyp 2 given side 2 . 

If the two sides are equal, side = hyp -t- -1.4142; or hyp X .7071, 

Area of a triangle given, to find base: Base = twice area -f- perpendicular 
height 

Area of a triangle given, to find height: Height = twice area -H base. 

Two sides and base given, to find perpendicular height (in a triangle in 
which both of the angles at the base are acute). 

RULE. As the base is to the sum of the sides, so is the difference of the 
sides to the difference of the divisions of the base made by drawing the per- 
pendicular. Half this difference being added to or subtracted from half 
the base will give the two divisions thereof. As each side and its opposite 



PLANE SURFACES. 



55 



t fvision of the base constitutes a right-angled triangle, the perpendicular is 
ascertained by the rule perpendicular = Vhyp 2 base 2 . 

Polygon. A plane figure having three or more sides. Regular or 
irregular, according as the sides or angles are equal or unequal. Polygons 
are named from the number of their sides and angles. 

To find the area of an Irregular polygon. Draw diagonals 
dividing the polygon into triangles, and mid the sum of the areas of these 
triangles. 

To find the area of a regular polygon : 

RULE. Multiply the length of a side by the perpendicular distance to the 
centre; multiply the product by the number of sides, and divide it by 2. 
Or, multiply half the perimeter by the perpendicular let fall from the centre 
on one of the sides. 

The perpendicular from the centre is equal to half of one of the sides of 
the polygon multiplied by the cotangent of the angle subtended by the half 
Side. 

The angle at the centre = 360 divided by the number of sides. 

TABLE OF REGULAR POLYGONS. 









Radius of Cir- 
















cumscribed 


'd 


1 




t 









Circle. 




^ls 




3 




fcJO 


II 






</j a) 


II 


| 


Is 


B 

ri 


s 


i 


2 II 




o > 


ss^ 


4> 


|| 


o5 


o 


02 


dS 






^3 r2 




^ 










II 




->-> W O 


<D 


(D J> 


c 


a 


OJ 


0*0) 


<x> 





5f .5 .^5 


'So 


1^ 


d 


rt 












c 




* 




< 


P* 


00 


? 


^ 


< 


<J 


3 


Triangle 


.4330127 


o 


.5773 


.2887 


1.732 


120 


60 


4 


Square 


1. 


I'AU 


.7071 


.5 


1.4142 


90 


90 


6 


Pentagon 


1.7204774 


1.238 


.8506 


.6882 


1.1756 


72 


108 


(5 

7 


Hexagon 
Heptagon 


2 5980762 
3.6339124 


1,155 
1.11 


1. 
1.1524 


.866 
1.0383 


18677 


60 
5126' 


120 

128 4-7 


8 


Octagon 


4.8284271 


1.083 


1.3066 


1.2071 


.7653 


45 


135 


9 


Nonagon 


6.1818242 


1.064 


1.4619 


1.3737 


.684 


40 


140 


10 


Decagon 


7.6942088 


1.051 


1.618 


1.5388 


.618 


36 


144 


11 


Undecagon 


9.3656399 


1.042 


1.7747 


1.T028 


.5634 


32 43' 


1473-11 


12 


Dodecagon 


11.1961524 


1.037 


1.9319 


1.866 


.5176 


30 


150 



To find the area of a regular polygon, when the length 
of a side only is given : 

RULE. Multiply the square of the side by the multiplier opposite to the 
name of the polygon in the table. 

To find the area of an ir- 
regular figure (Fig. 69). Draw or- 

dinates across its breadth at equal 
distances apart, the first and the last 
ordinate each being one half space 
from the ends of the figure. Find the 
average breadth by adding together 
the lengths of these lines included be- 
tween the boundaries of the figure, 
and divide by the number of the lines 
added ; multiply this mean breadth by 
the length. The greater the number 
of lines the nearer the approximation. 



1 1 2 3 4 5 6 7 8 9 j 10 



t-j Length. 4 

FIG. 69. 

In a figure of very irregular outline, as an indicator-diagram from a high- 
ipeed steam-engine, mean lines may be substituted for the actual lines of the 
figure, being so traced as to intersect the undulations, so that the total area 
of the spaces cut off may be compensated by that of the extra spaces in- 
closed. 



56 MENSURATION 

2d Method: THE TRAPEZOIDAL RULE. Divide the figure into any suffi- 
cient number of equal parts; add half the sum of the two end ordinates to 
the sum of all the other ordinates; divide by the number of spaces (that is, 
one less than the number of ordinates) to obtain the mean ordinate, and 
multiply this by the length to obtain the area. 

3d Method: SIMPSON'S RULE. Divide the length of the figure into an.y 
even number of equal parts, at the common distance D apart, and draw or- 
dinates through the points of division to touch the boundary lines. Add 
together the first and last ordinates and call the sum A; add together the 
even ordinates and call the sum B; add together the odd ordiuates, except 
the first and last, and call the sum C, Then, 

area of the figure = x D. 

o 

4th Method: DURAND'S RULE. Add together 4/W the sum of the first and 
last ordinates, 1 1/10 the sum of the second and the next to the last (or the 
penultimates), and the sum of all the intermediate ordinates, Multiply the 
sum thus gained by the common distance between the ordinates to obtain 
the area, or divide this sum by the number of spaces to obtain the mean or- 
dinate. 

Prof. Duraucl describes the method of obtaining his rule in Engineering 
News, Jan. 18, 1891. He claims that it is more accurate than Simpson's rule>> 
and practically as simple as the trapezoidal rule. He thus describes its ap- 
plication for approximate integration of differential equations. Any deft- 
nite integral may be represented graphically by an area. Thus, let 

Q = J*u dx 

be an integral in which u is some function of #, either known or admitting of 
computation or measurement. Any curve plotted with x as abscissa and u 
as ordinate will then represent the variation of u with x. and the area be- 
tween such curve and the axis Xwill represent the integral in question, no 
matter how simple or complex may be the real nature of the function u, 

Substituting in the rule as above given the word ' volume " for " area ** 
and the word '* section " for " ordinate," it becomes applicable to the deter- 
mination of volumes from equidistant sections as well as of areas from 
equidistant ordinates. 

Having approximately obtained an area by the trapezoidal rule, the area 
by Durand's rule may be found by adding algebraically to the sum of the 
ordinates used in the trapezoidal rule (that is, half the sum of the end ordi- 
nates -f- sum of the other ordinates) 1/10 of (sum of penultimates sum of 
first and last) and multiplying by the common distance between the ordi- 

5th 'Method Draw the figure on cross-section paper. Count the number 
of squares that are entirely included within the boundary; then estimate 



ruling o the cross-section paper the more accurate the result 

6th Method.- Use a planimeter. 

7th Method With a chemical balance, sensitive to one milligram, draw 
the figure on paper of uniform thickness and cut it out carefully; weigh the 
piece cut out, and compare its weight with the weight per square inch of the 
oaper as tested by weighing a piece of rectangular shape. 



THE CIRCLE. 



57 



THE CIRCLE. 

Circumference = diameter x 3.1416, nearly; more accurately, 3.14159265359. 

Approximations, ^ = 3.143; ~ = 3.1415929. 

7 Ho 

The ratio of circum. to diam. is represented by the symbol n- (called Pi). 



Multiples of TT. 
ITT= 3.14159265359 

277 = 6.28318530718 
377 = 9.42477796077 
47r = 12.56637061436 
577=15.70796326795 
6i7=18.84955592I54 
777 = 21.99 11 4857513 
877 = 25.13274122872 
977 = 28.27433388231 



Multiples of -. 
'77 = .7853982 

" x 2=1.5707963 
" x 3 = 2.3561945 
" x 4=3.1415927 
" x 5 = 3.9269908 
" x 6 = 4.7123890 
" x 7 = 5.4977871 
x 8 = 6. 2831853 
" x 9 = 7.0685835 



Ratio of diam. to circumference = reciprocal of 77 = 0.3183099. 



procal of ^77 = 1.27324. 


- = 2.22817 

77 


~jr = 0.261799 


i 


a 


77 


Multiples of -. 


- = 2.54648 

77 


~- = 0.0087266 


= .31831 


- = 2.86479 


? = 114.5915 


77 


77 


77 


= .63662 


- = 3.18310 


772 = 9.86960 


77 


77 




= .95493 


= 3.81972 


= 0.1 01321 


77 


77 


77^ 


= 1.27324 

77 


-^77 = 1.570796 


VTT = 1 772453 


5 


j 


-y/l = 0.564189 


= 1.59155 


-77 = 1.047197 


7T 


77 


3 




= 1.90986 


*77= 0.523599 


Log 77= 0.49714987 


77 


6 


Log \v-= 1.895090 
4 



Diam. in ins. = 13.5405 Varea in sq. ft. 

Area in sq. ft. = (diam. in inches) 2 x .0054542. 

D = diameter, R = radius, C = circumference, 

A A 



A = area. 



C =77Z>;= 277/2; = ~; = 2*77.4; = 3.545^ ; 



= Z> 2 x .7854 ; = = 4 



x .7854 ; = 7 



~. 



>=-; =0.31831(7; ;=2V-r; = 1.12838*^1; 



R = x. ; = 0.159155C; = V - ; = 0.564189 ^A. 
27r' w ' 

Areas of circles are to each other as the squares of their diameters. 
To find the length of an arc of a circle : 

RULE 1. As 360 is to the number of degrees in the arc, so is the circum- 
ference of the circle to the length of the arc. 

RULE 2. Multiply the diameter of the circle by the number of degrees in 
the arc, and this product by 0.0087266. 



58 MENSURATION. 

Relations of Arc, Chord, Chord of Half the Arc, 
Versed Sine, etc. 

Let R = radius, D = diameter, Arc length of arc, 
Cd = chord of the arc, ch = chord of half the arc, 
F = versed sine, or height of the arc, 

Sch Cd . Vcd* + 4F* x 10 T a 

Arc = __ (very nearly), = 15C ^ + 33Fa + 2ch < Deai>lv ' 

2ch x 10F , 

ArC = GQD-27V 4 ' Uea y ' 
Chord of the arc = 2 Vch*-V*; = VD* - (D- 2F)~2; = Sch - 3 Arc. 



= 2V J R2-(#-F) 2 ; _ 2 V(lT_ F) x F. 



Chord of half the arc, ch = \^ Cd* -f 4 F a ; = i 



ch* 
Diameter = -==; 



Versed sine 



1 . 

(or -(D + v Z) 2 - Cd 2 ), if F is greater than radius 



Half the chord of the arc is a mean proportional between the versed sine 
and diameter minus versed sine: Y^Cd |/F x (D - F) 

Length of the Chord subtending an angle at the centre = twice the 
sine of halt the angle. (See Table of Sines, p. 15?..) 

Length of a Circular Arc. Huyghens's Approximation. 

Let C represent the length of the chord ot the arc and c the length of the 
chord of half the arc; the length of the arc 






3 ' 

Professor Williamson shows that when the arc subtends an angle of 30, the 
radius being 100,000 feet (nearly 19 miles), the error by this formula is about 
two inches, or 1/600000 part of the radius. When the length of the arc is 
equal to the radius, i.e., when it subtends an angle of 57. 3, the error is less 
than 1/7680 part of the radius. Therefore, if the radius is 100,000 feet, the 

100000 
error is less than =13 feet. The error increases rapidly with the 

increase of the angle subtended. 

In the measurement of an arc which is described with a short radius the 
error is so small that it may be neglected. Describing an arc with a radius 
of 12 inches subtending an angle of 30, the error is 1/50000 of an inch. For 
57. 3 the error is less than 0".0015. 

In order to measure an arc when it subtends a large angle, bisect it and 
measure each half as before in this case making B = length of the chord of 
half the arc, and b = length of the chord of one fourth the arc ; then 
T _ 166 - 2B 

~F 

Relation of the Circle to its Equal, Inscribed, and Cir- 
cumscribed Squares. 

Diameter of circle x .88623 _ . , f , , miarp 

Circumference of circle x .28209 f = . equal squaie. 

Circumference of circle x 1.1284 = perimeter of equal square. 



THE ELLIPSE. 59 

Diameter of circle x .7071 J 

Circumference of circle x .22508 v = side of inscribed square. 

Area of circle x .90031n- diameter j 

Area of circle x 1.273-2 = area of circumscribed square. 

Area of circle x .63662 = area of inscribed square. 

Side of square x 1.4142 = diarn. of circumscribed circle. 

x 4.4428 = circum. " 

" x 1.1284 = d jam. of equal circle. 

" x 3.5449 = circum. " " 

Perimeter of square x 0.88623 = 
Square inches x 1.2732 = circular inches. 

Sectors and Segments. To find the area of a sector of a circle. 
RULE 1. Multiply the arc of the sector by half its radius. 
RULE 2. As 360 is to the number of degrees in the arc, so is the area of 
the circle to the area of the sector. 

RULE 3. Multiply the number of degrees in the arc by the square of the 
radius and by .008727. 

To find the area of a segment of a circle: Find the area of the sector 
which has the same arc, and also the area of the triangle formed by the 
chord of the segment and the radii of the sector. 

Then take the sum of these areas, if the segment is greater than a semi- 
circle, but take their difference if it is less. 

Another Method ; Area of segment = ^>#(arc - sin A), in which A is the 
central angle, R the radius, and arc the length of arc to radius J 

To find the area of a segment of a circle when its chord and height only 
are given. First find radius, as follows : 



1 P square of half the chord , 
radms = L height - + he ' 



2. Find the angle subtended by the arc, as follows: half chord -f- radius = 
sine of half the angle. Take the corresponding angle from_a table of sines, 
and double it to get the angle of the arc. 

3. Find area of the sector of which the segment is a part; 

area of sector = area of circle x degrees of arc -f- 360. 

4. Subtract area of triangle under the segment]: 

Area of triangle = half chord x (radius height of segment). 

The remainder is the area of the segment. 

When the chord, arc, and diameter are given, to find the area. From the 
length of the arc subtract the length of the chord. Multiply the remainder 
by the radius or one-half diameter; to the product add the chord multiplied 
by the height, and divide the sum by 2. 

Given diameter, d, and height of segment, h. 

When h is from to Yd, area = h ^1.766d/i -1>; _ 



_ 
" " " " Y 4 d to y%d, area =-. h Vo.017d 2 + 1.7d/i- A a 

(approx.). Greatest error 0.23#, when h y^d. 

To-find the chord: From the diameter subtract the height ; multiply the 
remainder by four times the height and extract the square root. 

When the chords of the arc and of half the arc and the rise are given: Tc 
the chord of the arc add four thirds of the chord of half the arc; multiply 
the sum by the rise and the product by .40426 (approximate). 

Circular King:. To find the area of a ring included between the cir- 
cumferences of two concentric circles; Take the difference between the areas 
of the two circles; or, subtract the square of the less radius from the square 
of the greater, and multiply their difference by 3.14159. 

The area of the greater circle is equal to nRV; 
and the area of the smaller, wr 2 . 

Their difference, or the area of the ring, is TrCR 2 - r 2 ). 

Tlie Ellipse. Area of an ellipse = product of its semi-axes x 3.14159 

= product of its axes x .785398. 

The Ellipse. Circumference (approximate) = 3.1416 V i , D and d 

being the two axes. 

Trautwine gives the following as more accurate: When the longer axis D 
is not more than five times the length of the shorter axis, d, 



60 MENSUKATIOJT. 

Circumference = 3.1416 



_ 

/i O.O 

When D is more than 5d, the divisor 8.8 is to be replaced by the following : 
ForD/d = 6 7 8 9 10 1:3 14 16 18 20 30 40 50 
Divisor* = 9 9.3 9.3 9.35 9.4 9.5 9.6 9.68 9.75 9.8 9.92 9.98 10 

/ AI A^ A^ 2*1/48 \ 

An accurate formula is O = (o + 6) (l + + - + ~~ + ~^ . . . ) , in 



which A = -r-y. Ingenieurs Taschenbuch, 1896. 

Carl G. Barth (Machinery, Sept., 1900) gives as a very close approximation 
to this formula 

_, , ... 64 - 3^4* 
<*="< + > 64^1035- 

^rea o/ a segment of an ellipse the base of which is parallel to one of 
the axes of the ellipse. Divide the height of the segment by the axis of 
which it is part, and ftnd the area of a circular segment, in a table of circu- 
lar segments, of which the height is equal to the quotient; multiply the area 
thus found by the product of the two axes of the ellipse. 

Cycloid. A curve generated by the rolling of a circle on a plane. 
Length of a cycloidal curve ~ 4 X diameter of the generating circle. 
Length of the base = circumference of the generating circle. 
Area of a cycloid = 3 X area of generating circle. 

Helix (Screw). A line generated by the progressive rotation of a 
point around an axis and equidistant from its centre. 

Length of a helix. -To the square of the circumference described by the 
generating-point add the square of the distance advanced in one revolution, 
and take the square root of their sum multiplied by the number of revolu- 
tions of the generating point. Or, 

V(c 2 + h"*)n = length, n being number of revolutions. 

Spirals. Lines generated by the progressive rotation of a point around 
a fixed axis, with a constantly increasing distance from the axis. 

A plane spiral is when the point rotates in one plane. 

A conical spiral is when the point rotates around an axis at a progressing 
distance from its centre, and advancing in the direction of the axis, as around 
a cone. 

Length of a plane spiral line. When the distance between the coils is 
uniform. 

RULE. Add together the greater and less diameters; divide their sum by 
2; multiply the quotient by 3.1416, and again by the number of revolutions. 
Or, take the mean of the length of the greater and less circumferences and 
multiply it by the number of revolutions. Or, 

length = irn -~ -, d and d' being the inner and outer diameters. 

Length of a conical spiral line. Add together the greater and less diam- 
eters; divide their sum by 2 and multiply the quotient by 3.1416. To the 
square of the product of this circumference and the number of revolutions 
of the spiral add the square of the height of its axis and take the square 
root of the sum. 



Or, length = + h*. 



SOLID BODIES. 

Xlie Prism. To find the surface of a right prism : Multiply the perim- 
eter of the base by the altitude for the convex surface. To this add the 
areas of the two ends when the entire surface is required. 

Volume of a prism = area of its base X its altitude. 

The pyramid. Convex surface of a regular pyramid = perimeter of 
its base X half the slant height. To this add area of the base if the whole 
surface is required. 

Volume of a pyramid = area of base X one third of the altitude. 



SOLID BODIES. 61 

To find the surface of a frustum of a regular pyramid : Multiply half the 
slant height by the sum of the perimeters of the two bases for the convex 
surface. To this add the areas of the two bases when the entire surface is 
required. 

To find the volume of a frustum of a pyramid : Add together the areas of 
the two bases and a mean proportional between them, and multiply the 
sum by one third of the altitude. (Mean proportional between two numbers 
= square root of their product.) 

Wedge. A wedge is a solid bounded by five planes, viz.: a rectangular 
base, two trapezoids, or two rectangles, meeting in an edge, and two tri- 
angular ends. The altitude is the perpendicular drawn from any point in 
the edge to the plane of the base. 

To find the volume of auiedge: Add the length of the edge to twice the 
length of the base, and multiply the sum by one sixth of the product of the 
height of the wedge and the breadth of the base. 

Rectangular prismoid. A rectangular prismoid is a solid bounded 
by six planes, of which the two bases are rectangles, having their corre- 
sponding sides parallel, and the four upright sides of the solids are trape- 
zoids. 

To find the volume of a rectangular prismoid : Add together the areas of 
the two bases and four times the area of a parallel section equally distant 
from the bases, and multiply the sum by one sixth of the altitude. 

Cylinder. Convex surface of a cylinder = perimeter of base X altitude. 
To this add the areas of the two ends when the entire surface is required. 
Volume of a cylinder = area of base X altitude. 

Cone. Convex surface of a cone = circumference of base X half the slant 
side. To this add the area of the base when the entire surface is required. 
Volume of a cone = area of base X one third of the altitude. 

To find the surface of a frustum of a cone: Multiply half the side by the 
sum 6t the circumferences of the two bases for the convex surface; to this 
add the areas of the two bases when the entire surface is required. 

To find the volume of a frustum of a cone : Add together the areas of the 
two bases and a mean proportional between them, and multiply the sum by 
one third of the altitude. Or, Vol. = 0.2618a(b a + c 3 + be) ; a = altitude : 
6 and c, diams. of the two bases. 

Sphere. To find the surface of a sphere : Multiply the diameter by the 
ciicumference of a great circle; or, multiply the square of the diameter by 
3.14159. 

Surface of sphere = 4 X area of its great circle. 

*' *' " = convex surface of its circumscribing cylinder. 

Surfaces of spheres are to each other as the squares of their diameters. 

To find the volume of a sphere : Multiply the surface by one third of the 
radius; or, multiply the cube of the diameter by 7r/6; that is, by 0.5236. 

Value of TT/O to 10 decimal places = .5235987756. 

The volume of a sphere = 2/3 the volume of its circumscribing cylinder. 

Volumes of spheres are to each other as the cubes of their diameters. 

Spherical triangle. To find the area of a splierical triangle : Com- 
pute the surface of the quadrantal triangle, or one eighth of the surface of 
the sphere. From the sum of the three angles subtract two right angles; 
divide the remainder by 90, and multiply the quotient by the area of the 
quadrantal triangle. 

Spherical polygon. To find the area of a spherical polygon: Com- 
pute the surface of the quadrantal triangle. From the sum of all the angles 
subtract the product of two right angles by the number of sides less two; 
divide the remainder by 90 and multiply the quotient by the area of the 
quadrantal triangle. 

The prismoid. The prismoid is a solid having parallel end areas, and 
may be composed of any combination of prisms, cylinders, wedges, pyra- 
mids, or cones or frustums of the same, whose bases and apices lie in the 
end areas. 

Inasmuch as cylinders and cones are but special forms of prisms and 
pyramids, and warped surface solids may be divided into elementary forms 
of them, and since frustums may also be subdivided into the elementary 
forms, it is sufficient to say that all prismoids may be decomposed into 
prisms, wedges, and pyramids. If a formula can be found which is equally 
applicable to all of these forms, then it will apply to any combination of 
them. Such a formula is called 



MENSURATION-. 

The Prismoidal Formula. 

Let A area of the base of a prism, wedge, or pyramid; 
%, Am = the two end and the middle areas of a prismoid, or of any of 
its elementary solids; 

h = altitude of the prismoid or elementary solid; 

V its volume; 



For a prism, A t , Am and A z are equal, = A ; V=-x6A = liA. 

For a wedge with parallel ends, A^ = 0, Am = -A l ; V ^(A l -f ZAJ = - 

For a cone or pyramid, A% = 0, Am = -.A^ V = ~(A } + AJ - ~. 

4 O O 

The prismoidal formula is a rigid formula for all prismoids. The only 
approximation involved in its use is in the assumption that the given solid 
may be generated by a right line moving over the boundaries of the end 
areas. 

The area of the middle section is never the mean of the two end areas if 
the prismoid contains any pyramids or cones among its elementary forms. 
When the three sections are similar in form the dimensions of the middle 
area are always the means of the corresponding end dimensions. This fact 
often enables the dimensions, and hence the area of the middle section, to 
be computed from the end areas. 

Polyedrons. A polyedron is a solid bounded by plane polygons. A 
regular pblyedrou is one whose sides are all equal regular polygons. 

To find the surface of a regular polyedron. Multiply the area of one of 
the faces by the number of faces ; or, multiply the square of one of the 
edges by the surface of a similar solid whose edge is unity. 

A TABLE OF THE REGULAR POLYEDRONS WHOSE EDGES ARE UNITY. 

Names. No. of Faces. Surface. Volume. 

Tetraedron .......................... 4 3.7320508 0.1178513 

Hexaedron. .' ......................... 6 6.0000000 3 .0000000 

Octaeclron ........................... 8 3. 4641016 0.4714045 

Dodecaedron ......................... 12 20.6457288 7.6631189 

Icosaedron ........................... 20 8.GCO-J540 2.1816950 

To find the volume of a regular polyedron. Multiply the 
surface by one third of the perpendicular let fall from the centre on one of 
the faces ; or, multiply the cube of one of the edges by the solidity of a 
similar polyedron whose edge is unity. 

Solid of revolution. The volume of any solid of revolution is 
equal to the product of the area of its generating surface by the length of 
the path of the centre of gravity of that surface. 

The convex surface of any solid of revolution is equal to the product of 
the perimeter of its generating surface by the length of path of its centre 
of gravity. 

Cylindrical ring. Let d = outer diameter ; d' inner diameter ; 

- (d d') = thickness = t\ -irf 2 = sectional area ; ~(d-\-d') = mean diam- 
eter = M ; TT t = circumference of section ; irM mean circumference of 
ring; surface = TT t X * M; = ^ 772 (d 2 - d' 2 ); = 9.86965 1 M- = 2.46741 (d 2 -d' 2 ); 

volume = 7 TT t* M TT; = 2.46741*2 M. 
4 

Spherical zone. Surface of a spherical zone or segment of a sphere 
= its altitude X the circumference of a great circle of the sphere. A great 
circle is one whose plane passes through the centre of the sphere. 

Volume of a zone of a sphere. To the sum of the squares of the radii 
of the ends add one third of the square of the height ; multiply the sum 
by the height and by 1.5708. 

Spherical segment. Volume of a spherical segment with one base. 



SOLID BODIES. 63 

i Multiply half the height of the segment by the area of the base, and the 
cube of the height by .5236 and add the two products. Or, from three times 
the diameter of the sphere subtract twice the height of the segment; multi- 
ply the difference by the square of the height and by .5236. Or, to three 
times the square of the radius of the base of the segment add the square of 
its height, and multiply the sum by the height and by .5236. 

Spheroid or ellipsoid. When the revolution of the spheroid is about 
: the transverse diameter it is prolate, and when about the conjugate it is 
oblate. 

Convex surface of a segment of a spheroid. Square the diameters of the 
: spheroid, arid take the square root of half their sum ; then, as the diameter 
; from which the segment is cut is to this root so is the height of the 
segment to the proportionate height of the segment to the mean diameter. 
; Multiply the product of the other diameter and 3.1416 by the proportionate 
? height. 

Convex surface of a frustum or zone of a spheroid. Proceed as by 
t previous rule for the surface of a segment, and obtain the proportionate 
I height of the frustum. Multiply the product of the diameter parallel to the 
base of the frustum and 3.1416 by the proportionate height of the frustum. 

Volume of a spheroid is equal to the product of the square of the revolving 
axis by the fixed axis and by .5236. The volume of a spheroid is two thirds 
i. of that of the circumscribing cylinder. 

Volume of a segment of a spheroid.]. When the base is parallel to the 

i revolving axis, multiply the difference between three times the fixed axis 

and twice the height of the segment, by the square of the height and by 

; .5236. Multiply the product by the square of the revolving axis, and divide 

| by the square of the fixed axis. 

2. When the base is perpendicular to the revolving axis, multiply the 

! difference between three times the revolving axis and twice the height of 

the segment by the square of the height and by .5236. Multiply the 

product by the 'length of the fixed axis, and divide by the length of the 

revolving axis. 

Volume of the middle frustum of a spheroid.]. When the ends are 
circular, or parallel to the revolving axis : To twice the square of the 
middle diameter add the square of the diameter of one end ; multiply the 
sum by the length of the frustum and by .2618. 

2. When the ends are elliptical, or perpendicular to the revolving axis: 
To twice the product of the transverse and conjugate diameters of the 
middle section add the product of the transverse and conjugate diameters 
of one end ; multiply the sum by the length of the frustum and by .2618. 

Spindles. Figures generated by the revolution of a plane area, when 
the curve is revolved about a chord perpendicular to its axis, or about its 
double ordinate. They are designated by the name of the arc or curve 
from which they are generated, as Circular, Elliptic, Parabolic, etc., etc. 

Convex surface of a circular spindle, zone, or segment of it Rule: Mul- 
tiply the length by the radius of the revolving arc; multiply this arc by the 
central distance, or distance between the centre of the spindle and centre 
of the revolving arc ; subtract this product from the former, double the 
remainder, and multiply it by 3.1416. 

Volume of a circular spindle. Multiply the central distance by half the 
area of the revolving segment; subtract the product from one third of the 
cube of half the length, and multiply the remainder by 12.5664. 

Volume of frustum or zone of a circular spindle. From the square of 
half the length of the whole spindle take one third of the square of half the 
length ofj the frustum, and multiply the remainder by the said half length 
of the frustum ; multiply the central distance by the revolving area which 
generates the frustum ; subtract this product from the former, and multi- 
ply the remainder by 6.2832. 

Volume of a segment of a circular spindle. Subtract the length of the 
segment from the half length of the spindle ; double the remainder and 
ascertain the volume of a middle frustum of this length ; subtract the 
result from the volume of the whole spindle and halve the remainder. 

Volume of a cycloidal spindle = five eighths of the volume of the circum- 
scribing cylinder. Multiply the product of the square of twice the diameter 
of the generating circle and 3.927 by its circumference, and divide this pro- 
duct by 8. 

Parabolic -conoid. Volume of a parabolic conoid (generated by the 
revolution of a parabola on its axis). Multiply the area of the base by half 
the height. 



64 MENSURATION. 

Or multiply the square of the diameter of the base by the height and by 

Volume of a frustum of a parabolic conoid. Multiply half the sum of 
the areas of the two ends by the height. 

Volume of a parabolic spindle (generated by the revolution of a parabola 
on its base). Multiply the square of the middle diameter by the length 
and by .4189. 

The volume of a parabolic spindle is to that of a cylinder of the same 
height and diameter as 8 to 15. 

Volume of the middle frustum of a parabolic spindle. Add together 
8 times the square of the maximum diameter, 3 times the square of the end 
diameter, and 4 times the product of the diameters. Multiply the sum by 
the length of the frustum and by .05236. 

This rule is applicable for calculating, the content of casks of parabolic 
form. 

Casks. To find the volume of a cask of any form. Add together 39 
times the square of the bung diameter, 25 times the square of the head 
diameter, and 26 times the product of the diameters. Multiply the sum by 
the length, and divide by 31,773 for the content in Imperial gallons, or by 
26,470 for U. S. gallons. 

This rule was framed by Dr. Hutton, on the supposition that the middle 
third of the length of the cask was a frustum of a parabolic spindle, and 
each outer third was a frustum of a cone. 

To find the ullage of a cask, the quantity of liquor in it when it is not full. 
1. For a lying cask : Divide the number of wet or dry inches by the bung 
diameter in inches. If the quotient is less than .5, deduct from it one 
fourth part of what it wants of .5. If it exceeds .5, add to it one fourth part 
of the excess above .5. Multiply the remainder or the sum by the whole 
content of the cask. The product is the quantity of liquor in the cask, in 
gallons, when the dividend is wet inches; or the empty space, if dry inches. 

2. For a standing cask : Divide the number of wet or dry inches by the 
length of the cask. If the quotient exceeds .5, add to it one tenth of its 
excess above .5; if less than .5, subtract from it one tenth of what it wants 
of .5. Multiply the sum or the remainder by the whole content of the cask. 
The product is the quantity of liquor in the cask, when the dividend is wet 
inches; or the empty space, if dry inches. 

Volume of cask (approxiimite) U. S. gallons = square of mean diam. 
X length in inches X .0034. Mean diam. = half the sum of the bung and. 
head diams. 

Volume of an irregular solid. Suppose it divided into parts, 
resembling prisms or other bodies measurable by preceding rules. Find 
the content of each part; the sum of the contents is the cubic contents of 
the solid. 

The content of a small part is found nearly by multiplying half the sum 
of the areas of each end by the perpendicular distance between them. 

The contents of small irregular solids may sometimes be found b)' im- 
mersing them under water in a prismatic or cylindrical vessel, and observ- 
ing the amount by which the level of the water descends when the solid is 
withdrawn. The sectional area of the vessel being multiplied by the descent 
of the level gives the cubic contents. 

Or, weigh the solid in air and in water; the difference is the weight of 
water it displaces. Divide the weight in pounds by 62.4 to obtain volume in 
cubic feet, or multiply it by 27.7 to obtain the volume in cubic inches. 

When the solid is very large and a great degree of accuracy is not 
requisite, measure its length, breadth, and depth in several ( itferent 
places, and take the mean of the measurement for each dimension, and 
multiply the three means together. 

When the surface of the solid is very extensive it is better to divide it 
into triangles, to find the area of each triangle, and to multiply it by the 
mean depth of the triangle for the contents of each triangular portion; the 
contents of the triangular sections are to be added together, 

The mean depth of a triangular section is obtained by measuring the 
depth at each angle, adding together the. three measurements, and taking 
one third of the sum. 



PLANE TRIGONOMETRY, 65 



PLANE TRIGONOMETRY. 

Trigonometrical functions. 

Every triangle has six parts three angles and three sides. When any 
three of these parts are given, provided one of them is a side, the other 
parts may be determined. By the solution of a triangle is meant the deter- 
mination of the unknown parts of a triangle when certain parts are given. 

The complement of an angle or arc is what remains after subtracting the 
angle or arc from 90. 

In general, if we represent any arc by A, its complement is 90 A. 
Hence the complement of an arc that exceeds 90 is negative. 

Since the two acute angles of a right-angled triangle are together equal to 
a right angle, each of them is the complement of the other. 

The supplement of an angle or arc is what remains after subtracting the 
angle or arc from 180. If A is an arc its supplement is 180 A. The sup- 
plement of an arc that exceeds 180 is negative. 

The sum of the three angles of a triangle is equal to 180. Either angle is 
the supplement of the other two. In a right-angled triangle, the right angle 
being equal to 90, each of the acute angles is the complement of the other. 

In all right-angled triangles having the same acute angle, the sides have 
to each other the same ratio. These ratios have received special names, as 
follows: 

If A is one of the acute angles, a the opposite side, b the adjacent side, 
and c the hypothenuse. 

The sine of the angle A is the quotient of the opposite side divided by the 

a 
hypothenuse. Sin. A = - 

The tangent of the angle A is the quotient of the opposite side divided by 
the adjacent side. Tang. A = 

The secant of the angle A is the quotient of the hypothenuse divided by 

c 
the adjacent side. Sec. A = jf 

The cosine, cotangent, and cosecant of an angle are respec- 
tively the sine, tangent, and secant of the complement of that angle. The 
terms sine, cosine, etc., are called trigonometrical functions. 

In a circle whose radius is unity, the sine of an arc, or of the angle at the 
centre measured by thai arc, is the perpendicular let fall from one extrem- 
ity of the arc upon the diameter passing through the other extremity. 

The tangent of an arc is the line which touches the circle at one extrem- 
ity of the arc, and is limited by the diameter ( produced) passing through 
the other extremity. 

The secant of an arc is that part of the produced diameter which is 
intercepted beticeen the centre and the tangent. 

The versed sine of an arc is that part of the diameter intercepted 
between the extremity of the arc and the foot of the sine. 

In a circle whose radius is not unity, the trigonometric functions of an arc 
will be equal to the lines here defined, divided by the radius of the circle. 

It 1C A (Fig. 70) is an angle in the first quadrant, and C F= radius, 

FG , Oft KF 

The sine of the angle = = r - r . Cos = ,-= r = - 
Rad. Had. Had. 

IA CI ' DL 

Tan S- = Rad/ Secant = Rad/ Cot = Ral." 
CL ._ . GA 
Rad/ 

If radius is 1, then Rad. in the denominator is 
omitted, and sine = F G, etc. 

The sine of an arc = half the chord of twice the 
arc. 

The sine of the supplement of the arc is the same 
as that of the arc itself. Sine of arc B D F = F G = 
sin arc FA. Fi. 70, 




66 



PLANE TRIGONOMETRY* 



The tangent of the supplement is equal to the tangent of the arc, but with 
a contrary sign. Tang. B D F = B M. 

The secant of the supplement is equal to the secant of the arc, but with a 
contrary sign. Sec. B D F = CM. 

Signs of the functions in the four quadrants. If we 
divide a circle into four quadrants by a vertical and a horizontal diame- 
ter, the upper right-hand quadrant is called the first, the upper left the sec- 
ond, the lower left the third, and the lower right the fourth. The signs of 
the functions in the four quadrants are as follows: 

First quad. Second quad. Third quad. Fourth quad. 
Sine and cosecant, + + 

Cosine and secant, + 

Tangent and cotangent, + + 

The values of the functions are as follows for the angles specified: 



























Ajigle..,.., , 





30 


45 


60 


90 


120 


135 


150 


180 


270 


360 






1 


1 


y o 




Vs 


1 


1 








Sine 











1 








(I 


-1 









2 


Vg 


2 




2 


^2 


2 












V3 


1 


1 




1 


1 


i/3 








Cosine 


1 
















_ 


-1 





1 






2 


^2 


2 






Vo 


2 








Tangent 





1 


1 


V* 


00 


-VI 




1 












V8 




j 








V3 





00 





Cotangent 






1 






1 
















*3 




Va 




~yo 


1 


~ | 7 3 


00 





00 


Secant 


1 


2 


^2 


8 


00 


-2 


i'o 


2 


-1 


OC 


1 






1/3 




o 




g 




1/3 








Cosecant 


QO 


2 


y 'o 


71 


1 




V2 


2 


oc 


-1 


CO 


l^ersed sine 





2- ^3 


4/o 1 


i 


j 


3 


V*+i 


2+1/3 













2 


2 


2 




2 


^ 


2 









TRIGONOMETRICAL. FORJNLULJE. 

The following relations are deduced from the properties of similar tri- 
angles (Radius = 1): 

cos A : sin A :: 1 : tan A, whence tan A = 



sin A : cos A :: 1 : cot A, 
cos A : 1 :: 1 : sec A, 



cos -4' 

cos^l 

cotan A = -. - 

sin A 



sec A = 



sin A : 1 



tan A : 1 



1 : cosec A, " cosec A = 



_ 

cos A 
1 



sin A* 



The sum of the square of the sine of an arc and the square of its c'osine 
equals unity. Sin 2 A -\- cos 2 A = 1. 

Also, 1 -ftan 2 ^ = sec* A: 1 + cot 2 A = cosec 5 A. 

Functions ot the sum and difference of two angles : 

Let the two angles be denoted by A and B, their sum A -f B = C, and 
their difference A - B by D. 

) = sin u4 cos B -\- cos A sin B; (1) 



TRIGONOMETRICAL FORMULAE. 67 

cos (A -f- J5) = cos A cos B sin .4 sin P; . . . . . (2) 

sin ( A B) = sin A cos 5 cos A sin 5; ..... (3) 

cos (A B) = cos A cos . + sm -4 sin P ...... (4) 

From these four formulae by addition and subtraction we obtain 

sin (A + B) -f sin (A - B) = 2 sin A cos B\ ..... (5) 

sin U + B) - sin U - 5) = 2cos ^ sin 5; ..... (6) 

cos (A + B) 4- cos (4 - B) = 2 cos ^. cos B; ..... (7) 

cos (A B) - cos (A + 5) = 2 sin A sin 5 ...... (8) 

If we put A 4- B = C, and ^ - P = A then ^ = ^74- >) and B = Y%(C - 
Z>), and we have 

sin C 4 sin D = 2 sin }(C 4- D) cos ^(C f - D); .... (9) 
sin C - sin D = 2 cos ^7 -f D) sin y 2 (C - D); . . . . (10) 
cosC-f cosZ) = 2cos^(C+D)cos^(<7 - D); . . . . (11) 
cos D - cos (7 = 2 sin ^((7 + D) sin ^(C - D) ..... (12) 



Equation (9) may be enunciated thus: The sum of the sines of any two 
angles is equal to twice the sine of half the sum of the angles multiplied by 
the cosine of half their difference. These formulae enable us to transform 
a sum or difference into a product. 

The sum of the sines of two angles is to their difference as the tangent of 
half the sum of those angles is to the tangent of half their difference. 
sin A 4 sin B _ 2 sin \fljA + B) cos y%(A - B) _ tan %(A -f- B) 
siu A - sin B ~ 2 cos %>(A + B) sin %>(A - B)~ tan %(A - B)' 
The sum of the cosines of two angles is to their difference as the cotangent 
of half the sum of those angles is to the tangent of half their difference. 
cos A + cos B _ 2 cos y 2 (A + B) cos \^(A - B) _ cot ^(A + B) 
eos B cos A ~~ 2 sin %(A + B) sin y^(A - B) ~ tan fflA - W 
The sine of the sum of two angles is to the sine of their difference as the 
sum of the tangents of those angles is to the difference of the tangents. 

sin (A + B) _ tan A + tan B m 
sin (A - B) tan A - tanJ5' ' 
= tan ^ | t tan .1 + tang 



cos A cos B 
sin (A - B) 
cos A cos B 
cos (A 4- B) 
cos ^1 cos B 
cos (A B) 
cos A cos .# 



= tan A - tan B; 



= 1 tan JL tan . 



= 1 + tan ^1 tan 5; 



tan (A-B) = 
cot U + JB) = 



1 tan A tan J?' 
tan A - tan B ^ 
1 4~ tan .4 tan B' 
cot ^4 cot j? 1 m 
cot #4- cot A ' 



cot U - B) = 
y 



cot J5 cot A ' 



Functions of t \vice an angle : 

sin 2 A = 2 sin A cos A ; 



tan 2A = 



2 tan A 
1 - tan 2 A* 



cos 2A = cos 2 .4 sin 2 A\ 



cot 2.4 = 



cot 2 A I 
2 cot .4 



Functions of naif an angle : 

sin \4>A = 



1 + cos J. ' 



cos \&A = 



68 PLANE TRIGONOMETRY. 

Solution oi Plane Right-angled Triangles. 

Let A and B be the two acute angles and C the right angle, and a, 6, and 
c the sides opposite these angles, respectively, then we have 

1. sin A = cosB ; 3. tan J. = cot = ; 

2. cos A = s'mB s" -: 4. cot jt = tan B = - 

c a 

1. In any plane right-angled triangle the sine of either of the acute angles 
is equal to the quotient of the opposite leg divided by the hypothenuse. 

2. The cosine of either of the acute angles is equal to the quotient of the 
adjacent leg divided by the hypothenuse. 

3. The tangent of either of the acute angles is equal to the quotient of the 
opposite leg divided by the adjacent leg. 

4. The cotangent of either of the acute angles is equal to the quotient of 
the adjacent leg divided by the opposite leg. 

5. The square of the hypothenuse equals the sum of the squares of the 
other two sides. 

Solution of Oblique-angled Triangles. 

The following propositions are proved in works on plane trigonometry. In 
any plane triangle 

Theorem 1. The sines of the angles are proportional to the opposite sides. 

Theorem 2. The sum of any two sides is to their difference as the tangent 
of half the sum of the opposite angles is to the tangent of half their differ- 
ence. 

Theorem 3. If from any angle of a triangle a perpendicular be drawn to 
the opposite side or base, the whole base will be to the sum of the other two 
sides as the difference of those two sides is to the difference of the segments 
of the base. 

CASE I. Given two angles and a side, to find the third angle and the other 
two sides. 1. The third angle = 180 sum of the two angles. 2. The sides 
may be found by the following proportion : 

The sine of the angle opposite the given side is to the sine of the angle op- 
posite the required side as the given sde is to the required side. 

CASE II. Given two sides and an angle opposite one of them, to find the 
third side and the remaining angles. 

The side opposite the given angle is to the side opposite the required angle 
as the sine of the given angle is to the sine of the required angle. 

The third angle is found by subtracting the sum of the other two from 180, 
and the third side is found as in Case I. 

CASE III. Given two sides and the included angle, to find the third side and 
the remaining angles. 

The sum of the required angles is found by subtracting the given angle 
from 180. The difference of the required angles is then found by Theorem 
II. Half the difference added to half the sum gives the greater angle, and 
half the difference subtracted from half the sum gives the less angle. The 
third side is then found by Theorem I. 

Another method : 

Given the sides c, b, and the included angle A, to find the remaining side a 
and the remaining angles B and G. 

From either of the unknown angles, as B, draw a perpendicular B e to the 
opposite side. 

Then 

Ae = ccosA, Be = csinA, eC=b - Ae, B e-t- e C = ten C. 

Or, in other words, solve B <?, A e and B e C as right-angled triangles. 

CASE IV. Given the three sides, to find the angles. 

Let fall a perpendicular upon the longest side from the opposite angle, 
dividing the given triangle into two right-angled triangles. The two seg- 
ments of the base may be found by Theorem III. There will then be given 
the bypothenuse and one side of a right-angled triangle to find the angles. 

For areas of triangles, see Mensuration. 




r 

V' 



ANALYTICAL GEOMETRY. 69 



ANALYTICAL GEOMETRY. 

Analytical geometry is that branch of Mathematics which has for 
its object the determination of the forms and magnitudes of geometrical 
magnitudes by means of analysis. 

Ordinates and abscissas. In analytical geometry two intersecting 
lines YY', XX' are used as coordinate axes^ 
XX' being the axis of abscissas or axis of Jf, 
and YY' the axis of ordinates or axis of Y. 
A. the intersection, is called the origin of co- 
ordinates. The distance of any point P from 
the axis of Y measured parallel to the axis of 
X is called the abscissa of the point, as AD or' 
CP, Fig. 71. Its distance from the axis of X, 
measured parallel to the axis of Y, is called 
the ordinate, as AC or PD. The abscissa and 
ordinate taken together are called the coor- 
dinates of the point P. The angle of intersec- 
tion is usually taken as a right angle, in which 
JT IG 7^ case the axes of X and Fare called rectangu- 

lar coordinates. 

The abscissa of a point is designated by the letter x and the ordinate by y. 
The equations of a point are the equations which express the distances of 
the point from the axis. Thus x = a,y = b are the equations of the point P. 
Equations referred to rectangular coordinates. The equa- 
tion of a line expresses the relation which exists between the coordinates of 
every point of the line. 

Equation of a straight line, y = ax b, in which a is the tangent of the 
angle the line makes with the axis of X, and b the distance above A in which 
the line cuts the axis of Y. 

Every equation of the first degree between two variables is the equation of 
ft straight line, as Ay 4- Bx 4- C = 0, which can be reduced to the form y = 
ax b. 
Equation of the distance between two points: 

in which x'y'^ x"y" are the coordinates of the two points. 
Equation of a line passing through a given point : 

y - y' = a(x #'), 

in which x'y' are the coordinates of the given point, a, the tangent of the 
angle the line makes with the axis of x, being undetermined, since any num- 
ber of lines may be drawn through a given point. 
Equation of a line passing through two given points : 



Equation of a line parallel to a given line and through a given point; 

y - y' = a(x - x'\ 

Equation of an angle V included between two given lines: 
a' a 

im % v =T+tt 

in which a and a' are the tangents of the angles the lines make with the 
axis of abscissas. 
If the lines are at right angles to each other tang V = oo, and 

1 + a'a - 0. 

Equation of an intersection of two lines, whose equations are 
y = ax -f b, and y = a'x + &', 

b - b' ab' - a'b 
x = and y = . 

a - a" * a - a ; ' 



70 ANALYTICAL GEOMETRY. 

Equation of a perpendicular from a given point to a given line: 



Equation of the length of the perpendicular Pi 
p _ y' - ax' - b 

yTT& 

The circle. Equation of a circle, the origin of coordinates being at the 
centre, and radius = R : 



If the origin is at the left extremity of the diameter, on the axis of X: 

y* = 2Rx - it* 2 . 

If the origin is at any point, and the coordinates of the centre are x'y' : 
(x - x')* + (y- 2/') 2 = &. 

Equation of a tangent to a circle, the coordinates of the point of tangency 
being x"y" and the origin at the centre, 



Tlie ellipse. Equation of an ellipse, referred to rectangular coordi- 
nates with axis at the centre: 

A i y * 4. #2^ = ^apa, 

in which A is half the transverse axis and B half the conjugate axis. 

Equation of the ellipse when the origin is at the vertex of the transverse 
axis: 



The eccentricity of an ellipse is the distance from the centre to either 
focus, divided by the semi-transverse axis, or 



The parameter of an ellipse is the double ordinate passing through the 
focus. It is a third proportional to the transverse axis and its conjugate, or 

2B* 
%A : 2B :: 2B : parameter; or parameter = . 

Any ordinate of a circle circumscribing an ellipse is to the corresponding 
ordinate of the ellipse as the semi-transverse axis to the semi-conjugate. 
Any ordinate of a circle inscribed in an ellipse is to the corresponding ordi- 
nate of the ellipse as the semi-conjugate axis to the semi-transverse. 

Equation of the tangent to an ellipse, origin of axes at the centre : 

A*yy" -f B^xx" = A*B*, 

y"x" being the coordinates of the point of tangency. 

Equation of the normal, passing through the point of tangency, and per- 
pendicular to the tangent: 



The normal bisects the angle of the two lines drawn from the point of 
tangency to the foci. 

The lines drawn from the foci make equal angles with the tangent. 

Tlae parabola. Equation of the parabola referred to rectangular 
coordinates, the origin being at the vertex of its axis, y* = 2px, in which 2p 
is the parameter or double ordinate through the focus. 



ANALYTICAL GEOMETRY. 71 

The parameter is a third proportional to any abscissa and its corresponding 
ordinate, or 

x :y :iy:2p. 
Equation of the tangent: 

yy" - p(x -f x"), 

y''x' f being coordinates of the point of tangency. 
Equation of the normal: 



The sub-normal, or projection of the normal on the axis, is constant, and 
equal to half the parameter. 

The tangent at any point makes equal angles with the axis and with the 
line drawn from the point of tangency to the focus. 

The hyperbola. Equation of the hyperbola referred to rectangular 
coordinates, origin at the centre: 

A*y* - B*x* = - -4 2 B 2 , 

in which A is the semi-transverse axis and B the semi-conjugate axis. 
Equation when the origin is at the right vertex of the transverse axis: 



Conjugate and equilateral hyperbolas. If on the conjugate 
axis, as a transverse, and a focal distance equal to \fA* -\- # 2 , we construct 
the two branches of a hyperbola, the two hyperbolas thus constructed are 
called conjugate hyperbolas. If the transverse and conjugate axes are 
equal, the hyperbolas are called equilateral, in which case y* # 2 = A* 
when A is the transverse axis, and # a 2/ 2 = B* when B is the trans- 
verse axis. 

The parameter of the transverse axis is a third proportional to the trans- 
verse axis and its conjugate. 

2A : 2B : : 2B : parameter. 

The tangent to a hyperbola bisects the angle of the two lines drawn from 
the point of tangency to the foci. 

The asymptotes of a tiyperbola are the diagonals of the rectangle 
described on the axes, indefinitely produced in both directions. 

In an equilateral hyperbola the asymptotes make equal angles with the 
transverse axis, and are at right angles to each other. 

The asymptotes continually approach the hyperbola, and become tangent 
to it at an infinite distance from the centre. 

Conic sections, Every equation of the second degree between two 
variables will represent either a circle, an ellipse, a parabola or a hyperbola. 
These curves are those which are obtained by intersecting the surface of a 
cone by planes, and for this reason they are called conic sections. 

Logarithmic curve. A logarithmic curve is one in which one of tho 
coordinates of any point is the logarithm of the other. 

The coordinate axis to v hich the lines denoting the logarithms are parallel 
is called the axis of logarithms, and the other the axis of numbers. If y is 
the axis of logarithms and x the axis of numbers, the equation of the curve 
is y = log x. 

If the base of a system of logarithms is a, we have a y = x, in which y is the 
logarithm of x. 

Each system of logarithms will give a different logarithmic curve. If y = 
0, x = 1. Hence every logarithmic curve will intersect the axis of numbers 
at a distance from the origin equal to 1. 



72 DIFFERENTIAL CALCULUS. 



DIFFERENTIAL CALCULUS. 

The differential of a variable quantity is the difference between any two 
of its consecutive values; hence it is indefinitely small. It is expressed by 
writing d before the quantity, as dx, which is read differential of x. 

The term - is called the differential coefficient of y regarded as a func- 

Q.X 

tion of x. 

The differential of a function is equal to its differential coefficient mul- 
tiplied by the differential of the independent variable; thus, -J^dx = dy. 

The limit of a variable quantity is that value to which it continually 
approaches, so as at last to differ from it by less than any assignable quan- 
tity. 

The differential coefficient is the limit of the ratio of the increment of the 
independent variable to the increment of the function. 

The differential of a constant quantity is equal to 0. 

The differential of a product of a constant by a variable is equal to the 
constant multiplied by the differential of the variable. 

If u = Av, du = Adv. 
In any curve whose equation is y=f(x), the differential coefficient 

= tan a; hence, the rate of increase of the function, or the ascension of 
dx 

the curve at any point, is equal to the tangent of the angle which the tangent 
line makes with the axis of .abscissas. 
All the operations of the Differential Calculus comprise but two objects: 

1. To find the rate of change in a function when it passes from one state 
of value to another, consecutive with it. 

2. To find the actual change in the function : The rate of change is the 
differential coefficient, and the actual change the differential. 

Differentials of algebraic functions. The differential of the 
sum or difference of any number of functions, dependent on the same 
variable, is equal to the sum or difference of their differentials taken sepa- 
rately : 

If u = y -{- z w, du dy -\- dz dw. 

The differential of a product of two functions dependent on the same 
variable is equal to the sum of the products of each by the differential of 
the other : 

d(uv) du , dv 
d(tti>) - vdu + udv. _-=_ + _. 

The differential of the product of any number of functions is equal to the 
sum of the products which arise by multiplying the differential of each 
function by the product of all the others: 

d(uts) = tsdu -f usdt 4- uids. 

The differential of a fraction equals the denominator into the differential 
of the numerator minus the numerator into the differential of the denom- 
inator, divided by the square of the denominator : 

/u\ vdu udv 
dt = d {J = . 

If the denominator is constant, dv = 0, and dt 5- = . 

v v 

If the numerator is constant, du = 0, and dt = 

The differential of the square root of a quantity is equal to the differen 
tial of the quantity divided by twice the square root of the quantity: 

If v u^, or v = 4/w, dv = ; 

2 Vu 



DIFFEREHTIAL CALCULUS. 73 

The differential of any power of a function is equal to the exponent multi- 
plied by the function raised to a power less one, multiplied by the differen- 
tial of the function, d(u n } = nu n - 1 du. 

Formula* for differentiating algebraic functions. 



1. d (a) = 0. 



2. d (ax) = adx. 

ry A i n yn\ _ ~)ix m dX. 

dx 



ydx - xdy 



5. d (xy) = xdy + ydx. 

To find the differential of the form u = (a + bx n ) m : 

Multiply the exponent of the parenthesis into the exponent of the varia- 
ble within the parenthesis, into the coefficient of the variable, into the bi- 
nomial raised to a power less 1, into the variable within the parenthesis 
raised to a power less 1, into the differential of the variable. 

du = d(a 4 bx n ) m = mnb(a + bx 1l ) m ~ 1 x n ~ l dx. 

To find the rate of change for a given value of the variable : 
Find the differential coefficient, and substitute the value of the variable in 
the second member of the equation. 

EXAMPLE. If x is the side of a cube and u its volume, u = x 9 , - = &e 2 . 

Hence the rate of change in the volume is three times the square of the 
edge. If the edge is denoted by 1, the rate of change is 3. 

Application. The coefficient of expansion by heat of the volume of a body 
is three times the linear coefficient of expansion. Thus if the side of a cube 
expands .001 inch, its volume expands .003 cubic inch. 1.001 s = 1.003003001. 

A partial differential coefficient is the differential coefficient of 
a function of two or more variables under the supposition that only one of 
them has changed its value. 

A partial differential is the differential of a function of two or more vari- 
ables under the supposition that only one of them has changed its value. 

The total differential of a function of any number of variables is equal to 
the sum of the partial differentials. 

If uf(xy\ the partial differentials are -^dx, ~dy. 

dx dy 

Itu = x* + y*-z,du = ^dx 4- d ~dy 4 ^dz\ = 2xdx + 3y*dy-dz. 
ax ay dz 

Integrals. An integral is a functional expression derived from a 
differential. Integration is the operation of finding the primitive function 
from the differential function. It is indicated by the sign /, which is read 
** the integral of." ThusfZxdx = x"* ; read, the integral of 2xdx equals x-. 

To integrate an expression of the form mx m ~ 1 dx or x m dx, add 1 to the 
exponent of the variable, and divide by the new exponent and by the differ- 
ential of the variable: f3x"*dx = x 3 . (Applicable in all cases except wheu 



1. 



Forjx dx see formula 2 page 78.) 



The integral of the product of a constant by the differential of a vari- 
able is equal to the constant multiplied by the integral of the differential: 

fax m dx = a/x m dx = a - x m + l . 
J m-f 1 

The integral of the algebraic sum of any number of differentials is equal to 
the algebraic sum of their integrals: 



du = 2ax*dx - bydy - z*dz; fda = ao; 3 - y* - . 

& 6 O 

Since the differential of a constant is 0, a constant connected with a vari- 
able by the sign + or - disappears in the differentiation; thus d(a + x) = 
dx m = mx m ~ l dx. Hence in integrating a differential expression we must 



74 DIFFERENTIAL CALCULUS. 

annex to the integral obtained a constant represented by C to compensate 
for the term which may have been lost in differentiation. Thus if we have 
dy = adx\ Jdy = afdx. Integrating, 

y = ax C. 

The constant (7, which is added to the first integral, must have such a 
value as to render the functional equation true for every possible value that 
may be attributed to the variable. Hence, after having found the first 
integral equation and added the constant C, if we then make the variable 
equal to zero, the value which the function assumes will be the true valus 
of C. 

An indefinite integral is the first integral obtained before the value of the 
constant C is determined. 

A particular integral is the integral after the value of Chas been found. 

A definite integral is the integral corresponding to a given value of the 
variable. 

Integration "between limits e Having found the indefinite inte- 
gral and the particular integral, the next step is to find the definite integral, 
and then the definite integral between given limits of the variable. 

The integral of a function, taken between two limits, indicated by given 
values of a?, is equal to the difference of the definite integrals correspond- 
ing to those limits. The expression 



/W / 

/ dy = a I dx 
Jx' J 



is read: Integral of the differential of ?/, taken between the limits x' and x"' 
the least limit, or the limit corresponding to the subtractive integral, being 
placed below. 

Integrate du = Qx^dx between the limits x = 1 and x = 3, u being equal tc 
81 when x = 0. fdu = fQx' 2 dx = 3# 3 + <?; C = 81 when x = 0, then 



-. 



= 3 

du = 3(3)3 _j_ gl, minus 3(1) 3 -f- 81 = 78. 

x = 1 



Integration of particular forms. 

To integrate a differential of the form du - (a-f- bx n ) m x n ~ *dx. 

1. If there is a constant factor, place it without the sign of the integral, 
and omit the power of the variable without the parenthesis and the differ 
ential; 

2. Augment the exponent of the parenthesis by 1, and then divide this 
quantity, with the exponent so increased, by the exponent of the paren- 
thesis, into the exponent of the variable within the parenthesis, into the co- 
efficient of the variable. Whence 



J. (m -f l)nb 

Tlie differential of an arc is the hypothenuse of a right-angle triangle of 
which the base is dx and the perpendicular dy. 



If z is an arc, dz = Vdx* + d?/ 2 

Quadrature of a plane figure. 

T/ie differential of the area of a plane surf ace is equal to the ordinate into 
the differential of the abscissa. 

da = ydx. 

To apply the principle enunciated in the last equation, in finding the area 
of any particular plane surface : 

Find the value of y in terms of x. from the equation of the bounding line; 
substitute this value in the differential equation, and then integrate between 
the required limits of x. 

Area of the parabola, Find the area of any portion of the com- 
mon parabola whose equation is 

yi = 2px' t whence y = ^2px. 



DIFFEKENTIAL CALCULUS. 75 

Substituting this value of y in the differential equation ds = ydx gives 

P 



/ ds = I \/2pxdx = |/^p / x^dx = ^ 



xl -f C\ 



Tf we estimate the area from the principal vertex, x = 0. y = 0, and (7=0; 






and denoting the particular integral by s', s' = r y. 

o 

That is, the area of any portion of the parabola, estimated from the ver- 
tex, is equal to % of the rectangle of the abscissa and ordinate of the extreme 
point. The curve is therefore quadrable. 

Quadrature of surfaces of revolution. The differential of a 
surface of revolution is equal to the circumference of a circle perpendicular 
to the axis into the differential of the arc of the meridian curve. 



ds = Ziry^d 

in which y is the radius of a circle of the bounding surface in a plane per- 
pendicular to the axis of revolution, and x is the abscissa, or distance of the 
plane from the origin of coordinate axes. 

Therefore, to find the volume of any surface of revolution: 

Find the value of y and dy from the equation of the meridian curve in 
terms of x and dx, then substitute these values in the differential equation, 
and integrate between the proper limits of x. 

By application of this rule we may find: 

The curved surface of a cylinder equals the product of the circumference 
of the base into the altitude*. 

The convex surface of a cone equals the product of the circumference of 
the base into half the slant height. 

The surface of a sphere is equal to the area of four great circles, or equal 
to the curved surface of the circumscribing cylinder. 

ubature of volumes of revolution. A volume of revolution 
is a volume generated by the revolution of a plane figure about a fixed line 
called the axis. 

If we denote the volume by F", dV iry^ dx. 

The area of a circle described by any ordinate y is iry*; hence the differ- 
ential of a volume of revolution is equal to the area of a circle perpendicular 
to the axis into the differential of the axis. 

The differential of a volume generated by the revolution of a plane figure 
about the axis of Y is irx*dy. 

To find the value of Ffor any given volume of revolution : 

Find the value of ?/ 2 in terms of x from the equation of the meridian 
curve, substitute this value in the differential equation, and then integrate 
between the required limits of x. 

By application of this rule we may find: 

The volume of a cylinder is equal to the area of the base multiplied by the 
altitude. 

The volume of a cone is equal to the area of the base into one third the 
latitude. 

The volume of a prolate spheroid and of an oblate spheroid (formed by 
ihe revolution of an ellipse around its transverse and its conjugate axis re- 
spectively) are each equal to two thirds of the circumscribing cylinder. 

If the axes are equal, the spheroid becomes a sphere and its volume = 
2 1 

yrR* x D = ~irD s i -R being radius and D diameter. 
o o 

The volume of a paraboloid is equal to half the cylinder having the same 
base and altitude. 

The volume of a pyramid equals the area of the base multiplied by one 
third the altitude. 

Second, third, etc., differentials, The differential coefficient 
being a function of the independent variable, it may be differentiated, and 
iv e thus obtain the second differential coefficient: 

d() = d . Dividing by dx, we have for the second differential coeffl- 
\dx/ dx 



76 DIFFEBEOTIAL CALCULUS. 



cient -r-^, which is read: second differential of u divided by the square of 

the differential of x (or dx squared). 

d 3 u 
The third differential coefficient ^ is read: third differential of u divided 

by dx cubed. 
The differentials of the different orders are obtained by multiplying the 

differential coefficients by the corresponding powers of dx; thus ^ dx 3 = 

third differential of u. 

Sign of the first differential coefficient. If we have a curve 
whose equation is y = /x, referred to rectangular coordinates, the curve 

will recede from the axis of X when - is positive, and approach the 

axis when it is negative, when- the curve lies within the first angle of the 
coordinate axes. For all angles and every relation of y and x the curve 
will recede from the axis of X when the ordinate and first differential co- 
efficient have the same sign, and approach it when they have different 
signs. If the tangent of the curve becomes parallel to the axis of X at any 

point -^ = 0. If the tangent becomes perpendicular to the axis of X at any 

dx 

dy 
point =co. 

dx 

Sign of the second differential coefficient. -The second dif- 
ferential coefficient has the same sign as the ordinate when the curve is 
convex toward the axis of abscissa and a contrary sign when it is concave. 
Maclaurin's Theorem. For developing into a series any function 
of a single variable as u = A -f- Bx -f Ox* -\- Dx 3 -f- Ex 4 , etc., in which A, B, 
<7, etc., are independent of x: 



In applying the formula, omit the expressions x = 0, although the coeffi- 
cients are always found under this hypothesis. 
EXAMPLES : 



1 J ___ * . ^ _ ^ . X* etc 

a -f- x ~~ a a 2 ^ a 3 a 4 ^ ' a n + i ' 

Taylor's Theorem. For developing into a series any function of the 
sum or difference of two independent variables, as u' = f(x y): 
. du , d^u y* . d 3 u y 3 ' 



in which u is what u' becomes when y = 0, is what becomes when 

dx dx 

y = 0. etc. 

Maxima and minima. To find the maximum or minimum value 
of a function of a single variable: 

1. Find the first differential coefficient of the function, place it equal to 0, 
and determine the roots of the equation. 

2. Find the second differential coefficient, and substitute each real root, 
in succession, for the variable in the second member of the equation. Each 
root which gives a negative result will correspond to a maximum value of 
the function, and each which gives a positive result will correspond to a 
minimum value. 

EXAMPLE. To find the value of x which will render the function y a 
maximum or minimum in the equation of the circle, y* + x z = R*' t 

-^ = - ; making - - = gives x = 0. 
dx y y 



DIFFERENTIAL CALCULUS, 77 

dM 

The second differential coefficient is: -=-^ -- 

When x = 0, ;; ^ R-, hence -^-| = , which being negative, y is a maxi- 

mum for R positive. 

In applying the rule to practical examples we first find an expression for 
the function which is to be made a maximum or minimum. 

2. If in such expression a constant quantity is found as a factor, it may- 
be omitted in the operation; for the product will be a maximum or a mini- 
mum when the variable factor is a maximum or a minimum. 

3. Any value of the independent variable which renders a function a max- 
imum or a minimum will render any power or root of that function 
maximum or minimum; hence we may square both members of an eo 
tion to free it of radicals before differentiating. 

By these rules we may find: 

The maximum rectangle which can be inscribed in a triangle is one whose 
altitude is half the altitude of the triangle. 

The altitude of the maximum cylinder which can be inscribed in a cone is 
one third the altitude of the cone. 

The surface of a cylindrical vessel of a given volume, open at the top, is a 
minimum when the altitude equals half the diameter. 

The altitude of a cylinder inscribed in a sphere when its convex surface is 
a maximum is r |/2. r = radius. 

The altitude of ajcylinder inscribed in a sphere when the volume is a 
maximum is 2r -*- V3. 

(For maxima and minima without the calculus see Appendix, p. 1070.) 

Differential of an exponential function. 

If u = a x . . , ............ (1) 

then du = da x = a x k dx t (2) 

in which fc is a constant dependent on a. 

The relation between a and k is eft = e\ whence a = e^ t ..... (3) 

in which e 2.7182818 . . . the base of the Naperian system of logarithms. 
logarithms. The logarithms in the Naperian system are denoted by 
Z, Nap. log or hyperbolic log, hyp. log, or log e ; and in the common system 
always by log. 

k Nap. log a, log a = k log e ....... (4) 

The common logarithm of e, = log 2.7182818 . . . = .4342945 . . . , is called 
the modulus of the common system, and is denoted by M. Hence, if we have 
the Naperian logarithm of a number we can find the 1 common logarithm of 
the same number by muliiplying by the modulus. Reciprocally, Nap. 
log com. log x 2 3025851. 

If in equation (4) we make a = 10, we have 

1 = k log e, or - = log e = M. 

That is, the modulus of the common system is equal to 1, divided by the 
Naperiau logarithm of the 'common base. 
From equation (2) we have 

du da x 
= = kdx. 
u a x 

If we make a =s 10, the base of the common system, x = log u t and 



That is, the differential of a common logarithm of a quantity is equal to the 
differential of the quantity divided by the quantity, into the modulus. 
If we make a =? e, the base of the Naperian system, x becomes the Nape- 



73 DIEFEBENTIAL CALCULUS. 

rian logarithm of w, and k becomes 1 (see equation (3)); hence M = 1, and 

du du 

d(Nap. log u) = dx ; = . 
a**' 

That is, the differential of a Naperian logarithm of a quantity is equal to the 
differential of the quantity divided by the quantity; and in the Naperian 
system the modulus is 1. 

Since k is the Naperian logarithm of a, du = a x I a dx. That is, the 
differential of a function of the form a x is equal to the function, into the 
Naperian logarithm of the base a, into the differential of the exponent. 

If we have a differential in a fractional form, in which the numerator is 
the differential of the denominator, the integral is the Naperian logarithm 
of the denominator. Integrals of fractional differentials of other forms are 
given helow: 

Differential forms which have known integrals; ex- 
ponential functions. (I = Nap. log.) 

1. / a x I a dx = a x -f- C\ 

o f*dx /\ 

* / = / dxx ~ L = lx + Cl 

J J 

3. / (xy x ~ l dy -f y x ly x dx) = y x -f C\ 

4. C dX = l(x + |/a; 2 a 2 ) + C; 
J yx* a* 

5. C _ d L = l(x a + y x i 2ax) 4- C; 

J MX* 2ax 



r-^= =i(=-t 

J x\/a* + x* \fVa + a;-f 

/> _ %adx fa - A/C&~~X 

/ - - = Zf _ 1 

J x y^- & \ a + -zrr 



i @. 

" 



+ 



Circular functions. Let 2 denote an arc in the first quadrant, y tts 
sine, x its cosine, v its versed sine, and t its tangent; and the following nota- 
tion be employed to designate an arc by any one of its functions, viz., 

sin ~ 1 y denotes an arc of which y is the sine 
cos" 1 x u " " " " x is the cosine, 
tan" 1 f " ' " " " t is the tangent 



DIFFERENTIAL CALCULUS. 



79 



<read "atv whose sine is ?/," etc.), we have the following differential forms 
which have known integrals (r = radius): 



cos z dz = sin z-\-C\ 
sin z dz = cos z -f C; 

/ *# _ _i 
/ dv _ 



C; 



= ver-sin ~" 1 v -f- (7; 



/ |/r 2 - 2/ 2 

/- rcte _ i 

= cos * x -f- C; 
|/r 2 - * 2 



f- 
rj, 

J cos 2 

rd_v 

y-&^+& = 



sin z dz = ver-sin 2; -f- C; 
=: tan -f C; 

In "W-f'Cj 



/, 
:r=- = sin ~~ * -f- O; 
|/a 2 - w 2 

/~ dtC __- = cos- 1 -4-C; 
|/a 2 - w* <* 

/ U = ver-sin ~ J - -f (7; 
|/^aw - ti 2 

/adit _ _ i, 

a2 + w2 a 



The cycloid. If a circle be rolled along a straight line, any point of 
the circumference, as P, will describe a curve which is called a cycloid. The 
circle is called the generating circle, and Pthe generating point. 

The transcendental equation of the cycloid is 

x ver-sin- l ~ \'%ry - 2/ 2 , 

ydx 
and the differential equation is dx = 4/0..- _ == 1' 

The area of the cycloid is equal to three times the area of the generating 
circle. 

The surface described by the arc of a cycloid when revolved about its base 
is equal to 64 thirds of the generating circle. 

The volume of the solid generated by revolving a cycloid about its base is 
equal to five eighths of the circumscribing cylinder. 

Integral calculus. In the integral calculus we have to return from 
the differential to the function from which it was derived A number of 
differential expressions are given above, each of which has a known in- 
tegral corresponding to it, and which being differentiated, will produce the 
given differential. 

In all classes of functions any differential expression may be integrated 
when it is reduced to one of the known forms; and the operations of the 
integral calculus consist mainly in making such transformations of given 
differential expressions as shall reduce them to equivalent ones whose in- 
tegrals are known. 

For methods of making these transformations reference must be made to 
Uie text-books on differential and integral calculus. 



80 



MATHEMATICAL TABLES. 



RECIPROCALS OF NUMBERS. 



No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


1 


1.00000000 


64 


.01562500 


127 


.00787402 


190 


.00526316 


253 


.00395257 


2 


.50000000 


5 


.01538461 


8 


.00781250 


1 


.00523560 


4 


.00393701 


3 


.33333333 


6 


.01515151 


9 


.00775191 


2 


.00520833 


5 


.00392157 


4 


.25000000 


7 


.01492537 


130 


.00709231 


3 


.00518135 


6 


.00390625 


5 


.20000000 


8 


.01470588 


1 


.00763359 


4 


.00515464 


ri 


.00389105 


6 


.16666667 


o 


.01449275 


2 


.00757576 


5 


.00512820 


8 


.00387597 


r- 


.14285714 


70 


.01428571 


3 


.00751880 


6 


.00510204 


9 


. 00386 100 


8 


.12500000 


1 


.01408451 


4 


.00746269 


7 


.00507614 


260 


.00384015 


9 


.11111111 


2 


.01388889 


5 


.00740741 


8 


.00505051 


1 


.00383142 


10 


.10000000 


3 


.01369863 


6 


.00735294 


9 


.00502513 


2 


.00381079 


11 


.09090909 


4 


.01351351 


7 


.00729927 


200 


.00500000 


3 


.00380228 


1-2 


.08338333 


^ 


.01333533 


8 


.00724638 


1 


.00497512 


4 


.00378188 


13 


.0769:2308 


6 


.01315789 


<j 


.00719424 


2 


.00495049 




.00377358 


14 


.07142857 


7 


.01298701 


140 


.00714286 


3 


.00492611 


6 


.00375940 


15 


.06666667 


8 


.01282051 


1 


.00709220 


4 


.00490196 


7 


.00374532 


16 


.06250000 


9 


.01265823 


2 


.00704225 


f. 


.00487805 


8 


.00373134 


17 


.05882353 


80 


.01250000 


t 


.00699301 


6 


.00485437 


9 


.00371717 


18 


. 05555556 


1 


.01234568 


4 


.00694444 


7 


.00483092 


270 


.0037(1370 


19 


.05263158 





.01219512 


5 


.00689655 


8 


.00480769 


j 


.00309004 


20 


.05000000 


8 


.01204819 


6 


.00681931 


9 


.00478469 


< 


.00367647 


1 


.04761905 


4 


.01190476 


r- 


.00680272 


210 


.00476190 


j 


.00300300 


2 


.04545455 


5 


.01176471 


8 


.00675676 


11 


.00473934 


L 


.00364963 


3 


.04347826 


6 


.01162791 


r 


.00671141 


12 


.00471698 


5 


.00363636 


4 


.04166667 


7 


.01149425 


150 


.00606667 


13 


.00469484 


( 


.00302319 


5 


.04000000 


8 


.01136364 


1 


.00662252 


14 


.00467290 


7 


.00361011 


6 


.03846154 


c 


.01123595 


o 


.00657895 


15 


.00465116 


h 


.00359712 


7 


.03703704 


90 


.01111111 


3 


.00653595 


16 


.00462963 


9 


.00358423 


8 


.03571429 


1 


.01098901 


4 


.00649351 


17 


.00460829 


280 


.00357143 


9 


.03448276 


5 


.01086956 


5 


.00645101 


18 


.00458716 


] 


.00355872 


30 


.01333333 


jj 


.01075269 


6 


.00641026 


19 


.00456621 


\ 


.00354610 


1 


.03225806 


4 


.01063830 


7 


.00636943 


220 


.00454545 


; 


.00353357 


2 


.03125000 


r 


.01052632 


8 


.00682911 


1 


.00452489 


i. 


.00352113 


3 


.03030303 


6 


.01041667 


9 


.00628931 


c 


.00450450 


5 


.00350877 


4 


.02941176 


7 


.01030928 


160 


.00625000 


J 


.00448430 


6 


.00349350 


5 


.02857143 


8 


.01020408 


1 


.00621118 


4 


.00446429 


7 


.00348432 


6 


.02777778 


g 


.01010101 


2 


.00617284 


5 


.00444444 


8 


.00347222 


7 


.02702703 


100 


.01000000 


? 


.00613497 


6 


.00442478 


1 


.00346021 


8 


.02631579 


1 


.00990099 


4 


.00609756 


7 


.00440529 


290 


.00344828 


9 


.02564103 


o 


.00980392 


5 


. 00606061 


8 


.00438596 




.00343613 


40 


.02500000 


3 


.00970874 


6 


.00602410 


c 


.00436681 


o 


.00342406 


1 


.02439024 


4 


.00961538 


7 


.00598802 


230 


.00434783 


3 


.00341297 


2 


.02380952 


5 


.00952381 


8 


.00595238 


1 


.00432900 


L 


.00340136 


3 


.02325581 


6 


.00943396 


9 


.00591716 


2 


.00431034 


5 


.00338983 


4 


.02272727 


7 


.00934579 


170 


'.00588235 


c 


.00429184 


( 


.00337S38 


5 


.02222222 


8 


.00925926 


1 


.00584795 


4 


.00427350 


7 


.00330700 


6 


.02173913 


9 


.00917431 


o 


.00581395 


^ 


.00425532 


8 


.00335570 


7 


.02127660 


110 


.00909091 


3 


'.00578035 


6 


.00423729 


9 


.00334448 


8 


.02083333 


11 


.00900901 


4 


.00574713 


n 


.00421941 


300 


.00333333 


9 


.02040816 


12 


.00892857 


5 


.00571429 


8 


.00420168 




.00332226 


50 


.02000000 


13 


.00884956 


6 


.00568182 


c 


.00418410 


jj 


.00331120 


1 


.01960784 


14 


.00877193 


7 


.00564972 


240 


.00416667 


i 


.00330033 


2 


.01923077 


15 


.00869565 


8 


.00561798 


1 


.00414938 


4 


.00328947 


3 


.01886792 


16 


.00862069 


c 


.00558659 


2 


.00413223 


5 


.00327809 


4 


.01851852 


17 


.00854701 


180 


.00555556 


3 


.00411523 


6 


.00320797 


5 


.01818182 


18 


.00847458 


1 


.00552486 


4 


.00409836 


r 


.00325733 


6 


.01785714 


19 


.00840336 


r 


.00549451 


ft 


.00408163 


8 


.00324670 


7 


.01754386 


120 


.00833333 


3 


.00546448 


e 


.00406504 


9 


.003.23625 


8 


.01724138 


1 


.00826446 


4 


.00543478 


7 


.00404858 


310 


.00322581 


9 


.01694915 


c 


.00819672 


r 


.00540540 


8 


.00403226 


11 


.00321543 


60 


.01666667 


3 


.00813008 


6 


.00537634 


9 


.00401606 


12 


.00320513 


1 


.01639344 


4 


.00806452 


7 


.00534759 


250 


.00400000 


13 


.00319489 


2 


.01612903 


i 


.00800000 


8 


.00531914 


1 


.00398406 


14 


.00318471 


3 


.01587302 


6 


.00793651 


c 


.00529100 


2 


.00396825 


15 


.00317460 



RECIPROCALS OF NUMBERS. 



81 



No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


316 


.00316456 


381 


.00262467 


446 


.0022-1215 


511 


.00195695 


576 


.00173611 


17 


.00315457 


2 


.00261780 


7 


.00223714 


12 


.00195312 


7 


.00173310 


18 


.00314465 


3 


.00261097 


8 


.00223214 


13 


.00194932 


8 


.00173010 


19 


.00313480 


4 


.00260417 


9 


.00222717 


14 


.00194552 


9 


.00172712 


320 


.00312500 


5 


.00259740 


450 


.00222222 


15 


.00194175 


580 


.00172414 


1 


.00311526 


6 


.00259067 


1 


.00221729 


16 


.00193798 


1 


.00172117 


2 


.00310559 


7 


.00258398 


g 


.00221239 


17 


.00193424 


2 


.00171821 


3 


.00309597 


8 


.00257732 


3 


.00220751 


18 


.00193050 


3 


.00171527 


4 


.00308642 


9 


.00257069 


4 


.00220264 


19 


.00192678 


4 


,001?J00 


5 


.00307692 


390 


.00256410 


5 


.002197'80 


520 


.00192308 


5 


.00170940 


6 


.00306748 


1 


.00255754 


6 


.00219298 


1 


.00191939 


6 


.00170648 




.00305810 


2 


.00255102 


7 


.00218818 


2 


.00191571 


7 


.00170358 


8 


.00304878 


3 


.00254453 


8 


.00218341 


3 


.00191205 


8 


.00170068 


9 


.00303951 


4 


.00253807 


9 


.00217865 


4 


.00190840 


9 


.00169779 


330 


.00303030 


5 


.00253165 


460 


.00217391 


5 


.00190476 


590 


.00169491 


1 


.00302115 


6 


.00252525 


1 


.00216920 


6 


.00190114 


1 


.00169205 


2 


.00301205 


7 


.00251889 


2 


.00216450 


7 


.00189753 


2 


.00168919 


3 


.00300300 


8 


.00251256 


3 


.00215983 


8 


.00189394 


3 


.00168634 


4 


.00299401 


g 


.00250627 


4 


.00215517 


9 


.00189036 


4 


.00168350 





.00298507 


400 


.0025000'! 


5 


.00215054 


530 


.00188679 


R 


.00168007 


ii 


.00297619 


1 


.00249377 


6 


.00214592 


1 


.00188324 


6 


.00167785 


7 


.00296736 


2 


.00248756 


7 


.00214133 


2 


.0018797'0 


7 


.00167504 


8 


.00295858 


3 


.00248131) 


8 


.00213675 


3 


.00187617 


8 


.00167224 


9 


.00294985 


4 


.00247525 


9 


.00213220 


4 


.00187-266 


9 


.00166945 


340 


.00294118 


5 


.00246914 


470 


.00212760 


5 


.00186916 


600 


.00166667 


1 


.00293255 


6 


.002-16305 


1 


.00212314 


6 


.00186567 


1 


.00166389 


c 


.00292398 


7 


.00245700 


2 


.00211864 


7 


.00186220 


2 


.00166113 


C 


.00291545 


8 


.00245098 


g 


.00211416 


8 


.00185874 





.00165837 


4 


.00290698 


9 


.00244490 


4 


.00210970 


9 


.00185528 


4 


.00165563 


5 


.00289855 


410 


.00243902 


5 


.00210526 


540 


.00185185 


5 


.00165289 


6 


.00289017 


11 


.00243309 


6 


.00210084 


1 


.00184^43 


6 


.001C5016 


r< 


.00288184 


12 


.00242718 


7 


.00209644 




.00184502 


7 


.00164745 


8 


.00287356 


13 


.00242131 


8 


.00200205 




.00184162 


8 


.00164474 


9 


.00286533 


14 


.00241546 


9 


.00208768 




.00183823 


9 


.00164204 


350 


. .00285714 


15 


.0021096-1 


480 


.00208333 




.00183486 


610 


.00163934 


] 


.00284900 


16 


.00240385 


1 


.00207900 




.00183150 


11 


.0016361)6 




.00784091 


17 


.00239808 


f 


.00207469 




.0018-2815 


12 


.001 64399 


f 


.00283286 


18 


.00-J39234 


3 


.00207039 




.0018248-2 


13 


.00163132 


4 


.00288486 


19 


.00238663 


4 


.00206612 




.00182149 


14 


.00162866 


5 


.00281690 


420 


.00238095 


e^ 


.00206186 


55 


.00181818 


15 


.00162602 


6 


.00280899 


1 


.00237530 


6 


.00205761 




.00181488 


16 


.00162338 


7 


.00280112 


2 


.00236967 


r- 


.00205339 




.00181159 


17 


.00162075 


8 


.00279330 


3 


.00286407 


8 


.00-204918 




.00180832 


18 


.00161812 


9 


.00278551 


4 


.00235849 


9 


.00204499 




.00180505 


19 


.00161551 


360 


.00277778 


5 


.00235294 


490 


.C0204082 




.00180180 


620 


.00161-290 


1 


.00277008 


6 


.00234742 


1 


.00-203666 




.CO 179856 


1 


.00161031 


2 


.00276243 


7 


.0023419!? 





.00203252 




.00179533 


f 


.00160772 





.00275482 


8 


.00283645 


c 


.0020-2840 


8 


.00170211 


3 


.00160514 


4 


.00274725 


9 


.00233100 


4 


.00202429 


9 


.00178891 


3 


.00160256 


f 


.00273973 


430 


.00232558 


5 


.00202020 


560 


.00178571 


c 


.00160000 


6 


.00273224 


1 


.00232019 


6 


.00201613 


1 


.00178253 


I 


.00159744 


r 


.00272480 


g 


.00231481 


7 


.00-201207 


2 


.00177936 


1 


.00159490 


8 


.00271739 


3 


.00230947 


8 


.00200803 


3 


.00177620 


8 


.00159-236 


9 


.00271003 


4 


.00230415 


t 


.00200401 


4 


.00177305 


9 


.00158982 


370 


.00270270 


5 


.00229885 


500 


.00200000 


5 


.00176991 


630 


.00158730 


1 


.00269542 


6 


.00229358 


1 


.00199601 


6 


.00176678 


1 


.0015847-9 


2 


.00268817 


7 


.00228833 


o 


.00199203 


7 


.00176367 


c 


.00158228 


: 


.00268096 


8 


.00228310 


'. 


.00198807 


8 


.00176056 


\ 


.00157978 


4 


.00267380 


9 


.00227790 


i 


.00198413 


9 


.00175747 


i 


.00157729 


5 


.00266667 


440 


.00227273 


r 


.00198020 


570 


.00175439 


f 


.00157480 


6 


.00265957 


1 


.00226757 


\ 


.00197628 


1 


.00175131 


I 


.00157233 


7 


.00265252 


2 


.00226244 


7 


.00197239 


2 


.00174825 


f t 


.00156986 


8 


.00264550 


3 


.00225734 


8 


.001968^0 


3 


.00174520 


8 


.00156740 


9 


.00263852 


4 


.00225225 


( 


.00196464 


4 


.00174216 


9 


.00156494 


380 


.00263158 


5 


! .00224719 


510 


.00196078 


5 


.00173913 


640 


.00156250 



MATHEMATICAL TABLES. 



No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Reci?iro- 
cal. 


No. 


Recipro- 
cal. 


641 


.00156006 


706 


.00141643 


771 


.00129702 1 


836 


.00119617 


901 


.00110988 


2 


.00155763 


7 


.00141443 


2 


.00129534 


7 


.00119474 


2 


.00110865 


3 


.001555-21 


8 


.00141243 


3 


.00129366 


8 


.00119332 


3 


.00110742 


4 


.00155279 


9 


.00141044 


4 


.00129199 


9 


.00119189 


4 


.00110619 


5 


.00155039 


710 


.00140845 


5 


.00129032 


840 


.00119048 


5 


.00110497 


. 6 


.00154799 


11 


.00140647 


6 


.00128866 


1 


.00118906 


6 


.00110375 


7 


.00154559 


12 


.00140449 


7 


.00128700 


2 


.00118765 


7 


.00110254 


8 


.00154321 


13 


.00140252 


8 


.00128535 


3 


.00118(524 


8 


.00110132 


9 


.00154083 


14 


.00140056 


9 


.00128370 


4 


.00118483 


9 


.00110011 


650 


.00153846 


15 


.00139860 


780 


.00128205 


5 


.00118343 


910 


.00109890 


1 


.00153610 


16 


.00139665 


1 


.00128041 


6 


.00118203 


11 


.00109769 


2 


.00153374 


17 


.00139470 


2 


.00127877 


7 


.00118064 


12 


.00109649 


g 


.00153140 


18 


.00139276 


g 


.00127714 


8 


.00117924 


13 


.00109529 


4 


.00152905 


19 


.00139082 


4 


.00127551 


9 


.00117786 


14 


.00109409 


5 


.00152672 


720 


.00138889 


e 


.00127388 


850 


.00117647 


15 


.00109290 


6 


.00152439 


1 


.00138696 


6 


.00127226 


1 


.00117509 


16 


.00109170 




.00152207 


2 


.00138504 


7 


.00127065 


2 


.00117371 


17 


.00109051 


8 


.00151975 





.00138313 


8 


.00126904 


3 


00117233 


18 


.00108932 


9 


.00151745 


4 


.00138121 


c 


.00126743 


4 


.00117096 


19 


.00108814 


660 


.00151515 


5 


.00137931 


790 


.00120582: 


5 


.00116959 


920 


.00108696 


1 


.00151286 


6 


.00137741 


1 


.001264221 


6 


.00116822 


1 


.00108578 




.00151057 


r 


.00137552 


2 


.00126263 


7 


.00116686 


2 


.00108460 


3 


.00150830 


8 


.00137363 


e 


.00126103 


8 


.00116550 


3 


.00108342 


i 


.00150602 


9 


.00137174 


L 


.00125945 


9 


.00116414 


4 


.00108225 


t 


.00150376 


730 


.00136986 


f 


.00125786 


860 


.00116279 


i 


.00108108 


( 


.00150150 




.00136799 


( 


.00125628 


] 


.00116144 


6 


.00107991 


j 


.00149925 


2 


.00136612 


1 


.00125470 


2 


.00116009 


ij 


.00107875 


8 


.00149701 





.00136426 


8 


.00125313 


3 


.00115875 


8 


.00107759 


] 


.00149477 


i 


.00136240 


9 


.00125156 


t 


.00115741 


c 


.00107643 


670 


.00149254 


5 


.00136054 


800 


.00125000 


5 


.00115607 


930 


.00107527 




.00149031 


1 


.00135870 




.00124844 


6 


.00115473 




.00107411 




.00148809 


' 


.00135685 


2 


.00124688 


7 


.00115340 


< 


.00107296 




.00148588 


j 


.00135501 


< 


.00124533 


8 


.00115207 


] 


.00107181 


i 


.00148368 




.00135318 





.00124378 


( 


.00115075 


i 


.00107066 




.00148148 


74i 


.00135135 


| 


.00124224 


870 


.00114942 


5 


.0010695* 




.00147929 




.00134953 


I 


.00124069 


; 


.00114811 





.0010683^ 




.00147710 




.0013477 


' 


.00123916 


J 


.00114679 


' 


.00106724 




.0014749: 




.00134589 




.00123762 




.00114547 


I 


001066 1C 




.00147275 


i 


.00134409 




.00123609 


t 


.00114416 




.00106496 


68 


.00147059 




.00134228 


8li 


.00123457 


j 


.00114286 


941 


.0010638? 




.00146843 




.00134048 


11 


.00123305 


1 


.00114155 




.0010627C 




.00146628 




.00133869 


12 


.00123153 


1 


.00114025 




.0010615" 




.00146413 




.00133690 


13 


.00123001 




.00113895 




.00106044 




.00146199 




.0013351 


14 


.00122850 




.00113766 


i 


.00105935 




.00145985 


75 


.0013333 


lo 


.00122699 


881 


.00113636 




.0010582C 




.00145773 




.00133156 


16 


.00122549 




.00113507 


i 


.0010570* 




.00145560 




.0013297 


r 


.00122399 




.00113379 


i 


.0010559, 




.00145349 




.0013280 


18 


.00122249 




.00113250 




.0010548? 




.00145137 




.0013262 


19 


.00122100 


< 


.00113122 




.00105374 


69 


.00144927 




.00132450 


82C 


.00121951 




.00112994 


95< 


.0010526c 




.00144718 




.0013227o 




.00121803 


i 


.00112867 




. 00105 15i 




.00144509 




.00132100 




.00121654 


r 


.00112740 




.0010504$ 




.00144300 




.00131926 




.00121507 


| 


.00112613 




.0010493^ 




.00144092 




.00131752 


i 


.00121359 




.00112486 


<. 


.00104822 




.00143885 


76' 


.00131579 


j 


.0012121? 


891 


.00112360 


1 


.00104712 




.00143678 




.00131406 


i 


.00121065 


; 


.00112233 


< 


.00104602 




.00143472 




.00131234 


r 


.00120919 


2 


.00112108 


' 


.00104493 




.00143266 




.00131062 


| 


.00120773 


j 


.00111982 


8 


.00104384 




.00143061 


t 


.00130890 




.00120627 


i 


.00111857 


j 


.00104275 


70 


.00142857 




.00130719 


i 831 


.00120482 


5 


.00111732 


960 


.0010416; 




.00142653 


( 


.00130548 




.00120337 


6 


.00111607 




.00104058 




.00142450 


1 


.00130378 


< 


.00120192 


\ 


.00111483 


j 


.00103950 




.00142247 


8 


.00130208 


j 


.00120048 


8 


.00111359 


j 


.00103842 


^ 


.00142045 


j 


.00130039 


i 


.00119904 


9 


.00111235 


^ 


.00103734 


; 


.00141844 


770 


.00129870 


5 


.00119760 


900 


.00111111 


t 


,00103627 



RECIPROCALS OF NUMBERS. 



83 



No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


966 


00103520 


1031 


000969932 


1096 


000912409 


1161 


.000861326 


1226 


.000815661 


7 


00103413 


2 


000968992 


7 


000911577 


2 


.000860585 




.000814996 


8 


00103306 


3 


000968054 


8 


000910747 


3 


.000859845 


8 


.000814332 


9 


00103199 


4 


000967118 


9 


000909918 


4 


.000859106 


9 


.000813670 


970 


00103093 


5 


000966184 


1100 


.000909091 


5 


.000858369 


1230 


.000813008 


1 


.00102987 


6 


000965251 


1 


.000908265 


6 


.000857633 


1 


.000812348 


2 


00102881 


7 


.000964320 


2 


.000907441 


7 


.000856898 


2 


.000811688 


3 


.00102775 


8 


.000963391 


3 


.000906618 


8 


.000856164 


3 


.000811030 


4 


.00102669 


9 


000962464 


4 


.000905797 


9 


.000855432 


4 


.000810373 


fc 


.00102564 


1040 


.000961538 


5 


.000904977 


1170 


.000854701 


5 


.000809717 


6 


.00102459 


1 


.000960615 


6 


.000904159 


1 


.000853971 


6 


.000809061 


7 


.00102354 


2 


.000959693 


n 


.000903342 


2 


.000853242 


7 


.000808407 


8 


.00102250 


3 


.000958774 


8 


.000902527 


3 


.000852515 


8 


.000807754 


9 


.00102145 


4 


.000957854 


c 


.000901713 


4 


.000851789 


9 


.000807102 


980 


.00102041 


g 


.000956938 


1110 


.000900901 


5| .000851064 


1240 


.000806452 


I 


.00101937 


I 


.000956023 


11 


000900090 


6 


.000850340 


1 


.000805802 


2 


.00101833 


7 


.000955110 


12 


.000899281 


7 


.000849618 


2 


.000805153 


3 


.00101729 


8 


.000954198 


13 


.000898473 


8 


.000848896 


3 


.000804505 


4 


.00101626 


9 


.000953289 


14 


.000897666 


9 


.000848176 


4 


.000803858 


5 


.00101523 


1050 


.000952381 


15 


.000896861 


1180 


.000847457 


5 


.000803213 


6 


.00101420 


1 


.000951475 


16 


.000896057 


1 


.000846740 


6 


.000802568 




.00101317 




.000950570 


17 


.000895255 


2 


.000846024 


7 


.000801925 


8 


.00101215 


\ 


.000949668 


18 


.000894454 


3 


.000845308 


8 


.000801282 


c 


.00101112 


i 


.000948767 


19 


.000893655 


4 


.000844595 


9 


.000800640 


990 


.00101010 


5 


.000947867 


1120 


.000892857 


5 


.000843882 


1250 


.000800000 




.00100908 


6 


.000946970 


3 


.000892061 


6 


.000843170 


1 


.000799360 


< 


.00100806 


\ 


.000946074 


< 


.000891266 


7 


.000842460 


2 


.000798722 


< 


.00100705 


8 


.000945180 


j 


.000890472 


S 


.000841751 


3 


.000798085 


c 


.00100604 


c 


.000944287 


<. 


.000889680 


9 


.000841043 


4 


.000797448 


t 


.00100502 


1060 


.000943396 


5 


.000888889 


1190 


.000840336 


5 


.000796813 


( 


.00100J02 




.000942507 


6 


.000888099 


1 


,000839631 


6 


.000796178 


" ( 


.00100301 





.000941620 


\ 


.000887311 


2 


.000838926 


"t 


.000795545 


8 


.00100200 


{ 


.000940734 


8 


.000886525 


3 


.000838222 


8 


.000794913 


< 


.00100100 


t 


.000939850 


9 


.000885740 


4 


.000837521 


9 


.000794281 


1000 


.00100000 


5 


.000938967 


1130 


.000884956 


5 


.000836820 


1260 


.000793651 




.000999001 


< 


.000938086 




.000884173 


6 


.000836120 


1 


.000793021 


2 


.000998004 


7 


.000937207 


< 


.000883392 


7 


.000835422 


* 


.000792393 


j 


.000997009 


8 


.000936330 


< 


.000882612 


8 


.000834724 


3 


.000791766 


t 


,000996016 


< 


.000935454 


i 


.000881834 


9 


.000834028 


4 


.000791139 


5 


.000995025 


1070 


.000934579 


5 


.000881057 


1200 


000833333 


5 


.000790514 


( 


.000994036 





.000933707 


6 


.000880282 


1 


.000832639 


6 


.000789889 





.000993049 


2 


.000932836 


r 


.000879508 


2 


.000831947 


7 


.000789266 


I 


.000992063 


j 


.000931966 


8 


.000878735 


3 


.000831255 


8 


.000788643 


I 


.000991080 


c 


.000931099 


j 


.000877963 


4 


.000830565 


9 


.000788022 


1010 


.000990099 


I 


.000930233 


1140 


.000877193 


5 


.000829875 


1270 


.000787402 


11 


.000989120 


6 


.000929368 




.000876424 


6 


.000829187 


1 


.000786782 


12 


.000988142 


t 


.000928505 


i 


.000875657 


7 


.000828500 


2 


.000786163 


13 


.000987167 


8 


.000927644 




.000874891 


8 


.000827815 


3 


.000785546 


14 


.000986193 


c 


.000926784 


t 


.000874126 


9 


.000827130 


i 


.000784929 


15 


.000985222 


1080 


.000925926 


5 


.000873362 


1210 


.000826446 


5 


.000784314 


16 


.000984252 





.000925069 


6 


.000872600 


11 


.000825764 


6 


.000783699 


1" 


.000983284 


j 


.000924214 


\ 


.000871840 


12 


.000825082 




.000783085 


18 


.000982318 


j 


.000923361 


8 


.000871080 


13 


.000824402 


8 


.000782473 


19 


.000981354 


t 


.000922509 


c 


.000870322 


14 


.000823723 


9 


.000781861 


1020 


.000980392 


i 


.000921659 


1150 


.0008695G5 


15 


.000823045 


1280 


.000781250 


; 


.000979432 


( 


.000920810 




.000868810 


16 


.000822368 


1 


.000780640 





000978474 


1 


.000919963 


< 


.000868056 


17 


.000821693 


2 


.000780031 


< 


.000977517 


8 


.000919118 


j 


.000867303 


18 


.000821018 


3 


.000779423 


t 


.000976562 


9 


000918274 


t 


.000866551 


19 


.000820344 


i 


.000778816 


t 


.000975610 


1090 


.000917431 


5 


.000865801 


1220 


.000819672 


5 


.000778210 


( 


.000974659 


] 


.000916590 


6 


.000865052 


1 


.000819001 


6 


.000777605 


\ 


.000973710 





.000915751 


1 


.000864304 


2 


.000818331 


j 


.000777001 


8 


.000972763 


< 


.000914913 


8 


.000863558 


3 


.000817661 


8 


.000776397 


9 


.000971817 


i 


.000914077 


( 


.000862813 


4 


.000816993 


9 


.000775795 


1030 


.000970874 


5 


.000913242 


1160 


.000862069 


5 


.000816326 


1290 


.000775194 



8"4 



MATHEMATICAL TABLES, 



No. 


Recipro- 
cal. 


No. 


! 

Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


1291 


.000774593 


1356 


.000737463 


1421 


.000703730 


1486 


.000672948 


1551 


.000644745 


2 


.00077391)4 


7 


000736920 


2 .000703235 


7 


.000672495 


2 


.000644330 


3 


.000773395 


8 


.000736377 


3 


.000702741 


8 


.000672043 


3 


.000643915 


4 


.000772797 


( 


.000735835 


4 


.000702247 


9 


.000671592 


4 


.000643501 


5 


.000772:201 


1360 


.000735294 


5 


.000701754 


1490 


.000671141 


5 


.000643087 


6 


.000771605 


1 


.000734754 


6 


.000701262 


1 


.000670691 


6 


.000642673 


7 


.000771010 


2 


.000734214 


7 


.000700771 


2 


.000670241 




.000642261 


8 


.000770416 


f 


.000733676 


8 


.000700280 


3 


.000669792 


8 


.000641848 


9 1.000769823 


4 


.000733138 


9 


.000699790 


4 


.000669344 


9 


.000641437 


1300 .000769231 


r 


.00073-2601 


1430 


.000699301 


5 


.000668896 


1560 


.000641026 


1 


.000768639 


e 


.000732064 


1 


.000698812 


6 


.000668449 


1 


.000640615 


2 


.000768049 




.000731529 


2 


.000698324 


7 


.000668003 


2 


.000640205 


3 


.000767459 


8 


.000730994 


3 


.000697'837 


8 


.000667557 


3 


.000639795 


4 


.000766871 


9 


.000730460 


4 


.000697350 


9 


.000667111 


4 


.000639386 


5 


.000766283 


1370 


.000729927 


5 


.000690864 


1500 


.000666667 


5 


.000638978 


6 


.000765697 


1 


.000729395 


6 


.000696379 


1 


.000666223 


6 


.000638570 


7 


.000765111 


c 


.000728863 


7 


.000695894 


2 


.000665779 


7 


.000638162 


8 


.000764526 


3 


.000728332 


8 


.000695410 


3 


.000665336 


8 


.000637755 


9 


.000763942 


4 


.000727802 


9 


.000694927 


4 


.000664894 


9 


.000637349 


1310 


.000763359 


s 


.000727273 


1440 


.000694444 


5 


.000664452 


1570 


.000636943 


11 


.000762776 


6 


.000726744 


1 


.000693962 


6 


.000664011 


1 


.000636537 


12 


.000762195 


7 


.000726216 


2 


.000693481 




.000663570 


2 .000636132 


13 


.000761615 


8 


.000725689 


3 


.000693001 


8 


.000663130 


3 ! . 000635728 


14 


.000761035' 


9 


.000725163 


4 


.000692521 


9 


.000662691 


4 


.000635324 


15 


. 000760456 ! 


1380 


.000724638 


51.000692041 


1510 


. 000662252 


5 


.000634921 


16 


.000759878! 


1 


.000724113 


6 


.000691563 


11 


.000661813 


6 


.000634518 


17 


.000759301 


2 


.000723589 


7 


.000691085 


12 


.000661376 


7 


.000634115 


18 


.000758725; 


g 


.000723066 


8 


.000690608 


13 


.000660939 


8 


.000633714 


19 


.000758150; 


4 


.000722543 


9 


.000690131 


14 


.000660502 


9 


.000633312 


1320 


.000757576! 




.000722022 


1450L 000689655 


15 


.000660066 


1580 


.000632911 


1 


.000757002 


6 


.000721501 


1 


.000689180 


16 


.000659631 


1 


.000632511 


2 


.000756430 


7 


.000720980 


a 


.000688705 


17 


.000659196 


2 


.000632111 


3 


.000755858 


8 


.000720461 


3 


.000688231 


18 


.000658761 


3 !. 000631712 


4 


.000755287 


9 


.000719942 


4 


.000687758 


19 


.000658328 


4 .000631313 


5 


.000754717 


1390 


.000719424 


5 


.000687285 


1520 


.000657895 


5 


.000630915 


6 


.000754148 


1 


.000718907 


6 


.000686813 


1 


.000657462 





.000630517 


7 


.000753579 


2 


000718391 




.000686341 


2 


.000657030 


7 


.000630120 


8 


.000753012 


3 


.000717875 


g 


.000685871 


3 


.000656598 


8 


.000629723 


9 


.000752445 


4 


.000717360 


9 


.000685401 


4 


.000656168 


9 


.000629327 


1330 


.000751880 


5 


.000716846 


1460 


.000684932 


5 


.0006557381 


1590 


.000628931 


1 


.000751315 


6 


.000716332 


1 


.000684463 


6 


.000655308 


1 


.000628536 


2 


.000750750 


7 


.000715820 


2 


.000683994 


7 


.000654879 


o 


.000628141 


3 


.000750187 


8 


.000715308 


3 


.000683527 


8 


.000654450 


3 


.000627746 


4 


.000749625 


9 


.000714796 


41.000683060 


9 


.000654022 


4 


.000627353 


5 


.000749064 


1400 


.000714286 


5 ! . 000682594 


1530 


.000653595 


5 


.000626959 


6 


.000748503 


1 


.000713776 


6 


.000682128 


1 


.000653168 


6 


.000626566 


7 


.000747943 


2 


.000713267 


7 


.000681663 


2 


.000652742' 


7 


.000626174 


8 


.000747384 


3 


.000712758 


8 


.000681199 


3 


.000652316' 


8 


.000625782 


9 


. 000746826 I 


4 


.000712251 


9 


.000680735 


4 


.000651890! 


9 


.000625391 


1340 


.000746269; 


5 


.000711744 


1470 


.000680272 


5 


. 000651466 ' 


1600 


.000625000 


1 


.000745712 


6 


.000711238 


1 


.000679810 


6 


.000651042 


2 


.000624219 


2 


.000745156 


7 


.000710732 


2 


.000679348 


7 


.000650618 


4 


.000623441 


3 


.000744602! 


8 


.000710227 


3 


.000678887 


8 


.000650195 


6 


.000622665 


4 


.000744048! 


9 


.000709723 


4 


.000678426 


9 


.000649773 


8 


.000621890 


5 


.0007434941 


1410 


.000709220 


5 


.000677966 


1540 


.000649351 


1610 


.000621118 


6 


.000742942 


11 


.000708717 


6 


.000677507 


1 


.000648929 


2 


.000620347 


7 


.000742390 


12 


.000708215 


7 


.000677048 


2 


.000648508 


4 


.000619578 


8 


.000741840 


13 


.000707714 


8 


.000676590 


3 


.000648088 


6 


.000618812 


9 


.000741290 


14 


000707214 


9 


.0006-76138 


4 


.000647668 


8 


.000618047 


1350 


.000740741J 


15 


000706714 


1480 


000675676 


5 


.000647249 


1620 


.000617284 


1 


.000740192; 


16 


.000706215 


1 


.00^675219 


6 


.000646830 


2 


.000616523 


2 


000739645 


17 


.000705716 


2 


.0006?4"64 




.000646412 


4 


.000615763 


3 


.000789098 


18 


.000705219 


3 


.000674309 


8 


.000645995 


6 


.000615006 


4 


.000738552 


19 


.000704722 


4 


.000673854 


9 


.000645578 


8 


.000614250 


5 


.000738007! 


1420 


.000704225 


5 


.0006^3*01 


1550 


.000645161 


1630 


.000613497 



RECIPROCALS OF NUMBERS. 



85 



No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


1632 


.000612745 


1706 


.000586166 


1780 


.000561798 


1854 


.000539374 


1928 


.000518672 


4 


.000611995 


8 


.000585480 


2 


.000561167 


6 


.000538793 


1930 


.000518135 


6 


.000611247 


1710 


.000584795 


4 


.000560538 


8 


.000538213 


2 


.000517599 


8 


.000610500 


12 


.000584112 


6 


.000559910 


1860 


.000537634 


4 


.000517063 


1640 


.000609756 


14 


.000583430 


8 .000559284 


2 


.000537057 


6 


.000516528 


2 


.000609013 


16 


.000582750 


1790 


.000558659 


4 


.000536480 


8 


.000515996 


4 


.000608272 


18 


.000582072| 


2 


.000558035 


6 


.000535905 


1940 


.000515464 


6 


.000607533 


1720 


.000581395! 


4 


.000557413 


8 


.000535332 


2 


.000514933 


8 


.000606796 


2 


.000580720 


6 


.000556793 


1870 


.000534759 


4 


.000514403 


1650 


.000606061 


4 


.000580046 


8 


.000556174 


2 


.000534188 


6 


.000513874 


2 


.000305327 


6 


.000579374 


1800 


.000555556 


4 


.000533618 


8 


.000513347 


4 


.000604595 


8 


.000578704 


2 


.000554939 


6 


.000533049 


1950 


.000512820 


6 


.000603865 


1730 


.000578035 


4 


.000554324 


8 


.000532481 


2 


.000512295 


8 


.000603136 


2 


.000577367 


6 


.000553710 


1880 


.000531915 


4 


.000511770 


1660 


.000602410 


4 


.000576701 


8 


.000553097 


2 


.000531350 


6 


.000511247 


o 


.000601685 


6 


.000576037 


1810 


.000552486 


4 


.000530785 


8 


.000510725 


4 


.OOD600962 


8 


.000575374 


12 


.000551876 


6 


.000530222 


1960 


.000510204 


6 


.000600240 


1740 


.000574713 


14 


.000551268 


8 


.000529661 


2 


.000509684 


8 


.000599520 


2 


.000574053 


16 


. 000550661 


1890 


.000529100 


4 


.000509165 


1670 


.000598802 


4 


.000573394 


18 


.000550055 


2 


.000528541 


6 


.000508647 


2 


.000598086 


6 


.000572737 


1820 


.000549451 


4 


.000527983 


8 


.000508130 


4 


.000597371 


8 


.000572082 


2 


.000548848 


6 


.000527426 


1970 


.000507614 


6 


.000596658 


1750 


.000571429 


4, 


.000548246 


8 


.000526870 


2 


.000507099 


8 


.000595947 


2 


.000570776 


6 


.000547645 


1900 


.000526316 


4 


.000506585 


1680 


.000595238 


4 


.000570125 


8 


000547046 


2 


.000525762 


6 


.000506073 


2 


.000594530 


6 


.000569476 


1830 


.000546448 


4 


.000525210 


8 


.000505561 


4 


.000593824 


8 


.000568828 


2 


.000545851 


6 


.000524659 


1980 


.000505051 


6 


.000593120 


1760 


.000568182 


4 


.000545255 


8 


.000524109 


2 


.000504541 


8 


.000592417 


2 


.000567537 


6 i 


.000544662 


1910 


.000523560 


4 


.000504032 


1690 


.000591716 


4 


.000566893 


8 


.000544069 


12 


000523012 


6 


.000503524 


2 


.000591017 


6 


.000566251 


1840 


.000543478 


14 


.000522466 


8 


.OOOoOSOlS 


4 


.000590319 


8 


.000565611 


2 


.000542888 


16 


.000521920 


1990 


.00050213 


6 


.000589622 


1770 


.000564972 


4 


.000542299 


18 


.000521376 


2 


.000502008 


8 


.000588928 


2 


000564334 


C 


.000541711 


1920 


.000520833 


4 


.000501504 


1700 


.000588235 


4 


.000563698 


8 


.000541125 


2 


.000520291 


6 


.000501002 


2 


.000587544 


6 


000563063 


1850 


.000540540 


4 


.000519750 


8 


.000500501 


4 


.000586854 


8 


.000562430 


2 


.000539957 6 


.000519211 2000 


000500000 



Use of reciprocals. Reciprocals may be conveniently used to facili- 
tate computations iu long division. Instead of dividing as usual, multiply 
the dividend by the reciprocal of the divisor. The method is especially- 
useful when many different dividends are required to be divided by the 
same divisor. In this case find the reciprocal of the divisor, and make a 
small table of its multiples up to 9 times, and use this as a multiplication- 
table instead of actually performing the multiplication in each case. 

EXAMPLE. 9871 and several other numbers are to be divided by 1638. The 
reciprocal of 1638 is .000610500. 
Multiples of the 



reciprocal : 



.0006105 
.0012210 
.0018315 
.0024420 
.0030525 



The table of multiples is made by continuous addition 
of 6105. The tenth line is written to check the accuracy 
of the addition, but it is not afterwards used. 
Operation: 

Dividend 9871 
Take from table 1 ... 
7... 



.0006105 
0.042735 
00.48S40 
005.4945 



.0042735 
.0048840 
.0054945 
10. .0061050 

Quotient.. 6.0262455 

Correct quotient by direct division 6.0262515 

The result will generally be correct to as many figures as there are signifi- 
cant figures in the reciprocal, less one, and the error of the next figure will in 
general not exceed one. In the above example the reciprocal has six sig 
nificant figures, 610500, and the result is correct to five places of figures. 



86 



MATHEMATICAL TABLES. 



SQUARES, CUBES, SQUARE ROOTS AND CUBE 
ROOTS OF NUMBERS FROM .1 TO 1600. 



No. 


Square. 


Cube. 


Sq. 

Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


.1 


.01 


.001 


.3162 


.4642 


3.1 


0.61 


29.791 


1.761 


1.458 


.15 


.0225 


.0034 


.3873 


.5313 


.2 


10.24 


32.768 


1.789 


1.474 


.2 


.04 


-008 


.4472 


.5848 


.3 


10.89 


35.937 


1.817 


1.489 


.25 


.0625 


0156 


.500 


.6300 


.4 


11.56 


39.304 


1.844 


1.504 


.3 


.09 


027 


.5477 


.6694 


.5 


12.25 


42.875 


1.871 


1.518 


.35 


.1225 


0429 


.5916 


.7047 


.6 


12.96 


46.656 


1.897 


1.533 


.4 


.16 


064 


.6325 


.7368 


7 


13.69 


50.653 


1.924 


1.547 


.45 


.2025 


.0911 


.6708 


.7663 


'.8 


14.44 


54.872 


1.949 


1.560 


.5 


.25 


125 


.7071 


.7937 


.9 


15.21 


59.319 


1.975 


1.574 


.55 


.3025 


.1664 


.7416 


.8193 


4. 


16. 


64. 


2. 


1.5874 


.6 


.36 


216 


.7746 


.8434 


.1 


16.81 


68.921 


2.025 


1.601 


.65 


.4225 


.2746 


.8062 


.8662 


.2 


17.64 


74.088 


2.049 


1.613 


.7 


.49 


.343 


.8367 


.8879 


.3 


18.49 


79.507 


2.074 


1.626 


.75 


.5625 


.4219 


.8660 


.9086 


.4 


19.36 


85.184 


2.098 


1.639 


.8 


.64 


.512 


.8944 


.9283 


.5 


20.25 


91.125 


2.121 


1.651 


.85 


.7225 


.6141 


.9219 


.9473 


.6 


21.16 


97.336 


2.145 


1.663 


.9 


.81 


.729 


.9487 


.9655 


.7 


22.09 


103.823 


2.168 


1.675 


.95 


.9025 


.8574 


.9747 


.9830 


.8 


23.04 


110.592 


2.191 


1.687 


1. 


1. 


1. 


1. 


1. 


.9 


24.01 


117.649 


2.214 


.698 


1.05 


1.1025 


1.158 


1.025 


1.016 


5. 


25. 


125. 


2.2361 


.7100 


1.1 


1.21 


1.331 


1.049 


1.032 


.1 


26.01 


132 651 


2.258 


.721 


1.15 


1.3225 


1.521 


1.072 


1.048 


.2 


27.04 


140.608 


2.280 


.732 


1.2 


1.44 


1.728 


1.095 


1.063 


.3 


28.09 


148.877 


2.302 


.744 


1.25 


1.5625 


1.953 


1.118 


1.077 


.4 


29.16 


157.464 


2.324 


.754 


1.3 


1.69 


2.197 


1.140 


1.091 


.5 


30.25 


166.375 


2.345 


.765 


.35 


1.8225 


2.460 


1.162 


1.105 


.6 


31.36 


175.616 


2.366 


.776 


.4 


1.96 


2.744 


1.183 


1.119 


7 


32.49 


185.193 


2 . 387 


.786 


.45 


2.1025 


3.049 


1.204 


1.132 


'.8. 


33.64 


195.112 


2.408 


.797 


.5 


2.25 


3.375 


1.2247 


1.1447 


.9 


34.81 


205.379 


2.429 


.807 


.55 


2.4025 


3.724 


1.245 


1.157 


6. 


36. 


216. 


2.4495 


.8171 


.6 


2.56 


4.096 


1.265 


1.170 


.1 


37.21 


226.981 


2.470 


.827 


.65 


2.7225 


4.492 


1.285 


1.182 


o 


38.44 


238.328 


2.490 


.837 


1.7 


2.89 


4.913 


1.304 


1.193 


.3 


39.69 


250.047 


2.510 


.847 


1.75 


3.0625 


5.359 


1.323 


1.205 


4 


40.96 


262.144 


2.530 


.857 


1.8 


3.24 


5.832 


1.342 


1.216 


.5 


42.25 


274.625 


2.550 


.866 


1.85 


3.4225 


6.332 


1.360 


1.228 


.6 


43.56 


287.496 


2.569 


.876 


1.9 


3.61 


6.859 


1.378 


1.239 


.7 


44 89 


300.763 


2.588 


.885 


1.95 


3.8025 


7.415 


1.396 


1.249 


.8 


46.24 


314.432 


2.608 


.895 


2. 


4. 


8. 


1.4142 


1.2599 


.9 


4? 61 


328.509 


2.627 


.904 


.1 


4.41 


9.261 


1.449 


1.281 


7. 


49. 


343. 


2.6458 


1.9129 


.2 


4.84 


10.648 


1.483 


1.301 


.1 


50.41 


357.911 


2.665 


1.922 


.3 


5.29 


12.167 


1.517 


1.320 


.2 


51.84 


373.248 


2.683 


1.931 


.4 


5.76 


13.824 


1.549 


1.339 


.3 


53.29 


389.017 


2.702 


1.940 


.5 


6.25 


15.625 


1.581 


1.357 


.4 


54.76 


405.224 


2.720 


1.949 


.6 


6.76 


17.576 


1.612 


1.375 


.5 


56.25 


421.875 


2.739 


1.957 


.7 


7 29 


19.683 


1.643 


1.392 


.6 


57.76 


438.976 


2.757 


1.966 


.8 


7^84 


21.952 


1.673 


1.409 


.7 


59.29 


456.533 


2.775 


1.975 


.9 


8.41 


24.389 


1.703 


1.426 


.8 


60.84 


474.552 


2 793 


1.983 


3. 


9. 


27. 


1.7321 


1.4422 


.9 


62.41 


493.039 


2.81J 


1.992 



SQUARES, CUBES, SQUARE AKD CUBE ROOTS. 87 



No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


8. 


64. 


512. 


2.8284 


2. 


45 


2025 


91125 


6.7082 


3.5569 


.1 


65.61 


531.441 


2.846 


2 008 


46 


2116 


97336 


6.7823 


3.5830 


.2 


67.24 


551.368 


2.864 


2.017 


47 


2:209 


103823 


6.8557 


3.6088 


.3 


68.89 


571.787 


2.881 


2 0^5 


48 


2304 


110592 


6.9282 


3.6342 


.4 


70.56 


592.704 


2.898 


2.033 


49 


2401 


117649 


7. 


3.6593 


.5 


72.25 


614.125 


2.915 


2.041 


50 


2500 


125000 


7.0711 


3.6840 


.6 


73.96 


636.056 


2.933 


2.049 


51 


2601 


132651 


7.1414 


3.7084 


.7 


75.69 


658.503 


2.950 


2.057 


52 


2704 


140608 


7 2111 


3.7325 


.8 


77.44 


681.472 


2.966 


2.065 


53 


2809 


148877 


7.2801 


3.7563 


9 


79.21 


704.969 


2.983 


2.072 


54 


2916 


157464 


7.3485 


3.7798 


9. 


81. 


729. 


3. 


2.0801 


55 


3025 


166375 


7.4162 


3.8030 


.1 


82.81 


753.571 


3.017 


2.088 


56 


3136 


175616 


7.4833 


3.8259 


.2 


84.64 


778.688 


3.033 


2.095 


57 


3249 


185193 


7.5498 


3.8485 


.3 


86.49 


804.35? 


3.050 


2.103 


58 


3364 


195112 


7.6158 


3.8709 


.4 


88.36 


830.584 


3.066 


2.110 


59 


3481 


205379 


7.6811 


3.8930 


.5 


90.25 


857.375 


3.082 


2.118 


60 


3600 


216000 


7.7460 


3.9149 


.6 


92.16 


884.736 


3.098 


2.125 


61 


3721 


226981 


7.8102 


3.9365 


.7 


94.09 


912.673 


3.114 


2 133 


62 


3844 


238328 


7.8740 


3.9579 


.8 


96.04 


941.192 


3.130 


2.140 


63 


3969 


250047 


7.9373 


3.9791 


.9 


98.01 


970.299 


3.146 


2.147 


64 


4096 


262144 


8. 


4. 


10 


100 


1000 


3.1623 


2.1544 


65 


4225 


274625 


8.0623 


4.0207 


11 


121 


1331 


3.3166 


2.2240 


66 


4356 


287496 


8.1240 


4.0412 


12 


144 


1728 


3.4641 


2.2894 


67 


4489 


300763 


8.1854 


4.0615 


13 


169 


2197 


3.6056 


2.3513 


68 


4624 


314432 


8.2462 


4.0817 


14 


196 


2744 


3.7417 


2.4101 


69 


4761 


3vJ8509 


8.3066 


4.1016 


15 


225 


3375 


3.8730 


2.4662 


70 


4900 


343000 


8.3666 


4.1213 


16 


256 


4096 


4. 


2.5198 


71 


5041 


357911 


8.4261 


4.1408 


17 


289 


4913 


4.1231 


2.5713 


72 


5184 


373248 


8.4853 


4.1602 


18 


324 


5832 


4.2426 


2.6207 


73 


5329 


389017 


8.5440 


4.1793 


19 


361 


6859 


4.3589 


2.6684 


74 


5476 


405224 


8.6023 


4.1983 


20 


400 


8000 


4.4721 


2./144 


75 


5625 


421875 


8.6603 


4.2172 


21 


441 


9261 


4.5826 


2 7589 


76 


5776 


438976 


8.7178 


4.2358 


22 


484 


10648 


4.6904 


2.8020 


77 


5929 


456533 


8.7750 


4.2543 


23 


529 


12167 


4.7958 


2.8439 


78 


6084 


474552 


8.8318 


4.2727 


24 


576 


13824 


4.8990 


2.8845 


79 


6241 


493039 


8.8882 


4.2908 


25 


625 


15625 


5. 


2.9240 


80 


6400 


512000 


8.9443 


4.3089 


26 


676 


17576 


5.0990 


2.9625 


81 


6561 


531441 


9. 


4.3267 


27 


729 


19683 


5.1962 


3. 


8-.' 


6724 


551368 


9.0554 


4.3445 


28 


784 


21952 


5.2915 


3 0366 


83 


6889 


571787 


9.1104 


4.3621 


29 


841 


24389 


5.3852 


3.0723 


84 


7056 


592704 


9.1652 


4.3795 


30 


900 


27000 


5.4772 


3.1072 


85 


7225 


614125 


9.2195 


4.3968 


31 


961 


29791 


5.5678 


3.1414 


86 


7396 


636056 


9.2736 


4.4140 


32 


1024 


32768 


5.6569 


3.1748 


87 


7569 


658503 


9 3276 


4.4310 


33 


1089 


35937 


5.7446 


3.2075 


88 


7744 


6S1472 


9.3808 


4.4480 


34 


1156 


39304 


5.8310 


3.2396 


89 


7921 


704969 


9.4340 


4.4647 


35 


1225 


42875 


5.9161 


3.2711 


90 


8100 


729000 


9.4868 


4.4814 


36 


1296 


46656 


6. 


3.3019 


91 


8281 


753571 


9.5394 


4.4979 


37 


1369 


50653 


6.0828 


3.3322 


92 


8464 


778688 


9.5917 


4.5144 


38 


1444 


54872 


6.1644 


3.3620 


93 


8649 


804357 


9 6437 


4.5307 


39 


1521 


59319 


6.2450 


3.3912 


94 


8836 


830584 


9.6954 


4.5468 


40 


1600 


64000 


6.3246 


3 4200 


95 


9025 


857375 


9 7468 


4.5629 


41 


1681 


689'>1 i 6. 4031 


3.4482 


96 


9216 


884736 


9.7980 


4.5789 


42 


1764 


74088 6.4807 


3.4760 


97 


9409 


912673 


9.8489 


4.5947 


43 


1849 


79507 6.5574 


3.5034 


98 


9604 


941192 


9.8995 


4.6104 


44 


1936 


85184 6.6332 


3.5303 


99 


9801 


970-299 


9.9499 


4.6261 



88 



MATHEMATICAL TABLES. 



No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube, 


Sq. 
Root. 


Cube 
Root. 


100 


10000 


1000000 


10. 


4.6416 


155 


24025 


3723875 


12.4499 


5.3717 


101 


10201 


1030301 


10.0499 


4.6570 


156 


24336 


3796416 


12.4900 


5.3832 


10:3 


10404 


1061208 


10.0995 


4.6723 


157 


24649 


3869893 


12.5300 


5.3947 


103 


10609 


1092727 


10.1489 


4.6875 


158 


24964 


3944312 


12.5698 


5.4061 


104 


10816 


1124864 


10.1980 


4.7027 


159 


25281 


4019679 


12.6095 


5.4175 


105 


11025 


1157625 


10.2470 


4.7177 


160 


25600 


4096000 


12.6491 


5.4288 


106 


11236 


1191016 


10.2956 


4.7326 


161 


25921 


4173281 


12.6886 


5.4401 


107 


11449 


1225043 


10.3441 


4.7475 


162 


26244 


4251528 


12.7279 


5 4514 


108 


11664 


1259712 


10.3923 


4.7622 


163 


26569 


4330747 


12.7671 


5.4626 


109 


11881 


1295029 


10.4403 


4.7769 


164 


26896 


4410944 


12.8062 


5.4737 


110 


12100 


1331000 


10.4881 


4.7914 


165 


27225 


4492125 


12.8452 


5.4848 


111 


12321 


1367631 


10.5357 


4.8059 


166 


27556 


4574-296 


12.8841 


5.4959 


112 


12514 


1404928 


10.5830 


4.8203 


167 


27889 


4657463 


12.9228 


5.5069 


113 


12769 


1442897 


10.6301 


4.8346 


168 


28224 


4741632 


12.9615 


5.5178 


114 


12996 


1481544 


10.6771 


4.8488 


169 


28561 


4826809 


13.0000 


5.5288 


115 


132-25 


1520875 


10.7238 


4.8629 


170 


28900 


4913000 


13.0384 


5.5397 


116 


13456 


156089(5 


10.7703 


4.8770 


171 


29241 


500021 1 


13.0767 


5.5505 


117 


13689 


1601613 


10.8167 


4.8910 


172 


29584 


5088448 


13.1149 


5.5613 


118 


13924 


1643032 


10.8628 


4.9049 


173 


29929 


5177717 


13.1529 


5.5721 


119 


14161 


1685159 


10.9087 


4.9187 


174 


30276 


5268024 


13.1909 


5.5828 


120 


14400 


1728000 


10.9545 


4.9324 


175 


30625 


5359375 


13.2288 


5.5934 


121 


14641 


1771561 


11.0000 


4.9461 


176 


30976 


5451776 


13.2665 


5.6041 


122 


14884 


1815848 


11.0454 


4.9597 


177 


31329 


5545233 


13.3041 


5.6147 


123 


15129 


1860867 


11.0905 


4.9732 


178 


31684 


5639752 


13.3417 


5.6252 


124 


15376 


1906624 


11.1355 


4.9866 


179 


32041 


5735339 


13.3791 


5.6357 


125 


15625 


1953125 


11.1803 


5.0000 


180 


32400 


583-2000 


13.4164 


5. 6402 


126 


15876 


2000376 


11.2250 


5.0133 


181 


32761 


5929741 


13.4536 


5.6567 


127 


16129 


2018383 


11.2694 


5 0265 


182 


33124 


6028568 


13.4907 


5.6671 


128 


16384 


2097152 


11.3137 


5.0397 


183 


33489 


6128487 


13.5277 


5.6774 


129 


16641 


2146689 


11.3578 


5.0528 


184 


33856 


6229504 


13.5647 


5.6877 


130 


16900 


2197000 


11.4018 


5.0658 


185 


342-25 


6331625 


13.6015 


5.6980 


131 


17161 


2248091 


11.4455 


5.0788 


186 


34596 


6434856 


13.6382 


5.7083 


132 


17424 


2299968 


11.4891 


5.0916 


187 


34969 


6539203 


13.6748 


5.7185 


133 


17689 


2352637 


11.5326 


5.1045 


188 


35344 


6644672 


13.7113 


5.T287 


134 


17956 


2406104 


11.5758 


5.1172 


189 


35721 


6751269 


13.7477 


5.7388 


135 


18225 


2460375 


11.6190 


5.1299 


190 


36100 


6859000 


13.7840 


5.7489 


136 


18496 


2515456 


11.6619 


5.1426 


191 


36481 


6967871 


13.8203 


5.7590 


137 


18769 


2571353 


11.7047 


5.1551 


192 


36864 


7077888 


13.8564 


5.7690 


138 


19044 


2628072 


11.7473 


5.1676 


193 


37249 


7189057 


13.8924 


5 7790 


139 


19321 


2685619 


11.7898 


5.1801 


194 


37636 


7301384 


13.9284 


5.7890 


140 


19600 


2744000 


11.8322 


5.1925 


195 


38025 


7414875 


3.9642 


5.7989 


141 


19881 


2803221 


11.8743 


5.2048 


196 


38416 


7529536 


14.0000 


5.8088 


142 


20164 


2863286 


11.9164 


5.2171 


197 


38809 


7645373 


14.0357 


5.8186 


143 


20449 


29-24207 


11.9583 


5.2293 


198 


39204 


7762392 


14.0712 


5.8285 


144 


20736 


2985984 


12.0000 


5.2415 


199 


39601 


7880599 


14.1067 


5.8383 


145 


21025 


3048625 


12.0416 


5.2536 


200 


40000 


8000000 


14.1421 


5.8480 


146 


21316 


3112136 


12.0830 


5.2656 


201 


40401 


8120601 


14.1774 


5.8578 


147 


21609 


31765-23 


12.1244 


5.2776 


202 


40804 


824-2408 


14.2127 


5.8675 


148 


21904 


3241792 


12.1655 


5.2896 


203 


41209 


8365427 


14.2478 


5.8771 


149 


22201 


3307949 


12.2066 


5.3015 


204 


41616 


8489664 


14.2829 


5.8868 


150 


22500 


3375000 


12.2474 


5.3133 


205 


42025 


8615125 


14.3178 


5.8964 


151 


2-2801 


3442951 


12.2882 


5.3251 


206 


42436 


8741816 


14.3527 


5.9059 


152 


23104 


3511808 


12.3288 


5.3368 


207 


42849 


8869743 


14.3875 


5.9155 


153 


23409 


3581577 


12.3603 


5 . :>, 185 


208 


43264 


8998912 


14.4222 


5.9250 


154 


23716 


3652264 1? *M>~ 


5.3C,0! 


209 


43681 


9129329 


14.4568 


5.9345 



SQUARES, CUBES, SQUARE AtfD CUBE ROOTS. 89 



No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


210 


44100 


9261000 


14.4914 


5.9439 


265 


70225 


18609625 


16.2788 


6.4232 


211 


44521 


9393931 


14.5258 


5.9533 


266 


70756 


18821096 


16.3095 


6.4312 


212 


44944 


9528128 


14.5602 


5.9627 


267 


71289 


19034163 


16.3401 


6.4393 


213 


45369 


9663597 


14.5945 


5.9721 


268 


71824 


19248832 


16.3707 


6.4473 


214 


45796 


9800344 


14.6287 


5.9814 


269 


72361 


19465109 


16.4012 


6.4553 


215 


46225 


9938375 


14.6629 


5 9907 


270 


72900 


19683000 


16 4317 


6.4633 


216 


46656 


10077696 


14.6969 


6.0000 


271 


73441 


19902511 


16.4621 


6.4713 


217 


47089 


10218313 


14.7309 


6.0092 


272 


73984 


20123648 


16.4924 


6.4792 


218 


47524 


10360232 


14.7648 


6 0185 


273 


74529 


20346417 


16.5227 


6.4872 


219 


47961 


10503459 


14.7986 


6 0477 


274 


75076 


20570824 


16.5529 


6.4951 


220 


48400 


10648000 


14.8324 


6.0368 


275 


75625 


20796875 


16.5831 


6.5030 


221 


48841 


10793861 


14.8661 


6.0459 


276 


76176 


21024576 


16.6132 


6.5108 


222 


49284 


10941048 


14.8997 


6 0550 


277 


76729 


21253933 


16.6433 


6.5187 


223 


49729 


11089567 


14.9332 


6.0641 


278 


77284 


21484952 


16.6733 


6.5265 


224 


50176 


112394^4 


14.9666 


6.0732 


279 


77841 


21717639 


16.7033 


6.5343 


2-25 


50625 


11390625 


15.0000 


6.0822 


280 


78400 


21952000 


16.7332 


6.5421 


226 


51076 


11543176 


15.0333 


6.0912 


281 


78961 


22188041 


16.7631 


6.5499 


227 


51529 


11697083 


15.0665 


6.1002 


282 


79524 


22425768 


16.7929 


6.5577 


228 


51984 


11852352 


15.0997 


6.1091 


283- 


80089 


22665187 


16.8226 


6.5654 


229 


52441 


12008989 


15.1327 


6.1180 


284 


80656 


22906304 


16.8523 


6.5731 


230 


52900 


12167000 


15.1658 


6.1269 


285 


81225 


23149125 


16.8819 


6.5808 


231 


53361 


12326:391 


15.1987 


6.1358 


:286 


81796 


23393656 


16.9115 


6.5885 


232 


53824 


12487168 


15.2315 


6.1446 


287 


82369 


23639903 


16.9411 


6.5962 


233 


54289 


12649337 


15.2643 


6.1534 


288 


82944 


23887872 


16.9706 


6.6039 


234 


54756 


12812904 


15.2971 


6.1622 


289 


83521 


24137569 


17.0000 


6.6115 


235 


55225 


12977875 


15.3297 


6.1710 


290 


84100 


24389000 


17.0294 


6.6191 


236 


55696 


13144256 


15.3623 


6.1797 


2.)1 


84681 


2464-J171 


17.0587 


6.6267 


237 


56169 


13312053 


15 3948 


6.1885 


292 


85264 


24897088 


17.0880 


6.6343 


238 


56644 


13481272 


15.4272 


6.1972 


293 


85849 


25153757 


17.1172 


6.6419 


239 


57121 


13651919 


15.4596 


6.2058 


294 


86436 


25412184 


17.1464 


6.6494 


240 


57600 


13824000 


15.4919 


6.2145 


295 


87025 


2567:2375 


17.1756 


6.6569 


241 


58081 


13997521 


15.5242 


6.2231 


296 


87616 


25934336 


17.2047 


6.6644 


242 


58564 


14172488 


15.5563 


6.2317 


297 


88-^09 


26198073 


17.2337 


6.6719 


243 


59049 


14348907 


15.5885 


6.2403 


r) 98 


88804 


26463592 


17.2627 


6.6794 


244 


59536 


14526784 


15.6205 


6.2488 


299 


89401 


26730899 


17.2916 


6.6869 


245 


60025 


14706125 


15.6525 


6.2573 


300 


90000 


27000000 


17.3205 


6.6943 


246 


60516 


14886936 


15.6844 


6.2658 


301 


90601 


27270901 


17.3494 


6.7018 


247 


61009 


15069^3 


15.7162 


6.2743 


30.2 


91204 


27543608 


17.3781 


6.7092 


248 


61504 


15252992 


15.7480 


6.2828 


303 


91809 


27818127 


17.4069 


6.7166 


249 


62001 


15438249 


15.7797 


6.2912 


304 


92416 


28094464 


17 4356 


6.7240 


250 


62500 


15625000 


15.8114 


6.2996 


305 


93025 


28372625 


17.4642 


6.7313 


251 


63001 


15813-J51 


15.8430 


6.3080 


306 


93636 


28652616 


17.4929 


6.7387 


252 


63504 


16003008 


15.8745 


6.3164 


307 


94249 


28934443 


17.5214 


6.7460 


253 


64009 


16194;77 


15.9060 


6.3247 


308 


94864 


29218112 


17.5499 


6.7533 


254 


64516 


16387064 


15.9374 


6.3330 


309 


95481 


29503629 


17.5784 


6.7606 


255 


65025 


16581375 


15.9687 


6.3413 


310 


96100 


29791000 


17.6068 


6.7679 


256 


65536 


16777216 


16.0000 


6.3496 


311 


96721 


30080231 


17.6352 


6 7752 


257 


66049 


16974593 


16.0312 


6.3579 


312 


97344 


30371328 


17.6635 


6.7824 


258 


66564 


17173512 


16.0624 


6.3661 


313 


97969 


30664297 


17.6918 


6.7897 


259 


67081 


17373979 


16.0935 


6.3743 


314 


98596 


30959144 


17.7200 


6.7969 


260 


67600 


17576000 


16.1245 


6.3825 


315 


99225 


31255875 


17.7482 


6.8041 


261 


68121 


17779581 


16.1555 


6.3907 


316 


99856 


31554496 


17.7764 


6.8113 


262 


68644 


17984728 


16.1864 


6.3988 


317 


100489 


31855013 


17.8045 


6.8185 


2G3 


69169 


18191447 


16.2173 


6.4070 


318 


101124 


32157432 


17.8326 


6.8256 


264 


69696 


18399744 


16.2481 


6.4151 


319 


101761 


324617o9 


17.8606 


6.8328 



90 



MATHEMATICAL TABLES. 



No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


320 


102400 


32768000 


17.8885 


6.8399 


375 


140625 


52734375 


19.3649 


7.2112 


321 


103041 


33076161 


17.9165 


6 8470 


376 


141376 


58157376 


19.3907 


7.2177 


322 


103684 


33386248 


17.9444 


6.8541 


377 


142129 


53582633 


19.4165 


7.2240 


323 


104329 


33698267 


17.9722 


6.8612 


378 


142884 


54010152 


19.4422 


7.2304 


324 


104976 


34012224 


18.0000 


6.8683 


379 


143641 


54439939 


19.4679 


7.23G8 


325 


105625 


34328125 


18.0278 


6.8753 


380 


144400 


54872000 


19.4936 


7.2432 


326 


106276 


34645976 


18.0555 


6.8824 


381 


145161 


55306341 


19.5192 


7.2495 


327 


106929 


34965783 


18.0831 


6.8894 


382 


145924 


55742968 


19.5448 


7.2558 


328 


107584 


35287552 


18.1108 


6.8964 


383 


146689 


56181887 


19.5704 


7.2622 


329 


108241 


35611289 


18.1384 


6.9034 


384 


147456 


50623104 


19.5959 


7.2685 


330 


108900 


35937000 


18.1659 


6.9104 


385 


148225 


57066625 


19.6214 


7.2748 


331 


109561 


36264691 


18.1934 


6.9174 


386 


148996 


57512456 


19.6469 


7.2811 


332 


110224 


36594368 


18.2209 


6.9244 


387 


149769 


57960603 


19.6723 


7.2874 


333 


110889 


36926037 


18.2483 


6.9313 


388 


150544 


58411072 


19.6977 


7.2936 


334 


111556 


37259704 


18.2757 


6.9382 


389 


151321 


58863869 


19.7231 


7.2999 


335 


112225 


37595375 


18.3030 


6.9451 


390 


152100 


59319000 


19.7484 


7 3061 


336 


112896 


37933056 


18.3303 


6.9521 


391 


152881 


59776471 


19.7737 


7.3124 


337 


113569 


38272753 


18.3576 


6.9589 


392 


153664 


60236288 


19.7990 


7.3186 


338 


114244 


38614472 


18.3848 


6.9658 


393 


154449 


60698457 


19.8242 


7.3248 


339 


114921 


38958219 


18.4120 


6.9727 


394 


155236 


61162984 


19.8494 


7.3310 


340 


115600 


39304000 


18.4391 


6.9795 


395 


156025 


61629875 


19.8746 


7.3372 


341 


116281 


39651821 


18.4662 


6.9864 


396 


156816 


62099136 


19.8997 


7.3434 


342 


116964 


40001688 


18.4932 


6 9932 


397 


157609 


62570773 


19.9249 


7.3496 


343 


117649 


40353607 


18.5203 


7.0000 


398 


158404 


63044792 


19.9499 


7.3558 


344 


118336 


40707584 


18.5472 


7.0068 


399 


159201 


63521199 


19.9750 


7.3619 


345 


119025 


41063625 


18.5742 


7.0136 


400 


160000 


64000000 


20 0000 


7.3681 


346 


119716 


41421736 


18.6011 


7.0203 


401 


160801 


64481201 


20 0250 


7.3742 


347 


120409 


41781923 


18.6279 


7.0271 


402 


161604 


64904808 


20.0499 


7.3803 


348 


121104 


42144192 


18.6548 


7.0338 


403 


162409 


65450827 


20 0749 


7.3864 


349 


121801 


42508549 


18.6815 


7.0406 


404 


163216 


65^39264 


20.0998 


7.3925 


350 


122500 


42875000 


18.7083 


7.0473 


405 


164025 


66430125 


20.1246 


7.3986 


351 


123201 


43243551 


18.7350 


7.0540 


406 


164836 


66923416 


20.1494 


7.4047 


352 


123904 


43614208 


18.7617 


7.0607 


407 


165649 


67419143 


20.1742 


7.4108 


353 


124609 


43986977 


18.7883 


7.0674 


408 


166464 


67917312 


20.1S90 


7.4169 


354 


125316 


44361864 


18.8149 


7.0740 


409 


167281 


68417'929 


20.2237 


7.4229 


355 


126025 


44738875 


18.8414 


7.0807 


410 


168100 


68921000 


20.2485 


7.4290 


356 


126736 


45118016 


18.8680 


7.0873 


411 


168921 


69426531 


20.2731 


7.4350 


357 


127449 


45499293 


18.8944 


7.0940 


412 


169744 


69934528 


20.2978 


7.4410 


358 


128164 


45882712 


18 9209 


7.1006 


413 


170569 


70444997 


20.3224 


7.4470 


359 


128881 


46268279 


18.9473 


7.1072 


414 


171396 


70957944 


20.3470 


7.4530 


360 


129600 


46056000 


18.9737 


7.1138 


415 


172225 


71473375 


20.3715 


7.4590 


361 


130321 


47045881 


19.0000 


7.1204 


416 


173056 


71991296 


20.3961 


7.4650 


362 


131044 


47437928 


19.0263 


7.1269 


417 


173889 


72511713 


20.4206 


7.4710 


363 


131769 


47832147 


19.0526 


7.1335 


418 


174724 


73034632 


20.4450 


7.4770 


364 


132496 


48228544 


19.0788 


7.1400 


419 


175561 


73500059 


20.4695 


7.4829 


365 


133225 


48627125 


19.1050 


7.1466 


420 


176400 


74088000 


20.4939 


7.4889 


366 


133956 


49027896 


19.1311 


7.1531 


421 


177241 


74618461 


20.5183 


7.4948 


367 


134689 


49430863 


19.1572 


7.1596 


422 


178084 


75151448 


20.5426 


7.5007 


368 


135424 


49836032 


19.1833 


7.1661 


423 


178929 


75686967 


20.5670 


7.5067 


369 


136161 


50243409 


19.2094 


7.1726 


424 


179776 


76225024 


20.5913 


7.5126 


370 


136900 


50653000 


19.2354 


7.1791 


425 


180625 


76765625 


20.6155 


7.5185 


371 


137641 


51064811 


19.2614 


7.1855 


426 


181476 


77308776 


20.6398 


7.5244 


372 


138384 


51478848 


19.2873 


7.1920 


427 


182329 


77854483 


20.6640 


7.5302 


373 


139129 


51895117 


19.3132 


7.1984 


428 


183184 


78402752 


20.6882 


7.5361 


374 


139876 


52313624 


19.3391 


7.2048 


429 


184041 


78953589 


20.7123 


7.5420 



SQUARES, CUBES, SQUARE AND CUBE ROOTS. 91 



No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


430 


184900 


79507000 


20.7364 


7.5478 


485 


235225 


114084125 


22.0227 


7.8568 


431 


185761 


80062991 


20.7605 


7 . 5537 


486 


236196 


114791256 


22.0454 


7.8622 


432 


186624 


80621568 


20.7846 


7.5595 


487 


237169 


115501303 


22.0681 


7.8676 


433 


187489 


SI 182737 


20.8087 


7.5654 


488 


238144 


116214.272 


22.0907 


7.8730 


434 


188356 


81746504 


20.8327 


7.5712 


489 


239121 


116930169 


22.1133 


7.8784 


435 


189225 


82312875 


20.8567 


7.5770 


490 


240100 


117649000 


22.1359 


7.8837 


436 


190096 


82881856 


20.8806 


7.5828 


491 


241081 ,118370771 


22.1585 


7.8891 


437 


190969 


83453453 


20.9045 


7.5886 


492 


242064 1119095488 


22.1811 


7.8944 


438 


191844 


84027672 


20.9284 


7.5944 


493 


243049 1119823157 


22.2036 


7.8998 


439 


192721 


84604519 


20.9523 


7.6001 


494 


244036 


120553784 


22.2261 


7.9051 


440 


193600 


85184000 


20.9762 


7.6059 


495 


245025 


121287375 


22.2486 


7.9105 


441 


194481 


85766121 


21.0000 


7.6117 


496 


246016 


122023936 


22.2711 


7.9158 


442 


195364 


86350888 


21.0238 


7.6174 


497 


247009 


122763473 


22.2935 


7.9211 


443 


196249 


86938307 


21.0476 


7.6232 


498 


248004 


123505992 


22.3159 


7.9264 


444 


197136 


87528384 


21.0713 


7.6289 


499 


249001 


124251499 


22 3383 


7.9317 


445 


198025 


88121125 


21.0950 


7.6346 


500 


250000 


125000000 


22.3607 


7.9370 


446 


198916 


88716536 


21.1187 


7.6403 


501 


251001 


125751501 


22.3830 


7.9423 


447 


199809 


89314623 


21.1424 


7.6460 


502 


252004 


126506008 


22.4054 


7.9476 


448 


200704 


8991539-3 


21.1660 


7.6517 


503 


253009 


127263527 


22.4277 


7.9528 


449 


201601 


90518849 


21.1896 


7.6574 


504 


254016 


128024064 


22.4499 


7.9581 


450 


202500 


91125000 


21.2132 


7.6631 


505 


255025 


128787625 


22.4722 


7.9634 


451 


203401 


91733851 


21.2368 


7.6688 


506 


256036 


129554216 


22.4944 


7.9686 


452 


204804 


92345408 


21.2603 


7.6744 


507 


257049 


130323843 


22.5167 


7.9739 


453 


205209 


92959677 


21.2838 


7.6800 


508 


258064 


131096512 


22.5389 


7.9791 


454 


206116 


93576664 


21.3073 


7.6857 


509 


259081 


131872229 


22.5610 


7.9843 


455 


207025 


94196375 


21.3307 


7.6914 


510 


260100 


132651000 


22.5832 


7.9896 


456 


207936 


94818816 


21.3542 


7.6970 


511 


261121 


133432831 


22.6053 


7.9948 


457 


208849 


95443993 


21.3776 


7.7026 


512 


262144 


134217728 


22.6274 


8.0000 


458 


209764 


96071912 


21.4009 


7.7082 


513 


263169 


135005697 


22.6495 


8.0052 


159 


210681 


96702579 


21.4243 


7.7138 


514 


264196 


135796744 


22.6716 


8.0104 


460 


211600 


97336000 


21.4476 


7.7194 


515 


265225 


136590875 


22 6936 


8.0156 


iQl 


212521 


97972181 


21.4709 


7.7250 


516 


266256 


137388096 


22.7156 


8.0208 


462 


213444 


98611128 


21.4942 


7.7306 


517 


267289 1138188413 


22.7376 


8.0260 


463 


214369 


99252847 


21.5174 


7.7362 


518 


268324 


138991832 


22.7596 


8.0311 


464 


215296 


99897344 


21.5407 


7.7418 


519 


269361 


139798359 


22.7816 


8.03G3 


465 


216225 


100544625 


21.5639 


7 . 7473 


520 


270400 


140608000 


22.8035 


8.0415 


466 


217156 


101194696 


21 5870 


7.7529 


521 


271441 1141420761 


22.8254 


8.0466 


467 


218089 


101847563 


21.6102 


7.7584 


522 


272484 142236648 


22.8473 


8.0517 


468 


219024 


102503232 


21.6333 


7.7639 


523 


273529 143055667 


22.8692 


8.0569 


469 


219961 


103161709 


21.6564 


7.7695 


524 


274576 


143877824 


22.8910 


8.0620 


470 


220900 


103823000 


21.6795 


7.7750 


525 


275625 


144703125 


22.9129 


8.0671 


471 


221841 


104487111 


21.7025 


7.7805 


526 


276676 145531576 


22.9347 


8.0723 


472 


222784 


105154048 


21.7256 


8.7860 


527 


277729 146363183 


22.9565 


8.0774 


473 


223729 


105823317 


21.7486 


7.7915 


528 


278784 147197952 


22.9783 


8.0825 


474 


224676 


106496424 


21.7715 


7.7970 


529 


279841 


148035889 


23.0000 


8.0876 


475 


225625 


107171875 


21.7945 


7.8025 


530 


280900 


148877000 


23.0217 


8.0927 


476 


226576 


107850176 121.8174 


7. 8079 1 531 


281961 149721291 


23.0434 


8.0978 


477 


227529 


108531333 


21.8403 


7. 8184 1 582 


283024 150568768 


23.0651 


8.1028 


478 


228484 


109215352 


21 8632 


7.8188 533 


284089 151419437 


23.0368 


8.1079 


479 


229441 


109902239 


21.8861 


7.8243 


534 


285156 


152273304 


23.1084 


8.1130 


480 


230400 


110592000 


21 9089 


7.8297 


535 


286225 


153130375 


23.1301 


8.1180 


481 


231361 


111284641 


21.9317 


7.8352 


536 


287296 153990656 


23.1517 


8.1231 


482 


232324 


111980168 


21.9545 


7.8406 


537 


288369 154854153 


23.1733 


8.1281 


483 


233-89 


112678587 


21.9773 


7.8460 


538 


289444 155720872 


23.1948 


8.1332 


484 


234256 


113379904 


22.0000 


7.8514 


539 


290521 156590819 


23.2164 


8.1382 



92 



MATHEMATICAL TABLES. 



No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Root. 


Cube 
Root. 


540 


291600 


157464000 


23.2379 


8.1433 


595 


354025 


210644875 


24.3926 


8.4108 


541 


292681 


158340421 


23.2594 


8.1483 


596 


355216 


211708736 


24.4131 


8.4155 


542 


293764 


159220088 


23.2809 


8.1533 


597 


356409 


212776173 


24.4336 


8.4202 


543 


294849 


160103007 


23.3024 


8.1583 


598 


357604 


213847192 


24.4540 


8.4249 


544 


295936 


160989184 


23.3238 


8.1633 


599 


358801 


214921799 


24.4745 


8.4296 


545 


297025 


161878625 


23.3452 


8.1683 


6CO 


360000 


216000000 


24.4949 


8 4343 


546 


298116 


162771336 


23.3666 


8.1733 


601 


301201 


217081801 


24.5153 


8.4390 


547 


299209 


163667323 


23.3880 


8.1783 


602 


362404 


218167208 


24.5357 


8.4437 


548 


300304 


164566592 


23.4094 


8.1833 


603 


363609 


219256227 


24.5561 


8.4484 


549 


301401 


165469149 


23.4307 


8.1882 


604 


364816 


220348864 


24.5764 


8.4530 


550 


302500 


166375000 


23.4521 


8.1932 


605 


366025 


221445125 


24.5967 


8.4577 


551 


303601 


167284151 


23.4734 


8.1982 


606 


367236 


22254501 ( 


24.6171 


8.4623 


552 


304704 


168196H08 


23.4947 


8.2031 


607 


368449 


223648543 


24.6374 


8.4670 


553 


305809 


169112377 


23.5160 


8.2081 


608 


369664 


224755712 


24.6577 


8.4716 


554 


306916 


170031464 


23.5372 


8.2130 


609 


370881 


225866529 


24.6779 


8.4763 


555 


308025 


170953875 


23.5584 


8.2180 


610 


372100 


226981000 


24 6982 


8.4809 


556 


309136 


171879616 


23.5797 


8.2229 


611 


373321 


228099131 


24.7184 


8.4856 


557 


310249 


172808693 


23.6008 


8.2278 


612 


374544 


229220928 


24 7386 


8.4902 


558 


311364 


173741112 


.23.6220 


8.2327 


613 


375769 


230346397 


24.7588 


8.4948 


559 


312481 


174676879 


23.6432 


8.2377 


614 


376996 


231475514 


24.7790 


8.4994 


560 


313600 


175616000 


23.6643 


8.2426 


615 


378225 


232608375 


24.7992 


8.5040 


561 


314721 


176558481 


23.6854 


8.2475 


616 


379456 


233744896 


24.8193 


8.5086 


562 


315844 


177504328 


23.7065 


8 2524 


617 


380689 


234885113 


24.8395 


8.5132 


563 


316969 


178453547 


23.7276 


8.2573 


618 


381924 


236029032 


24.8596 


8.5178 


564 


318096 


179406144 


23.7487 


8.2621 


619 


383161 


237176659 


24.8797 


8.5224 


565 


319225 


180362125 


23.7697 


8.2670 


620 


384400 


238328000 


24.8998 


8.5270 


566 


320356 


181321496 


23.7908 


8.2719 


621 


385641 


239483061 


24.9199 


8.5316 


567 


321489 


182284263 


23.8118 


8.2768 


622 


3*6884 


240641848 


24 . 9399 


8.5362 


568 


322624 


183250432 


23.8326 


8.2816 


623 


388129 


241804367 


24.9600 


8.5408 


569 


323761 


184220009 


23.8537 


8.2865 


624 


389376 


242970624 


24.9800 


8.5453 


570 


324900 


185193000 


23.8747 


8.2913 


625 


390625 


244140625 


25.0000 


8.5499 


571 


326041 


186169411 


23.8956 


8.2962 


626 


391876 


245314376 


25.0200 


8.5544 


572 


327184 


187149248 


23.9165 


8.3010 


627 


393129 


246491883 


25.0400 


8.5590 


573 


328329 


188132517 


23.9374 


8.3059 


628 


394384 


247673152 


25.0599 


8.5635 


574 


329476 


189119224 


23.9583 


8.3107 


629 


395641 


248858189 


25.0799 


8.5681 


575 


330625 


190109375 


23.9792 


8.3155 


630 


396900 


250047000 


25.0998 


8.5726 


576 


331776 


191102976 


24.0000 


8.3203 


631 


398161 


251239591 


25.1197 


8.5772 


577 


332929 


19-2100033 


24.0208 


8.3251 


632 


399424 


252435968 


25.1396 


8.5817 


578 


334084 


193100552 


24.0416 


8.3300 


633 


400689 


253636137 


25.1595 


8.5862 


579 


335241 


194104539 


24.0624 


8.3348 


634 


401956 


254840104 


25.1794 


8.5907 


580 


336400 


195112000 


24.0832 


8.3396 


635 


403225 


256047875 


25.1992 


8.5952 


581 


337561 


196122941 


24.1039 


8.3443 


636 


404496 


257259456 


25.2190 


8 5997 


582 


338724 


197137368 


24.1247 


8.3491 


637 


405769 


258474853 


25.2389 


8.6043 


583 


339889 


198155287 


24.1454 


8.3539 


638 


407044 


259694072 


25.2587 


8.6088 


584 


341056 


199176704 


24.1661 


8.3587 


639 


408321 


260917119 


25.2784 


8.6132 


5S5 


342225 


200201625 


24.1868 


8.3634 


640 


409600 


262144000 


25.2982 


8.6177 


586 


343396 


201230056 


24.2074 


8.3682 


641 


410881 


263374721 


25.3180 


8.6222 


587 


344569 


202262003 


24.2281 


8.3730 


642 


412164 


264609288 


25.3377 


8.6267 


588 


345744 


203297472 


24.2487 


8.3777 


643 


413449 


265847707 


25.3574 


8.6312 


589 


346921 


204336469 


24.2693 


8.3825 


644 


414736 


267089984 


25.3772 


8.6357 


590 


348100 


205379000 


24.2899 


8.3872 


645 


416025 


268836125 


25.3969 


8.6401 


591 


349281 


206425071 


24 3105 


8.3919 


646 


417316 


269586136 


25.4165 


8 6446 


592 


350464 


207474688 


24.3311 


8.3967 


647 


418609 


270840023 


25.4362 


8.6490 


593 


351649 


208527857 


24.3516 


8.4014 


648 


419904 


272097792 


25.4558 


8.G535 


594 


352836 


209584584 


24.3721 


8.4061 


649 


421201 


273359449 


25.4755 


8.6579 



SQUARES, CUBES, SQUARE AND CUBE ROOTS. 93 



No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


650 


422500 


274625000 


25.4951 


8.6624 


705 


497025 


350402625 


26.5518 


8.9001 


651 


423801 


275894451 


25.5147 


8.66(58 


706 


498436 


351895816 


26.5707 


8.9043 


652 


425104 


277167808 


25.5343 


8.6713 


707 


499849 


353393243 


26.5895 


8.9085 


653 


426409 


278445077 


25.5539 


8.6757 


708 


501264 


354894912 


26.6083 


8.9127 


654 


427716 


279726264 


25.5734 


8.6801 


709 


502681 


356400829 


26.6271 


8.9169 


655 


429025 


281011375 


25.5930 


8.6845 


710 


504100 


357911000 


26.6458 


8.9211 


656 


430336 


282300416 


25.6125 


8.6890 


711 


505521 


359425431 


26.6646 


8.9253 


65? 


431649 


283593393 


25.6320 


8.6934 


712 


506944 


360944128 


J6.6833 


8.9295 


658 


432964 


284890312 


25.6515 


8.6978 


713 


508369 


362467097 


26.7021 


8.9337 


659 


434281 


286191179 


25.6710 


8.7022 


714 


509796 


363994344 


26.7208 


8.9378 


660 


435600 


287496000 


25.6905 


8.7066 


715 


511225 


365525875 


26.7395 


8.9420 


661 


436921 


288804781 


25.7099 


8.7110 


716 


512656 


367061696 


26.7582 


8.9462 


662 


438244 


290117528 


25 7294 


8.7154 


717 


514089 


368601813 


26.7769 


8.9503 


663 


439569 


291434247 


25.7488 


8.7198 


718 


515524 


370146232 


26.7955 


8.9545 


664 


440896 


292754944 


25.7682 


8.7241 


719 


516961 


371694959 


26.8142 


8.9587 


665 


442225 


294079625 


25.7876 


8.7285 


720 


518400 


373248000 


26.8328 


8.9628 


666 


443556 


295408296 


25.8070 


8.7329 


721 


519841 


374805361 


26.8514 


8.9670 


667 


444889 


296740963 


25.8263 


8.7373 


722 


521284 


376367048 


26.8701 


8.9711 


668 


446224 


298077632 


25.8457 


8.7416 


723 


522729 


377933067 


26.8887 


8.9752 


669 


447561 


299418309 


25.8650 


8.7460 


724 


524176 


379503424 


26.9072 


8.9794 


670 


448900 


300763000 


25.8844 


8.7503 


725 


525625 


381078125 


26.9258 


8.9835 


671 


450241 


302111711 


25.9037 


8.7547 


726 


527076 


382657176 


26.9444 


8.9876 


672 


451584 


303464448 


25.9230 


8.7590 


727 


528529 


384240583 


26.9629 


8.9918 


673 


452929 


304821217 


25.942". 


8.7G34 


728 


529984 


385828352 


26.9815 


8.9959 


674 


454276 


306182024 


25.9615 


8.7677 


729 


531441 


387420489 


27.0000 


9.0000 


675 


455625 


307546875 


25.9808 


8.7721 


730 


532900 


389017000 


27 0185 


9.0041 


676 


456976 


308915776 


26.0000 


8.7764 


731 


534361 


390617891 


27.0370 


9.0082 


677 


458329 


310288733 


26.0192 


8.7807 


732 


535824 


392223168 


27.0555 


9.0123 


678 


459684 


311665752 


26.0384 


8.7850 


733 


537289 


393832837 


27.0740 


9.0164 


679 


461041 


313046839 


26.0576 


8.7893 


734 


538756 


395446904 


27.0924 


9.0205 


680 


462400 


314432000 


26.0768 


8.7937 


735 


540225 


397065375 


27.1109 


9.0246 


681 


463761 


315821241 


26.0960 


8.7980 


736 


541696 


398688256 


27.1293 


9.0287 


682 


465124 


317214568 


26.1151 


8.8023 


737 


543169 


400315553 


27.1477 


9.0328 


683 


466489 


318611987 


26.1343 


8.8066 


738 


544644 


401947272 


27.1662 


9.0369 


684 


467856 


320013504 


26.1534 


8.8109 


739 


546121 


403583419 


27.1846 


9.0410 


685 


469225 


321419125 


26.1725 


8.8152 


740 


547600 


405224000 


27.2029 


9.0450 


686 


470596 


322828856 


26.1910 


8.8194 


741 


549801 


406869021 


27.2213 


9.0491 


687 


471969 


324242703 


26.2107 


8.8237 


742 


550564 


408518488 


27.2397 


9.0532 


688 


473344 


325660672 


26.2298 


8.8280 


743 


552049 


410172407 


27.2580 


9.0572 


689 


474721 


327082769 


26.2488 


8.8323 


744 


553536 


411830784 


27.2764 


9.0613 


690 


476100 


328509000 


26.2679 


8.8366 


745 


555025 


413493625 


27.2947 


9.0654 


691 


477481 


329939371 


26.2869 


8.8408 


746 


556516 


415160936 


27.3130 


9.0694 


692 


478864 


331373888 


26.3059 


8.8451 


747 


558009 


416832723 


27.3313 


9.0735 


693 


480249 


332812557 


26.3249 


8.8493 


748 


559504 


418508992 


27.3496 


9.0775 


694 


481636 


334255384 


26.3439 


8.8536 


749 


561001 


420189749 


27.3679 


9.0816 


695 


483025 


335702375 


26.3629 


8.8578 


750 


562500 


421875000 


27.3861 


9.0856 


696 


484416 


337153536 


26.3818 


8.8621 


751 


564001 


423564751 


27.4044 


9.0896 


697 


485809 


338608873 


26.4008 


8.8663 


752 


565504 


425259008 


27.4226 


9.0937 


698 


487204 


340068392 


26.4197 


8.8706 


753 


567009 


426957777 


27.4408 


9.0977 


699 


488601 


341532099 


26.4386 


8.8748 


754 


568516 


428661064 


27.4591 


9.1017 


700 


490000 


343000000 


26.4575 


8.8790 


755 


570025 


430368875 


27.4773 


9.1057 


701 


491401 


344472101 


26.4764 


8.8833 


756 


571536 


432081216 


27.4955 


9.1098 


702 


492804 


345948408 


26.4953 


8.8875 


757 


573049 


433798093 


27.5136 


9.1138 


703 


494209 


347428927 


26.5141 


8.8917 


758 


574564 


435519512 


27.5318 


9.1178 


704 


495616 


348913664 


26.5330 


8.8959 


759 


576081 


437245479 


27.55001 9.1&18 



MATHEMATICAL TABLES. 



No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


760 


577600 


438976000 


27.5681 


9.1258 


815 


664225 


541343375 


28.5482 


9.3408 


761 


579121 


440711081 


27.5862 


9.1298 


81 6 j 665856 


543338496 


28.5657 


9.3447 


762 


580644 


442450728 


27.6043 


9.1338 


817 667489 


545338513 


28.5832 


9.3485 


763 


582169 


444194947 


27.6225 


9.1378 


818 669124 


547343432 


28.6007 


9.3523 


764 


583696 


445943744 


27.6405 


9.1418 


819 670761 


549353259 


28.6182 


9.3561 


765 


585225 


447697125 


27.6586 


9.1458 


820 672400 


551368000 


28.6356 


9.3599 


766 


586756 


449455096 


27.6767 


9.1498 


821! 674041 


553387661 


28.6531 


9.3637 


767 


588289 


451217663 


27.6948 


9.1537 


822 


67'5684 


555412248 


28 . 6705 


9.3675 


768 


589824 


452984832 


27.7128 


9.1577 


823 


677329 


557441767 


28.6880 


9.3713 


769 


591361 


454756609 


27.7308 


9.1617 


824 


678976 


559476224 


28.7054 


9.3751 


770 


592900 


456533000 


27.7489 


9.1657 


825 


680625 


561515625 


28.7228 


9.3789 


771 


594441 


458314011 


27.7669 


9.1696 


826 


682276 


563559976 


28.7402 


9.3827 


772 


595984 


460099648 


27.7849 


9.1736 


827 


683929 


565609283 


28.7576 


9.3865 


773 


597529 


461889917 


27.8029 


9.1775 


828 


685584 


567663552 


28.7750 


9.3902 


774 


599076 


463684824 


27.8209 


9.1815 


829 


687241 


569722789 


28.7924 


9.3940 


775 


600625 


465484375 


27.8388 


9.1855 


830 


688900 


571787000 


28.8097 


9.3978 


776 


602176 


467288576 


27.8568 


9.1894 


831 


690561 


573856191 


28.8271 


9.4016 


777 


603729 


469097433 


27.8747 


9.1933 


832 


692224 


575930368 


28.8444 


9.4053 


778 
779 


605284 
606841 


470910952 
472729139 


27.8927 
27.9106 


9.1973 
9.2012 


883 
834 


693889 
695556 


578009537 
580093704 


28.8617 
28.8791 


9.4091 
9.4129 


780 


608400 


474552000 


27.9285 


9.2052 


835 


697'225 


582182875 


28.8964 


9.4166 


781 


609961 


476379541 


27.9464 9.2091 


836 


698896 


584277056 


28.9137 


9.4204 


782 


611524 


478211768 


27.9643 9.2130 


837 


700569 


586376253 


28.9310 


9.4241 


783 


613089 


480048687 


27.9821! 9.2170 


838 


702244 


588480472 


28.9482 


9.4279 


784 


614656 


481890304 


28.0000 


9.2209 


839 


703921 


590589719 


28.9655 


9.4316 


785 


616225 


483736625 


28.0179 


9.2248 


840 


705600 


592704000 


28.9828 


9.4354 


786 


617796 


485587656 


28.03571 9.2287 


841 


707281 


594823321 


29.0000 


9.4391 


787 


619369 


487443403 


28.0535 9.2326 


842 


708964 


596947688 


29.0172 


9.4429 


788 


620944 


489303872 


28.0713! 9.2365 


843 


710649 


599077107 


29.0345 


9.4466 


789 


622521 


491169069 


28.0891 9.2404 


844 


712336 


601211584 


29.0517 


9.4503 


790 


624100 


493039000 


28.1069 1 9.2443 


845 


714025 


603351125 


29.0689 


9.4541 


791 


625681 


494913671 


28.1247. 9.2482 


846 


715716 


605495736 


29.0861 


9.4578 


792 


627264 


496793088 


28.14251 9.2521 


847 


717409 


607645423 


29.1033 


9.4615 


793 


628849 


498677257 


28.1603 9.2560 


848 


719104 


609800192 


29.1204 


9.4652 


794 


630436 


500566184 


28.1780 


9.2599 


849 


720801 


611960049 


29.1376 


9.4690 


795 


632025 


502459875 


28.1957 


9.2638 


850 


722500 


614125000 


29.1548 


9.4727 


796 


633616 


504358336 


28.2135 


9.2677 


851 


724201 


616295051 


29.1719 


9.4764 


797 


635209 


50626157'3 


28.2312! 9.2716 


852 


725904 


618470208 


29.1890 


9.4801 


798 


636804 


508169592 


28.2489 


9.2754 


853 


727609 


620650477 


29.2062 


9.4838 


799 


638401 


510082399 


28.2666 


9.2793 


854 


729316 


622835864 


29.2233 


9.4875 


800 


640000 


512000000 


28.2843 


9.2832 


855 


731025 


625026375 


29.2404 


9.4912 


801 


641601 


513922401 


28.3019 


9.2870 


856 


732736 


627222016 


29.2575 


9.4949 


802 


643204 


515849608 


28.3196 


9.2909 


857 


734449 


629422793 


29.2746 


9.4986 


803 


644809 


517781627 


28.3373 


9.2948 


858 


736164 


631628712 


29.2916 


9.5023 


804 


646416 


519718464 


28.3549 


9.2986 


859 


737881 


633839779 


29.3087 


9.5060 


805 


648025 


521660125 


28.3725 


9.3025 


860 


739600 


636056000 


29.3258 


9.5097 


806 


649636 


523606616 


28.3901 


9.3063 


861 


741321 


C38277381 


29.3428 


9.5134 


807 


651249 


525557943 


28.4077 


9.3102 


862 


743044 


640503928 


29.3598 


9.5171 


808 


652864 


5275141 12 '28. 4253 


9.3140 


863 


744769 


642735647 


29.3769 


9.5207 


809 


654481 


529475129 


28.4429 


9.3179 


864 


746496 


644972544 


29.3939 


9.5244 


810 


656100 


531441000 


28.4605 


9.3217 


865 


748225 


647214625 


29.4109 


9.5281 


811 


657721 


53341173128.4781 9.3255 


866 


749956 


649461896 


29.4279 


9.5317 


812 


659344 


53538732828.4956 9.3294 


867 


751689 


651714363 


29.4449 


9.5354 


813 


660969 


53736779728.5132 9.3332 


868 


753424 


653972032 


29.4618 


9.5391 


814 


662596 


53935314428.5307 9.3370 


869 


755161 


656234909 


29.4788 


9.5427 



SQUARES, CUBES, SQUARE AND CUBE ROOTS. 95 



No. 


Square. 


Cube. 


Sq. 
Root, 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


870 


756900. 


658503000 


29.4958 


9.5464 


925 


855625 


791453125 


30.4138 


9.7435 


871 


758641 


060776311 


29.5127 


9.5501 


926 


857476 


794022776,30.4302 


9.7470 


872 


760384 


663054848 


29.5296 


9.5537 


927 


859329 


796597983 '30. 4467 


9.7505 


873 


762129 


665338617 


29.5466 


9.5574 


928 


861184 


799178752 30.4631 


9 7540 


874 


763876 


667627624 


29.5635 


9.5610 


929 


863041 


801765089 


30.4795 


9.7575 


875 


765625 


669921875 


29.5804 


9.5647 


930 


864900 


804357000 


30.4959 


9.7610 


876 


767376 


672221376 


29.5973 


9.5683 


931 


866761 


806954491 


30.5123 


9.7645 


877 


769129 


674526133 


29.6142 


9.5719 


932 


868624 


809557568 


30.5287 


9.7680 


878 


770884 


676836152 


29.6311 


9.5756 


933 


870489 


812166237 


30.5450 


9.7715 


879 


772641 


679151439 


29.6479 


9.5792 


934 


872356 


814780504 


30.5614 


9.7750 


880 


774400 


681472000 


29.6648 


9.5828 


935 


874225 


817400375 


30.5778 


9.7785 


881 


776161 


683797841 


29.6816 


9.5865 


936 


876096 


820025856 


30.5941 


9.7819 


882 


777924 


686128968 


29.6985 


9.5901 


937 


877969 


822656953 


30.6105 


9.7854 


883 


779689 


688465387 


29.7153 


9.5937 


938 


879844 


825293672 


30.6268 


9.7880 


884 


781456 


690807104 


29.7321 


9.5973 


939 


881721 


827936019 


30.6431 


9.7924 


885 


783225 


693154125 


29.7489 


9.6010 


940 


883600 


830584000 


30.6594 


9.7959 


886 


784996 


695506456 


29.7658 


9.6046 


941 885481 


833237621 


30.6757 


9.799S 


887 


786769 


697864103129.7825 


9.6082 


942 887364 


835896888 


30.6920 


9.8028 


888 


788544 


70022707229.7993 


9.6118 


943 889249 


838561807 


30.7083 


9.8063 


889 


790321 


702595369 


29.8161 


9.6154 


944 


891136 


841232384 


30.7246 


9.8097 


890 


792100 


704969000 


29.8329 


9.6190 


945 


893025 


843908625 


30.7409 


9.8132 


891 


793881 


707347971 


29.8496 


9.6226 


946i 894916 


846590536 


30.7571 


9.8167 


892 


795664 


709732288 


29.8664 


9.6262 


947 1 896809 


849278123 


30.7734 


9.8201 


893 


797449 


712121957 


29.8831 


9.6298 


948 898704 


851971392 


30.7896 


9.8236 


894 


799236 


714516984 


29.8998 


9.6334 


949, 900601 


854670349 


30.8058 


9.8270 


895 


801025 


716917375 


29.9166 


9.6370 


950 902500 


857375000 


30.8221 


9.8305 


896 802816 


719323136 


29.9333 


9.6406 


951 904401 


860085351 


30.8383 


9.8339 


897 


804609 


721734273 


29.9500 


9.6442 


952 906304 


862801408 


30.8545 


9.8374 


898 


806404 


724150792 


29.9666 


9.6477 


953 908209 


865523177 


30.8707 


9.8408 


899 


808201 


726572699 


29.9833 


9 6513 


954 910116 


868250664 


30.8869 


9.8443 


900 


810000 


729000000 


30 0000 


9.6549 


955' 912025 


870983875 


30.9031 


9.8477 


901 


811801 


731432701 30.0167 


9.6585 


956 913936 


873722816 


30.9192 


9.8511 


902 I 813604 


733870808 


30.0333 


9.6620 


957 i 915849 


876467493 


30.9354 


9.8546 


9031 815409 


736314327 


30.0500 


9.6656 


958 


917764 


879217912 


30.9516 


9.8580 


904 


817216 


738763264 


30.0666 


9.6692 


959 


919681 


881974079 


30.9677 


9.8614 


905 


819025 


741217625 


30.0832 


9.6727 


960 


921600 


884736000 


30.9839 


9.8648 


906 


820836 


743677416 


30.0998 


9.6763 


961 


923521 


887503681 


31.0000 


9.8683 


907 


822649 


746142643 


30.1164 


9.6799 


962 


925444 


890277128 


31.0161 


9.8717 


908 


824464 


748613312 


30.1330 


9.6834 


963 


927369 


893056347 


31.0322 


9.8751 


909 


826281 


751089429 


30.1496 


9.6870 


964 


929296 


895841344 


31.0483 


9.8785 


910 


828100 


753571000 


30.1662 


9.6905 


965 


931225 


898632125 


31.0644 


9.8819 


911 


829921 


756058031 


30.1828 


9.6941 


966 


933156 


901428696 


31.0805 


9.8854 


912 


831744 


758550528 


30.1993 


9.6976 


967 


935089 


904231063 


31.0966 


9.8888 


913 


833569 


761048497 


30.2159 


9.7012 


968 


937024 


907039232 


31.1127 


9.8922 


914 


835396 


763551944 


30.2324 


9.7047 


969 


938961 


909853209 


31.1288 


9.8956 


915 


837225 


766060875 


30.2490 


9.7082 


970 


940900 


912673000 


31.1448 


9.8990 


916 


839056 


768575296 


30.2655 


9.7118 


971 


942841 


915498611 


31.1609 


9.9024 


917 


840889 


771095213 


30.2820 


9.7153 


972 


944784 


918330048 


31.1769 


9.9058 


918 


842724 


773620632 


30.2985 


9.7188 


973 


946729 


921167317 


31.1929 


9.9092 


919 


844561 


776151559 


30.3150 


9.7224 


974 


948676 


924010424 


31.2090 


9.9126 


920 


846400 


778688000 


30.3315 


9.7259 


975 


950625 


926859375 


31.2250 


9.9160 


921 


848241 


781229961 


30.3480 


9.7294 


976 


952576 


929714176 


31.2410 


9.9194 


922 


850084 


783777448 


30.3645 


9.7329 


977 


954529 


932574833 


31.2570 


9 9227 


923 


851929 786330467 


30.3809 


9.7364 


978 


956484 


935441352 


31.2730 


9.9261 


924 


853776 ! 788889024 


30.3974 


9.7400 


979 


958441 


938313739 


31.2890 


9.9295 



96 



MATHEMATICAL TABLES. 



No. 

980 

981 
982 
983 
984 


Square. 


Cube. 


Sq. 
Root. 


Cube. 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cub6 
Root, 


960400 
962361 
964324 
966289 
968256 


941192000 
944076141 
946966168 
949862087 
952763904 


31.3050 
31.3209 
31.3369 
31.3528 

31.3688 


9.9329 
9.9363 
9.9396 
9.9430 
9.9464 


1035 
1036 
1037 
1038 
1039 


1071225 
1073296 
1075369 
1077444 
1079521 


1108717875 
1111934656 
1115157653 
1118386872 
1121622319 


32.1714 
32.1870 
32.2025 
32.2180 
32.2335 


10.1153 
10.1186 
10.1218 
10.1251 
10.1283 


985 
986 
987 
988 
989 


970225 
972196 
974169 
976144 
978121 


955671625 
958585256 
961504803 
964430272 
967361669 


31.3847 
31.4006 
31.4166 
31.4325 
31.4484 


9.9497 
9.9531 
9.9565 
9.9598 
9.9632 


1040 
1041 
1042 
1043 
1044 


1081600 
1083681 
1085764 
1087849 
1089936 


1124864000 
1128111921 
1131366088 
1134626507 
1137893184 


32.2490 
32.2645 
32.2800 
32.2955 
32.3110 


10.1316 
10.1348 
10.1381 
10.1413 
10.1446 


990 
991 
992 
993 
994 


980100 
982081 
984064 
986049 
988036 


970299000 
973242271 
976191488 
979146657 
982107784 


31.4643 
31.4802 
31.4960 
31.5119 

31.5278 


9.9666 
9.9699 
9.9733 
9.9766 
9.9800 


1045 
1046 
1047 
1048 
1049 


1092025 
1094116 
1096209 
1098304 
1100401 


1141166125 
1144445336 
1147730823 
1151022592 
1154320649 


32.3265 
32.3419 
32.3574 
32.3728 
32.3883 


10.1478 
10.1510 
10.1543 
10.1575 
10.1607 


995 
996 
997 
998 
999 


990025 
992016 
994009 
996004 
998001 


985074875 
988047936 
991026973 
994011992 
997002999 


31.5436 
31.5595 
31.5753 
31.5911 
31.6070 


9.9833 
9.9866 
9.9900 
9 9933 
9.9967 


1050 
1051 
1052 
1053 
1054 


1102500 
1104601 
1106704 
1108809 
1110916 


1157625000 
1160935651 
1164252608 
1167575877 
1170905464 


32.4037 
32.4191 
32.4345 
32.4500 
32.4654 


10.1640 
10.1672 
10.1704 
10.1736 
10.1769. 


1000 
3001 
1002 
1003 
1004 


1000000 
1002001 
1004004 
1006009 
1008016 


1000000000 
1003003001 
1006012008 
1009027027 
1012048064 


31.6228 
31.6386 
31.6544 
31.6702 
31.6860 


10.0000 
10.0033 
10.0067 
10.0100 
10.0133 


1055 
1056 
1057 
1058 
1059 


1113025 
1115136 
1117249 
1119364 
1121481 


1174241375 
1177583616 
1180932193 
1184287112 
1187648379 


32.4808 
32.4962 
32.5115 
32.5269 
32.5423 


10.1801 
10.1833 
10.1865 
10.1897 
10.1929 


1005 
1006 
1007 
1008 
1009 


1010025 
1012036 
1014049 
1016064 
1018081 


1015075125 
1018108216 
1021147343 
1024192512 
1027243729 


31.7017 
31.7175 
31.7333 
31.7490 
31.7648 


10.0166 
10.0200 
10.0233 
10.0266 
10.0299 


1060 
1061 
1062 
1063 
1064 


1123600 
1125721 
1127844 
1129969 
1132096 


1191016000 
1194389981 
1197770328 
1201157047 
1204550144 


32.5576 
32.5730 
32.5883 
32.6036 
32.6190 


10 1961 
10.1993 
10.2025 
10.2057 
10.2089 


1010 
1011 
1012 
1013 
1014 


1020100 
1022121 
1024144 

1026169 
1028196 


1030301000 
1033364331 
1036433728 
1039509197 
1042590744 


31.7805 
31.7962 
31.8119 
31.8277 
31.8434 


10.0332 
10.0365 
10.0398 
10.0431 
10.0465 


1065 
1066 
1067 

1068 
1069 


1134225 
1136356 
1138489 
1140624 
1142761 


1207949625 
1211355496 
1214767763 
1218186432 
1221611509 


32.6343 
32.6497 
32.6650 
32.6803 
32.6956 


10.2121 
10.2153 
10.2185 
10.2217 
10.2249 


1015 
1016 
1017 
1018 
1019 


1030225 
1032256 
1034289 
1036324 
1038361 


1045678375 
1048772096 
1051871913 
1054977832 
1058089859 


31.8591 
31.8748 
31.8904 
31.9061 
31.9218 


10.0498 
10.0531 
10.0563 
10.0596 
10.0629 


1070 
1071 
1072 
1073 
1074 


1144900 
1147041 
1149184 
1151329 
1153476 


1225043000 
1228480911 
1231925248 
1235376017 
1238833224 


32.7109 
32.7261 
32.7414 
32.7567 
32.7719 


10.2281 
10.2313 
10.2345 
10.2376 
10.2408 


1020 
1021 
1022 
1023 
1024 


1040400 
1042441 
1044484 
1046529 
1048576 


1061208000 
1064332261 
1067462648 
1070599167 
1073741824 


31.9374 
31.9531 
31.9687 
31.9844 
32.0000 


10.0662 
10.0695 
10.0728 
10.0761 
10.0794 


1075 
1076 
1077 
1078 
1079 


1155625 
1157776 
1159929 
1162084 
1164241 


1242296875 
1245766976 
1249243533 
1252726552 
1256216039 


32.7872 
32.8024 
32.8177 
32.8329 
32.8481 


10.2440 
10.2472 
10.2503 
10.2535 
10.2567 


1025 
1026 
1027 
1028 
1029 


1050625 
1052676 
1054729 
1056784 
1058841 


1076890625 
1080045576 
1083206683 
1086373952 
1089547389 


32.0156 
32.0312 
32.0468 
32.0624 
32.0780 


10.0826 
10.0859 
10.0892 
10.0925 
10.0957 


1080 
1081 
1082 
1083 
1084 


1166400 
1168561 
1170724 
1172889 
1176056 


1259712000 
1263214441 
1266723368 
1270238787 
1273760704 


32.8634 
32.8786 
32.8938 
32.9090 
32.9242 


10.2599 
10.2630 
10.2662 
10.2693 
10.2725 


1030 
1031 
1032 
1033 
1034 


1060900 
1062961 
1065024 
1067089 
1069156 


1092727000 
1095912791 
1099104768 
1102302937 
1105507304 


32.0936 
32.1092 
32.1248 
32.1403 
32.1559 


10.0990 
10.1023 

10.1055 
10.1088 
10.1121 


1085 
1086 
1087 
1088 
1089 


1177225 
1179396 
1181569 
1183744 
1185921 


1277289125 
1280824056 
1284365503 
1287913472 
1291467969 


32.9393 
32.9545 
32.9697 
32.9848 
33.0000 


10.2757 
10.2788 
10.2820 
10.2851 
10.2883 



SQUARES, CUBES, SQUARE AKD CUBE ROOTS. 9? 



No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


1090 


1188100 


1295029000 


33.0151 


10.2914 


1145 


1311025 


1501123625 


33.8378 


10.4617 


1091 


1190281 


1298596571 


33.0303 


10.2946 


1146 


1313316 


1505060136 


33.8526 


10.4647 


1092 


1192464 


1302170688 


33.0454 


10.2977 


1147 


1315609 


1509003523 


33.8674 


10.4678 


1093 


1194649 


1;05751357 


33.0606 


10.3009 


1148 


1317904 


1512953792 


33.8821 


10.4708 


1094 


1196836 


1309338584 


33.0757 


10.3040 


1149 


1320201 


1516910949 


33.8969 


10.4739 


1095 


1199025 


1312932375 


33.0908 


10.3071 


1150 


1322500 


1520875000 


33.9116 


10.4769 


1096 


1201216 


1316532736 


33.1059 


10.3103 


1151 


1324801 


1524845951 


33.9264 


10.4799 


1097 


1203409 


1320139673 


33.1210 


10.3134 


1152 


1327104 


1528823808 


33.9411 


10.4830 


1098 


1205604 


1323753192 


33.1361 


10.3165 


1153 


1329409 


1532808577 


33.9559 


10.4860 


1099 


1207801 


1327373299 


33.1512 


10.3197 


1154 


1331716 


1536800264 


33.9706 


10.4890 


1100 


1210000 


1331000000 


33.1662 


10.3228 


1155 


1334025 


1540798875 


33.9853 


10.4921 


1101 


1212201 


1334633301 


33.1813 


10.3259 


1156 


1336336 


1544804416 


34.0000 


10.4951 


1103 


1214404 


1338273208 


33.1964 


10.3290 


1157 


1338649 


1548816893 


34.0147 


10.4981 


1103 


1216609 


1341919727 


33.2114 


10.3322 


1158 


1340964 


1552836312 


34.0294 


10.5011 


1104 


1218816 


1345572864 


33.2264 


10.3353 


1159 


1343281 


1556862879 


34.0441 


10.5042 


1105 


1221025 


1349232625 


33.2415 


10.3384 


1160 


1345600 


1560896000 


34.0588 


10.5072 


1106 


1223236 


1352899016 


33.2566 


10.3415 


1161 


1347921 


1564936281 


34.0735 


10.5102 


1107 


1225449 


1356572043 


33.2716 


10.3447 


1162 


1350244 


1568983528 


34.0881 


10.5132 


1108 


1227G64 


1360251712 


33.2866 


10.3478 


1163 


1352569 


1573037747 


34.1028 


10.5162 


1109 


1229881 


1363938029 


33.3017 


10.3509 


1164 


1354896 


1577098944 


34.1174 


10.5192 






















1110 


1232100 


136?631000'33.3167 


10.3540 


1165 


1357225 


1581167125 


34.1321 


10 5223 


1111 


1234321 


1371330631 33.3317 


10.3571 


1166 


1359556 


1585242296 


34.1467 


10.5253 


1112 


1236544 


137503692833.3467 


10.3602 


1167 


1361889 


1589324463 


34.1614 


10.5283 


1113 


1238769 


137874989733.3617 


10.3633 


1168 


1364224 


1593413632 


34.1760 


10.5313 


111* 


1240996 


1382469544 


33.3766 


10.3664 


1169 


1366561 


1597509809 


34.1906 


10.5343 


1115 


1243225 


1386195875 


33.3916 


10 3695 


1170 


1368900 


1601613000 


34.2053 


10.5373 


1116 


1245456 


138992889633.4066 


10.3726 


1171 


1371241 


1605723211 


34.2199 


10.5403 


1117 


1247689 


139366861338.4215 


10.3757 


1172 


1373584 


1609840448 


34.2345 


10.5433 


1118 


1249924 


139741503233.4365 


10.3788 


1173 


1375929 


1613964717 


34.2491 


10.5463 


1119 


1252161 


1401 168159|33. 4515 


10.3819 


1174 


1378276 


1618096024 


34.2637 


10.5493 


1120 


1254400 


140492800033.4664 


10.3850 


1175 


1380625 


1622234375 


34.2783 


10.5523 


11<J1 


1256641 


1408694561 33.4813 


10.3881 


1176 


1382976 


1626379776 


34.2929 


10.5553 


1122 


1258884 


141246784833.4963 


10.3912 


1177 


1385329 


1630532233 


34.3074 


10.5583 


1123 


1261129 


141624786733.5112 


10.3943 


1178 


1387684 


1634691752 


34.3220 


10.5612 


1124 


1263376 


142003462433.5261 


10.3973 


1179 


1390041 


1638858339 


34.3366 


10.5642 


1125 


1265625 


1423828125 33 5410 


10.4004 


1180 


1392400 


1643032000 


34.3511 


10.5672 


1126 


1267876 


142762837633.5559 


10.4035 


1181 


1394761 


1647212741 


34.3657 


10.5702 


1127 


1270129 


143143538333.5708 


10.4066 


1182 


1397124 


1651400568 


34.3802 


10.5732 


1128 


1272384 


143524915233.5857 


10.4097 


1183 


1399489 


1655595487 


34.3948 


10 5762 


1129 


1274641 


1439069689 33.6006 


10.4127 


1184 


1401856 


1659797504 


34.4093 10.5791 






| 














1130 


1276900 


144289700033.6155 


10.4158 


1185 


1404225 


1664006625 


34.4238 


10.5821 


1131 


1279161 


1446731091 


33.6303 


10.4189 


1186 


1406596 


1668222856 


34.4384 


10.5851 


1132 


1281424 


1450571968 


33.6452 


10.4219 


1187 


1408969 


1672446203 


34.4529 


10.5881 


1133 


1283689 


1454419637 


33.6601 


10.4250 


1188 


1411344 


1676676672 


34.4674 


10.5910 


1134 


1285956 


1458274104 


33.0749 


10.4281 


1189 


1413721 


1680914269 


34.4819 


10.5940 


1135 


1288225 


1462135375 


33.6898 


10.4311 


1190 


1416100 


1685159000 


34.4964 


10.5970 


1136 


1290496 


1466003456 


33.7046 


10.4342 


1191 


1418481 


1689410871 


34.5109 


10.6000 


1137 


1292769 


1469S78353 


33.7174 


10.4373 


1192 


1420864 


1693669888 


34.5254 


10.6029 


1138 


1295044 


1473760072 


33.7342 


10.4404 


1193 


1423249 


1697936057 


34.5398 


10.6059 


1139 


1297321 


1477648619 


33.7491 


10.4434 


1194 


1425636 


1702209384 


34.5543 


10.6088 


1140 


1299600 


1481544000 


33.7639 


10.4464 


1195 


1428025 


1706489875 


34.5688 


10.6118 


1141 


1301881 


1485446221 


33.7787 


10.4495 


1196 


1430416 


1710777536 


34.5832 


10.6148 


1142 


1304164 


1489355288 


33.7935 


10.4525 


1197 


1432809 


1715072373 


34.5977 


10 6177 


1143 


1306449 


1493271207 


33.8083 


10.4556 


1198 


1435204 


1719374392 


34.6121 


10.6207 


1144 


1308736 


1497198984 


33.8231 


10.4586 


1199 


1437601 


1723683599 


34.6266 


10.6230 



98 



MATHEMATICAL TABLES. 



No. 

1200 
1201 
1202 
1203 
1304 


Square. 


Cube. 


Sq. 
Boot, 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


1440000 
1442401 
1444804 
1447209 
1449616 


172800000034.6410 
1732323601 34.6554 
173665440834.6699 
174099242734.6843 
174533766434.6987 


10.6266 
10.6295 
10.6325 
10.6354 
10.6384 


1255 
1256 
1257 
1258 
1259 


1575025 
1577536 
1580049 
1582564 
1585081 


1976656375 35.4260 
1981385216 35.4401 
1986121593 35.4542 
199086551235.4683 
199561697935.4824 


10.7865 
10.7894 
10.7922 
10.7951 
10.7980 


1205 
1206 
1207 
1208 
1209 


1452025 
1454436 
1456849 
1459264 
1461681 


1749690125 
1754049816 
1758416743 
1762790912 
1767172329 


34.7131 
34.7275 
34.7419 
34.7563 
34.7707 


10.6413 
10.6443 
10.6472 
10.6501 
10.6530 


1260 
1261 
1262 
1263 
1264 


1587600 
il 590 121 
1592644 
1595169 
1597696 


2000376000 35.4965 
2005142581 35.5106 
200991672835.5246 
201469844735.5387 
2019487744,35.5528 


10.8008 
10.8037 
10.8065 
10.8094 
10.8122 


1210 
1211 
1212 
1213 
1214 


1464100 
1466521 
1468944 
1471369 
1473796 


1771561000 
1775956931 
1780360128 
1784770597 
1789188344 


34.7851 
34.7994 
34.8138 
34.8281 
34.8425 


10.6560 
10.8590 
10.6619 
10.6648 
10.6678 


1265 
1266 
1267 
1268 
1269 


1600225 
1602756 
1605289 
1607824 
1610361 


2024284625 
2029089096 
2033901163 
2038720832 
2043548109 


35.5668 
35.5809 
35.5949 
35.6090 
35.6230 


10.8151 
10.8179 
10 8208 
10.8236 
10.8265 


1215 
1216 

1217 
1218 
1219 


1476225 
1478656 
1481089 
1483524 
1485961 


1793613375 
1798045696 
1802485313 
1806932232 
1811386459 


34.8569 
34.8712 
34.8855 
34.8999 
34.9142 


10.6707 
10.6736 
10.6765 
10.6795 
10.6824 


1270 
1271 
1272 
1273 
1274 


1612900 
1615441 
1617984 
1620529 
1623076 


2048383000 
2053225511 
2058075648 
2062933417 
2067798824 


35.6371 
35.6511 
35.6651 
35.6791 
35.6931 


10.8293 
10.8322 
10.8350 
10 8378 
10.8407 


1220 
1221 
1222 
1223 
1224 


1488400 
1490841 
1493284 
1495729 
1498176 


1815848000 
1820316861 
1824793048 
1829276567 
1833767424 


34.9285 
34.9428 
34.9571 
34.9714 
34.9857 


10.6853 
10.6882 
10.6911 
10.6940 
10.6970 


1275 
1276 
1277 
1278 
1279 


1625625 
1628176 
1630729 
1633284 
1635841 


2072671875 
2077552576 
2082440933 
2087336952 
2092240639 


35.7071 
35.7211 
35.7351 
35.7491 
35.7631 


10.8435 
10.8463 
10.8492 
10.8520 
10.8548 


1225 

1226 
12:27 
1228 
1229 


1500625 
1503076 
15055-*) 
1507984 
1510441 


1838265625 
1842771176 

1847284083 
1851804352 
1856331989 


35.0000 
35.0143 
35.0286 
35.0428 
35.0571 


10.6999 
10.7028 
10.7057 
10.7086 
10.7115 


1280 
1281 
1282 
1283 
1284 


1638400 
1640961 
1643524 
1646089 
1648656 


2097152000 
2102071041 
2106997768 
2111932187 
2116874304 


35.7771 
35.7911 
35.8050 
35.8190 
35.8329 


10.8577 
10.8605 
10.8633 
10.8661 
10.8690 


1230 
1231 
1232 
1233 
1234 


1512900 
1515361 
1517824 
1520289 
1522756 


1860867000 
1865409391 
1869959168 
1874516337 
1879080904 


35.0714 
35.0856 
35.0999 
35.1141 
35.1283 


10.7144 
10.7173 
10.7202 
10.7231 
10.7260 


1285 
1286 

1287 
1288 
1289 


1651225 
1653796 
1656369 
1658944 
1661521 


2121824125 
2126781656 
2131746903 
2136719872 
2141700569 


35.8469 
35.8608 
35.8748 
35.8887 
35.9026 


10.8718 
10.8746 
10.8774 

10.8802 
10.8831 


1235 
1236 
1237 
1238 
1239 


1525225 
1527696 
1530169 
1532644 
1535121 


1883652875 
1888232256 
1892819053 
1897413272 
1902014919 


35.1426 
35.1568 
35.1710 
35.1852 
35.1994 


10.7289 
10.7318 
10.7347 
10.7376 
10.7405 


1290 
1291 
1292 
1293 
1294 


1664100 
1666681 
1669264 
1671849 
1674436 


2146689000 
2151685171 
2156689088 
2161700757 
2166720184 


35.9166 
35.9305 
35.9444 
35.9583 
35.9722 


10.8859 
10.8887 
10.8915 
10.8948 
10.8971 


1240 
1241 
1242 
1243 
1244 


1537600 
1540081 
1542564 
1545049 
1547536 


1906624000 
1911240521 
1915864488 
1920495907 
1925134784 


35.2136 
35.2278 
35.2420 
35.2562 
35.2704 


10.7434 
10.7463 
10.7491 
10.7520 
10.7549 


1295 
1296 
1297 
1298 
1299 


1677025 
1679616 
1682209 
1684804 
1687401 


2171747375 
2176782336 
2181825073 
2186875592 
2191933899 


35.9861 
36.0000 
36.0139 
36.0278 
36.0416 


10.8999 
10.9027 
10.9055 
10.9083 
10.9111 


1245 
1246 
1247 
1248 
1249 


1550025 
1552516 
1555009 
1557504 
1560001 


1929781125 
1934434936 
1939096223 
1943764992 
19^8441249 


35.2846 
35.2987 
35.3129 
35.3270 
35.3412 


10.7578 
10.7607 
10.7635 
10.7664 
10.7693 


1300 
1301 
1302 
1303 
1304 


1690000 
1692601 
1695204 
1697809 
1700416 


2197000000 
2202073901 
2207155608 
2212245127 
2217342464 


36.0555 
36.0694 
36.0832 
36.0971 
36.1109 


10.9139 
10.9167 
10.9195 
10.9223 
10.9251 


1250 
1261 
1252 
1253 
1254 


1562500 
1565001 
1567504 
1570009 
1572516 


19531 25000 
1957816251 
1962515008 
1967221277 
1971935064 


35.3553 
35.3695 
35.3836 
35.3977 

35.4119 


10.7722 
10.7750 
10.7779 
10 7808 
10.7837 


1305 
1306 
1307 
1308 
1309 


1703025 
1705636 
1708249 
17108G4J 
1713481! 


2222447625 
2227560616 
2232681443 
2237810112 
2242946629 


36.1248 
36.1386 
36.1525 
36.1663 
36.1801 


10.9279 
10.9307 
10.9335 
10.9363 
10.939 



SQUARES, CUBES, SQUARE AKD CUBE ROOTS, 



1)9 



STo. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


310 
311 
312 
313 
314 


17161002248091000 
1718721 2253243231 
1721344 2258403328 
1723969 2263571297 
1726596 2268747144 


36.1939 
36. 2077 
36.2215 
36.2353 
36.2491 


10.9418 
10.9446 
10.9474 
10.9502 
10.9530 


1365 
1366 
1367 
1368 
1369 


1863225 
1865956 
1868689 
1871424 
1874161 


2543302125 
2548895896 
2554497863 
2560108032 
2565726409 


36.9459 
36.9594 
36.9730 
36.9865 
37.0000 


11.0929 
11.0956 
11.0983 
11.1010 
11.1037 


315 

31(5 
317 

318 
319 


1729225 2273930875 
1731856 2279122496 
17344892284322013 
1737124 2289529432 
1739761 2294744759 


36.2629 
36.2767 
36.2905 
36.3043 
36.3180 


10.9557 
10.9585 
10.9613 
10.9640 
10.9668 


1370 
1371 
1372 
1373 
1374 


1876900 
1879641 

1882384 
1885129 
1887876 


2571353000 
2576987811 
2582630848 
2588282117 
2593941624 


37.0135 
37.0270 
37.0405 
37.0540 
37.0675 


11.1064 
11.1091 
11.1118 
11.1145 
11.1172 


3-20 
321 
322 
823 

324 


1742400 
1745041 
1747684 
1750329 
1752976 


2299968000 
2305199161 
2310438248 
2315685267 
2320940224 


36.3318 
36.3456 
36.3593 
36.3731 
36.3868 


10.9696 
10.9724 
10.9752 
10.9779 
10.9807 


1375 
1376 
1377 
1378 
1379 


1890625 
1893376 
1896129 
1898884 
1901641 


2599609375 
2605285376 
2610969633 
2616662152 
2622362939 


37.0810 
37 0945 
37.1080 
37.1214 
37.1349 


11.1199 
11.1226 
11.1253 
11.1280 
11.1307 


325 
326 
38? 

32S 
329 


1755625 
1758276 
1760929 
1763584 
1766241 


2326203125 
2331473976 
2336752783 
2342039552 
2347334289 


36.4005 
36.4143 
36.4280 
36.4417 
36.4555 


10.9834 
10.9862 
10.9890 
10.9917 
10.9945 


1380 
1381 
1382 
1383 
1384 


1904400 
1907161 
1909924 
1912689 
1915456 


2628072000 
2633789341 
2639514968 

2645248887 
2650991104 


37.1484 
37.1618 
37.1753 
37.1887 
37.2021 


11.1334 
1 1 . 1361 
11.1387 
11.1414 
11.1441 


330 
331 

332 
333 
33 1 


1768900 
1771561 
1774224 

1776889 
1779556 


2352637000 
2357947691 
2363266368 
2368593037 
2373927704 


36.4692 
36.4829 
36.4966 
36.5103 
36.5240 


10.9972 
11.0000 
11.0028 
11.0055 
11.0083 


1385 
1386 

1387 
13S8 
1389 


1918225 
1920996 
1923769 
1926544 
1929321 


2656741625 
2662500456 
2668267603 
2674043072 
2679826869 


37.2156 
37.2290 
37.2424 
37.2559 
37.2693 


11.1468 
11.1495 
11.1522 
11.1548 
11.1575 


335 
836 
337 
388 

339 


1782225 
1784896 
1787569 
1790244 
1792921 


2379270375 
2384621056 
2389979753 
231)5346472 
2400721219 


36.5377 
36.5513 
36.5650 
36.5787 
36.5923 


11.0110 
11.0138 
11.0165 
11 0193 
11.0220 


1390 
1391 
1392 
1393 
1394 


1932100 
1934881 
1937664 
1940449 
1943236 


2685619000 
2691419471 
2697228288 
2703045457 
2708870984 


37.2827 
37.2961 
37.3095 
37.3229 
37.3363 


11.1602 
11.1629 
11.1655 
11.1682 
11.1709 


340 
341 
312 
343 
344 


1795600 

1798281 
1800964 
1803649 
1806336 


2406104000 
2411494821 
2416893688 
2422300607 

2427715584 


36.6060 
36.6197 
36.6333 
36.6469 
36.6606 


11.0247 
11.0275 
11.0302 
11.0330 
11.0357 


1395 
1396 
1397 
1398 
1399 


1946025 

1948816 
1951609 
1954404 
1957201 


2714704875 
2720547136 
2726397773 
2732256792 
2738124199 


37.3497 
37.3631 
37.3765 
37.3898 
37.4032 


11.1736 
11.1762 
11.1789 
11.1816 
11.1842 


345 
340 
347 
34H 
349 


1809025 
1811716 
1814409 
1817104 
1819801 


2433138625 
2438569736 
2444008923 
2449456192 
2454911549 


36.6742 

36.6879 
36.7015 
36.7151 
36.7287 


11.0384 
11.0412 
11.0439 
11.0466 
11.0494 


1400 
1401 
1402 
1403 
1404 


1960000 
1962801 
1965604 
1968409 
1971216 


2744000000 
2749884201 
2755776808 
2761677827 
2767587264 


37.4166 
37.4299 
37.4433 
37.4566 
37.4700 


11.1869 
11.1896 
11.192? 
11.1949 
11.1975 


350 

351 
352 
353 
354 


1822500 
1825201 
1827904 
1830609 
1833316 


2460375000 
2465846551 
2471326208 
2476813977 
2482309864 


36.7423 
36.7560 
36.7696 
36.7831 
36.7967 


11.0521 
11.0548 
11.0575 
11.0603 
11.0630 


1405 
1406 
1407 
1408 
1409 


1974025 
1976836 
1979649 
1982464 
1985281 


2773505125 
2779431416 
2785366143 
2791309312 
2797260929 


37.4833 
37.4967 
37.5100 
37.5233 
37.5366 


11.2002 
11.2028 
11.2055 
11.2082 
11.2108 


355 
356 
357 

35S 
359 


1836025 
1838736 
1841449 
1844164 
1846881 


2487813875 
2493326016 
2498846293 
2504374712 
2509911279 


36.8103 
36.8239 
36.8375 
36.8511 
36.8646 


11.0657 
11.0684 
11.0712 
11.0739 
11.0766 


1410 
1411 
1412 
1413 
1414 


1988100 
1990921 
1993744 
1996569 
1999396 


2803221000 
2809189531 
2815166528 
2821151997 
2827145944 


37.5500 
37.5633 
37.5766 
87.5899 
37.6032 


11.2135 
11.2161 
11.2188 
11.2214 
11.2240 


360 
361 
362 
:!t)3 
364 


1849600 
1852321 
1855044 
1857769 
1860496 


2515456000 

2521008881 
2526569928 
2532139147 
2537716544 


36.8782 
36.8917 
36.9053 
36.9188 
36.9324 


11.0793 
11.0820 
11.0847 
11.0875 
11.0902 


1415 
1416 
1417 
1418 
1419 


2002225 
2005056 
2007889 
2010724 
2013561 


2833148375 
2839159296 
2845178713 
2851206632 
2857243059 


37.6165 
37.6298 
37.6431 
37.6563 
37.6696 


11.2267 
11 2293 
11.2320 
11.2346 
11.2373 



100 



MATHEMATICAL TABLES. 



No. 

1420 
1421 
1422 
1423 
1424 


Square, 


Cube. 


Sq. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


2016400 
2019241 
2022084 
2024929 

2027776 


286328800037.6829 
2869341461 37.6962 
287540344837.7094 
288147396737.7227 
2887553024 37.7359 


11.2399 
11.2425 
11.2452 
11.2478 
11.2505 


1475 
1476 
1477 
1478 
1479 


2175625 
2178576 
2181529 
2184484 
2187441 


3209046875 
3215578176 
3222118333 
3228667352 
3235225239 


38.4057 
38.4187 
38.4318 
38.4448 
38.4578 


11.3832 
11.3858 
11.3883 
11.3909 
11.3935 


1425 
1426 
1427 

1428 
1429 


2030625 
2033476 
2036329 
2039184 
2042041 


289364062537.7492 
289973677637.7624 
290584148337.7757 
291195475237.7889 
291807658937.8021 


11.2531 
11.2557 
11.2583 
11.2610 
11.2636 


1480 
1481 
1482 
1483 
1484 


2190400 
2193361 
2196324 
2199289 
2202256 


3241792000 
3248367641 
3254952168 
3261545587 
3268147904 


38.4708 
38.4838 
38.4968 
38.5097 
38.5227 


11.3960 
11.3986 
11.4012 
11.4037 
11.4063 


1430 
1431 
1432 
1433 
1434 


2044900 
2047761 
2050624 
2053489 
2056356 


292420700037.8153 
2930345991 37.8286 
2936493568 37.8418 
294264973737.8550 
2948814504 ' 37. 8682 


11.2662 
11.2689 
11.2715 
11.2741 

11.2767 


1485 
1.486 

1487 
1488 
1489 


2205225 
2208196 
2211169 
2214144 
2217121 


3274759125 
3281379256 
3288008303 
3294646272 
3301293169 


38.5357 

38.5487 
38.5616 
38.5746 
38.5876 


11.4089 
11.4114 
11.4140 
11.4165 
11.4191 


1435 
1436 
1437 
1438 
1439 


2059225 
2062096 
2064969 
2067844 
2070721 


295498787537.8814 
2961169856 37.8946 
296736045337.9078 
297355967237.9210 
2979767519 37.9342 


11.2793 
11.2820 
11.2846 
11.2872 
11.2898 


1490 
1491 
1492 
1493 
1494 


2220100 
2223081 
22.20064 
2229049 
2232036 


3307949000 
3314613771 
3321287488 
3327970157 
3334661784 


38.6005 
38.6135 
38.6264 
38.6394 
38.6523 


11.4216 
11.4242 
11.4268 
11.4203 
11.4319 


1440 
1441 
1442 
1443 
1444 


2073600 
2076481 
2079364 
2082249 
2085136 


298598400037.9473 
299220912137.9605 
299844288837.9737 
300468530737.9868 
301093638438.0000 


11.2924 
11.2950 
11.2977 
11.3003 
11.3029 


1495 
1496 
1497 
1498 
1499 


2235025 
2238016 
2241009 
2244004 
2247001 


3341362375 
3348071936 
3354790473 
3361517992 
3368254499 


38.6652 
38.6782 
38.6911 
38.7040 
38.7169 


11.4344 
11.4370 
1 1 . 4395 
11.4421 
11.4446 


1445 
1446 
1447 
1448 
1449 


2088025 
2090916 
2093809 
2096704 
2099601 


3017196125 
3023464536 
3029741623 
3036027392 
3042321849 


38.0132 
38.0263 
38.0395 
38.0526 
38.0657 


11.3055 
11.3081 
11.3107 
11.3133 
11.3159 


1500 2250000 
1501 2253001 
1502 ; 2256004 
1503 2259009 
1504 2262016 


3375000000 
3381754501 
3388518008 
3395290527 
3402072064 


38.7298 
38.7427 
38.7556 
38.7685 
38.7814 


11.4471 
11.4497 
11.4522 
11.4548 
11.4573 


1450 
1451 
1452 
1453 
1454 


2102500 
2105401 
2108304 
2111209 
2114116 


3048625000 
3054936851 
3061257408 
3067586677 
3073924664 


38.0789 
38.0920 
38.1051 
38.1182 
38.1314 


11.3185 
11.3211 
11.3237 
11.3263 
11.3289 


1505 
1506 
1507 
1508 
1509 


2265025 
2268036 
2271049 
2274064 
2277081 


3408862625 
3415662216 
3422470843 
3429288512 
3436115229 


38.7943 
38.8072 
38.8201 
38.8330 

38.8458 


11.4598 
11.4624 
11.4649 
11.4675 
11.4700 


1455 
1456 
1457 
1458 
,1459 


2117025 
2119936 
2122849 
2125764 
2128681 


3080271375 
3086626816 
3092990993 
3099363912 
3105745579 


38.1445 
38.1576 
38.1707 
38.1838 
38.1969 


11.3315 
11.3341 
11.3367 
11.3393 
11.3419 


1510 
1511 
1512 
1513 
1514 


2280100 
2283121 
2286144 
2289169 
2292196 


3442951000 
3449795831 
3456649728 
3463512697 
3470384744 


38.8587 
38.8716 
38 8844 
38.8973 
38.9102 


11.4725 
11.4751 
11.4776 
11.4801 
11.4820 


1460 
1461 
1462 
1463 
1464 


2131600 
2134521 
2137444 
2140369 
2143296 


3112136000 
3118535181 
3124943128 
3131359847 
3137785344 


38.2099 
38.2230 
38.2361 
38.2492 
38.2623 


11.3445 
11.3471 
1 1 . 3496 
11.3522 
11.3548 


1515 
1516 
1517 
1518 
1519 


2295225 
2298256 
2301289 
2304324 
2307361 


3477265875 
3484156096 
3491055413 
3407963832 
3504881359 


38.9230 
38.9358 
38.9487 
38.9615 
38.97'44 


11.485-3 
11.4877 
11.4902 
11.4927 
11.4953 


1465 
1466 
1467 
1468 
1469 


2146225 
2149156 
2152089 
2155024 
2157961 


3144219625 
3150662696 
3157114563 
3163575232 
3170044709 


38.2753 
38.2884 
38.3014 
38.3145 
38.3275 


11.3574 
11.3600 
11.3626 
11.3652 
11.3677 


1520 
1521 
1522 
1523 
1524 


2310400 
2313441 
2316484 
2319529 
2322576 


3511808000 
3518743761 
3525688648 
3532642667 
3539605824 


38.9872 
39.0000 
39.0128 
39.0256 
39.0384 


11.4978 
11.5003 
11.5028 
11.5054 
11.5079 


1470 
1471 
1472 
1473 
1474 


2160900 
2163841 
2166784 
2169729 
2172676 


3176523000 
3183010111 
3189506048 
3196010817 
3202524424 


38.3406 
38.3536 
38.3667 
38 . 3797 
88.3SWT 


11.3703 
11.3729 
11.3755 
11.3780 
11.3806 


1525 
1526 
1527 

1528 
1529 


2325625 

2328676 
2331729 
2334784 
2337841 


3546578125 
3553559576 
3560550183 
3567549952 

357455889 


39.0512 
39.0640 
39.0768 
39.0896 
39.1024 


11.5104 
11.5129 
11.5154 
11.5179 
11.5204 



SQUARES, CUBES, SQUARE AND CUBE R<JOTS. 101 



No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


1530 
1531 
1532 
1533 
1534 


2340900 
2343961 
2347024 
2350089 
2353156 


3581577000 
3588604291 
3595640768 
3602080437 
3609741304 


39.1152 
39.1280 
39.1408 
39.1535 
39.1663 


11.5230 
11.5255 
11.5280 
11.5305 
11.5330 


1565 
1566 
1567 
1568 
1569 


2449225 3833037125 
2452356,3840389496 
24554893847751263 
2458624 3855123432 
2461761 3862503009 


39.5601 
39.5727 
39.5854 
39.5980 
39.6106 


11.6102 
11.6126 
11.6151 
11.6176 
11.6200 


1535 
1536 
1537 
1538 
1539 


2356225 
2359296 
2362369 
2365444 
2368521 


3616805375 
3623878656 
3630961153 
3638052872 
3645153819 


39.1791 
39.1918 
39.2046 
39.2173 
39.2301 


11.5355 
11.5380 
11.5405 
11.5430 
11.5455 


1570 
1571 
1572 
1573 
1574 


2464900 3869893000 
2468041 3877292411 
24711 84 384701 248 
24743293892119517 
24774763899547224 


39.6232 
39.6358 
39.6485 
39.6611 
39.6737 


11.6225 
11.6250 
11.6274 
11.6299 
11.6324 


1540 
1541 
1542 
1543 
1544 


2371600 
2374681 
2377764 
2380849 
2383936 


3652264000 
3659383421 
3666512088 
3673650007 
2680797184 


39.2428 
39.2556 

39.2683 
39.2810 
39.2938 


11.5480 
11.5505 
11.5530 
11.5555 
11.5580 


1575 
1576 
1577 
1578 
1579 


2480625 3906984375 
2483770 3914430976 
2486929 3921887033 
2490084 :^9^9352552 
249324 lj 3936827539 


39.6863 
39.6989 
39.7115 
39.7240 
39.7366 


11.6348 
11.6373 
11.6398 
11.6422 
11.0447 


1545 
1546 
1547 
1548 
1519 


2387025 
2390116 
2393209 
2396304 
2399401 


3687953625 
3695119336 
3702294323 
3709478592 
3716672149 


39.3065 
39.3192 
39.3319 
39.3446 
39.3573 


11.5605 
11.5630 
11.5655 
11.5680 
11.5705 


1580 

1581 
1582 
1583 
1584 


2496400 3944312000 
2499561 3651805941 
2502724 3959309368 
2505889 3960822287 
25090563974344704 


39.7492 
39.7618 
39.7744 
39.7869 
39.7995 


11.6471 
11.6496 
11.6520 
11.6545 
11,6570 


1550 
1551 
1552 
1553 
1554 


2402500 
2405601 
2408704 
2411809 
2414916 


3723875000 
3731087151 
3738308608 
3745539377 
3752779464 


39.3700 
39.3827 
39.3954 
39.4081 
39.4208 


11.5729 
11.5754 
11.5779 
11.5804 
11.5829 


1585 
1586 
1587 

1588 
1589 


2512225 3981876625 
2515396 3989418056 
2518569 3996969003 
2521744 4004529472 
2524921 4012099469 


39.8121 11.6594 
39.824611.6619 
39.837211.6643 
39.849711.6608 
39.862311.6692 


1555 
1556 
1557 
1558 
1559 


2418025 
2421136 
2424249 
2427364 

2430481 


3760028875 
3767287616 
3774555693 
3781833112 
3789119879 


39.4335 11.5854 
39.4462 11.5879 
39.4588 11.5903 
39.4715 11.5928 
39.4842 11.5953 


1590 
1591 
1592 
1593 
1594 


2528100 4019679000 
2531281 4027268071 
2534464 4034866688 
2537649 4042474857 
2540836 4050092584 


39.874811.6717 
39.887311.6741 
39.8999 11.6765 
39.9124 11.6790 
39.924911.6814 


1560 
1561 
1562 
1563 
1564 


2433600 
2436721 
2439844 
2442969 
2446096 


3796416000 
3803721481 
3811036328 
3818360547 
3825694144 


39 496811.5978 
39.5095111.6003 
39.5221 11.6027 
39. 5348' 11. 6052 
39.547411.6077 


1595 
1590 
1597 
1598 
1599 


2544025 4057719875 
2547216 4065356736 
2550409 4073003173 
2553604 4080659192 
2556801 4088324799 


39.937511.7839 
39.950011.6863 
39. 9625 jll. 6888 
39.9750 11.6912 
39.9875 11.6936 












1600 


2560000 


4096000000 


40.000011.6961 



SQUARES AND CUBES OF 



No. 


Square. 


Cube. 


No. 


Square. 


Cube. 


No. 


Square. 


Cube. 


.1 


.01 


.001 


.01 


.0001 


.000 001 


.001 


.00 00 01 


.000 000 001 


.2 


.04 


.008 


.02 


.0001 


.000 008 


.002 


.00 00 04 


.000 000 008 


3 


.09 


.027 


.03 


.0009 


.000 027 


.003 


.00 00 09 


.000 000 027 


4 


.16 


.064 


.04 


.0016 


.000 064 


.004 


.00 00 16 


.000 000 064 


.5 


.25 


.125 


.05 


.0025 


.000 125 


.005 


00 00 25 


.000 000 125 


.6 


.36 


.216 


.06 


.0036 


.000 216 


.006 


.00 00 36 


.000 000 216 


.7 


.49 


.343 


.07 


.0049 


.000 343 


007 


.00 00 49 


.OOO.OOQ 343 


.8 


.64 


.512 


.08 


.0064 


.000 512 


.008 


; .00 00 64 


.000' 000 512 


.9 


.81 


.729 


.09 


.0081 


.000 729 


009 


> .00 0'.) 81 


.000 COu 729 


1.0 


1.00 


1.000 


.10 


.0100 


.001 000 


.010 


.^X) 01 00 


.000 '001 000 


1.2 


1.44 


1.728 


.12 


.0144 


.001,728 


.012 


,00 M *4 


.ooa qo 1 . 728 



Note that the square has twice as many xlecirf al places ; an'i 
times as many decimal places, as the root. 



102 



MATHEMATICAL TABLES. 



FIFTH ROOTS AND FIFTH 

(Abridged from TRAUTWINE.) 



II 


Power. 


II 


Power. 


II 


Power. 


ji 


Power. 


t, . 

II 


Power. 


.10 


.000010 


3.7 


693.440 


9.8 


90392 


21.8 


4923597 


40 


102400000 


.15 


.000075 


3.8 


792.352 


9.9 


95099 


22.0 


5153632 


41 


115856201 


.20 


.000320 


3.9 


902.242 


10.0 


100000 


22.2 


5392186 


42 


130691232 


.25 


.000977 


4.0 


1024.00 


10 2 


110408 


22.4 


5639493 


43 


147008443 


.30 


.002430 


4.1 


1158.56 


10.4 


121665 


22.6 


5895793 


44 


164916224 


.35 


.005252 


4.2 


1306.91 


10.6 


133823 


22.8 


6161327 


45 


184528125 


.40 


.010240 


4.3 


1470.08 


10.8 


146933 


23 


6436343 


46 


205962976 


.45 


.018453 


4.4 


1649.16 


11.0 


161051 


23^2 


6721093 


47 


229345007 


.50 


.031250 


4.5 


1845.28 


11.2 


176234 


23.4 


7015834 


48 


254803968 


.55 


.050328 


4.6 


2059.63 


11.4 


192541 


23.6 


7320825 


49 


282475249 


.60 


.077760 


4.7 


2293.45 


11.6 


210084 


23.8 


7636332 


50 


312500000 


.65 


.116029 


4.8 


2548.04 


11.8 


228776 


24.0 


7962624 


51 


345025251 


.70 


168070 


4.9 


2824.75 


12.0 


248832 


24.2 


8299976 


52 


380204032 


.75 


,237305 


5.0 


3125.00 


12.2 


270271 


24.4 


8648666 


53 


418195498 


.80 


.327680 


5.1 


3450.25 


12.4 


293163 


24.6 


9008978 


54 


459165024 


.85 


.443705 


5.2 


3802 04 


12. G 


317580 


24.8 


9381200 


55 


503284375 


.90 


.590490 


5.3 


4181.95 


12.8 


343597 


25.0 


9765625 


56 


550731776 


.95 


.773781 


5.4 


4591.65 


13.0 


371293 


25.2 


10162550 


57 


601692057 


1.00 


1.00000 


5.5 


5032.84 


13.2 


400746 


25.4 


10572278 


58 


656356768 


1.05 


1.27628 


5.6 


5507.32 


13.4 


432040 


25.6 


10995116 


59 


714924299 


1.10 


1.61051 


5.7 


6016.92 


13 6 


465259 


25.8 


11431377 


60 


777600000 


1.15 


2.01135 


5.8 


6563.57 


13.8 


500490 


26.0 


11881376 


61 


844596301 


1.20 


2.48832 


5.9 


7149.24 


14.0 


537824 


26.2 


12345437 


62 


916132832 


1.25 


3.05176 


6.0 


7776.00 


34.2 


577353 


26.4 


1282388G 


63 


992436543 


1.30 


3.71293 


6.1 


8445.96 


14.4 


619174 


26.6 


13317055 


64 


1073741824 


1.35 


4.48403 


6.2 


9161.33 


'A 6 


663383 


26.8 


13825281 


65 


1160290625 


1.40 


5.37824 


6.3 


9924.37 


14.8 


710082 


27.0 


14348907 


66 


1252332576 


1.45 


6.40973 


6 4 


10737 


15.0 


759375 


27.2 


14888280 


67 


1350125107 


1.50 


7.59375 


6.5 


11603 


15 2 


811368 


27.4 


15443752 


68 


1453933568 


1.55 


8.94661 


6.6 


12523 


15.4 


866171 


27.6 


16015681 


69 


1564031349 


1.60 


10.4858 


6.7 


13501 


15. G 


923896 


27.8 


16604430 


70 


168070000C 


1.65 


12.2298 


6.8 


14539 


15.8 


984658 


28.0 


17210368 


71 


1804229351 


1.70 


14.1986 


6.9 


15G40 


16.0 


1048576 


28.2 


17633868 


72 


1934917632 


1.75 


16.4131 


7.0 


16807 


16.2 


1115771 


28.4 


18475309 


73 


2073071593 


1.80 


18.8957 


7.1 


18042 


16.4 


118G367 


28.6 


19135075 


74 


2219006G24 


1.85 


21.6700 


7.2 


19349 


16.6 


1260493 


28.8 


19813557 


75 


2373046875 


1.90 


24.7610 


7.3 


20731 


16 8 


1338278 


29.0 


20511149 


76 


2535525376 


1.95 


28.1951 


7.4 


22190 


17.0 


1419857 


29.2 


21228253 


77 


2706784157 


2.00 


32.0000 


7.5 


23730 


17.2 


1505366 


29.4 


21965275 


78 


2887174368 


2.05 


36.2051 


7.6 


25355 


17.4 


1594947 


29.6 


22722628 


79 


3077056399 


2.10 


40.8410 


7.7 


27'068 


17.6 


1688742 


29 8 


3500728 


80 


3276800000 


2.15 


45.9401 


7.8 


28872 


17.8 


17'86899 


30.0 


24300000 


81 


3486784401 


2 20 


51.5363 


7.9 


30771 


18.0 


1889568 


30.5 


26393634 


82 


3707398432 


2.25 


57.6650 


8.0 


32768 


18.2 


199G903 


31.0 


28G29151 


83 


3939040643 


2.30 


64.3634 


8.1 


34868 


18.4 


2109061 


31.5 


31013642 


84 


4182119424 


2.35 


71.6703 


8.2 


37074 


18.6 


2226203 


32.0 


33554432 


85 


4437053125 


2.40 


79.6262 


8.3 


39390 


18.8 


2348493 


32.5 


36259082 


86 


4704270176 


2.45 


88.2735 


8.4 


41821 


19.0 


2476099 


33.0 


39135393 


87 


4984209207 


2.50 


97.6562 


8.5 


44371 


19.2 


2609193 


33.5 


42191410 


88 


5277319168 


2.55 


107.820 


8.6 


47043 


19.4 


2747949 


34.0 


45435424 


89 


5584059449 


2.60 


118.814 


8.7 


49842 


19.6 


2892547 


31.5 


4S875980 


90 


5904900000 


2.70 


143.489 


8.8 


52773 


19.8 


3043168 


35.0 


52521875 


91 


6240321451 


2.80 


172.104 


8.9 


55841 


20.0 


3200000 


35.5 


56382167 


92 


6590815232 


2.90 


205.111 


9.0 


59049 


20.2 


3363232 


36.0 


G0466176 


93 


6956883693 


3 00 


243.000 


9.1 


62403 


20.4 


3533059 


36 5 


64783487 


94 


7339040224 


3,10 


286.292 


9.2, 


05998- 


20.6 


3709677 


37.0 


69343957 


95 


7737809375 


3 f*0 


335.544 


a. 3 


'58509 


20.8 


3893289 


37.5 


74157715 


96 


815372G976 


3^30 


391.354 


9.4 


V3390 


21.0 


4084101 


38.0 


79235168 


97 


8587340257 


ft. 40 


454 . 54 


,9,5, 


77378 


.21.2 


4282322 


38.5 


84587005 


98 


90392079G8 


3..M) 525 19 


96-1 81237- , 


21.4 


4488166 


39.0 


90224199 


99 


950990U499 


Sieol OC4.'662 


9";7j 8,5873--' 


21.6 


4701850 


39.5 


96158012 







CIRCUMFERENCES AND AREAS OF CIRCLES, 103 



CIRCUMFERENCES AND AREAS OF CIRCLES. 



Piam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


1 


3.1416 


0.7854 


65 


204.20 


3318.31 


129 


405.27 


13069.81 


2 


6.2832 


3.1416 


66 


207.34 


3421 . 19 


130 


408.41 


13273.23 


3 


9.4248 


7.0686 


67 


210.49 


3525.65 


131 


411.55 


13478.22 


4 


12.5664 


12.5664 


68 


213.63 


3631.68 


132 


414.69 


13684 78 


5 


15.7080 


19.635 


69 


216.77 


3739.28 


133 


417.83 


13892.91 


6 


18.850 


28 274 


70 


219.91 


3848.45 


134 


420.97 


14102.61 


7 


21.991 


38.485 


71 


223.05 


3959.19 


135 


424.12 


14313.88 


8 


25.133 


50.266 


72 


226.19 


4071.50 


136 


427.26 


14526.72 


9 


28.274 


63.617 


73 


229.34 


4185.39 


137 


430.40 


14741.14 


ilO 


31.416 


78.540 


74 


232.48 


4300 84 


138 


433.54 


14957.12 


11 


34.558 


95.033 


75 


235.62 


4417.86 


139 


436.68 


15174.68 


12 


37.699 


113.10 


76 


238.76 


4536.46 


140 


439.82 


15393.80 


13 


40.841 


132.73 


77 


241.90 


4656.63 


141 


442.96 


15614.50 


14 


43.982 


153.94 


78 


245.04 


4778.36 


142 


446.11 


15836.77 


15 


47.124 


176.71 


79 


248.19 


4901.67 


143 


449.25 


16060.61 


16 


50.265 


201.06 


80 


251.33 


5026.55 


144 


452.39 


16286.02 


17 


53.407 


226.98 


81 


254.47 


5153.00 


145 


455.53 


16513.00 


18 


56.549 


254.47 


82 


257.61 


5281.02 


146 


458.67 


16741.55 


10 


59.690 


283.53 


83 


260.75 


5410.61 


147 


461.81 


16971.67 


20 


62.832 


314.16 


84 


263.89 


5541.77 


148 


464.96 


17203.36 


21 


65.973 


346.36 


85 


267.04 


5674 50 


149 


468.10 


17436.62 


22 


69.115 


380.13 


86 


270.18 


5808.80 


150 


471.24 


17671.46 


23 


72.257 


415.48 


8?' 


273.32 


5944.68 


151 


474.38 


17907 86 


24 


75.398 


452.39 


88 


276.46 


6082.12 


152 


477.52 


18145.84 


25 


78.540 


490.87 


89 


279.60 


6221.14 


153 


480.66 


18385.39 


26 


81.681 


530.93 


90 


282.74 


6361.73. 


154 


483.81 


18626.50 


27 


84.823 


572.56 


91 


285.88 


6503.88' 


155 


486.95 


18869.19 


28 


87.965 


615.75 


92 


289.03 


6647.61 


156 


490.09 


19113.45 


29 


91.106 


660.52 


93 


292.17 


6792.91 


157 


493.23 


19359.28 


30 


94.248 


706.86 


94 


295.31 


6939.78 


158 


496.37 


19606.68 


31 


97.389 


754 . 77 


95 


298.45 


7088.22 


159 


499.51 


19855.65 


32 


100.53 


804.25 


96 


301.59 


7238.23 


160 


502.65 


20106.19 


33 


103.67 


855.30 


97 


304.73 


7389.81 


161 


505.80 


20358.31 


34 


106.81 


907.92 


98 


307.88 


7542.96 


162 


508.94 


20611.99 


35 


109.96 


962.11 


99 


311.02 


7697.69 


163 


512.08 


20867.24 


36 


113.10 


1017.88 


100 


314.16 


7853.98 


164 


515.22 


21124.07 


37 


116.24 


1075.21 


101 


317.30 


8011.85 


165 


518.36 


21382.46 


38 


119.38 


1134.11 


102 


320.44 


8171.28 


166 


521.50 


21642.43 


39 


122.52 


1194.59 


103 


323.58 


8332.29 


167 


524.65 


21903 97 


40 


125.66 


1256.64 


104 


326.73 


8494.87 


168 


527.79 


22167 08 


41 


128.81 


1320 25 


105 


329 87 


8659.01 


169 


530.93 


22431.76 


42 


131.95 


1385.44 


106 


333.01 


8824.73 


170 


534.07 


22698.01 


43 


135.09 


1452.20 


107 


336.15 


8992.02 


171 


537.21 


22965.83 


44 


138.23 


'1520.53 


108 


339.29 


9160.88 


172 


540.35 


23235.22 


45 


141.37 


1590.43 


109 


342.43 


9331.32 


173 


543.50 


23506.18 


46 


144.51 


1661.90 


110 


345.58 


9503.32 


174 


546.64 


23778.71 


47 


147.65 


1734.94 


111 


348.72 


9676.89 


175 


549.78 


24052.82 


48 


150.80 


1809.56 


112 


351.86 


9852.03 


176 


552.92 


24328.49 


49 


153 94 


1885.74 


113 


355.00 


10028.75 


177 


556.06 


24605.74 


50 


157.08 


1963.50 


114 


358.1.4 


10207.03 


178 


559.20 


24884.56 


51 


160.22 


2042.82 


115 


361.28 


10386 89 


179 


562.35 


25164.94 


53 


163.36 


2123.72 


116 


364.42 


10568.32 


180 


565.49 


25446.90 


53 


166.50 


2206.18 


117 


367.57 


10751.32 


181 


568.63 


25730.43 


54 


169.65 


2290 22 


118 


370.71 


10935.88 


182 


571.77 


26015.53 


55 


172.79 


2375.83 


119 


373.85 


11122.02 


183 


574.91 


26302.20 


56 


175.93 


2463 01 


120 


376.99 


11309.73 


184 


578.05 


26590.44 


57 


179.07 


2551.76 


121 


380.13 


11499.01 


185 


581.19 


26880.25 


58 


182.21 


2642.08 


122 


383.27 


11689.87 


186 


584.34 


27171.63 


59 


185.35 


2733.97 


123 


386.42 


11882.29 


187 


587.48 


27464.59 


60 


188.50 


2827.43 


124 


389.56 


12076.28 


188 


590.62 


27759.11 


61 


191.64 


2922.47 


125 


392.70 


12271.85 


189 


593.76 


28055.21 


6-4 


194.78 


3019.07 


126 


395.84 


12468.98 


190 


596.90 


28352 87 


63 


197.92 


3117.25 


127 


398.98 


12(567.69 


191 


600.04 


28652.11 


64 


201.06 


3216.99 


128 


402.12 


12867.96 


192 


603.19 


28952.92 



104 



MATHEMATICAL TABLES. 



Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


193 


606.33 


29255.30 


260 


816.81 


53092.92 


327 


1027.30 


83981.84 


194 


609.47 


29559.25 


261 


819.96 


53502.11 


328 


1030.44 


84496.28 


195 


612.61 


29864.77 


262 


823.10 


53912.87 


329 


1033.58 


85012.28 


196 


615.75 


30171.86 


263 


826.24 


54325.21 


330 


1036.73 


85529.86 


197 


618.89 


30480.52 


264 


829.38 


54739.11 


331 


1039.87 


86049.01 


198 


622.04 


30790.75 


265 


832.52 


55154.59 


332 


1043.01 


86569.73 


199 


625.18 


31102.55 


266 


835.66 


55571.63 


333 


1046.15 


87092.02 


200 


628.32 


31415.93 


267 


838.81 


55990.25 


334 


1049.29 


87615.88 


201 


631.46 


31730.87 


268 


841 . 95 


56410.44 


335 


1052.43 


88141.31 


202 


634.60 


32047.39 


269 


845.09 


56832.20 


336 


1055.58 


88668.31 


203 


637.74 


32365.47 


270 


848.23 


57255.53 


337 


1058.72 


89196.88 


204 


640.88 


32685.13 


271 


851.37 


57680.43 


338 


1061.86 


89^27.03 


205 


644.03 


33006.36 


272 


854.51 


58106.90 


339 


1065.00 


90258.74 


206 


647.17 


33329.16 


273 


57.65 


58534.94 


340 


1068.14 


90792.03 


207 


650.31 


33653.53 


274 


860.80 


58964.55 


341 


1071.28 


91326.88 


208 


653.45 


33979.47 


275 


863.94 


59S95.74 


342 


1074.42 


91863.31 


209 


656.59 


34306.98 


276 


867.08 


59828.49 


343 


1077.57 


92401 . 31 


210 


659.73 


34G36.06 


277 


870.22 


60262.82 


344 


1080.71 


92940.88 


211 


662.88 


34966.71 


278 


873.36 


60698.71 


345 


1083.85 


93482.02 


212 


066.02 


35298.94 


279 


876.50 


61136.18 


346 


1086.99 


94024.73 


213 


669.16 


35632.73 


280 


879.65 


61575.22 


347 


1090.13 


94569.01 


214 


672.30 


35968.09 


281 


882.79 


62015.82 


348 


1093.27 


95114.86 


215 


675.44 


36305.03 


282 


885.93 


62458.00 


349 


1096.42 


95662.28 


216 


678.58 


36643.54 


283 


889.07 


62901.75 


350 


1099.56 


96211.28 


217 


681.73 


36983.61 


284 


892.21 


63347.07 


351 


1102.70 


96761.84 


218 


684.87 


37325.26 


285 


895.35 


63793.97 


352 


1105.84 


97313.97 


219 


688.01 


37668.48 


286 


898.50 


64242.43 


353 


1108.98 


97867.68 


220 


691.15 


38013.27 


287 


901.64 


64692.46 


354 


1112.12 


98422.96 


221 


694.29 


38359.63 


288 


904.78 


65144.07 


355 


1115.27 


98979.80 


222 


697.43 


88707.56 


289 


907.92 


65597.24 


356 


1118.41 


99538.22 


223 


700.58 


39057.07 


290 


911.06 


66051.99 


357 


1121.55 


100098.21 


224 


703.72 


39408.14 


291 


914.20 


66508.30 


358 


1124.69 


100659.77 


225 


706.86 


39760.78 


292 


917.35 


66966.19 


359 


1127.83 


101222.90 


226 


710.00 


40115.00 


293 


920.49 


67425.65 


360 


1130.97 


101787.60 


227 


713.14 


40470.78 


294 


923.63 


67886.68 


361 


1134.11 


102353.87 


228 


716.28 


40828.14 


295 


926.77 


68349.28 


362 


1137.26 


102921.72 


229 


719.42 


41187.07 


296 


929.91 


68813.45 


363 


1140.40 


103491.13 


230 


722.57 


41547.56 


297 


933.05 


69279.19 


364 


1143.54 


104062.12 


231 


725.71 


41909.63 


298 


936.19 


69746.50 


365 


1146.68 


104634.67 


232 


728.85 


42273.27 


299 


939.34 


70215.38 


306 


1149.82 


105208.80 


233 


731.99 


42638.48 


300 


942.48 


70685.83 


367 


1152.96 


105784.49 


234 


735.13 


43005.26 


301 


945.62 


71157.86 


368 


1156.11 


106361.76 


235 


738.27 


43373.61 


3J33 


948.76 


71631.45 


369 


1159.25 


106940.60 


236 


741.42 


43743.54 


303 


951.90 


72106.62 


370 


1162.39 


107521.01 


237 


744.56 


44115.03 


304 


955.04 


72583.36 


371 


1165.53 


108102.99 


238 


747.70 


44488.09 


305 


958.19 


73061.66 


372 


1168.67 


108686.54 


239 


750.84 


44862.73 


306 


961.33 


73541.54 


373 


1171.81 


109271.66 


240 


753.98 


45238.93 


307 


964.47 


74022.99 


374 


1174.96 


109858.35 


241 


757.12 


45616.71 


308 


967.61 


74506.01 


375 


1178.10 


110446.62 


242 


760.27 


45996.06 


309 


970.75 


74990.60 


376 


1181.24 


111036.45 


243 


763.41 


46376.98 


310 


973.89 


75476.76 


377 


1184.38 


111627.86 


244 


766.55 


46759.47 


311 


977.04 


75964.50 


378 


1187.52 


112220.83 


245 


769.69 


47143.52 


312 


980.18 


76453.80 


379 


1190.66 


112815.38 


246 


772.83 


47529.16 


313 


983.32 


76944.67 


380 


1193.81 


113411.49 


247 


775.97 


47916.36 


314 


986.46 


77437.12 


381 


1196.95 


114009.18 


248 


779.11 


48305.13 


315 


989.60 


77931.13 


382 


1200.09 


114608.44 


249 


782.26 


'48695.47 


316 


992.74 


784-26.72 


383 


1203.23 


115209.27 


250 


785.40 


49087.39 


317 


995.88 


78923.88 


384 


1206.37 


115811.67 


251 


788.54 


49480.87 


318 


999.03 


79422.60 


385 


1209.51 


116415.64 


252 


791.68 


49875.92 


319 


1002.17 


79922.90 


386 


1212.65 


117021.18 


253 


794.82 


50272.55 


320 


1005.31 


80424.7? 


387 


1215.80 


117628.30 


254 


797.96 


50670.75 


321 


1008.45 


80928.21 


388 


1218.94 


118236.98 


255 


801.11 


51070.52 


322 


1011.59 


81433 22 


389 


1222.08 


118847.24 


256 


804.25 


51471.85 


323 


1014.73 


81939.80 


390 


1225.22 


119459.06 


257 


807.39 


51874.76 


324 


1017.88 


82447.96 


391 


1228.36 


120072.46 


258 


810.53 


52279.24 


325 


1021.02 


82957.68 


392 


1231.50 


120687.42 


259 


813.67 


52685.29 


326 


1024.16 


83468.98 


393 


1234.65 


121303.96 



CIRCUMFERENCES AND AREAS OF CIRCLES. 



Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


394 


1237.79 


121922.07 


461 


1448.27 


166913.60 


528 


1658.76 


218956.44 


395 


1240.93 


122541.75 


462 


1451.42 


167638.53 


529 


1661.90 


219786.61 


396 


1244.07 


123163.00 


463 


1454.56 


168365.02 


530 


1665.04 


220618.34 


397 


1247.21 


123785.82 


464 


1457.70 


169093.08 


531 


1668.19 


221451.65 


398 


1250.35 


124410.21 


465 


1460.84 


169822.72 


532 


1671.33 


222286.53 


399 


1253.50 


125036.17 


466 


1463.98 


170553.92 


533 


1674.47 


223122.98 


400 


1256.64 


125663.71 


467 


1467.12 


171286.70 


534 


1677.61 


223961.00 


401 


1259.78 


126292.81 


468 


1470.27 


172021.05 


535 


1680.75 


224800.59 


402 


1262.92 


126923.48 


469 


1473.41 


172756.97 


536 


1683.89 


225641.75 


403 


1266.06 


127555.73 


470 


1476.55 


173494.45 


537 


1687.04 


226484.48 


404 


1269.20 


128189.55 


471 


1479.69 


174233.51 


538 


1690.18 


227328.79 


405 


1272.35 


128824.93 


472 


1482.83 


174974.14 


539 


1693.32 


228174.66 


406 


1275.49 


129461.89 


473 


1485.97 


175716.35 


540 


1696.46 


229022.10 


407 


1278.63 


130100.42 


474 


1489.11 


176460.12 


541 


1699.60 


229871.12 


408 


1281.77 


130740.52 


475 


1492.26 


177205.46 


542 


1702.74 


230721.71 


409 


1284.91 


131382.19 


476 


1495.40 


177952.37 


543 


1705.88 


231573.86 


410 


1288.05 


132025.43 


477 


1498.54 


178700.86 


544 


1709.03 


232427.59 


411 


1291.19 


132670.24 


478 


1501.68 


179450.91 


545 


1712.17 


233282.89 


412 


1294.34 


133316.63 


479 


1504.82 


180202.54 


546 


1715.31 


234139.76 


413 


1297.48 


133964.58 


480 


1507.96 


180955.74 


547 


1718.45 


234998720 


414 


1300.62 


134614.10 


481 


1511.11 


181710.50 


548 


1721.59 


235858.21 


415 


1303.76 


135265.20 


482 


1514.25 


182466.84 


549 


1724.73 


236719.79 


416 


1306.90 


135917.86 


483 


1517.39 


183224.75 


550 


1727.88 


237582.94 


417 


1310.04 


136572.10 


484 


1520.53 


183984.23 


551 


1731.02 


238447.67 


418 


1313.19 


137227.91 


485 


1523.67 


184745.28 


552 


1734.16 


239313.96 


419 


1316.33 


137885.29 


486 


1526.81 


185507 90 


553 


1737.30 


240181.83 


420 


1319.47 


138544.24 


487 


1529.96 


186272.10 


554 


1740.44 


241051.26 


421 


1322.61 


139204.76 


488 


1533.10 


187037.86 


555 


1743.58 


241922.27 


422 


1325.75 


139866.85 


489 


1536.241 187805.19 


556 


1746.73 


242794.85 


423 


1328.89 


140530.51 


490 


1539.38 


188574.10 


557 


1749.87 


243668.99 


424 


1332.04 


141195.74 


491 


1542.52 


189344.57 


558 


1753.01 


244544.71 


425 


1335.18 


141862 54 


492 


1545.66 


190116.62 


559 


1756.15 


245422.00 


426 


1338.32 


142530.92 


493 


1548.81 


190890.24 


560 


1759.29 


246300.86 


427 


1341.46 


143200.86 


494 


1551.95 


191665.43 


561 


1762.43 


247181.30 


428 


1344.60 


143872.38 


495 


1555 09 


192442.18 


562 


1765.58 


248063.30 


429 


1347.74 


144545.46 


496 


1558.23 


193220.51 


563 


1768.72 


248946.87 


430 


1350.88 


145220.12 


497 


1561.37 


194000.41 


564 


1771.86 


249832.01 


431 


1354.03 


145896.35 


498 


1564.51 


194781.89 


565 


1775.00 


250718 73 


432 


1357.17 


146574.15 


499 


1567.65 


195564.93 


566 


1778.14 


251607.01 


433 


1360.31 


147253.52 


500 


1570.80 


196349.54 


567 


1781.28 


252496.87 


434 


1363.45 


147934.46 


501 


1573.94 


197135.72 


568 


1784.42 


253388.30 


435 


1366.59 


148616.97 


502 


1577.08 


197923.48 


569 


1787.57 


254281.29 


436 


1369.73 


149301.05 


503 


1580.22 


198712.80 


570 


1790.71 


355175.86 


437 


1372.88 


149986.70 


504 


1583.36 


199503.70 


571 


1793.85 


256072.00 


438 


1376.02 


150673.93 


505 


1586 50 


200296.17 


572 


1796.99 


256969.71 


439 


1379.16 


151362.72 


506 


1589.65 


201090.20 


573 


1800.13 


257868.99 


440 


1382.30 


152053.08 


507 


1592.79 


201885.81 


574 


1803.27 


258769.85 


441 


1385.44 


152745.02 


508 


1595.93 


202682.99 


575 


1806.42 


259672.27 


442 


1388.58 


153438.53 


509 


1599.07 


203481.74 


576 


1809.56 


260576.26 


443 


1391.73 


154133.60 


510 


1602.21 


204282.06 


577 


1812.70 


261481.83 


444 


1394.87 


154830.25 


511 


1605.35 


205083.95 


578 


1815 84 


262388.90 


445 


1398.01 


155528.47 


512 


1608.50 


205887.42 


579 


1818.98 


263297.67 


446 


1401.15 


156228.26 


513 


1611.64 


206692.45 


580 


1822.12 


264207.94 


447 


1404.29 


156929.62 


514 


1614.78 


207499.05 


581 


1825.27 


265119.79 


448 


1407.43 


157632.55 


515 


1617.92 


208307.23 


582 


1828.41 


266033.21 


449 


1410.58 


158337,06 


516 


1621.06 


209116.97 


583 


1831.55 


266948.20 


450 


1413.72 


159043.13 


517 


1624.20 


209928.29 


584 


1834.69 


267864.76 


451 


1416.86 


159750.77 


518 


1627.34 


210741.18 


585 


1837.83 


268782.89 


452 


1420.00 


160459.99 


519 


1630.49 


211555.63 


586 


1840.97 


269702.59 


453 


1423.14 


161170.77 


520 


1633.63 


212371.66 


587 


1844.11 


270623.86 


454 


1426.28 


161883.13 


521 


1636.77 


213189.26 


588 


1847.26 


271546.70 


455 


1429.42 


162597.05 


522 


1639.91 


214008.43 


589 


1850.40 


272471.12 


456 


1432.57 


163312.55 


523 


1643.05 


214829.17 


590 


1853.54 


273397.10 


457 


1435.71 


164029.62 


524 


1646.19 


215651.49 


591 


1856.68 


274324.60 


458 


1438.85 


164748.26 


525 


1649.34 


216475.37 


592 


1859.82 


275253.78 


459 


1441.99 


165468.47 


526 


1652.48 


217300.82 


593 


1862.96 


276184.48 


460 


1445.13 


166190.25 


527 


1655.62 


_218127.85 


594 


1866.11 


277116.75 



106 



MATHEMATICAL TABLES. 



Diarn. 


Circum. 


Area. 


Diam- 


Circum. 


Area. 


Diana- JCircum. 


Area. 


595 


1869.25 


278050.58 


663 


2082.88 


345236.69 


731 ! 2296.50 


419686.15 


596 


1872.39 


278985.99 


664 


2086.02 


346278.91 


732 2299.65 


420835.19 


597 


1875.53 


279922.97 


665 


2089.16 


347322.70 


733 


2302.79 


421985.79 


598 


1878.67 


280861.52 


666 


2092.30 


348368.07 


734 


2305.93 


423137.97 


599 


1881.81 


281801 65 


667 


2095.44 


349415.00 


735 


2309.07 


424291.72 


600 


1884.96 


282743.34 


668 


2098.58 


350463.51 


736 


2312.211 425447.04 


601 


1888.10 


283686.60 


669 


2101.73 


351513.59 


737 


2315.35 


426603.94 


602 


1891.24 


284631.44 


670 


2104.87 


352565.24 


738 


2318.50 


427762.40 


603 


1894.38 


285577.84 


671 


2108.01 


353618.45 


739 


2321.64 


428922.43 


604 


1897.52 


286525.82 


672 


2111.15 


354673 24 


740 


2324.78 


430084.03 


605 


1900.6(5 


287475.36 


673 


2114.29 


355729.60 


741 


2327.92 


431247.21 


606 


1903.81 


288426.48 


674 


2117.43 


356787.54 


742 


8981.06 


432411.95 


60? 


1906.95 


289379.17 


675 


2120.58 


357847.04 


743 


2334.20 


433578.27 


608 


1910.09 


290333.43 


676 


2123.72 


358908.11 


744 


2337.34 


434746.16 


609 


1913.23 


291289.26 


677 


2126.86 


359970.75 


745 


2340.49 


435915.62 


610 


1916.37 


292246.66 


678 


2130.00 


361034.97 


746 


2343.63 


437086.64 


611 


1919 51 


293205.63 


679 


2133.14 


362100.75 


747 


2346.77 


438259.24 


612 


1922.65 


294166.17 


680 


2136.28 


363168.11 


748 


2349.91 


439433.41 


613 


1925.80 


295128.28 


681 


2139.42 


364237.04 


749 


2353.05 


440609 16 


614 


1928.94 


296091.97 


682 


2142.57 


365307.54 


750 


2356.19 


441786.47 


615 


1932.08 


297057.22 


683 


2145.71 


366379.60 


751 


2359.34 


442965.35 


616 


1935.22 


298024.05 


684 


2148.85 


367453.24 


752 


2362.48 


444145.80 


617 


1938.36 


298992.44 


685 


2151 99 


368528.45 


753 


2365.62 


445327.83 


618 


1941.50 


299962.41 


686 


2155.13 


369605.23 


754 


2368.76 


446511.42 


619 


1944.65 


300933.95 


687 


2158.27 


370683.59 


755 


2371.90 


447696.59 


620 


1947.79 


301907.05 


688 


2161.42 


371763.51 


756 


2375.04 


448883.32 


6-21 


1950.93 


302881.73 


689 


2164.56 


372845.00 


757 


2378.19 


450071.63 


622 


1954.07 


303857.98 


690 


2167.70 


373928.07 


758 


2381.33 


451261.51 


623 


1957.21 


304835.80 


691 


2170.84 


375012.70 


759 


2384.47 


452452.96 


624 


1960.35 


305815.20 


692 


2173.98 


376098.91 


760 


2387.61 


453645.98 


625 


1963.50 


306796.16 


693 


2177.12 


377186.68 


761 


2390.75 


454840.57 


626 


1966.64 


307778.69 


694 


2180.27 


378276.03 


762 


2393.89 


456036.73 


627 


1969.78 


308762.79 


695 


2183.41 


379366.95 


763 


S397.04 


457234.46 


628 


1972.92 


309748.47 


696 


2186.55 


380459.44 


764 


SJ400.18 


458433.77 


629 


1976.06 


310735.71 


697 


2189.69 


381553.50 


765 


2403.32 


459634.64 


630 


1979.20 


311724.53 


698 


2192.83 


382649.13 


766 


2406.46 


460837.08 


631 


1982.35 


312714.92 


699 


2195.97 


383746.33 


767 


2409.60 


462041.10 


632 


1985.49 


313706.88 


700 


2199.11 


384845.10 


768 


2412.74 


463246.69 


633 


1988.63 


314700.40 


701 


2202.26 


385945.44 


769 


2415.88 


464453.84 


634 


1991.77 


315695.50 


7'02 


2205.40 


387047.36 


770 


2419.03 


465662.57 


635 


1994.91 


316692.17 


703 


2208.54 


388150.84 


771 


2422.17 


466872.87 


636 


1998.05 


317690.42 


704 


2211.68 


389255.90 


772 


2425.31 


468084.74 


637 


2001.19 


318690.23 


705 


2214.82 


390362.52 


773 


2428.45 


469298.18 


638 


2004.34 


319691.61 


706 


2217.96 


391470.72 


774 


2431.59 


470513.19 


639 


2007.48 


320694.56 


707 


2221.11 


392580.49 


775 


2434.73 


471729.77 


640 


2010.62 


321699.09 


708 


2224,25 


393691.82 


776 


2437.88 


472947.92 


641 


2013.76 


322705.18 


709 


2227.39 


394804.73 


777 


2441.02 


474167.65 


642 


2016.90 


323712.85 


710 


2230.53 


395919.21 


778 


2444.16 


475388.94 


643 


2020.04 


324722.09 


711 


2233.67 


397035.26 


779 


2447.30 


476611.81 


644 


2023.19 


325732.89 


712 


2236.81 


398152.89 


780 


2450.44 


477836.24 


645 


2026.33 


326745.27 


713 


2239.96 


399272.08 


781 


2453.58 


479062.25 


646 


2029.47 


327759.22 


714 


2243.10 


400392.84 


782 


2456.73 


480289.83 


647 


2032.61 


328774.74 


715 


2246.24 


401515.18 


788 


2459.8? 


481518.97 


648 


2035.75 


329791.83 


716 


2249.38 


402639.08 


784 


2463.01 


482749.69 


649 


2038.89 


330810.49 


717 


2252.52 


403764.56 


785 


2466.15 


483981.98 


650 


2042. 04 


331830.72 


718 


2255.66 


404891.60 


786 


2469.29 


485215.84 


651 


2045.18 


332852.53 


719 


2258.81 


406020.22 


787 


2472.43 


486451.28 


652 


2048.32 


333875.90 


720 


2261.95 


407150.41 


788 


2475.58 


487688.28 


653 


2051.46 


334900.85 


721 


2265.09 


408282.17 


789 


2478.72 


488926.85 


654 


2054.60 


335927.36 


722 


2268.23 


409415.50 


790 


2481.86 


490166.99 


655 


2057.74 


336955.45 


723 


2271.37 


410550.40 


791 


2485.00 


491408.71 


656 


2060.88 


337985.10 


724 


2274.51 


411686.87 


792 


2488.14 


492651.99 


657 


2064.03 


339016.33 


725 


2277.65 


412824.91 


793 


2491.28 


493896.85 


658 


2067.17 


340049.13 


726 


2280.80 


413964.52 


794 


2494.42 


495143.28 


659 


2070.31 


341083,50 


727 


22S3.94 


415105.71 


795 


2497.57 


496391.27 


660 


2073.45 


342119.44 


728 


2287.08 


416248.46 


796 


2500.71 


497640.84 


661 


2076.59 


343156.95 


729 


2290.22 


417392.79 


797 


2503.85 


498891.98 


662 


2079.73 


344196.03 


730 


2293.36 


418538.68 


798 1 2506.99 


500144.69 



CIRCUMFERENCES AND AREAS OF CIRCLES. 107 



Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


Diam. 


Circum. Area. 


799 


2510.13 


501398.97 


867 


2723.76 


590375.16 


935 


2937.39: 686614.71 


800 


2513.27 


502654.82 


868 


2726.90 


591737.83 


936 


2940.53 


688084.19 


801 


2516.42 


503912.25 


869 


2730.04 


593102.06 


937 


2943.67 


689555.24 


802 


2519.56 


505171.24 


870 


2733.19 


594467.87 


938 


2946 81 


691027.86 


803 


2522.70 


506431.80 


871 


2736.33 


595835.25 


939 


2949.96 


692502.05 


804 


2525.84 


507693.94 


872 


2739.47 


597204.20 


940 


2953.10 


693977.82 


805 


2528.98 


508957.64 


873 


2742.61 


598574.72 


941 


2956.24 


695455.15 


806 


2532.12 


510222.92 


874 


2745.75 


599946.81 


942 


2959.38 


696934.06 


807 


2535.27 


511489.77 


875 


2748.89 


601320.47 


943 


2962.52 


698414.53 


808 


2538.41 


512758.19 


876 


2752.04 


602695.70 


944 


2965.66 


699896.58 


809 


2541.55 


514028.18 


877 2755.18 


604072.50 


945 


2968.81 


701380.19 


810 


2544.69 


515299.74 


878 2758.32 


605450.88 


946 


2971.95 


702865.38 


811 


2547.83 


516572.87 


879 


2761.46 


606830.82 


947 


2975.09 


704352.14 


812 


2550.97 


517847.57 


880 


2764.60 


608212.34 


948 


2978.23 


705840.47 


813 


2554.11 


519123 84 


881 


2767.74 


609595.42 


949 


2981.37 


707330.37 


814 


2557.26 


520401.68 


882 


2770.88 


610980.08 


950 


2984.51 


708821.84 


815 


2560.40 


521681.10 


883 


2774.03 


612366.31 


951 


2987.65 


710314.88 


816 


2563.54 


522982.08 


884 


2777.17 


613754.11 


952 


2990.80 


711809.50 


817 


2566.68 


524244.63 


885 


2780.31 


615143.48 


953 


2993.94 


713305.68 


818 


2569.82 


525528.76 


886 


2783.45 


616534.42 


954 


2997.08 


714803.43 


819 


2572 96 


526814.46 


887 


2786.59 


617926.93 


955 


3000.22 


716302.76 


820 


2576.11 


528101.73 


888 


2789.73 


619321.01 


956 


3003.36 


717803.66 


821 


2579.25 


529390.56 


889 


2792.88 


620716.66 


957 


3006.50 


719306.12 


822 


2582.39 


530680.97 


890 


2796.02 


622113.89 


958 


3009.65 


720810.16 


823 


2585.53 


531972.95 


891 


2799.16 


623512.68 


959 


3012.79 


722315.77 


824 


2588.67 


533266.50 


892 


2802.30 


624913.04 


960 


3015.93 


723822.95 


825 


2591.81 


534561.62 


893 


'2805.44 


626314.98 


961 


3019.07 


725331.70 


826 


2594.96 


535858.32 


894 


2808.58 


627718.49 


962 


3022.21 


726842.02 


827 


2598.10 


537156.58 


895 


2811.73 


629123.56 


963 


3025.35 


728353.91 


828 


2601.24 


538456.41 


896 


2814.87 


630530.21 


964 


3028.50 


729867.37 


829 


2604.38 


539757.82 


897 


2818.01 


631938.43 


965 


3031.64 


731382.40 


830 


2607.52 


541060.79 


898 


2821.15 


633348.22 


966 


3034.78 


732899.01 


831 


2610.66 


542365.34 


899 


2824.29 


634759.58 


967 


3037.92 


734417.18 


832 


2613.81 


543671.46 


900 


2827.43 


636172.51 


968 


8041.06 


735936.93 


833 


2616.95 


544979.15 


901 


2830.58 


637587.01 


969 


3044.20 


737458.24 


834 


2620.09 


546288.40 


902 


2833.72 


639003.09 


970 


3047.34 


738981.13 


835 


2623.23 


547599.23 


903 


2836.86 


640420.73 


971 


3050.49 


740505.59 


836 


2626.37 


548911.63 


904 


2840.00 


641839.95 


972 


3053.63 


742031.62 


837 


2629.51 


550225.61 


905 


2843.14 


643260.73 


973 


3056.77 


743559.22 


838 


2632.65 


551541.15 


906 


2846.28- 


644683.09 


974 


3059.91 


745088.39 


839 


2635.80 


552858.26 


907 


2849.42 


646107.01 


975 


3063.05 


746619.13 


840 


2638.94 


554176.94 


908 


2852.57 


647532.51 


976 


3066.19 


748151.44 


841 


2642.08 


555497.20 


909 


2855.71 


648959.58 


977 


3069.34 


749685.32 


842 


2645.22 


556819.02 


910 


2858.85 


650388.22 


978 


3072.48 


751220.78 


843 


2648.36 


558142.42 


911 


2861.99 


651818.43 


979 


3075.62 


752757.80 


844 


2651.50 


559467.39 


912 


2865.13 


653250.21 


980 


3078.76 


754296.40 


845 


2654.65 


560793.92 


913 


2868.27 


654683.56 


981 


3081.90 


755836.56 


846 


2657.79 


562122.03 


914 


2871.42 


656118.48 


982 


3085.04 


757378.30 


847 


2660.93 


563451.71 


915 


2874.56 


657554.98 


983 


3088.19 


758921.61 


848 


2664.07 


564782.96 


916 


2877.70 


658993.04 


984 


3091.33 


760466.48 


849 


2667.21 


566115.78 


917 


2880.84 


600432.68 


985 


3094.47 


762012.93 


850 


2670.35 


567450.17 


918 


2883.98 


661873.88 


986 


3097.61 


763560.95 


851 


2673.50 


568786.14 


919 


2887.12 


663316.66 


987 


3100.75 


765110.54 


852 


2676.64 


570123.67 


920 


2890.27 


664761.01 


988 


3103.89 


766661.70 


853 


2679.78 


571462.77 


921 


2893.41 


666206.92 


989 


3107.04 


768214.44 


854 


2682.92 


572803.45 


922 


2896.55 


667654.41 


990 


3110.18 


769768.74 


855 


2686.06 


574145.69 


923 


2899.69 


669103.47 


991 


3113.32 


771324.61 


856 


2689.20 


575489.51 


924 


2902.83 


670554.10 


992 


3116.46 


772882.06 


857 


2692.34 


576834.90 


925 


2905.97 


672006.30 


993 


3119.60 


774441.07 


H58 


2695.49 


578181.85 


926 


2909.11 


673460.08 


994 


3122.74 


776001.66 


859 


2698.63 


579530.38 


927 


2912.26 


674915.42 


995 


31.25.88 


777563.82 


860 


2701.77 


580880.48 


928 


2915.40 


676372.33 


996 


3129.03 


779127.54 


861 


2704.91 


582232.15 


929 


2918.54 


677830.82 


997 


3132.17 


780692.84 


862 


2708.05 


583585.39 


930 


2921.68 


679290.87 


998 


3135.31 


782259.71 


863 


2711.19 


584940.20 


931 


2924.82 


680752.50 


999 


3138.45 


783828.15 


864 


2714.34 


586296.59 


932 


2927.96 


682-215.69 


1000 


3141.59 


785398 16 


865 


2717.48 


587654.54 


933 


2931.11 


683680.46 








866 


2720 62 


589014.07 


934 


2934.25 


685146.80 









108 



MATHEMATICAL TABLES. 



CIRCUMFERENCES AND AREAS OF CIRCLES 

Advancing: by Eighths. 



Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


1/64 


.04909 


.00019 


2 % 


7.4613 


4.4301 


6 H 


19.242 


29.465 


1/32 


.09818 


.00077 


7/16 


7.6576 ' 


4.6664 


H 


19.635 


30.680 


3/64 


.14726 


.00173 


H 


7.8540 


4.9087 


% 


20.028 


31.919 


1/16 


.19635 


.00307 


9/16 


8.0503 


5.1572 




20.420 


33.183 


3/33 


.29452 


.00690 


% 


8.2467 


5.4119 


% 


20.813 


34.472 


Ys 


.39270 


.0122? 


11/16 


8.4430 


5.6727 


M 


21 206 


35.785 


5/32 


.49087 


.01917 


n 


8.6394 


5.9396 


% 


21.598 


37.122 


3/16 


.58905 


.02761 


13/16 


8.8357 


6.2126 


7. 


21.991 


38.485 


7/32 


.08722 


.03758 


Vs 


9.0321 


6.4918 


ix 


22.384 


39.871 








15/16 


9.2284 


6.7771 


/4 


22.776 


41.282 


y 


.78540 


.04909 








ax 


23.169 


42.718 


pa 


.88357 


.06213 


3. 


9.4248 


7.0686 


i^ 


23.562 


44.179 


5/16 


.98175 


.07670 


1/16 


9.6211 


7.3662 


% 


23.955 


45 664 


11/33 


1.0799 


.09281 


Hi 


9.8175 


7.6699 


% 


24.347 


47.173 


% 


1.1781 


.11045 


3/16 


10.014 


7.9798 


7X 


24.740 


48.707 


13/32 


1.2763 


.12962 


y* 


10.210 


8.2958 


8. 


25.133 


50.265 


7/16 


1.3744 


.15033 


5/16 


10.407 


8.6179 


/^ 


25.525 


51.849 


15/32 


1.4726 


. 17257 


% 


10.603 


8.9462 


f4 


25.918 


53.456 








7/16 


10.799 


9.2806 


% 


26.311 


55.088 


^ 


1 5708 


.19635 


H 


10.996 


9.6211 


y* 


26.704 


56.745 


17/32 


1.6690 


.22166 


9/16 


11.192 


9.9678 


% 


27.096 


58.426 


9/16 


1.7671 


.24850 


% 


11.388 


10.321 


M 


27.489 


60.132 


19/32 


1.8653 


.27688 


11/16 


11.585 


10.680 


H 


27.882 


61.862 


% 


1.9635 


.30680 


H 


11.781 


11.045 


9. 


28.274 


63.617 


21/32 


2.0617 


.33824 


13/16 


11.977 


11.416 


/^j 


28.667 


65.397 


11/16 


2.1598 


.37122 


% 


12.174 


11.793 


^4 


29.060 


67.201 


23/32 


2.2580 


.40574 


15/16 


12.370 


12.177 


% 


29.452 


69.029 








4. 


12.566 


12.566 


L 


20.845 


70.882 


% 


2.3562 


.44179 


1/16 


12.763 


12.962 


% 


30.238 


72.760 


25/32 


2.4544 


.47937 


H 


12.959 


13.364 


3 


30.631 


74.662 


13/16 


2.5525 


.51849 


3/16 


13.155 


13.772 


/^O 


31.023 


76.589 


27/32 


2.6507 


.55914 


H 


13.352 


14.186 


10. 


31.416 


78.540 


K 


2.7489 


.60132 


5/16 


13.548 


14.607 


H 


31.809 


80.516 


29/32 


2.8471 


.64504 


% 


13.744 


15.033 


$ 


32.201 


82 516 


15/16 


2.9452 


.69029 


7/16 


13.941 


15.466 




32.594 


84.541 


31/32 


3.0434 


.73708 


H 


14.137 


15.904 


i^ 


32.987 


86.590 








9/16 


14.334 


16.349 


% 


33.379 


88.664 


I. 


3.1416 


.7854 


% 


14.530 


16.800 


% 


33.772 


90.703 


1/16 


3.3379 


.8860 


11/16 


14.726 


17.257 


% 


34.165 


92.886 


y& 


3.5343 


.9940 


H 


14.923 


17.721 


11 


34.558 


95.033 


3/16 


3.7306 


1.1075 


13/16 


15.119 


18.190 




34.950 


97.205 


k 


3.9270 


1.2272 


% 


15.315 


18.665 


M 


35.343 


99.402 


5/16 


4.1233 


1.3530 


15/16 


15 512 


19.147 


78 


35.736 


101.62 


% 


4.3197 


1.4849 


5. 


15.708 


19.635 


^ 


36.128 


103.87 


7/16 


4.5160 


1.6230 


1/16 


15.904 


20.129 


% 


36.521 


106.14 


H 


4.7124 


1.7671 


H 


16.101 


20.629 


% 


36.914 


108.43 


9/16 


4.9087 


1.9175 


3/16 


16.297 


21.135 


% 


37.306 


110.75 


% 


5.1051 


2.0739 


M 


16.493 


21.648 


12 


37.699 


113.10 


11/16 


5.3014 


2.2365 


5/16 


16.690 


22.166 


/^ 


38.092 


115.47 


H 


5.4978 


2.4053 


% 


16.886 


22.691 


M 


38.485 


117.86 


13/16 


5.6941 


2.5802 


7/16 


17.082 


23.221 


a| 


38.877 


120.28 


% 


5.8905 


2.7612 


y% 


17.279 


23.758 


L^ 


39.270 


122.72 


15/16 


6.0868 


2.9483 


9/16 


17.475 


24.301 


Kg 


39.663 


125.19 








% 


17.671 


24.850 


% 


40.055 


127.68 


2. 


6.2832 


3.1416 


11/16 


17.868 


25.406 


% 


40.448 


130.19 


1/16 


6.4795 


3.3410 


H 


18.064 


25.967 


13. 


40.841 


132.73 


M 


6.6759 


3.5466 


13-16 


18.261 


26.535 


ix 


41.233 


135.30 


3/16 


6.8722 


3.7583 


% 


18.457 


27.109 


M 


41.626 


137.89 


k 


7.0686 


3.9761 


15-16 


18.653 


27.688 


% 


42.019 


140.50 


5/16 


7.2649 


4.2000 


fi 


18.850 


28.274 


^ 


42.412 


143.14 



CIRCUMFERENCES AND AREAS OF CIRCLES. 109 



Diam. 


Circum . 


Area. 


Diam. 


Circum u 


Area. 


Diam. 


Circum. 


Area. 


n% 


42.804 


145.80 


21% 


68.722 


375.83 


301/6 


94.640 


712.76 


% 


43.197 


148.49 


22. 


69.115 


380.13 


M 


95.033 


718.69 


% 


43.590 


151.20 


/^ 


69.508 


384.46 


% 


95.426 


724 64 


14. 


43.982 


153.94 


ix 


69.900 


388.82 




95.819 


730.62 




44.375 


156.70 


% 


70.293 


393.20 


% 


96.211 


736.62 


IX 


44.768 


159.48 


12 


70.686 


397.61 


M 


96.604 


742.64 


az 


45.160 


162.30 


% 


71.079 


402.04 


To 


96.997 


748.69 


i/ 


45.553 


165.13 


'M 


71.471 


406.49 


31 


97.389 


754.77 


% 


45.946 


167.99 


% 


71.864 


410.97 


/^ 


97.782 


760.87 


SX 


46.338 


170.87 


23 


72.257 


415.48 


J4 


98.175 


766.99 


Yi 


46.731 


173.78 


H 


72.649 


420.00 


% 


98.567 


773.14 


15 


47.124 


176.71 


& 


73.042 


424.56 


/^ 


98.960 


779.31 


^ 


47.517 


179.67 


3X 


73.435 


429.13 


% 


99.353 


785.51 


| 


47.909 ' 


182.65 


ix: 


73.827 


433.74 


% 


99.746 


791.73 


ax 


48.302 


185.66 


ft/. 


74.220 


438.36 


% 


100.138. 


797.98 


ix 


48.695 


188.69 


M 


74.613 


443.01 


32. 


100.531 


804.25 


% 


49.087 


191.75 


% 


75.006 


447.69 


I/, 


100.924 


810.54 


% 


49.480 


194.83 


24. 


75.398 


452.39 


% 


101.316 


816.86 


% 


49.873 


197.93 


/^ 


75.791 


457.11 


% 


101.709 


823.21 


1 i 


50.265 


201.06 


IX: 


76.184 


461.86 


\fa 


102.102 


829.58 




50.658 


204.22 


a 


76.576 


466.64 


% 


102.494 


835.97 


ix 


51.051 


207.39 


IX 


76.969 


471.44 


% 


102.887 


842.39 


ty 


51.444 


210.60 


% 


77.362 


476.26 


% 


103.280 


848.83 


/^ 


51.836 


213.82 


% 


77.754 


481.11 


33. 


103.673 


855.30 


5X 


52.229 


217.08 


% 


78.147 


485.98 


YB 


104.065 


861.79 


sx 


52.622 


220.35 


25. 


78.540 


490.87 




104.458 


868.31 


% 


53.014 


223.65 




78.933 


495.79 


% 


104.851 


874.85 


17 


53.407 


226.98 ' 


24 


79.325 


500.74 


/^ 


105.243 


881.41 


/6 


53.800 


230.33 


% 


79.718 


505.71 


% 


105.636 


888.00 




54.192 


233.71 


Lj 


80.111 


510.71 


M 


106.029 


894.62 


% 


54.585 


237.10 


?B 


80.503 


515.72 


Ys 


106.421 


901.26 


HJ 


54.978 


240.53 


3X 


80.896 


520.77 


34 


106.814 


907.92 


% 


55.371 


243.98 


% 


81.289 


525.84 


i^ 


107.207 


914.61 


% 


55.763 


247.45 


26. 


81.681 


530.93 


/4 


107.600 


921.32 


% 


56.156 


250.95 


i^ 


82.074 


536.05 


% 


107.992 


928.06 


18 


56.549 


254.47 


IX 


82.467 


541.19 


ix. 


108.385 


934.82 


/^ 


56.941 


258.02 


a2 


82.860 


546.35 


% 


108.778 


941.61 


/4 


57.334 


261.59 


/^3 


83.252 


551.55 


!% 


109.170 


948.42 


% 


57.727 


265.18 


5X. 


83.645 


556.76 


78 


109.563 


955.25 


V& 


58.119 


268.80 


M 


84.038 


562.00 


35. 


109.956 


962.11 


% 


58.512 


272.45 


72 


84.430 


567.27 


^ 


110.348 


969.00 


ax 


58.905 


276.12 


27. 


84.823 


572.56 


y 


110.741 


975.91 


% 


59.298 


279.81 




85.216 


577.87 




111.134 


982.84 


19. 


59.690 


283.53 


\A 


85.608 


583.21 


Xsjj 


111.527 


989.80 


ii 


60.083 


287.27 


s/. 


86.001 


588.57 


% 


111.919 


996. 7'8 




60.476 


291.04 


/"* 


86.394 


593.96 


M 


112.312 


1003.8 


% 


60.868 


294.83 


% 


86.786 


599.37 


% 


112.705 


1010.8 


v& 


61.261 


298.65 


M 


87.179 


604.81 


36. 


113.097 


1017.9 


% 


61.654 


302.49 


xo 


87.572 


610.27 


^ 


113.490 


1025.0 


$4 


62.046 


306.35 


28 


87.965 


615.75 


H 


113.883 


1032.1 


7X 


62.439 


310.24 




88.357 


621.26 


% 


114.275 


1039.2 


20. 


62.832 


314.16 


IX 


88.750 


626.80 




114.668 


1046.3 


H 


63.225 


318.10 


a^. 


89.143 


632.36 


% 


115.061 


1053.5 


IX 


63.617 


322.06 


^> 


89.535 


637.94 


M 


115.454 


1060.7 


5s 


64.010 


326.05 


5? 


89.928 


643.55 


% 


115.846 


1068.0 


l^ 


64.403 


330.06 


% 


90.321 


649.18 


37 


116.239 


1075.2 


7& 


64.795 


334.10 


7X 


90.713 


654.84 




116.632 


1082.5 


% 


65.188 


338.16 


29. 


91.106 


660.52 


ix 


117.024 


1089.8 


% 


65.581 


342.25 


^ 


91.499 


666.23 


% 


117.417 


1097.1 


21. 


65.973 


346.36 


H 


91.892 


671.96 


ix 


117.810 


1104.5 


^ 


66.366 


350.50 


a| 


92.284 


677.71 


% 


118.202 


1111.8 


\A 


66.759 


354.66 


VZ 


92.677 1683.49 


ax 


118.596 


1119.2 


$ 


67.152 


358.84 


% 


93.070 689.30 


% 


118.988 


1126.7 


^ 


67.544 


363.05 


% 


93.462 695.13 


38. 


119.381 


1134.1 


7& 


67.937 


367.28 


so 


93.855 700.98 


^ 


119.773 


1141.0 


s % 


68.330 


371.54 


30. 


94.248 706.86 


| 


120.166 


1149.1 



MATHEMATICAL TABLES. 



Diam. 


Circum. 


Area, 


Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area 


38% 


120.559 


1156.6 


46% 


146.477 


1707.4 


54% 


172.395 


2365.0 




120.951 


1164.2 


% 


146.869 


1716.5 


55. 


172.788 


2375.8 


% 


121.344 


1171.7 


% 


147.262 


1725.7 


H 


173.180 


2386.6 


M 


121.737 


1179.3 


47 


147.655 


1734.9 


H 


173.573 


2397.5 


/o 


122.129 


1186.9 


/4 


148.048 


1744.2 


% 


173.966 


2408 3 


89 


122.522 


1194.6 


/4 


148.440 


1753.5 




174.358 


2419.2 


x*6 


122.915 


1202.3 


% 


148.833 


1762.7 


% 


174.751 


2430.1 


% 


123.308 


1210.0 


\& 


149.226 


1772.1 


M 


175.144 


2441.1 


% 


123.700 


1217.7 


% 


149.618 


1781.4 


Vs 


175.536 


2452.0 


$& 


124.093 


1225.4 


M 


150.011 


1790.8 


56 


175.929 


2463.0 


% 


124.486 


1233.2 


xo 


150.404 


1800.1 


*/8 ' 


176.322 


2474.0 


% 


124.878 


1241.0 


48 


150.796 


1809.6 


M 


176.715 


2485.0 


% 


125.271 


1248.8 




151.189 


1819.0 


% 


177.107 


2496.1 


40. 


125.664 


1256.6 


/4 


151.582 


1828.5 




177.500 


2507.2 


K 


126.056 


1264.5 


J^B 


151.975 


1837.9 


% 


177.893 


2518.3 


H 


126.449 


1272.4 


L 


152.367 


1847.5 


<X 


178.285 


2529.4 


% 


126.842 


1280.3 


% 


152.760 


1857.0 


% 


178.678 


2540.6 


^ 


127.235 


1288.2 


3x 


153.153 


1866.5 


57 


179.071 


2551.8 


% 


127.627 


1296.2 


% 


153.545 


1876.1 


Vs 


179.463 


2563.0 


$4 


128.020 


1304.2 


49 


153.938 


1885.7 


IX 


179.856 


2574.2 


% 


128.413 


1312.2 


3^ 


154.331 


1895.4 


% 


180.249 


2585.4 


41. 


128.805 


1320.3 


x4 


154.723 


1905.0 


IX 


180.642 


2596.7 


H 


129.198 


1328.3 


% 


155.116 


1914.7 


% 


181.034 


2608.0 


J4 


129.591 


1336.4 


i^ 


155.509 


1924.4 


34 


181.427 


2619.4 


i 


129.983 


1344.5 


% 


155.902 


1934.2 


7 /8 


181.820 


2630.7 




130.376 


1352.7 


M 


156.294 


1943.9 


58. 


182.212 


2642.1 


ax 


130.769 


1360.8 


7X 


156.687 


1953.7 




182.605 


2653.5 


M 


131.161 


1309.0 


50. 


157.080 


1963.5 


IX- 


182.998 


2664.9 


% 


131.554 


1377.2 




157.472 


1973.3 


% 


183.390 


2676.4 


42. 


131.947 


1385.4 


\A 


157.865 


1983.2 


x"l3 


183.783 


2687.8 


! 


132.340 


1393.7 


% 


158.258 


1993.1 


% 


184.176 


2699.3 


M 


132.732 


1402.0 


L/j 


158.650 


2003.0 


34 


184.569 


2710.9 


% 


133.125 


1410.3 


% 


159.043 


2012.9 


% 


184.961 


2722.4 


IX 


133.518 


1418.6 


M 


159.436 


2022.8 


59. 


185.354 


2734.0 


K^ 


133.910 


1427.0 


% 


159.829 


2032.8 


Ys 


185.747 


2745.6 


M 


134.303 


1435.4 


51 


160.221 


2042.8 


x4 


186.139 


2757.2 


% 


134.696 


1443.8 


H 


160.614 


2052 .8 


a2 


186.532 


2768.8 


43 


135.088 


1452.2 


M 


161.007 


2062.9 


x-4 


186.925 


2780.5 


^ 


135.481 


1460.7 


| 


161.399 


2073.0 


% 


187.317 


2792.2 


i 


135.874 


1469.1 




161.792 


2083.1 


M 


187.710 


2803.9 


% 


136.267 


1477.6 


% 


162.185 


2093.2 


% 


188.103 


2815.7 




136.659 


1486.2 


94 


162.577 


2103.3 


60. 


188.496 


2827.4 


K/ 


137.052 


1494.7 


7X 


162.970 


2113.5 


Ys 


188.888 


2839.2 


% 


137.445 


1503.3 


53. 


163.363 


2123.7 


M 


189.281 


2851.0 





137.837 


1511.9 




163.756 


2133.9 


% 


189.674 


2862.9 


44.? 


138.230 


1520.5 


14 


164.148 


2144.2 


/^ 


190.066 


2874.8 




138.623 


1529.2 


% 


164.541 


2154.5 


% 


190.459 


2886.6 


IX 


139.015 


1537.9 


jx 


164.934 


2164. H 


M 


190.852 


2898.6 


% 


139.408 


1546.6 


% 


165.326 


2175.1 


/o 


191.244 


2910.5 


IX 


139.801 


1555.3 


ax 


165.719 


2185.4 


61 


191.637 


2922.5 


KX 


140 194 


1564.0 


% 


166.112 


2195.8 


*6 


192.030 


2934.5 


3X 


140.586 


1572.8 


53. 


106.504 


2206.2 


H 


192.423 


2946.5 


7X 


140.979 


1581.6 




166.897 


2216.6 


% 


192.815 


2958.5 


45. 


141.372 


1590.4 


IX 


167.290 


2227.0 




193.208 


2970.6 




141.764 


1599.3 


% 


167.683 


2237.5 


% 


193.601 


2982.7 


IX 


142.157 


1608.2 


x^> 


168.075 


2248.0 


M 


193.993 


2994.8 


KX 


142.550 


1617.0 


RX 


168.468 


2258.5 


% 


194.386 


3006.9 


IX 


142.942 


1626.0 


ax 


168.861 


2269.1 


62 


194.779 


3019.1 


KX 


143.335 


1634.9 


xo 


109.253 


2279.6 


x6 


195.171 


3031.3 


3X 


143.728 


1643.9 


54 


169.646 


2290.2 


y. 


195.564 


3043.5 


% 


144.121 


1652.9 




170.039 


2300.8 


% 


195.957 


3055.7 


46 


144.513 


1661.9 


IX 


170.431 


2311.5 


/12 


196.350 


3068.0 




144.906 


1670.9 


a/j 


170.824 


2322.1 


% 


196.742 


3080.3 


IX 


145.299 


1680.0 


IX 


171.217 


2332.8 


a^ 


197.135 


3092.6 


az 


145.691 


1689.1 


% 


171.609 


2343.5 


yQ 


197.528 


3104.9 


H 


146.084 


1698.2 


M 


172.002 


2354.3 


63 


197.920 


3117.2 



CIRCUMFERENCES AND AREAS OF CIRCLES. Ill 



Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


63^ 


198.313 


3129.6 


71 % 


224.231 


4001.1 


79% 


250.149 


4979.5 


i% 


198.706 


3142.0 




224.624 


4015.2 


M 


250.542 


4995.2 


% 


199.098 


3154.5 


% 


225.017 


4029.2 


% 


250.935 


5010.9 


ix 


199.491 


3166.9 


ax 


225.409 


4043.3 


80. 


251.327 


5026.5 


% 


199.884 


3179.4 


% 


225.802 


4057.4 


/^ 


251.720 


5042.3 


3X 


200.277 


3191.9 


72 * 


226.195 


4071.5 


/4 


252.113 


5058.0 


% 


200.669 


3204.4 


Y 


226.587 


4085.7 


^8 


252.506 


5073.8 


64. 


201.062 


3217.0 


(? 


226.980 


4099.8 


1^2 


252.898 


5089.6 


H 


201.455 


3229.6 


% 


227.373 


4114.0 


% 


253.291 


5105.4 


3 


201.847 


3242.2 


ix 


227.765 


4128.2 


ax 


253.684 


5121.2 


% 


202.240 


3254.8 


% 


228.158 


4142.5 


YH 


254.076 


5137.1 


ix 


202.633 


3267.5 


ax 


228.551 


4156.8 


81. 


254.469 


5153.0 


% 


203.025 


3280.1 


7X 


228.944 


4171.1 


^ 


254.862 


5168.9 


ax 


203.418 


3292.8 


73. 


229.336 


4185.4 


! 


255.254 


5184.9 


VB 


203.811 


3305.6 


^ 


229.729 


4199.7 


% 


255.647 


5200.8 


65. 


204.204 


3318.3 




230.122 


4214.1 


^ 


256.040 


5216.8 


H 


204.596 


3331.1 


% 


230.514 


4228.5 


% 


256.433 


5232.8 


8 


204.989 


3343.9 


V& 


230.907 


4242.9 


M 


256.825 


5248.9 


% 


205.382 


3356.7 


% 


231.300 


4257.4 


TO 


257.218 


5264.9 




205.774 


3369.6 


ax 


231.692 


4271.8 


82 


257.611 


5281.0 


% 


206.167 


3382.4 


% 


232.085 


4286.3 




258.003 


5297.1 


M 


206.560 


3395.3 


74. 


232.478 


4300.8 


4 


258.396 


5313.3 


% 


206.952 


3408.2 




232.871 


4315.4 


% 


258.789 


5329.4 


66 


207.345 


3421.2 


/4 


233.263 


4329.9 


1^ 


259.181 


5345.6 


YB 


207.738 


3434.2 


9s 


233.656 


4344.5 


% 


259.574 


5361.8 


/4 


208.131 


3447.2 


LX 


234.049 


4359.2 


ax 


259.967 


5378.1 


&x 


208.523 


3460.2 


5^j 


234.441 


4373.8 


% 


260.359 


5394.3 


Via 


208.916 


3473.2 


ax 


234.834 


4388.5 


83 


260.752 


5410 6 


% 


209.309 


3486.3 


% 


235.227 


4403.1 


H 


261.145 


5426.9 


M 


209.701 


3499.4 


75. 


235.619 


4417.9 


xl 


261.538 


5443.3 


% 


210.094 


3512.5 




236.012 


4432.6 


% 


261.930 


5459.6 


67. 


210.487 


3525 7 


M 


236.405 


4447.4 


% 


262.323 


5476.0 


K 


210.879 


3538.8 


% 


236.798 


4462.2 


% 


262.716 


5492.4 


H 


211.272 


3552.0 


/^ 


237.190 


4477.0 


ax^ 


263.108 


5508.8 


% 


211.665 


3565.2 


% 


237.583 


4491.8 


xo 


263.501 


5525.3 




212.058 


3578.5 


M 


237.976 


4506.7 


84 


263.894 


5541.8 


% 


212.450 


3591.7 


/a 


238.368 


4521.5 


/^ 


264.286 


5558.3 


M 


212.843 


3C05.0 


76 


238.761 


4536.5 


IX- 


264.679 


5574.8 


% 


213.236 


3618.3 


^6 


239.154 


4551.4 


% 


265.072 


5591.4 


48 


213.628 


3631.7 


/4 


239.546 


4566.4 


IX 


265.465 


5607.9 


M 


214.021 


3645.0 


% 


239.939 


4581.3 


5^ 


265.857 


5624.5 


M 


214.414 


3658.4 


Hi 


240.332 


4596.3 


ax 


266.250 


5641.2 




214.806 


3671.8 


% 


240.725 


4611.4 


TO 


266.643 


5657.8 


/^ 


215.199 


3685.3 


M 


241.117 


4626.4 


85 


267.035 


5674.5 


% 


215.592 


3698.7 


% 


241.510 


4641.5 




267.428 


5691.2 


^4 


215.984 


3712.2 


77. 


241.903 


4656.6 


/4 


267.821 


5707.9 


% 


216.377 


3725.7 




242.295 


4671.8 


a| 


268.213 


5724.7 


69. 


216.770 


3739.3 


/4 


242.688 


4686.9 


ix 


268.606 


5741.5 


Ml 


217.163 


3752.8 


% 


243.081 


4702.1 


&x 


268.999 


5758.3 


H 


217.555 


3766.4 


Vi> 


243.473 


4717.3 


ax 


269.392 


5775.1 


% 


217.948 


3780.0 


^i 


243.866 


4732.5 


% 


269.784 


5791.9 


H 


218.341 


3793.7 


ax 


244.259 


4747.8 


86. 


270.177 


5808.8 


% 


218.733 


3807.3 


% 


244.652 


4763.1 




270.570 


5825.7 


M 


219.126 


3821.0 


78 


245.044 


4778.4 


M 


270.962 


5842.6 


% 


219.519 


3834.7 


/^ 


245.437 


4793.7 


% 


271.355 


5859.6 


so. 


219.911 


3848.5 


/4 


245.830 


4809.0 


^ 


271.748 


5876.5 


H 


220.304 


3862.2 


% 


246.222 


4824.4 


% 


272.140 


5893.5 




220.697 


3876.0 


i^ 


246.615 


4839.8 


ax 


272.533 


5910.6 


% 


221.090 


3889.8 


% 


247.008 


4855.2 


TO 


272.926 


5927.6 


/^ 


221.482 


3903.6 


M 


247.400 


4870.7 


87 


273.319 


5944.7 


% 


221.875 


3917.5 


7^ 


247.793 


4886.2 




273.711 


5961.8 


M 


222.268 


3931.4 


79. 


.248.186 


4901.7 


IX 


274.104 


5978.9 


% 


222.660 


3945.3 




248.579 


4917.2 


% 


274.497 


5996.0 


71. 


223.053 


3959.2 


M 


248.971 


4932.7 


/^ 


274.889 


6013.2 


H 


223.446 


3973.1 


% 


249.364 


4948.3 


KX 


275.282 


6030.4 


i 


223.838 


3987.1 


^ 


249.757 


4963.9 


M 


275.675 


6047.6 



112 



MATHEMATICAL TABLES. 



Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


87% 


276.067 


6064.9 


92. 


289.027 


6647.6 


96^ 


301.986 


7257.1 


88. 


276.460 


6082.1 


X 


289.419 


6665.7 


y 


302.378 


7276.0 


YB 


276.853 


6099.4 


M 


289.812 


6683.8 


% 


302.771 


7294.9 


1? 


277.246 


6116.7 


% 


290.205 


6701.9 


r 


303.164 


7313.8 


% 


277.638 


6134.1 


/^ 


290.597 


67^0.1 


% 


303.556 


7332.8 


/^3 


278.031 


6151.4 


% 


290.990 


6738.2 


M 


303.949 


7351.8 


% 


278.424 


6168.8 


M 


291.383 


6756 . 4 


% 


304.342 


7370.8 


M 


278.816 


6186.2 


sn 


291.775 


0774. 7 


97 


304.734 


7389.8 


7A 


279.209 


6203.7 


93 


292.168 


6792.9 


H 


305.127 


7408.9 


89. 


279.602 


6221 . 1 


H 


292.561 


6811.2 


M 


305.520 


7428.0 


H 


279.994 


6238.6 


H 


292.954 


6829.5 


% 


305.913 


7447.1 


Y4. 


280.387 


6256.1 


8 


293.316 


6847.8 


iz 


306.305 


7466.2 


g 


280.780 


6273.7 




293.739 


6866.1 


5^ 


306.698 


7485.3 




281.173 


6291.2 


% 


294.132 


6884.5 


H 


307.091 


7504 5 


% 


281.565 


6 08.8 


M 


294.524 


6902.9 


Vs 


307.483 


7523.7 


M 


281.958 


6326.4 


% 


294.917 


6921.3 


98 


307.876 


7543.0 


% 


282.351 


6344.1 


94. 


295.310 


6939.8 


YB 


308.269 


7562.2 


90, 


282.743 


6361.7 


^ 


295.702 


6958.2 


M 


308.661 


7581.5 


x^J 


283.136 


6379.4 


! 


296.095 


6976.7 


% 


309.054 


7600.8 


^4 


283.529 


6397.1 


% 


296.488 


6995.3 


/^ 


309.447 


7620.1 


% 


283.921 


6414.9 


Yi 


296.881 


7013.8 


5X 


309.840 


7639.5 


/^ 


284.314 


6432.6 


% 


297.273 


703^.4 


M 


310.232 


7658.9 


% 


284.707 


6450.4 


M 


297.666 


7051 .0 


% 


310.625 


7678.3 


% 


285.100 


6468.2 


*2 


298.059 


7069.6 


99.^ 


311.018 


7697.7 


To 


285.492 


6486.0 


95 


298.451 


7088.2 




311.410 


7717.1 


91 


285.885 


6503.9 




298.844 


7106.9 


IX 


311.803 


7736.6 


/^ 


286.278 


6521.8 


M 


299.237 


7125.6 


3X 


312.196 


7756.1 


M 


286.670 


6539.7 


% 


299.629 


7144.3 


\ 


312.588 


7775.6 


2 


287.063 


6557 . 6 


\^ 


300.022 


7163.0 


% 


312.981 


?795.2 


L 


287.456 


6575.5 


% 


300.415 


7181.8 


3/ 


313.374 


7814.8 


% 


287.848 


6593.5 


M 


300.807 


7200.6 


% 


313.767 


7834.4 


M 


288.241 


6611.5 


% 


301.200 


7219.4 


100. 


314.159 


7854.0 


I/B 


288.634 


6629.6 


96 


301.593 


7238.2 









DECIMALS OF A FOOT EQUIVALENT TO INCHES 
AND FRACTIONS OF AN INCH. 



Inches. 





Ys 


H 


% 


K 


YB 


H 


VB 








.01042 


.02083 


.03125 


.04167 


.05208 


.06250 


.07292 


1 


.0833 


.0938 


.1042 


.1146 


.1250 


.1354 


.1458 


.1563 


2 


.1667 


.1771 


.1875 


.1979 


.2083 


.2188 


.2292 


.2396 


3 


.2500 


.2604 


.2708 


.2813 


.2917 


.3021 


3125 


,3229 


4 


.3333 


.3438 


.3542 


.3646 


.3750 


.3854 


.3958 


.4063 


5 


.4167 


.4271 


.4375 


.4479 


.4583 


.4688 


.4793 


.4896 


6 


.5000 


.5104 


.5208 


.5313 


.5417 


.5521 


.5625 


.5729 


7 


.5833 


.5938 


.6042 


.6146 


.6250 


.6354 


.6458 


.6563 


8 


.6667 


.6771 


.6875 


.6979 


.7083 


.7188 


.7292 


.7396 


9 


.7500 


.7604 


.7708 


.7813 


.7917 


.8021 


.8125 


8229 


10 


.8333 


.8438 


.8542 


.8646 


.8750 


.8854 


.8958 


.9063 


11 


.9167 


.9271 


.9375 


.9479 


.9583 


.9688 


.9792 


.9896 



CIRCUMFERENCES OF CIRCLES. 



113 






M O O W CO O - 



Vf\?*NflCvx si -rx xx 

e$\r>XTH\l>\-N-Kl>\aSWS .. _ . _ 

>i iOOiOOJOOT-i-i lCOiOCOaOOr-ir-lCO^ 



h 



i e* o o* c* o QO i- -5P gp r-i -3" t^ o co o o eo CD OB g} *o GO i-t o oo e 3! fc: QS? 5r ff ^ 

T-.T-iT-i(?4^0<^5COCOrfT}i-riOiOOiO35COJ.-J.-J>QOOOQ00500JOO 





M JO & 30 O O T O 



rT ' 1 -" 

r^coio?oacooT-icoiooaDO' ' i ICC 



^ 

( O 




114 



MATHEMATICAL TABLES. 



LENGTHS OF CIRCULAR ARCS. 

(Degrees being given. Radius of Circle = 1 .) 

FORMULA. Length of arc - <OA X radius X number of degrees. 

loU 

RULE. Multiply the factor in table for any given number of degrees by 
the radius. 

EXAMPLE. Given a curve of a radius of 55 feet and an angle of 78 20'. 
What is the length of same in feet ? 

Factor from table for 78 1.3613568 

Factor from table for 20' .0058178 

Factor 1.3671746 

1.3671746 X 55 = 75.19 feet. 



Degrees. 



1 


.0174533 


61 


1.0646508 


121 


2.1118484 


1 


.0002909 


2 


.0349066 


62 


1.0821041 


122 


2.1293017 


2 


.0005818 


3 


.0523599 


63 


1.0995574 


123 


2.1467550 


3 


.0008727 


4 


.0698132 


64 


1.1170107 


124 


2.1642083 


4 


.0011636 


5 


.0872665 


65 


1.1344640 


125 


2.1816616 


5 


.0014544 


6 


.1047198 


66 


1.1519173 


126 


2.1991149 


6 


.0017453 


7 


.1221730 


67 


1.1693706 


127 


2.2165682 


7 


.0020362 


8 


.1396263 


68 


1.1868239 


128 


2.2340214 


8 


.0023271 


9 


.1570796 


69 


1.2042772 


129 


2.2514747 


9 


.0026180 


10 


.1745329 


70 


1.2217305 


130 


2.2689280 


10 


.0029089 


11 


.1919862 


71 


1.2391838 


131 


2.2863813 


11 


.0031998 


12 


.2094395 


72 


1.2566371 


132 


2.3038346 


12 


.0034907 


13 


.2268928 


73 


1.2740904 


133 


2.3212879 


13 


.0037815 


14 


.2443461 


74 


1.2915436 


134 


2.3387412 


14 


.0040724 


15 


.2617994 


75 


1.3089969 


135 


2.3561945 


15 


.0043633 


16 


.2792527 


76 


1.3264502 


136 


2.3736478 


16 


.0046542 


17 


.2967060 


77 


1.3439035 


137 


2.3911011 


17 


.0049451 


18 


.3141593 


78 


1.3613568 


138 


2.4085544 


18 


.0052360 


19 


.3316126 


79 


1.3788101 


139 


2.4260077 


19 


.0055269 


20 


.3490659 


80 


1.3962634 


140 


2.4434610 


20 


.0058178 


21 


.3665191 


81 


1.4137167 


141 


2.4609142 


21 


.0061087 


22 


.3839724 


82 


1.4311700 


142 


2.4783675 


22 


.0063995 


23 


.4014257 


83 


1.4486233 


143 


2.4958208 


23 


.0066904 


24 


.4188790 


84 


1.4660766 


144 


2.5132741 


24 


.0069813 


25 


.4363323 


85 


1.4835299 


145 


2.5307274 


25 


.0072722 


26 


.4537856 


86 


1.5009832 


146 


2.5481807 


26 


.0075631 


27 


.4712389 


87 


1.5184364 


147 


2.5656340 


27 


.0078540 


28 


.4886922 


88 


1.5358897 


148 


2.5830873 


28 


.0081449 


29 


.5061455 


89 


1.5533430 


149 


2.6005406 


29 


.0084358 


30 


.5235988 


90 


1.5707963 


150 


2.6179939 


30 


.0087266 


31 


.5410521 


91 


1.5882496 


151 


2.6354472 


31 


.0090175 


32 


.5585054 


92 


1.6057029 


152 


2.6529005 


32 


.0093084 


33 


.5759587 


93 


1.6231562 


153 


2.6703538 


33 


.0095993 


34 


.5934119 


94 


1.6406095 


154 


2.6878070 


34 


.0098902 


35 


.6108652 


95 


1.6580628 


155 


2.7052603 


35 


.0101811 


36 


.6283185 


96 


1.6755161 


156 


2.7227136 


36 


.0104720 


37 


.6457718 


97 


1.6929694 


157 


2.7401669 


37 


.0107629 


38 


.6632251 


98 


1.7104227 


158 


2.7576202 


38 


.0110538 


39 


.6806784 


99 


1.7278760 


159 


2.7750735 


39 


.0113446 


40 


.6981317 


100 


1.7453293 


160 


2.7925268 


40 


0116355 


41 


.7155850 


101 


1.7627825 


161 


2.8099801 


41 


.0119264 


42 


.7330383 


102 


1.7802358 


162 


2.8274334 


42 


.0122173 


43 


.7504916 


103 


1.7976891 


163 


2.8448867 


43 


.0125082 


44 


.7679449 


104 


1.8151424 


164 


2.8623400 


44 


.0127991 


45 


.7853982 


105 


1.8325957 


165 


2.8797933 


45 


.0130900 


46 


.8028515 


106 


1.8500490 


166 


2.8972466 


46 


.0133809 


47 


.8203047 


107 


1.8675023 


167 


2.9146999 


47 


.0136717 


48 


.8377580 


108 


1.8849556 


168 


2.9321531 


48 


.0139626 


49 


.8552113 


109 


1.9024089 


169 


2.9496064 


49 


.0142535 


50 


.8726646 


110 


1.9198622 


170 


2.9670597 


50 


.0145444 


51 


.8901179 


111 


1.9373155 


171 


2.9845130 


51 


.0148358 


52 


.9075712 


112 


1.9547688 


172 


3.0019663 


52 


.0151262 


53 


.9250245 


113 


1.9722-221 


173 


3.0194196 


53 


.0154171 


54 


.9424778 


114 


1.9896753 


174 


3.0368729 


54 


.0157080 


55 


.9599311 


115 


2.0071286 


175 


3.0543262 


55 


.0159989 


56 


.9773844 


116 


2.0245819 


176 


3.0717795 


56 


.0162897 


57 


.9948377 


117 


2 042o:;:.:> 


177 


3.0892328 


57 


.0165806 


58 


1.0122910 


118 


2.0594885 


178 


3.1066861 


58 


.0168715 


59 


1.0297443 


119 


2.0769418 


179 


3.1241394 


59 


.0171624 


60 


1.0471976 


120 


2.0943951 


180 


3.1415927 


60 


.0174533 



LENGTHS OF CIRCULAR ARCS. 



115 



LENGTHS OF CIRCULAR ARCS. 

(Diameter = 1. Given tlie Chord and Height of the Arc.) 

RULE FOR USE OP THE TABLE. Divide the height by the chord. Find in the 
column of heights the number equal to this quotient. Take out the corre- 
sponding number from the column of lengths. Multiply this last number 
by the length of the given chord; the product will be length of the arc. 

If the arc is greater than a semicircle, first find, the diameter from the 
formula, Diam. (square of half chord -*- rise) -f rise; the formula is true 
whether the arc exceeds a semicircle or not. Theji find the circumference. 
From the diameter subtract the given height of arc, the remainder will be 
height of the smaller arc of the circle; find its length according to the rule, 
arid subtract it from the circumference. 



Hgts. 


Lgths. 


Hgts. 


Lgths. 


Hgts. 


Lgths. 


Hgts. 


Lgths. 


Hgts. 


Lgths. 


.001 


1.00002 


.15 


1.05896 


.238 


1.14480 


.326 


1.26288 


.414 


1.40788 


.005 


1.00007 


.152 


1.06051 


.24 


1.14714 


.328 


1.26588 


.416 


1.41145 


.01 


1.00087 


.154 


1.06209 


.242 


1.14951 


.33 


1.26892 


.418 


1.41503 


.015 


1.00061 


.156 


1.06368 


.244 


1.15189 


.332 


1.27196 


.42 


1.41861 


.02 


1.00107 


.158 


1.06530 


.246 


1.15428 


.331 


1.27502 


.422 


1.42221 


.025 


1.00167 


.16 


1.06693 


.248 


1.15C70 


.336 


1.27810 


.424 


1.42583 


.03 


1.00240 


.162 


1.06858 


.25 


1.15912 


.338 


1.28118 


.426 


1.42945 


.035 


1.00327 


.164 


1.07025 


.252 


1.16156 


.34 


1.28428 


.428 


1.43309 


.04 


1.00426 


.166 


.07194 


.254 


1.16402 


.342 


1.28739 


.43 


1.43673 


.045 


1.00539 


.168 


.07365 


.256 


1.16650 


.344 


1.29052 


.432 


.44039 


.05 


1.00665 


.17 


.07537 


.258 


1.16899 


.346 


1.29366 


.434 


.44405 


.055 


1.00805 


.172 


.07711 


.26 


1.17150 


.348 


1.29681 


.436 


.44773 


.06 


1.00957 


.174 


.07888 


.262 


1.17403 


.35 


1.29997 


.438 


.45142 


.065 


1.01123 


.176 


.08066 


.264 


1.17657 


.352 


1.30315 


.44 


.45512 


.07 


1.01302 


.178 


.08246 


.266 


1.17912 


.354 


1.30634 


.442 


.45883 


.075 


1.01493 


.18 


1.08428 


.268 


1.18169 


.356 


1.30954 


.444 


.46255 


.08 


1.01698 


.182 


1.08611 


.27 


1.18429 


.358 


1.31276 


.446 


.46628 


.085 


1.01916 


.184 


1.08797 


.272 


1.18689 


.36 


1.31599 


.448 


.47002 


.09 


1.02146 


.186 


1.08984 


.274 


1.18951 


.362 


1.31923 


.45 


.47377 


.095 


1.02389 


.188 


1.09174 


.276 


1.19214 


.364 


1.32249 


.452 


.47753 


.10 


1.02646 


.19 


1.09365 


.278 


1.19479 


.366 


1.32577 


.454 


.48131 


.102 


1.02752 


.192 


1.09557 


.28 


1.19746 


.368 


1.32905 


.456 


.48509 


.104 


1.02860 


.194 


1.09752 


.282 


1.20014 


.37 


1.33234 


.458 


.48889 


.106 


1.02970 


.196 


1.09949 


.284 


1.20284 


.372 


1.33564 


.46 


.49269 


.108 


1.03082 


.198 


1.10147 


.286 


1.20555 


.374 


1.33896 


.462 


.49651 


.11 


1.03196 


.20 


1.10347 


.288 


1.20827 


.376 


1.34229 


.464 


.50033 


.112 


1.03312 


.202 


1.10548 


.29 


1.21102 


.378 


1.34563 


.466 


.50416 


.114 


1.03430 


.204 


1.10752 


.292 


1.21377 


.38 


1.34899 


.468 


.50800 


.116 


1.03551 


.206 


1.10958 


.294 


1.21654 


.382 


1.35237 


.47 


.51185 


.118 


1.03672 


.208 


1.11165 


.296 


1.21933 


.384 


1.35575 


.472 


.51571 


.12 


1.03797 


.21 


1.11374 


.298 


1.22213 


.386 


1.35914 


.474 


.51958 


,122 


1.03923 


.212 


1.11584 


.30 


1.22495 


.388 


1.36254 


.476 


.52346 


.124 


1.04051 


.214 


1.11796 


.302 


1.22778 


.39 


1.36596 


.478 


.52736 


.126 


1.04181 


.216 


1.12011 


.304 


1.23063 


.392 


1.36939 


.48 


.53126 


.128 


1.04313 


.218 


1.12225 


.306 


1.23349 


.394 


1.37283 


.482 


.53518 


.13 


1.04447 


.22 


1.12444 


.308 


1.23636 


.396 


1.37628 


.484 


.53910 


.132 


1.04584 


.222 


1.12664 


.31 


1.23926 


.398 


1.37974 


.486 


.54302 


.134 


1.04722 


.224 


1.12885 


.312 


1.24216 


.40 


1.38322 


.488 


.54696 


.136 


1.04862 


.226 


1.13108 


.314 


1.24507 


.402 


1.38671 


.49 


.55091 


.138 


1.05003 


.228 


1.13331 


.316 


1.24801 


.404 


1.39021 


.492 


.55487 


-14 


1.05147 


.23 


1.13557 


.318 


1.25095 


.406 


1.39372 


.494 


.55854 


.142 


1.05293 


.232 


1.13785 


.32 


1.25391 


.408 


1.39724 


.496 


.56282 


.144 


1.05441 


.234 


1.14015 


,322 


1.25689 


.41 


1.40077 


.498 


.56681 


146 


1.05591 


.236 


1.14247 


.324 


1.25988 


.412 


1.40432 


.50 


1.57080 


.148 


1.05743 



















116 



MATHEMATICAL TABLES. 



AREAS OF THE: SEGMENTS OF A 

(Diameter = 1; Rise or Height in parts of Diameter being 
given.) 

RULE FOR USB OF THE TABLE. Divide the rise or height of the segment by 
the diameter. Multiply the area in the table corresponding to the quotient, 
thus found by the square ot the diameter. 

If the segment exceeds a semicircle its area is area of circle area of seg 
ment whose rise is (diam. of circle rise of given segment) 

Given chord and rise, to find diameter. Diam = (square of half chord *- 
rise) ~\- rise The half chqrd is a mean proportional between the two parts 
into which the chord divides the diameter which is perpendicular to it. 



Rise 

-5- 

Diam. 


Area, 


Rise 
Diam 


Area 


Rise 

-5- 

Diam. 


Area. 


Rise 
Diam 


Area. 


Rise 
Diam 


Area. 


.001 


.00004 


.054 


.01646 


.107 


.04514 


.16 


.08111 


.213 


.12235 


.002 


.00012 


.055 


.01691 


.108 


.04576 


.161 


.08185 


.214 


.12317 


.003 


.00022 


.056 


.01737 


.109 


.04638 


.162 


.08258 


.215 


. 12399 


.004 


.00034 


.057 


.01783 


.11 


.04701 


.163 


.08332 


.216 


.12481 


.005 


.00047 


.058 


.01830 


.111 


.04763 


.164 


.08406 


.217 


.12563 


.006 


.00062 


.059 


.01877 


.112 


.04826 


.165 


.08480 


.218 


.12646 


.007 


.00078 


.06 


.01924 


.113 


.04889 


.166 


.08554 


.219 


.12729 


.008 


.00095 


.061 


.01972 


.114 


.04953 


.167 


.08629 


.22 


.12811 


.009 


.00113 


.062 


.02020 


.115 


.05016 


.168 


.08704 


.221 


.12894 


.01 


.00133 


.063 


.02068 


.116 


.05080 


.169 


.08779 


.222 


.12977 


.011 


.00153 


.064 


.02117 


.117 


.05145 


.17 


.08854 


.223 


.13060 


.012 


.00175 


.065 


.02166 


.118 


.05209 


.171 


.08929 


.224 


.13144 


.013 


.00197 


.066 


.02215 


.119 


.05274 


.172 


.09004 


.225 


.13227 


.014 


.0022 


.067 


,02265 


.12 


.05338 


.173 


.09080 


.226 


.13311 


.015 


.00244 


.068 


.02315 


.121 


.05404 


.174 


.09155 


.227 


.13395 


.016 


.00268 


.069 


.02366 


.122 


.05469 


.175 


.09231 


.228 


.13478 


.017 


.00294 


.07 


.02417 


.123 


.05535 


.176 


.09307 


.229 


.13562 


.018 


.0032 


.071 


.02468 


.124 


.05600 


.177 


.09384 


.23 


.13646 


.019 


.00347 


.072 


.02520 


.125 


.05666 


.178 


.09460 


.231 


.13731 


.02 


.00375 


.073 


.02571 


.126 


.05733 


.179 


.09537 


.232 


.13815 


.021 


.00403 


.074 


.02624 


.127 


.05799 


.18 


.09613 


.233 


.13900 


.02-2 


.00432 


.075 


.02676 


.128 


.05866 


.181 


.09690 


.234 


.13984 


.023 


.00462 


.076 


.02729 


.129 


.05933 


.182 


.09767 


.235 


.1406S 


.024 


.00492 


.077 


.02782 


.13 


.06000 


.183 


.09845 


.236 


.14154 


.025 


.00523 


.078 


.02836 


.131 


.06067 


.184 


.09922 


.237 


.14239 


.026 


.00555 


.079 


.02889 


.132 


.06135 


.185 


.10000 


.238 


.14324 


.027 


.00587 


.08 


.02943 


.133 


.06203 


.186 


.10077 


.239 


.14409 


.028 


.00619 


.081 


.02998 


.134 


.06271 


.187 


.10155 


.24 


.14494 


.029 


.00653 


.082 


.03053 


.135- 


.06339 


.188 


.10233 


.241 


.14580 


.03 


.00687 


.083 


.03108 


.136 


.06407 


.189 


.10312 


.242 


.14666 


.031 


.00721 


.084 


.03163 


.137 


.06476 


.19 


. 10390 


.243 


. 14751 


.032 


.00756 


.085 


.03219 


.138 


.06545 


.191 


.10469 


.244 


.14837 


.033 


.00791 


.086 


.03275 


.139 


.06614 


.192 


.10547 


.245 


.14923 


.034 


.00827 


.087 


.03331 


.14 


.06683 


.193 


.10626 


.246 


-.15009 


.035 


.00864 


.088 


.03387 


.141 


.06753 


.194 


.10705 


.247 


.15095 


,036 


.00901 


.089 


.03444 


.142 


.06822 


.195 


.10784 


.248 


.15182 


.037 


.00938 


.09 


.03501 


.143 


.06892 


.196 


.10864 


.249 


. 15268 


038 


.00976 


.091 


.03559 


.144 


.06963 


.197 


.10943 


.25 


.15355 


.039 


.01015 


.092 


.03616 


.145 


.07033 


.198 


.11023 


.251 


.1,5441 


.04 


.01054 


.093 


.03674 


.146 


.07103 


.199 


.11102 


.252 


.15528 


.041 


.01093 


.094 


.03732 


.147 


.07174 


.2 


.11182 


.253 


.15615 


.042 


.01133 


.095 


.03791 


.148 


.07245 


.201 


.11262 


.254 


.15702 


.043 


.01173 


.096 


.03850 


.149 


.07316 


.202 


.11343 


.255 


.15789 


.044 


.01214 


.097 


.03909 


.15 


.07'387 


.203 


.11423 


.256 


.15876 


.045 


.01255 


.098 


.03968 


.151 


.07459 


.204 


.11504 


.257 


.15964 


.046 


.01297 


.099 


.04028 


.152 


.07531 


.205 


.11584 


.258 


.16051 


.047 


.01339 


.1 


.04087 


.153 


.07603 


.206 


.11665 


.259 


.16139 


.048 


.01382 


.101 


.04148 


.154 


.07675 


.207 


.11746 


.26 


.16226 


.049 


.01425 


.102 


.04208 


.155 


.07747 


.208 


.11827 


.261 


.16314 


.05 


.01468 


.103 


.04269 


.156 


.07819 


.209 


.11908 


.262 


.16402 


.051 


.01512 


.104 


.04330 


.157 


.07892 


.21 


.11990 


.263 


.16490 


.052 


.01556 


.105 


.04391 


.158 


.07965 


.211 


.12071 


.264 


.16578 


.053 


.01601 


.106 


.04452 


.159 


.08038 


.212 


.12153 


.265 


.16666 



AREAS OF THE SEGMENTS OF A CIRCLE. 



117 



Rise 

-5- 

Diam 


Area. 


Rise 

-i- 
Diam. 


Area. 


Rise 

-r- 

Diara. 


Area. 


Rise 
-f- 
Diam. 


Area 


Rise 
Diam 


Area. 


.266 


.16755 


.313 


.21015 


.36 


.25455 


.407 


.30024 


.454 


.34676 


.267 


.16843 


.314 


.21108 


.361 


.25551 


.408 


.30122 


.455 


.34776 


.268 


.16932 


.315 


.21201 


.362 


.25647 


.409 


.30220 


.456 


.34876 


.269 


.17020 


.316 


.21294 


.363 


.25743 


.41 


.30319 


.457 


.34975 


.27 


.17109 


.317 


.21387 


.364 


.25839 


.411 


.30417 


.458 


.35075 


.271 


.17198 


.318 


.21480 


.365 


.25936 


.412 


.30516 


.459 


.35175 


272 


.17287 


.319 


.21573 


.366 


.26032 


.413 


.30614 


.46 


.35274 


i273 


.17376 


.32 


.21667 


.367 


.26128 


.414 


.30712 


.461 


.35374 


.274 


.17465 


.321 


.21760 


.368 


.26225 


.415 


.30811 


.462 


.35474 


.275 


.17554 


,32~i 


.21853 


.369 


.26321 


.416 


.30910 


.463 


.35573 


.276 


.17644 


.323 


.21947 


.37 


.26418 


,417 


.31008 


.4C4 


.35673 


.277 


.17733 


.324 


.22040 


.371 


.26514 


.418 


.31107 


.465 


.35773 


.278 


.17823 


.325 


,22134 


.372 


.26611 


.419 


.31205 


.466 


.35873 


.279 


.17912 


.326 


22228 


.373 


.26708 


.42 


.31304 


.467 


.35972 


.28 


.18002 


.327 


,22322 


.374 


.26805 


.421 


.31403 


.468 


.36072 


.281 


.18092 


.328 


.82415 


.375 


.26901 


.422 


.31502 


.469 


.36172 


.282 


.18182 


.329 


.22509 


.376 


.26998 


.423 


.31600 


.47 


.36272 


.283 


.18272 


.33 


.22603 


.377 


.27095 


.424 


.31699 


.471 


.36372 


.284 


.18362 


.331 


.22697 


.378 


.27192 


.425 


.31798 


.472 


.36471 


.285 


.18452 


.332 


.22792 


.379 


.27289 


.426 


.31897 


.473 


.36571 


.286 


.18542 


.333 


.28886 


.38 


.27386 


.427 


.31996 


.474 


.36671 


.287 


.18633 


.334 


.25J980 


.381 


.27483 


.428 


.32095 


.475 


.36771 


.288 


.18723 


.335 


.28074 


.382 


.27580 


.429 


.32194 


.476 


.36871 


.289 


.18814 


.336 


.23169 


.383 


.27678 


.43 


.32293 


.477 


.36971 


.29 


.18905 


.337 


.23263 


.384 


.27775 


.431 


.32392 


.478 


.37071 


.291 


.18996 


.338 


.23358 


.385 


.27872 


.432 


.32491 


.479 


.37171 


.292 


.19086 


.339 


.33453 


.386 


.27969 


.433 


.32590 


.48 


.37270 


.293 


.19177 


.34 


.513547 


.387 


.28067 


.434 


.32689 


.481 


.37370 


.294 


.19268 


.341 


.^8642 


.388 


.28164 


.435 


.32788 


.482 


.37470 


.295 


.19360 


.342 


.'23737 


.389 


.28262 


.436 


.32837 


.483 


.37570 


.296 


.19451 


.343 


,23832 


.39 


.28359 


.437 


.32987 


.484 


.37670 


.297 


.19542 


.344 


,,23927 


.391 


.28457 


.438 


.33086 


.485 


.37770 


.298 


.19634 


.345 


.4022 


.392 


.28554 


.439 


.33185 


.486 


.37870 


.299 


.19725 


.346 


.24117 


.393 


.28652 


.44 


.33284 


.487 


.37970 


.3 


.19817 


.347 


,24212 


.394 


.28750 


.441 


.33384 


.488 


.38070 


.301 


.19908 


.34S 


,24307 


.395 


.28848 


.442 


.33483 


.489 


.38170 


.302 


.20000 


.349 


,24403 


.396 


.28945 


.443 


.33582 


.49 


.38270 


.303 


.20092 


.35 


,24498 


.397 


.29043 


.444 


.33682 


.491 


.38370 


.304 


.20184 


.351 


.24593 


.398 


.29141 


.445 


.33781 


.492 


.38470 


.305 


.20276 


.352 


.24689 


.399 


.29239 


.446 


.33880 


.493 


.38570 


,306 


.20368 


,353 


.24784 


.4 


.29337 


.447 


.33980 


.494 


.38670 


.307 


.20460 


.854 


.24880 


.401 


.29435 


.448 


.34079 


.495 


.38770 


,308 


.20553 


855 


.24976 


.402 


.29533 


.449 


.34179 


.496 


.38870 


.309 


.20845 


.356 


.25071 


.403 


.29631 


.45 


.34278 


.497 


.38970 


.31 


.20738 


.357 


.25167 


.404 


.29729 


.451 


.34378 


.498 


.39070 


11 


.20830 


.358 


.25263 


.405 


.29827 


.452 


.34477 


.499 


.39170 


^312 


,99^3 


.359 


.25359 


.406 


.29926 


.453 


.34577 


.5 


.39270 



Fof rrttes for finding the area of a segment see Mensuration, page 59. 



MATHEMATICAL TABLES. 

SPHERES. 

(Some errors of 1 in the last figure only. From TRAUTWINE.) 



Diam. 


Sur- 
face. 


Vol- 
ume. 


Diam. 


Sur- 
face. 


Vol- 
ume. 


Diam. 


Sur- 
face. 


Vol- 
ume. 


1-32 


.00307 


.00002 


3 M 


33.183 


17.974 


9 Vs 


306.36 


504.21 


1-16 


.01227 


.00013 


5-16 


34.472 


19.031 


10. 


314.16 


523.60 


3-32 


.02761 


.00043 




35.784 


20.129 




322.06 


543.48 


t* 


.04909 
.07670 


.00102 
.00200 


7-16 

y* 


37.122 

38.484 


21.268 
22.449 


?! 


330.06 
338.16 


563.86 
584.74 


3-16 


.11045 


.00345 


9-16 


39.872 


23.674 


L 


346.36 


606.13 


7-32 


.15033 


.00548 


% 


41.283 


24.942 


% 


354.66 


628.04 




.19635 


.00818 


11-16 


42.719 


26.254 


4 


363.05 


650.46 


9-32 


.24851 


.01165 


M 


44.179 


27.611 


7 


371.54 


673.42 


5-16 


.30680 


.01598 


13-16 


45.664 


29.016 


11. 


380.13 


696.91 


11-32 


.37123 


.02127 


Vs 


47.173 


30.466 




388.83 


720.95 




.44179 


.02761 


15-16 


48.708 


31.965 


M 


397.61 


745.51 


13-32 


.51848 


.03511 


4. 


50.265 


33.510 


% 


406.49 


770.64 


7-16 


.60132 


.04385 




53.456 


36.751 


x^2 


415.48 


796.33 


15-32 


.69028 


.05393 


/4 


56.745 


40.195 


% 


424.50 


822.58 


L 


.78540 


.06545 


% 


60.133 


43.847 


54 


433.73 


849.40 


9-16 


.99403 


.09319 


i^ 


63.617 


47.713 


Xo 


443.01 


876.79 


% 


1.2272 


.12783 


% 


67.201 


51.801 


12. 


452.39 


904.78 


11-16 


1.4849 


.17014 


%: 


70.883 


56.116 


^ 


471.44 


962.52 




1.7671 


.22089 


% 


74.663 


60.663 


i^ 


490.87 


1022.7 


13-16 


2.0739 


.28084 


5. 


78.540 


65.450 


a/ 


510.71 


1085.3 


Vs 


2.4053 


.35077 




82.516 


70.482 


13. 


530.93 


1150.3 


15-16 


2.7611 


.43143 


M 


86.591 


75.757 




551.55 


1218.0 


1. 


3.1416 


.52360 


% 


90.763 


81.308 


i^ 


572.55 


1288.3 


1-16 


3.5466 


.62804 


Y& 


95.033 


87.113 


% 


593.95 


1361.2 




3.9761 


.74551 


5^ 


99.401 


93.189 


14. 


615.75 


1436.8 


3-16 


4.4301 


.87681 


M 


103.87 


99.541 




637.95 


1515.1 


M 


4.9088 


1.0227 


% 


108.44 


106.18 


Lj 


660.52 


1596.3 


5-16 


5.4119 


1.1839 


6. 


113.10 


113.10 


3^ 


683.49 


1680.3 




5.9396 


1.3611 




117.87 


120.31 


15. 


706 85 


1767.2 


7-16 


6.4919 


1.5553 


J4 


122.72 


127.83 


y* 


730.63 


1857.0 




7.0686 


1.7671 


% 


127.68 


135.66 




754.77 


1949.8 


9-16 


7.6699 


1.9974 


L 


132.73 


143.79 


% 


779.32 


2045.7 


% 


8.2957 


2.2468 


% 


137.89 


152.25 


16. 


804.25 


2144.7 


11-16 


8.9461 


2.5161 


M 


143.14 


161.03 




829.57 


2246.8 


H 


9.6211 


2.8062 


% 


148.49 


170.14 


L 


855.29 


2352.1 


13-16 


10.321 


3.1177 


7. 


153.94 


179.59 


ax 


881.42 


2460.6 


VB 


11.044 


3.4514 


K 


159.49 


189.39 


17. 


907.93 


2572.4 


15-16 


11.793 


3.8083 




165.13 


199.53 


/4 


934.83 


2687.6 


2. 


12.566 


4.1888 


% 


170.87 


210.03 


i^ 


962.12 


2806.2 


1-16 


13.364 


4.5939 


i 


176.71 


220.89 


ax 


989.80 


2928.2 




14.186 


5.0243 


% 


182.66 


232.13 


18. 


1017.9 


3053.6 


3-16 


15.033 


5.4809 


54 


188.69 


243.73 


/4 


1046.4 


3182.6 


14 


15.904 


5.9641 


% 


194.83 


255.72 


L/j 


1075.2 


3315.3 


5-16 


16.800 


6.4751 


8. 


201.06 


268.08 


% 


1104.5 


3451.5 




17.721 


7.0144 




207.39 


280.85 


19. 


1134.1 18681.4 


7-16 


18.666 


7.5829 


IX- 


213.82 


294.01 


i/ 


1164.2 J3735.0 


/^ 


19.635 


8.1813 


% 


220.36 


307.58 


Jl2 


1194.6 


3882.5 


9-16 


20.629 


8.8103 


L 


226.98 


321.56 


3 


1225.4 


4033.7 


% 


21.648 


9.4708 


R/. 


233.71 


335.95 


20. 


1256.7 


4188.8 


11-16 


22.691 


10.164 


3/ 


240.53 


350.77 


/4 


1288.3 


4347.8 




23.758 


10.889 


/o 


247.45 


360.02 


1Z 


1320.3 


4510.9 


13-16 


24.850 


11.649 


9. 


254.47 


381.70 


3^ 


1352.7 


4677.9 


% 


25.967 


12.443 




261.59 


397.83 


21. 


1385.5 


4849.1 


15-16 


27.109 


13.272 


/4 


268.81 


414.41 


/4 


1418.6 


5024.3 


3. 


28.274 


14.137 


% 


270.12 


431.44 


^ 


1452.2 


5203.7 


1-16 


29.465 


15.039 


Y% 283.53 


448.92 


% 


1486.2 


5387.4 


% 30.680 


15.979 


% 1291.04 


466.87 


22. 


1520.5 


5575.3 


3-16 .31.919 


16.957 


% i 298. 65 


485.31 


M 


1555.3 15767.6 



SPHERES. 
SPHERES (Continued.) 



119 



Diam. 


Sur- 
face. 


Vol- 
ume. 


Diam. 


Sur- 
face. 


Vol- 
ume 


Diam. 


Sur- 
face. 


Vol. 
ume. 


22 % 


159C.4 


5964.1 


40 54 


5153.1 


34783 


70 Yz 


15615 


183471 


n 


1626.0 


6165.2 


41. 


5281.1 


36087 


.71. 


15837 


187402 


23. 


1661.9 


6370.6 


54 


5410.7 


37423 


Yz 


16061 


191389 


/4 


1698.2 


6580.6 


42. 


5541.9 


38792 


72. 


16286 


195433 


54 


1735.0 


6795.2 


^ 


5674.5 


40194 


Yz 


16513 


199532 


M 


1772.1 


7014.3 


43. 


5808.8 


41630 


73. 


16742 


203689 


24. 


1809.6 


7238.2 


54 


5944.7 


43099 


Yz 


16972 


207903 


54 


1847.5 


7466.7 


44. 


6082.1 


44602 


74. 


17204 


212175 


/4 


1885.8 


7700.1 


M 


6221.2 


46141 


K 


17437 


216505 


% 


1924.4 


7938.3 


45. 


6361.7 


47713 


75. 


17672 


220894 


25. 


1963.5 


8181.3 


H 


6503.9 


49321 


y* 


17908 


225341 


54 


2002.9 


8429.2 


46. 


6647.6 


50965 


76. 


18146 


229848 


^2 


2042.8 


8682.0 


fcf 


6792.9 


52645 


Yz 


18386 


234414 


M 


2083.0 


8939.9 


47. 


6939.9 


54362 


77. 


18626 


239041 


26. 


2123.7 


9202.8 


^ 


7088.3 


56115 


Yz 


18869 


243728 


M 


2164.7 


9470.8 


48. 


7238.3 


57906 


78. 


19114 


248475 




2206.2 


9744.0 


K 


7389.9 


59734 


Y* 


19360 


253284 


M 


2248.0 


10022 


49. 


7543.1 


61601 


79. 


19607 


258155 


27. 


2290.2 


10306 


K 


7697.7 


63506 


54 


19856 


263088 


M 


2332.8 


10595 


50. 


7854.0 


65450 


80. 


20106 


268083 




2375.8 


10889 


K 


8011.8 


67433 


H 


20358 


273141 


M 


2419.2 


11189 


51. 


8171.2 


69456 


81. 


20612 


278263 


28. 


2463.0 


11494 


H 


8332.3 


71519 


54 


20867 


283447 


ix 


2507.2 


11805 


52. 


8494.8 


73622 


82. 


21124 


288696 


Jij 


2551.8 


12121 


54 


8658.9 


75767 


54 


21382 


294010 


M 


2596.7 


12443 


53. 


8824.8 


77952 


83. 


21642 


299388 


29. 


2642.1 


12770 


54 


8992.0 


80178 


54 


21904 


304831 


54 


2687.8 


13103 


54. 


9160.8 


82448 


84. 


22167 


310340 


/^ 


2734.0 


13442 


K 


9331.2 


84760 


54 


22432 


315915 


M 


2780.5 


13787 


55. 


9503.2 


87114 


85. 


22698 


321556 


30. 


2827.4 


14137 


54 


9676.8 


89511 


H 


22966 


327264 


54 


2874.8 


14494 


56. 


9852.0 


91953 


86. 


23235 


333039 


54 


2922.5 


14856 


54 


10029 


94438 


54 


23506 


338882 


a/ 


2970.6 


15224 


57. 


10207 


96967 


87. 


23779 


344792 


81. 


3019.1 


15599 


^ 


10387 


99541 


K 


24053 


350771 


^4 


3068.0 


15979 


58. 


10568 


102161 


88. 


24328 


356819 




3117.3 


16366 


54 


10751 


104826 


54 


24606 


362935 


34 


3166.9 


16758 


59. 


10936 


107536 


89. 


24885 


369122 


32. 


3217.0 


17157 


K 


11122 


110294 


54 


25165 


375378 


54 


3267.4 


17563 


60. 


11310 


113098 


90. 


25447 


381704 


% 


3318.3 


17974 


Y% 


11499 


115949 


54 


25730 


388102 


M 


3369.6 


18392 


61. 


11690 


118847 


91. 


26016 


394570 


33. 


3421.2 


18817 


54 


11882 


121794 


54 


26302 


401109 


ix 


3473.3 


19248 


62. 


12076 


124789 


92. 


26590 


407721 


Yz 


3525.7 


19685 


H 


12272 


127832 


54 


2(5880 


414405 


M 


3578.5 


20129 


63. 


12469 


130925 


93. 


27172 


421161 


34. 


3631.7 


20580 


H 


12668 


134067 


54 


27464 


427991 


54 


3685.3 


21037 


64. 


12868 


137259 


94. 


27759 


434894 


I/, 


3739.3 


21501 


54 


13070 


140501 


54 


28055 


441871 


35. " 


3848.5 


22449 


65. 


13273 


143794 


95. 


28353 


448920 


N 


3959.2 


23425 


54 


13478 


147138 


54 


28652 


456047 


36. 


4071.5 


24429 


66. 


13685 


150533 


96. 


28953 


463248 


34 


4185.5 


25461 


H 


13893 


153980 


54 


29255 


470524 


37. 


4300.9 


26522 


67. 


14103 


157480 


97. 


29559 


477874 


& 


4417.9 


27612 


54 


14314 


161032 


54 


29865 


485302 


38. 


4536.5 


28731 


68. 


14527 


164637 


98. 


30172 


492808 


54 


4656.7 


29880 


Ya 


14741 


168295 


54 


30481 


500388 


39. 


4778.4 


31059 


69. 


14957 


172007 


99. 


30791 


508047 


34 


4901.7 


32270 


^ 


15175 


175774 


54 


31103 


515785 


40. 


5026.5 


33510 


70. 


15394 


179595 


100. 


31416 


523598 



120 



MATHEMATICAL TABLES. 



CONTENTS IN CUBIC FEET AND U. S. GALLONS OF 
PIPES AND CYLINDERS OF VARIOUS DIAMETERS 
AND ONE FOOT IN LENGTH. 

1 gallon = 231 cubic inches. 1 cubic foot = 7.4805 gallons. 





For 1 Foot in 




For 1 Foot in 




For 1 Foot in 


a 


Length. 


jd 

t- . 


Length. 


a 


Length. 


Diameter 
Inches. 


Cubic Ft. 
also Area 
in Sq. Ft. 


U.S. 
Gals., 
231 
Cu. In. 


Diametei 
Inches 


Cubic Ft. 
also Area 
in Sq. Ft. 


U.S. 
Gals., 
231 
Cu. In. 


Diamete] 
Inches, 


Cubic Ft. 
also Area 
in Sq. Ft. 


U.S. 
Gals., 
231 
Cu. In. 


H 


.0003 


.0025 


fA 


.2485 


1.859 


19 


1.969 


14.73 


5-18 


.0005 


.004 




.2673 


1.999 


1014 


2.074 


15.51 


% 


.0008 


.0057 


714 


.28<57 


2.145 


20 


2.182 


16.32 


7^16 


.001 


.0078 




.3068 


2.295 


20^ 


2.292 


17.15 


H 


,0014 


.0102 


7% 


.3276 


2.45 


21 


2. -105 


17.99 


9-16 


.0017 


.0129 


8 


.3491 


2.611 


2H/2 


2.521 


18.86 


% 


.0021 


.0159 


8J4 


.3712 


2.777 


22 


8.640 


19.75 


11-16 


.00-20 


.0193 


gL 


.3941 


2'. 9 48 


221^ 


2.761 


20.UO 


H 


.0031 


.0230 


8% 


.4176 


3.125 


23 


2.885 


21.58 


13-16 


.0030 


.0269 


9 


.4418 


3.305 


23^ 


3.012 


22.53 


% 


.0042 


.0312 


9J4 


.4667 


3.491 


24 


3.142 


23.50 


15-16 


.0048 


.0359 


9^ 


.4922 


3.682 


25 


3.409 


25.50 


1 


.0055 


.0408 


9% 


.5185 


3.879 


26 


3.087 


27.58 




.0085 


.0638 


10 


.5454 


4.08 


27 


3.970 


29.74 


% 


.0123 


.0918 


IOM 


.5730 


4.286 


28 


4.276 


31.99 


1% 


.0167 


.1249 


10^ 


.6013 


4.498 


29 


4.587 


34.31 


2 4 


.0218 


.1632 


10% 


.6303 


4.715 


30 


4.909 


36.72 


2J4 


.0276 


.2066 


11 


.66 


4.937 


31 


5.241 


39.21 


giz 


.0341 


.2550 


11^4 


.6903 


5.164 


32 


5.585 


41.78 


2% 


.0412 


.3085 


11^ 


.7213 


5.396 


33 


5.940 


44.43 


3 


.0491 


.3672 


11% 


.7530 


5.633 


34 


6.305 


47.16 


314 


.0576 


.4309 


12 


.7854 


5.875 


35 


6.681 


49.98 




.0608 


.4998 


12*6 


.8522 


6.375 


36 


7.069 


52.88 


3M 


.0767 


.5788 


13 


.9218 


6.895 


37 


7.467 


55.86 


4 


.0873 


.6528 


13fc 


.994 


7.436 


38 


7.876 


58.92 


VA 


.0985 


.7369 


14 


1.069 


7.997 


39 


8.296 


62.06 


41 


.1104 


.8263 


14^ 


1 147 


8.578 


40 


8.727 


65.28 


4% 


.1231 


.9200 


15 


1.227 


9.180 


41 


9.168 


68.58 


5 


.1364 


1.020 


15J4 


1.310 


9.801 


42 


9.621 


71.97 


5^ 


.1503 


1.125 


16 


1.396 


10.44 


43 


10.085 


75.44 


5^ 


.1650 


1.234 


16J4 


1.485 


11.11 


44 


10.559 


78.99 


5% 


.1803 


1.349 


17 


1.576 


11.79 


45 


11.045 


82.62 


6 


.1963 


1.469 


17^ 


1.670 


12.49 


46 


11.541 


86.33 


6^ 


.2131 


1.594 


18 


1.788 


13.22 


47 


12.048 


90.13 


6^ 


.2304 


1.724 


18J* 


1.867 


13.96 


48 


12.566 


94.00 



To find the capacity of pipes greater than the largest given in the table, 
look in the table for a pipe of one half the given size, and multiply its capac- 
ity by 4; or one of one third its size, and multiply its capacity by 9, etc. 

To find the weight of water in any of the given sizes multiply the capacity 
in cubic feet by 62*4 or the gallons by 8^, or, if a closer approximation is 
required, by the weight of a cubic foot of water at the actual temperature in 
the pipe. 

Given the dimensions of a cylinder in inches, to find its capacity in U. S. 
gallons: Square the diameter, multiply by the length and by .0034. If d ~ 

diameter, I = length, gallons = d * X 54 - = .0034cW. 



CAPACITY OF CYLINDRICAL VESSELS. 



121 



CYLINDRICAL VESSELS, TANKS, CISTERNS, ETC. 

Diameter in Feet and Indies, Area in Square Feet, and 
U. S. Gallons Capacity for One Foot in Depth. 

1 gallon = 231 cubic inches = 1 C u b Ot = 0.13368 cubic feet. 



Diam. 


Area. 


Gals. 


Diam. 


Area. 


Gals. 


Diam. 


Area. 


Gals. 


Ft. In. 


Sq. ft. 


1 foot 
depth. 


Ft. In. 


Sq. ft. 


1 foot 
depth. 


Ft. In. 


Sq. ft. 


1 foot 
depth. 


1 


.785 


5.87 


5 8 


25.22 


188.66 


19 


283.53 


2120.9 


1 1 


.922 


6.89 


5 9 


25.97 


194.25 


19 3 


291.04 


2177.1 


2 


1.069 


8.00 


5 10 


26.73 


199.92 


19 6 


298.65 


2234.0 


' 3 


1.227 


9.18 


5 11 


27.49 


205.67 


19 9 


306.35 


2291.7 


4 


1.396 


10.44 


6 


28.27 


211.51 


20 


314.16 


2350.1 


5 


1.576 


11.79 


6 3 


30.68 


229.50 


20 3 


322.06 


2409.2 


6 


1.767 


13.22 


6 6 


33.18 


248.23 


20 6 


330.06 


2469.1 


7 


1.969 


14.73 


6 9 


35.78 


267.69 


20 9 


338.16 


2529.6 


8 


2.182 


16.32 


7 


38.48 


287.88 


21 


346.36 


2591.0 


9 


2.405 


17.99 


7 3 


41.28 


308.81 


21 3 


354.66 


2653.0 


10 


2.640 


19.75 


7 6 


44.18 


330.48 


21 6 


363.05 


2715.8 


11 


2.885 


21.58 


7 9 


47.17 


352.88 


21 9 


371.54 


2779.3 


2 


3.142 


23.50 


8 


50.27 


376.01 


22 


380.13 


2843.6 


2 1 


3.409 


25.50 


8 3 


53.46 


399.88 


22 3 


388.82 


2908.6 


2 2 


3.687 


27.58 


8 6 


56.75 


424.48 


22 6 


397.61 


2974.3 


2 3 


3.976 


29.74 


8 9 


60.13 


449.82 


22 9 


406.49 


3040.8 


2 4 


4.276 


31.99 


9 


63.62 


475.89 


23 


415.48 


3108.0 


2 5 


4.587 


3431 


9- 3 


6720 


502.70 


23 3 


424.56 


3175.9 


2 6 


4.909 


36.72 


9 6 


70.88 


53024 


23 6 


433.74 


3244.6 


2 7 


5.241 


39.21 


9 9 


74.66 


558.51 


23 9 


443.01 


33140 


2 8 


5.585 


41.78 


10 


78.54 


587.52 


24 


452.39 


3384.1 


2 9 


5.940 


44.43 


10 3 


82.52 


617.26 


24 3 


461.86 


3455.0 


2 10 


6.305 


47.16 


10 6 


86.59 


647.74 


24 6 


471.44 


3526.6 


2 11 


6.681 


49.98 


10 9 


90.76 


678.95 


24 9 


481.11 


3598.9 


3 


7.069 


52.88 


11 


95.03 


710.90 


25 


490.87 


3672.0 


3 1 


7.467 


55.86 


11 3 


99.40 


743.58 


25 3 


500.74 


3745.8 


3 2 


7.876 


58.92 


11 6 


103.87 


776.99 


25 6 


510.71 


38203 


3 3 


8.296 


62.06 


11 9 


108.43 


811.14 


25 9 


520.77 


3895.6 


3 4 


8.727 


65.28 


12 


113.10 


846.03 


26 


530.93 


3971.6 


3 5 


9.168 


68.58 


12 3 


117.86 


881.65 


26 3 


541.19 


4048.4 


3 6 


9.621 


71.97 


12 6 


122.72 


918.00 


26 6 


551.55 


4125.9 


3 7 


10.085 


75.44 


12 9 


127.68 


955.09 


26 9 


562.00 


4204.1 


3 8 


10.559 


78.99 


13 


132.73 


992.91 


27 


572.56 


4283.0 


3 9 


11.045 


82.62 


13 3 


137.89 


1031.5 


27 3 


583.21 


4362.7 


3 10 


11.541 


86.33 


13 6 


143.14 


1070.8 


27 6 


593.96 


4443.1 


3 11 


12.048 


90.13 


13 9 


148.49 


1110.8 


27 9 


604.81 


4524.3 


4 


12.566 


94.00 


14 


153.94 


1151.5 


28 


615.75 


4606.2 


4 1 


13.095 


97.96 


14 3 


159.48 


1193.0 


28 3 


626.80 


4688.8 


4 2 


13.685 


102.00 


14 6 


165.13 


1235.3 


28 6 


637.94 


4772.1 


4 3 


14.186 


106.12 


14 9 


170.87 


1278.2 


28 9 


649.18 


4856.2 


4 4 


14.748 


110.32 


15 


176.71 


1321.9 


29 


660.52 


4941.0 


4 5 


15.321 


114.61 


15 3 


182.65 


1366.4 


29 3 


671.96 


5026.6 


4 G 


15.90 


118.97 


15 6 


188.69 


1411.5 


29 6 


683.49 


5112.9 


4 7 


16.50 


123.42 


15 9 


194.83 


1457.4 


29 9 


695.13 


5199.9 


4 8 


17.10 


127.95 


16 


201.06 


1504.1 


30 


706.86 


5287.7 


4 9 


17.72 


132.56 


16 3 


207.39 


1551.4 


30 3 


718.69 


5376.2 


4 10 


18.35 


137.25 


16 6 


21382 


1599.5 


30 6 


730.62 


5465.4 


4 11 


18.99 


142.02 


16 9 


220.35 


1648.4 


30 9 


742.64 


5555.4 


ft 


19.63 


146.88 


17 


226.98 


1697.9 


31 


754.77 


5646.1 


5 1 


20.29 


151.82 


17 3 


233.71 


1748.2 


31 3 


766.99 


5737.5 


5 2 


20.97 


156.83 


17 6 


240.53 


1799.3 


31 6 


779.31 


5829.7 


5 3 


21.65 


161.93 


17 9 


247.45 


1851.1 


31 9 


791.73 


5922.6 


5 4 


22.34 


167.12 


18 


254.47 


1903.6 


32 


80425 


6016.2 


5 5 


23.04 


172.38 


18 3 


261.59 


1956.8 


32 3 


816.86 


6110.6 


5 6 


23.76 


177.72 


18 6 


268.80 


2010.8 


32 6 


829.58 


6205.7 


5 7 


24.48 


183.15 


18 9 


276.12 


2065.5 


32 9 


842.39 


6301.5 



122 



MATHEMATICAL TABLES. 



GALLONS AND CUBIC FEET. 
United States Gallons in a given Number of Cubic Feet. 

1 cubic foot = 7.480519 U. S. gallons; 1 gallon = 231 cu. in. = .13368056 cu. ft. 



Cubic Ft. 


Gallons. 


Cubic Ft. 


Gallons. 


Cubic Ft. 


Gallons. 


0.1 


0.75 


50 


374.0 


8,000 


59,844.2 


0.2 


1.50 


60 


448.8 


9,000 


67,324.7 


0.3 


2.24 


70 


523.6 


10,000 


74,805.2 


0.4 


2.99 


80 


598.4 


20,000 


149,610.4 


0.5 


3.74 


90 


673.2 


30,000 


224,415.6 


0.6 


4.49 


100 


748.0 


40,000 


299,220.8 


0.7 


5.24 


200 


1,496.1 


50,000 


374,025.9 


0.8 


5.98 


300 


2,244.2 


60,000 


448,831.1 


0.9 


6.73 


400 


2,992.2 


70,000 


523,636.3 


1 


7.48 


500 


3,740.3 


80,000 


598,441.5 


2 


14.96 


600 


4,488.3 


90,000 


673,246.7 


3 


22.44 


700 


5,236.4 


100,000 


748,051.9 


4 


29.92 


800 


5,984.4 


200,000 


1,496,103.8 


5 


37.40 


900 


6,732.5 


300,000 


2,244,155.7 


6 


44.88 


1,000 


7,480.5 


400,000 


2,992,207.6 


7 


52.36 


2,000 


14,961.0 


500,000 - 


3,740,259.5 


8 


59.84 


3,000 


22,441.6. 


600,000 


4,488,311.4 


9 


67.32 


4,000 


29,922.1 


700,000 


5,236,363.3 


10 


74.80 


5,000 


37,402.6 


800,000 


5,984,415.2 


20 


149.6 


6,000 


44,883.1 


900,000 


6,732,467.1 


30 


224.4 


7,000 


52,363.6 


1,000,000 


7,480,519.0 


40 


299.2 











Cubic Feet in a given Number of Gallons. 



Gallons. 


Cubic Ft. 


Gallons. 


Cubic Ft. 


Gallons. 


Cubic Ft. 


1 


.134 


1,000 


133.681 


1,000,000 


133,680.6 


2 


.267 


2,000 


267.361 


2,000,000 


267,361.1 


3 


.401 


3,000 


401.042 


3,000,000 


401,041.7 


4 


.535 


4,000 


534.722 


4,000,000 


534,722.2 


5 


.668 


5,000 


668.403 


5,000,000 


668,402.8 


6 


.802 


6,000 


802.083 


6,000,000 


802,083.3 


7 


.936 


7,000 


935.764 


7,000,000 


935,763.9 


8 


1.069 


8,000 


1,069.444 


8,000,000 


1,069,444.4 


9 


1.203 


9,000 


1,203.125 


9,000,000 


1,203,125.0 


10 


1.337 


10,000 


1,336.806 


10,000,000 


1,336,805.6 



NUMBER OF SQUARE FEET IK PLATES. 



123 



NUMBER OF SQUARE FEET IN PLATES 3 TO 32 
FEET LONG, AND 1 INCH WIDE. 

For other widths, multiply by the width in inches. 1 sq. in. .0069$ sq. ft. 



Ft. and 
In. 

Long. 


Ins. 
Long. 


Square 
Feet. 


Ft. and 
Ins. 
Long. 


Ins. 
Long. 


Square 
Feet. 


Ft. and 
Ins. 

Long. 


Ins. 
Long. 


Square 
Feet. 


8. 


36 


.25 


7.10 


94 


.6528 


13.8 


152 


.056 




37 


.2569 


11 


95 


.6597 


9 


153 


.063 


2 


38 


.2639 


8. 


96 


.6667 


10 


154 


.069 


3 


39 


.2708 


1 


97 


.6736 


11 


155 


.076 


4 


40 


.2778 


2 


98 


.6806 


13.0 


156 


.083 


5 


41 


.2847 


3 


99 


.6875 


1 


157 


.09 


6 


42 


.2917 


4 


100 


.6944 


2 


158 


.097 


7 


43 


.2986 


5 


101 


.7014 


3 


159 


.104 


8 


44 


.3056 


6 


102 


.7083 


4 


160 


.114 


9 


45 


.3125 




103 


.7153 


5 


161 


.118 


10 


46 


.3194 


8 


104 


.7222 


6 


162 


1.125 


11 


47 


.3264 


9 


105 


.7292 


7 


163 


1.132 


4. 


48 


.3333 


10 


106 


.7361 


8 


164 


1.139 


1 


49 


.3403 


11 


107 


.7431 


9 


165 


1.146 


2 


50 


.3472 


9. 


108 


.75 


10 


166 


1.153 


a 


51 


.3542 


1 


109 


.7569 


11 


167 


1.159 


4 . 


52 


.3611 


2 


110 


.7639 


14.0 


168 


1.167 


5 


53 


.3681 


3 


111 


.7708 


1 


169 


1.174 


6 


54 


.375 


4 


112 


.7778 


2 


170 


1.181 


7 


55 


.3819 


5 


113 


.7847 


3 


171 


1.188 


8 


56 


.3889 


6 


114 


.7917 


4 


172 


1.194 


9 


57 


.3958 


7 


115 


.7986 


5 


173 


1.201 


10 


58 


.4028 


8 


116 


.8056 


6 


174 


1.208 


11 


59 


.4097 


9 


117 


.8125 


7 


175 


1.215 


5. 


60 


.4167 


10 


118 


.8194 


8 


176 


1.222 


1 


61 


.4236 


11 


119 


.8264 


9 


177 


1.229 


2 


62 


.4306 


10.0 


120 


.8333 


10 


178 


1.236 


3 


63 


.4375 


1 


121 


.8403 


11 


179 


1.243 


4 


64 


.4444 


2 


122 


.8472 


15.0 


180 


1.25 


5 


65 


.4514 


3 


123 


.8542 


1 


181 


1.257 


6 


66 


.4583 


4 


124 


.8611 


2 


182 


1.264 


7 


67 


.4653 


5 


125 


.8681 


3 


183 


1.271 


8 


68 


.4722 


6 


126 


.875 


4 


184 


1.278 


9 


69 


.4792 


7 


127 


.8819 


5 


185 


1.285 


10 


70 


.4861 


8 


128 


.8889 


6 


186 


1.292 


11 


71 


.4931 


9 


129 


.8958 


7 


187 


1.299 


6. 


72 


.5 


10 


130 


.9028 


8 


188 


1.306 


1 


73 


.5069 


11 


131 


.9097 


9 


189 


1.313 


2 


74 


.5139 


11.0 


132 


.9167 


10 


190 


1.319 


3 


75 


.5208 


1 


133 


.9236 


11 


191 


1.326 


4 


76 


.5278 


2 


134 


.9306 


16.0 


192 


1.333 


5 


77 


.5347 


3 


135 


.9375 


1 


193 


1.34 


6 


78 


.5417 


4 


136 


.9444 


2 


194 


1.347 


7 


79 


.5486 


5 


137 


.9514 


3 


195 


1.354 


8 


80 


.5556 


6 


138 


.9583 


4 


196 


1 361 


9 


81 


.5625 


7 


139 


.9653 


5 


197 


1.368 


10 


82 


.5694 


8 


140 


.9722 


6 


198 


1.3T5 


11 


83 


.5764 


9 


141 


.9792 


7 


199 


1.382 


7. 


84 


.5834 


10 


142 


.9861 


8 


200 


1.389 


1 


85 


.5903 


11 


143 


.9931 


9 


201 


1.396 


2 


86 


.5972 


12.0 


144 


1.000 


10 


202 


1.403 


3 


87 


.6042 


1 


145 


1.007 


11 


203 


1.41 


4 


88 


.6111 


2 


146 


1.014 


17.0 


204 


1.417 . 


5 


89 


.6181 


3 


147 


1.021 


1 


205 


1.424 


6 


90 


.625 


4 


148 


1.028 


2 


206 


1.431 


7 


91 


.6319 


5 


149 


1.035 


3 


207 


1.438 


8 


92 


.6389 


6 


150 


1.042 


4 


208 


1.444 


9 


93 


.6458 


7 


151 


1.049 


5 


209 


1.451 



MATHEMATICAL TABLES. 



SQUARE: FEET IN 



Ft. and 
Ins. 
Long. 


Ins. 
Long. 


Square 
Feet. 


Ft. and 
Ins. 
Long. 


Ins. 
Long. 


Square 

Feet. 


Ft. and 
Ins. 
Long. 


Ins. 

Long. 


Square 
Feet. 


17.6 


210 


1.458 


22.5 


269 


1.868 


27.4 


328 


2.278 




211 


1.465 


6 


270 


1.875 


5 


329 


2.285 


8 


21 2 


1.472 


7 


271 


1.882 


6 


330 


2.292 


9 


213 


1.479 


8 


272 


1.889 


7 


331 


2.299 


10 


214 


1.486 


9 


273 


1.896 


8 


332 


2.306 


11 


215 


1.493 


10 


274 


1.903 


9 


3&3 


2.313 


18.0 


216 


1.5 


11 


275 


1.91 


10 


334 


2.319 


1 


217 


1.507 


38. 


276 


1.917 


11 


335 


2.326 


2 


218 


1.514 


1 


277 


1.924 


28.0 


336 


2.333 


3 


219 


1.521 


2 


278 


1.931 


1 


337 


2.34 


4 


220 


1.528 


3 


279 


1.938 


2 


3:38 


2.347 


5 


221 


1.535 


4 


280 


1.944 


3 


339 


2.354 


6 


222 


1.542 


5 


281 


1.951 


4 


340 


2.361 


7 


223 


1.549 


6 


282 


1.958 


5 


341 


2.368 


8 


224 


1.556 


7 


283 


1.965 


6 


342 


2.375 


9 


225 


1.563 


8 


284 


1.972 


7 


343 


2.382 





226 


1.569 


9 


285 


1.979 


8 


344 


2.389 


11 


227 


1.576 


10 


286 


1.986 


9 


345 


2.396 


19.0 


228 


1.583 


11 


287 


1.993 


10 


346 


2.403 


1 


229 


1.59 


24.0 


288 


2 


11 


347 


2.41 


2 


230 


1.597 


1 


289 


2.007 


29. 


348 


2.417 


3 


231 


1.604 


2 


290 


2.014 


1 


349 


2.424 


4 


232 


1.611 


3 


291 


2.021 


2 


350 


2.431 


5 


233 


1.618 


4 


292 


2 028 


3 


351 


2.438 


6 


234 


1.6-25 


5 


293 


2.035 


4 


352 


2.444 


7 


235 


1.632 


6 


294 


2.042 


5 


353 


2.451 


8 


236 


1.639 


7 


295 


2.049 


6 


354 


2.458 


9 


237 


1.645 


8 


296 


2.056 


7 


355 


2.465 


10 


238 


1.653 


9 


297 


2.0fi3 


8 


356 


2.472 


11 


239 


1.659 


10 


298 


2.069 


9 


357 


2.47d 


20.0 


240 


1.667 


11 


299 


2.076 


10 


358 


2.486 




241 


1.674 


25.0 


300 


2.083 


11 


359 


2.493 


2 


242 


1.681 


1 


301 


2.09 


30.0 


360 


2.5 


3 


243 


1.688 


2 


302 


2.097 


1 


361 


2.507 


4 


244 


1.694 


3 


303 


2.104 


2 


362 


2.514 


5 


245 


1.701 


4 


304 


2.111 


3 


363 


2.521 


6 


246 


1.708 


5 


305 


2.118 


4 


364 


2.528 


7 


247 


1.715 


6 


306 


2.125 


5 


365 


2.535 


8 


248 


1.722 


7 


307 


2.132 


6 


366 


2.542 


9 


249 


1.729 


8 


308 


2.139 


7 


367 


2.549 


10 


250 


1.736 


9 


309 


2.146 


8 


368 


2.556 


11 


251 


1.743 


10 


310 


2.153 


9 


369 


2.563 


21.0 


252 


1.75 


11 


311 


2.16 


10 


370 


2.569 


1 


253 


1 .757 


26.0 


312 


2.167 


11 


371 


2.576 


2 


254 


1.764 


1 


313 


2.174 


31.0 


372 


2.583 


3 


255 


1.771 


2 


314 


2.181 


1 


' 373 


2.59 


4 


256 


1.778 


3 


315 


2.188 


2 


374 


2.597 


5 


257 


1.785 


4 


316 


2.194 


3 


375 


2.604 


6 


258 


1.792 


5 


317 


2.201 


4 


376 


2.611 


7 


259 


1.799 


6 


318 


2.208 


5 


377 


2.618 


8 


260 


1.806 


7 


319 


2.215 


6 


378 


2.625 


9 


261 


1.813 


8 


3*0 


2.222 


7 


379 


2.632 


10 


262 


1.819 


9 


321 


2.229 


8 


380 


2.639 


11 


263 


1.826 


10 


322 


2.236 


9 


381 


2.646 


22.0 


264 


1.833 


11 


323 


2.243 


10 


382 


2.653 


1 


265 


1.84 


27.0 


324 


2.25 


11 


383 


2.66 


2 


266 


1.847 


1 


325 


2.257 


32. 


384 


2.667 


3 


267 


1.854 


2 


326 


2.264 


1 


385 


2.674 


4 


268 


1.861 


3 


327 


2.271 


2 


386 


2.681 



CAPACITY OF KECTAHGULAR TAHKS. 



125 



CAPACITIES OF RECTANGULAR TANKS IN U. S. 
GALLONS, FOR EACH FOOT IN DEPTH. 

1 cubic foot = 7.4805 U. S. gallons. 



Width 
of 
Tank. 


Length of Tank. 


feet. 
2 


ft. in. 
2 6 


feet. 
3 


ft. in. 
3 6 


feet. 
4 


ft. in. 
4 6 


feet. 
5 


ft. in. 
5 6 


feet. 
6 


ft. in. 
6 6 


feet. 

7 


ft, in. 
2 
2 6 
3 
3 6 
4 

4 6 

5 
5 6 
6 
6 6 

7 


29.92 


37.40 
46.75 


44.88 
56.10 
67.32 


52.36 
65.45 
78.54 
91.64 


59.84 
74.80 
89.77 
104.73 
119.69 


67.32 
84.16 
100.99 
117.82 
134.65 

151.48 


74.81 
93.51 
112.21 
130.91 
149.61 

168.31 
187.01 


82.29 
102.86 
123.43 
144.00 
164.57 

185.14 
205.71 

226.28 


89.7" 
112.21 
134.6J 
15701 
179.5; 

201.9' 
224.4 1 

246. 8( 
269.3( 


* 97.25 
121.56 
> 145.87 
) 170.18 
J 194.49 

* 218.80 
243.11 
) 267.43 
) 291.74 
316.05 


104.73 
130.91 
157.09 
183.27 
209.45 

235.63 
261.82 
288.00 
314.18 
340.36 

366.54 


































































































Width 
of 
Tank. 


Length of Tank. 


ft. in. 
7 6 


feet. 
8 


ft. in. 
8 6 


feet. 
9 


ft. in. 
9 6 


feet. 
10 


ft. in. 
10 6 


feet. 
11 


ft. in. 
11 6 


feet. 
12 


ft. in. 
2 
2 6 
3 
3 6 
4 

4 6 
5 
5 6 
6 
6 6 

7 3 
8 
8 6 
9 

9 6 
10 
10 6 
11 
11 6 

12 


112.21 
140.26 
168.31 
196.36 
224.41 

252.47 
280.52 
308.57 
336.62 
364.67 

392.72 

420.78 


119.69 
149.61 
179.53 
209.45 
239.37 

269.30 
299.22 
329.14 
359.06 

388.98 

418.91 
448.83 
478.75 


127.17 
158.96 
190.75 
222.54 
254.34 

286.13 
317.92 
349.71 
381.50 
413.30 

44509 
476.88 
508.67 
540.46 


134.65 
168.31 
202.97 
235.63 
269.30 

302.96 
336.62 
370.28 
403.94 
437.60 

471.27 
504.93 
538.59 
572.25 
605.92 


14213 
177.66 
213.19 
248.73 
284.26 

319.79 
355.32 
390.85 
426.39 
461.92 

497.45 
532.98 
568.51 
604.05 
639.58 

675.11 


149.61 
187.01 
22441 
261.82 
299.22 

336.62 
374.03 
411.43 
448.83 
486.23 

523.64 
561.04 
598.44 
635.84 
673.25 

710.65 
748.05 


157.09 
196.36 
235.63 
274.90 
314.18 

353.45 
392.72 
432.00 
471.27 
510.54 

549.81 
589.08 
628.36 
66763 
706.90 

746.17 
785.45 
824.73 


164.57 
205.71 
246.86 
288.00 
329.14 

370.28 
411.43 
452.57 
493.71 
534.85 

575.99 
617.14 
658.28 
699.42 
740.56 

781.71 
822.86 
864.00 
905.14 


172.05 

215.06 
258.07 
301.09 
344.10 

387.11 
430.13 
473.14 
516.15 
559.16 

602.18 
645.19 
688.20 
731.21 
774.23 

817.24 
860.26 
903.26 
946.27 
989.29 


179.53 
224.41 
269.30 
314.18 
359.06 

403.94 
448.83 
493.71 
538.59 
583.47 

628.36 
673.24 
718.12 
763.00 
807.89 

852.77 
897.66 
942.56 
987.43 
1032.3 

1077.2 





























































































126 



MATHEMATICAL TABLES. 



NUMBER OF BARRELS (31 1-2 GALLONS) IN 
CISTERNS AND TANKS. 



1 Barrel = 31^ gallons = 



31.5 X 231 



= 4.21094 cubic fret. Reciprocal = .237477. 



Depth 


Diameter in Feet. 


in 
Feet. 


5 


6 


7 


8 


9 


10 


11 


12 


13 


14 


1 


4.663 


6.714 


9.139 


11.93' 


' 15.108 


18.652 


22.569 


26.859 


31.522 


36.557 


5 


23.3 


33.6 


45.7 


59.7 


75.5 


93.3 


112.8 


134.3 


157.6 


182.8 


6 


28.0 


40.3 


54.8 


71.6 


90.6 


111.9 


135.4 


161.2 


189.1 


219 3 


7 


32.6 


47.0 


64.0 


83.6 


105.8 


130.6 


158.0 


188.0 


220.7 


255.9 


8 


37.3 


53.7 


73.1 


95.5 


120.9 


149.2 


180.6 


214.9 


252.2 


292.5 


9 


42.0 


60.4 


82.3 


107.4 


136.0 


167.9 


203.1 


241.7 


283.7 


329.0 


10 


46.6 


67.1 


91.4 


119.4 


151.1 


186.5 


225.7 


268.6 


315.2 


365.6 


11 


51.3 


73.9 


100.5 


131.3 


166.2 


205.2 


248 3 


295.4 


346.7 


402.1 


12 


56.0 


80.6 


109.7 


143.2 


181.3 


223.8 


270.8 


322.3 


378.3 


438.7 


13 


60.6 


87.3 


118.8 


155.2 


196.4 


242.5 


293.4 


349.2 


409.8 


475 2 


14 


65.3 


94.0 


127.9 


167.1 


211.5 


261.1 


316.0 


376.0 


441.3 


511.8 


15 


69.9 


100.7 


137.1 


179.1 


226.6 


289.8 


338.5 


402.9 


472.8 


548.4 


16 


74.6 


107.4 


146.2 


191.0 


241.7 


298.4 


361.1 


429.7 


504.4 


584.9 


17 


79.3 


114.1 


155.4 


202.9 


256.8 


317.1 


383.7 


456.6 


535.9 


621 .5 


18 


83.9 


120.9 


164.5 


214.9 


271.9 


335.7 


406.2 


483.5 


567.4 


658.0 


19 


88.6 


127.6 


173.6 


226.8 


287.1 


354.4 


428.8 


510.3 


598.9 


694.6 


20 


93 3 


134.3 


182.8 


238.7 


302.2 


373.0 


451.4 


537.2 


630.4 


731.1 


Depth 


Diameter in Feet. 


in 
Feet. 


15 


16 


17 


18 


19 


20 


21 


22 


1 


iijwc 


47.748 


53.903 


60.431 


67.332 


74.606 


82.253 


90.273 


5 


209.8 


238.7 


269.5 


302.2 


336.7 


373.0 


411.3 


451.4 


6 


251.8 


286.5 


323.4 


362 6 


404.0 


447.6 


493.5 


541.6 


7 


293.8 


334.2 


377.3 


423 


471.3 


522.2 


575.8 


631.9 


8 


335.7 


382.0 


431.2 


483.4 


538.7 


596.8 


658.0 


722.2 


9 


377.7 


429.7 


485.1 


543.9 


606.0 


671.5 


740.3 


812.5 


10 


419.7 


477.5 


539.0 


604.3 


673.3 


746.1 


822.5 


902.7 


11 


481.6 


525.2 


592.9 


664.7 


740.7 


820.7 


904.8 


993.0 


12 


503.6 


573.0 


646.8 


725.2 


808.0 


895.3 


987.0 


1083.3 


13 


545.6 


620.7 


700.7 


785.6 


875.3 


969.9 


1069.3 


1173.5 


14 


587.5 


668.5 


754.6 


846.0 


942.6 


1044.5 


1151.5 


1263.8 


15 


629.5 


716.2 


808.5 


906.5 


1010.0 


1119.1 


1233.8 


1354.1 


16 


671.5 


764.0 


862.4 


966.9 


1077.3 


1193.7 


1316.0 


1444.4 


17 


713.4 


811.7 


916.4 


1027.3 


1144.6 


1268.3 


1398.3 


1534.5 


18 


755.4 


859.5 


970.3 


1087.8 


1212.0 


1342.9 


1480.6 


1624.9 


19 


797.4 


907.2 


1024.2 


1148.2 


1279.3 


1417.5 


1562.8 


1715.2 


20 


839.3 


955.0 


1078.1 


1208.6 


1346.6 


1492.1 


1645.1 


1805.5 



LOGARITHMS. 



12? 



NUMBER OF BARRELS (31 1-2 GALLONS) IN 
CISTERNS AND TANKS. Continued. 



Depth 
in 
Feet. 


Diameter in Feet. 


23 


24 


25 


26 


27 


28 


29 


30 


1 


98.666 


107.432 


116.571 


126.083 


135.968 


146.226 


157.858 


167. 86S 


5 


493.3 


537.2 


582.9 


630.4 


679.8 


731.1 


784.3 


839.3 


6 


592.0 


644.6 


699.4 


756.5 


815.8 


877.4 


941.1 


1007.2 


7 


690.7 


752.0 


816.0 


882.6 


951.8 


1023.6 


1098.0 


1175 


8 


789.3 


859.5 


933.6 


1008.7 


1087.7 


1169.8 


1254 9 


1342.9 


9 


888.0 


966.9 


1049.1 


1134.7 


1223.7 


1316.0 


1411.7 


1510.8 


10 


986.7 


1074.3 


1165.7 


1260.8 


1359.7 


1462.2 


1568.6 


1678.6 


11 


1085.3 


1181.8 


1282.3 


1386.9 


1495.6 


1608.5 


1725.4 


1846.5 


1'2 


1184.0 


1289.2 


1398.8 


1513.0 


1631.6 


1754.7 


1882.3 


2014.4 


13 


1282.7 


1396.6 


1515.4 


1639.1 


1767.6 


1900.9 


2039.2 


2182.2 


14 


1381.3 


1504 


1632.0 


1765.2 


1903.6 


2047.2 


2196.0 


2350.1 


.15 


1480.0 


1611.5 


1748.6 


1891.2 


2039.5 


2193.4 


2352.9 


2517.9 


16 


1578.7 


1718.9 


1865.1 


2017.3 


2175.5 


2339.6 


2509.7 


2685.8 


17 


1677.3 


1826.3 


1981.7 


2143.4 


2311.5 


2485.8 


2666.6 


2853.7 


18 


1776.0 


1933.8 


2098.3 


2269.5 


2447.4 


2632.0 


2823.4 


3021.5 


19 


1874.7 


2041.2 


2214.8 


2395.6 


2583.4 


2778.3 


2980.3 


3189.4 


20 


1973.3 


2148.6 


2321.4 


2521.7 


2719.4 


2924.5 


3137.2 


3357.3 



LOGARITHMS. 

Logarithms (abbreviation log}. The log of a number is the exponent 
of the power to which it is necessary to raise a fixed number to produce the 
given number. The fixed number is called the base. Thus if the base is 10, 
the log of 1000 is 3, for 10 3 = 1000. There are two systems of logs in general 
use, the common, in which the base is 10, and the Naperian, or hyperbolic, 
in which the base is 2.718281828 .... The Naperian base is commonly de- 
noted by e, as in the equation e y = x, in which y is the Nap. log of x. 

In any system of logs, the log of 1 is 0; the log of the base, taken in that 
system, is 1. In any system the base of which is greater than 1, the logs of 
all numbers greater than 1 are positive and the logs of all numbers less than 
1 are negative. 

The modulus of any system is equal to the reciprocal of the Naperian log 
of the base of that system. The modulus of the Naperian system is 1, that 
of the common system is .4342945. 

The log of a number in any system equals the modulus of that system X 
the Naperian log of the number. 

The hyperbolic or Naperian log of any number equals the common log 
X 2.3025851. 

Every log consists of two parts, an entire part called the characteristic, or 
index, and the decimal part, or mantissa. The mantissa only is given in the 
usual tables of common logs, with the decimal point omitted. The charac- 
teristic is found by a simple rule, viz., it is one less than the number of 
figures to the left of the decimal point in the number whose log is to be 
found. Thus the characteristic of numbers from 1 to 9.99 + is 0, from 10 to 
99.99 + is 1, from 100 to 999 + is 2, from .1 to .99 -f is - 1, from .01 to .099 -** 
is - 2, etc. Thus 






log of 2000 is 3.30103; 
" " 200 " 2.30103; 
' " 20 " 1.30103; 
* " 2 " 0.30103; 



log of 



.2 is - 1.30103; 

.02 " - 2.30103; 

.002 " - 3.30103; 

.0002 " - 4.30103. 



MATHEMATICAL TABLES. 

The minus sign is frequently written above the characteristic thus : 
log .002 = 3 .30103. The characteristic only is negative, the decimal part, or 
mantissa, being always positive. 

When a log consists of a negative index and a positive mantissa, it is usual 
to write the negative sign over the index, or else to add 10 to the index, and 
to indicate the subtraction of 10 from the resulting logarithm. 

Thus log .2 = Y- 3 9 10 3. and this may be written 9.30103 - 10. 

In tables of logarithmic sines, etc., the 10 is generally omitted, as being 
understood. 

Rules for use of the table of Logarithms. To find the 
log of any whole number. For 1 to 100 inclusive the log is given 
complete in the small table on page 129. 

For 100 to 999 inclusive the decimal part of the log is given opposite the 
given number in the column headed in the table (including the two figures 
to the left, making six figures). Prefix the characteristic, or index, 2. 

For 1000 to 9999 inclusive : The last four figures of the log are found 
opposite the first three figures of the given number and in the vertical 
column headed with the fourth figure of the given number ; prefix the two 
figures under column 0, and the index, which is 3. 

For numbers over 10,000 having five or more digits : Find the decimal part 
pf the log for the first four digits as above, multiply the difference figure 
in the last column by the remaining digit or digits, and divide by 10 if there 
be only one digit more, by 100 if there be two more, and so on ; add the 
quotient to the log of the first four digits and prefix the index, which is 4 
if there are five digits, 5 if there are six digits, and so on. The table of pro- 
portional parts may be used, as shown below. 

To find the log of a decimal fraction or of a whole 
number and a decimal. First find the log of the quantity as if there 
were no decimal point, then prefix the index according to rule ; the index is 
one less than the number of figures to the left of the decimal point. 

Required log of 3.141593. 

log of 3.141 =0.497068. Diff. = 138 

From proportional parts 5 = 690 

09 = 1242 

** " * 003 = 041 



log 3.141593 0.4971498 

To find the number corresponding to a given log. Find 

in the table the log nearest to the decimal part of the given log and take the 
first four digits of the required number from the column N and the top or 
foot of the column containing the log which is the next less than the given 
log. To find the 5th and 6th digits subtract the log in the table from the 
given log, multiply the difference by 100, and divide by the figure in the 
Diff. column opposite the log ; annex the quotient to the four digits already 
found, and place the decimal point according to the rule ; the number *l 
figures to the left of the decimal point is one greater than the index. 

Find number corresponding to the log 0.497150 

Next lowest log in table corresponds to 3141 497068 

Diff. = 82 

Tabular diff. = 138; 82 -* 138 = .59 -f 

The Index being 0, the number is therefore 3.14159 -f. 

To multiply two numbers by the use of logarithms,-- 

Add together the logs of the two numbers, and find the number whose log 
is the sum. 

To divide two numbers. Subtract the log of the divisor from 
the log of the dividend, and find the number whose log is the difference. 

To raise a number to any given power. Multiply the log of 
the number by the exponent of the power, and find the number whose log in 
the product. 

To find any root of a given number. Divide the log of the 
Dumber by the index of the root. The quotient is the log of the root. 

To find the reciprocal of a number. -Subtract the decimal 
part of the log of the number from 0, add 1 to the index and change the sign 
pf, the index. The result is the log of the reciprocal. 



LOGARITHMS. 



129 



Required the reciprocal of 3.141593. 

Log of 3.141593, as found above 0.4971498 

Subtract decimal part from gives 0.5028502 

Add 1 to the index, and changing sign of the index gives.. T.5028502 
which is the log of 0.31831. 

To find the fourth term of a proportion by logarithms. 
Add the logarithms of tJ*e second and third terms, and from their sum 
subtract the logarithm of the first term. .. 

When one logarithm is to be subtracted from another, it may be more 
convenient to convert the subtraction into an addition, which may be done 
by first subtracting tLo given logarithm from 10, adding the difference to the 
other logarithm, and afterwards rejecting the 10. 

The difference between a given logarithm and 10 is called its arithmetical 
complement, or cologarithm. 

To subtract one logarithm from another is the same as to add its comple- 
ment and then reject 10 from the result. For a b = 10 b + a, 10. 

To work a proportion, then, by logarithms, add the complement of the 
logarithm of the first term to the logarithms of the second and third terms. 
The characteristic must afterwards be diminished by 10. 

Example In logarithms with a negative Index. Solve by 
7686V" 



logarithms 



\101l7 



, which means divide 526 by 1011 and raise the quotient 



to the 2.45 power. 



log 526 = 
log 1011 = 



2.720986 
3.004751 



Jog of quotient = - 1.716235 
Multiply by 2.45 



- 2.581175 
- 2.8 64940 

- 1.43 2470 

- 1.30 477575 = .20173, Ans. 



In multiplying - 1.7 by 5, we say: 5 x 7 - 35, 3 to carry; 5 x 1 = 5 less 
4- 3 carried = 2. In adding -2-f-8-f3-fl carried from previous column, 
we say: 1 4- 3 + 8 = 12, minus 2 = 10, set down and carry 1; 1 -f 4 2 = 3. 

LOGARITHMS OF NUMBERS FROM 1 TO 100. 



N. 


Log. 


N. 


Log. 


N. 


Log. 


N. 


Log. 


N. 


Log. 


1 


0.000000 


21 


1.322219 


41 


1.612784 


61 


1.785330 


81 


1.908485 


2 


0.301030 


22 


1.342423 


42 


1.623249 


62 


1.792392 


82 


1.913814 


3 


0.477121 


23 


1.361728 


43 


1.633468 


63 


1.799341 


83 


1.919078 


4 


0.602060 


24 


1.380211 


44 


1.643453 


64 


1.806180 


84 


1.924279 


5 


0.698970 


25 


1.397940 


45 


1.653213 


65 


1.812913 


85 


1.929419 


6 


0.778151 


26 


1.414973 


46 


1.662758 


66 


1.819544 


86 


1.934498 


7 


0.845098 


27 


1.431364 


47 


1.672098 


67 


1.826075 


87 


1.939519 


8 


0.903090 


28 


1.447158 


48 


1.681241 


68 


1.832509 


88 


1.944483 


9 


0.954243 


29 


1.462398 


49 


1.690196 


69 


1.838849 


89 


1.949390 


10 


1.000000 


30 


1.477121 


50 


1.698970 


70 


1.845098 


90 


1.954243 


11 


1.041393 


31 


1.491362 


51 


1.707570 


71 


1.851258 


91 


1.959041 


12 


1.079181 


32 


1.505150 


52 


1.716003 


72 


1.857332 


92 


1.963788 


13 


1.113943 


33 


1.518514 


53 


1.724276 


73 


1.863323 


93 


1.968483 


14 


1.146128 


34 


1.531479 


54 


1.732394 


74 


1.869232 


94 


1.973128 


15 


1.176091 


35 


1.544068 


55 


1.740363 


75 


1.875061 


95 


1.977724 


16 


1.204120 


36 


1.556303 


56 


1.748188 


76 


1.880814 


96 


1.982271 


17 


1.230449 


37 


1.568202 


57 


1.755875 


77 


1.886491 


97 


1.986772 


18 


1.255273 


38 


1.579784 


58 


1.763428 


78 


1.892095 


98 


1.991226 


19 


1.278754 


39 


1.591065 


59 


1.770852 


79 


1.897627 


99 


1.995635 


20 


1.301030 


40 


1.602060 


60 


1.778151 


80 


1.903090 


100 


2.000000 



LOGARITHMS OF LUMBERS. 



No. 


100 L. 000.] 








[No. lu9 L. 040. 


*: 





1 


2 


8 4 


6 


6 


7 


8 


9 


Diff. 


100 


000000 


0434 


0868 


1301 1734 


2166 


2598 


3029 


3461 


3891 


432 


i 


4321 


4751 


5,181 


5609 6038 


6466 


6894 


7321 


7748 


8174 


438 




8600 


9026 


9451 


9876 






















0300 


0724 


1147 


1570 


1993 


2415 


/KM 


3 


012837 


3259 


3680 


4100 4521 


4940 


5360 


5779 


6197 


6616 


4K4 

420 


4 


7033 


7451 


7868 


8284 8700 


9116 


9532 


9947 
























0361 


0775 


. , _ 


5 


021189 


1603 


2016 


2428 2841 


3252 


3664 


4075 


4486 


4896 


412 


6 


5306 


5715 


6125 


6533 6942 


7350 


7757 


8164 


8571 


8978 


408 


7 


9384 


9789 
























0195 


0600 1004 


1408 


1812 


2216 


261S 


3021 


4A1 


8 


033424 


3826 


4227 


4628 5029 


5430 


5830 


6230 


6629 


7028 


400 


9 


7426 


7825 


8223 


8620 9017 


9414 


9811 












04 












0207 


0602 


0998 


397 


PROPORTIONAL PARTS. 


Diff. 


1 


2 




3 


4 


5 


6 


7 


8 


9 


434 




43.4 


86.8 




130.2 


173.6 


217.0 


260.4 


3( 


)3.8 


347.2 


390.6 


433 




43.3 


86.6 




129.9 


173.2 


216.5 


259 


8 


at 


)3.1 


346.4 


389.7 


432 




43.2 


86.4 




12 


3.6 


172.8 


216.0 


259 


2 


3( 


)2.4 


345.6 


388.8 


431 




43.1 


86.2 




129.3 


172.4 


215.5 


258 


6 


301.7 


344.8 


387.9 


430 




43.0 


86.0 




129.0 


172.0 


215.0 


258.0 


301.0 


344.0 


387.0 


429 




42.9 


85.8 




12 


8.7 


171.6 


214.5 


257 


4 


3( 


K).3 


343.2 


386.1 


428 




42.8 


85.6 




128.4 


171.2 


214.0 


256 


8 


2 ( . 


)9.6 


342.4 


385.2 


427 




42.7 


85.4 




128.1 


170.8 


213.5 


256.2 


21 


)8.9 


341.6 


384.3 


426 




42.6 


85.2 




127.8 


170.4 


213.0 


255.6 


298.2 


340.8 


383.4 


425 




42.5 


85.0 




127.5 


170.0 


212.5 


255 





297.5 


340.0 


382.5 


424 




42.4 


84.8 




127 2 


169.6 


212.0 


254 


4 


296.8 


339.2 


381.6 


423 




42.3 


84.6 




12 


6.9 


169.2 


211.5 


253 


8 


2J 


)6.1 


338.4 


380.7 


422 




42.2 


84.4 




126.6 


168.8 


211.0 


253.2 


295.4 


337.6 


379.8 


421 




42:1 


84.2 




126.3 


168.4 


210.5 


252.6 


294.7 


336.8 


378.9 


420 




42.0 


84.0 




12 


6.0 


168.0 


210.0 


252 





% 


)4.0 


336.0 


378.0 


419 




43.9 


83.8 




125.7 


167.6 


209.5 


251 


4 


20*. 3 


335.2 


377.1 


418 




41.8 


83.6 




12 


5.4 


167.2 


209.0 


250 


8 


2< 


)2.6 


334.4 


37'6.2 


417 




41.7 


as. 4 




125.1 


166.8 


208.5 


250 


8 


291.9 


333.6 


375.3 


416 




41.6 


83.2 


124.8 


166.4 


208.0 


249.6 


291.2 


332.8 


374.4 


415 


41.5 


83.0 


124.5 


166.0 


207.5 


249.0 


290,5 


332.0 


373.5 


414 




41.4 


82.8 


124.2 


165.6 


207.0 


248 


4 


289.8 


331.2 


372.6 


413 




41.3 


82.6 




12 


3.9 


165.2 


206.5 


247 


8 


2 


39.1 


330.4 


371.7 


412 




41.2 


82.4 




12 


3.6 


164.8 


206.0 


247 


2 


2 


38.4 


329.6 


370.8 


411 




41.1 


82.2 


123.3 


164.4 


205.5 


246 


6 


287.7 


328.8 


309. 9 


410 


41.0 


82.0 


123.0 


164.0 


205.0 


246.0 


21 


37.0 


328.0 


369.0 


409 


40.9 


81.8 


122.7 


163.6 


204.5 


245 


.4 


286.3 


327.2 


368.1 


408 




40.8 


81.6 




12 


2.4 


163.2 


204.0 


244 


.8 


21 


35.6 


326.4 


367.2 


407 


40.7 


81.4 


122.1 


162.8 


203.5 


244 


.2 


284.9 


325. 


366.3 


406 




40.6 


81.2 




12 


1.8 


162.4 


203.0 


243 


6 


2! 


34.2 


324.8 


365.4 


405 


40.5 


81.0 


121.5 


162.0 


202.5 


243.0 


2 


33.5 


324.0 


364.5 


404 


40.4 


80.8 


121.2 


161.6 


202.0 


242 


.4 


282.8 


323.2 


363.6 


403 


40.3 


so.e 




120.9 


161.2 


201.5 


241 


.8 


282.1 


322.4 


362.7 


40$ 




40.2 


80.4 


I 


1$ 


!0.6 


160.8 


201.0 


241 


2 


21 


31.4 


321.6 


361.8 


401 


40.1 


80.2 


120.3 


160.4 


200.5 


240 


.6 


280.7 


320.8 


360.9 


400 


40.0 


80-0 


120.0 


160.0 


200.0 


240 


.0 


280.0 


320.0 


360.0 


39< 


1 


39.9 


79 


J 


11 


9.7 


159.6 


199.5 


239 


.4 


2 1 


79.3 


319.2 


359.1 


39* 


\ 


39.8 


79!6 


119.4 


159.2 


199.0 


238.8 


278.6 


318.4 


358.2 


39 r 


r 


39.7 


79.^ 


( 


11 


9.1 


158.8 


198.5 


238 


.2 


2 


77.9 


317.6 


357.3 


396 


39.6 


79. $ 


> 


118.8 


158.4 


198.0 


237 


.6 


2 


77.2 


316.8 


356.4 


39 


. 39.5 


79.0 118.5 


158.0 


197.5 237.0 276.5 316.0 355.5 



LOGARITHMS OF NUMBERS. 



No. 110 L. 041.] 








[No. 119 L. 078. 


N. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


110 041393 


1787 


2182 


2576 


2969 


3362 


3755 


4148 


4540 


4932 


393 


1 5323 


5714 


6105 


6495 


6885 


7275 


7664 


8053 


8442 


8830 


390 


2 9218 


9606 


9993 
























0380 


0766 


1153 


1538 


1924 


2309 


2694 


OQC 


3 053078 


3463 


3846 


4230 


4613 


4996 


5378 


5760 


6142 


6524 


oOD 

383 


4 6905 


7286 


7666 


8046 


8428 


8805 


9185 


9563 


9942 
























0320 


379 


5 060698 


1075 


1452 


1829 


2206 


2582 


2958 


sass 


3709 


4083 


376 


6 4458 


4832 


5206 


5580 


5953 


6326 


6699 


7071 


7443 


7815 


373 


7 8186 


8557 


8928 


9298 


9668 
























0038 


0407 


0776 


1145 


1514 


370 


8 071862 


2250 


2617 


2985 


3352 


3718 


4085 


4451 


4816 


5182 


366 


9 5547 


5912 


6276 


6640 


7004 


7368 


7731 


8094 


8457 


8819 


363 


PROPORTIONAL PARTS. 


Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


395 
394 


39.5 
39.4 


79.0 

78.8 


118.5 
118.2 


158.0 
157.6 


197.5 
197.0 


237.0 
236.4 


276.5 
275.8 


316.0 
815.2 


355.5 
354.6 


393 


39.3 


78.6 




11 


7.9 


157.2 


196.5 


235 


.8 


2 


75.1 


314.4 


353.7 


392 


39.2 


78.4 


11 


7.6 


156.8 


196.0 


235.2 


274.4 


313.6 


352.8 


391 


39.1 


78.2 


117.3 


156.4 


195.5 


234 


.0 


273.7 


312.8 


351.9 


390 


39.0 


78.0 




11 


7.0 


156.0 


195.0 


234 


.0 


2 


73.0 


312.0 


351.0 


389 


38.9 


77.8 


116.7 


155.6 


194.5 


233 


.4 


272.3 


311.2 


350.1 


388 


38.8 


77.6 




11 


6.4 


155.2 


194.0 


232 


.8 


2 


71.6 


310.4 


349.2 


387 


38.7 


77.4 


116.1 


154.8 


193.5 


232.2 


270.9 


309.6 


348.3 


386 


38.6 


77.2 




11 


5.8 


154.4 


198.0 


231 


.e 


2 


70.2 


308.8 


347.4 


385 


38.5 


77.0 


115.5 


154.0 


192.5 


231 


.0 


269.5 


308.0 


346.5 


384 


38.4 


76.8 


115.2 


153.6 


192.0 


230.4 


268.8 


307.2 


345.6 


383 


38.3 


76.6 


114.9 


153.2 


191.5 


229.8 


2 


68.1 


306.4 


344.7 


382 


38.2 


76.4 


[ 


11 


4.6 


152.8 


191.0 


228 


.2 


2 


67.4 


305.6 


343.8 


381 


38.1 


76.2 


114.3 


152.4 


190.5 


228.6 


266.7 


304.8 


342.9 


380 


38.0 


76.0 


) 


11 


4.0 


152.0 


190.0 


228 


.0 


2 


66.0 


304.0 


342.0 


379 


37.9 


75. 


! 


11 


3.7 


151.6 


189.5 


227 


.4 


2 


65.3 


303.2 


341.1 


378 


37.8 


75.6 


113.4 


151.2 


189.0 


226.8 


264.6 


302.4 


340.2 


377 


37.7 


75.4 


1 


11 


3.1 


150.8 


188.5 


226 


.2 


2 


63.9 


301.6 


339.3 


376 


37.6 


75.2 


112.8 


150.4 


188.0 


225.6 


263.2 


300.8 


338.4 


375 


37.5 


75.0 


112.5 


150.0 


187.5 


225.0 


I 


62.5 


300.0 


337.5 


374 


37.4 


74.8 


112.2 


149.6 


187.0 


224.4 


261.8 


299.2 


336! 6 


373 


37.3 


74. ( 


1 


11 


1.9 


149.2 


186.5 


22? 


.8 


2 


61.1 


298.4 


335.7 


372 


37.2 


74.4 


111.6 


148.8 


186.0 


22c 


.2 


260.4 


297.6 


334.8 


371 


37.1 


74. $ 


J 


11 


1.3 


148.4 


185.5 


22$ 


.6 


2 


59.7 


296.8 


333.9 


370 


37.0 


74.0 


111.0 


148.0 


185.0 


22$ 


.0 


259.0 


296.0 


333.0 


369 


36.9 


73.* 


J 


11 


0.7 


147-.6 


184.5 


221 


.4 


2 


58.3 


295.2 


332.1 


368 


36.8 


73.6 


110.4 


147.2 


184.0 


220.8 


257.6 


294.4 


331.2 


367 


36.7 


73.' 


1 


11 


LO.l 


146.8 


183.5 


22C 


).2 


i 


56.9 


293.6 


830.3 


366 


36.6 


73.2 


109.8 


146.4 


183.0 


219.6 


256.2 


292.8 


329.4 


365 


36.5 


73.0 


109.5 


146.0 


182.5 


219.0 


255.7 


292.0 


328.5 


364 


36.4 


72.8 


109.2 


145.6 


182.0 


218.4 


254.8 


291.2 


327.6 


363 


36.3 


72. ( 


3 


1( 


)8.9 


145.2 


181.5 


217 


.8 


$ 


S54.1 


290.4 


326.7 


362 


36.2 


72.4 


108.6 


144.8 


181.0 




.2 


X 


53.4 


289.6 


325.8 


361 


36.1 


72.$ 


2 


1( 


)8.3 


144.4 


180.5 


2ie 


.6 


1 


52.7 


288.8 


324.9 


360 


36.0 


72. 


3 


1( 


)8.0 


144.0 


180.0 


2ie 


.0 


2 


52.0 


288.0 


324.0 


359 


35.9 


71.8 


1( 


)7.7 


143.6 


179.5 


215.4 


251.3 


287.2 


323.1 


358 


35.8 


71.6 


107.4 


143.2 


179.0 


214.8 


250.6 


286.4 


322.2 


357 


35.7 


71. 


I 


1( 


)7.1 


142.8 


178.5 


214 


.2 


2 


49.9 


285.6 


321.3 


356 


35.6 


71.2 


106.8 


142.4 


178.0 


213.6 


249.2 


284.8 


320.4 



LOGARITHMS OF NUMBERS. 



No. 120 L. 079.] [No. 134 L. 130. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


120 


079181 


9543 


9904 


















0266 


0626 II 0987 


1347 


1707 


2067 


2426 


360 


1 


082785 


3144 


3503 


3861 


4219 


4576 


4934 


5291 


5647 


6004 


357 


2 
3 


6360 
9905 


6716 


7071 


7426 


7781 


| 8136 


8490 


8845 


9198 


9552 


355 






0258 


0611 


0963 


1315 


1667 


2018 


2370 


2721 


3071 


352 


4 


093422 


3772 


4122 


4471 


4820 


5169 


5518 


5866 


6215 


6562 


349 


5 


6910 


7257 


7604 


7951 


8298 


8644 


8990 


9335 


9681 


























0026 


346 


6 


100371 


0715 


1059 


1403 


1747 


2091 


2434 


2777 


3119 


3462 


343 


7 


3804 


4146 


4487 


4828 


5169 


5510 


5851 


6191 


6531 


6871 


341 


8 


7210 


7549 


7888 


8227 


8565 


8903 


9241 


9579 


9916 


0253 


338 


9 


110590 


0926 


1263 


1599 


1934 


2270 


2605 


2940 


3275 


3609 


335 


130 


3943 


4277 


4611 


4944 


5278 


5611 


5943 


6276 


6608 


6940 


333 


1 


7271 


7603 


7934 


8265 


8595 


8926 


9356 


9586 


9915 


























0245 


330 


2 


120574 


0903 


1231 


1560 


1888 


2216 


2544 


2871 


3198 


3525 


328 


3 


3852 


4178 


4504 


4&30 


5156 


5481 


5806 


6131 


6456 


6781 


325 


4 


7105 


7429 


7753 


8076 


8399 


8722 


9045 


9368 


9690 








13 


















0012 


323 


PROPORTIONAL PARTS. 


Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


355 


a5.5 


71.0 


106.5 


142.0 


177.5 


213 


248.5 


284.0 


319.5 


354 


35.4 


70.8 


106.2 


141.6 


177.0 


212.4 


247.8 


283.2 


318.6 


353 


35.3 


70.6 


105.9 


141.2 


176.5 


211.8 


247.1 


282.4 


317.7 


352 


35.2 


70.4 


105.6 


140.8 


176.0 


211.2 


246.4 


281.6 


316.8 


351 


35.1 


70.2 


105.3 


140.4 


175.5 


210.6 


245.7 


280.8 


315.9 


350 


35.0 


70.0 


105.0 


140.0 


175.0 


210.0 


245.0 


280.0 


315,0 


349 


34.9 


69.8 


104.7 


139.6 


174.5 


209.4 


244.3 


279.2 


314.1 


348 


34.8 


69.6 


104.4 


139.2 


174.0 


208.8 


243.6 


278.4 


313.2 


847 


34.7 


69.4 


104.1 


138.8 


173.5 


208.2 


242.9 


277.6 


312.3 


346 


34.6 


69.2 


103.8 


138.4 


173.0 


207.6 


242.2 


276.8 


311.4 


345 


34.5 


69.0 


103.5 


138.0 


172.5 


207.0 


241.5 


276.0 


310.5 


344 


34.4 


68.8 


103.2 


137.6 


172.0 


206.4 


240.8 


275.2 


309.6 


343 


34.3 


68.6 


102.9 


137.2 


171.5 


205.8 


240.1 


274.4 


308.7 


342 


34.2 


68.4 


102.6 


136.8 


171.0 


205.2 


239.4 


273.6 


307.8 


341 


34.1 


68.2 


102.3 


136.4 


170.5 


204.6 


238.7 


272.8 


306.9 


340 


34.0 


68.0 


102.0 


136.0 


170.0 


204.0 


238.0 


272.0 


306.0 


339 


33.9 


67.8 


101.7 


135.6 


169-. 5 


203.4 


237.3 


271.2 


305 J 


338 


33.8 


67.6 


101.4 


135.2 


169.0 


202.8 


236.6 


270.4 


304.2 


337 


33.7 


67.4 


101.1 


134.8 


1G8.5 


202.2 


235.9 


269.6 


303.3 


336 


33.6 


67.2 


100.8 


134.4 


168.0 


201.6 


235.2 


268.8 


302.4 


335 


33.5 


67.0 


100.5 


134.0 


167.5 


201.0 


234.5 


268.0 


301.5 


334 


33.4 


66.8 


100.2 


133.6 


167.0 


200.4 


233.8 


267.2 


300.6 


333 


33.3 


66.6 


99.9 


133.2 


166.5 


199.8 


233.1 


266.4 


299.7 


332 


33.2 


664 


99.6 


132.8 


166.0 


199.2 


232.4 


265.6 


298.8 


331 


33.1 


66,2 


99.3 


132.4 


165.5 


198.6 


231.7 


264.8 


297.9 


330 


33.0 


66.0 


99.0 


132.0 


165.0 


198.0 


231.0 


264.0 


297.0 


329 


32.9 


65.8 


98.7 


131.6 


164.5 


197.4 


230.3 


263.2 


296.1 


328 


32.8 


65.6 


98.4 


131.2 


164.0 


196.8 


229.6 


262.4 


295.2 


327 


32.7 


65.4 


98.1 


130.8 


163.5 


196.2 


228.9 


261.6 


294.3 


326 


32.6 


65.2 


97.8 


130.4 


163.0 


195.6 


228.2 


260.8 


293.4 


325 


32.5 


65.0 


97.5 


130.0 


162.5 


195.0 


227.5 


260.0 


292.5 


324 


32.4 


64.8 


97.2 


129.6 


162.0 


194.4 


226.8 


259.2 


291.6 


323 


32.3 


64.6 


96.9 


129.2 


161.5 


193.8 


226.1 


258.4 


290.7 


3S2 


32.2 


64.4 


96.6 


128.8 


161.0 


193.2 


225.4 


257.6 


289.8 



LOGARITHMS OE NUMBERS. 



No. 135 L. 130.] 


[No. 149 L. 175. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


135 


130334 


0655 


0977 


1298 


1619 


1939 


2260 


2580 


2900 


3219 


321 


6 


3539 


3858 


4177 


4496 


4814 


5133 


5451 


5769 


6086 


6403 


318 


7 
g 


6721 
9879 


7037 


7354 


7671 


7987 


8303 


8618 


8934 


9249 


9564 


316 






0194 


0508 


0822 


1136 


1450 


1763 


2076 


2389 


2702 


314 


9 


143015 


3327 


3639 


3951 


4263 


4574 


4885 


5196 


5507 


5818 


311 


140 
j 


6128 
9219 


6438 
9527 


6748 
9835 


7058 


7367 


7676 


7985 


8294 


8603 


8911 


309 










0142 


0449 


0756 


1063 


1370 


1676 


1982 


307 


2 


152288 


2594 


2900 


3205 


3510 


3815 


4120 


4424 


4728 


5032 


305 


3 


5336 


5640 


5943 


6246 


6549 


6852 


7154 


7457 


7759 


8061 


303 


4 


8362 


8664 


8965 


9266 


9567 


9868 


























0168 


0469 


0769 


1068 


301 


5 


161368 


1667 


1967 


2266 


25G4 


2863 


3161 


3460 


3758 


4055 


299 


6 


4353 


4650 


4947 


5244 


5541 


5838 


6134 


6430 


6726 


7022 


297 


7 


7317 


7613 


7908 


8203 


8497 


8792 


9086 


9380 


9674 


9968 


295 


8 


170262 


0555 


0848 


1141 


1434 


1726 


2019 


2311 


2603 


2895 


293 


9 


3186 


3478 


3769 


4060 


4351 


4641 


4932 


5222 


5512 


5802 


291 


PROPORTIONAL PARTS. 


Diff. ' 


1 


2 


3 


4 


5 


6 


7 


8 


9 


321 


32.1 


64.2 


96.3 


128.4 


160.5 


192.6 


2$ 


4.7 


256.8 


288.9 


320 


32.0 


64.0 


96.0 


128.0 


160.0 


192.0 


224.0 


256.0 


288.0 


319 


31.9 


63.8 


95.7 


127.6 


159.5 


191.4 


2$ 


53.3 


255.2 


287.1 


318 


31.8 


63.6 


95.4 


127.2 


159.0 


190.8 


222.6 


254.4 


286.2 


317 


31.7 


63.4 


95.1 


126.8 


158.5 


190.2 


& 


J1.9 


253.6 


285.3 


316 


31.6 


63.2 


94.8 


126.4 


158.0 


189.6 


221.2 


252.8 


284.4 


315 


31.5 


63.0 


94.5 


126.0 


157.5 


189.0 


2$ 


JO. 5 


252.0 


283.5 


314 


31.4 


62.8 


94.2 


125.6 


157.0 


188.4 


219.8 


251.2 


282.6 


313 


31.3 


62.6 


93.9 


125.2 


156.5 


187.8 


21 


9.1 


250.4 


281.7 


312 


31.2 


62.4 


93.6 


124.8 


156.0 


187.2 


218.4 


249.6 


280.8 


311 


31.1 


62.2 


93.3 


124.4 


155.5 


186.6 


217.7 


248.8 


279.9 


310 


31.0 


62.0 


93.0 


124.0 


155.0 


186.0 


21 


7.0 


248.0 


279.0 


309 


30.9 


61.8 


92.7 


123.6 


154.5 


185.4 


2] 


6.3 


247.2 


278.1 


308 


30.8 


61.6 


92.4 


123.2 


154.0 


184.3 


215.6 


246.4 


277.2 


307 


30.7 


61.4 


92.1 


122.8 


153.5 


184.2 


214.9 


245.6 


276.3 


306 


30.6 


61.2 


91.8 


122.4 


153.0 


183.6 


21 


4.2 


244.8 


275.4 


305 


30.5 


61.0 


91.5 


122.0 


152.5 


183.0 


21 


3.5 


244.0 


274,5 


304 


30.4 


60.8 


91.2 


121.6 


152.0 


182.4 


212.8 


243.2 


273.6 


303 


30.3 


60.6 


90.9 


121.2 


151.5 


181.8 


2] 


L2.1 


242.4 


272.7 


302 


30.2 


60.4 


90.6 


120.8 


151.0 


181.2 


211.4 


241.6 


271.8 


301 


30.1 


60.2 


90.3 


120.4 


150.5 


180.6 


210.7 


240.8 


270.9 


300 


30.0 


60.0 


90.0 


120.0 


150.0 


180.0 


21 


LO.O 


240.0 


270.0 


299 


29.9 


59.8 


89.7 


119.6 


149.5 


179.4 


209.3 


239.2 


269.1 


298 


29.8 


59.6 


89.4 


119.2 


149.0 


178.8 


2( 


)8.6 


238.4 


268.2 


297 


29.7 


59.4 


89.1 


118.8 


148.5 


178.2 


207.9 


237.6 


267.3 


296 


29.6 


59.2 


88.8 


118.4 


148.0 


177.6 


2( 


)7.2 


236.8 


266.4 


295 


29.5 


59.0 


88.5 


118.0 


147.5 


177.0 


206.5 


'236.0 


265:5 


294 


29.4 


58.8 


88.2 


117.6 


147.0 


176.4 


2( 


)5.8 


235.2 


264.6 


293 


29.3 


58.6 


87.9 


117.2 


146.5 


175.8 


205.1 


234.4 


263.7 


292 


29.2 


58.4 


87.6 


116.8 


146.0 


175.2 


204.4 


233.6 


262.8 


291 


29.1 


58.2 


87.3 


116.4 


145.5 


174.6 


203.7 


232.8 


261.9 


290 


29.0 


58.0 


87.0 


116.0 


145.0 


174.0 


2< 


)3.0 


232.0 


261.0 


289 


28.9 


57.8 


86.7 


115.6 


144.5 


173.4 


2( 


)2.3 


231.2 


260.1 


288 


28.8 


57.6 


86.4 


115.2 


144.0 


172.8 


2( 


)1.6 


230.4 


259.2 


287 


28.7 


57.4 


86.1 


114.8 


143.5 


172.2 


200.9 


229.6 


258.3 


286 


28.6 


57.2 


85.8 


114.4 


143.0 


171.6 


200.2 


228.8 


257.4 



134 



LOGARITHMS OF KUMBERS. 



No. 150 L,. 176.] [No. 169 L. 230. 1 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


150 


176091 


6381 


6670 


6959 7248 


7536 


7825 


8113 


8401 


8689 


289 


j 


8977 


9264 


9552 


OH SO 






















mo 


0413 


0699 


0986 


1272 


1558 


287 


2 


181844 


2129 


2415 


2700 


2985 


3270 


3555 


3839 


4123 


4407 


285 


3 


4691 


4975 


5259 


5542 


5825 


6108 


6391 


6674 


6956 


7239 


283 


4 


7521 


7803 


8084 


8366 


8647 


8928 


9209 


9490 


9771 


























0051 


281 


5 


190332 


0612 


0892 


1171 


1451 


1730 


2010 


2289 


2567 


2846 


279 


6 


3125 


3403 


3681 


3959 


4237 


4514 


4792 


5069 


5346 


5623 


378 


7 


5900 


6176 


6453 


6729 


7005 


7281 


7556 


7832 


8107 


8382 


276 


g 


8657 


8932 


9206 


9481 


9755 


























0029 


0303 


0577 


0850 


1124 


274 


9 


201397 


1670 


1943 


2216 


2488 


2761 


3033 


3305 


3577 


3848 


272 


160 


4120 


4391 


4663 


4934 


5204 


5475 


5746 


6016 


6286 


6556 


271 


1 


6826 


7096 


7365 


7634 


7904 


8173 


8441 


8710 


8979 


9247 


269 


2 


9515 


9783 


























0051 


0319 


0586 


0853 


1121 


1388 


1654 


1921 


267 


3 


212188 


2454 


2720 


2986 


3252 


3518 


3783 


4049 


4314 


4579 


266 


4 


4844 


5109 


5373 


5638 


5902 


6166 


6430 


6694 


6957 


7221 


264 


5 


7484 


7747- 


8010 


8273 


8536 


8798 


9060 


9323 


9585 


9846 


262 


6 


220108 


0370 


0631 


0892 


1153 


1414 


1675 


1936 


2196 


2456 


261 


7 


2716 


2976 


3236 


3496 


3755 


4015 


4274 


4533 


4792 


5051 


259 


8 


5309 


5568 


5826 


6084 


6342 


6600 


6858 


7115 


7372 


7630 


258 


9 


7887 


8144 


8400 


8657 


8913 


9170 


9426 


9682 


9938 









23 


















0193 


256 


PROPORTIONAL PARTS. 


Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


285 


28.5 


57.0 


85.5 


114.0 


142.5 


171.0 


199.5 


228.0 


256.5 


284 


28.4 


56.8 


85.2 


113.6 


142.0 


170.4 


198.8 


227.2 


255.6 


283 


28.3 


56.6 


84.9 


113.2 


141.5 


169.8 


198.1 


226.4 


254.7 


282 


28.2 


56.4 


84.6 


112.8 


141.0 


169.2 


197.4 


225.6 


253.8 


281 


28.1 


56.2 


84.3 


112 4 


140.5 


168.6 


196.7 


224.8 


252.9 


280 


28.0 


56.0 


84.0 


112.0 


140.0 


168.0 


196.0 


224.0 


252.0 


279 


27.9 


55.8 


83.7 


111.6 


139.5 


167.4 


195.3 


223.2 


251.1 


278 


27.8 


55.6 


83.4 


111.2 


139.0 


166.8 


194.6 


222.4 


250.2 


277 


27.7 


55.4 


83.1 


110.8 


138.5 


166.2 


193.9 


221.6 


249.3 


276 


27.6 


55.2 


82.8 


110.4 


138.0' 


165.6 


193.2 


220.8 


248.4 


275 


27.5 


55.0 


82.5 


110.0 


137.5 


165.0 


192.5 


220.0 


247.5 


274 


27.4 


54.8 


82.2 


109.6 


137.0 


164.4 


191.8 


219.2 


246.6 


273 


27.3 


54.6 


81.9 


109.2 


136.5 


163.8 


191.1 


218.4 


245.7 


272 


27.2 


54.4 


81.6 


108.8 


136.0 


163.2 


190.4 


217.6 


244.8 


271 


27.1 


54.2 


81.3 


108.4 


135.5 


162.6 


189.7 


216.8 


243.9 


270 


27.0 


54.0 


81.0 


108.0 


135.0 


162,0 


189.0 


216.0 


243.0 


269 


26.9 


53.8 


80.7 


107.6 


134.5 


161.4 


188.3 


215.2 


242.1 


268 


26.8 


53.6 


80.4 


107.2 


134.0 


160.8 


187.6 


214.4 


241.2 


267 


26.7 


53.4 


80.1 


106.8 


133.5 


160.2 


186.9 


213 6 


240.3 


266 


26.6 


53.2 


79.8 


106.4 


133.0 


159.6 


186.2 


212.8 


239.4 


265 


26.5 


53.0 


79.5 


106.0 


132.5 


159.0 


185.5 


212.0 


238.5 


264 


26.4 


52.8 


79.2 


105.6 


132.0 


158.4 


184.8 


211.2 


237.6 


263 


26.3 


52.6 


78.9 


105.2 


131.5 


157.8 


184.1 


210.4 


236.7 


262 


26.2 


52.4 


78.6 


104.8 


131.0 


157.2 


183.4 


209.6 


235.8 


261 


26.1 


52.2 


78.3 


104.4 


130.5 


156.6 


182.7 


208.8 


234.9 


260 


26.0 


52.0 


78.0 


104.0 


130.0 


156.0 


182.0 


208.0 


234.0 


259 


25.9 


51.8 


77.7 


103.6 


129.5 


155.4 


181.3 


207.2 


233.1 


258 


25.8 


51.6 


77.4 


103.2 


129.0 


154.8 


180.6 


206.4 


232.2 


257 


25.7 


51.4 


77.1 


102.8 


128.5 


154.2 


179.9 


205.6 


231.3 


256 


25.6 


51.2 


76.8 


102.4 


128.0 


153.6 


179.2 


204.8 


230.4 


255 


25.5 


51.0 


76.5 


102.0 


17.5 


153.0 


178.5 


204.0 


229.5 



LOGARITHMS OF LUMBERS. 



No. 170 L. 230.] [No. 189 L. 278. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


170 


230449 


0704 


0960 


1215 


1470 


1724 


1979 


2234 


2488 


2742 


255 


1 


2996 


3250 


3504 


3757 


4011 


4264 


4517 


4770 


5023 


5276 


253 


2 


5528 


5781 


6033 


6285 


6537 


6789 


7041 


7292 


7544 


7795 


252 




8046 


8297 


8548 


8799 


9049 


9299 


9550 


9800 


























0050 


0300 


250 


4 


240549 


0799 


1048 


1297 


1546 


1795 


2044 


2293 


2541 


2790 


249 


5 


3038 


3286 


3534 


3782 


4030 


4277 


4525 


4772 


5019 


5266 


248 


6 


5513 


5759 


6006 


6252 


6499 


6745 


6991 


7237 


7482 


7728 


246 




7973 


8219 


8464 


8709 


8954 


9198 


9443 


9687 


9932 


























0176 


245 


8 


250420 


0664 


0908 


1151 


1395 


1638 


1881 


2125 


2368 


2610 


243 


9 


2853 


3096 


3338 


3580 


3822 


4064 


4306 


4548 


4790 


5031 


242 


180 


5273 


5514 


5755 


5996 


6237 


6477 


6718 


6958 


7198 


7439 


241 


1 


7679 


7918 


8158 


8398 


8637 


8877 


9116 


9355 


9594 


9833 


239 


2 


260071 


0310 


0548 


0787 


1025 


1263 


1501 


1739 


1976 


2214 


238 


3 


2451 


2688 


2925 


3162 


3399 


3636 


3873 


4109 


4346 


4582 


237 


4 


4818 


5054 


5290 


5525 


5761 


5996 


6232 


6467 


6702 


6937 


235 


5 


7172 


7406 


7641 


7875 


8110 


8344 


8578 


8812 


9046 


9279 


234 




9513 


9746 


9980 


























213 


0446 


0679 


0912 


1144 


1377 


1609 


233 


7 


271842 


2074 


2306 


2538 


2770 


3001 


3233 


3464 


3696 


3927 


232 


8 


4158 


4389 


4620 


4850 


5081 


5311 


5542 


5772 


6002 


6232 


230 


9 


6462 


6692 


6921 


7151 


7380 


7609 


7838 


8067 


8296 


8525 


229 


PROPORTIONAL PARTS. 


Diff. 


1 


2 


3 


4 


5 


6 


7 




8 


9 


255 


25,5 


51,0 


76,5 


102.0 


127,5 


153,0 


17S.5 


204.0 


229,5 


254 


25.4 


50.8 


76.2 


101.6 


127.0 


152.4 


177.8 


203.2 


228.6 


253 


25.3 


50.6 


75.9 


101.2 


126.5 


151.8 


177.1 


202.4 


227.7 


252 


25.2 


50.4 


75.6 


100.8 


126.0 


151.2 


176.4 


201.6 


226.8 


251 


25.1 


50.2 


75.3 


100.4 


125.5 


150.6 


175.7 


200.8 


225.9 


250 


25 


50.0 


75.0 


100.0 


125.0 


150.0 


175.0 


200.0 


225.0 


249 


24.9 


49.8 


74.7 


99.6 


124.5 


149.4 


174.3 


199.2 


224.1 


248 


24.8 


49.6 


74.4 


99.2 


124.0 


148.8 


173.6 


198.4 


223.2 


247 


24.7 


49.4 


74.1 


98.8 


123.5 


148.2 


172.9 


197.6 


222.3 


246 


24.6 


49.2 


73.8 


98.4 


123.0 


147.6 


172.2 


196.8 


221.4 


245 


24.5 


49.0 


73.5 


98.0 


122.5 


147.0 


171.5 


196.0 


220.5 


244 


24.4 


48.8 


73.2 


97.6 


122.0 


146.4 


170.8 


195.2 


219.6 


243 


24.3 


48.6 


72.9 


97.2 


121.5 


145.8 


170.1 


194.4 


218.7 


242 


24.2 


48.4 


72.6 


96.8 


121.0 


145.2 


169.4 


193.6 


217.8 


241 


24.1 


48.2 


72.3 


96.4 


120.5 


144.6 


168.7 


192.8 


216.9 


240 


24.0 


48.0 


72.0 


96.0 


120.0 


144.0 


168.0 


192.0 


216.0 


239 


23.9 


47.8 


71.7 


95.6 


119.5 


143.4 


167.3 


191.2 


215.1 


238 


23.8 


47.6 


71.4 


95.2 


119.0 


142.8 


166.6 


190.4 


214.2 


237 


23.7 


47.4 


71.1 


94.8 


118.5 


142.2 


165.9 


189.6 


213.3 


236 


23.6 


47.2 


70.8 


94.4 


118.0 


141.6 


165.2 


188.8 


212.4 


235 


23.5 


47.0 


70.5 


94.0 < 


117.5 


141.0 


164.5 


188.0 


211.5 


234 


23.4 


46.8 


70.2 


93.6 


117.0 


140.4 


163.8 


187.2 


210.6 


233 


23.3 


46.6 


69.9 


93.2 


116.5 


139.8 


163.1 


186.4 


209.7 


232 


23.2 


46.4 


69.6 


92.8 


116.0 


139.2 


162.4 


185.6 


208.8 


.231 


23.1 


46.2 


69.3 


92.4 


115.5 


138.6 


161.7 


184.8 


207.9 


230 


23.0 


46.0 


69.0 


92.0 


115.0 


138.0 


161.0 


184.0 


207.0 


229 


22.9 


45.8 


68.7 


91.6 


114.5 


137.4 


160.3 


183.2 


206.1 


228 


22.8 


45.6 


68.4 


91.2 


114.0 


136.8 


159.6 


182.4 


205.2 


227 


22.7 


45.4 


68.1 


90.8 


113.5 


136.2 


158.9 


181.6 


204.8 


226 


22.6 


45.2 


67.8 


90.4 


113.0 


135.6 


158 2 


180.8 


203.4 



LOGARITHMS OF HUMBEK& 



No. 190 L. 278.] [No. 214 L. 332. 


N. 





1 


f 


3 


4 


5 


6 


7 


8 


9 


Diff. 


190 


278754 


8982 


9211 


9439 


9667 


9895 












0123 


0351 


0578 


0806 


228 


1 


281033 


1261 


1488 


1715 


1942 


2169 


2396 


2022 


2849 


3075 


227 


2 


3301 


3527 


3753 


3979 


4205 


4431 


4656 


4882 


5107 


5332 


226 


3 


5557 


5782 


6007 


6232 


6456 


6681 


6905 


7130 


7354 


7578 


225 


4 


7802- 


8026 


8249 


8473 


8096 


8920 


9143 


9366 


9589 


9812 


223 


5 


290035 


0257 


0480 


0702 


0925 


1147 


1369 


1591 


1813 


2034 


222 


6 


2256 


2478 


2699 


2920 


3141 


3303 


3584 


3804 


4025 


4246 


221 


7 


4466 


4687 


4907 


5127 


5347 


5567 


5787 


6007 


6226 


6446 


220 


8 


6665 


68&4 


7104 


7323 


7542 


7761 


7979 


8198 


&416 


8635 


219 


9 


8853 


9071 


9289 


9507 


9725 


9943 


























flifii 


0378 


0595 


0813 


218 


200 


301030 


1247 


1464 


1681 


1898 


2114 


UJ.UJ. 

2331 


2547 


2764 


2980 


217 


1 


3196 


3412 


3628 


3844 


4059 


4275 


4491 


4706 


4921 


5136 


216 


2 


5351 


5566 


5781 


5996 


6211 


6425 


6639 


6854 


7068 


7282 


215 


3 


7496 


7710 


7924 


8137 


8351 


85G4 


8778 


8991 


9204 


9417 


213 


4 


9630 


9843 


























0056 


0268 


/VjQI 


flfiOQ 


flQftfi 


1 1 18 


iQQfj 


1 *vd.9 


919 


5 


311754 


1966 


2177 


2389 


Viol 

2600 


UOiAJ 

2812 


uyuo 
3023 


111O 

3234 


loou 

3445 


1O4X 

3656 


KOI 

211 


6 


3867 


4078 


4289 


4499 


4710 


4920 


5130 


5340 


5551 


5760 


210 


7 


5970 


6180 


6390 


6599 


6809 


7018 


7227 


7436 


7646 


7854 


209 


8 


8063 


8272 


8481 


8689 


8898 


9106 


9314 


9522 


9730 


9938 


208 


9 


320146 


0354 


0562 


0769 


0977 


1184 


1391 


1598 


1805 


2012 


207 


210 


2219 


2426 


2633 


2839 


3046 


3252 


3458 


3665 


3871 


4077 


206 


1 


4282 


4488 


4694 


4899 


5105 


5310 


5516 


5721 


5926 


6131 


205 


2 


6336 


6541 


6745 


6950 


7155 


7359 


7563 


7767 


7972 


8176 


204 


3 


8380 


8583 


8787 


8991 


9194 


9398 


9601 


9805 


























0008 


0211 


203 


4 


330414 


0617 


0819 


1022 


1225 


1427 


1630 


1832 


2034 


2236 


202 


PROPORTIONAL PARTS. 


Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


225 


22.5 


45.0 


67.5 


90.0 


112.5 


135.0 


157.5 


180.0 


202.5 


224 


22.4 


44.8 


67.2 


89.6 


112.0 


134.4 


156.8 


179.2 


201.6 


223 


22.3 


44.6 


66.9 


89.2 


111.5 


133.8 


156.1 


178.4 


200.7 


222 


22.2 


44.4 


66.6 


88.8 


111.0 


133.2 


155.4 


177.6 


199.8 


221 


22.1 


44.2 


66.3 


88.4 


110.5 


132.6 


154.7 


176.8 


198.9 


220 


22.0 


44.0 


66.0 


88.0 


110.0 


132.0 


154.0 


176.0 


198.0 


219 


21.9 


43.8 


65.7 


87.6 


109.5 


131.4 


153.3 


175.2 


197.1 


218 


21.8 


43.6 


65.4 


87.2 


109.0 


130.8 


152.6 


174.4 


196.2 


217 


21.7 


43.4 


65.1 


86.8 


108.5 


130.2 


151.9 


173.6 


195.3 


216 


-21.6 


43.2 


64.8 


86.4 


108.0 


129.6 


151.2 


172.8 


194.4 


215 


21.5 


43.0 


64.5 


86.0 


107.5 


129.0 


150.5 


172.0 


193.5 


214 


21.4 


42.8 


64.2 


85.6 


107.0 


128.4 


149.8 


171.2 


192.6 


213 


21.3 


42.6 


63.9 


85.2 


106.5 


127.8 


149.1 


170.4 


191.7 


212 


21.2 


42.4 


63.6 


84.8 


106.0 


127.2 


148.4 


169.6 


190.8 


211 


21.1 


42.2 


63.3 


84.4 


105.5 


126.6 


147.7 


168.8 


189.9 


210 


21.0 


42.0 


63.0 


84.0 


105.0 


126.0 


147.0 


168.0 


189.0 


209 


20.9 


41.8 


62.7 


83.6 


104.5 


125.4 


146.3 


167.2 


188.1 


208 


20.8 


41.6 


62.4 


83.2 


104.0 


124.8 


145.6 


1664 


187.2 


207 


20.7 


41.4 


62.1 


82.8 


103.5 


124.2 


144.9 


165.6 


186.3 


206 


20.6 


41.2 


61.8 


82.4 


103.0 


123.6 


144.2 


164.8 


185.4 


205 


20.5 


44.0 


61.5 


82.0 


102.5 


123.0 


143.5 


164.0 


184.5 


204 


20.4 


40.8 


61.2 


81.6 


102.0 


122.4 


142.8 


163.2 


183.6 


203 


20.3 


40.6 


60.9 


81.2 


101.5 


121.8 


142.1 


162.4 


182.7 


202 


20.2 


40.4 


60.6 


/0,8 


101.0 


121.2 


141.4 


161.6 


181.8 



LOGARITHMS OF NtTMBEHS. 



No. 215 L. 832.] [No. 239 L. 380. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


215 


332438 


2640 


2842 


3044 


3246 


3447 


3649 


3850 


4051 


4253 


202 


6 


4454 


4655 


4856 


5057 


5257 


5458 


5658 


5859 


6059 


6260 


201 


7 


6460 


6660 


6860 


7060 


7260 


7459 


7659 


7858 


8058 


8257 


200 


g 


8456 


8656 


8855 


9054 


9253 


9451 


9650 


9849 


























0047 


0246 


100 


9 


340444 


0642 


0841 


1039 


1237 


1435 


1632 


1830 


2028 


2225 


JUJJ 

198 


220 


2423 


2620 


2817 


3014 


3212 


3409 


3606 


3802 


3999 


4196 


197 


1 


4392 


4589 


4785 


4981 


5178 


5374 


5570 


5766 


5062 


6157 


196 


2 


6353 


6549 


67'44 


6939 


7135 


7330 


7525 


7720 


7915 


8110 


195 


3 


8305 


8500 


8694 


8889 


9083 


9278 


9472 


9666 


9860 


























0054 


194 


4 


350248 


0442 


~0636~ 


0829 


1023 


1216 


1410 


1603 


1796 


1989 


193 


5 


2183 


2375 


2568 


2761 


2954 


3147 


3339 


3532 


3724 


3916 


193 


6 


4108 


4301 


4493 


4685 


4876 


5068 


5260 


5452 


5643 


5834 


192 


7 


6026 


6217 


6408 


6599 


6790 


6981 


7172 


7363 


7554 


7744 


191 


8 

Q 


7935 
9835 


8125 


8316 


8506 


8696 


8886 


9076 


9266 


9456 


9646 


190 






0025 


0215 


0404 


0593 


0783 


0972 


1161 


1350 


1539 


1ftQ 


230 


361728 


1917 


2105 


2294 


2482 


2671 


2859 


3048 


3236 


3424 


ioy 

188 


1 


3612 


3800 


3988 


4176 


4363 


4551 


4739 


4926 


5113 


5301 


188 


2 


5488 


5675 


5862 


6049 


6236 


6423 


6610 


6796 


6983 


7169 


187 


3 


7356 


7542 


7729 


7915 


8101 


8287 


8473 


8659 


8845 


9030 


186 


4 


9216 


9401 


9587 


9772 


9958 


























O1 AQ 


nqoQ 


AK-jO 


OfiQft 


AQOQ 


1QK 


5 


371068 


1253 


1437 


1622 


1806 


U1<*O 

1991 


UcWo 

2175 


UO1O 

2360 


UOoO 

2544 


UOOO 

2728 


100 

184 


6 


2912 


3096 


3280 


3464 


8647 


3831 


4015 


4198 


4382 


4565 


184 


7 


4748 


4932. 


5115 


5298 


5481 


5664 


5846 


6029 


6212 


6394 


183 


8 


6577 


6759 


6942 


7124 


7306 


7488 


7670 


7852 


8034 


8216 


182 


9 


8398 


8580 


8761 


8943 


9124 


9306 


9487 


9668 


9849 








38 


















0030 


181 


PROPORTIONAL PARTS. 


Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


202 
201 


20.2 
20.1 


40.4 
40.2 


60.6 
60.3 


80.8 
80.4 


101.0 

100.5 


121.2 
120.6 


141.4 
140.7 


161.6 
160.8 


181.8 
180.9 


200 


20.0 


40.0 


60.0 


80.0 


100.0 


120.0 


140.0 


160.0 


180.0 


199 


19.9 


39.8 


59.7 


79.6 


99.5 


119.4 


139.3 


159.2 


179.1 


198 


19.8 


39.6 


59.4 


79.2 


99.0 


118.8 


138.6 


158.4 


178.2 


197 


19.7 


89.4 


59.1 


78.8 


98.5 


118.2 


137.9 


157.6 


177.3 


196 


19.6 


39.2 


58.8 


78.4 


98.0 


117.6 


137.2 


156.8 


176.4 


195 


19.5 


39.0 


58.5 


78.0 


97.5 


117.0 


136.5 


156.0 


175.5 


194 


19.4 


38.8 


58.2 


77.G 


97.0 


116.4 


135.8 


156.2 


174.6 


193 


19.3 


38.6 


57.9 


77.2 


96.5 


115.8 


135.1 


154.4 


173.7 


192 


19.2 


38.4 


57.6 


76.8 


96.0 


115.2 


134.4 


153.6 


172.8 


191 


19.1 


38.2 


57.3 


76.4 


95.5 


114.6 


133.7 


152.8 


171.9 


190 


19.0 


38.0 


57.0 


76.0 


95.0 


114.0 


133.0 


152.0 


171.0 


189 


18.9 


37.8 


56.7 


75.6 


94.5 


113.4 


132.3 


151.2 


170.1 


188 


18.8 


37.6 


56.4 


75.2 


94.0 


112.8 


131.6 


150.4 


169.2 


187 


18.7 


374 


56.1 


74.8 


93.5 


112.2 


130.9 


149.6 


168.3 


186 


18.6 


37.3 


55.8 


74.4 


93.0 


111.6 


130.2 


148.8 


167.4 


185 


18.5 


37.0 


55.5 


74.0 


92.5 


111.0 


129.5 


148.0 


166.5 


184 


18.4 


36.8 


55.2 


73.6 


92.0 


110.4 


128.8 


147.2 


165.6 


183 


18.3 


36.6 


54.9 


73.2 


91.5 


109.8 


128.1 


146.4 


164.7 


182 


18.2 


36.4 


54.6 


72.8 


91.0 


109.2 


127.4 


145.6 


163.8 


181 


18.1 


36.2 


54.3 


72.4 


90.5 


108.6 


126.7 


144.8 


162.9 


380 


18.0 


36.0 


54.0 


72.0 


90.0 


108.0 


126.0 


144.0 


162.0 


179 


17.9 


35.8 


53.7 


71.6 


89.5 


107.4 


125.3 


143.3 


161.1 



138 



LOGARITHMS OF NUMBERS. 



No. 240 L. 380.] 




[No. 269 L. 431. 


N. 





1 


| 


3 


4 


5 


6 


7 


8 


9 


Difl. 


240 
1 
2 
3 
4 
5 

6 
7 
8 
9 

250 

1 

2 

3 
4 
5 
6 

7 

8 
9 

260 
1 
2 
3 

4 
5 

6 

7 
8 
9 


380211 
2017 
3815 
5606 
7390 
9166 

390935 
2697 
4452 
6199 

7940 
9674 

401401 
3121 
4834 
6540 
8240 
9933 


0392 
2;97 
3995 
5785 
756.8 
9343 


0573 
2377 
4174 
5964 
7746 
9520 


0754 
2557 
4353 
6142 
7924 
9698 


0934 
2737 
4533 
6321 
8101 
9875 


1115 
2917 
4712 
6499 

8279 


1296 
3097 
4891 
6677 
8456 


1476 
3277 
5070 
6856 
8634 


1656 
345G 
5249 
7034 
8811 


1837 
3636 
428 
7212 
8989 


181 

180 
179 

178 

178 

177 
176 
176 
175 
174 
173 

173 
172 
171 
171 

170 
169 

169 
168 
167 

167 
166 
165 

165 
164 
164 
163 
162 
162 

161 


0051 
1817 
3575 
5326 
7071 

8808 


0228 
1993 
3751 
5501 
7245 

8981 


0405 
2169 
3926 
5676 
7419 

9154 


0582 
2345 
4101 
5850 
7593 

9328 


0759 
2521 
4277 
6025 
7766 
9501 


1112 
2873 
4627 
6374 

8114 

9847 


1288 
3048 
4802 
6548 

8287 


1464 
3224 
4977 
6722 

8461 


1641 
3400 
5152 
6896 

8634 


0020 
1745 
3464 
5176 

6881 
8579 


0192 
1917 
3635 
5346 
7051 
8749 


0365 
2089 
.3807 
5517 
7221 
8918 


0538 
2261 
3978 
5688 
7391 
9087 


0711 
2433 
4149 
5858 
7561 
9257 


0883 
2605 
4320 
6029 
7731 
9426 


1056 
2777 
4492 
6199 
7901 
9595 


1228 
2949 
4663 
6370 
8070 
9764 


1573 

3292 
5005 
6710 
8410 


0102 
1788 
3467 

5140 

6807 
8467 


0271 
1956 
3635 

5307 
6973 
8633 


0440 
2124 

3803 

5474 
7139 

8798 


0609 
2293 
3970 

5641 

7306 
8964 


0777 
2461 
4137 

5808 
7472 
91^9 


0946 
2629 
4305 

5974 
7638 
9295 


1114 
2796 

4472 

6141 
7804 
9460 


1283 
2964 
4639 

6308 
7970 
9625 


1451 
3132 

4806 

6474 
8135 
9791 


411620 
3300 

4973 
6841 
8301 
9956 

421604 
3246 
4882 
6511 
8135 
9752 
43 


0121 
1768 
3410 
5045 
6674 
8297 
9914 


0286 
1933 
3574 
5208 
6836 
8459 

0075 


0451 
2097 
3737 
5371 
6999 
8621 

0236 


0616 
2261 
3901 
5534 

7161 

8783 


i 0781 
2426 
4065 
5697 
7324 
8944 


0945 
2590 
4228 
5860 
7486 
9106 


1110 
2754 
4392 
6023 
7648 
9268 


'1275 
2918 
4555 
6136 
7811 
9429 


1439 
3082 
4718 
6349 
7973 
9591 


0398 


0559 


0720 


0881 


1042 


1203 


PROPORTIONAL PARTS. 


Diff. 1 , 


2 3 


4 


5 


6 

106.8 
106.2 
105.6 
105.0 
104.4 
103.8 
103.2 
102.6 
102.0 

101.4 
100.8 
100.2 
99.6 
99.0 
98.4 
97.8 
97.2 
96.6 


7 


8 


9 


178 17.8 
177 17.7 
176 17.6 
175 17.5 
174 17.4 
173 17.3 
172 17.2 
171 17.1 
170 17.0 

169 16.9 
168 16.8 
167 16.7 
166 16.6 
165 16.5 
164 16.4 
163 16.3 
162 16.2 
161 16.1 


35.6 53.4 
35.4 53.1 
35.2 52.8 
35.0 52.5 
34.8 52.2 
34.6 51.9 
34.4 51.6 
34.2 51.3 
34.0 51.0 

33.8 50.7 
33.6 50.4 
33.4 50.1 
33.2 49.8 
33.0 49.5 
32.8 49.2 
32.6 48.9 
32.4 48.5 
32.2 48.3 


71.2 
70.8 
70.4 
70.0 
69.6 
69.2 
68.8 
68.4 
68.0 

67.6 
67.2 
66.8 
66.4 
66.0 
65.6 
65.2 
64.8 
64.4 


89.0 
88.5 
88.0 
87.5 
87.0 
86.5 
86.0 
85.5 
85.0 

84.5 
84.0 
83.5 
83.0 

82.5 
82.0 
81.5 
81.0 
80.5 


124.6 
123.9 
123.2 
122.5 
121.8 
121.1 
120.4 
119.7 
119.0 

118.3 
117.6 
116.9 
116.2 
115.5 
114.8 
114.1 
113.4 
112.7 


142.4 
141.6 
140.8 
140.0 
139.2 
138.4 
137.6 
136.8 
136.0 
135.2 
134.4 
133.6 
132.8 
132.0 
131.2 
130.4 
129.6 
128.8 


160.2 
159.3 
158.4 
157.5 
156.6 
155.7 
154.8 
153.9 
153.0 

152.1 
151.2 
150.3 
149.4 
148.5 
147.6 
146.7 
145.8 
144.9 



LOGARITHMS OF NUMBERS. 



No. 270 L. 431.] [No. 299 L. 476. 


N. 





1 


2 


3 


4 


5 





7 


8 


9 


Diff. 


270 


431364 


1525 


1685 


1848 


2007 


2167 


2328 


2488 


2649 


2809 


161 


1 


2969 


3130 


3290 


3450 


3610 


3770 . 


3930 


4090 


4249 


4409 


160 


2 


4569 


4729 


4888 


5048 


5207 


5367 


5526 


5685 


5844 


6004 


159 


3 


6163 


6322 


6481 


6640 


6799 


6957 


7116 


7275 


7433 


7592 


159 


4 


7751 


7909 


8067 


8226 


8384 


8542 


8701 


8859 


9017 


9175 


158 


5 


9333 


9491 


9648 ' Qftnft 


9964 


























0122 


0279 


0437 


0594 


0752 


158 


6 


440909 


1066 


1224 


1381 


isSf 


1695 


1852 


2009 


2166 


2323 


157 


7 


2480 


2637 


2793 


2950 


3106 


3263 


3419 


3576 


3732 


3889 


157 


8 


4045 


4201 


4357 


4513 


4669 


4825 


4981 


5137 


5293 


5449 


156 


9 


5604 


5760 


5915 


6071 


6226 


6382 


6537 


6692 


6848 


7003 


155 


280 


7158 


7313 


7468 


7623 


7778 


7933 


8088 


8242 


8397 


8552 


155 


1 


,8706 


8861 


9015 


9170 


9324 


9478 


9633 


9787 


9941 


























0095 


154 


2 


'450249 


0403 


0557 


0711 


0865 


1018 


117'2 


1326 


1479 


1633 


154 


3 


1786 


1940 


2093 


2247 


2400 


2553 


2706 


2859 


3012 


3165 


153 


4 


3318 


3471 


3624 


3777 


3930 


4082 


4235 


4387 


4540 


4692 


153 


5 


4845 


4997 


5150 


5302 


5454 


5606 


5758 


5910 


6062 


6214 


152 


6 


6366 


6518 


6670 


6821 


6973 


7125 


7276 


7428 


7579 


7731 


152 


7 


7882 


8033 


8184 


8336 


8487 


8638 


8789 


8940 


9091 


9242 


151 


g 


9392 


9543 


9694 


9845 


9995 


























0146 


0296 


0447 


AKQ7 


0748 


JK-f 


9 


460898 


1048 


1198 


1348 


1499 


1649 


1799 


1948 


uoy< 
2098 


U(4O 

2248 


1O1 

150 


290 


2398 


2548 


2697 


2847 


2997 


3146 


3296 


3445 


3594 


3744 


150 


1 


3893 


4042 


4191 


4340 


4490 


4639 


4788 


4936 


5085 


5234 


149 


2 


5383 


5532 


5680 


5829 


5977 


6126 


6274 


6423 


6571 


6719 


149 


3 


6868 


7016 


7164 


7312 


7460 


7608 


7756 


7904 


8052 


8200 


148 


4 


8347 


8495 


8643 


8790 


8938 


9085 9233 


9380 


9527 


9675 


148 


5 


9822 


9969 


























0116 


0G3 


0410 


0557 


0704 


0851 


0998 


1145 


147 


6 


471292 


1438 


1585 


1732 


1878 


2025 


2171 


2318 


2464 


2610 


itt 
146 


7 


2756 


2903 


3049 


3195 


3341 


3487 


3633 


3779 


3925 


4071 


146 


8 


4216 


4362 


4508 


4653 


4799 


4944 


5090 


5235 


5381 


5526 


146 


9 


5671 


5816 


5962 


6107 


6252 


6397 6542 


6687 


6832 


6976 


145 
























PROPORTIONAL PARTS. 


Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


161 


16.1 


32.2 


48.3 


64.4 


80.5 


96.6 


112.7 


128.8 


144.9 


160 


16.0 


32.0 


48.0 


64.0 


80.0 


96.0 


112.0 


128.0 


144.0 


159 


15.9 


31.8 


47.7 


63.6 


79.5 


95.4 


111.3 


127.2 


143.1 


158 


15.8 


31.6 


47.4 


63.2 


79.0 


94.8 


110.6 


126.4 


142.2 


157 


15.7 


31.4 


47.1 


62.8 


78.5 


94.2 


109.9 


125.6 


141.3 


156 


15.6 


31.2 


46.8 


62.4 


78.0 


93.6 


109.2 


124.8 


140.4 


155 


15.5 


31.0 


46.5 


62.0 


77.5 


93.0 


108.5 


124.0 


139.5 


154 


15.4 


30.8 


46.2 


61.6 


77.0 


92.4 


107.8 


123.2 


138.6 


153 


15.3 


30.6 


45.9 


61.2 


76.5 


91.8 


107.1 


122.4 


137.7 


152 


15.2 


30.4 


45.6 


60.8 


76.0 


91.2 


106.4 


121.6 


136.8 


151 


15.1 


30.2 


45.3 


60.4 


75.5 


90.6 


105.7 


1208 


135.9 


150 


15.0 


30.0 


45.0 


60.0 


75.0 


90.0 


105.0 


120.0 


135.0 


149 


14.9 


29.8 


44.7 


59 6 


74.5 


89.4 


104.3 


119.2 


134.1 


148 


14.8 


29.6 


44.4 


59.2 


74.0 


88.8 


103.6 


118.4 


133.2 


147 


14.7 


29.4 


44.1 


58.8 


73.5 


88.2 


102.9 


117.6 


132.3 


146 


14 6 


29.2 


43.8 


58.4 


73.0 


87.6 


102.2 


116.8 


131.4 


145 


14.5 


29.0 


43.5 


58.0 


72.5 


87.0 


101.5 


116.0 


130.5 


144 


14.4 


28.8 


43.2 


57.6 


72.0 


86.4 


100.8 


115.2 


129.6 


143 


14.3 


28.6 


42.9 


57.2 


71.5 


85.8 


100.1 


114.4 


128.7 


142 


14.2 


28.4 


42.6 


56.8 


71.0 


85.2 


99.4 


113.6 


127.8 


141 


14.1 


28.2 


42.3 


56.4 


70.5 


84.6 


98 '1 


112.8 


126.9 


140 


14.0 


28.0 


42.0 


56.0 


70.0 


84.0 


98.0 


112.0 


126.0 



LOGARITHMS OP NUMBERS. 



No. 300 L, 4vr.] 




|No. 339 L. 531. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


300 
1 

2 
3 
4 
5 
6 

r 

8 
9 

310 
1 
2 
3 
4 
5 
6 

8 
9 

320 
1 
2 
3 

4 
5 
6 

7 
8 
9 

330 
1 

2 
3 
4 
5 
6 
7 
8 

9 


477121 
8566 


7266 
8711 


7411 
8855 


7555 
8999 


7700 
9143 


7844 
9287 


7989 
9431 


8133 
9575 


8278 
9719 


8422 
9863 


145 
144 

144 
143 
143 
142 
142 
141 
141 

140 

140 
139 
139 
139 

138 
138 

137 
137 
136 
13G 

136 
135 
135 

134 
134 
133 
133 
133 
132 
132 

181 

131 
131 
130 
130 
129 
129 
129 

128 
128 


480007 
1443 
2874 
4300 
5721 
7138 
8551 
9958 


0151 
1586 
3016 
4442 

5863 
7280 
8692 


0294 
1729 
3159 
4585 
6005 
7421 
8833 


0438 
1872 
3302 
4727 
6147 
7563 
8974 


0582 
2016 
3445 
4869 
6289 
7704 
9114 


0725 
2159 
3587 
5011 
6430 
7845 
9255 


0869 
2302 
3730 
5153 
657'2 
7986 
9396 


1012 
2445 

3872 
5295 
6714 
8127 
9537 


1156 
2588 
4015 
5437 
6855 
8269 
9677 


1299 
2731 
4157 
5579 
6997 
8410 
9818 


0099 

1502 
2900 
4294 
5683 
7068 
8448 
9824 


0239 

1642 
3040 
4433 

5822 
7206 
8586 
9962 


0380 

1782 
3179 
4572 
5960 
7344 
8724 


0520 

1922 
3319 
4711 

6099 

7483 
8862 


0661 

2062 
3458 
4850 
6238 
7621 
8999 

0374 
1744 
3109 
4471 

5828 
7181 
8530 
9874 


0801 

2201 
3597 
4989 
6376 
7759 
9137 


0941 

2341 

3737 
5128 
6515 
5897 
9275 


1081 

2481 
S876 
5267 
6653 
8035 
9412 

1)785~ 
2154 

3518 
487'8 

6234 
7586 
8934 


1222 

2621 
4015 
5406 
6791 
6173 
9550 


491362 
2760 
4155 
5544 
6930 
8311 
9687 


0099 
1470 
2837 
4199 

5557 
6911 
8260 
9606 


0236 
1607 
297'3 
4335 

5693 
7046 
8395 
9740 


0511 

1880 
3246 
4607 

5964 
7316 
8G64 


0648 
2017 
3382 
4743 

6099 
7451 
8799 


0922 
2291 
8655 
5014 

6370 

7721 
9068 


501059 
2427 
3791 

5150 
6505 
7856 
9203 


1196 
2564 
3927 

5286 
6640 
7991 
9337 


1333 

2700 
4063 

5421 
6776 
8126 
9471 


0009 
1349 

9684 
4016 
5344 

6668 

79S7 

9303 


0143 
1482 
2818 
4149 
5476 
6800 
8119 

9434 

0745 
2053 
3356 
4656 
5951 
7243 
8531 
9815 


0277 
1616 
2951 
4282 
5609 
6932 
8251 

9566 


0411 
1750 
3084 
4415 
5741 
7064 
8S82 

9697 


510545 

1883 
3218 
4548 
5874 
7196 

8514 

9828 

521138 
2444 
3746 
5045 
G339 
7630 
8917 


0679 
2017 
3351 
4681 
6006 
7328 

8646 
9959 


0813 
2151 
3484 
4813 
6139 
7460 

8777 


0947 
2284 
3617 
4946 
6271 
7592 

8909 


1081 
2418 
3750 
5079 
6403 
7724 

9040 


1215 
2551 
3883 
5211 
6535 
7855 

9171 


0090 
1400 
2705 
4006 
5304 
6598 
7888 
9174 


0221 
1530 
2835 
4136 
5434 
6727 
8016 
9302 


0353 
1661 
2966 
4266 
5563 
6856 
8145 
9430 


0484 
1792 
3096 
4396 
5693 
6985 
8274 
9559 


0615 
1922 
3226 
4526 
5822 
7114 
8402 
9687 


0876 
2183 
3486 
4785 
6081 
7372 
8060 
9943 


1007 
2314 
3616 
4915 
6210 
7501 
8788 


1289 
2575 
3876 
5174 
6469 
7759 
9045 


0072 
1351 


530200 


0328 5 0456 


0584 


0712 


0840 


0968 


1096 


1223 


PROPORTIONAL PARTS. 


Diff. 1 


2 3 


4 


5 


6 


7 


8 


9 


139 13.9 
138 13.8 
137 13.7 
136 13.6 
135 13.5 
134 13.4 
133 13.3 
132 13.2 
131 131 
130 130 
129 12.9 
128 12.8 
127 12 7 


27.8 41.7 
27.6 41.4 
27.4 41.1 
27.2 40.8 
27.0 40.5 
26.8 40.2 
26.6 39.9 
26.4 39.6 
26.2 89.3 
26.0 89.0 
25.8 38.7 
25.6 38.4 
25.4 38.1 


55.6 
55.2 
54.8 
54.4 
54.0 
53.6 
53.2 
52.8 
52.4 
52.0 
51.6 
51.2 
50.8 


69.5 
69.0 
68.5 
68.0 
67.5 
67.0 
66.5 
66.0 
65.5 
65.0 
64.5 
64.0 
63.5 


83.4 
82.8 
82.2 
81.6 
81.0 
80.4 
79.8 
79.2 
78.6 
78.0 
77.4 
76.8 
76.2 


97.3 
96.6 
95.9 
95.2 
94.5 
93.8 
93.1 
92.4 
91.7 
91.0 
90.3 i 
89.6 
88.9 


111.2 
110.4 
109.6 
108.8 
108.0 
107.2 
106.4 
105.3 
104.8 
104.0 
103.2 
102.4 
101.6 


125.1 
124.2 
123.3 
122.4 
121.5 
120.6 
119.7 
118.8 
117.9 
117.0 
116.1 
115.2 
114.3 



kOGAKITHMS OP NUMBERS. 



No. 340 L. 531.] 


[No. 379 L.5,9. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 

128 
127 
127 
126 
126 
126 

125 
125 
125 
124 

124 
124 
123 
123 

123 
122 
122 
121 
121 
121 

120 
120 
120 

119 
119 
119 
119 
118 
118 
118 

117 

117 
117 
116 
116 
116 
115 
115 
115 
114 


340 
1 
2 
3 
4 
5 
6 

7 
8 
9 

350 
1 
2 
3 
4 

5 

6 

7 
8 
9 

360 
1 
2 
3 

4 
5 

6 

7 
8 
9 

370 

1 

2 
3 

4 
5 

6 

7 
8 
9 


531479 
2754 
4026 
5294 
6558 
7819 
9076 


1607 
2882 
4153 
5421 
6685 
7945 
9202 


1734 

3009 
4280 
5547 
6811 
8071 
9327 


1862 
3136 
4407 
567'4 
6937 
8197 
9452 


1990 
3264 
4534 
5800 
7063 
8322 
9578 


2117 
3391 
! 4001 
5927 
7189 
8448 
9703 


2245 

3518 
4787 
6053 
7315 
8574 
9829 


2372 
3645 
4914 
6180 
7441 
8699 
9954 


2500 
3772 
5041 
6306 

7567 

8825 


2627 
3899 
5167 
6432 
7693 
8951 


0079 
1330 
2576 
3820 

5060 
6296 
7529 

8758 
9984 


0204 
1454 
2701 
3944 

5183 
6419 
7652 

8881 


540329 
1579 

2825 

4068 
5307 
6543 

7775 
9003 


0455 
1704 
2950 

4192 
5431 
6666 
7898 
9126 


0580 
1829 
3074 

4316 
5555 
6789 

8021 
9249 


0705 
1953 
3199 

4440 
5078 
6913 
8144 
9371 


0830 
2078 
3323 

4564 

5802 
7036 
8267 
9494 


0955 
2203 
3447 

4688 
5925 
7159 
8:389 
9616 


1080 
2327 
3571 

4812 
6049 
7282 
8512 
9739 


1205 
2452 
3096 

4936 
6172 
7405 
8635 
9861 


0106 
1328 
2547 
3762 
4973 
6182 

7387 
a589 
9787 


550228 
1450 
2668 
3883 
5094 

6303 
7507 
8709 
9907 


0351 
1572 
2790 
4004 
5215 

6423 

7627 

8829 


0473 
1094 
2911 
4126 
5336 

6544 

7?'48 
8948 


0595 
1816 
3033 
4247 
5457 

6664 

7868 
90G8 


0717 
1938 
3155 
4368 
5578 

6785 
7988 
9188 


0840 
2000 
3276 
4489 
5699 

6905 
8108 
9308 


0962 
2181 
3398 
4610 
5820 

7026 
8228 
9428 


1084 
2303 
3519 
4731 
5940 

7146 
8349 
9548 


1206 
2425 
3640 
4852 
6061 

7267 
8469 
9007 


0026 
12^1 
2412 
3600 
4784 
5906 
7144 

8319 
9491 


0146 
1340 
2531 
3718 
4903 
6084 
7262 

8436 
9008 


0265 
1459 
2050 
3837 
5021 
6202 
7379 

8554 

9725 


0385 
1578 
2769 
3955 
5139 
6320 
7497 

8671 
9842 


0504 
1698 
2887 
4074 
5257 
6437 
7614 

8788 
9959 


0624 
1817 
3006 
4192 
5376 
6555 
7732 

8905 


0743 
1936 
3125 
4311 
5494 
6673 
7849 

9023 


0803 
2055 
3244 
4429 
5612 
6791 
7967 

9140 


0982 
2174' 
3362 
4548 
5730 
6909 
8084 

9257 


561101 
2293 
3481 
4666 
5848 
7026 

8202 
9374 


0076 
1243 

2407 
3568 
4726 
5880 
7032 
8181 
9326 


0193 
1359 
2523 
3684 
4841 
5996 
7147 
8295 
9441 


0309 
1476 
2639 
3800 
4957 
6111 
7262 
8410 
9555 


0426 
1592 
2755 
3915 
5072 
6226 
7377 
8525 
9669 


570543 
1709 
2872 
4031 
5188 
6341 
7492 
8639 


0660 
1825 
2988 
4147 
5303 
6457 
7607 
8754 


0776 
1942 
3104 
42o3 
5419 
6572 
7722 
8868 


0893 
2058 
3220 
4379 
5534 
6687 
7836 
8983 


1010 
2174 
3336 
4494 
5650 
6802 
7951 
9097 


1126 
2291 
3452 
4610 
5765 
6917 
8066 
9212 


PROPORTIONAL PARTS. 


Diff. 1 


2 


3 


4 


5 


6 


7 


8 


9 


128 12.8 
127 12.7 
126 12.6 
125 12.5 
124 12.4 
123 12.3 
122 12.2 
121 12.1 
120 12.0 
119 11.9 


25.6 
25.4 
25.2 
25.0 
24.8 
24.6 
24.4 
24.2 
24.0 
23.8 


38.4 
38.1 . 
37.8 
37.5 
37.2 
36.9 
36.6 
36.3 
86.0 
35.7 


51.2 
50.8 
50.4 
50.0 
49.6 
49.2 
48.8 
48.4 
48.0 
47.6 


64.0 
63.5 
63.0 
62.5 
62.0 
61.5 
61.0 
60.5 
60.0 
59.5 


76.8 
76.2 
75.6 
75.0 
74.4 
73.8 
73.2 
72.6 
72.0 
71.4 


89.6 
88.9 
88.2 
87.5 
86.8 
86.1 
85.4 
84.7 
84.0 
83.3 


102.4 
101.6 
100.8 
100.0 
99.2 
98.4 
97.6 
96.8 
96.0 
95.2 


115.2 
114.3 
113.4 
112.5 
111.6 
110.7 
109.8 
108.9 
108.0 
107.1 



LOGARITHMS OF LUMBERS. 



No. 380. I, 579.] 






[No. 414 L. 617. 


N. 

380 

1 
2 
3 
4 
5 
6 
7 
8 
9 

390 
1 
2 
3 
4 
5 
G 
7 
8 

9 

400 
1 
2 
3 
4 
5 
6 
7 

8 
9 

410 

1 
2 
3 
4 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


579784 


9898 




1 




"0469" 

1608 
2745 
3879 
5009 
6137 
7262 
8384 
9503 








0012 
1153 

2291 
3426 
4557 

5686 
6812 
7935 
9056 


0126 

1207 
2404 
3539 
4670 
5799 
6925 
8047 
9167 


0241 
1381 
2518 
3652 
4783 
5912 
7037 
8100 
9279 


0355 
1495 
2031 
3765 

4896 
6024 
7149 

8272 
9391 


0583 

1722 
2858 
3992 
5122 
6250 
7374 
8496 
9615 


0697 

1836 
297'2 
4105 
5235 
6362 
7486 
8608 
9726 


0811 
1950 
3085 
4218 
5348 
6475 
7599 
8720 
9838 

0953 

2066 
3175 

4282 
5386 
6487 
7586 
8681 
9774 


114 

113 
112 

111 
110 
109 

108 
107 

106 
105 


580925 
2063 
3199 
4331 
5461 
6587 
7711 
8832 
9950 


1039 

2177 
3312 
4444 
5574 
6700 
7823 
8944 


0061 

1176 

2288 
3397 
4503 
5606 
6707 
7805 
8900 
9992 

~1082~ 

2169 
3253 
4334 
5413 
6489 
7562 
8633 
9701 


0173 

1287 
2339 
3508 
4614 
5717 
6817 
7914 
9009 


0284 

1399 
2510 
3018 
47.24 
5827 
6927 
8024 
9119 

0210 
1299 

2386 
3469 
4550 

5628 
6704 
7777 
8847 
9914 


0396 

1510 

2621 
3729 
4834 
5937 
7037 
8134 
9228 


0507 

1621 
2732 
3840 
4945 
6047 
7146 
8343 
9337 


0619 

1732 
2843 
3950 
5055 
6157 
7256 
8353 
9446 


0730 

1843 
2954 
4001 
5165 
6*67 
7306 
8462 
9556 


0842 

1955 

3004 
4171 
5270 
6377 
7476 
8572 
9665 

0755 
1843 

2928 
4010 
5089 
6166 
7241 
8312 
9381 


591065 
2177 
3286 
4393 
5496 
6597 
7695 
8791 
9883 

600973 

2060 
3144 
4226 
5305 
6381 
7455 
8526 
9594 


0101 
1191 

2277 
3361 
4442 
5521 
6596 
7669 
8740 
9808 


C319 
1408 

2494 

3577 
4G58 
5736 
6811 

7884 
8954 


0428 
1517 

2603 
3686 
4706 
5844 
6919 
7991 
9001 


0537 
1625 

2711 
3794 
4874 
5951 
7026 
8098 
9167 


0646 
1734 

2819 
3902 
4982 
6059 
7133 
8205 
9274 


0804 
1951 

3036 
4118 
5197 
6274 
7348 
8419 
9488 


0021 
1086 
2148 

3207 
4264 
5319 
6370 
7420 


0128 
1192 
2254 

3313 
4370 
5424 
6476 

7525 


0234 
1298 
2360 

3419 
4475 
5529 
6581 
7629 


0341 
1405 
2466 

3525 

4581 
5634 
6686 

7734 


0447 
1511 
2572 

3630 
4686 
5740 
6790 
7839 


0554 
1617 
2678 

8736 

4792 
5845 
6895 
7943 


010660 
1723 

2784 
3842 
4897 
5950 
7000 


0767 
1829 

2890 
3947 
5003 
6055 
7105 


0873 
1936 

2996 
4053 
5108 
6160 
7210 


0979 
2042 

3102 
4159 
5213 
6265 
7315 


PROPORTIONAL PARTS. 


Diff. 1 


2 


3 


4 


5 


S 


7 


8 


9 


118 11.8 
117 11.7 
116 11.6 
115 11.5 
114 11.4 
113 11.3 
113 11.2 

111 11.1 
110 11.0 

109 10.9 
108 10.8 
107 10.7 
106 10.6 
105 10.5 
104 10.4 


23.6 
23.4 
23.2 
23.0 
22.8 
22.6 
22.4 

22.2 

22.0 
21.8 
21.6 
21.4 
21.2 
21.0 
20.8 


35.4 
35.1 
34.8 
34.5 
84.2 
33.9 
33.6 

33.3 

33.0 
32.7 
32.4 
32.1 
31.8 
.81.5 
31.2 


47.2 
46.8 
46.4 
46.0 
45.6 
45.2 
44.8 

44.4 

44.0 
43.6 
43.2 
42.8 
42.4 
42.0 
41.6 


59.0 
58.5 
58.0 
57.5 
57.0 
56.5 
56.0 

55.5 
55.0 
54.5 
54.0 
53.5 
53.0 
52.5 
52.0 


70.8 
70.2 
69.6 
69.0 
68.4 
67.8 
67.2 

66.6 
66.0 
65.4 
64.8 
64.2 
63.6 
63.0 
62.4 


82.6 
81.9 

81.2 
80.5 
79.8 
79.1 

78.4 

77.7 
77.0 
76.3 
75.6 
74.9 
74.2 
73.5 
72.8 


94.4 
93.6 
92,8 
92.0 
91.2 
90.4 
89.6 

88.8 
88.0 
87.2 
86.4 
85.6 
84.8 
84.0 
83.2 


100.2 
105.3 
104.4 
103.5 
102.6 
101.7 
100.8 

99.9 
99.0 
98.1 
97.2 
96.3 
95.4 
94.5 
93.0 



LOGARITHMS OF NUMBERS. 



143 



No. 415 L. 618.] INo. 459 L. 662 


N. 

415 
6 

7 
8 
9 

420 
1 
2 
3 
4 
5 
6 

7 
8 
9 

430 
1 
2 
3 
4 
5 
6 

r 

8 
9 

440 
1 

2 
3 
4 
5 
6 

7 
8 
9 

450 
1 
2 

3 
4 
5 
6 

8 
9 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


618048 
9093 


8153 
9198 


8257 
9302 


8362 
9406 


8466 
9511 


8571 
9615 


8676 
9719 

0760 
1799 
2835 

3869 
4901 
5929 
6956 
7980 
9002 


8780 
9824 


8884 
9928 

~0968~ 
2007 
3042 

4076 
5107 
6135 
7161 
8185 
9206 


8989 

0032 
107'2 
2110 
3146 

4179 

5210 
6238 
7263 
8287 
9308 


105 
104 

103 
102 

101 
100 

99 
98 

97 
96 

95 


620136 
1176 
2214 

3249 

4282 
5312 
6340 
7366 
8389 
9410 


0240 
1280 
2318 

3353 
4385 
5415 
6443 
7468 
8491 
9512 

0580 
1545 
2559 

3569 

4578 
5584 
6588 
7590 
8589 
9586 


0344 

1384 
2421 

3456 
4488 
5518 
6516 
7571 
8593 
9613 


0448 
1488 
2525 

3559 
4591 
5621 
6648 
7673 
8695 
9715 


0552 
1592 
2628 

3663 
4695 
5724 
6751 

7775 
8797 
9817 


0656 
1695 
2732 

3766 
4798 
5827 
6853 
7878 
8900 
9919 


0864 
1903 
2939 

3973 
5004 
6032 
7058 
8082 
9104 


0021 
1088 
2052 
3064 

4074 
5081 
6087 
7089 
8090 
9088 


0123 
1139 
2153 
3165 

4175 
5182 
6187 
7189 
8190 
9188 

~0183~ 
1177 
2168 
3156 

4143 
5127 
6110 

7089 
8067 
9043 


0224 
1241 
2255 
3266 

4276 
5283 
6287 
7290 
8290 
9287 

0283 
1276 
2267 
3255 

4342 

5226 
6208 
7187 
8165 
9140 


0326 
1342 
2356 
3367 

4376 
5383 
6388. 
7'390 

8389 
9387 


630428 
1444 
2457 

3468 
4477 

5484 
6488 
7490 
8489 
9486 


0631 
1647 
2660 

3670 
4679 
5685 
6688 
7690 
8689 
9686 


0733 
1748 
2761 

3771 
4779 

5785 
6789 
7790 
8789 
9785 


0835 
1849 
2862 

3872 
4880 
5886 
6889 
7890 
8888 
9885 


0936 
1951 
2963 

3973 
4981 
5986 
6989 
7990 
8988 
9984 


0084 
1077 
2069 
3058 

4044 
5029 
6011 
6992 
7969 
8945 
9919 


0382 
1375 
2366 
3354 

4340 
5324 
6306 

7285 
8262 
9237 


640481 
1474 
2465 

3453 
4439 
5422 
6404 
7383 
8360 
9335 


0581 
1573 
2563 

3551 

4537 
5521 
6502 
7481 
8458 
9432 


0680 
1672 
2662 

3650 
4636 
5619 
6600 
7579 
8555 
9530 


0779 
1771 

2761 

3749 
4734 
5717 
6698 
7676 
8653 
9627 


0879 
1871 
2860 

3847 

4832 
5815 
6796 
777'4 
8750 
9724, 


0978 
1970 
2959 

3946 
4931 
5913 
6894 

7872 
8848 
9821 


0016 
0987 
1956 
2923 

3888 
4850 
5810 
6769 
7725 
8679 
9631 

0581 
1529 
2475 


0113 
1084 
2053 
3019 

3984 
4946 
5906 
6864 
7820 
8774 
9726 

0676 
1623 
2569 


0210 
1181 
2150 
3116 

4080 
5042 
6002 
6960 
7916 
8870 
9821 

~vm 

171fi 
2663 


650308 
1278 
2246 

3213 
4177 
5138 
6098 
7056 
8011 
8965 
9916 


0405 
1375 
2343 

3309 
4273 
5235 
6194 
7152 
8107 
9060 


0502 
1472 
2440 

3405 
4369 
5331 
6290 

7247 
8202 
9155 


0599 
1569 
2536 

3502 
4465 
5427 
6386 
7343 
8298 
9250 


0696 
1666 
2633 

3598 
4562 
5523 
6482 
7438 
8393 
9346 


0793 
17'62 
2730 

3695 
4658 
5619 
6577 
7534 
8488 
9441 


0890 
1859 
2826 

3791 
4754 
5715 
6673 
7629 
8584 
9536 


0011 
0960 
1907 


0106 
1055 

2002 


0201 
1150 
2096 


0296 
1245 
2191 


0391 
1339 

2286 


0486 
1434 
2380 


660865 
1813 


PROPORTIONAL PARTS. 


Diff. 1 


234 


5 


6 7 


8 


9 


105 10.5 
104 10.4 
103 10.3 
102 10.2 
101 10.1 
100 10.0 
99 9.9 


21.0 31.5 42.0 
20.8 31.2 41.6 
20.6 30.9 41.2 
20.4 30.6 40.8 
20.2 30.3 40.4 
20.0 30.0 40.0 
19.8 29.7 39.6 


52.5 
52.0 
51.5 
51.0 
50.5 
50.0 
49.5 


63.0 73.5 
62.4 72 8 
61.8 721 
61.2 714 
60.6 70 7 
60.0 70 
59.4 69.3 


84.0 
83.2 
82.4 
81.6 
80.8 
80.0 
79.2 


94.5 
93.6 
92.7 
91.8 
90.9 
90.0 
89.1 



LOGARITHMS OF LUMBERS. 



No. 4GO L. 662.] 




[No. 499 L. 698.' 


N, 





1 


2 


8 


4 


5 


C 


7 


8 


9 


Diff. 


460 


662758 


2852 


2947 


3041 


3135 


3230 


3324 


3418 3512 


3607 




1 


3701 


3795 


3889 


398 


3 


4078 


4172 


4266 


4 




4454 


4548 




2 


4642 


4736 


4830 


4924 


5018 


5112 


5206 


5299 


5393 


5487 


94 


3 


5581 


5675 


5769 


586 


2 


5956 


6050 


6143 


6$ 


5:-i7 


6331 


6424 




4 


6518 


6612 


6705 


6799 


6892 


6986 


7079 


7173 


7266 


7360 




5 


7453 


7546 


7640 


773 


3 


7826 


7920 


8013 


81 


08 


8199 


8293 




6 


8386 


8479 


8572 


866 


5 


8759 


8852 


8945 


9038 


9131 


9224 






9317 


9410 


9503 


959 


ft 


9689 


9782 


9875 


91 


)t\7 


























0060 


0153 


93 


8 


670246 


0339 


0431 


0524 


0617 


0710 


0802 


0895 


0988 


1080 




9 


1173 


1265 


1358 


1451 


1543 


1636 


1728 


1821 


1913 


2005 




470 


2098 


2190 


2283 


2375 


2467 


2560 


2652 


2744 


2836 


2929 




1 


3021 


3113 


3205 


329 


7 


3390 


3482 


3574 


3( 




3758 


3850 




2 


3942 


4034 


4126 


4218 


4310 


4402 


4494 


4586 


4677 


4769 


92 


3 


4861 


4953 


5045 


5137 


5228 


5320 


5412 


5503 


5595 


5687 




4 


5778 


5870 


5962 


605 


3 


6145 


6236 


6328 


fr 


U9 


6511 


6602 




5 


6694 


6785 


6876 


6968 


7059 


7151 


7242 


7333 


7424 


7516 




6 


7607 


7698 


7789 


788 


1 


7972 


8063 


8154 




nr> 


8336 


8427 




7 


8518 


8609 


8700 


8791 


8882 


8973 


9064 


9155 


9246 


9337 


91 


g 


9428 


9519 


9610 


970 


in 


9791 


9882 


9973 




























0063 


0154 


0245 




9 


680336 


0426 


0517 


0607 


0698 


0789 


0879 


0970 


10GO 


1151 




480 


1241 


1332 


1422 


1513 


1603 


1693 


1784 


1874 


1964 


2055 




1 


2145 


2235 


2326 


241 


6 


2506 


2596 


2686 


2 


77 


2867 


2957 




2 


3047 


3137 


3227 


3317 


3407 


3497 


3587 


3677 


37G7 


3857 


90 


3 


3947 


4037 


4127 


4217 


4307 


4396 


4486 


4576 


4666 


4756 




4 


4845 


4935 


5025 


511 


4 


5204 


5294 


5383 


fr 


173 


5563 


5652 




5 


5742 


5831 


5921 


6010 


6100 


6189 


6279 


6 


J68 


6458 


6547 




6 


6636 


6726 


6815 


69C 


4 


6994 


7083 


7172 


7, 


861 


7351 


7440 




7 


7529 


7618 


7707 


7796 


7886 


T975 


8064 


8153 


8242 


8331 


89 


8 


8420 


8509 


8598 


868 


7 


8776 


8865 


8953 


9( 


m 


9131 


9220 




9 


9309 


9398 


9486 


957 


5 


9664 


9753 


9841 


Q 


)30 






























0019 


0107 




490 


690196 


0285 


0373 


0462 


0550 


0639 


0728 


0816 


0905 


0993 




1 


1081 


1170 


1258 


134 




1435 


1524 


1612 


ji 


'00 


1789 


1877 




2 


1965 


2053 


2142 


2:230 


2318 


2406 


2494 


2583 


2671 


2759 




3 


2847 


2935 


3023 


311 


1 


3199 


3287 


3375 


& 


63 


3551 


3639 


88 


4 


3727 


3815 


3903 


3991 


4078 


4166 


4254 


4342 


4430 


4517 




5 


4605 


4693 


4781 


48C 


8 


4956 


5044 


5131 


5219 


5307 


5394 




6 


5482 


5569 


5657 


574 


4 


5832 


5919 


6007 


6( 


)94 


6182 


6269 




7 


6356 


6444 


6531 


6618 


6706 


6793 


6880 


6968 


7055 


7142 




8 


7229 


7317 


7404 


74[ 


1 


7578 


7665 


7752 


7* 


89 


7926 


8014 




9 


8100 


8188 


8275 


8362 


8449 


8535 


8622 


8709 


8796 


8883 


87 


PROPORTIONAL PARTS. 


Diff. 1 


2 3 


4 


5 


6 


7 


8 


9 


98 9.8 


19.6 29.4 


39.2 


49.0 


58.8 


68.6 


78.4 


88.2 


9? 9.7 


19.4 29.1 


38.8 


48.5 


58.2 


67.9 


77.6 


87.8 


96 9.6 


19.2 28.8 


38.4 


48.0 


57.6 


67.2 


76.8 


86.4 


95 9.5 


19.0 28.5 


38.0 


47.5 


57.0 


66.5 


76.0 


85.5 


94 9.4 


18.8 28.2 


37.6 


47.0 


56.4 


65.8 


75.2 


84.6 


93 9.3 


18.6 27.9 


37.2 


46.5 


55.8 


65.1 


74.4 


83.7 


92 9.2 


18.4 27.6 


36.8 


46.0 


55.2 


64.4 


73.6 


82.8 


91 9.1 


18.2 27.3 


36.4 


45.5 


54.6 


63.7 


72.8 


81.9 


90 9.0 


18.0 27.0 


36.0 


45.0 


54.0 


63.0 


72.0 


81.0 


89 8.9 


17.8 26.7 


35.6 


44.5 


53.4 


62.3 


71.2 


80.1 


88 8.8 


17.6 26.4 


35.2 


44.0 


52.8 


61.6 


70.4 


79.2 


87 8.7 17.4 26.1 


34.8 


43.5 


52.2 


60.9 


69.6 


78.3 


86 8.6 


17.2 25.8 


34.4 


43.0 


51.6 


60.2 


68.8 


77.4 



LOGARITHMS OP NUMBERS. 



145 



[ No. 500 L. 698.] [No. 544 L. 736. 


N. 





1 


2 


3 


4 


6 


6 


7 


8 


9 


Diff. 


500 


698970 


9057 


9144 


9231 


9317 


9404 


9491 


9578 


9664 


9751 




1 


9838 


9924 


























0011 


0098 


0184 


0271 


0358 


0444 


0531 


0617 




2 


700704 


0790 


0877 


0963 


1050 


1136 


1222 


1309 


1395 


1482 




3 


1568 


1654 


1741 


1827 


1913 


1999 


2086 


2172 


2258 


2344 





4 


2431 


2517 


2603 


2689 


2775 


2861 


2947 


3033 


3119 


3205 




5 


3291 


3377 


3463 


3549 


3635 


3721 


3807 


3893 


3979 


4065 


86 


6 


4151 


4236 


4322 


4408 


4494 


4579 


4665 


4751 


4837 


4922 




7 


5008 


5094 


5179 


5265 


5350 


5436 


5522 


5607 


5693 


5778 




8 


5864 


5949 


6035 


6120 


62G6 


6291 


6376 


6462 


6547 


6632 




9 


6718 


6803 


6888 


6974 


7059 


7144 


7229 


7315 


7400 


7485 




510 


7570 


7655 


7740 


7826 


7911 


7996 


8081 


8166 


8251 


8336 




1 


8421 


8506 


8591 


8676 


8761 


8846 


8931 


9015 


9100 


9185 


85 


2 


9270 


9355 


9440 


9524 


9609 


9694 


9779 


9863 


9948 


























0033 




3 


710117 


0202 


0287 


0371 


0456 


0540 


0625 


0710 


0794 


0879 




4 


0963 


1048 


1132 


1217 


1301 


1385 


1470 


1554 


1639 


1723 




5 


1807 


1892 


1976 


2060 


2144 


2229 


2313 


2397 


2481 


2566 




6 


2650 


2734 


2818 


2902 


2986 


3070 


3154 


3238 


3323 


3407 


ft4 


7 


3491 


3575 


3659 


3742 


3826 


3910 


3994 


4078 


4162 


4246 


04 


8 


4330 


4414 


4497 


4581 


4665 


4749 


4833 


4916 


5000 


5084 




9 


5167 


5251 


5335 


5418 


5502 


5586 


5669 


5753 


5836 


5920 




520 


6003 


6087 


6170 


6254 


6337 


6421 


6504 


6588 


6671 


6754 




1 


6838 


6921 


7004 


7088 


7171 


7254 


7338 


7421 


7504 


7587 




2 


7671 


7754 


7837 


7920 


8003 


8086 


8169 


8253 


8336 


8419 




3 


8502 


8585 


8668 


8751 


8834 


8917 


9000 


9083 


9165 


9248 


83 


4 


9331 


9414 


9497 


9580 


9663 


9745 


9828 


9911 


9994 


























0077 




5 


720159 


0242 


0325 


0407 


0490 


0573 


0655 


0738 


0821 


0903 




6 


0986 


1068 


1151 


1233 


1316 


1398 


1481 


1563 


1646 


1728 




7 


1811 


1893 


1975 


2058 


2140 


2222 


2305 


2387 


2469 


2552 




8 


2634 


2716 


2798 


2881 


2963 


3045 


3127 


3209 


3291 


3374 




9 


3456 


3538 


3620 


3702 


3784 


3866 


3948 


4030 


4112 


4194 


82 


530 


4276 


4358 


4440 


4522 


4604 


4685 


4767 


4849 


4931 


5013 




1 


5095 


5176 


5258 


5340 


5422 


5503 


5585 


5667 


5748 


5830 




2 


5912 


'5993. 


6075 


6156 


6238 


6320 


6401 


6483 


6564 


6646 




3 


6727 


6809 


6890 


697'2 


7053 


7134 


7216 


7297 


7379 


7460 




4 


7541 


7623 


7704 


7785 


7'866 


7948 


8029 


8110 


8191 


8273 




5 


8354 


8435 


8516 


8597 


8678 


8759 


8841 


8922 


9003 


9084 




6 


9165 


9246 


9327 


9408 


9489 


9570 


9651 


9732 


9813 


9893 


81 


7 


9974 


























0055 


0136 


0217 


0298 


0378 


0459 


0540 


0621 


0702 




8 


730782 


0863 


0944 


1024 


1105 


1186 


1266 


1347 


1428 


1508 




9 


1589 


1669 


1750 


1830 


1911 


1991 


2072 


2152 


2233 


2313 




540 


2394 


2474 


2555 


2635 


2715 


2796 


2876 


2956 


3037 


3117 




1 


8197 


3278 


3358 


3438 


3518 


3598 


3679 


3759 


3839 


3919 




2 


3999 


4079 


4160 


4240 


4320 


4400 


4480 


4560 


4640 


4720 


on 


3 


4800 


4880 


4960 


5040 


5120 


5200 


5279 


5359 


5439 


5519 


w 


4 


5599 


5679 


5759 


5838 


5918 


5998 


6078 


6157 


6237 


6317 




PROPORTIONAL PARTS. 


Diff. 1 


234 


5 


678 


9 


87 8.7 


17.4 26.1 34.8 


43.5 


52.2 60.9 69.6 


78.3 


86 8.6 


17.2 25.8 34.4 


43.0 


51.6 60.2 68.8 


77.4 


85 8.5 


17.0 25.5 34.0 


42.5 


51.0 59.5 68.0 


76.5 


84 8.4 


16.8 25.2 33.6 


42.0 


50.4 58.8 67.2 


75.6 



LOGARITHMS OP CUMBERS. 



No. 545 L. 736.] 


[No. 584 L. 767. 


N. 





1 


2 


8 


4 


5 


6 


7 


8 


9 


Diff. 


545 


736397 


6476 


6556 


6635 


6715 1 6795 


6874 


6954 


7034 


7113 




6 


7193 


7272 


7352 


7431 


7511 


7590 


7670 


7749 


7829 


7908 




7 


7987 


8067 


8146 


8S& 


25 


8305 


8384 


8463 


8543 


862 


2 


8701 




8 


8781 


8860 


8939 


9018 


9097 


9177 


9256 


9335 


9414 


9493 




9 


9572 


9651 


9731 


98 


10 


9889 


9968 






























0047 


O19A 


H90 




C\9RA 




550 


740363 


0442 


0521 


0600 


0678 


0757 


0836 


U1/*D 

0915 


\J4\JO 

0994 


\)4& 

1073 


29 


1 


1152 


1230 


1309 


13* 


S8 


1467 


1546 


1624 


1703 


178 


2 


1860 




2 


1939 


2018 


2096 


2175 


2254 


2332 


2411 


2489 


2568 


2647 




3 


2725 


2804 


2882 


29t 


51 


3039 


! 3118 


3196 


3275 


335 


3 


3431 




4 


3510 


3588 


3667 


3745 


3823 


3902 


3980 


4058 


4136 


4215 




5 


4293 


4371 


4449 


4528 


4606 


4684 


4762 


4840 


4919 


4997 




6 


5075 


5153 


5231 


53( 


)9 


5387 


; 5465 


5543 


5621 


569 


) 


5777 


78 


7 


5855 


5933 


6011 


601 


S9 


6167 


6245 


6323 


6401 


647 


) 


6556 




8 


6634 


6712 


6790 


68( 


58 


6945 


7023 


7101 


7179 


7256 


7334 




9 


7412 


7489 


7567 


7645 


7722 


7800 


7878 


7955 


8oa 


J 


8110 




560 


8188 


8266 


8343 


8421 


8498 


1 8576 


8653 


8731 


880 


* 


8885 




1 


8963 


9040 


9118 


9195 


9272 


| 9350 


9427 


9504 


9582 


9659 




2 


9736 


9814 


9891 


99( 


58 




























0045 


0123 


0200 


O977 


AOR 


4 


fMOl 




3 


750508 


0586 


0663 


0740 


0817 


0894 


0971 


\)/ii t 

1048 


UoO^ 

1125 


U4ol 

1202 




4 


1279 


1356 


1433 


1510 


1587 


1664 


1741 


1818 


1895 


1972 




5 


2048 


2125 


2202 


2279 


2356 


2433 


2509 


2586 


2663 


2740 


77 


6 


2816 


2893 


2970 


3fr 


17 


3123 


3200 


3277 


3a r >3 


343 


) 


3506 




7 


3583 


3660 


3736 


3813 


3889 


3966 


4042 


4119 


4195 


4272 




8 


4348 


4425 


4501 


45' 


'8 


4654 


4730 


4807 


4883 


496 


3 


5036 




9 


5112 


5189 


5265 


5341 


5417 


i 5494 


5570 


5646 


5722 


5799 




5TO 


5875 


5951 


6027 


6103 


6180 


1 6256 


6332 


6408 


6484 


6560 




1 


6636 


6712 


6788 


6864 


6940 


7016 


7092 


7168 


7244 


7320 


76 


2 


7396 


7472 


7548 


7ft 


24 


7700 


i 7775 


7851 


7927 


800 


3 


8079 




3 


8155 


8230 


8306 


8382 


8458 


> 8533 


8609 


8685 


87'61 


8836 




4 


8912 


8988 


9063 


91, 


39 


9214 


i 9290 


9366 


9441 


951 


I 


9592 




5 


9668 


9743 


9819 


DM 


)4 


997'0 






























0045 


0121 


0196 


097 


-> 


fiQ/17 




6 


760422 


0498 


0573 


0649 


0724 


0799 


0875 


0950 


\}4ii & 
1025 


Uo4< 

1101 




7 


1176 


1251 


1326 


1402 


1477 


1552 


1627 


1702 


1778 


1853 




8 


1928 


2003 


2078 


21, 


33 


2228 


i 2303 2378 


2453 


252 


1 


2604 




9 


2679 


2754 


2829 


2904 


29?'8 


3053 3128 


3203 


3278 


3353 


73 


580 


3428 


3503 


3578 


3653 


3727 


3802 


3877 


3952 


4027 


4101 




1 


4176 


4251 


4326 


44( 


)0 


4475 


4550 


4624 


4699 


477 


i 


4848 




2 


4923 


4998 


5072 


5147 


5221 


5296 


5370 


5445 


5520 


5594 




3 


5669 


5743 


5818 


58< 


K 


5966 


6041 


6115 


6190 


62fr 


1 


6338 




4 


6413 


6487 


6562 


6636 


6710 


6785 


6859 


6933 


7007 


7082 




PROPORTIONAL PARTS. 


Diff. 1 


2 3 


4 


5 


6 7 


8 
66.4 


9 

74.7 


83 8.3 


16.6 24.9 


33.2 


41.5 


49.8 58.1 


82 8.2 


16.4 24.6 


32.8 


41.0 


49.2 57.4 


65.6 


73.8 


81 8.1 


16.2 24.3 


32.4 


40.5 


48.6 56.7 


64.8 


72.9 


80 8.0 


16.0 24.0 


32.0 


40.0 


48.0 56.0 


64.0 


72.0 


79 7.9 


15.8 23.7 


31.6 


39.5 


47.4 55.3 


63.2 


71.1 


78 78 


15.6 23.4 


31.2 


39.0 


46.8 54.6 


62.4 


70.2 


77 7.7 


15.4 23.1 


30.8 


38.5 


46.2 53.9 


61.6 


69.3 


76 7.6 


15.2 22.8 


30.4 


38.0 


45.6 53.2 


60.8 


68.4 


75 7.5 


15.0 22.5 


30.0 


37.5 


45.0 52.5 


60.0 


67.5 


74 7.4 


14.8 22.2 


29.6 


37.0 


44.4 51.8 


59.2 


66.6 



LOGARITHMS OF NUMBERS. 



147 



No. 585 L. 767.1 




[No. 629 L. 799. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


585 


767156 


7230 


7304 


7379 


7453 


7527 


7601 


7675 


7749 


7823 




6 


7898 


7972 


8046 


8120 


8194 


8268 


8342 


8416 


8490 


8564 


74 


7 


8638 


8712 


8786 


88t 





8934 


9008 


9082 


9 


156 


9230 


9303 




g 


9377 


9451 


9525 


95 


9 


9673 


9746 


9820 


9 


894 


9968 






























0042 




9 


770115 


0189 


0263 


0336 


0410 


0484 


0557 


0631 


0705 


0778 




590 


0852 


0926 


0999 


1073 


1146 


1220 


1293 


1367 


1440 


1514 




1 


1587 


1661 


1734 


180 


8 


1881 


1955 


2028 


2 


102 


2175 


2248 




2 


2322 


2395 


2468 


2542 


2615 


2688 


2762 


2835 


2908 


2981 




3 


3055 


3128 


3201 


327 


4 


3348 


3421 


3494 


a 


567 


3640 


3713 




4 


3786 


3860 


3933 


4006 


4079 


4152 


4225 


4298 


4371 


4444 


73 


5 


4517 


4590 


4663 


473 


6 


4809 


4882 


4955 


& 


108 


5100 


5173 




6 


5246 


5319 


5392 


5465 


5538 


5610 


5683 


5756 


5829 


5902 




7 


5974 


6047 


6120 


619 


3 


6265 


6338 


6411 


6- 


188 


6556 


6629 




8 


6701 


6774 


6846 


6919 


6992 


7064 


7137 


7209 


7282 


7354 




9 


7427 


7499 


7572 


7644 


7717 


7789 


7862 


7934 


8006 


8079 




600 


8151 


8224 


8296 


8368 


8441 


8513 


8585 


8 


.58 


8730 


8802 




1 


8874 


8947 


9019 


909 


1 


9163 


9236 


9308 


9 


M) 


9452 


9524 




2 


9596 


9669 


9741 


981 


3 


9885 


9957 






























0029 


O 


mi 


0173 


0245 




3 


780317 


0389 


0461 


0533 


0605 


0677 


0749 


0821 


0893 


0965 


72 


4 


1037 


1109 


1181 


1253 


1324 


1396 


1468 


1540 


1612 


1684 




5 


1755 


1827 


1899 


197 


1 


2042 


2114 


2186 


2 1 


358 


2329 


2401 




6 


2473 


2544 


2616 


268 


8 


2759 


2831 


2902 


2 


J?4 


3046 


3117 




7 


3189 


3260 


3332 


3403 


3475 


3546 


3618 


8 


>s<) 


3761 


3832 




8 


3904 


3975 


4046 


4118 


4189 


4261 


4332 


4403 


4475 


4546 




9 


4617 


4689 


4760 


483 


1 


4902 


4974 


5045 


5116 


5187 


5259 




610 


5330 


5401 


5472 


5543 


5615 


5686 


5757 


5 


328 


5899 


5970 




1 


6041 


6112 


6183 


625 


4 


6325 


6396 


6467 


6 


588 


6609 


6680 


71 


2 { 6751 


6822 


6893 


6964 


7035 


7106 


7177 


7248 


7319 


7390 




3 7460 


7531 


7602 


767 


a 


7744 


7815 


7885 


7 


)56 


8027 


8098 




4 


8168 


8239 


8310 


838 


i 


8451 


8522 


8593 


8 


>r>3 


8734 


8804 




5 


8875 


8946 


9016 


908 


7 


9157 


9228 


9299 


9 


369 


9440 


9510 






9581 


9651 


9722 


979 





9863 


9933 






























0004 


Ot 


"74 


0144 


0215 




7 


790285 


0356 


0426 


0496 


0567 


0637 


0707 


0778 


0848 


0918 




8 


0988 


1059 


1129 


119 


9 


1269 


1340 


1410 


1 


ISO 


1550 


1620 




9 


1691 


1761 


1831 


1901 


1971 


2041 


2111 


2181 


2252 


2322 




620 


2392 


2462 


2532 


2602 


2672 


2742 


2812 


2! 


m 


2952 


3022 


70 


1 


3092 


3162 


3231 


330 


1 


3371 


3441 


3511 


a 


581 


3651 


3721 




2 


3790 


3860 


3930 


400 





4070 


4139 


4209 


4 


279 


4349 


4418 




3 


4488 


4558 


4627 


4697 


4767 


4836 


4906 


4976 


5045 


5115 




4 


5185 


5254 


5324 


539 


3 


5463 


5532 


5602 


5 


>'~ 


5741 


5811 




5 


5880 


5949 


6019 


6088 


6158 


6227 


6297 


6366 


6436 


6505 




6 


6574 


6644 


6713 


678 


8 


6852 


6921 


6990 


7( 


MJO 


7129 


7198 




7 


7268 


7337 


7406 


7475 


7545 


7614 


7683 


7752 


7821 


7890 




8 


7960 


8029 


8008 


816 


7 


8236 


8305 


8374 


& 


i43 


8513 


8582 




9 


8651 


8720 


8789 


8858 


8927 


8996 


9065 


9134 


9203 


927.8 


69 


PROPORTIONAL PARTS. 


DiflP. 1 


2 3 


4 


5 


6 


7 8 


9 


75 7.5 


15.0 22.5 


30.0 


37.5 


45.0 


52.5 60.0 


67.5 


74 7.4 


14.8 22.2 


29.6. 


37.0 


44.4 


51.8 59.2 


66.6 


73 7.3 


14.6 21.9 


29.2 


36.5 


43.8 


51.1 58.4 


65.7 


72 7.2 


14.4 21.6 


28.8 


36.0 


43.2 


50.4 57.6 


64.8 


71 7.1 


14.2 21.3 


28.4 


35.5 


42.6 


49.7 56.8 


63.9 


70 7.0 


14.0 21.0 


28.0 


35.0 


42.0 


49.0 56.0 


63.0 


69 6.9 


13.8 20.7 


27.6 


34.5 


41.4 


48.3 55.2 


62.1 



148 



LOGARITHMS OP KUMBERS. 



No, 630 L. 799.] [No. 674 L. 829. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


630 


799341 


9409 


9478 


9547 


9616 


9685 


9754 


9823 


9892 


9961 




1 


800029 


0098 


0167 


0236 


0305 


0373 


0442 


0511 


0580 


0640 




2 


0717 


0786 


0854 


0923 


0992 


1061 


1129 


1198 


1266 


1335 




3 


1404 


1472 


1541 


1609 


1678 


1747 


1815 


1884 


1952 


2021 




4 


2089 


2158 


2226 


2295 


2363 


2432 


2500 


2568 


2637 


2705 




5 


2774 


2842 


2910 


2979 


3047 


3116 


3184 


3252 


3321 


3389 




6 


3457 


3525 


3594 


3662 


3730 


3798 


3867 


3935 


4003 


4071 




7 


4139 


4208 


4276 


4344 


4412 


4480 


4548 


4616 


4685 


4753 




8 


4821 


4889 


4957 


5025 


5093 


5161 


5229 


5297 


5365 


5433 


68 


9 


5501 


5569 


5637 


5705 


5773 


5841 


5908 


5976 


6044 


6112 




640 


806180 


6248 


6316 


63&4 


6451 


6519 


6587 


6655 


6723 


6790 




1 


6858 


6926 


6994 


7061 


7129 


7197 


7264 


7332 


7400 


7467 




2 


7535 


7603 


7670 


7738 


7806 


7873 


7941 


8008 


8076 


8143 




3 


8211 


8279 


8346 


8414 


8481 


8549 


8616 


8684 


8751 


8818 




4 


8886 


8953 


9021 


9088 


9156 


9223 


9290 


9358 


9425 


9492 




5 


9560 


9627 


9694 


9762 


9829 


9896 


9964 


























0031 


0098 


0165 




6 


810233 


0300 


0367 


0434 


0501 


0569 


0636 


0703 


0770 


0837 




7 


0904 


0971 


1039 


1106 


1173 


1240 


1307 


1374 


1441 


1508 


67 


8 


1575 


1642 


1709 


1776 


1843 


1910 


1977 


2044 


2111 


2178 




9 


2245 


2812 


2379 


2445 


2512 


2579 


2646 


2713 


2780 


2847 




650 


2913 


2980 


3047 


3114 


3181 


3247 


3314 


3381 


3448 


3514 




1 


3581 


3648 


3714 


3781 


3848 


3914 


3981 


4048 


4114 


4181 




2 


4248 


431-4 


4381 


4447 


45*4 


4581 


4647 


4714 


4780 


4847 




3 


4913 


4980 


5046 


5113 


5179 


5246 


5312 


5378 


5445 


5511 




4 


5578 


5644 


5711 


5777 


5843 


5910 


5976 


6042 


6109 


6175 




I 


6241 


6308 


6374 


6440 


6506 


6573 


6639 


6705 


6771 


6838 




6 


6904 


6970 


7036 


7102 


7169 


7235 


T301 


7367 


7433 


7499 




7 


75G5 


7631 


7698 


7764 


7830 


7896 


7962 


8028 


8094 


8160 




8 


8226 


8292 


8358 


8424 


8490 


8556 


8622 


8688 


8754 


8820 




9 


8885 


8951 


9017 


9083 


9149 


9215 


9281 


9346 


9412 


9478 


Co 


660 


9544 


9610 


9676 


9741 


9807 


9873 


9969 


0004 


0070 


0136 




1 


820201 


0267 


0333 


0399 


0464 


0530 


0595 


0661 


0727 


0792 




2 


0858 


0924 


0989 


1055 


1120 


1186 


1251 


1317 


1382 


1448 




3 


1514 


1579 


1645 


1710 


1775 


1841 


1906 


1972 


2037 


2103 




4 


2168 


2233 


2299 


2364 


2430 


2495 


2560 


2626 


2691 


2756 




5 


2822 


2887 


2952 


3018 


3083 


3148 


3213 


3279 


3344 


3409 




6 


3474 


3539 


3605 


3670 


37'35 


3800 


3865 


3930 


3996 


4061 




7 


4126 


4191 


4256 


4321 


4386 


4451 


4516 


4581 


4646 


4711 


fiK 


8 


4776 


4841 


4906 


4971 


5036 


5101 


5166 


5231 


5296 


5361 


DO 


9 


5426 


5491 


5556 


5621 


5686 


5751 


5815 


5880 


5945 


6010 




670 


6075 


6140 


6204 


6269 


6334 


6399 


6464 


6528 


6593 


6658 




1 


6723 


6787 


6852 


6917 


6981 


7046 


7111 


7175 


7240 


7305 




2 


7369 


7434 


7499 


7563 


7628 


7692 


7757 


7821 


7886 


7951 




3 


8015 


8080 


8144 


8209 


8273 


8338 


8402 


8467 


8531 


8595 




4 


8660 


8724 


8789 


8853 


8918 


8982 


9046 


9111 


9175 


9239 




PROPORTIONAL PARTS. 


Diff 


1 


2 


3 4 


5 


678 


9 


68 


6.8 


13.6 


20.4 27.2 


34.0 


40.8 47.6 544 


61.2 


67 


6.7 


13.4 


20.1 26.8 


33.5 


40.2 46.9 53,6 


60.3 


66 


6.6 


13.2 


19.8 26.4 


33.0 


39.6 46.2 52.8 


59.4 


65 


6.5 


13.0 


19.5 26.0 


32.5 


39.0 45.5 52.0 


58.5 


64 


6.4 


1.8 


19.2 25. Q 


32.0 


38,4 44.8 51.2 


57.6 



LOGARITHMS OF NUMBERS. 



U9 



No. 675 L. 829.? [No. 719 L. 857. 


N. 





1 


2 


8 


4 


6 


6 


7 


8 


9 


Diff. 


675 


829304 


9368 


9432 


9497 


9561 


9625 


9690 


9754 


9818 


9882 




g 


9947 


























0011 


0075 


0139 


0204 


0268 


0332 


0396 


0460 


0525 




7 


830589 


0653 


0717 


0781 


0845 


0909 


0973 


1037 


1102 


1166 




8 


1230 


1294 


1358 


1422 


1486 


1550 


1614 


1678 


1742 


1806 


64 


9 


1870 


1984 


1998 


2062 


2126 


2189 


2253 


2317 


2381 


2445 




680 


2509 


2573 


2637 


2700 


2764 


2828 


2892 


2956 


3020 


3083 




1 


3147 


3211 


3275 


3338 


3402 


3466 


3530 


3593 


3657 


3721 




2 


3784 


3848 


3912 


3975 


4039 


4103 


4166 


4230 


4294 


4357 




3 


4421 


4484 


4548 


4611 


4675 


4739 


4802 


4866 


4929 


4993 




4 


5056 


5120 


5183 


5247 


5310 


5373 


5437 


5500 


5564 


5627 




5 


5691 


5754 


5817 


5881 


5944 


6007 


6071 


6134 


6197 


6261 




6 


6324 


6387 


6451 


6514 


6577 


6641 


6704 


6767 


6830 


6894 




7 


6957 


7020 


7083 


7146 


7210 


7273 


7336 


7399 


7462 


7525 




8 


7588 


7652 


7715 


7778 


7841 


7904 


7967 


8030 


8093 


8156 




9 


8219 


8282 


8345 


8408 


8471 


8534 


8597 


8660 


8723 


8786 


63 


690 


8849 


8912 


8975 


9038 


9t01 


9164 


9227 


9289 


9&52 


9415 




1" 


9478 


9541 


9604 


9667 


97'29 


9792 


9855 


9918 


9981 


























0043 




2 


840106 


0169 


0232 


0294 


0357 


0420 


0482 


0545 


0608 


0671 




3 


0733 


0796 


0859 


0921 


0984 


1046 


1109 


1172 


1234 


1297 




4 


1359 


1422 


1485 


1547 


1610 


1672 


1735 


1797 


1860 


1922 




'5 


1985 


2047 


2110 


2172 


2235 


2297 


2360 


2422 


2484 


2547 




6 


2609 


2672 


2734 


2796 


2859 


2921 


2983 


3046 


3108 


3170 




7 


3233 


3295 


3357 


3420 


3482 


3544 


3606 


3669 


3731 


3793 




8 


3855 


3918 


3980 


4042 


4104 


4166 


4229 


4291 


4353 


4415 




9 


447.7 


4539 


4601 


4664 


4726 


4788 


4850 


4912 


4974 


5036 




700 


5098 


5160 


5222 


5284 


5346 


5408 


5470 


5532 


5594 


5656 


62 


1 


5718 


5780 


5842 


5904 


5966 


6028 


6090 


6151 


6213 


6275 




2 


6337 


6399 


6461 


6523 


6585 


6646 


6708 


6770 


6832 


6894 




3 


6955 


7017 


7079 


7141 


7202 


7264 


7326 


7388 


7449 


7511 




4 


7573 


7634 


7696 


7758 


7819 


7881 


7943 


8004 


8066 


8128 




5 


8189 


8251 


8312 


8374 


8435 


8497 


8559 


8620 


8682 


8743 




6 


8805 


8866 


8928 


8989 


9051 


9112 


9174 


9235 


9297 


9358 




7 


9419 


9481 


95^ 


9604 


9665 


9726 


9788 


9849 


9911 


9972 




8 


850033 


0095 


0156 


0217 


0279 


0340 


0401 


0462 


0524 


0585 




9 


0646 


0707 


0769 


0830 


.0891 


0952 


1014 


1075 


1136 


1197 




710 


1258 


1320 


1381 


1442 


1503 


1564 


1625 


1686 


1747 


1809 




1 


1870 


1931 


1992 


2053 


2114 


2175 


2236 


2297 


2358 


2419 




2 


2480 


2541 


2602 


2663 


2724 


2785 


2846 


2907 


2968 


3029 


61 


3 


3090 


3150 


3211 


3272 


3333 


3394 


3455 


3516 


3577 


3637 




4 


3698 


3759 


3820 


3881 


3941 


4002 


4063 


4124 


4185 


4245 




5 


4306 


4367 


4428 


4488 


4549 


4610 


4670 


4731 


4792 


4852 




6 


4913 


4974 


5034 


5095 


5156 


5216 


5277 


5337 


5398 


5459 




7 


5519 


5580 


5640 


5701 


5761 


5822 


5882 


5943 


6003 


6064 




8 


6124 


6185 


6245 


6306 


6366 


6427 


6487 


6548 


6608 


6668 




9 


6729 


6789 


6850 


6910 


6970 


7031 


7091 


7152 


7212 


7272 




PROPORTIONAL PARTS. 


Diff 


. 1 


234 


5 


678 


9 


65 


6.5 


13.0 19.5 26.0 


32.5 


39.0 45.5 52.0 


58.5 


64 


6.4 


12.8 19.2 25.6 


32.0 


38.4 44.8 51.2 


57.6 


63 


6.3 


12.6 18.9 25.2 


31.5 


37.8 44.1 50.4 


56.7 


62 


6.2 


12.4 18.6 24.8 


31.0 


37.2 43.4 49.6 


55 8 


61 


6.1 


12.2 18.3 24.4 


30.5 


36.6 42.7 48,8 


54.9 


60 


6.0 


12.0 18.0 24.0 


30.0 


36.0 42.0 48.0 


54.0 



150 



LOGARITHMS OF K UMBERS. 



No. 720 L. 857.] [No. 764 L. 883. 


























. 





















9 


Diff. 


720 


857332 


7393 


7453 


7513 


7574 


7634 


7694 


7755 


7815 


7875 




1 


7935 


7995 


8056 


8116 


8176 


8236 


8297 


8357 


8417 


8477 




2 


8537 


8597 


8657 


8718 


8778 


8838 


8808 


8958 


9018 


9078 




3 


9138 


9198 


9258 


9318 


9879 


9439 


9499 


9559 


9619 


9679 


60 


4 


9739 


9799 


9859 


9918 


9978 


























0038 


0098 


0158 


0218 


0278 




5 


860338 


0398 


0458 


0518 


0578 


0637 


0697 


0757 


0817 


0877 




6 


0937 


0996 


1056 


1116 


1176 


1236 


1295 


1355 


1415 


1475 




7 


1534 


1594 


1654 


1714 


1773 


1833 


1893 


1952 


2012 


2072 




8 


2131 


2191 


2251 


2310 


2370 


2430 


2489 


2549' 


2608 


2668 




9 


2728 


2787 


2847 


2906 


2966 


3025 


3085 


31 J4 


3204 


3263 




730 


3323 


3382 


3442 


3501 


3561 


3620 


3680 


3739 


3799 


3858 




1 


3917 


3977 


4036 


4096 


4155 


4314 


4274 


4333 


4392 


4452 




2 


4511 


4570 


4630 


4689 


4748 


4808 


4867 


4926 


4985 


5045 




3 v 5104 


5163 


5222 


5282 


5341 


5400 


5459 


5519 


5578 


5637 




4 


5696 


5755 


5814 


5874 


5933 


5992 


6051 


6110 


6169 


6228 




5 


6287 


6346 


6405 


6465 


6524 


6583 


G642 


6701 


6760 


6819 




6 


6878 


6937 


6996 


7055 


7114 


7173 


7232 


7291 


7350 


7409 


59 


7 


7467 


7526 


7585 


7644 


7703 


7762 


7821 


7880 


7939 


7998 




8 


8056 


8115 


8174 


8233 


8292 


8350 


8409 


8468 


8527 


8586 




9 


8644 


8703 


8762 


8821 


8879 


8938 


8997 


9056 


9114 


9173 




740 


9232 


9290 


9349 


9408 


9466 


9525 


9584 


9642 


9701 


9760 




1 


9818 


9877 


9935 


9994 


























0053 


0111 


0170 


0228 


O987 


O'3AK 




2 


870404 


0462 


0521 


0579 


0638 


0696 


Ul t\J 
0755 


0813 


Uo< 

0872 


Uo40 

0930 




3 


0989 


1047 


1106 


1164 


1223 


1281 


1339 


1398 


1456 


1515 




4 


1573 


1631 


1690 


1748 


1806 


1865 


1923 


1981 


2040 


2008 




5 


2156 


2215 


2273 


2331 


2389 


2448 


2506 


2564 


2622 


2681 




6 


2739 


2797 


2855 


2913 


2972 


3030 


3088 


8146 


3204 


3262 




7 


3321 


3379 


3437 


3495 


3553 


3611 


3669 


3727 


3785 


3844 




8 


3902 


3960 


4018 


4076 


4134 


4192 


4250 


4308 


4366 


4424 


58 


9 


4482 


4540 


4598 


4656 


4714 


4772 


4830 


4888 


4945 


5003 




750 


5061 


5119 


5177 


5235 


5293 


5351 


5409 


5466 


5524 


5582 




1 


5640 


5698 


5756 


5813 


5871 


5929 


5987 


6045 


6102 


6160 




2 


6218 


6276 


6333 


6391 


6449 


6507 


6564 


6622 


6680 


6737 




3 


6795 


6853 


6910 


6968 


7026 


7083 


7141 


7199 


7256 


7314 




4 


7371 


7429 


7487 


7544 


7602 


7659 


7717 


7774 


7832 


7889 




5 


7947 


8004 


8062 


8119 


8177 


8234 


8292 


8349 


8407 


8464 




6 


8522 


8579 


8637 


8694 


8752 


8809 


8866 


8924 


8981 


9039 




7 


9096 


9153 


9211 


9268 


9325 


9383 


9440 


9497 


9555 


9612 




g 


9669 


9726 


9784 


9841 


9898 


9956 


























0013 


0070 


0127 


0185 




9 


880242 


0299 


0356 


0413 


0471 


0528 


0585 


0642 


0699 


0756 




760 


0814 


0871 


0928 


0985 


1042 


1099 


1156 


1213 


1271 


1328 




1 


1385 


1442 


1499 


1556 


1613 


1670 


1727 


1784 


1841 


1898 




2 


1955 


2012 


2069 


2126 


2183 


2240 


2297 


2354 


2411 


2468 


57 


3 


2525 


2581 


2638 


2695 


2752 


2809 


2866 


2923 


2980 


3037 




4 


3093 


3150 


3207 


3264 


3321 


3377 


3434 


3491 


3548 


3605 




PROPORTIONAL PARTS. 


Diff 


1 


2 


3 4 


5 


678 


9 


59 


5.9 


11.8 


17.7 23.6 


29.5 


35.4 41.3 47.2 


53.1 


58 


5.8 


11.6 


17.4 23.2 


29.0 


S4.8 40.6 46.4 


52.2 


57 


5.7 


11.4 


17.1 22.8 


28.5 


34.2 39.9 45.6 


51.3 


56 


5.6 


11.2 


16.8 22.4 


28.0 


33.6 39.2 44.8 


50.4 



LOGARITHMS OF NUMBERS. 



151 



No. 765 L. 883.] [No. 809 L. 908. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


765 


883661 


3718 


3775 


3832 


3888 


3945 


4002 


4059 


4115 


4172 




6 


4229 


4285 


4342 


4399 


4455 


4512 


4569 


4625 


4682 


4739 




7 


4795 


4852 


4909 


4965 


5022 


5078 


5135 


5192 


5248 


5305 




8 


5361 


5418 


5474 


5531 


5587 


5644 


5700 


5757 


5813 


5870 




9 


5926 


5983 


6039 


6096 


6152 


6209 


6265 


6321 


6378 


6434 




770 


6491 


6547 


6604 


6660 


6716 


6773 


6829 


6885 


6942 


6998 




1 


7054 


7111 


7167 


7223 


7280 


7336 


7392 


7449 


7505 


7561 




2 


7617 


7674 


7730 


7786 


7842 


7898 


7955 


8011 


8067 


8123 




3 


8179 


8236 


8292 


8348 


8404 


8460 


8516 


8573 


8629 


8685 




4 


8741 


8797 


8853 


8909 


8965 


9021 


9077 


9134 


9190 


9246 




5 


9302 


9358 


9414 


9470 


9526 


9582 


9638 


9694 


9750 


9806 


56 


g 


9862 


9918 


9974 


























0030 


0086 


0141 


0197 


0253 


0309 


0365 




7 


890421 


0477 


0533 


0589 


0645 


0700 


0756 


0812 


0868 


0924 




8 


0980 


1035 


1091 


1147 


1203 


1259 


1314 


1370 


1426 


1482 




9 


1537 


1593 


1649 


1705 


1760 


1816 


1872 


1928 


1983 


2039 




780 


2095 


2150 


2206 


2262 


2317 


2373 


2429 


2484 


2540 


2595 




1 


2651 


2707 


2762 


2818 


2873 


2929 


2985 


3040 


3096 


3151 




2 


3207 


3262 


3318 


3373 


3429 


3484 


3540 


3595 


3651 


3706 




3 


3762 


3817 


3873 


3928 


3984 


4039 


4094 


4150 


4205 


4261 




4 


4316 


4371 


4427 


4482 


4538 


4593 


4648 


4704 


4759 


4814 




5 


4870 


4925 


4980 


5036 


5091 


5146 


5201 


5257 


5312 


5367 




6 


5423 


5478 


z:m 


5588 


5644 


5699 


5754 


5809 


5864 


5920 




7 


5975 


6030 


6085 


6140 


6195 


6251 


6306 


6361 


6416 


6471 




8 


6526 


6581 


6636 


6692 


6747 


6802 


6857 


6912 


6967 


7022 




9 


7077 


7132 


7187 


7242 


7297 


7352 


7407 


7462 


7517 


7572 




790 


7627 


7682 


7737 


7792 


7847 


7902 


7957 


8012 


8067 


8122 


55 


1 


8176 


8231 


8286 


8341 


8396 


8451 


8506 


8561 


8615 


8670 




2 


8725 


8780 


sass 


8890 


8944 


8999 


9054 


9109 


9164 


9218 




3 
4 


9273 

9821 


9328 
9875 


9383 
9G30 


9437 

9985 


9492 


9547 


9G02 


9656 


9711 


9766 














0039 


0094 


0149 


0203 


0258 


0312 




5 


900367 


0422 


0476 


0531 


0586 


0640 


0695 


0749 


0804 


0859 




6 


0913 


09G8 


1022 


1077 


1131 


1186 


1240 


1295 


1349 


1404 




7 


1458 


1513 


1567 


1622 


1676 


1731 


1785 


1840 


1894 


1948 




8 


2003 


2057 


2112 


2166 


2221 


2275 


2329 


2384 


2438 


2492 




9 


2547 


2601 


2655 


2710 


2764 


2818 


2873 


2927 


2981 


3036 




800 


3090 


3144 


3199 


3253 


3307 


3361 


8416 


3470 


35^ 


3578 




1 


3633 


3687 


3741 


3795 


3849 


3904 


3958 


4012 


4066 


4120 




2 


4174 


4229 


4283 


4337 


4391 


4445 


4499 


4553 


4607 


4661 




3 


4716 


4770 


4824 


4878 


4932 


4986 


5040 


5094 


5148 


5202 


54 


4 


5256 


5310 


5364 


5418 


5472 


5526 


5580 


5634 


5688 


5742 




5 


5796 


5850 


5904 


5958 


6012 


6066 


6119 


6173 


6227 


6281 




6 


6335 


6389 


6443 


6497 


6551 


6604 


6658 


6712 


6766 


6820 




7 


6874 


6927 


6981 


7035 


7089 


7143 


7196 


7250 


7304 


7358 




8 


7411 


7465 


7519 


7573 


7626 


7680 


7734 


7787 


7841 


7895 




9 


7949 


8002 


8056 


8110 


8163 


8217 


8270 


8324 


8378 


8431 




PROPORTIONAL PARTS. 


Diff. 1 


234 


5 


6 7 


8 


9 


57 5.7 


11.4 17.1 22.8 


28.5 


34.2 39.9 


45.6 


51.3 


56 5.6 


11.2 16.8 22.4 


28.0 


33.6 39.2 


44.8 


50.4 


55 5.5 


11.0 16.5 22.0 


27.5 


33.0 38.5 


44.0 


49.5 


54 5.4 


10.8 16.2 21.6 


27.0 


32.4 37.8 


43.2 


48.6 



LOGARITHMS OF HUMBERS. 



No. 8K) L. 908.] [No. 854 L. 931. 


N. 





1 


2 


3 


4 


6 


6 


7 


8 


9 


Diff. 


810 


908485 


8539 


8592 


8646 


8699 


8753 


8807 


8860 


8914 


8967 




1 


9021 


9074 


9128 


9181 


9235 


9289 


9342 


9396 


9449 


9503 




2 


9556 


9610 


9663 


9716 


9770 


9823 


9877 


9930 


9984 


























0037 




3 


910091 


0144 


0197 


0251 


0304 


0358 


0411 


0464 


0518 


0571 




4 


0624 


0678 


0731 


0784 


0838 


0891 


0944 


0998 


1051 


1104 




5 


1158 


1211 


1264 


1317 


1371 


1424 


1477 


1530 


1584 


1637 




6 


1690 


1743 


1797 


1850 


1903 


1956 


2009 


2063 


2116 


2169 




7 


2222 


2275 


2328 


2381 


2435 


2488 


2541 


2594 


2647 


2700 




8 


2753 


2806 


2859 


2913 


2966 


3019 


3072 


3125 


3178 


3231 




9 


3284 


3337 


3390 


3443 


3496 


3549 


3602 


3655 


3708 


3761 


53 


820 


3814 


3867 


3920 


3973 


4026 


4079 


4132 


4184 


4237 


4290 




1 


4343 


4396 


4449 


4502 


4555 


4608 


4660 


4713 


4766 


4819 




2 


4872 


4925 


4977 


5030 


5083 


5136 


5189 


5241 


5294 


5347 




3 


5400 


5453 


5505 


5558 


5611 


5664 


5716 


5769 


5822 


5875 




4 


5927 


5980 


6033 


6085 


6138 


6191 


6243 


6296 


6349 


6401 




5 


6454 


6507 


6559 


6012 


6664 


6717 


6770 


6822 


6875 


6927 




6 


6980 


7033 


7085 


7138 


7190 


7243 


7295 


7348 


7400 


7453 




7 


7506 


7558 


7611 


7063 


7716 


7708 


7820 


7873 


7925 


7978 




8 


8030 


8083 


8135 


8188 


8240 


8293 


8345 


8397 


8450 


8502 




9 


8555 


8607 


8659 


8712 


8764 


8816 


8869 


8921 


8973 


9026 




830 


9078 


9130 


9183 


9235 


9287 


9340 


9392 


9444 


9496 


9549 




1 


9601 


9653 


9706 


9758 


9810 


9802 


9914 


9967 
























n/vm 


0071 




2 


920123 


0176 


0228 


0280 


0,332 


0384 


0436 


0489 


0541 


0598 




3 


0645 


0697 


0749 


0801 


0853 


0906 


0958 


1010 


1062 


1114 




4 


1166 


1218 


1270 


1322 


1374 


1426 


1478 


1530 


1582 


1634 


52 


5 


1686 


1738 


1790 


1842 


1894 


1946 


1998 


2050 


2102 


2154 




6 


2206 


2258 


2310 


2362 


2414 


2466 


2518 


2570 


2622 


2674 




7 


2725 


2777 


2829 


2881 


2933 


2985 


3037 


3089 


3140 


3192 




8 


3244 


3296 


3348 


3399 


3451 


3503 


3555 


3607 


3658 


3710 




9 


3762 


3814 


3865 


3917 


3969 


4021 


4072 


4124 


4176 


4228 




840 


4279 


4331 


4383 


4434 


4486 


4538 


4589 


4641 


4693 


4744 




1 


4796 


4848 


4899 


4951 


5003 


5054 


5106 


5157 


5209 


5261 




2 


5312 


5364 


6415 


5467 


5518 


5570 


5621 


5673 


5725 


5776 




3 


5828 


5879 


6931 


5982 


6034 


6085 


6137 


6188 


6240 


6291 




4 


6342 


6394 


6445 


6497 


6548 


6600 


6651 


6702 


6754 


6805 




5 


6857 


6908 


6959 


7011 


7062 


7114 


7165 


7216 


7268 


7319 




6 


7370 


7422 


7473 


7524 


7576 


7627 


7678 


7730 


7781 


7832 




7 


7883 


7935 


7986 


8037 


8088 


8140 


8191 


8242 


8293 


8345 




8 


8396 


8447 


8498 


8549 


8601 


8652 


8703 


8754 


8805 


8857 




9 


8908 


8959 


9010 


9061 


9112 


9163 


9215 


9266 


9317 


9368 




850 


9419 


9470 


9521 


9572 


9623 


9674 


9725 


9776 


9827 


9879 




1 


9930 


9981 


















51 








0032 


0083 


0134 


0185 


0236 


0287 


0338 


0389 




2 


930440 


0491 


0542 


0592 


0643 


0694 


0745 


0796 


0847 


0898 




3 


0949 


1000 


1051 


1102 


1153 


1204 


1254 


1305 


1356 


1407 




4 


1458 


1509 


1560 


1610 


1661 


1712 


1763 


1814 


1865 


1915 




PROPORTIONAL PARTS. 


Diff. 1 


234 


5 


678 


9 


53 5.3 


10.6 15.9 21.2 


26.5 


31.8 37.1 42.4 


47.7 


52 5.2 


10.4 15.6 20.8 


26.0 


31.2 36.4 41.6 


46.8 


51 5.1 


10.2 15.3 20.4 


25.5 


30.6 35.7 40.8 


45.9 


50 5.0 


10.0 15.0 20.0 


25.0 


30.0 35.0 40.0 


45.0 



LOGARITHMS OF NUMBERS. 



153 



No. 855 L. 931.] [No. 899 L. 954. 


N. 





1 


2 


3 


4 


6 


6 


7 


8 


9 


Diff. 


855 


931966 


2017 


2068 


2118 


2169 


2220 


2271 


2322 


2372 


2423 




6 


2474 


2524 


2575 


2626 


2677 


2727 


2778 


2829 


2879 


2930 




7 


2981 


3031 


3082 


3133 


3183 


3234 


3285 


3335 


3386 


3437 




8 


3487 


3538 


3589 


3639 


3690 


3740 


3791 


3841 


3892 


3943 




9 


3993 


4044 


4094 


4145 


4195 


4246 


4296 


4347 


4397 


4448 




860 


4498 


4549 


4599 


4650 


4700 


4751 


4801 


4852 


4902 


4953 




1 


5003 


5054 


5104 


5154 


5205 


5255 


5306 


5356 


5406 


5457 




2 


5507 


5558 


5608 


5658 


5709 


5759 


5809 


5860 


5910 


5960 




3 


6011 


6061 


6111 


6162 


6212 


6262 


6313 


6363 


6413 


6463 




4 


6514 


6564 


6614 


6665 


6715 


6765 


6815 


6865 


6916 


6966 




5 


7016 


7066 


7116 


7167 


7217 


7267 


7317 


7367 


7418 


74i8 




6 


7518 


7568 


7618 


7668 


7718 


7769 


7819 


7869 


7919 


7969 




7 


8019 


8069 


8119 


8169 


8219 


8269 


8320 


8370 


8420 


8470 


50 


8 


8520 


8570 


8620 


8670 


8720 


8770 


8820 


8870 


8920 


8970 




9 


9020 


9070 


9120 


9170 


9220 


9270 


9320 


9369 


9419 


9469 




870 


9519 


9569 


9619 


9669 


9719 


9769 


9819 


9869 


9918 


9968 




1 


940018 


0068 


0118 


0168 


0218 


0267 


0317 


0367 


0417 


0467 




2 


0510 


0566 


0616 


0666 


0716 


0765 


0815 


0865 


0915 


0964 




3 


1014 


1064 


1114 


1163 


1213 


1263 


1313 


1362 


1412 


1462 




4 


1511 


1561 


1611 


1GGO 


1710 


1760 


1809 


1859 


1909 


1958 




5 


2008 


2058 


2107 


2157 


2207 


2256 


2306 


2355 


2405 


2455 




6 


2504 


2554 


2603 


2653 


2702 


2752 


2801 


2851 


2901 


2950 




7 


3000 


3049 


3099 


3148 


3198 


3247 


3297 


3346 


3396 


3445 




8 


3495 


3544 


3593 


3643 


3692 


3742 


3791 


3841 


3890 


3939 




9 


3989 


4038 


4088 


4137 


4186 


4236 


4285 


4335 


4384 


4433 




880 


4483 


4532 


4581 


4631 


4680 


4729 


4779 


4828 


4877 


4927 




1 


4976 


5025 


5074 


5124 


5173 


5222 


5272 


5321 


5370 


5419 




2 


5469 


5518 


5567 


5616 


5665 


5715 


5764 


5813 


5862 


5912 




3 


5961 


6010 


6059 


6108 


6157 


6207 


6256 


6305 


6354 


6403 




4 


6452 


6501 


6551 


6600 


6649 


6698 


6747 


6796 


6845 


6894 




5 


6943 


6992 


7041 


7090 


7139 


7189 


7238 


7287 


7336 


7385 




6 


7434 


7483 


7532 


7581 


7630 


7679 


7728 


7777 


7826 


7875 


49 


7 


7924 


7973 


8022 


8070 


8119 


8168 


8217 


8266 


8315 


8364 




8 


8413 


8462 


8511 


8560 


8608 


8657 


8706 


8755 


8804 


8853 




9 


8902 


8951 


8999 


9048 


9097 


9146 


9195 


9244 


9292 


9341 




890 


9390 


9439 


9488 


9536 


9585 


9634 


9683 


9731 


9780 


9829 




1 


9878 


9926 


9975 


























0024 


0073 


0191 


0170 


0219 


O9fl7 


nq-jft 




2 


950365 


0414 


0462 


0511 


0560 


VIXl 

0608 


0657 


0706 


U/SJlM 

6754 


UolO 

0803 




3 


0851 


0900 


0949 


0997 


1046 


1095 


1143 


1192 


1240 


1289 




4 


1338 


1386 


1435 


1483 


1532 


1580 


1629 


1677 


1726 


1775 




5 


1823 


1872 


1920 


1969 


2017 


2066 


2114 


2163 


2211 


2260 




G 


2308 


2356 


2405 


2453 


2502 


2550 


2599 


2647 


2696 


2744 




7 


2792 


2841 


2889 


2938 


2986 


3034 


3083 


3131 


3180 


3228 




8 


3276 


3325 


3373 


3421 


3470 


3518 


3566 


3615 


3663 


3711 




9 


3760 


3808 


3856 


3905 


3953 


4001 


4049 


4098 


4146 


4194 




PROPORTIONAL PARTS. 


Diff. 


1 


2 


3 4 


5 


6 


7 8 


9 


51 
50 
49 
48 


5.1 
5.0 
4.9 
4.8 


10.2 
10.0 
9.8 
9.6 


15.3 20.4 
15.0 20.0 
14.7 19.6 
14.4 19.2 


25.5 
25.0 
24.5 
24.0 


30.6 
30.0 
29.4 

28.8 


35.7 40.8 
35.0 40.0 
34.3 39.2 
33.6 38.4 


45.9 
45.0 
44.1 
43.2 



LOGARITHMS OP KUMBERS. 



No 900 L. 954.1 [No. 944 L. 975. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


900 


954243 


4291 


4339 


4387 


4435 


4484 


4532 


4580 


4628 


4677 




1 


4725 


4773 


4821 


4869 


4918 


4966 


5014 


5062 


5110 


5158 




2 


5207 


5255 


5303 


5351 


5399 


5447 


5495 


5543 


5592 


5640 




3 


5688 


5736 


5784 


5832 


5880 


5928 


5976 


6024 


6072 


6120 




4 


6168 


6216 


6265 


6313 


6361 


6409 


6457 


6505 


6553 


6601 




5 


6649 


6697 


6745 


6793 


6840 


6888 


6936 


6984 


7032 


7080 


48 


6 


7128 


7176 


7224 


7272 


7320 


7368 


7416 


7464 


7512 


7559 




7 


7607 


7655 


7703 


7751 


7799 


7847 


7894 


7942 


7990 


8038 




8 


8086 


8134 


8181 


8229 


8277 


8325 


8373 


8421 


8468 


8516 




9 


8564 


8612 


8659 


8707 


8755 


8803 


8850 


8898 


8946 


8994 




910 


9041 


9089 


9137 


9185 


9232 


9280 


9328 


9375 


9423 


9471 




1 
2 


9518 
9995 


9566 


9614 


9661 


9709 


9757 


9804 


9852 


9900 


9947 








0042 


0090 


0138 


0185 


0233 


AOQA 


0328 


AQ7 


ryoQ 




3 


960471 


0518 


0566 


0613 


0661 


0709 


UcoU 

0756 


0804 


Uo<O 

0851 


U4/*O 

0899 




4 


0946 


0994 


1041 


1089 


1136 


1184 


1231 


1279 


1326 


1374 




5 


1421 


1469 


1516 


1563 


1611 


1658 


1706 


1753 


1801 


1848 




6 


1895 


1943 


1990 


2038 


2085 


2132 


2180 


2227 


2275 


2322 




7 


2369 


2417 


2464 


2511 


2559 


2606 


2653 


2701 


2748 


2795 




8 


2843 


2890 


2937 


2985 


3032 


3079 


3126 


3174 


3221 


3268 




! 9 


3316 


3363 


3410 


3457 


3504 


3552 


3599 


3646 


3693 


3741 




[920 


3788 


3835 


3882 


3929 


3977 


4024 


4071 


4118 


4165 


4212 




1 


4260 


4307 


4354 


4401 


4448 


4495 


4542 


4590 


4637 


4684 




2 


4731 


4778 


4825 


4872 


4919 


4966 


5013 


5061 


5108 


5155 




3 


5202 


5249 


5296 


5343 


5390 


5437 


5484 


5531 


5578 


5625 




4 


5672 


5719 


5766 


5813 


5860 


5907 


5954 


6001 


6048 


6095 


47 


5 


6142 


6189 


6236 


6283 


6329 


6376 


6423 


6470 


6517 


6564 




6. 


6611 


6658 


6705 


6752 


6799 


6845 


6892 


6939 


6986 


7033 




7 


7080 


7127 


7173 


7220 


7267 


7314 


7361 


7408 


7454 


7501 




8 


7548 


7595 


7642 


7688 


7735 


7782 


7829 


7875 


7922 


7969 




9 


8016 


8062 


8109 


8156 


8203 


8249 


8296 


8343 


8390 


8436 




930 


8483 


8530 


8576 


8623 


8670 


8716 


8763 


8810 


8856 


8903 




1 


8950 


8996 


9043 


9090 


9136 


9183 


9229 


9276 


9323 


93G9 




2 


9416 


9463 


9509 


9556 


9602 


9649 


9695 


9742 


9789 


9835 




3 


9882 


9928 


9975 


























0021 


0068 


0114 


0161 


0207 


0254 


0300 




4 


970347 


0393 


0440 


0486 


0533 


0579 


0626 


0672 


0719 


0765 




5 


0812 


0858 


0904 


0951 


0997 


1044 


1090 


1137 


1183 


1229 




6 


1276 


1322 


1369 


1415 


1461 


1508 


1554 


1601 


1647 


1693 




7 


1740 


1786 


1832 


1879 


1925 


1971 


2018 


20G4 


2110 


2157 




8 


2203 


2249 


2295 


2342 


2388 


2434 


2481 


2527 


2573 


2619 




9 


2666 


2712 


2758 


2804 


2851 


2897 


2943 


2989 


3035 


3082 




940 


3128 


3174 


3220 


3266 


3313 


3359 


3405 


3451 


3497 


3543 




1 


3590 


3636 


3682 


3728 


3774 


3820 


3866 


3913 


3959 


4005 




2 


4051 


4097 


4143 


4189 


4235 


4281 


4327 


4374 


4420 


4466 




3 


4512 


4558 


4604 


4650 


4696 


4742 


4788 


4834 


4880 


4926 




4 


4972 


5018 


5064 


5110 


5156 


5202 


5248 


5294 


5340 


5386 


46 


PROPORTIONAL PARTS. 


Diff. 1 


234 


5 


678 


9 


47 4.7 


9.4 14.1 18.8 


23.5 


28.2 32.9 37.6 


42.3 


46 4.6 


9.2 13.8 18.4 


23.0 


27.6 32.2 36.8 


41.4 



LOGARITHMS OF NUMBERS. 



155 



No. 945 L. 975.] [No. 989 L. 995. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


945 


975432 


5478 


5524 


5570 


5616 


5662 


5707 


5753 


5799 


5845 




6 


5891 


5937 


5983 


6029 


6075 


6121 


6167 


6212 


6258 


6304 




7 


6350 


6396 


6442 


6488 


6533 


6579 


6625 


6671 


6717 


6763 




8 


6808 


6854 


6900 


6946 


6992 


7037 


7083 


7129 


7175 


7220 




9 


7266 


7312 


7358 


7403 


7449 


7495 


7541 


7586 


7632 


7678 




950 


7724 


7769 


7815 


7861 


7906 


7952 


7998 


8043 


8089 


8135 




1 


8181 


8226 


8272 


8317 


8363 


8409 


8454 


8500 


8546 


8591 




2 


8637 


8683 


8728 


8774 


8819 


8865 


8911 


8956 


9002 


9047 




3 


9093 


9138 


9184 


9230 


9275 


9321 


93G6 


9412 


9457 


9503 




4 


9548 


9594 


9639 


9685 


9730 


9776 


9821 


9867 


9912 


9958 




5 


980003 


0049 


0094 


0140 


0185 


0231 


0276 


0322 


0367 


0412 




G 


0458 


0503 


0549 


0594 


0640 


0685 


0730 


0776 


0821 


0867 




7 


0912 


0957 


1003 


1048 


1093 


1139 


1184 


1229 


1275 


1320 




8 


1366 


1411 


1456 


1501 


1547 


1592 


1637 


1683 


1728 


1773 




9 


1819 


1864 


1909 


1954 


2000 


2045 


2090 


2135 


2181 


2226 




960 


2271 


2316 


2362 


2407 


2452 


2497 


2543 


2588 


2633 


2678 




1 


2723 


2769 


2814 


2859 


2904 


2949 


2994 


3040 


3085 


3130 




2 


3175 


3220 


3265 


3310 


3356 


3401 


3446 


3491 


3536 


3581 




3 


3626 


3671 


3716 


37'62 


3807 


3852 


3897 


3942 


3987 


4032 




4 


4077 


4122 


4167 


4212 


4257 


4302 


4347 


4392 


4437 


4482 




5 


4527 


4572 


4617 


4662 


4707 


4752 


4797 


4842 


4887 


4932 


45 


6 


4977 


5022 


5067 


5112 


5157 


5202 


5247 


5292 


5337 


5382 




7 


5426 


5471 


5516 


5561 


5606 


5651 


5696 


5741 


5786 


5830 




8 


5875 


5920 


5965 


6010 


6055 


6100 


6144 


6189 


6234 


6279 




9 


6324 


6369 


6413 


6458 


6503 


6548 


6593 


6637 


6682 


6727 




970 


6772 


6817 


6861 


6906 


6951 


6996 


7040 


7085 


7130 


7175 




1 


7219 


7264 


7309 


7353 


7398 


7443 


7488 


7532 


7577 


7622 




2 


7666 


7711 


7756 


7800 


7845 


7890 


7934 


7979 


8024 


8068 




3 


8113 


8157 


8202 


8247 


8291 


8336 


8381 


8425 


8470 


8514 




4 


8559 


8604 


8648 


8693 


8737 


8782 


8826 


8871 


8916 


8960 




5 


9005 


9049 


9094 


9138 


9183 


9227 


9272 


9316 


9361 


9405 




6 

7 


9450 
9895 


9494 
9939 


9539 
9983 


9583 


9628 


9672 


9717 


9761 


9806 


9850 












0028 


0072 


0117 


0161 


0206 


0250 


0294 




8 


990339 


0383 


0428 


0472 


0516 


0561 


0605 


0650 


0694 


0738 




9 


0783 


0827 


0871 


0916 


0960 


1004 


1049 


1093 


1137 


1182 




980 


1226 


1270 


1315 


1359 


1403 


1448 


1492 


1536 


1580 


1625 




1 


1669 


1713 


1758 


1802 


1846 


1890 


1935 


1979 


2023 


2067 




2 


2111 


2156 


2200 


2244 


2288 


2333 


2377 


2421 


2465 


2509 




3 


2554 


2598 


2642 


2686 


2730 


2774 


2819 


2863 


2907 


2951 




4 


2995 


3039 


3083 


3127 


3173 


3216 


3260 


3304 


3348 


3392 




5 


3436 


3480 


3524 


3568 


3613 


3657 


3701 


3745 


3789 


3833 




6 


3877 


3921 


3965 


4009 


4053 


4097 


4141 


4185 


4229 


4273 




7 


4317 


4361 


4405 


4449 


4493 


4537 


4581 


4625 


4669 


4713 


44 


8 


4757 


4801 


4845 


4889 


4933 


4977 


5021 


5065 


5108 


5152 




9 


5196 


5240 


5284 


5328 


5373 


5416 


5460 


55U4 


5547 


5591 




PROPORTIONAL PARTS. 


Diff 1 


234 


5 


678 


9 


46 4.6 


9.2 13.8 18.4 


23.0 


27.6 32.2 36.8 


41.4 


45 4.5 


9.0 33.5 18.0 


22.5 


27.0 31.5 86 


40.5 


44 4.4 


8.8 13. 2 17.6 


22.0 


26.4 30.8 35.2 


39.6 


43 4.3 


8.6 12.9 17.2 


yi.5 


25.8 30.1 34.4 


38.7 



156 

No. 990 L. 995.] 



MATHEMATICAL TABLES. 



[No. 999 L. 999. 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


990 


995635 


5679 


5723 


5767 


5811 


5854 


5898 


5942 


5986 


6030 




1 


6074 


6117 


6161 


6205 


6249' 


6293 


6337 


6380 


6424 


6468 


44 


2 


6512 


6555 


6599 


6643 


6687 


6731 


6774 


6818 


6862 


6906 




3 


6949 


6993 


7037 


7080 


7124 


7168 


7212 


7255 


7299 


7343 




4 


7386 


7430 


7474 


7517 


7561 


7C05 


7648 


7692 


7736 


7779 




5 


7823 


7867 


7910 


7954 


7998 


8041 


8085 


8129 


8172 


8216 




6 


8259 


8303 


8347 


8390 


8434 


8477 


8521 


8564 


8608 


8652 




7 


8695 


8739 


8782 


8826 


8869 


8913 


8956 


9000 


9043 


9087 




8 


9131 


9174 


9218 


9261 


9305 


9348 


9392 


9435 


9479 


9522 




9 


9565 


9609 


9652 


9696 


9739 


9783 


9826 


9870 


9913 


9957 


43 



HYPERBOLIC LOGARITHMS. 



No. 


Log. 


No. 


Log. 


No. 


Log. 


No. 


Log. 


No. 


Log. 


1.01 


.0099 


1.45 


.3716 


1.89 


.6366 


2.33 


.8458 


2.77 


1.0188 


1.02 


.0198 


1,46 


.3784 


1.90 


.6419 


2.34 


.8502 


2.78 


1.0225 


1.03 


.0296 


1.47 


.3853 


1.91 


.6471 


2.35 


.8544 


2.79 


.0260 


1.04 


.0392 


1.48 


.3920 


1.92 


.6523 


2.36 


.8587 


2.80 


.0296 


1.05 


.0488 


1.49 


.3988 


1.93 


.6575 


2.37 


.8629 


2.81 


.0332 


1.06 


.0583 


1.50 


.4055 


1.94 


.6627 


2.38 


.8671 


2.82 


.0367 


1.07 


.0677 


1.51 


.4121 


1.95 


.6678 


2.39 


.8713 


2.83 


.0403 


1.08 


.0770 


1.52 


.4187 


1.96 


.6729 


2.40 


.8755 


2.84 


.0438 


1.09 


.0862 


1.53 


.4253 


1.97 


.6780 


2.41 


.8796 


2.85 


.0473 


1.10 


.0953 


1.54 


.4318 


1.98 


.6831 


2.42 


.8838 


2.86 


.0508 


1.11 


.1044 


1.55 


.4383 


1.99 


.6881 


2.43 


.8879 


2.87 


.0543 


1.12 


.1133 


1.56 


.4447 


2.00 


.6931 


2.44 


.8920 


2.88 


.0578 


1.13 


.1222 


1.57 


.4511 


2.01 


.6981 


2.45 


.8961 


2.89 


.0613 


1.14 


.1310 


1.58 


.4574 


2.02 


.7031 


2.46 


.9002 


2.90 


.0647 


1.15 


.1398 


1.59 


.4637 


2.03 


.7080 


2.47 


.9042 


o gi 


.0682 


1.16 


.1484 


1.60 


.4700 


2.04 


.7129 


2.48 


.9083 


2^92 


.0716 


1.17 


.1570 


1.61 


.4762 


2.05 


.7178 


2.49 


.9123 


2.93 


.0750 


1.18 


.1655 


1.62 


.4824 


2.06 


.7227 


2.50 


.9163 


2.94 


.0784 


1.19 


.1740 


1.63 


.4886 


2.07 


.7275 


2.51 


.9203 


2.95 


.Obis 


1.20 


.1823 


1.64 


.4947 


2.08 


.7324 


2.52 


.9243 


2.96 


.0852 


1.21 


.1906 


1.65 


,5008 


2.09 


.7372 


2.53 


.9282 


2.97 


.0886 


1.22 


.1988 


1.66 


.5068 


2.10 


.7419 


2.54 


.9.322 


2.98 


.0919 


1.23 


.2070 


1.67 


.5128 


2.11 


.7467 


2.55 


.9361 


2.99 


.0953 


1.24 


.2151 


1.68 


.5188 


2.12 


.7514 


2.56 


.9400 


3.00 


.0986 


1.25 


.2231 


1.69 


.5247 


2.13 


.7561 


2.57 


.9439 


3.01 


.1019 


1.26 


.2311 


1.70 


.5306 


2.14 


.7608 


2.58 


.9478 


3.02 


.1053 


1.27 


.2390 


1.71 


.5365 


2.15 


.7655 


2.59 


.9517 


3.03 


.1086 


1.28 


.2469 


.72 


.5423 


2.13 


.7701 


2.60 


.9555 


3.04 


.1119 


1.29 


.2546 


.73 


.5481 


2.17 


.7747 


2.61 


.9594 


3.05 


.1151 


1.30 


.2624 


.74 


.5539 


2.18 


.7793 


2.62 


.9632 


3.06 


.1184 


1.31 


.2700 


.75 


.5596 


2.19 


.7839 


2.63 


.9670 


3.07 


.1217 


1.32 


.2776 


.76 


.5653 


2.20 


.7885 


2.64 


.9708 


3.08 


.1249 


1.33 


.2852 


.77 


.5710 


2.21 


.7930 


2.65 


.9746 


3.09 


.1282 


1.34 


.2927 


.78 


.5766 


2.22 


.7975 


2.66 


.9783 


3.10 


.1314 


1.35 


.3001 


.79 


.5822 


2.23 


.8020 


2.67 


.9821 


3.11 


.1346 


1.36 


.3075 


.80 


.5878 


2.24 


.8065 


2.68 


.9858 


3.12 


.1378 


1.37 


.3148 


.81 


.5933 


2.25 


.8109 


2.69 


.9895 


3.13 


.1410 


1.38 


.3221 


.82 


.5988 


2.26 


.8154 


2.70 


.9933 


3.14 


.1442 


1.88 


.3293 


.83 


.6043 


2.27 


.8198 


2.71 


.9969 


3.15 


.1474 


1.40 


.3365 


.84 


.6098 


2.28 


.8242 


2.72 


1.0006 


3 16 


.1506 


1.41 


.3436 


1.85 


.6152 


2.29 


.8286 


2.73 


1.0043 


3.17 


.1537 


1.42 


.3507 


1.86 


.6206 


2.30 


.8329 


2.74 


1.0080 


3.18 


.1569 


1.43 


.3577 


1.87 


.6259 


2.31 


.8372 


2.75 


1.0116 


3.19 


.1600 


1.44 


.3646 


1.88 


.6313 


2.32 


.8416 


2.76 


1.0152 


3.20 


.1632 



HYPERBOLIC LOGARITHMS. 



157 



No, 


Log. 


No. 


Log. 


No. 


Log. 


No. 


Log. 


No. 


Log. 


3.21 


1.1663 


3.87 


1.3533 


4.53 


1.5107 


5.19 


1.6467 


5.85 


1.7664 


3.22 


1.1694 


3.88 


1.3558 


4.54 


1.5129 


5.20 


1.6487 


5.86 


1.7681 


3.23 


1.1725 


3.89 


1.3584 


4.55 


1.5151 


5.21 


1.6506 


5.87 


1.7699 


3.24 


1.1756 


3.90 


1.3610 


4.56 


1.5173 


5.22 


1.6525 


5.88 


1.7716 


3.25 


1.1787 


3.91 


1.3635 


4.57 


1.5195 


5.23 


1.6544 


5.89 


1.7733 


3.26 


1.1817 


3.92 


1.3661 


4.58 


1.5217 


5.24 


1.6563 


5.90 


1.7750 


3.27 


1.1848 


3.93 


1.3686 


4.59 


1.5239 


5.25 


1.6582 


5.91 


1.7766 


3.28 


1.1878 


3.94 


1.3712 


4.60 


1.5261 


5.26 


1.6601 


5.92 


1.7783 


3.29 


1.1909 


3.95 


1.3737 


4.61 


1.5282 


5.27 


1.6620 


5.93 


1.7800 


3.30 


1.1939 


3.96 


1.3762 


4.62 


1.5304 


5.28 


1.6639 


5.94 


1.7817 


3.31 


1.1969 


3.97 


1.3788 


4.63 


1.5326 


5.29 


1.6658 


5.95 


1.7834 


3.32 


1.1999 


3.98 


1.3813 


4.64 


1.5347 


5.30 


1.6677 


5.96 


1.7851 


3.33 


1.2030 


3.99 


1.3838 


4.65 


1.5369 


5.31 


1.6696 


5.97 


1.7867 


3.34 


1.2060 


4.00 


1.3863 


4.66 


1.5390 


5.32 


1.6715 


5.98 


1.7884 


3.35 


1.2090 


4.01 


1.3888 


4.67 


1.5412 


5.33 


1.6734 


5.99 


1.7901 


3.36 


1.2119 


4.02 


1.3913 


4.68 


1.5433 


5.34 


1.6752 


6.00 


1.7918 


3.37 


1.2149 


4.03 


1 .3938 


4.69 


1.5454 


5.35 


1.6771 


6.01 


1.7934 


3.38 


1.2179 


4.04 


1.3962 


4.70 


.5476 


5.36 


1.6790 


6.02 


1.7951 


3.39 


1.2208 


4.05 


1.3987 


4.71 


.5497 


5.37 


1.6808 


6.03 


1.7967 


3.40 


1.2238 


4.06 


1.4012 


4.72 


.5518 


5.38 


1.6827 


6.04 


1.7984 


3.41 


1.2267 


4.07 


1.4036 


4.73 


.5539 


5.39 


1.6845 


6.05 


1.8001 


3.42 


1.2296 


4.08 


1.4061 


4.74 


.5560 


5.40 


1.6864 


6.06 


1.8017 


3.43 


1.2326 


4.09 


1.4085 


4.75 


.5581 


5.41 


1.6882 


6.07 


1.8034 


3.44 


1.2355 


4.10 


1.4110 


4.76 


.5602 


5.42 


1.6901 


6.08 


1.8050 


3.45 


1.2384 


4.11 


1.4134 


4.77 


.5623 


5.43 


1.6919 


6.09 


1.8066 


3.46 


1.2413 


4.12 


1.4159 


4.78 


1.5644 


5.44 


1.6938 


6.10 


1.8083 


3 47 


1.2442 


4.13 


1.4183 


4.79 


1.5665 


5.45 


1.6956 


6.11 


1.8099 


3.48 


1.2470 


4.14 


1.4207 


4.80 


1.5686 


5.46 


1.6974 


6.12 


1.8116 


8.49 


1.2499 


4.15 


1.4231 


4.81 


1.5707 


5.47 


1.6993 


6.13 


1.8132 


3.50 


1.2528 


4.16 


1.4255 


4.82 


1.5728 


5.48 


1.7011 


6.14 


1.8148 


3.51 


1.2556 


4.17 


1.4279 


4.83 


1.5748 


5.49 


1.7029 


6.15 


1.8165 


3.52 


1.2585 


4.18 


1.4303 


4.84 


1.5769 


5.50 


1.7047 


6.16 


1.8181 


3.53 


1.2613 


4.19 


1.4327 


4.85 


1.5790 


5.51 


1.7066 


6.17 


1.8197 


3.54 


1.2641 


4.20 


1.4351 


4.86 


1.5810 


5.52 


1.7084 


6.18 


1-8213 


3.55 


1.3669 


4.21 


1.4375 


4.87 


1.5831 


5.53 


1.7102 


6.19 


1.8229 


3.56 


1.2698 


4.22 


1.4398 


4.88 


1.5851 


5.54 


1.7120 


6.20 


1.8245 


3.57 


1.2726 


4.23 


1.4422 


4.89 


.5872 


5.55 


1.7138 


6.21 


1.8262 


3.58 


1.2754 


4.24 


1.4446 


4.90 


.5892 


5.56 


1.7156 


6.22 


1.8278 


3.59 


1.2782 


4.25 


1.4469 


4.91 


.5913 


5.57 


1.7174 


6.23 


1.8294 


3.60 


1.2809 


4.26 


1.4493 


4.92 


.5933 


5.58 


1.7192 


6.24 


1.8310 


3.61 


1.2837 


4.27 


1.4516 


4.93 


.5953 


5.59 


1.7210 


6.25 


1.8326 


3.62 


1.2865 


4.28 


1.4540 


4.94 


.5974 


5.60 


1.7228 


6.26 


1.8342 


3.63 


1.2892 


4.29 


1.4563 


4.95 


.5994 


5.61 


1.7246 


6.27 


1.8358 


3.64 


1 .2920 


4.30 


1.4586 


4.96 


.6014 


5.62 


1.7263 


6.28 


1.8374 


3.65 


1.2947 


4.31 


1.4609 


4.97 


.6034 


5.63 


1.7281 


6.29 


1.8390 


3.66 


1.2975 


4.32 


1.4633 


4.98 


.6054 


5.64 


1.7299 


6.30 


1.8405 


3.67 


1.3002 


4.33 


.4656 


4.99 


.6074 


5.65 


1.7317 


6.31 


1.8421 


3.68 


1.3029 


4.34 


.4679 


5.00 


.6094 


5.66 


1.7334 


6.32 


1.8437 


3.69 


1.3056 


4.35 


.4702 


5.01 


.6114 


5.67 


1.7352 


6.33 


1.8453 


3.70 


1.3083 


4.36 


.4725 


5.02 


.6134 


5.68 


1.7370 


6.34 


1.8469 


3.71 


1.3110 


4.37 


.4748 


5.03 


.6154 


5.69 


1.7387 


6.35 


1.8485 


3.72 


1.3137 


4.38 


.4770 


5.04 


.6174 


5.70 


1.7405 


6.36 


1.8500 


3.73 


1.3164 


4.39 


.4793 


5.05 


.6194 


5.71 


1.7422 


6.37 


1.8516 


3.74 


1.3191 


4.40 


.4816 


5.06 


.6214 


5.72 


1.7440 


6.38 


1.8532 


3.75 


1.3218 


4.41 


.4839 


5.07 


.6233 


5.73 


1.7457 


6.39 


1.8547 


3.76 


1.3244 


4.42 


.4861 


5.08 


.6253 


5.74 


1.7475 


6.40 


1.8563 


3.77 


1.3271 


4.43 


.4884 


5.09 


.6273 


5.75 


1.7492 


6.41 


1,8579 


3.78 


1.3297 


4.44 


.4907 


5.10 


.6292 


5.76 


1.7509 


6.42 


1.8594 


3.79 


1.3324 


4.45 


.4929 


5.11 


.6312 


5.77 


1.7527 


6.43 


1.8610 


3.80 


1.3350 


4.46 


.4951 


5.12 


.6332 


5.78 


1.7544 


6.44 


1.8625 


8.81 


1.3376 


4.47 


.4974 


5.13 


.6351 


5.79 


1.7561 


6.45 


1.8641 


3.82 


1.3403 


4.48 


.4996 


5.14 


.6371 


5.80 


1.7579 


6.46 


1.8656 


3.83 


1.3429 


4.49 


.5019 


5.15 


.6390 


5.81 


1.7596 


6.47 


1.8672 


3.84 


1.3455 


4.50 


.5041 


5.16 


.6409 


5.82 


1.7613 


6.48 


1.8687 


8.85 


1.3481 


4.51 


.5063 


5.17 


.6429 


5.83 


1 .7630 


6.49 


1.8703 


3.86 


1.3507 


4.52 


.5085 


5.18 


1.6448 


5.84 


1.7647 


6.50 


1.8718 



158 



MATHEMATICAL TABLES. 



No. 


Log. 


No. 


Log. 


No. 


Log. 


No. 


Log. 


No. 


Log. 


6.51 


1.8733 


7.15 


.9671 


7.79 


2.0528 


8.6(5 


2.1587 


9.94 


2.2966 


6.52 


1.8749 


7.16 


.9685 


7.80 


2.0541 


8.68 


2.1610 


9.96 


2.2986 


6.53 


1.8764 


7.17 


.9699 


7.81 


2.0554 


8.70 


2.1633 


9.98 


2.3006 


6.54 


1.8779 


7.18 


.9713 


7.82 


2.0567 


8.72 


2.1656 


10.00 


2.3026 


6.55 


1.8795 


7.19 


.9727 


7.83 


2.0580 


8.74 


2.1679 


10.25 


2.3279 


6.56 


1.8810 


7.20 


.9741 


7.84 


2.0592 


8.76 


2.1702 


10.50 


2.3513 


6.57 


1.8825 


7.21 


.9754 


7.85 


2.0605 


8.78 


2.1725 


10.75 


2.3749 


6.58 


1.8840 


7.22 


.9769 


7.86 


2.0618 


8.80 


2.1748 


11.00 


2.3979 


6.59 


.8856 


7.23 


.9782 


7.87 


2.0631 


8.82 


2.1770 


11.25 


2.4201 


6.60 


.8871 


7.24 


.9796 


7.88 


2.0643 


8.84 


2.1793 


11.50 


2.4430 


6.61 


.8886 


7.25 


.9810 


7.89 


2.0656 


8.86 


2.1815 


11.75 


2.4636 


6.62 


.8901 


7.26 


1.9824 


7.90 


2.0669 


8.88 


2.1838 


12.00 


2.4849 


6.63 


.8916 


7.27 


1.9838 


7.91 


2.0681 


8.90 


2.1861 


12.25 


2.5052 


6.64 


.8931 


7.28 


1.9851 


7.92 


2.0694 


8.92 


2.1883 


12.50 


2.5262 


6.65 


.8946 


7.29 


1.9865 


7.93 


2.0707 


8.94 


2.1905 


12.75 


2.5455 


6.66 


.8961 


7.30 


1.9879 


7.94 


2.0719 


8.96 


2.1928 


13.00 


2.5649 


6.67 


.8976 


7.31 


1.9892 


7.95 


2.0732 


8.98 


2.1950 


13.25 


2.5840 


6.68 


.8991 


7.32 


1.9906 


7.96 


2.0744 


9.00 


2.1972 


13.50 


2.6027 


6.69 


.9006 


7.33 


1.9920 


7.97 


2.0757 


9.02 


2.1994 


13.75 


2.6211 


6.70 


.9021 


7.34 


1.9933 


7.98 


2.0769 


9.04 


2.2017 


14.00 


2.6391 


6.71 


.9086 


7.35 


1.9947 


7.99 


2.0782 


9.06 


2.2039 


14.25 


2.6567 


6.72 


.9051 


7.36 


1.9961 


8-00 


2.0794 


9.08 


2.2061 


14.50 


2.6740 


6.73 


.9066 


7.37 


1.9974 


8.01 


2.0807 


9 10 


2.2083 


14.75 


2.6913 


6.74 


.9081 


7.38 


1.9988 


8.02 


2.0819 


9.12 


2.2105 


15.00 


2.7081 


6.75 


.9095 


7.39 


2.0001 


8.03 


2.0832 


9.14 


2.2127 


15.50 


2.7408 


6.76 


.9110 


7.40 


2.0015 


8.04 


2.0844 


9.1-6 


2.2148 


16.00 


2.7726 


6.77 


.9125 


7.41 


2.0028 


8.05 


2.0857 


9.18 


2.2170 


16.50 


2.8034 


6.78 


1.9140 


7.42 


2.0041 


8-06 


2.0869 


9.20 


2.2192 


17.00 


2.8332 


6.79 


1.9155 


7.43 


2-0055 


8.07 


2.0882 


9.22 


2.2214 


17.50 


2.8621 


6.80 


1.9169 


7.44 


2-0069 


8.08 


2.0894 


9.24 


2.2235 


18.00 


2.8904 


6.81 


1.9184 


7.45 


2.0082 


8-09 


2.0906 


9.26 


2.2257 


18.50 


2.9178 


6.82 


1.9199 


7.46 


2.0096 


8.10 


2.0919 


9.28 


2.2279 


19.00 


2.9444 


6.83 


1.9213 


7.47 


2.0108 


8.11 


2.0931 


9.30 


2.2300 


19.50 


2.9703 


6.84 


1.9228 


7.48 


2.0122 


8.12 


2.0943 


9.32 


2.2322 


20.00 


2.9957 


6.85 


1.9242 


7.49 


2.0136 


8.13 


2.0958 


9.34 


2.2343 


21 


3.0445 


6.86 


1.9257 


7.50 


2.0149 


8.14 


2.0968 


9.36 


2.2364 


22 


3.0910 


6.87 


1.9272 


7.51 


2.0162 


8.15 


2.0980 


9.38 


2.2386 


23 


3.1355 


6.88 


1.9286 


7.52 


2.0176 


8.16 


2.0992 


9.40 


2.2407 


24 


3.1781 


6.89 


1.9301 


7.53 


2.0189 


8-17 


2.1005 


9.42 


2.2428 


25 


3.2189 


6.90 


1.9315 


7.54 


2.0202 


8.18 


2.1017 


9.44 


2.2450 


26 


3.2581 


6.91 


1.9330 


7.55 


2.0215 


8-19 


2.1029 


9.46 


2.2471 


27 


3.2958 


6.92 


1.9344 


7.56 


2.0229 


8.20 


2.1041 


9.48 


2.2492 


28 


3.3322 


6.93 


1.9359 


7.57 


2.0242 


8.22 


2.1066 


9.50 


2.2513 


29 


3.3673 


6.94 


1.9373 


7.58 


2.0255 


8.24 


2.1090 


9.52 


2.2534 


30 


3.4012 


6.95 


1.9387 


7.59 


2.0268 


8.26 


2.1114 


9.54 


2.2555 


31 


3.4340 


6.96 


1.9402 


7.60 


2.0281 


8.28 


2.1138 


9.56 


2.2576 


32 


3.4657 


6.97 


1.9416 


7.61 


2.0295 


8.30 


2.1163 


9.58 


2.2597 


33 


3.4965 


6.98 


.9430 


7.62 


2.0308 


8.32 


2.1187 


9.60 


2.2618 


34 


3.5263 


6.99 


.9445 


7.63 


2.0321 


8.34 


2.1211 


9.62 


2.2638 


35 


3.5553 


7.00 


.9459 


7.64 


2.0334 


8.36 


2.1235 


9.64 


2.ii659 


36 


3.5835 


7.01 


.9473 


7.65 


2.0347 


8.38 


2.1258 


9.66 


2.2680 


37 


3.6109 


7.02 


.9488 


7.66 


2.0360 


8.40 


2.1282 


9.68 


2.2701 


38 


3.6376 


7.03 


.9502 


7.67 


2.0373 


8.42 


2.1306 


9.70 


2.2721 


39 


3.6636 


7.04 


.9516 


7.68 


2.0386 


8.44 


2.1330 


9.72 


2.2742 


40 


3.6889 


7.05 


.9530 


7.69 


2.0399 


8.46 


2.1353 


9.74 


2.2762 


41 


3.7136 


7.06 


.9544 


7.70 


2.0412 


8.48 


2.1377 


9.76 


2.2783 


42 


3.7377 


7.07 


.9559 


7.71 


2.0425 


8.50 


2.1401 


9.78 


2.2803 


43 


3.7612 


7.08 


.9573 


7.72 


2.0438 


8.52 


2.1424 


9.80 


2.2824 


44 


3.7842 


7.09 


.9587 


7.73 


2.0451 


8.54 


2.1448 


9.82 


2.2844 


45 


3.8067 


7.10 


.9601 


7.74 


2.0464 


8.56 


2.1471 


9.84 


2.2865 


46 


3.8286 


7.11 


.9615 


7.75 


2.0477 


8.58 


2.1494 


9.86 


2.2885 


47 


3.8501 


7.12 


.9629 


7.76 


2.0490 


8.60 


2.1518 


9.88 


2.2905 


48 


3.8712 


7.13 


.9643 


7.77 


2.0503 


8.62 


2.1541 


9.90 


2.2925 


49 


3.8918 


7.14 


.9657 


7.78 


2.0516 


8.64 


2.1564 


9.92 


2.2946 


50 


3.9120 



NATURAL TRIGONOMETRICAL FUNCTIONS. 



159 



NATURAL TRIGONOMETRICAL FUNCTIONS. 






M. 


Sine. 


Co-Vers. 


Cosec. 


Tang. 


Cotn. Secant. 


r er. Sin. 


CoHine. 






o 





00000 


.0000 


nfinite 


00000 


Infinite 1.0000 


.00000 


1.0000 


90 







15 


00436 


.99564 


229.18 


00436 


229.18 ! 1.0000 


.00001 


.99999 




45 




30 


00873 


.99127 


14.59 


00873 


114.59 


1.0000 


.00004 


.99996 




30 




45 


01309 


.98691 


76.397 


01309 


76.390 


1.0001 


.00009 


.99991 




15 


1 





01745 


.98255 


57.299 


01745 


57.290 


1.0001 


.00015 


.99985 


89 







15 


02181 


.97819 


45.840 


02182 


45.829 


1.0002 


.00024 


.99976 


45 




30 


02618 


.97382 


38.202 


02618 


38.188 


1.0003 


.00034 


.99966 


30 




45 


03054 


.96946 


32.746 


03055 


32.730 


1.0005 


.00047 


.99953 




15 


2 





03490 


.96510 


28.654 


03492 


28.636 


1.0006 


.00061 


.99939 88 







15 


03926 


.96074 


25.471 


03929 


25.452 


1.0008 


.00077 


.99923 




45 




30 


04362 


.95638 


22.926 


04366 


22.904 


1.0009 


.00095 


.99905 




30 




45 


04798 


.95202 


20.843 


04803 


20.819 


1.0011 


.00115 


.99885 




15 


3 





05234 


.94766 


19.107 


05241 


19.081 


1.0014 


.00137 


.99863 


87 







15 


05669 


.94331 


17.639 


05678 


17.611 


1.0016 


.00161 


.99839 




45 




30 


06105 


.93895 


16.380 


06116 


16.350 


1.0019 


.00187 


.99813 




30 




45 


06540 


.93460 


15.290 


06554 


15.257 


1.0021 


.00214 


.99786 




15 


4 





06976 


.93024 


14.336 


06993 


14.301 I 1.0024 


.00244 


.99756 


86 







15 


07411 


.92589 


13.494 


07431 


13.457 1 1.0028 


.00275 


.99725 




45 




30 


07846 


.92154 


12.745 


07870 


12.706 


1.0031 


.00308 


.99692 




30 




45 


08231 


.91719 


12.076 


08309 


12.035 


1.0034 


.00343 


.99656 




15 


5 





08716 


.91284 


11.474 


08749 


11.430 


1.0038 


.00381 


.99619 


85 







15 


09150 


.90850 


10.929 


09189 


10.883 


1.0042 


.00420 


.99580 




45 




30 


09585 


.90415 


10.433 


09629 


10.385 


1.0046 


.00460 


.99540 




30 




45 


10019 


.89981 


9.9812 


10069 


9.9310 


1.0051 


.00503 


.99497 




15 


6 





10453 


.89547 


9.5668 


10510 


9.5144 


1.0055 


.00548 


.99452 


84 







15 


10887 


.89113 


9.1855 


10952 


9.1309 1.0060 


.00594 


.99406 




45 




30 


11320 


.88680 


8.8337 


11393 


8.7769 1.0065 


.00643 


.99357 




30 




45 


11754 


.88246 


8.5079 


11836 


8.4490: 1.0070 


.00693 


.99307 




15 


7 





12187 


.87813 


8.2055 


12278 


8.1443 1.0075 


.00745 


.99255 


83 







15 


12620 


.87380 


7.9240 


12722 


7.8606 1.0081 


.00800 


.99200 




45 




30 


13053 


.86947 


7.6613 


13165 


7.5958 1.0086 


.00856 


.99144 




30 




45 


13485 


.86515 


7.4156 


13609 


7.3479 1.0092 


.00913 


.99086 




15 


8 





13917 


.86083 


7.1853 


14054 


7.1154 1.0098 


.00973 


.99027 


82 







15 


14349 


.85651 


6.9690 


14499 


6.8969; 1.0105 


.01035 


.98965 




45 




30 


14781 


.85219 


6.7655 


14945 


6.6912 1.0111 


.01098 


.98902 




30 




45 


15212 


.84788 


6.5736 


15391 


6.4971 


1.0118 


.01164 


.98836 




15 


9 





15643 


.84357 


6.3924 


15838 


6.3138 


1.0325 


.01231 


.98769 


81 







15 


16074 


.83926 


6.2211 


16286 


6.1402 


1.0132 


.01300 


.98700 




45 




30 


.16505 


.83495 


6.0589 


16734 


5.9758 


1.0139 


.01371 


.98629 




30 




45 


.16935 


.83065 


5.9049 


.17183 


5.8197 


1.0147 


.01444 


.98556 




15 


10 





.17365 


.82635 


5.7588 


.17633 


5.6713 


1.0154 


.01519 


.98481 


80 







15 


.17794 


.82206 


5.6198 


.18083 


5.5301 


1.0162 


.01596 


.98404 




45 




30 


.18224 


.81776 


5.4874 


.18534 


5.3955 


1.0170 


.01675 


.98325 




30 




45 


.18652 


.81348 


5.3612 


.18986 


5.2672 


1.0179 


.01755 


.98245 




15 


11 





.19081 


.80919 


5.2408 


.19438 


5.1446 


1.0187 


.01837 


.98163 


79 







15 


.19509 


.80491 


5.1258 


.19891 


5.0273 


1.0196 


.01921 


.98079 




45 




30 


.19937 


.80063 


5.0158 


.20345 


4.9152 


1.0205 


.02008 


.97992 




30 




45 


.20364 


.79636 


4.9106 


.20800 


4.8077 


1.0214 


.02095 


.97905 




15 


12 





.20791 


.79209 


4.8097 


.21256 


4.7046 


1.0223 


.02185 


.97815 


78 







15 


.21218 


.78782 


4.7130 


.21712 


4.6057 


1.0233 


.02277 


.97723 




45 




30 


.21644 


.78356 


4.6202 


.22169 


4.5107 


1.0243 


.02370 


.97630 




30 




45 


.22070 


.77930 


4.5311 


.22628 


4.4194 


1.0253 


.02466 


.97534 




15 


13 





.22495 


.77505 


4.4454 


23087 


4.3315 


1.0263 


.02563 


.97437 


77 







15 


.22920 


.77080 


4.3630 


23547 


4.2468 


1.0273 


.02662 


.97338 




45 




30 


.23345 


.76655 


4.2837 


.24008 


4.1653 


1.0284 


.02763 


.97237 




30 




45 


.23769 


.76231 


4.2072 


.24470 


4.0867 


1.0295 


.02866 


.97134 




15 


14 





.24192 


.75808 


4.1336 


.24933 


4.0108 


1.0306 


.02970 


.97030 


76 







15 


.24615 


.75385 


4.0625 


.25397 


3.9375 


1.0317 


.03077 


.96923 




45 




30 


.25038 


.74962 


3 993! 


.25862 


3.8667 


1.0329 


.03185 


.96815 




30 




45 


.25460 


.7454C 


3*.92r 


.26328 


3.7983 


1.034 


.03295 


.96705 




15 


15 





.25882 


.74118 


3.8637 


.2679P 


3.732C 


1.0353 


.03407 


.96593 


75 









Cosine 


Ver. Sin 


Secant. 


Cotan. 


Tang. 


Cosec. 


Co-Vers 


Sine. 





M. 



From 75 to 90 read from bottom of table upwards. 



1GO 



MATHEMATICAL TABLES. 






M. 


Sine. 


Co-Vers. 


Cosec. 


Tang. 


Cotan. 


Secant. 


Ver. Sin. 


Cosine. 






15 





.25882 


.74118 


3.8637 


.26795 


3.7320 


1.0353 


.03407 


.96593 


75 







15 


.26303 


.73697 


3.8018 


.27263 


3.6680 


1.0365 


.03521 


.96479 




45 




30 


.26724 


.73276 


3.7420 


.27732 


3.6059 


1.0377 


.03637 


.96363 




at 




45 


.27144 


.72856 


3.6840 


.28203 


3.5457 


1.0390 


.03754 


.96246 




15 


16 





.27564 


.72436 


3.6280 


.286?4 


3.4874 


1.0403 


.03874 


.96126 


74 







15 


.27983 


.72017 


3.5736 


.29147 


3.4308 


1.0416 


.03995 


.96005 




45 




30 


.28402 


.71598 


3.5209 


.29621 


3.3759 


1.0429 


.04118 


.95882 




30 




45 


.28820 


.71180 


3.4699 


.30096 


3.3226 


1.0443 


.04243 


.95757 




15 


17 





.29237 


.70763 


3.4203 


.30573 


3.2709 


1.0457 


.04370 


.95630 


7* 







15 


.29654 


.70346 


3.3722 


.31051 


3.2205 


1.0471 


.04498 


.95502 




45 




30 


.30070 


.69929 


3.3255 


.31530 


3.1716 


1.0485 


.04628 


.95372 




30 




45 


.30486 


.69514 


3.2801 


.32010 


3.1240 


1.0500 


.04760 


.95240 




15 


18 





.30902 


.69098 


3.2361 


.32492 


3.0777 


1.0515 


.04894 


.95106 


72 







15 


.31316 


.68684 


3.1932 


.32975 


3.0326 


1.0530 


.05030 


.94970 




45 




30 


.31730 


.68270 


3.1515 


.33459 


2.9887 


1.0545 


.05168 


.94832 




30 




45 


.32144 


.67856 


3.1110 


.33945 


2.9459 


1.0560 


.05307 


.94693 




15 


19 





.32557 


.67443 


3.0715 


.34433 


2.9042 


1 .0576 


.05448 


.94552 


71 







15 


.32969 


.67031 


3.0331 


.34921 


2.8636 


1.0592 


.05591 


.94409 




45 




30 


.33381 


.66619 


2.9957 


.35412 


2.8239 


1.0608 


.05736 


.94264 




30 




45 


.33792 


.66208 


2.9593 


.35904 


2.7852 


1.0625 


.05882 


.94118 




15 


20 





.3420:2 


.65798 


2.9238 


.36397 


2.7475 


1 .0642 


.06031 


.93969 


70 







15 


.34612 


.65388 


2.8892 


.36892 


2.7106 


1.0659 


.06181 


.93819 




45 




30 


.35021 


.64979 


2.8554 


.37388 


2.6746 


1.0676 


.06333 


.93667 




30 




45 


.35429 


.64571 


2.8225 


.37887 


2.6395 


1.0694 


.06486 


.93514 




15 


21 





.35837 


.64163 


2.7904 


.38386 


2.6051 


1.0711 


.06642 


.93358 


69 







15 


.36244 


.63756 


2.7591 


.38888 


2-5715 


1.0729 


.06799 


.93201 




45 




30 


.36650 


.63350 


2.7285 


.39391 


2.5386 


1.0748 


.06958 


.93042 




30 




45 


.37056 


.62944 


2.6986 


.39896 


2.5065 


1.0766 


.07119 


.92881 




15 


22 





.37461 


.62539 


2.6695 


.40403 


2-4751 


1.0785 


.07282 


.92718 


68 







15 


.37865 


.62135 


2.6410 


.40911 


2 4443 


1.0804 


.07446 


.92554 




45 




30 


.38268 


.61732 


2.6131 


.41421 


2-4142 


1.0824 


.07612 


.92388 




30 




45 


.38671 


.61329 


2.5859 


.41933 


2-3847 


1.0844 


.07780 


.92220 




15 


23 





.39073 


.60927 


2.5593 


.42447 


2-3559 


1.0864 


.07950 


.92050 


67 







15 


.39474 


.60526 


2.5333 


.42963 


2-3276 


1.0884 


.08121 


.91879 




45 




30 


.39875 


60125 


2.5078 


.43481 


2-2998 


1.0904 


.08294 


.91706 




30 




45 


.40275 


.59725 


2.4829 


.44001 


2 2727 


1.0925 


.08469 


.91531 




15 


24 





.40674 


.59326 


2.4586 


.44523 


2-2460 


1.0946 


.08645 


.91355 


66 







15 


.41072 


.58928 


2.4348 


.45047 


2.2199 


1.0968 


.08824 


.91176 




45 




30 


.41469 


.58531 


2.4114 


.45573 


2.1943 


1.0989 


.09004 


.90996 




30 




45 


41866 


.58134 


2.3886 


.46101 


2.1692 


1.1011 


.09186 


.90814 




15 


25 





.42262 


.57738 


2.3662 


.46631 


2.1445 


1.1034 


.09369 


.90631 


65 







15 


.42657 


.57343 


2.3443 


.47163 


2-1203 


1.1056 


.09554 


.90446 




45 




30 


.43051 


.56949 


2.3228 


.47697 


2-0965 


1.1079 


.09741 


.90259 




30 




45 


.43445 


.56555 


2.3018 


.48234 


2.0732 


1.1102 


.09930 


.90070 




15 


26 





.43837 


.56163 


2.2812 


.48773 


2-0503 


1.1126 


.10121 


.89879 


64 







15 


.44229 


.55771 


2.2610 


.49314 


2.0278 


1.1150 


.10313 


.89687 




45 




30 


.44620 


.55380 


2.2412 


.49858 


2.0057 


1.1174 


.10507 


.89493 




30 




45 


.45010 


.54990 


2.2217 


.50404 


.9840 


1.1198 


.107'02 


.89298 




15 


27 





.45399 


.54601 


2.2027 


.50952 


.9626 


1.1223 


.10899 


.89101 


63 







15 


.45787 


.54213 


2.1840 


.51503 


.9415 


1.1248 


.11098 


.88902 




45 




30 


.46175 


.53825 


2.1657 


.52057 


.9210 


1.1274 


.11299 


.88701 




30 




45 


.46561 


.53439 


2.1477 


.52612 


.9007 


1.1300 


.11501 


.88499 




15 


2$ 





.46947 


.53053 


2.1300 


.53171 


.8807 


1.1326 


.11705 


.88295 


62 







15 


.47332 


.52668 


2.1127 


.53732 


.8611 


1.1352 


.11911 


.88089 




45 




30 


.47716 


.52284 


2.0957 


.54295 


.8418 


1.1379 


.12118 


.87882 




30 




45 


.48099 


.51901 


2.0790 


.54862 


.8228 


1.1400 


.12327 


.87673 




15 


29 





.48481 


.51519 


2.0627 


.55431 


.8040 


1.1433 


.12538 


.87462 


61 







15 


.48862 


.51138 


2.0466 


.56003 


.7856 


1.1461 


.12750 


.87250 




45 




30 


.49242 


.50758 


2.0308 


.56577 


.7675 


1.1490 


.12964 


.87036 




30 




45 


. 49622 


.50378 


2.0152 


.57155 


.7496 


1.1518 


.13180 


.86820 




15 


30 





.50000 


.50000 


2.0000 


.57735 


1.7320 


1.1547 


.13397 


.86603 


60 









Cosine. 


Ver. Sin. 


Secant. 


Cotan. 


Tang. 


Cosec. 


Co-Vers. 


Sine. 


' 


M. 



From 60 to 75 read from bottom of table upwards. 



NATURAL TRIGONOMETRICAL FUNCTIONS. 



161 






M. 


Sine. 


Co-Vers. 


Cosec. 


Tang. 


Cotan. 


Secant. 


Ver. Sin. 


Cosine. 






80 





.50000 


.50000 


2.0000 


.57735 


.7320 


1.1547 


.13397 


.86603 


60 







15 


.50377 


.49623 


.9850 


.58318 


.7147 


1.1576 


.13616 


.86384 




45 




30 


.50754 


.49246 


.9703 


.58904 


.6977 


1.1606 


.13837 


.86163 




30 




45 


.51129 


.48871 


.9558 


.59494 


.6808 


1.1636 


.14059 


.85941 




15 


81 





.51504 


.48496 


.9416 


.60086 


.6643 


1.1666 


.14283 


.85717 


59 







15 


.51877 


.48123 


.9276 


.60681 


.6479 


.1697 


.14509 


.85491 




45 




30 


.52250 


.47750 


.9139 


.61280 


.6319 


.1728 


.14736 


.85264 




30 




45 


.52621 


.47379 


.9004 


.61882 


1.6160 


.1760 


.14965 


.85035 




15 


32 





.52992 


.47008 


.8871 


.62487 


1.6003 


.1792 


.15195 


.84805 


58 







15 


.53361 


.46639 


.8740 


.63095 


1.5849 


.1824 


.15427 


.84573 




45 




30 


.53730 


.46270 


.8612 


.63707 


1.5697 


.1857 


.15661 


.84339 




30 




45 


.54097 


.45903 


.8485 


.64322 


1.5547 


.1890 


.15896 


.84104 




15 


33 





.54464 


.45536 


.8361 


.64941 


1.5399 


1924 


.16133 


.83867 


57 







15 


.54829 


.45171 


.8238 


.65563 


1.5253 


.1958 


.16371 


.83629 




45 




30 


.55194 


.44806 


.8118 


.66188 


1.5108 


.1992 


.16611 


.83389 




30 




45 


.55557 


.44443 


.7999 


.66818 


1.4966 


.2027 


.16853 


.83147 




15 


34 





.55919 


.44081 


.7883 


.67451 


1.4826 


.2062 


.17096 


.82904 


56 







15 


.56280 


.43720 


.7768 


.68087 


1.4687 


.2098 


.17341 


.82659 




45 




30 


;56641 


.43359 


.7655 


.68728 


1.4550 


.2134 


.17587 


.82413 




30 




45 


.57000 


.43000 


.7544 


. 69372 


1.4415 


.2171 


.17835 


.82165 




15 


85 





.57358 


.42642 


.7434 


.70021 


1.4281 


.2208 


.18085 


.S1915 


55 







15 


.57715 


.42285 


.7327 


.70673 


1.4150 


.2245 


.18336 


.81664 




45 




30 


.58070 


.41930 


.72x>0 


.71329 


1.4019 


.2283 


.18588 


.81412 




30 




45 


.58425 


.41575 


.7116 


.71990 


1.3891 


.2322 


.18843 


.81157 




15 


36 





.58779 


.41221 


.7013 


.72654 


1.3764 


.2361 


.19098 


.80902 


54 







15 


.59131 


.40869 


.6912 


.73323 


1.3638 


.2400 


.19356 


.80644 




45 




30 


.59482 


.40518 


.6812 


.73996 


1.&514 


.2440 


.19614 


.80386 




30 




45 


.59832 


.40168 


.6713 


.74673 


1.3392 


.2480 


.19875 


.80125 




15 


37 





.60181 


.39819 


.6616 


.75355 


1.3270 


.2521 


.20136 


.79864 


53 







15 


.60529 


.39471 


.6521 


.76042 


1.3151 


.2563 


.20400 


.79600 




45 




30 


.60876 


.39124 


.6427 


.76733 


1.3032 


.2605 


.20665 


.79335 




30 




45 


.81222 


.38778 


.6334 


. 77428 


1.2915 


.2647 


.20931 


.79069 




15 


38 





.61566 


.38434 


.6243 


.78129 


1.2799 


.2690 


.21199 


.78801 


52 







15 


.61909 


.38091 


.6153 


.78834 


1.2685 


1.2734 


.21468 


.78532 




45 




30 


.62251 


.37749 


.6064 


.79543 


1.2572 


1.2778 


.21-739 


.78261 




3" 




45 


.62592- 


.37408 


.5976 


.80258 


1.2460 


1.2822 


.22012 


.77988 




15 


39 





.62932 


.37068 


.5890 


.80978 


1.2349 


1.2868 


.22285 


.77715 


51 







15 


.63271 


.36729 


.5805 


.81703 


1.2239 


1.2913 


.22561 


.77'439 




45 




30 


.63608 


.36392 


.5721 


.82434 


1.2131 


1.2960 


.22838 


.77162 




30 




45 


.63944 


.36056 


.5639 


.83169 


1.2024 


1.3007 


.23116 


.76884 




15 


40 





.64279 


.35721 


.5557 


.83910 


1.1918 


1.3054 


. .23396 


.76604 


50 







15 


.64612 


.35388 


.5477 


.84656 


1.1812 


1.3102 


.23677 


.76323 




45 




30 


.64945 


.35055 


.5398 


.85408 


1.1708 


1.3151 


.23959 


.76041 




30 




45 


.65276 


.34724 


1.5320 


.86165 


1.1606 


1.3200 


.24244 


.75756 




15 


41 





.65606 


.34394 


1.5242 


.86929 


1.1504 


1.3250 


.24529 


.75471 


49 







15 


.65935 


.34065 


1.5166 


.87698 


1.1403 


1.3301 


.24816 


.75184 




45 




30 


.66262 


.33738 


1 5092 


.88472 


1.1303 


1.3352 


.25104 


. 7489( 




30 




45 


.66588 


.33412 


1.5018 


.89253 


1.1204 


1.3404 


.25394 


.74606 




15 


42 





.66913 


.33087 


1.4945 


.90040 


1.1106 


1.3456 


.25686 


.74314 


48 







15 


.67237 


.32763 


1.4873 


.90834 


1.1009 


1.3509 


.25978 


.7402? 




45 




30 


.67559 


.32441 


1.4802 


.91633 


1.0913 


1.3563 


.2627'2 


.73728 




30 




45 


.67830 


.32120 


1.4732 


.92439 


1.0818 


1.3618 


.26568 


.73432 




15 


43 





.68200 


.31800 


1.4663 


.93251 


1.0724 


1.3673 


.26865 


.73135 


47 







15 


.68518 


.31482 


1.4595 


.94071 


1.0630 


1.3729 


.27163 


.72837 




45 




30 


.68835 


.31165 


1.4527 


.94896 


1.0538 


1 .3786 


.27463 


.72537 




30 




45 


.69151* 


.30849 


1.4461 


.95729 


1.044G 


1.3843 


.27764 


.72236 




15 


44 





.69466 


.30534 


1.43*96 


.96569 


1.0355 


1.3902 


.28066 


.71934 


46 







15 


.69779 


.30221 


1.4331 


.97416 


1.0265 


1.3961 


.28370 


.71630 




45 




30 


.70091 


.29909 


1.4267 


.98270 


1.0176 


1.4020 


.28675 


.71325 




30 




45 


.70401 


.29599 


1.4204 


.99131 


1.0088 


1.4081 


.28981 


.71019 




15 


45 





.70711 


.29289 


1.4142 


1.0000' 


1.0000 


1.4142 


.29289 


.70711 


45 









Cosine. 


Ver. Sin. 


Secant. 


Cotan. 


Tang. 


Cosec. 


Co-Vers. 


Sine. 


o 


M. 



From 45 to 60 read from bottom of table upwards, 



162 



MATHEMATICAL TABLES. 
LOGARITHMIC SINES, ETC. 



Deg. 


Sine. 


Cosec. 


Versin. 


Tangent. 


Cotan. 


Covers. 


Secant. 


Cosine. 


Deg. 





In.Neg. 


Infinite. 


In.Neg. 


In.Ne?. 


Infinite. 


10.00000 


10.00000 


10.00000 


90 


1 


8.24186 


11.75814 


0.18271 


8.24192 


11.75808 


9.99235 


10.00007 


9.99993 


89 


2 


8.54282 


11.45718 


6.78474 


8.54308 


11.45692 


9.98457 


10.00026 


9.99974 


88 


3 


8.71880 


11.28120 


7.13687 


8.71940 


11.28060 


9.97665 


10.00060 


9.99940 


87 


4 


8.84358 


11.15642 


7.38667 


8.84464 


11.15536 


9.96860 


10.06106 


9.99894 


86 


5 


8.94030 


11.05970 


7.58039 


8.94195 


11.05805 


9.96040 


10.00166 


9.99834 


85 


6 


9.01923 


10.98077 


7.73863 


9.02162 


10.97838 


9.95205 


10.00239 


9.99761 


84 


7 


9.08589 


10.91411 


7.87238 


9.08914 


10.91086 


9.94356 


10.00325 


9.99675 


83 


8 


9.14356 


10.85644 


7.98820 


9.14780 


10.85220 


9.93492 


10.00425 


9.99575 


82 


9 


9.19433 


10.80567 


8.09032 


9.19971 


10.80029 


9.92612 


10.00538 


9.99462 


81 


10 


9.23967 


10.76033 


8.18162 


9.24632 


10.75368 


9.91717 


10.00665 


9.99335 


80 


11 


9.28060 


10.71940 


8.26418 


9.28865 


10.71135 


9.90805 


10.00805 


9.99195 


79 


12 


9.31788 


10.68212 


8.33950 


9.32747 


10.67253 


9.89877 


10.00960 


9. 99f)40 


78 


13 


9.35209 


10.64791 


8.40875 


9.36336 


10.636G4 


9.88933 


10.01128 


9.98872 


77 


14 


9.38368 


10.61632 


8.47282 


9.39677 


10.60323 


9.87971 


10.01310 


9.98690 


76 


15 


9.41300 


10.58700 


8.53243 


9.42805 


10.57195 


9.86992 


10.01506 


9.98494 


75 


16 


9.44034 


10.55966 


8.58814 


9.45750 


10.54250 


9.85996 


10.01716 


9.98284 


74 


17 


9.46594 


10.53406 


8.64043 


9.48534 


10.51460 


9.84981 


10.01940 


9.98060 


73 


18 


9.48998 


10.51002 


8.68969 


9.51178 


10.48822 


9.83947 


10.02179 


9.97821 


72 


19 


9.51264 


10.48736 


8.73625 


9.53697 


10.46303 


9.82894 


10.02433 


9.97567 


71 


20 


, 9. 53405 


10.46595 


8.78037 


9.56107 


10.43893 


9.81821 


10.02701 


9.97299 


70 


21 


9.55433 


10.44567 


8.82230 


9.58418 


10.41582 


9.807*9 


10.02985 


9.97015 


69 


22 


9.57358 


10.42642 


8.86223 


9.60641 


10.39359 


9.79615 


10.03283 


9.96717 


68 


23 


9.59188 


10.40812 


8.90034 


9.62785 


10.37215 


9.78481 


10.03597 


9.96403 


67 


24 


9.60931 


10.39069 


8.93679 


9.64858 


10.35142 


9.77325 


10.03927 


9.96073 


66 


25 


9.62595 


10.37405 


8.97170 


9.66867 


10.33133 


9.7614C 


10.04272 


9.95728 


65 


26 


9.64184 


10.35816 


9.00521 


9.68818 


10.31182 


9.74945 


10.04634 


9.95366 


64 


27 


9.65705 


10.34295 


9.03740 


9.70717 


10.29283 


9.73720 


10.05012 


9.94988 


63 


28 


9.67161 


10.32839 


9.06838 


9.72567 


10.27433 


9.72471 


10.05407 


9.94593 


62 


29 


9.68557 


10.31443 


9.09823 


9.74375 


10.25625 


9.71197 


10.05818 


9.94182 


61 


30 


9.69897 


10.30103 


9.12702 


9.76144 


10.23856 


9.69897 


10.06247 


9.93753 


60 


31 


9.71184 


10.28816 


9.15483 


9.77877 


10.22123 


9.68571 


10.06693 


9.93307 


59 


32 


9.72421 


10.27579 


9.18171 


9.79579 


10.20421 


9.67217 


10.07158 


9.92842 


58 


33 


9.73611 


10.26389 


9.20771 


9.81252 


10.18748 


9.65836 


10.07641 


9.92359 


57 


34 


9.74756 


10.25244 


9.23290 


9.82899 


10.17101 


9.64425 


10.08143 


9.91857 


56 


35 


9.75859 


10.24141 


9.25731 


9.84523 


10.15477 


9.62984 


10.08664 


9.91336 


55 


36 


9.76922 


10.23078 


9.28099 


9.86126 


10.13874 


9.61512 


10.09204 


9.90796 


54 


SP- 


9.77946 


10.22054 


9.30398 


9.87711 


10.12289 


9.60008 


10.09765 


9.90235 


53 


SS 


9.78934 


10.21066 


9.32631 


9.89281 


10.10719 


9.58471 


10.10347 


9.89653 


52 


39 


9.79887 


10.20113 


9.34802 


9.90837 


10.09163 


9.56900 


10.10950 


9.89050 


51 


40 


9.80807 


10.19193 


9.36913 


9.92381 


10.07619 


9.55293 


10.11575 


9.88425 


50 


41 


9.81694 


10.18306 


9.38968 


9.93916 


10.06084 


9.53648 


10.12222 


9.87778 


49 


42 


9.82551 


10.17449 


9.40969 


9.95444 


10.04556 


9.51966 


10.12893 


9.87107 


48 


43 


9.83378 


10.16622 


9.42918 


9.96966 


10.03034 


9.50243 


10.13587 


9.86413 


47 


44 


9.84177 


10.15823 


9.44818 


9.98484 


10.01516 


9.48479 


10.14307 


9.85693 


46 


^45 


9.84949 


10.15052 


9.46671 


10.00000 


10.00000 


9.46671 


10.15052 


9.84949 


45 




Cosine. 


Secant. 


Covers. 


Cotan. 


Tangent. 


Versin. 


Cosec. 


Sine. 





From 45 to 90 read from bottom of table upwards. 



SPECIFIC GRAVITY. 



MATERIALS. 

THE CHEMICAL ELEMENTS. 

Common Elements (42). 



1o* 




'i 


a-- 




0^ 


"3-; 






If 


Name. 


si 

S'S 


It 


Name. 


s*S 

$% 


|| 


Name. 


fl bJO 

'3 


gco 




4fje 


o^ 




<$ 


go> 




<: 


Al 


Aluminum 


27.1 


F 


Fluorine 


19. 


Pd 


Palladium 


106. 


Sb 


Antimony 


120.4 


Au 


Gold 


197.2 


P 


Phosphorus 


31. 


As 


Arsenic 


75.1 


H 


Hydrogen 


1.01 


Pt 


Platinum 


194.9 


Ba 


Barium 


137.4 


I 


Iodine 


126.8 


K 


Potassium 


39.1 


Bi 


Bismuth 


208.1 


Ir 


I rid in m 


193.1 


Si 


Silicon 


28.4 


B 


Boron 


10.9 


Fe 


Iron 


56. 


Ag 


Silver 


107.9 


Br 


Bromine 


79.9 


Pb 


Lead 


206.9 




Sodium 


23. 


Cd 


Cadmium 


111.9 


Li 


Lithium 


7.03 


S? 


Strontium 


87.6 


Ca 


Calcium 


40.1 


Mg 


Magnesium 


24.3 


S 


Sulphur 


32.1 


C 


Carbon 


12. 


Mn 


Manganese 


55. 


Sn 


Tin 


119. 


Cl 


Chlorine 


35.4 


Hg 


Mercury 


200. 


Ti 


Titanium 


48.1 


Cr 


Chromium 


52.1 


Ni 


Nickel 


58.7 


W 


Tungsten 


184.8 


Co 


Cobalt 


59. 


N 


Nitrogen 


14. 


Va 


Vanadium 


51.4 


Cu 


Copper 


63.6 





Oxygen 


16. 


Zn 


Zinc 


65.4 



j-tureu to \j = JD ttuu n = i.vuo. vvneii n. is ttt.i4.cii tt i, v^ = lu.o/y, uiiu 1110 

other figures are diminished proportionately. (See Jour. Am. Chem. Soc., 

TV. o i.^V. -i OflC \ 




The Rare Elements (27). 



Beryllium, Be. 
Caesium, Cs. 
Cerium, Ce. 
Didymium, D. 
Erbium, E. 
Gallium, Ga. 
Germanium, Ge. 


Glucinum, G. 
Indium, In. 
Lanthanum, La. 
Molybdenum, Mo. 
Niobium, Nb. 
Osmium, Os. 
Rhodium, R. 


Rubidium, Rb. 
Ruthenium, Ru. 
Samarium, Sm. 
Scandium, Sc. 
Selenium, Se. 
Tantalum, Ta. 
Tellurium, Te. 


Thallium, Tl. 
Thorium, Th. 
Uranium, U. 
Ytterbium, Yr. 
Yttrium, Y. 
Zirconium, Zr. 



SPECIFIC GRAVITY. 

The specific gravity of a substance is its weight as compared with the 
weight of an equal bulk of pure water. 
To find the specific gravity of a substance. 

W = weight of body in air; w = weight of body submerged in water. 

W 
Specific gravity = w _ w - 

If the substance be lighter than the water, sink it by means of a heavier 
substance, and deduct the weight of the heavier substance. 

Specific-gravity determinations are usually referred to the standard of the 
weight of water at 62 F., 62.355 Ibs. per cubic foot. Some experimenters 
have used 60 F. as the standard, and others 32 and 39.1 F. There is no 
general agreement. 

Given sp. gr. referred to water at 39.1 F., to reduce it to the standard of 
62 F. multiply it by 1.00112. 

Given sp. gr. referred to water at 62 F., to find weight per cubic foot mul- 
tiply by 62.355. Given weight per cubic foot, to find sp. gr. multiply by 
0.016037. Given sp. gr., to find weight per cubic inch multiply by .036085. 



164 



MATERIALS. 



Weight and Specific Gravity of Metals. 





Specific Gravity. 
Range accord- 
ing to 
several 
Authorities. 


Specific Grav- 
ity. Approx. 
Mean Value, 
used in 
Calculation of 
Weight. 


Weight 
per 
Cubic 
Foot, 
Ibs. 


Weight 
per 
Cubic 
Inch, 
Ibs. 






2.56 to 2.71 
6.66 to 6.86 
9.74 to 9.90 

7.8 to 8.6 

8.52 to 8.96 

8.6 to 8.7 
1.58 
5.0 
8.5 to 8.6 
19.245 to 19.361 
8.69 to 8.92 
22.38 to 23. 
6.85 to 7.48 
7.4 to 7.9 
11.07 to 11.44 
7. to 8. 
1.69 to 1.75 
13.60 to 13.62 
13.58 
13.37 to 13.38 
8.279 to 8.93 
20.33 to 22.07 
0.865 
10.474 to 10.511 
0.97 
7.69* to 7.932t 
7.291 to 7.409 
5.3 
17. to 17.6 
6.86 to 7.20 


2.67 
6.76 
9.82 

rs.eo 

J8.40 
1 8.36 
[8.20 

8.853 
8.65 

19.258 
8.853 

7.218 
7.70 
11.38 
8. 
1.75 
13.62 
13.58 
13.38 
8.8 
21.5 

10.505 

7.854 
7.350 

7.00 


166.5 
421.6 
612.4 

536.3 
523.8 
521.3 
511.4 

552. 
539. 

1200.9 
552. 
1396. 
450. 
480. 
709.7 
499. 
109. 
849.3 
846.8 
834.4 
548.7 
1347.0 

655.1 

489.6 

458.3 

436.5 


.0963 
.2439 
.3544 

.3103 
.3031 
.3017 
.2959 

.3195 
.3121 

.6949 
.3195 

.8076 
.2604 
.2779 
.4106 
.2887 
.0641 
.4915 
.4900 
.4828 
.3175 
.7758 

.3791 

.2834 
.2652 

.2526 


Antimony 




Bismuth . ... 




Brass: Copper 4- Zinc 1 
80 20 I 
70 30 >-.. 
60 40 
50 50 J 
Rron7p jCopper,95to80> 
onze lTin, 5 to 20 f 
Cadmium 


Calcium 








Cobalt 




Gold pure 




Copper 








Iron Cast 




" Wrought 




Lead 




Manganese 




Magnesium.. .. 




Mercury 
Nickel 


j 32 

....-< 60 
(212 


Platinum 




Potassium 




Silver 




Sodium 




Steel 




Tin 




Titanium 




Tungsten 




Zinc . 





* Hard and burned. 

t Very pure and soft. The sp. gr. decreases as the carbon is increased. 

In the first column of figures the lowest are usually those of cast metals, 
which are more or less porous; the highest are of metals finely rolled or 
drawn into wire. 

Specific Gravity of Liquids at 60 F. 



Acid, Muriatic 1.200 

" Nitric 1.217 

" Sulphuric 1.849 

Alcohol, pure 794 

" 95 per cent 816 

" 50 " " 934 

Ammonia, 27.9 per cent 891 

Bromine 2.97 

Carbon disulphide 1 .26 

Ether, Sulphuric 72 

Oil, Linseed 94 

Compression of tlse following Fluids under a Pressure of 
15 Ibs. per Square Inch. 

Water 00004663 I Ether 00006158 

Alcohol 0000216 [Mercury 00000265 



Oil, Olive 92 

Palm 97 

Petroleum 78 to .88 

Rape 92 

Turpentine 87 

Whale 92 

Tar 1. 

Vinegar 1.08 

Water 1. 

" sea 1.026 tol.O 



SPECIFIC GRAVITY. 



165 



The Hydrometer, 

The hydrometer is an instrument for determining the density of liquids. 
It is usually made of glass, and consists of three parts: (1) the upper part, 
a graduated stem or fine tube of uniform diameter; (2) a bulb, or enlarge- 
ment of the tube, containing air ; and (3) a small bulb at the bottom, con- 
taining shot or mercury which causes the instrument to float in a vertical 
position. The graduations are figures representing either specific gravities, 
or the numbers of an arbitrary scale, as in Baume's, Twaddell's, Beck's, 
and other hydrometers. 

There is a tendency to discard all hydrometers with arbitrary scales and 
;o use only those which read in terms of the specific gravity directly. 

Baume's Hydrometer and Specific Gravities Compared. 



Degrees 
Baume. 


Liquids 
Heavier 
than 
Water, 
sp. gr. 


Liquids 
Lighter 
than 
Water, 
sp. gr. 


Degrees 
Baume. 


Liquids 
Heavier 
than 
Water, 
sp. gr. 


Liquids 
Lighter 
than 
Water, 
sp. gr. 


Degrees 
Baum6. 


Liquids 
Heavier 
than 
Water, 
sp. gr. 


Liquids 
Lighter 
than 
Water, 
sp. gr. 




1 


1.000 
1.007 




19 
20 


1.143 
1.152 


.942 
.936 


38 
8P 


1.333 
1.345 


.839 
.834 





1 013 




21 


1 160 


.930 


40 


1 357 


.830 


3 
4 
5 


1.020 
1.027 
1.034 





22 
23 
24 


1.169 
1.178 
1.188 


.924 
.918 
.913 


41 
42 

44 


1.369 
1.382 
1.407 


.825 
.820 
.811 


fi 


1 041 




25 


1 197 


.907 


46 


1.434 


.802 


7' 


1.048 




96 


1.206 


.901 


48 


1.462 


.794 


8 
q 


1.05G 
1 063 




27 

98 


1.216 
1.226 


.896 
.890 


50 

53 


1.490 
1.520 


.785 

.777 


10 

11 

12 
13 
14 
15 


1.070 
1.078 
1.086 
1.094 
1.101 
1.109 


1.000 
.993 
.986 
.980 
.973 
.967 


29 
30 
31 
32 
33 
84 


1.236 
1.246 
1.256 
1.267 
1.277 
1.288 


.885 
.880 
.874 
.869 
.864 
.859 


54 

56 
58 
60 
65 
70 


1.551 

.583 
.617 
.652 

.747 
.854 


.768 
,760 
.753 
.745 


16 


1 118 


960 


35 


1 299 


.854 


75 


1 974 




17 


1.126 


.954 


36 


1 310 


.849 


76 


2.000 




18 


1.134 


.948 


37 


1.322 


.844 









Specific Gravity and Weight of Wood. 





Specific Gravity. 


Weight 
per 
Cubic 
Foot. 
Ibs. 




Specific Gravity. 


Weight 
per 
Cubic 
Foot, 
Ibs. 


Alder 


Avge. 
0.56 to 0.80 .68 


42 


Hornbeam. . . 


Avge. 

.76 .76 


47 


Apple 
A.sh 


.73 to .79 .76 
.60 to .84 .72 


47 
45 


Juniper 
Larch 


.56 .56 
.56 .56 


35 
35 


Bamboo.. . . 
Beech . 


.31 to .40 .35 
.62 to .85 .73 


22 

46 


Lignum vitse 
Linden 


.65 to 1.33 1.00 
.604 


62 

37 


Birch 


.56 to .74 .65 


41 


Locust 


.728 


46 


Box, 


.91 to 1.33 1.12 


70 


Mahogany. .. 


.56 to 1.06 .81 


51 


Cedar ... . 


.49 to .75 .62 


39 


Maple 


.57 to .79 .68 


42 


Cherry 
Chestnut 
Cork 


.61 to .72 .66 
.46 to .66 .56 
.24 .24 


41 
35 
15 


Mulberry 
Oak, Live 
44 White 


.56 to .90 .73 
.96 to 1.26 1.11 

.69 to .86 .77 


46 
69 
48 


Cypress.... 
Dogwood . . . 
Ebony 
Elm . . 


.41 to .66 .53 
.76 .76 
1.13 to 1.33 1.23 
.55 to .78 .61 


33 

47 
76 
38 


44 Red.... 
Pine, White. . 
44 Yellow. 
Poplar .... 


.73 to .75 .74 
.35 to .55 .45 
.46 to .76 .61 
.38 to .58 .48 


46 
28 
-38 
30 


Fir 
Gum 
Hackmatack 


.48 to .70 .59 
.84 to 1.00 .92 
.59 .59 


37 
57 
37 


Spruce. 
Sycamore.... 
Teak 


.40 to .50 .45 
.59 to .62 .60 
.66 to .98 .82 


28 
37 
51 


Hemlock . . . 
Hickory 


.36 to .41 .38 
69 to .94 .77 


24 

48 


Walnut 
Willow. 


.50 to .67 .58 
.49 to .59 .54 


36 
34 


Holly....'!.!! 


.76 .76 


47 









166 



MATE-KLAUS. 



Weight and Specific Oravlty of Stones, Brick, 
Cement, etc. 



Pounds per 
Cubic Foot. 



Specific 
Gravity. 



Asphaltum 87 

Brick, Soft 100 

" Common 112 

41 Hard 125 

" Pressed 135 

" Fire 140 to 150 

Brickwork in mortar 100 

" cement 112 

Cement, Rosendale, loose 60 

*' Portland, " 78 

Clay 120 to 150 

Concrete 120 to 140 

Earth, loose 72 to 80 

rammed 90 to 110 

Emery 250 

Glass 156 to 172 

" flint 180tol96 

Gneiss I , A , ^ n 

Granite p ' 160 to 170 

Gravel 100 to 120 

Gypsum 130 to 150 

Hornblende 200 to 220 

Lime, quick, in bulk 50 to 55 

Limestone 170 to 200 

Magnesia, Carbonate 150 

Marble 160 to 180 

Masonry, dry rubble 140 to 160 

" dressed 140 to 180 

Mortar 90 to 100 

Pitch 72 

Plaster of Paris 74 to 80 

Quartz 165 

Sand 90 to 110 

Sandstone 140 to 150 

Slate 170tol80 

Stone, various 135 to 200 

Trap 170 to 200 

Tile 110 to 120 

Soapstone 166 to 175 



1.39 
1.6 
1.79 
2.0 
2.16 

2.24 to 2.4 
1.6 
1.79 
.96 
1.25 

1.92 to 2.4 
1.92 to 2. 24 

1.15 to 1.28 
1.44 to 1.76 
4. 

2.5 to 2.73 
2. 88 to 3. 14 

2.56 to 2.72 

1.6 to 1.92 
2. 08 to 2. 4 
3.2 to 3. 52 
> .8 to .88 
2. 72 to 3. 2 
2.4 

2. 56 to 2. 88 

2. 24 to 2. 56 

2.24 to 2.88 

1.44 to 1.6 

1.15 

1.18 to 1.28 

2.64 

1.44 to 1.76 

2.24 to 2.4 

2. 72 to 2. 88 

2.16 to 3. 4 
2. 72 to 3.4 
1.76 to 1.92 
2.65 to 2.8 



Specific Gravity and Weight of Oases at Atmospheric 
Pressure and 32 F. 

(For other temperatures and pressures see pp. 459, 479.) 



Density, 
Air ='l. 



Air 

Oxygen, O 

Hydrogen, H 

Nitrogen, N 

Carbon monoxide, CO... 

Carbon dioxide, CO 2 

Methane, marsh-ga s, CH 4 

Eihylene, C 2 H 4 

Acetylene, C 2 H 2 

Ammonia, NH 3 

Water vapor, H 2 O 



1.0UOO 
1.1052 
0.0692 
0.9701 
0.9671 
1.5197 
0.5530. 
0.9674 
0.8982 
0.5889 
0.6218 



Density, 
H = 1. 

14.444 
15.963 

1.000 
14.012 
13.968 
21 .950 

7.987 
13.973 
12.973 

8.506 

8.981 



per Litre. 



1.2931 
1.4291 
0.0895 
1.2544 
1.2505 
1.9650 
0.7150 
1.2510 
1.1614 
0.7615 
0.8041 



Lbs. per Cubic Ft. 



Cu. Ft, 



.080753 

.08921 

.00559 

.07831 

.07807 

.12567 

.04464 

.07809 

.07251 

.04754 

.05020 



per Lb. 



12.388 
11.209 
178.931 
12.770 
12.810 
8.152 
22.429 
12.805 
13.792 
21.036 
19.922 



PROPERTIES OF THE USEFUL METAXS. 16? 

PROPERTIES OF THE USEFUL METALS. 

Aluminum, Al. Atomic weight 27.1. Specific gravity 2.6 to 2.7. 
The lightest of all the useful nietals except magnesium. A soft, ductile, 
malleable metal, of a white color, approaching silver, but with a bluish cast. 
Very non-corrosive. Tenacity about one third that of wrought-iron. For- 
merly a rare metal, but since 1890 its production and use have greatly in- 
creased on account of the discovery of cheap processes for reducing it from 
the ore. Melts at about 1160 F. For further description see Aluminum, 
under Strength of Materials. 

Antimony (Stibium), Sb. At. wt. 120.4. Sp. gr. 6.7 to 6.8. A brittle 
metal of a bluish-white color and highly crystalline or laminated structure. 
Melts at 842 F. Heated in the open air it burns with a bluish-white flame. 
Its chief use is for the manufacture of certain alloys, as type metal (anti- 
mony 1, lead 4), britannia (antimony 1, tin 9), and various anti-friction 
metals (see Alloys). Cubical expansion by heat from 92 to 212 F., 0.0070. 
Specific heat .050. 

Bismuth, Bi. At. wt. 208.1. Bismuth is of a peculiar light reddish 
color, highly crystalline, and so brittle that it can readily be pulverized. It 
melts at 510 F., and boils at about 2300 F. Sp. gr. 9.823 at 54 F., and 
10.055 just above the melting-point:. Specific heat about .0301 at ordinary 
temperatures. Coefficient of cubical expansion from 32 to 212, 0.0040. Con- 
ductivity for heat about 1/56 and for electricity only about 1/80 of that of 
silver. Its tensile strength is about 6400 Ibs. per square inch. Bismuth ex- 
pands in cooling, and Tribe has shown that this expansion does not take 
place until after solidification. Bismuth is the most diamagrietic element 
known, a sphere of it being repelled by a magnet. 

Cadmium, Cd. At. wt. 112. Sp. gr. 8.6 to 8.7. A bluish-white metal, 
lustrous, with a fibrous fracture. Melts below 500 F. and volatilizes at 
about 680 F. It is used as an ingredient in some fusible alloys with lead, 
tin, and hismuth. Cubical expansion from 32 to 212 F., 0.0094. 

Copper, Cu. At. wt. 63.2. Sp. gr. 8.81 to 8.95. Fuses at about 1930 
F. Distinguished from all other metals by its reddish color. Very ductile 
and malleable, and its tenacity is next to iron. Tensile strength 20,000 to 
30,000 Ibs. per square inch. Heat conductivity 73. 6# of that of silver, and su- 
perior to that of other metals. Electric conductivity equal to that of gold 
and silver. Expansion by heat from 32 to 212 F., 0.0051 of its volume. 
Specific heat .093. (See Copper under Strength of Materials: also Alloys.) 

Gold (Aurum). Au At. wt. 197.2. Sp. gr., when pure and pressed in a 
die, 19.34. Melts at about 1915 F. The most malleable and ductile of all 
metals. One ounce Troy may be beaten so as to cover 160 sq. ft. of surface. 
The average thickness of gold-leaf is 1/282000 of an inch, or 100 sq. ft. per 
ounce. One grain may be drawn into a wire 500 ft. in length. The ductil- 
ity is destroyed by the presence of 1/2000 part of lead, bismuth, or antimony. 
Gold is hardened by the addition of silver or of copper. In U. S. gold coin 
there are 90 parts gold and 10 parts of alloy, which is chiefly copper with a 
little silver. By jewelers the fineness of gold is expressed in carats, pure 
gold being 24 carats, three fourths fine 18 carats, etc. 

Iridium. Indium is one of the rarer metals. It has a white lustre, re- 
sembling that of steel; its hardness is about equal to that of the ruby; in 
the cold it is quite brittle, but at a white heat it is somewhat malleable. It 
is one of the Heaviest of metals, having a specific gravity of *jy.3S. It is ex- 
tremely infusible and almost absolutely inoxiclizable. 

For uses of iridium, methods of manufacturing it, etc., see paper by W. D. 
Dudley on the "Iridium Industry," Trans. A. I. M. E. 1884. 

Iron (Ferrum), Fe. At. wt. 56. Sp. gr.: Cast, 6.85 to 7.48; Wrought, 
7.4 to 7.9. Pure iron is extremely infusible, its melting point being above 
3000 F , but its fusibility increases with the addition of carbon, cast iron fus* 
ing about 2500 F. Conductivity for heat 11.9, and for electricity 12 to 14.8, 
silver being 100. Expansion in bulk by heat: cast iron .0033, and wrought iron 
.0035, from 32 to 212 F. Specific heat: cast iron .1298, wrought iron .1138, 
steel .1165. Cast iron exposed to continued heat becomes permanently ex- 
panded 1^ to 3 per cent of its length. Grate-bars should therefore be 
allowed about 4 per cent play. (For other properties see Iron and Steel 
under Strength of Materials.) 

Lead (Plumbum), JPb. At. wt. 208.9. Sp. gr. 11.07 to 11.44 by different 
authorities. Melte at about 625 F., softens and becomes pasty at about 
617 F. If broken by a sudden blow when just below the melting-point it ia 
quite brittle and the fracture appears crystalline. Lead is very malleable 



168 'MATERIALS. 

and ductile, but its tenacity is such that it can be drawn into wire with great 
difficulty. Tensile strength, 1600 to 2400 Ibs. per square inch. Its elasticity is 
very low, and the metal flows under very slight strain. Lead dissolves to 
some extent in pure water, but water containing carbonates or sulphates 
forms over it a film of insoluble salt which prevents further action. 

Magnesium, Mg. At. wt. 24. Sp. gr. 1.69 to 1.75. Silver-white, 
brilliant, malleable, and ductile. It is one of the lightest of metals, weighing 
only about two thirds as much as aluminum. In the form of filings, wire, 
or thin ribbons it is highly combustible, burning with a light of dazzling 
brilliancy, useful for signal-lights and for flash-lights for photographers. It 
is nearly non-corrosive, a thin film of carbonate of magnesia forming on ex- 
posure to damp air, which protects it from further corrosion. It may be 
alloyed with aluminum, 5 per cent Mg added to Al giving about as much in- 
crease of strength and hardness as 10 per cent of copper. Cubical expansion 
by heat 0.0083, from 32 to 212 F. Melts at 1200 F. Specific heat .25. 

Manganese, Mn. At. wt. 55. Sp. gr. 7 to 8. The pure metal is not 
used iu tne arts, but alloys of manganese and iron, called spiegeleisen when 
containing below 25 per cent of manganese, and ferro-manganese when con- 
taining from 25 to 90 per cent, are used in the manuf ,cture of steel. Metallic 
manganese, when alloyed with iron, oxidizes rapidly in the air, and its func* 
tion in steel manufacture is to remove the oxygen from the bath of steel 
whether it exists as oxide of iron or as occluded gas. 

Mercury (Hydrargyrum), Hg. At. wt. 199.8. A silver-white metal, 
liquid at temperatures above 39 F., and boils at 680 F. Unchangeable as 
gold, silver, and platinum in the atmosphere at ordinary temperatures, but 
oxidizes to the red oxide when near its boiling-point. Sp.gr.: when liquid 
13.58 to 13.59, when frozen 14.4 to 14.5. Easily tarnished by sulphur fumes, 
also by dust, from which it may be freed by straining through a cloth. No 
metal except iron or platinum should be allowed to touch mercury. The 
smallest portions of tni, lead, zinc, and even copper to a less extent, cause it 
to tarnish and lose its perfect liquidity. Coefficient of cubical expansion 
from 32 to 212 F. .0182; per deg. .000101. 

Nickel, Ni. At. wt. 58.3. Sp. gr. 8.27 to 8.93. A silvery- white metal 
with a strong lustre, not tarnishing on exposure to the air. Ductile, hard, 
and as tenacious as iron. It is attracted to the magnet and may be made 
magnetic like iron. Nickel is very difficult of fusion, melting at about 
3000* F. Chiefly used in alloys with copper, as german-silver, nickel-silver, 
etc., and recently in the manufacture of steel to increase its hardness and 
strength, also for nickel-plating. Cubical expansion from 32 to 212 F., 
0.0038. Specific heat .109. 

Platinum, Pt. At. wt. 195. A whitish steel-gray metal, malleable, 
very ductile, and as unalterable by ordinary agencies as gold. When fused 
and refined it is as soft as copper. Sp. gr. 21.15. It is fusible only by the 
pxyhydrogen blowpipe or in strong electric currents. When combined with 
indium it forms an alloy of great hardness, which has been used for gun- 
vents and for standard weights and measures. The most important uses of 
platinum in the arts are for vessels for chemical laboratories and manufac- 
tories, and for the connecting wires in incandescent electric lamps. Cubical 
expansion from 32 to 212 F., 0.0027, less than that of any other metal. ex- 
cept the rare metals, and almost the same as glass. 

Silver (Argentum), Ag. At. wt. 107.7. Sp. gr. 10.1 to 11.1, according to 
condition and purity. It is the whitest of the metals, very malleable and 
ductile, and in hardness intermediate between gold and copper. Melts at 
about 1750 F. Specific heat .056. Cubical expansion from 32 to 212 F., 
0.0058. As a conductor of electricity it is equal to copper. As a conductor 
of heat it is superior to all other metals. 

Tin (Stannum) Sn. At. wt. 118. Sp. gr. 7.293. White, lustrous, soft : 
malleable, of little strength, tenacity about 3500 Ibs. per square inch. Fuses 
at 442 F. Not sensibly volatile when melted at ordinary heats. Heat con- 
ductivity 14.5, electric conductivity 12.4; silver being 100 in each case. 
Expansion of volume by heat .0069 from 32 to 212 F. Specific heat .055. Its 
chief uses are for coating of sheet-iron (called tin plate) and for making 
alloys with copper and other metals. 

Zinc, Zn. At. wt. 65. Sp. gr. 7.14. Melts at 780 F. Volatilizes and 
burns in the air when melted, with bluish-white fumes of zinc oxide. It is 
ductile and malleable, but to a much less extent than copper, and its tenacity, 
about 5000 to 6000 Ibs. per square inch, is about one tenth that of wrought 
iron. It is practically non-corrosive in the atmosphere, a thin film of car- 
bonate of zinc forming upon it. Cubical expansion between 32 and 212 F., 



MEASURES AKD WEIGHTS OF VARIOUS MATERIALS. 169 



0.0088. Specific heat .096. Electric conductivity 29, heat conductivity 36, 
silver being 100. Its principal uses are for coating iron surfaces, called 
" galvanizing," and for making brass and other alloys. 

Table Showing the Order of 
Malleability. Ductility. Tenacity. Infusitoility. 



Gold 

Silver 

Aluminum 

Copper 

Tin 

Lead 

Zinc 

Platinum 

Iron 



Platinum 

Silver 

Iron 



Aluminum 

Zinc 

Tin 

Lead 



Iron 

Copper 

Aluminum 

Platinum 

Silver 

Zinc 

Gold 

Tin 

Lead 



Platinum 

Iron 

Copper 

Gold 

Silver 

Aluminum 

Zinc 

Lead 

Tin 



WEIGHT OF RODS, BARS, PLATES, TUBES, AND 
SPHERES OF DIFFERENT MATERIALS. 

Notation : b = breadth, t = thickness, s = side of square, d = external 
Diameter, d l = internal diameter, all in inches. 

Sectional areas : of square bars s 2 ; of flat bars = bt\ of round rods 
,7854d a ; of tubes = .7854(d a - d, a ) = 3.1410(d* - f 2 ). 

Volume of 1 foot in length :" of square bars = 12s 2 ; of flat bars = 126 ; of 
round bars = 9.4248d a ; ot tubes = 9.4248(<i 2 - d-ft = 37.699(<i ? 2 ), in cu. in. 

Weight per foot length = volume X weight per cubic inch of the material. 
Weight of a sphere = diam. 3 X .5236 X weight per cubic inch. 



Material. 



Cast iron 

Wrought Iron 

Steel 

Copper & Bronze I 
(copper and tin) f 



Lead 

Aluminum 

Glass 

Pine Wood, dry . . . 



8.855 



11.38 
2.G7 
2.62 
0.481 



7.218450. 
7.7 480. 
7.854489.6 



552. 



8.393523.2 



709.6 
166.5 
163.4 
30.0 



37.5 

40. 
40. 



Sit 

W 



83 



$&! 

.4s' 



46. 

43.63.633s 2 

59.1 4. 93s 2 
13.91.16s 2 
13.61.13s* 
2.50.21s 2 



3.46* 
3.8336* 

3.6336* 

4.936* 
1.166* 
1.136* 
0.216* 




,2604 

,27791. 

28331.02 



.31951.15 

30291.09 
,41061.48 



3470 



09630. 
09450.34 
0174 1-16 



15-16 2. 454d 2 
2.618d 2 
2.670d 2 



3. Olid 2 

2.854d 2 
3.870d 2 



0.164d 2 



.0091d 



Weight per cylindrical in., 1 in. long, = coefficient of d 2 in ninth col. -v- 12. 

For tubes use the coefficient of d 2 in ninth column, as for rods, and 
multiply it into (d 2 c?r); or multiply it by 4(dt l 2 ). 

For hollow spheres use the coefficient of d 3 in the last column and 
multiply it into (d 3 rfj 3 ). 

For hexagons multiply the weight of square bars by 0.866 (short 
diam. of hexagon == side of square). For octagons multiply by 0.8284. 

MEASURES AND WEIGHTS OF VARIOUS 
MATERIALS (APPROXIMATE). 

Brickworlt. Brickwork is estimated by the thousand, and for various 
thicknesses of wall runs as follows: 



8*4-in. wall, or 1 brick in tbickness v 14 bricks per superficial feet. 
12% ** ** " 1U " * 21 '* " " 

17 ' 2 



21 
28 

35 



An ordinary brick measures about $4X4X2 inches, which is equal to 66 
cubic inches, or 26.2 bricks to a cubic foot. The average weight is % Ibs. 



170 



MATERIALS. 



Fuel. A bushel of bituminous coal weighs 76 pounds and contains 2688 
cubic inches = 1.554 cubic feet. 29 .47 bushels = 1 gross ton. 

A bushel of coke weighs 40 Ibs. (35 to 42 Ibs.). 

One acre of bituminous coal contains 1600 tons of 2240 Ibs. per foot of 
thickness of coal worked. 15 to 25 per cent must be deducted for waste in 
mining. 

41 to 45 cubic feet bituminous coal when broken down = 1 ton, 2240 Ibs. 

34 to 41 " " anthracite, prepared for market = 1 ton, 2240 Ibs. 

123 " ' of charcoal .= 1 ton, 2240 Ibs. 

70.9 ** " "coke = 1 ton, 2240 Ibs. 

1 cubic foot of anthracite coal (see also page 625) = 55 to 66 Ibs. 

1 *' "bituminous 4 * , = 50 to 55 Ibs. 

1 " '* Cumberland coal. = 53 Ibs. 

1 " " Cannel coal = 50.3 Ibs. 

1 " " charcoal (hardwood) = 18.5 Ibs. 

1 " " " (pine) =181bs. 

A bushel of charcoal. In 1881 the American Charcoal-Iron Work- 
ers' Association adopted for use in its official publications for the standard 
bushel of charcoal 2748 cubic inches, or 20 pounds. A ton of charcoal is to 
be taken at 2000 pounds. This figure of 20 pounds to the bushel was taken 
as a fair average of different bushels used throughout the country, and it 
has since been established by law in some States. 

Ores, Earths, etc. 

13 cubic feet of ordinary gold or silver ore, in mine = 1 ton = 2000 Ibs. 

20 " " " broken quartz = 1 ton = 2000 Ibs. 

18 feet of gravel in bank =1 ton. 

27 cubic feet of gravel when dry = 1 ton. 

25 " *' "sand = 1 ton. 

18 ' " earth in bank = 1 ton. 

27 ** " ** " when dry = 1 ton. 

17 " clay =lton. 

Cement. English Portland, sp. gr. 1.25 to 1.51, per bbl 400 to 430 Ibs. 

Rosendale, U. S., a struck bushel 62 to 70 Ibs. 

liime. A struck bushel 72 to 75 Ibs. 

Grain. A struck bushel of wheat = 60 Ibs.; of corn = 56 Ibs. : of oats = 
30 Ibs. 

Salt. A struck bushel of salt, coarse, Syracuse, N. Y. = 56 Ibs. ; Turk's 
Island = 76 to 80 Ibs. 

Weight of Earth Filling. 
(From Howe's " Retaining Walls.") 

Average weight in 
Ibs. per cubic foot. 

Earth, common loam, loose 72 to 80 

" shaken 82 to 92 

4 * rammed moderately 90 to 100 

Gravel 90 to 106 

Sand 90tol06 

Soft flowing mud 104 to 120 

Sand, perfectly wet 118 to 129 

COMMERCIAL SIZES OF IRON BARS. 

Flats. 



Width. Thickness. 



Width. Thickness. Width. Thickness. 





5* 



WEIGHTS OF WROUGHT IRON BARS. 



171 



Rounds : H to \% inches, advancing by 16ths, and \% to 5 inches by 
8ths. 

Squares : 5/16 to 1J4 inches, advancing by 16ths, and 1J4 to 3 inches by 
8ths. 

Half rounds: 7/16, %, %, 11/16, %, 1, % 1^, % 1%, 2 inches. 

Hexagons : % to 1^ inches, advancing by 8ths. 

Ovals : y% X y, % X 5/16, % x %, Vs X 7/16 inch. 

Half ovals : ^ X & % X 5/32, % X 3/16, % X 7/32, 1^ X H, 1% X %, 
1% X % inch. 

Round-edge flats : 1^ X J4 1% X %, 1% X % inch. 

Rands : }4 to \y% inches, advancing by 8ths, 7 to 16 B. W. gauge. 

1J4 to 5 inches, advancing by 4ths, 7 to 16 gauge up to 3* inches, 4 to 14 
gauge, 3J4 to 5 inches. 

WEIGHTS OF SQUARE AND ROUND RARS OF 
WROUGHT IRON IN POUNDS PER LINEAL FOOT. 

Iron weighing 480 Ibs. per cubic foot. For steel add 2 per cent. 



Thickness or 
Diameter 
in Inches. 


Weight of 
Square Bar 
One Foot 
Long. 


Weight of 
Round Bar 
One Foot 
Long. 


Thickness or 
Diameter 
in Inches. 


Weight of 
Square Bar 
One Foot 
Long. 


Weight of 
Round Bar 
One Foot 
Long. 


Thickness or 
Diameter 
in Inches. 


Weight of 
Square Bar 
One Foot 
Long. 


Weight of 
Round Bar 
One Foot 
Long. 









11/16 


24.08 


18.91 


% 


96.30 


75.64 


1/16 


.013 


.010 


M 


25.21 


19.80 


7/16 


98.55 


77.40 


M 


.052 


.041 


13/16 


26.37 


20.71 


H 


100.8 


79.19 


3/16 


.117 


.092 




27.55 


21.64 


9/16 


103.1 


81.00 


M 


.208 


.164 


15/16 


28.76 


22.59 


% 


105.5 


82.83 


5/16 


.326 


.256 


3 


30.00 


23.56 


11/16 


107.8 


84.69 


% 


.469 


.368 


1/16 


31.26 


24.55 


H 


110.2 


86.56 


7/16 


.638 


.501 


\& 


32.55 


25.57 


13/16 


112.6 


88.45 




.833 


.654 


3/16 


33.87 


26.60 




115.1 


90.36 


9/16 


1.055 


.828 


\A 


35.21 


27.65 


15/16 


117.5 


92.29 


Ys 


1.302 


1.023 


5/16 


36.58 


28.73 


6 


120.0 


94.25 


11/16 


1.576 


1.237 




37.97 


29.82 




125.1 


98.22 


82 


1.875 


1.473 


7/16 


39.39 


30.94 


^4 


130.2 


102.3 


13/16 


2.201 


1.728 




40.83 


32.07 


3X 


135.5 


106.4 


% 


2.552 


2.004 


9/16 


42.30 


33.23 


L 


140.8 


110.6 


15/16 


. 2.930 


2.301 


% 


43.80 


34.40 


To 


146.3 


114.9 




3.333 


2.618 


11/16 


45.33 


35.60 


% 


151.9 


119.3 


1/16 


3.763 


2.955 




46.88 


36.82 


VR 


157.6 


123.7 




4.219 


3.313 


13/16 


48.45 


38.05 


7 


163.3 


128.3 


3/16 


4.701 


3.692 




50.05 


39.31 




169.2 


132.9 


/4 


5.208 


4.091 


15/16 


51.68 


40.59 


x4 


175.2 


137.6 


5/16 


5.742 


4.510 




53.33 


41.89 


s2 


181.3 


142.4 




6.302 


4.950 


1/16 


55.01 


43.21 


i^ 


187.5 


147.3 


7/16 


6.888 


5.410 




56.72 


44.55 


% 


193.8 


152.2 




7.500 


5.890 


3/16 


58.45 


45.91 


% 


200.2 


157.2 


9/16 


8.138 


6.392 




60.21 


47.29 


y& 


206.7 


162.4 


9& 


8.802 


6.913 


5/16 


61.99 


48.69 


8 


213.3 


167.6 


11/16 


9.492 


7.455 


'n. 


63.80 


50.11 




226.9 


178.2 


M 


10.21 


8.018 


7/16 


65.64 


51.55 


vh 


240.8 


189.2 


13/16 


10.95 


8.601 


y 


67.50 


53.01 


3^ 


255.2 


200.4 


H 


11.72 


9.204 


9/16 


69.39 


54.50 


9 


270.0 


212.1 


15/16 


12.51 


9.828 




71.30 


56.00 




285.2 


224.0 


2 


13.33 


10.47 


11/16 


73.24 


57.52 


/^ 


300.8 


236.3 


1/16 


14.18 


11.14 


M 


75.21 


59.07 


*M 


316.9 


248.9 


^0 


15.05 


11.82 


18/16 


77.20 


60.63 


10 


333.3 


261.8 


3/16 


15.95 


12.53 




79.22 


62.22 




350.2 


275.1 


U 


16.88 


13.25 


15/16 


81.26 


63.82 


V% 


367.5 


288.6 


5/16 


17.83 


14.00 


5 


83.33 


65.45 


% 


385.2 


302.5 


sk 


18.80 


14.77 


1/16 


85.43 


67.10 


11 


403.3 


316.8 


7/16 


19.80 


15.55 




87.55 


68.76 




421.9 


331.3 




20.83 


16.36 


3/16 


89.70 


70.45 


^ 


440.8 


346.2 


9/16 


21.89 


17.19 


k 


91.88 


72.16 


34 


460.2 


361.4 




22.97 


18.04 


5/16 


94.08 


73.89 


12 


480. 


377. 



172 



MATERIALS. 



g: 

* 

i 

If 

' ~***^^^z2zsz2s^z%%z%%$%% 

^ ^ioot~t-ooo>ooWw^^oorcoo^Hg|g3^o 

N 

tt 

h * 



N -33 

g 

^3 
g 

P 

1^ 

W 

* 1 * 
g 

b ^ " ' "rH^T^(^WC^COCCCOTj<T}<Tj<JOlO*n)ODOOt>J>b-OOOOQOOSOSO5O 

pa 

O 9S?^ 

S^, SSS 
' TH " r H T ^iHCi(NC^NooeococOT^r^^T^inooo)oi>i>{><>ocQO 

A<I.S 

S 9c4i 



WEIGHTS OF FLAT WROUGHT IKON. 173 



ft 




8888 



X*> 



If 

at 



1* 



174 



MATERIALS. 



WEIGHT OF IRON AND STEEL SHEETS. 

Weights per Square Foot. 

(For weights by Decimal Gauge, see page 32.) 



Thickness by Birmingham Gauge. 


Thickness by American (Brown and 
Sharpe's) Gauge. 


No. of 

Gauge. 


Thick- 
ness in 
Inches. 


Iron. 


Steel. 


No. of 
Gauge. 


Thick- 
ness in 
Inches. 


Iron. 


Steel. 


0000 


.454 


18.16 


18.52 


0000 


.46 


18.40 


18.77 


000 


.425 


17.00 


17.34 


000 


.4096 


16.38 


16.71 


00 


.38 


15.20 


15.50 


00 


.3648 


14.59 


14.88 





.34 


13.60 


13.87 





.3249 


13.00 


13.26 


1 


.3 


12.00 


12.24 


1 


.2893 


11.57 


11.80 


2 


.284 


11.36 


11.59 


2 


.2576 


10.30 


10.51 


3 


.259 


10.36 


10.57 


3 


.2294 


9.18 


9.36 


4 


.238 


9.52 


9.71 


4 


.2043 


8.17 


8.34 


5 


.22 


8.80 


8.98 


5 


.1819 


7.28 


7.42 


6 


.203 


8.12 


8.28 


6 


.1620 


6.48 


6.61 


7 


.18 


7.20 


7.34 


7 


.1443 


5.77 


5.89 


8 


.165 


6.60 


6.73 


8 


.1285 


5.14 


5.24 


9 


.148 


5.92 


6.04 


9 


.1144 


4.58 


4.67 


10 


.134 


5.36 


5.47 


10 


.1019 


4.08 


4.16 


11 


.12 


4.80 


4.90 


11 


.0907 


3.63 


3.70 


12 


.109 


4.36 


4.45 


12 


.0808 


3.23 


3.30 


13 


.095 


3.80 


3.88 


13 


.0720 


2.88 


2.94 


14 


.083 


3.32 


3.39 


14 


.0641 


2.56 


2.62 


15 


.072 


2.88 


2.94 


15 


.0571 


2.28 


2.33 


16 


.065 


2.60 


2.65 


16 


.0508 


2.03 


2.07 


17 


.058 


2.32 


2.37 


17 


.0453 


1.81 


.85 


18 


.049 


.96 


2.00 


18 


.0403 


1.61 


.64 


19 


.042 


.68 


1.71 


19 


.0359 


1.44 


.46 


20 


.035 


.40 


1.43 


20 


.0320 


1.28 


.31 


21 


.032 


.28 


1.31 


21 


.0,285 


1.14 


.16 


22 


.028 


.12 


1.14 


22 


.0253 


1.01 


.03 


23 


.025 


.00 


1.02 


23 


.0226 


.904 


.922 


24 


.022 


.88 


.898 


24 


.0201 


.804 


.820 


25 


.02 


.80 


.816 


25 


.0179 


.716 


.730 


26 


.018 


.72 


.734 


26 


.0159 


.636 


.649 


27 


.016 


.64 


.653 


27 


.0142 


.568 


.579 


28 


.014 


.56 


.571 


28 


.0126 


.504 


.514 


29 


.013 


.52 


.530 


29 


.0113 


.452 


.461 


30 


.012 


.48 


.490 


30 


.0100 


.400 


.408 


31 


.01 


.40 


.408 


31 


.0089 


.356 


.363 


32 


.009 


.36 


.367 


32 


.0080 


.320 


.326 


33 


.008 


.32 


.326 


33 


.0071 


.284 


.290 


34 


.007 


.28 


.286 


34 


.0063 


.252 


.257 


35 


.005 


.20 


.204 


35 


.0056 


.224 


.228 


Specific gravity . 


I 


ron. Steel. 

.7 7.854 
4SQ fi 


fnnt . 480 


" g P " " inch 2778 .2833 



As there are many gauges in use differing from each other, and even the 
thicknesses of a certain specified gauge, as the Birmingham, are not assumed 
the same by all manufacturers, orders for sheets and wires should always 
state the weight per square foot, or the thickness in thousandths of an inch. 



WEIGHT OF PLATE IRON. 



175 



8 
iQ^S^SSS^SSgfiSSSSSSSSiSS^^SSSfiSSS 

THf-HT-H-IT-.l-H-frHT-.^T-11-lT-I^Hl-H^ 

^ _ < ^oopT-.(?'roincot-aooT-iOJcoincoi>ocoinQOOecinQOOccnooocoin 
JL r o 09 p 

" OO Tf -<3< "tf 

^ *?" : o =5 ? 05 ^' id co ^j 3; i^ p ? * o oo r ^ tr ??' os d o co" ?i op r? 

fa J, ^'ini^ooooc^rtiios<Nnt-ocoinooT^oc>?t^(>?QOcoosTf<oinoOTHt-.oi 

g 5 TOoaw^TjiTTTrininino^ooi-^i-i-ooaooosoo^^MTOcc^Tfinino 

5"S 8inpoS8Spo8S8SpSpS8p888ooppooopppp 

I! 

KQ, ^2 oi-occ^ooiTJinQO^Tft'-o^inooi i-^fi coosinocooJoo^ocowcooo 

f 

53 

bM 

* 

5_ 

gfe ^3 JL 2?S:Sffa?o r? co t Qt r c5^coinoQQ 

gl 

J . 

Djf 

g* * 
* 2 S 

^ H ^ ^^ojotof^wwoiTOWOTWWccro^^^^^ooo^'oofic^^^Qddooo 

!l 

H* 

1 

few 

ll 

gl 

*8 

2 

H^ 
^ 

^SSS^SJSi 



176 



MATERIALS. 



WEIGHTS OF STK12L BLOOMS. 

Soft steel. 1 cub ; c inch = 0.284 Ib. 1 cubic foot = 490.75 Ibs. 



Sizes. 


Lengths. 


1" 


6" 


12" 


18" 


24" 


30" 


36" 


42" 


48" 


54" 


60" 


66" 


12" x 4" 


13.63 


82 


164 


245 


327 


409 


491 


573 


654 


736 


818 


900 


11 x 6 


18.75 


113 


225 


338 


450 


563 


675 


788 


900 


1013 


1125 


1238 


x 5 


15.62 


94 


188 


281 


375 


469 


562 


656 


750 


843 


937 


1031 


x 4 


12.50 


75 


150 


225 


300 


375 


450 


525 


600 


675 


750 


825 


10 x 7 


19.88 


120 


239 


358 


477 


596 


715 


835 


955 


1074 


1193 


1312 


x 6 


17.04 


102 


204 


307 


409 


511 


613 


716 


818 


920 


1022 


1125 


x 5 


14.20 


85 


170 


256 


341 


426 


511 


596 


682 


767 


852 


937 


x 4 


11.36 


68 


136 


205 


273 


341 


409 


477 


546 


614 


682 


750 


x 3 


8.52 


51 


102 


153 


204 


255 


306 


358 


409 


460 


511 


562 


9 x 7 


17.89 


107 


215 


322 


430 


537 


644 


751 


859 


966 


1073 


1181 


x 6 


15.34 


92 


184 


276 


368 


460 


552 


644 


736 


828 


920 


1012 


x 5 


12.78 


77 


153 


230 


307 


383 


460 


537 


614 


690 


767 


844 


x 4 


10.22 


61 


123 


.184 


245 


307 


368 


429 


490 


552 


613 


674 


8 x 8 


18.18 


109 


218 


327 


436 


545 


655 


764 


873 


982 


1091 


1200 


x 7 


15.9 


95 


191 


286 


382 


477 


572 


668 


763 


859 


954 


1049 


x 6 


13.63 


82 


164 


245 


327 


409 


491 


573 


654 


736 


818 


900 


x 5 


11.36 


68 


136 


205 


273 


341 


409 


477 


546 


614 


682 


750 


x 4 


9.09 


55 


109 


164 


218 


273 


327 


382 


436 


491 


545 


600 


7 x 7 


13.92 


83 


167 


251 


334 


418 


501 


585 


668 


752 


835 


919 


x 6 


11.93 


72 


143 


215 


286 


358 


430 


501 


573 


644 


716 


,788 


x 5 


9.94 


60 


119 


179 


238 


298 


358 


417 


477 


536 


596 


656 


x 4 


7.95 


48 


96 


143 


191 


239 


286 


334 


382 


429 


477 


525 


x 3 


5.96 


36 


72 


107 


143 


179 


214 


250 


286 


322 


358 


393 


6^x W/ 2 


12. 


72 


144 


216 


388 


360 


432 


504 


576 


648 


720 


792 


x 4 


7.38 


44 


89 


133 


177 


221 


266 


310 


354 


399 


443 


487 


6 x 6 


10.22 


61 


123 


184 


245 


307 


368 


429 


490 


551 


613 


674 


x 5 


8.52 


51 


102 


153 


204 


255 


307 


358 


409 


460 


511 


562 


x 4 


6.82 


41 


82 


123 


164 


204 


245 


286 


327 


368 


409 


450 


x 3 


5.11 


31 


61 


92 


123 


153 


184 


214 


245 


276 


307 


337 


5^x sy 2 


8.59 


52 


103 


155 


206 


258 


309 


361 


412 


464 


515 


567 


x 4 


6.25 


37 


75 


112 


150 


188 


225 


262 


300 


337 


375 


412 


5 x 5 


7.10 


43 


85 


128 


170 


213 


256 


298 


341 


383 


426 


469 


x 4 


5.68 


34 


68 


102 


136 


170 


205 


239 


273 


307 


341 


375 


4^ x 4^ 


5.75 


35 


69 


104 


138 


173 


207 


242 


276 


311 


345 


380 


x 4 


5.11 


31 


61 


92 


123 


153 


184 


215 


246 


276 


307 


338 


4 x 4 


4.54 


27 


55 


80 



109 


136 


164 


191 


218 


246 


272 


300 


x 3}4 


3.97 


24 


48 


72 


96 


119 


143 


167 


181 


215 


238 


262 


x 3 


3.40 


20 


41 


61 


82 


102 


122 


143 


163 


184 


204 


224 


3^ x*3J^ 


3.48 


21 


42 


63 


84 


104 


125 


146 


167 


188 


209 


230 


x 3 


2.98 


18 


36 


54 


72 


89 


107 


125 


14-3 


161 


179 


197 


3 x 3 


2.56 


15 


31 


46 


61 


77 


92 


108 


123 


138 


154 


169 



SIZES AKD WEIGHTS OF STRUCTURAL SHAPES. 177 



SIZES AND WEIGHTS OF STRUCTURAL SHAPES. 

Minimum, Maximum, and Intermediate Weights and 
Dimensions of Carnegie Steel I-Beams. 



Sec- 
tion 
Index 


Depth 
of 
Beam. 


Weight 
per 
Foot, 


Flange 
Width- 


Web 
Thick- 
ness. 


Sec- 
tion 
Index 


Depth 
of 
Beam. 


Weight 
pei- 
Foot. 


Flange 
Width. 


Web 
Thick- 
ness. 




ins. 


Ibs. 


ins. 


ins. 




ins. 


Ibs. 


ins. 


ins. 


Bl 


24 


100 


7.25 


0.75 


B19 


6 


17.25 


3.58 


0.48 


44 


44 


95 


7.19 


0.69 


44 


44 


14.75 


3.45 


0.35 


44 


< 4 


90 


7.13 


0.03 


44 


44 


12.25 


3.33 


0.23 


44 


44 


85 


7.07 


0.57 


B21 


5 


14.75 


3.29 


0.50 


44 


44 


8 9 


7.00 


0.50 


44 


44 


12.25 


3.15 


0.36 


B3 


20 


75 


6.40 


0.65 


44 


44 


9.75 


3.00 


0.21 






70 


6.33 


0.58 


B23 


4 


1015 


2.88 


0.41 


4k 


44 


65 


6.25 


0.50 


44 


44 


9.5 


2.81 


0.34 


B80 


18 


70 


6.26 


0.72 


44 


44 


8:5 


2.73 


26 


44 


44 


65 


6.18 


0.64 


44 


44 


7.5 


2.66 


0.19 


44 


44 


60 


6.10 


0.56 


B77 


3 


7.5 


2.52 


0.36 


. it 


44 


55 


6.00 


0.46 


44 


44 


6.5 


2.42 


0.26 


B7 


15 


55 


5.75 


0.06 


44 


44 


5,5 


2 33 


0.17 




44 


50 


5.65 


0.56 


B2 


20 


100 


^.28 


0.88 


44 


44 


45 


5.55 


0.46 


44 


44 


95 


*" 21 


0.81 


44 


u 


42 


5.50 


0.41 


44 


44 


90 


".U 


0.74 


B9 


12 


35 


5.09 


0.44 


44 


44 


85 


*"' 06 


0.66 




44 


31.5 


5.00 


0.35 


44 


44 


80 


".00 


0.60 


Bll 


10 


40 


5.10 


0.75 


B4 


15 


100 


6.77 


1.18 






35 


4.95 


0.60 


44 


44 


95 


6.68 


1.09 


44 


44 


30 


4.81 


0.46 


44 


44 


90 


6.58 


0.99 


44 


* 4 


20 


4.66 


0.31 


44 


44 


85 


6.48 


0.89 


B13 


9 


35 


4.77 


0.73 


44 


44 


80 


6.40 


0.81 


44 


44 


30 


4.61 


0.57 


B5 


15 


75 


6.29 


0.88 


44 


44 


25 


4.45 


0.41 




44 


70 


6.19 


0.78 


44 


44 


21 


4.33 


0.29 


44 


44 


65 


6.10 


0.69 


B15 


8 


25.5 


4.27 


0.54 


4k 


44 


60 


6.00 


0.59 


44 


44 


23 


4.18 


45 


B8 


12 


55 


5.61 


0.82 


44 


44 


20.5 


4.09 


0.36 


44 


44 


50 


5.49 


0.70 


44 


44 


18 


4.00 


27 


44 


44 


45 


5.37 


0.58 


B17 


7 


20 


3.87 


0.46 


* 4 


44 


40 


5.25 


0.46 


tt 


H 


17.5 


3.76 
3.66 


0.35 
0.25 


Sections B2, B4, B5, and B8 are 












" special ^ beams, the others are 












"standard." 



Sectional area = weight in Ibs. per ft. -*- 3.4, or X 0.2941. 
Weight in Ibs. per foot = sectional area X 3.4. 

Maximum and Minimum Weights and Dimensions of 
Carnegie Steel Deck Beams. 



Section 


Depth 
of 


Weight per 
Foot, Ibs. 


Flange Width. 


Web 
Thickness. 


Increase of 
Web and 
Flange per 


* 


, ' 














Ib. increase 






Min. 


Max. 


Min. 


Max. 


Min. 


Max. 


of Weight. 


B100 


10 


27.23 


35.70 


5.25 


5.50 


.38 


.63 


.029 


B101 


9 


26.00 


30.00 


4.91 


5.07 


.44 


57 


.033 


BIOS 


8 


20.15 


24.48 


5.00 


5.16 


31 


.47 


.037 


B103 


7 


18.11 


23.46 


4.87 


5.10 


31 


.54 


.042 


BIOS 


6 


15.30 


18.36 


4.38 


4.53 


.28 


.43 


.049 



178 



MATERIALS. 



Minimum, Maximum, and Intermediate Weights and 
Dimensions of Carnegie Standard Channels. 



1 I 


^"3 


! 




!s 


p 

t-H 


^ ' 


froS 


_c ^ 


! j 


a 

sd 


I s " 


s g l 

.SPo 

*& 


0>S'J 

il 

5 


Web Th 
ness. 
Inchei 


Section 
dex. 


o.;R a 

& 


4J O '-^ 

. ^^ 


A 


WebTh 
ness. 
Inche! 


01 


15 


55 


3.82 


0.82 


05 


8 


16.25 


2.44 


0.40 





44 


50 


3.72 


0.72 


44 


44 


13.75 


2.35 


0.31 


4 * 


44 


45 


3.62 


0.62 


44 


44 


11.25 


2.26 


0.22 


44 


44 


40 


3.52 


0.52 


06 


7 


19.75 


2^51 


0.63 


44 


44 


35 


3.43 


0.43 


44 


44 


17.25 


2.41 


0.53 


44 


44 


33 


3.40 


0.40 


44 


44 


14.75 


2.30 


0.42 


02 


12 


40 


3.42 


0.76 


44 


44 


12.25 


2.20 


0.32 


4 - 


44 


35 


3.30 


0.64 


44 


i4 


9.75 


2.09 


0.21 


44 


11 


30 


3.17 


0.51 


07 


6 


15.50 


2.28 


0.56 


4 ! 


44 


25 
20.5 


3.05 
2.94 


0.39 

0.28 


u 


,4 


13 
10.50 


2.16 
2.04 


0.44 
0.32 


03 


10 


35 


3.18 


0.82 





44 


8 


1.92 


0.20 


44 


44 


30 


3.04 


0.68 


08 


5 


11.50 


2.04 


0.48 


44 


44 


25 


2.89 


0.53 


44 


** 


9 


1.89 


0.33 


41 


44 


20 


2.74 


0.38 


* 4 


44 


6.50 


1.75 


0.19 


** 


* 


15 


2.60 


0.24 


09 


4 


7 . 25 


1.73 


0.33 


04 


9 


25 


2.82 


0.62 


44 


44 


6^25 


1.65 


0.25 




44 


20 


2.65 


0.45 


44 


44 


5.25 


1.58 


0.18 


44 


44 


15 


2.49 


0.29 


072 


3 


6 


1.60 


0.36 


! 


11 


13.25 


2.43 


0.23 


* 


44 


5 


1.50 


0.26 


05 


8 


21.25 


2.62 


0.58 


M 


44 


4 


1.41 


0.17 


' 


" 


18.75 


2.53 


0.49 













Weight* and Dimensions of Carnegie Steel Z-Bars. 







Size. 








Size. 






02 "3 








a 






Section 
Index. 


$ -2 

i! 

2 o 

H 


Flanges. 


43 

4> 


il 

vft 
P 


Section 
Index. 


Thicknes 
of Meti 


Flanges. 


1 


Weight. 
Pounds 


Zl 


% 


3 K 


6 


15.6 


Z6 


H 


3 5/16 


5 1/16 


26.0 


* 


7{1Q 


39/16 


6 1/16 


18.3 


44 


13/16 


3 % 


5 X 


28.3 


" 


& 


3 % 


6 Ys 


21.0 


Z7 


Y4 


3 1/16 


4 


8.2 


Z2 


9/16 


3 i^ 


6 


22.7 


44 


5/16 


3 y& 


4 1/16 


10.3 





N 


T 9/16 


6 1/16 


25.4 


44 


% 


3 3/16 


4 K 


12.4 


* 


11/16 


3 % 


6 Ys 


28.0 


Z8 


7/16 


3 1/16 


4 


13.8 


Z3 


13/16 


3 fc 

3 9/1 G 


6 
6 1/16 


29.3 
32.0 


;; 


$, 


3 K 

3 3/16 


4 1/16 
4 K 


15.8 
17.9 


* l 


% 


3 % 


6 Ys 


34.6 


Z9 


% 


3 1/16 


4 


18.9 


Z4 


5/16 


3 H 


5 


11.6 


44 


11/16 


3 X 


4 1/16 


20.9 


* 


% 


3 5/16 


5 1/16 


13.9 


14 


% 


3 3/16 


4 J4 


22.9 


M 


7/16 


3 % 


5 X 


16.4 


Z10 


l /4 


2 11/16 


3 


6.7 


Z5 


K 


3 M 


5 


17.8 


" 


5/16 


2 % 


3 1/16 


8.4 


" 


9/16 


3 5/16 


5 1/16 


20.2 


Zll 


% 


2 11/18 


3 


9.7 


" 


% 


3 % 


5 y s 


22.6 


4t 


7/16 


2 % 


3 1/16 


11.4 


Z6 


11/16 


3 J4 


5 


23.7 


Z12 


Vk 


2 11/16 


a 


12.5 












' 


9/16 


2 M 


3 1/16 


14.2 



SIZES AND WEIGHTS OF STRUCTURAL SHAPES. 179 



Pencoyd Steel Angles. 

EVEN LEGS. 



Size in 
Inches. 


Approximate Weight in Pounds per Foot for Various 
Thicknesses in Inches. 


































.125 


3/16 
.1875 


H 

.25 


5/16 
.3125 


3 4 


7/16 
.4375 


& 


9/16 
.5625 


% 
.625 


11/16 
.6875 


3 A 
.75 


13/16 
.8125 


.75 


15/16 
.9375 


1 
1.00 


8x8 














26.4 


29.8 


33.2 


36.6 


39.0 


42.4 


45.8 


49.3 


52.8 


6x6 










14.8 


17.3 


19.7 


22.0 


24.4 


26.5 


28.8 


31.0 


33.4 


35.9 




5 x5 










12.3 


14.3 


16.3 


18.2 


20.1 


22.0 


23.8 


25.6 


27.4 


29.4 




4 x4 








8.2 


9.8 


11.3 


12.8 


14.5 


15.8 


17.2 


18.6 










31*2 x 3Jrfjj 








7.1 


8.5 


9.8 


11.1 


12.4 


13.7 














3 x3 






4.0 


6.1 


7.2 


8.3 


9.4 


10.4 


11.5 














2% x 2f> 






4.5 


5.5 


6.6 


7 7 


8.6 


















gi^j x 2^ 




3.1 


4.1 


5.0 


5.9 


6.9 


7.8 


















2^x2^4 




2.7 


3.6 


4.5 


5.4 






















2 x2 




2.5 


3.2 


4.0 


4.8 






















1% x 1% 




2.1 


2.8 


3.5 


4.1 






















1V6 x ll/ 


1.2 


1.8 


2.4 


2.9 


3.5 






















1)4 x 1/4 


1.0 


1.5 


2.0 


























1 x 1 


0.8 


1.2 


1.5 



























UNEVEN LEGS. 



Size in 


Approximate Weight in Pounds per Foot for Various 
Thicknesses in Inches. 


Inches. 


































H 




i/^ 


5/16 


% 


7/16 


i^ 


9/16 


% 


11/16 


$4 


13/16 


% 


15/16 


1 




.186 


.1875 


.25 


.3125 


.375 


.4375 


.50 


.5625 


.625 


.6875 


.75 


.8125 


.875 


.9375 


1.00 


8 x6 














23.0 


25.8 


28.7 


31.7 


33.8 


36.6 


39.5 


42.5 


45.6 
















17.0 


19.0 


21.0 


23.0 


24.8 


26.7 


28.6 


30.5 


32.5 


6J^x4 










12.9 


15.0 


17.0 


19.0 


21.2 


23.4 


25.6 


27.8 


29.8 


31.9 




6 x4 










12.2 


14.3 


16.3 


18.1 


20.1 


22.0 


23.8 


25.6 


27.4 


29.4 




6 x3J4 










11.6 


13.6 


15.5 


17.1 


19.0 


20.8 


22.6 


24.5 


26.5 


28.6 




5L x 314 










11.0 


12.8 


14.6 


16.2 


17.9 














5 x4 










11.0 


12.8 


14.6 


16.2 


17.9 


19.6 


21.3 










5 x3^ 








8.7 


10.3 


12.0 


13.6 


15.2 


16.8 


18.4 


20.0 










5x3" 








8.2 


9.7 


11.2 


12.8 


14.2 


15.7 


17.2 


18.7 










4^x3 








7.7 


9.1 


10.5 


11.9 


13.3 


14.7 


16.0 


17.4 










4 x3^ 








7.7 


9.1 


10.5 


11.9 


13.3 


14.7 


16.0 


17.4 










4 x3 " 








7.1 


8.5 


9.8 


11.1 


12.4 


13.8 














3^x3 








6.6 


7.8 


9.1 


10.3 


11.6 


12.9 




















4 9 


6.1 


7.2 


8.3 


9.4 


















gi/ x 2 






I* 


5.5 


6.6 






















Q 21^C 






4.' J 


5.5 


6.6 


7.7 


8.7 


















3 x2 






4.1 


5.0 


5.9 


6.9 


7.9 


















2^x2 




2.7 


3. 


4.5 


5.4 


6.2 


7.0 






















2.3 




3.7 


4.4 






















2 xl}4 




2.1 


> 9 


3.6 


4.3 






















2 xl}4 




1.9 


2.'6 


3.3 


3.9 























ANGLE-COVERS. 



Siz in 
Inches. 


3/16 


y* 


5/16 


% 


7/16 


X 


9/16 


% 


3 x3 




4.8 


5.9 


7.1 


8.2 


9.3 


10.4 


11.5 


2^x2% 

2^x2^ 
2*4 x 2J4 


3.0 
2.6 


4.4 
4.0 
3.5 


5.5 
5.0 

4.4 


6.6 
6.0 
5.3 


7.7 
7.0 


8.8 
8.1 






2 x2 


2.4 


3.2 


4.0 


4.8 











180 



MATERIALS. 



SQUARE-ROOT ANGLES. 



Size in 
Inches. 


Approximate Weight in Pounds 
per Foot for Various Thicknesses 
in Inches. 


Size in 
Inches. 


Approximate Weight in 
Pounds per Foot for 
Various Thicknesses 
in Inches. 


I 


5/16 
.3125 


.375 


7/16 
.4375 


^ 
.50 


9/16 
.5625 


% 
.625 


Ys 
.125 


3/16 

.1875 


1 


5/16 
.3125 


% 
.375 


4 x4 

3^x3^ 
3 x3 
2%x2% 
2^x2^ 
8*4 x2fc 


4.9 

4.5 
4.1 
3.6 


7.1 
6.1 
5.6 
5.1 
4.5 


9.8 

8.5 
7.2 
6.7 
6.1 
5.4 


11.4 
9.9 
8.3 

7.8 
7.1 


13.0 
11.4 
9.4 
8.9 
8.2 


14.6 


16.2 


2 x2 

l%xl% 
l^xl^ 

iMxi^ 

1 xl 


0.82 


1.80 
1.53 
1.16 


3.3 
2.9 
2.4 
2.04 
1.53 


4.1 
3.6 
3.0 
2.55 


4.9 
4.4 



Pencoyd Tees. 



Section 
Number. 


Size 
in Inches. 


Weight 
per Foot. 


Section 
Number. 


Size 
in Indies. 


Weight 
per Foot. 


EVEN TEES. 


UNEVEN TEES. 


440T 
441T 


4 x4 
4 x4 


10.9 
13.7 


43T 


4 x3 


9.0 


335T 


3^3 x 3Jx> 


7.0 


44T 


4 x3 


10.2 


336T 


3J^x3>J 


9.0 


45T 


4 x4^ 


13.5 


337T 


3>Jx3^ 


11.0 


38T 


31^x3 


7.0 


330T 


3 x3 


6.5 


39T 


3^x3 


8.5 


33 IT 


3 x3 


7.7 


SOT 


3 xlU 


4.0 


225T 


2^x2^ 


5.0 


31T 


3 x2J4 


5.0 


226T 




5.8 


32T 


3 x2^ 


6.0 


227T 


2L x ;;>ij? 


6.6 


33T 


3 x2^J 


7.0 


222T 


2^4 x 214 


4.0 


34T 


3 x2U 


8.0 


223T 


2J4 x 2*4 


4.0 


35T 


3 x3i| 


8.3 


220T 


2 x2 


3.5 


36T 


3 X3U 


9.5 


117T 


l%xl% 


2.4 


28T 


2%x 1% 


6.6 


115T 


l^xl^ 


2.0 


29T 


OHX x 2 


7.2 


112T 


1*4*1*4 


1.5 


25T 


2^x114 


3.3 


HOT 


1 xl 


1.0 ' 


26T 


2^x2% 


5.7 








27T 


2^x3 


6.0 




24T 


2J4x 9/16 


2.2 


UNEVEN TEES. 


20T 
2 9 T 


2 x 9/16 
2 xl 1/16 


2.0 
2.0 








21T 


2 xl 


2.5 


64T 
65T 


6x4 
6x5J4 


17.4 
39.0 


23T 
17T 


2 xl^ 

l%xl 1/16 


3.0 
1.9 


53T 


5x3^ 


17.0 


1ST 


l^xl^j 


3.5 


54T 


5x4 


15.3 


15T 


\Y 2 x 15/16 


1.4 


42T 


4x2 


6.5 


12T 


1^4 x 15/16 


1.2 



Pencoyd Miscellaneous Shapes. 



Section 
Number. 


Section. 


Size in Inches. 


Weight per Foot 
in Pounds. 


217M 
210M 
260M 


Heavy rails. 
Floor-bars. 


3 1/16x4x3 


6 

i/i6x*4 to y 2 


50.0 
7.1 to 14.3 
9.8 to 14.7 



SIZES AND WEIGHTS OF KOOFIHG MATERIALS. 181 



SIZES AND WEIGHTS OF HOOFING MATERIALS. 

Corrugated Iron. (The Cincinnati Corrugating Co.) 

SCHEDULE OF WEIGHTS. 



. aJ 

02 to 

"1 


Thickness in 
decimal parts 
of an inch. 
Flat. 


Weight per 
100 sq. ft. 
Flat, Pain ted. 


Weight per 
100 sq. ft. 
Corrugated 
and Painted. 


Weight per 
100 sq. ft. 
Corrugated 
and 
Galvanized; 


Weight in oz. 
per sq. ft. 
Flat, Galvan- 
ized. 


No. 28 
No. 26 
No. 24 
No. 22 
No. 20 
No. 18 
No. 16 


.015625 
.01875 
.025 
.03125 
.0375 
.05 
.0625 


62^ Ib 
75 
100 
125 
150 
200 
250 


s. 


70 Ib 
84 
111 
138 
165 
220 
275 


s. 


86 Ib 
99 
127 

154 
182 
286 
291 


s. 


12^0 
14J4 

isU 

22^3 
26^ 

342 

42^ 


z. 



The above table is on the basis of sheets rolled according to the U. S. 
Standard Sheet-metal Gauge of 1893 (see page 31). It is also on the basis of 
' " K ' in. corrugations. 



To estimate the weight per 100 sq. ft. on the roof when lapped one corru- 
gation at sides and 4 in. at ends, add approximately 12^$ to the weights per 
100 sq. ft., respectively, given above. 

Corrugations 2^ in. wide by ^ or % in. deep are recognized generally as 
the standard size for both roofing and siding; sheets are manufactured 
usually in lengths 6, 7, 8, 9, and 10 ft., and have a width of 26^ or 26 in. out- 
side width ten corrugations, and will cover 2 ft. when lapped one corruga- 
tion at sides. 

Ordinary corrugated sheets should have a lap of 1^6 or 2 corrugations side- 
lap for roofing in order to secure water-tight side seams; if the roof is 
rather steep 1^ corrugations will answer. 

Some manufacturers make a special high-edge corrugation on sides of 
sheets (The Cincinnati Corrugating Co.), and thereby are enabled to secure 
a water-proof side-lap with one corrugation only, thus saving from 6$ to 12% 
of material to cover a given area. 

The usual width of flat sheets used for making the above corrugated 
material is 28J4 inches. 

No. 28 gauge corrugated iron is generally used for applying to wooden 
buildings; but for applying to iron framework No. 24 gauge or heavier 
should be adopted. 

Few manufacturers are prepared to corrugate heavier than No. 20 gauge, 
but some have facilities for corrugating as heavy as No. 12 gauge. 

Ten feet is the limit in length of corrugated sheets. 

Galvanizing sheet iron adds about 2% oz. to its weight per square foot. 

Corrugated Arches. 

For corrugated curved sheets for floor and ceiling construction in fire- 
proof buildings, No. 16, 18, or 20 gauge iron is commonly used, and sheets 
may be curved from 4 to 10 in. rise the higher the rise the stronger the 
arch. 

By a series of tests it has been demonstrated that corrugated arches give 
the most satisfactory results with a base length not exceeding 6 ft., and ff 
ft. or even less is preferable where great strength is required. 

These corrugated arches are usually made with 2^ X % i". corrugations, 
and in same width of sheet as above mentioned. 

Terra-Cotta. 

Porous terra-cotta roofing 3" thick weighs 16 Ibs, per square foot and 2" 
thick, 12 Ibs. per square foot. 
Ceiling made of the same material 2" thick weighs 11 Ibs. per square foot. 

Tiles. 

Flat tiles 6M" X 10J4" X % ff weigh from 1480 to 1850 Ibs. per square of 
roof (100 square feet), the lap being one-half the length of the tile. 

Tiles with grooves and fillets weigh from 740 to 925 Ibs. per square of roof. 
Pantiles 1%" X 10^ /x laid 10" to the weather weigh 850 Ibs. per square. 



182 



MATERIALS. 



Tin Plate Tinned Sheet Steel. 

The usual sizes for roofing tin are 14" X 20" and 20" X 28". Without 
allowance for lap or waste, tin roofing weighs from 50 to 62 Ibs. per square. 
Tin on the roof weighs from 62 to 75 Ibs. per square. 

Roofing plates or terne plates (steeJ plates coated with an alloy of tin 
and lead) are made only in 1C and IX thicknesses (29 and 27 Birmingham 
gauge). "Coke" and "charcoal' 1 tin plates, old names used when iron 
made with coke and charcoal was used for the tinned plate, are still used in 
the trade, although steel plates have been substituted for iron; a coke plate 
now commonly meaning one made of Bessemer steel, and a charcoal plate 
one of open-hearth steel. The thickness of the tin coating on the plates 
varies with different " brands. 1 ' 

For valuable information on Tin Roofing, see circulars of Merchant & Co., 
Philadelphia. 

The thickness and weight of tin plates were formerly designated in the 
trade, both in the United States and England, by letters, such as I.C., D.C., 
I.X., D.X., etc. A new system was introduced in the United States in 1898, 
known as the " American base-box system." The base-box is a package 
containing 32,000 square inches of plate. The actual boxes used in the trade 
contain 60, 120, or 240 sheets, according to the size. The number of square 
inches in any given box divided by 32,000 is known as the " box ratio." This 
ratio multiplied by the weight or price of the base-box gives the weight or 
price of the given box. Thus the ratio of a box of 120 sheets 14 X 20 in. is 
33,600 -*- 32,000 = 1.05, and the price at $3.00 base is $3.00 X 1.05 = $3.15. The 
following tables are furnished by the American Tin Plate Co., Chicago, 111. 
Comparison of Gauges and Weights of Tin Plates. 
(Based on U. S. standard Sheet-metal Gauge.) 

ENGLISH BASE-BOX. 

(31,360 sq. in.) 
Gauge. Weight. 

No. 38. 00 54. 44 Ibs. 

37.00 57.84 

36.00 61.24 

35.00 68.05 

34.00... 74.85 

33.24 ... 80.00 

32.50 85/00 

31.77 90.00 

31.04 95.00 

30.65 100.00 

30.06 108.00 

28.74 126.00 

28.00 136.00 

26.46 157.00 

25.46 178.00 

24.68 199.00 

23.91 220.00 

23.14 241.00 

22.37 262.00 

21.60 283.00 

27.86 139.00 

25.38 180.00 

24.24 211.00 

23.12 212.00 

22.00 273.00 



Weig 
55 Ib 
60 ' 

65 * 
70 ' 
75 ' 
80 4 
85 ' 
90 * 
95 ' 
100 ' 
110 ' 
130 ' 
140 ' 
160 ' 
180 ' 
200 ' 
220 ' 
240 l 
260 l 
280 ' 
140 ' 
180 ' 
220 ' 
240 ' 
280 ' 


AMERICAN BASE-BO 
(32,000 sq. in.) 
ht. ( 
s M 


X. 

3auge. 
o. 38.00 
k 36.72 
' 35.64 
1 34.92 
' 34.20 
' 33.48 
' 32 76 
















* 32.04 
' 31.32 






' 30.80 
' 30.08 
' 28.64 
* 27.9^ 
1 26.48 
' 25.5-) 
' 24.85 
1 24.0 8 
' 23.3 6 
' 22.64 
' 21.9a 
' 27.9-2 
* 25.52 
' 24.08 
1 23.36 
1 21.92 

































I.C.L. 

I.C. 

IX.L. 

IX. 

I.2X. 

I.3X. 

I.4X. 

I. 5X. 

I.6X. 

I.7X. 

I. 8X. 

D.C. 

D.X. 

D. 2X. 

D. 3X. 

D. 4X. 



American Packages Tin Plate. 



Inches 
Wide. 


Length. 


Sheets 
per Box 


Inches 
Wide. 


Length. 


Sheets 
per Box 


9 to 16% 
17 * 25% 
26 ' 30 
9 ' 10% 
11 1 11% 

12 ' 12% 


Square. 
Square. 
Square. 
All lengths. 
To 18 in. long, incl. 
18J4 and longer. 
To 17 in. long, incl. 


240 
120 
60 
240 
240 
120 
240 


13 " 13% 
13 to 13% 
14 " 14% 
14 " 14% 
15 " 25% 
26 " 30 


17*4 and longer. 
To 16 in. long, incl. 
16*4 and longer. 
To 15 in. long, incl. 
15J4 and longer. 
All lengths. 
All lengths. 


120 
240 
120 
240 
120 
120 
60 


Small sizes of light base weights will be packed in double Tboxes. 



SIZES AND WEIGHTS OF ROOFING MATERIALS* 183 



Slate. 

Number and superficial area of slate required for one square of roof. 
(1 square = 100 square feet.) 



Dimensions 


Number 


Superficial 


Dimensions 


Number 


Superficial 


in 


per 


Area in 


in 


per 


Area in 


Inches. 


Square. 


Sq. Ft. 


Inches. 


Square. 


Sq. Ft. 


6x12 


533 


267 


12x18 


160 


240 


7x12 


457 




10x20 


169 


235 


8x12 


400 




11 x20 


!54 




9x12 


355 




12x20 


141 




7x14 


374 


254 


14x20 


121 




8x14 


327 




16x20 


137 




9x14 


291 




12x22 


126 


231 


10x14 


261 




14x22 


108 




8x16 


277 


246 


12x24 


114 


228 


9x16 


246 




14x24 


98 




10x16 


221 




16 x 24 


86 




9x18 


213 


240 


14 x 26 


89 


225 


10x18 


192 




16x26 


78 

















As slate is usually laid, the number of square feet of roof covered by one 
slate can be obtained from the following formula : 



width x (length 3 inches) 


e number of square feet of roof covered. 
s and thicknesses required for one square 


Weight of slate of various length 
of roof : 


Length 
in 
Inches. 


Weight in Pounds per Square for the Thickness. 


w 












M" 


V 




M 


% 


M 


% 


12 
14 
16 
18 
20 
22 
24 
26 


483 
460 
445 
434 
425 
418 
412 
407 


724 

688 
667 
650 
637 
626 
617 
610 


967 

920 
890 
869 
851 
836 
825 
815 


1450 
1379 
1336 
1303 
1276 
1254 
1238 
1222 


1936 
1842 
1784 
1740 
1704 
1675 
1653 
1631 


2419 
2301 
2229 
2174 
2129 
2093 
2066 
2039 


2902 
2760 
2670 
2607 
2553 
2508 
2478 
2445 


3872 
3683 
3567 
3480 
3408 
3350 
3306 
3263 



The weights given above are based on the number of slate required for one 
square of roof, taking the weight of a cubic foot of slate at 175 pounds. 

Pine Shingles. 

Number and weight of pine shingles required to cover one square of 
roof : 



Number of 


Number of 


Weight in 




Inches 
Exposed to 
Weather. 


Shingles 
per Square 
of Roof. 


Pounds of 
Shingle on 
One-square 


Remarks. 






of Roofs. 




4 


900 


216 


The number of shingles per square is 


4}x> 


800 


192 


for common gable-roofs. For hip 


ly^ 


720 
655 


-173 
157 


roofs add five per cent, to these figures. 
The weights per square are based on 


6 


600 


144 


the number per square. 



184 



MATERIALS. 



Skylight Glass. 

The weights of various sizes and thicknesses of fluted or rough plate-glass 
required for one square of roof. 



Dimensions in 
Inches. 



Thickness in 
Inches. 



Area 
in Square Feet. 



Weight in Lbs. per 
Square of Roof. 



12x48 
15x60 
20x100 
94x156 



3.997 

6.246 

13.880 

101.768 



250 

350 
500 
700 



In the above table no allowance is made for lap. 

If ordinary window-glass is used, single thick glass (about 1-16") will weigli 
about 82 Ibs. per square, and double thick glass (about %") will weigh about 
164 Ibs. per square, no allowance being made for lap. A box of ordinary 
window-glass contains as nearly 50 square feet as the size of the panes will 
admit of. Panes of any size are made to order by the manufacturers, but a 
great variety of sizes are usually kept in stock, ranging from 6x8 inches to 
36 x 60 inches. 

APPROXIMATE WEIGHTS OF VARIOUS ROOF- 
COVERINGS. 

For preliminary estimates the weights of various roof coverings maybe 
taken as tabulated below (a square of roof = 10 ft. square = 100 sq. ft.); 



Name. 



Weight in Lbs. per 
Square of Roof. 



Cast-iron plates (%" thick) 1500 

Copper 80-125 

Felt and asphalt 100 

Felt and gravel 800-1000 

Iron, corrugated 100-375 

Iron, galvanized, flat 100- 350 

Lath and plaster 900-1000 

Sheathing, pine, 1" thick yellow, northern .. 300 

" southeru.. 400 

Spruce, 1" thick 200 

Sheathing, chestnut or maple, V thick 400 

" ash, hickory, or oak, 1" thick.... 500 

Sheet iron (1-16" thick) 300 

" and laths 500 

Shingles, pine 200 

Slates W thick) 900 

, Skylights (glass 3--16" to J" thick) . . .. 250- 700 

Sheet lead 500- 800 

Thatch ; 650 

Tin 70-125 

Tiles, flat 1500-2000 

(grooves and fillets) 700-1000 

pan 1000 

" with mortar 2000-3000 

Zinc ..... 100-200 

Approximate Loads per Square Foot for Roofs of Span.* 
under 75 Feet, Including Weight of Truss. 

(Carnegie Steel Co.) 

Roof covered with corrugated sheets, unboarded 8 Ibs. 

Roof covered with corrugated sheets, on boards. 11 

Roof covered with slate, on laths 13 

Same, on boards, 1*4 in. thick 16 

Roof covered with shingles, on laths : 10 

Add to above if plastered below rafters 10 

Snow, light, weighs per cubic foot . ... 5 to 12 

For spans over 75 feet add 4 Ibs. to the above loads per square foot. 

It is customary to add 30 Ibs. per square foot to the above for gnow and 
when separate calculations are not made, 



WEIGHT OF CAST-IRON PIPES OR COLUMKS. 185 



WEIGHT OF CAST-IRON PIPES OR COLUMNS. 

In L.bs. per Lineal Foot. 

Cast iron = 450 Ibs. per cubic foot. 



Bo*\i. 


Thick, 
of 
Metal. 


Weight 
per Foot. 


Bore. 


Thick, 
of 
Metal. 


Weight 
per Foot. 


Bore. 


Thick, 
of 
Metal. 


Weight 
per Foot. 


Ins. 


Ins. 


Lbs. 


Ins. 


Ins. 


Lbs. 


Ins. 


Ins. 


Lbs. 


3 


% 


12.4 


10 


% 


79.2 


22 


94 


167.5 




/^| 


17.2 


10}r<J 


i^ 


54.0 




% 


196.5 




% 


22.2 ' 




% 


68 2 


23 


94 


174.9 


3^2 


% 


14.3 




M 


82.8- 




% 


205.1 




v& 


19.6 


11 


x^ 


56.5 




l 


235.6 




% 


25.3 




% 


71.3 


24 


94 


182.2 


4 


% 


16.1 




% 


86.5 






213.7 




^ 


22.1 


\\}/f> 


L<2 


58.9 




1 8 


245.4 




% 


28.4 




7& 


74.4 


25 


94 


189.6 


4^3 


% 


17.9 




g 


90.2 




% 


222.3 




1< 


24.5 


12 




61.3 




l 


255.3 




% 


31.5 




E^ 


77.5 


26 


94 


197.0 


5 


% 


19.8 




^4 


93.9 




% 


230.9 




L 


27.0 


JO1Z 


/^ 


63.8 




l 


265.1 




2 


34.4 




% 


80.5 


27 




204.3 


5}^> 


^ 


21.6 




% 


97.6 




v4 


239.4 




l2 


29.4 


13 


/^ 


66 3 




1 


274.9 




76 


37.6 




% 


83.6 


28 


94 


211.7 


6 


% 


23.5 




94 


101.2 




% 


248.1 




1 " 


31.8 


14 


/^ 


71.2 




l 


284.7 




6^ 


40.7 




% 


89.7 


29 


94 


219.1 


gi^ 


7& 


25.3 







108.6 






256.6 




LJJJ 


34.4 


15 


% 


95.9 




i 8 


294.5 




Y8 


43.7 




4 


116.0 


30 




265.2 


7 


% 


27.1 




% 


136.4 




i 8 


304.3 




1^3 


36.8 


16 


% 


102.0 






343 7 




% 


46.8 




4 


123.3 


31 


% 


273.8 


71^ 


% 


29.0 




% 


145.0 




i 


314.2 




LJ 


89.3 


17 


% 


108.2 




\\/. 


354.8 




% 


49.9 




94 


130.7 


32 


% 


282.4 


8 


% 


30.8 




% 


153.6 




l 


324.0 




Hi 


41.7 


18 


% 


114.3 




i/"6 


365.8 




% 


52.9 




94 


138.1 


33 


% 


291.0 


8J^ 


^ 


44.2 




% 


162.1 




l 


333.8 




% 


56.0 


19 


% 


120.4 




i/^ 


376.9 




94 


68.1 




94 


145.4 


34 


% 


299.6 


9 


M 


46.6 




% 


170.7 




i 


343.7 




% 


59.1 


20 


% 


126.6 




ji^j 


388.0 




94 


71.8 




94 


152.8 


35 


% 


308.1 


9^ 


^ 


49.1 




% 


179.3 




l 


353.4 




% 


62.1 


21 


% 


132.7 




\\ 


399.0 




M 


75.5 




94 


160.1 


36 


% 


316 6 


10 


/^ 


51.5 




% 


187.9 




1 


363.1 




% 


65.2 


22 


% 


138.8 




^ 


410.0 



The weight of the two flanges may be reckoned = weight of one foot- 



186 



MATERIALS. 



WEIGHTS OF CAST-IRON PIPE TO LAY 12 FEET 
LENGTH. 

Weights are Gross Weights, including Hull. 

(Calculated by F. H. Lewis.) 



Thickness. 


Inside Diameter. 


Inches. 


Equiv. 
Decimals. 


4 // 


6" 


8" 


10" 


12" 


14" 


16" 


18" 


20" 

1640 
1810 
1980 
2152 
2324 
2498 
2672 
3024 

saso 

3739 


,&, 

7-16 
15-32 

17-32 
9-16 
19-32 

11--16 

& 

% 
15-16 
1 
1J| 


.375 
.40625 
.4375 
.4687 
.5 
.53125 
.5625 
.59375 
.625 
.6875 
.75 
.8125 
.875 
.9375 
1. 
1.125 
1.25 
1.375 


209 
228 
247 
266 
286 
306 
327 


304 
331 
358 
386 
414 
442 
470 
498 


400 
435 
470 
505 
541 
577 
613 
649 
686 


581 
624 
668 
712 
756 
801 
845 
935 
1026 


692 
744 
795 
846 
899 
951 
1003 
1110 
1216 
1324 
1432 


804 
863 
922 
983 
1043 
1103 
1163 
1285 
1408 
1531 
1656 
1783 
1909 


1050 
1118 
1186 
1254 
1322 
1460 
1598 
1738 
1879 
2021 
2163 


1177 
1253 
1329 
1405 
1481 
1635 
1789 
1945 
2101 
2259 
2418 
2738 
3062 
3389 
















































































Thickness. 


Inside Diameter. 


Inches. 


Equiv. 
Decimals. 


22" 


24" 


27" 


30" 


33" 


36" 


42" 


48" 


60" 

9742 
10740 
11738 
12744 
13750 
14763 
15776 
17821 
19880 
21956 


11-16 

H 
13-16 

J M 

I 

1^4 

f 


.625 
.6875 

'.8125 

.875 
.9375 

'125 
.25 
.375 
.5 
.625 
.75 
.875 
2. 
2.25 
2.5 
2.75 


1799 
1985 
2171 
2359 
2547 
2737 
2927 
3310 
3698 


2160 
2362 
2565 
2769 
2975 
3180 
3598 
4016 
4439 


2422 

2648 
2875 
3103 
3332 
3562 
4027 
4492 
4964 
5439 


2934 
3186 
3437 
3690 
3942 
4456 
4970 
5491 
6012 
6539 


3221 
3496 
3771 
4048 
4325 
4886 
5447 
6015 
6584 
7159 
7737 


3507 
3806 
4105 
4406 
4708 
5316 
5924 
6540 
7158 
7782 
8405 


4426 
4773 
5122 
5472 
6176 
6880 
7591 
8303 
9022 
9742 
10468 
11197 


5442 
5839 
6236 
7034 
7833 
8640 
9447 
10260 
11076 
11898 
12725 
14385 











































































CAST-IRON PIPE FITTINGS. 



187 



CAST-IRON PIPE FITTINGS. 

Approximate Weight. 

(Addyston Pipe and Steel Co., Cincinnati, Ohio.) 



Size in 
Inches. 


Weight 
in Lbs. 


Size in 
Inches. 


Weight 
in Lbs 


Size in 
Inches. 


Weight 
in Lbs. 


Size in 
Inches. 


Weight 
in Lbs. 


CROSSES. 


TEES. 


SLEEVES. 


REDUCERS. 


2 
3 

3x2 
4 
4x3 

4x2 
6 
6x4 
6x3 

8 
8x6 
8x4 
8x3 
10 
10x8 
10x6 
10x4 
10x3 
12 
12x10 
12x8 
12x6 
- 12 x 4 
12x3 
14 x 10 
14x8 
14x6 
16 
16xl 4 
16xl 2 
16xl Q 
16x8 
16x6 
16x4 
18 
20 
20x12 
20x10 
20x8 
20x6 
20x4 
24 
24x20 
24x6 
30x20 
30x12 
30x8 


40 
110 
90 
120 
114 
90 
200 
160 
160 
325 
280 
265 
225 
575 
415 
450 
390 
350 
740 
650 
620 
540 
525 
495 
750 
635 
570 
1100 
1070 
1000 
1010 
825 
700 
650 
1560 
1790 
1370 
1225 
1000 
1000 
1000 
2400 
2020 
1340 
2C35 
2250 
1995 


8x4 
8x3 
10 
10x8 
10x6 
10x4 
10x3 
12 
12 x 10 
12x8 
12x6 
12x4 
14x12 
14x10 
14x8 
14x6 
14x4 
14x3 
16 
16x14 
16x12 
16x10 
16x8 
16x6 
16x4 
18 
20 
20 x 16 
20x12 
20x10 
20x8 
20x6 
20x4 
20x10 
24 
24x12 
24x8 
24x6 
30 
30x24 
30 x 20 
30x12 
30x10 
30x6 
36 
36x30 
36x12 


250 
220 
390 
330 
370 
350 
310 
600 
555 
515 
550 
525 
650 
650 
575 
545 
525 
490 
790 
850 
850 
850 
755 
680 
655 
1235 
1475 
1115 
1025 
1090 
900 
875 
845 
1465 
2000 
1425 
1375 
1450 
3025 
2640 
2200 
2035 
2050 
1825 
5140 
4200 
4050 


2 
3 
4 
6 

8 
10 
12 
14 
16 
18 
20 
24 
30 
36 


10 
25 
45 

65 
80 
140 
190 
208 
350 
375 
500 
710 
965 
1200 


8x3 
10x8 
10x6 
10x4 * 
12x10 
12x8 
12x6 
12x4 
14 x 12 
14x 10 
14x8 
14x6 
16 x 12 
16x10 
20x16 
20x14 
20x12 
20x8 
24x20 
30x24 
30x18 
36x30 


116 
212 
170 
160 
320 
250 
250 
250 
475 
440 
390 
285 
475 
435 
690 
575 
540 
400 
990 
1305 
1385 
1730 


90 ELBOWS. 


3 
4 
6 
8 
10 
12 
14 
16 
18 
20 
24 
30 


14 
34 
55 
120 
150 
260 
370 
450 
660 
850 
900 
1400 
3000 


ANGLE REDUC- 
ERS FOR GAS. 


6x4 
6x3 


95 
70 


S PIPES. 


^ or 45 BENDS. 


I 


105 
190 


3 
4 
6 
8 
10 
12 
16 
18 
20 
24 
30 


30 
70 
95 
150 

200 
290 
510 
580 
780 
1425 
2000 


PLUGS. 


2 
3 
4 
6 
8 
10 
12 
14 
16 
18 
20 
24 
30 


3 

10 
10 
15 
30 
46 
66 
90 
100 
130 
150 
185 
370 


1/16 or 2214 
BENDS. 


6 

8 
10 
12 
16 
24 
30 


150 
155 
205 
260 
450 
1280 
2000 


CAPS. 


TEES. 


45 BRANCH 
PIPES. 


3 
4 
6 

8 
10 
12 


20 
25 
60 
75 
100 
120 


2 
3 

3x2 
4 
4x3 
4x2 
6 
6x4 
6x3 
6x2 
8 
8x6 


2H 
80 
76 
100 
90 
87 
150 
145 
145 
75 
300 
270 


3 
4 
6 
6x6x4 
8 
8x6 
24 
24 x 24 x 20 
30 
36 


90 
125 
205 
145 
330 
330 
2765 
2145 
4170 
10300 


REDUCERS. 


3x2 
4x3 
4x2 

6x4 
6x3 
8x6 

8x4 


25 
42 
40 
95 
70 
126 
116 


DRIP BOXES. 


4 
6 

8 
10 
20 


295 
330 
375 
875 
1420 



188 



MATERIALS. 



WEIGHTS OF CAST-IRON WATER- AND GAS-PIPE C 

(Addyston Pipe and Steel Co., Cincinnati, Ohio.) 



at 


Standard Water-pipe. 


*i 


Standard Gas -pipe. 


v 

N U 

m$ 


Per Foot. 


Thick- 
ness. 


Pei- 
Length. 


li 

MM 


Per Foot. 


Thick- 
ness. 


Pei- 
Length. 


2 
3 


15 


5/16 


63 

180 


2 
3 


6 
12K 


5/16 


48 
150 


3 


17 


^ 


204 










4 


22 


L 


264 


4 


17 


% 


204 


6 


33 


^ 


396 


6 


30 


7/16 


360 


8 


42 


/^ 


504 


8 


40 


7'/l 6 


480 


8 


45 


^ 


540 










10 


60 


9/16 


720 


10 


50 


7/16 


600 


12 


75 


9/16 


900 


12 


70 


y 


840 


14 


117 


H 


1400 


14 


84 


9/16 


1000 


16 


125 


n 


1500 


16 


100 


9/16 


1200 


18 


167 


% 


2000 


18 


134 


11/16 


1600 


20 


200 


15/16 


2400 


20 


150 


11/16 


1800 


24 


250 


1 


3000 


24 


184 


% 


2200 


30 


350 


ji^ 


4-,>00 


30 


250 


% 


3000 


36 


475 


1% 


5700 


36 


350 


/& 


4200 


42 


600 


1% 


7200 


42 


417 


15/16 


5000 


48 


775 


1L 


9300 


48 


542 


\\ 


6500 


60 


1330 


2 


15960 


60 


900 


1% 


10800 


72 


1835 


2M 


22020 


72 


1250 


m 


15000 



THICKNESS OF CAST-IRON WATER-PIPES. 

P. H. Baermann, in a paper read before the Engineers' Club of Phila- 
delphia in 1882, gave twenty different formulas for determining the thick- 
ness of cast-iron pipes under pressure. The formulas are of three classes: 

1. Depending upon the diameter only. 

2. Those depending upon the diameter and head, and which add a con- 
stant. 

3. Those depending upon the diameter and head, contain an additive >i 
subtractive term depending upon the diameter, and add a constant. 

The more modern formulas are of the third class, and are as follows: 

t= .OOOOS/id -f .01 d + .36 Shedd, No. 1. 

t = .00006/id -f .0133d -f .296 Warren Foundry, No. 2. 

t = .000058/id -f .0152d -f- .312 Francis, No. 3. 

t= .000048/i<2 + .013^4- .32 ...Dupuit, No. 4. 

t- .00004/td 4- .1 |/d-f.l5 Box, No. 5. 

t = .000135/id 4- .4 .OOlld Whitman, No. 6. 

t = .00006(/i 4- 230)d -f .333 - .0033d Fanning, No. 7. 

t = .00015/id 4- .25 - '.0052d Meggs, No. 8. 

In which t = thickness in inches, h = head in feet, d = diameter in inches. 

Rankine, "Civil Engineering," p. 721, says: "Cast-iron pipes should be 
made of a soft and tough quality of iron. Great attention should be paid 
to moulding them correctly, so that the thickness may be exactly uniform all 
round. Each pipe should be tested for Jr-bubMes and flav s by ringing it 
with a hammer, and for strength by exposing "t to *ou ie tlL intended 
greatest working pressure. " The rule for competing the Jiickness of a pipe 

to resist a given working pressure is t = -4-, where r is the radius in inches, 

p the pressure in pounds per square inch, and / the tenacity of the iron x>er 
square inch. When/ = 18000, and a factor of safety of 5 is used, the above 
expressed in terms of d and h becomes 

.5rf.4887t dh nnnnp 
*" "3600" = 16628 = 00006d/l 

"There are limitations, however, arising from difficulties in casting, and 
by the strain produced by shocks, which cause the thickness to be made 
greater than that given by the above formula." 



THICKHESS OF CAST-IROK PIPE. 



189 



Thickness of Metal and Weight per Length for Different 
Sizes of Cast-iron Pipes under Various Heads of Water. 

(Warren Foundry and Machine Co.) 





50 

Ft. Head. 


100 

Ft. Head. 


150 

Ft. Head. 


200 

Ft. Head. 


250 

Ft. Head. 


300 

Ft. Head. 


Size. 


Thickness 
of Metal. 


Weight 
i per Length. 


Thickness 
of Metal. 


Weight 
per Length. 


Thickness 
of Metal. 


Weight 
per Length. 


Thickness 
of Metal. 


Weight 
per Length. 


Thickness 
of Metal. 


Weight 
per Length. 


Thickness 
of Metal. 


Weight 
per Length. 


3 


.344 


144 


.353 


149 


.862 


153 


.371 


157 


.380 


161 


.390 


166 


4 


.361 


197 


.373 


204 


.385 


211 


.397 


218 


.409 


226 


.421 


235 


5 


.378 


254 


.393 


265 


.408 


275 


.423 


286 


.438 


298 


.453 


309 


6 


.393 


315 


.411 


330 


.429 


345 


.447 


361 


.465 


377 


.483 


393 


8 


.422 


445 


.450 


475 


.474 


502 


.498 


529 


.522 


557 


.546 


584 


10 


.459 


600 


.489 


641 


.519 


682 


.549 


723 


.579 


766 


.609 


808 


12 


.491 


768 


.527 


826 


.563 


885 


.599 


944 


.635 


1004 


.671 


1064 


14 


.524 


952 


.566 


1031 


.608 


1111 


.650 


1191 


.692 


1272 


.734 


1352 


16 


.557 


1152 


.604 


1253 


.652 


1360 


.700 


1463 


.748 


1568 


.796 


1673 


18 


.589 


1370 


.643 


1500 


.697 


1630 


.751 


1761 


.805 


1894 


.859 


2026 


20 


.622 


1603 


.682 


1763 


.742 


1924 


.802 


2086 


.862 


2248 


.922 


2412 


24 


.687 


2120 


.759 


2349 


.831 


2580 


.903 2811 


.975 


3045 


1.047 


3279 


30 


.785 


3020 


.875 


3376 


.965 


3735 


1.055 4095 


1.145 


4458 


1.235 


4822 


36 


.882 


4070 


.990 


4581 


1.098 


5096 


1.206 5613 


1.314 


6133 


1.422 


6656 


42 


.980 


5265 


1.106 


5958 


1.232 


6657 


1.358 7360 


1.484 


8070 


1.610 


8804 


48 


1.078 


6616 


1.222 


7521 


1.366 


8431 


1.510 9340 


1.654 


10269 


1.798 


11195 



All pipe cast vertically in dry sand; the 3 to 12 inch in lengths of 12 feet, 
all larger sizes in lengths of 12 feet 4 inches. 

Safe Pressures and Equivalent Heads of Water for Cast- 
iron Pipe of Different Sizes and Thicknesses, 

(Calculated by F. H. Lewis, from Fanning's Formula.) 
Size of Pipe. 



Thick- 
ness. 



k 

*s 



10" 



16" 



18" 






20" 



112 
140 
168 



116 
141 
166 



190 



MATERIALS. 



Safe Pressures, etc., for Cast-iron Pipe. (Continued.) 



Thick- 
ness. 


Size of Pipe. 


22" 


24" 


27" 


80" 


33" 


36" 


42" 


48" 


Pressure I 
in Pounds, j 1 


Head in * i 
Feet. 1 1 


Pressure 
in Pounds. 


Head in 
Feet. 


! 

H C 


Head in 
Feet. 


Pressure 
in Pounds. 


Head in 
Feet. 


Pressure 
in Pounds. 


Head in 
Feet. 


Pressure 
In Pounds. 


ii 
|h 


Pressure 
in Pounds. 


Head in 
Feet. 


Pressure 
in Pounds. 


Head in 
Feet. 


Pressure 
in Pounds. 


Is 

^ 


11-16 
3-4 
13-16 
7-8 
15-16 
1 
1 1-8 
1 1-4 
1 3-8 
1 1-2 
1 5-8 
1 3-4 
1 7-8 
2 
2 1-8 
2 1-4 
21-2 
23-4 


40 
60 
80 
101 
121 
142 
182 
224 


92 
138 
184 
233 
279 
327 
419 
516 


30 
49 
68 
86 
105 
124 
161 
199 
237 


69 
113 
157 
198 
242 
286 
371 
458 
546 


19 
36 
52 
69 
85 
102 
135 
169 
202 
236 


64 
83 
120 
159 
196 
S55 
311 
389 
465 
544 


24 
39 
54 
69 
84 
114 
144 
174 
204 
234 


55 
90 
124 
159 
194 
263 
332 
401 
470 
538 


42 

55 
69 
96 
124 
1^1 


97 
127 
159 
221 
286 
348 
410 
472 
537 


32 
44 

57 
82 
1(1? 
132 
157 
182 
207 


74 

101 
131 
189 

247 
304 
362 
419 

477 


38 
59 
81 
103 
124 
145 
167 
188 
210 


88 
136 
187 
237 
286 
334 
385 
433 
484 


24 
43 
62 
81 
99 
118 
136 
155 
174 
193 
212 


55 
99 
143 
187 
228 
272 
313 
357 
401 
445 
488 


Si 
49 
64 
79 
94 
109 
124 
139 
154 
134 
214 


78 
113 
147 

182 
217 
251 
286 
320 
355 
424 
482 






178 
206 
233 

















































































































































































































NOTE. The absolute safe static pressure which may be 

2T S 
put upon pipe is given by the formula P = ~=r X -z-, in 

which formula P is the pressure per square inch.; T, the 
thickness of the shell; S, the ultimate strength per square 
inch of the metal in tension; and D, the inside diameter of 
the pipe. In the tables S is taken as 18000 pounds per 
square inch, with a working strain of one fifth this amount 
or 3600 pounds per square inch. The formula for the 

7200 T 
absolute safe static pressure then is: P = . 

It is, however, usual to allow for "water-ram" by in- 
creasing tho thickness enough to provide for 100 pounds 
additional static pressure, ana, to insure sufficient metal for 
good casting and for wear and tear, a further increase 

equal to .333 (l JQQ)- 
The expression for the thickness then becomes: 

(P-flOO)D 883(l~^ 
7200 M V 100/' 

and for safe working pressure 



The additional section provided as above represents an 
increased value under static pressure for the different sizes 
of pipe as follows (see table in margin). So that to test 
the pipes up to one fifth of the ultimate strength of the 
material, the pressures in the marginal table should be 
added to the pressure-values given in the table above. 



Size 

of 

Pipe. 



RIVETED HYDRAULIC PIPE. 



191 



RIVETED HYDRAULIC PIPE. 

(Pel ton Water Wheel Co.) 
Weight per foot with safe head for various sizes of double-riveted pipe. 



-S 


gj' 




1 


_ | _j- 


a 


3| 




d 

+= a 





o-g 


il 


43 


OJ 


t- ^j 


*o"o 


11 


+3 CO 


$_ , -S 


*" O T 


i- a 


qu-2 . 








t- a 










031 1 

i, of 


.^& 


"3-gJ 


. ^3 


^1 o 


u aT 


^w 6 
tj 


111 


S 1? 


S'l 5 


s ^ 


"O 05 


"3-3 C 


IS-S^ 


.Sf-S^ 


S3 


U 5 


. '~ Q 


T3 Pi"-* 


.s&s^ 


ft 


H 


T' 


w 


gHl. 


s 


*S H^O 


H Hl " 


l^cc 


^,p. 


3 


18 


.05 


810 


2)4 


18 


12 


.109 


295 


2514 


4 


18 


.05 


607 


3 


18 


11 


.125 


337 


29 


4 
5 


16 

18 


.062 
.05 


760 

485 


3% 
3% 


18 
18 


10 

8 


.14 
.171 


378 
460 


5* 


5 


16 


.062 


605 


4^3 


20 


16 


.062 


151 


16 


5 


14 


.078 


757 


5% 


20 


14 


.078 


189 


19% 


6 


18 


.05 


405 


4J4 


20 


12 


.109 


265 




6 


16 


.062 


505 


5J4 


20 


11 


.125 


304 


3)i^ 


6 


14 


.078 


630 


6^ 


20 


10 


.14 


340 


35 


7 


18 


.05 


346 


4% 


20 


8 


.171 


415 


4514 


7 


16 


.062 


433 


6 


22 


16 


.062 


138 


17% 


7 


14 


.078 


540 


7J><2 


22 


14 


.078 


172 


22 


8 


16 


.062 


378 


7 


22 


12 


.109 


240 


30^3 


8 


14 


.078 


472 


8% 


22 


11 


.125 


276 


34(1 


8 


12 


.109 


660 


12 


22 


10 


.14 


309 


39 


9 


16 


.062 


336 


7V*> 


22 


8 


.171 


376 


50 


9 


14 


.078 


420 


9J4 


24 


14 


.078 


158 


23% 


9 


12 


.109 


587 


12% 


24 


12 


.109 


220 


32 


10 


16 


.062 


307 




24 


11 


.125 


253 


37^ 


10 


14 


.078 


378 


10)4 


24 


10 


.14 


283 


42 


10 


12 


.109 


530 


14J4 


24 


8 


.171 


346 


50 


10 


11 


.125 


607 


16J4 


24 


6 


.20 


405 


59 


10 


10 


.14 


680 


18J4 


26 


14 


.078 


145 


25V* 


11 


16 


.062 


275 


9 


26 


12 


.109 


203 


351^ 


11 


14 


.078 


344 


11 


26 


11 


.125 


233 


39^ 


11 


12 


.109 


480 


15/4 


26 


10 


.14 


261 


44^ 


11 


11 


.125 


553 


17V& 


26 


8 


.171 


319 


54 


11 


10 


.14 


617 


19V6 


26 


6 


.20 


373 


64 


12 


16 


.062 


252 


10 " 


28 


14 


.078 


135 


27^ 


12 


14 


.078 


316 




28 


12 


.109 


188 


38 


12 


12 


.109 


442 


17 4 


28 


11 


.125 


216 




12 


11 


.125 


506 


19^ 


28 


10 


.14 


242 


47V^ 


12 


10 


.14 


567 


21% 


28 


8 


.171 


295 


58 


13 


16 


.062 


233 


10i| 


28 


6 


.20 


346 


69 


13 


14 


.078 


291 




30 


12 


.109 


176 


39^ 


13 


12 


.109 


407 


18 


30 


11 


.125 


202 


45 


13 


11 


.125 


467 


20L 


30 


10 


.14 


226 


50^ 


13 


10 


.14 


522 


23 


30 


8 


.171 


276 


61% 


14 


16 


.062 


216 


H/4 


30 


6 


.20 


323 


73 


14 


14 


.078 


271 


14 


30 


H 


.25 


404 


90 


14 


12 


.109 


378 




36 


11 


.125 


168 


54 


14 


11 


.125 


433 


22*4 


36 


10 


.14 


189 


60^ 


14 
15 


10 
16 


.14 
.062 


485 
202 


25 


36 

36 


i 


.187 
.25 


252 
337 


81 
109 


15 


14 


.078 


252 


14% 


36 




.312 


420 


135 


15 


12 


.109 


352 


20^4 


40 


10 


.14 


170 


67^ 


15 


11 


.125 


405 


23 J4 


40 




.187 


226 


90 


15 


10 


.14 


453 


26 


40 


M 


.25 


303 


120 


16 


16 


.062 


190 


13 


40 


j> 


.312 


378 


150 


16 


14 


.078 


237 


16 


40 


% 


.375 


455 


180 


16 


12 


.109 


332 




42 


10 


.14 


162 


71 


16 


11 


.125 


379 


24/^1 


42 


A 


.187 


216 


94^ 


16 


10 


.14 


425 


28^j 


42 


M 


.25 


289 


126 


18 


16 


.062 


168 


14% 


42 


T B 5 


.312 


360 


158 


18 1 


14 


.078 


210 


18H 


42 


% 


.375 


435 


190 



192 



MATERIALS. 



STANDARD PIPE FLANGES. 

Adopted August, 1894, at a conference of committees of the American 
Society of Mechanical Engineers, and the Master Steam and Hot Water Fit- 
ters' Association, with representatives of leading manufacturers and users 
of pipe. Trans. A. S. M. E., xxi. 29. (The standard dimensions given have 
not yet, 1901, been adopted by some manufacturers on account of their un- 
willingness to make a change in their patterns.) 

The list is divided into two groups; for medium and high pressures, the 
first ranging up to 75 Ibs. per square inch, and the second up to 200 Ibs. 



a 

K 



6 

7 

8 

9. 
10 
12 
14 
15 
16 
18 
20 
22 
24 
26 
28 
30 
36 
42 





j\ 6090 



NOTES. Sizes up to 24 inches are designed for 200 Ibs. or less. 

Sizes from 24 to 48 inches are divided into two scales, one for 200 Ibs., the 
other for less. 

The sizes of bolts given are for high pressure. For medium pressures the 
diameters are % in. less for pipes 2 to 20 in. diameter inclusive, and % in. 
less for larger sizes, except 48-in. pipe, for which the size of bolt is 1% in. 

When two lines of figures occur under one heading, the single columns are 
for both medium and high pressures. Beginning with 24 inches, the left-hand 
columns tire for medium and the right-hand lines are for high pressures. 

The sudden increase in diameters at 16 inches is due to the possible inser- 
tion of wrought-iron pipe, making with a nearly constant width of gasket a 
greater diameter desirable. 

When wrought-iron pipe is used, if thinner flanges than those given are 
sufficient, it is proposed that bosses be used to bring the nuts up to the 
standard lengths. This avoids the use of a reinforcement around the pipe. 

Figures in the 3d, 4th, 5th, and last columns refer only to pipe for high 
pressure. 

In drilling valve flanges a vertical line parallel to the spindles should be 
midway between two holes on the upper side of the flanges. 



CAST-IRON" PIPE AKD PIPE FLANGES. 



193 



FLANGE DIMENSIONS, ETC., FOR EXTRA HEAVY 
PIPE FITTINGS (UP TO 250 LBS. PRESSURE). 

Adopted by a Conference of Manufacturers, June 28, 1901. 



Size of 
Pipe. 


Diam. of 
Flange. 


Thickness 
of Flange. 


Diameter of 
Bolt Circle. 


Number of 
Bolts. 


Size of 
Bolts. 


Inches. 


Inches. 


Inches. 


Inches. 




Inches. 


2 


6^ 


% 


5 


4 


% 


% 


7V 


1 


5^ 


4 


% 


3 

3^ 


SJ4 

9 


It* 




8 
8 


1 


4 

4^ 


10 

ttM 


PU 


If 


8 

8 


i 


5 


11 


1% 


9J4 


8 


8 


6 


12^ 


1 7-16 


10% 


13 


M 


7 


14 


1^4 


31% 


12 


y& 


^8 


15 


1% 


13 


12 


% 


9 


16 


1M 


14 


12 


7^ 


10 
12 


g* 


3* 


Wi 

im 


16 
16 


1 


14 
15 


22U 


2^ 
2 3-16 


20 
21 


20 
20 


ft 


16 


25 


214 


22^ 


20 




18 


27 


2% 


1V& 


24 




20 


29^ 


2Va 


26^ 


24 


^ 


22 


81JJ 


2% 


28% 


28 


/^ 


24 


84 


234 


31% 


28 






DIMENSIONS OF PIPE FLANGES AND CAST-IRON 
PIPES. 

(J. E. Codman, Engineers 1 Club of Philadelphia, 1889.) 



4 
5 
6 
8 
10 
12 
14 
16 
18 
20 
22 
24 



32 
34 



40 
42 
44 

46 

48 



.So 

5 



3 O 



Is 



8 

8 
10 
12 
14 
16 
16 
18 
20 
22 
24 
24 
26 
28 
30 
32 
32 
34 
34 

2 1-16 
40 2^ 



Thickness 
of Pipe. 



Frac. Dec. 



13-32 
7-16 
7-16 
15-32 



19-32 
21-32 
11-16 

Has 

27-32 



31-32 
1 

1 1-16 
1 

l'5-32 
1 3-16 

1 5-16 
111-32 

1T-16 



.373 

.396 

.420 

.443 

.466 

.511 

.557 

.603 

.649 

.695 

.741 

.787 

.833 

.879 

.925 

.971 

1.017 

1.063 

1.109 

1.155 

1.201 

1.247 

1.293 

1.339 

1.385 

1.431 



Cfe.-g o3 
JUS 



6.96 

11.16 

15.84 

21.00 

26.64 

39.36 

54.00 

70.56 

89.04 

109.44 

131.76 

156.00 

182.16 

210.24 

240.24 

272.16 

306.00 

341.76 

379.44 

419.04 

460.56 

504.00 

549.36 

596.64 

645.84 

696.96 



4.41 

5.93 

7.66 

9.63 

11.82 

16.91 

23.00 

30.13 

38.34 

47.70 

58.23 

70.00 

83.05 

97.42 

113.18 

130.35 

149.00 

169.17 

190.90 

214.26 

239.27 

266.00 

294 49 

324.78 

356.94 

391.00 



D = Diameter of pipe. All dimensions in inches. 
FORMUUE. Thickness of flange = 0.033D -f- 0.56 ; thickness of pipe 

0'23D -4- 0.327* r*irrVif f\f i"kii-ck r-xjii. -F/-W/-W*- n f>A TY1 I O 7~ . ,, ^:_Ui xii _._ 

' 



194 



MATERIALS. 



Df Perf ".2J2S9?2:*SeSJ52}2P<P 
Thread! 

No. of 
Threads 
aerlnch. 

. Tfioco^i-ieooDOiccot^i-toiocxtco^TEcoTttoo oo T~co T~ei ^ 

per 

Lin. Ft. S "~~^~'~^;S;^Sw33$SSo^oco 
of Pipe. ___^ r-,^-.t-i ^ 

Weight .TtCJOT*iNl~TtOOi-i^^OOTt<Oimr~GOOeoojer<-->^r-,<-->r-> - 

of Pipe 

per -ii-tS'totcoiot>osoW^GC2oc6coono:T 

Li ^ Ft i-^i^fr^co^Tr^ioo 

^ __ .^ JyQ 

oyp'jpl i- c '"ssss-ssssnssigsg ^ 

i\u nt f?. %^ e 

^ IPJ.llliillliSSSISIilgligigSlS ^ 

- 1 ^ ^ i-^ i-^ I-' o*' iy| eo 

I 

g 

_ ^ ^ 

CC ' ' "T^^H'^CJUJ'C 

o^ 

02 " ^* T-I <ri y-t & a ai *& >+* 

Length j ^ OQOI>OOTt( ^ OQOO:)V ^ irao;)Tt ,j > j > ^ Heot , tOO 

! Outside' fa os t- 10 -^wojcjoj i-^i-^d B& 

Surface.j 

?rl?t ^Sg'g2Slgi?^^^^^^^^^^^^"^^^^ SSS ^ 
Inside fa -^ o' i> to *? oo ci oi i-i i-i TH ^ d ^ 

Surface. ^ 

ence. 

Internal .f3!S!iba 
Circum- S . '-J *. >. w 

f erence. l ~ l ^-< '-> c 

Thick- . GO oo as c 

A otual _^?^cocs'fjc^ : ^GOr : Hcbo : 3"55oTfoc$6cco^oSoioiocioioio ^sp 

^.^1/UlO.i gg^J^^^QOQJlQ^^^CjJ^Q^^l^X^QS^^^^^^JQ^^J^^g^ ^\ 

Inside H ^ 

Diam. T-II-H i-< ri ^ 1-1 r-i 1-1 1-1 1-1 1-1 ?? g< g 

Dm e - ' ^"^^^^^^^sdSsasftftii* g- 

Nominal j 
Inside fl w 
Diam. ' 



WROUGHT-IROH PIPE. 



195 



For discussion of the Briggs Standard of Wrought-iron Pipe Dimensions, 
see Report of the Committee of the A. S. M. E. in " Standard Pipe and Pipe 
Threads," 1886. Trans., Vol. VIII, p. 29. The diameter of the bottom of 

the thread is derived from the formula J> (0.05D + 1.9) x , in which 

D = outside diameter of the tubes, and n the number of threads to the 
inch. The diameter of the top of the thread is derived from the formula 

0.8 X 2 -f d, or 1.6 }- d, in which d is the diameter at the bottom of the 

thread at the end of the .pipe. 

The sizes for the diameters at the bottom and top of the thread at the end 
of the pipe are as follows: 



Diam. 


Diam. 


Diam. 


Diam. 


Diam. 


Diam. 


Diam. 


Diam. 


Diam. 


of Pipe, 
Nom- 


at Bot- 
tom of 


at Top 
of 


of Pipe, 
Nom- 


at Bot- 
tom of 


at Top 
of 


of Pipe, 
Nom- 


at Bot- 
tom of 


at Top 
of 


inai. 


Thread. 


Thread. 


inal. 


Thread. 


Thread. 


inal. 


Thread. 


Thread. 


in. 


in. 


in. 


In. 


in. 


in. 


in. 


in. 


in. 




.334 


.393 


**4 


2.620 


2.820 


8 


8.334 


8.534 


M 


.438 


.522 


3 


3.241 


3.441 


9 


9.327 


9.527 


% 


.568 


.658 


3^ 


3.738 


3.938 


10 


10.445 


10.645 


Ut 


.701 


.815 


4 


4.234 


4.434 


11 


11.439 


11.639 


% 


.911 


1.025 


4^ 


4.731 


4.931 


12 


12.433 


12.633 


I 


1.144 


1.283 


5 


5.290 


5.490 


13 


13.675 


13.875 


w 


1.488 


1.627 


6 


6.346 


6 546 


14 


14.669 


14.869 


i*i 


1.727 


1.866 


7 


7.340 


7.540 


15 


15.663 


15.863 


8 


2.223 


2.339 
















Having the taper, length of full-threaded portion, and the sizes at bottom 
and top of thread at the end of the pipe, as given in the table, taps and dies 
can be made to secure these points correctly, the length of the imperfect 
threaded portions on the pipe, aud the length the tap is run into the fittings 
beyond the point at which the size is as given, or, in other words, beyond 
the end of the pipe, having no effect upon the standard. The angle of the 
thread is 60, and it is slightly rounded off at top and bottom, so that, instead 
of its depth being 0.866 its pitch, as is the case with a full V-thread, it is 
4/5 the pitch, or equal to 0.8 -*- n, n being the number of threads per inch. 

Taper of conical tube ends, 1 in 32 to axis of tube = 2 inch to the foot 
total taper. 



L96 



MATERIALS. 



WROUGHT-IRON WELDED TUBES, EXTRA STRONG. 

Standard Dimensions. 



Nominal 
Diameter. 


Actual Out- 
side 
Diameter. 


Thickness, 
Extra 
Strong. 


Thickness, 
Double 
Extra 
Strong. 


Actual Inside 
Diameter, 
Extra 
Strong. 


Actual Inside 
Diameter, 
Double Extra 
Strong. 


Inches. 


Inches. 


Inches. 


Inches. 


Inches. 


Inches. 


V6 


0.405 


0.100 




0.205 




\* 


0.54 


0.123 




294 




Z 


0.675 


0.127 




0.421 




H 


0.84 


0.149 


0.298 


0.542 


0.244 


fi 


1.05 


0.157 


0.314 


0.736 


0.422 


1 


1.315 


0.182 


0.364 


0.951 


0.587 


1/4 


1.66 


0.194 


0.388 


1.272 


0.884 


Ii2 


1.9 


0.203 


0.406 


1.494 


1.088 


2 


2.375 


0.221 


0.442 


1.933 


1.491 


2^ 


2.875 


0.280 


0.560 


2.315 


1.755 


3 


3.5 


0.304 


0.608 


2.892 


2.284 


3^ 


4.0 


0.321 


0.642 


3.358 


2.716 


4 


4.5 


0.341 


0.682 


3.818 


3.136 



STANDARD SIZES, ETC., OF LAP-WELDED CHAR- 
COAL-IRON BOILER-TUBES. 

(National Tube Works.) 



I 


A 




" 


, . 






IJ 


-o'Si 


oy 




* 


















^^M 


-,^D 


5 


s 


I 


O 


al 


Internal 


External 


E^.jj 


S^g 


0^1 


E 












Area. 


Area. 










1^ 


I 1 ' 


| H 


pS 

I 3 


11 
fl* 






fll 


fill 


ftS 


1 
hj 


in. 


in. 


in. 


in. 


in. 


sq. in. 


sq.ft. 


sq. in. 


sq.ft. 


ft. 


ft. 


ft. 


Ibs. 


1 


.810 


.095 


2.545 


3.142 


.515 


.0036 


.785 


.0055 


4.479 


3.820 


4.149 


.90 


1 1-4 


1.060 


.095 


3.330 


3.927 


.882 


.0061 


1.227 


.0085 


3.604 


3.056 


3.330 


1.15 


1 1-2 


1.310 


.095 


4.115 


4.712 


1.348 


.0094 


1.767 


.0123 


2.916 


2.547 


2.732 


1.40 


13-4 


1.560 


.095 


4.901 


5.498 


1.911 


.0133 


2.405 


.0167 


2.448 


2.183 


2.316 


1.65 


2 


1.810 


.095 


5.686 


6.283 


2.573 


.0179 


3.142 


.0218 


2.110 


1.910 


2.010 


1.91 


2 1-4 


2.060 


.095 


6.472 


7.069 


3.333 


.0231 


3.976 


0276 


1.854 


1.698 


1.776 


2.16 


2 1-2 


2.282 


.109 


7.169 


7.854 


4.090 


.0284 


4.909 


.0341 


1.674 


1.528 


1.601 


2.75 


23-4 


2.532 


.109 


7.955 


8.639 


6.035 


.0350 


5.940 


.0412 


1.508 


1.389 


1.449 


3.04 


3 


2.782 


.109 


8.740 


9.425 


6.079 


.0422 


7.069 


.0491 


1.373 


1.273 


1.322 


3.33 


3 1-4 


3.010 


.120 


9.456 


10.210 


7.116 


.0494 


8.296 


.0576 


1.269 


1.175 


1.222 


3.96 


31-2 


3.260 


.120 


10.242 


10,996 


8.347 


.0580 


9.621 


.0668 


1.172 


1.091 


1.132 


4.28 


33-4 


8.510 


120 


11.027 


11.781 


9.676 


.0672 


11.045 


07G7 


1.088 


1.019 


1.054 


4.60 


4 


3.732 


.134 


11.724 


J 2.566 


10.939 


.0760 


12.566 


0873 


1.024 


.955 


.990 


5.47 


41-2 


4.232 


.134 


13.295 


14.137 


14.066 


.0977 


15.904 


.1104 


.903 


.849 


.876 


6.17 


5 


4.704 


.148 


14.778 


15.708 


17.379 


.1207 


19.635 


.1364 


.812 


.764 


.788 


7.58 


6 


5.670 


.165 


17.813 


18.850 


25.250 


.1750 


28.274 


.1963 


.674 


.637 


.656 


10.16 


7 


6.670 


.165 


20.954 


21.991 


34,942 


.2427 


38.485 


.2673 


.573 


.546 


.560 


11.90 


8 


7.670 


.165 


24.096 


25.133 


46.204 


.3209 


50.266 


.3491 


.498 


.477 


.488 


13.65 


9 


8.640 


.180 


27.143 


28.274 


58.630 


.4072 


63.617 


.4418 


.442 


.424 


.433 


16.76 


10 


9.594 


.203 


30.141 


31.416 


72.292 


.5020 


78.540 


-5454 


.398 


.382 


.390 


21.90 


11 


10.560 


.220 


33.175 


34.558 


87.583 


.6082 


95.033 


.6600 


.362 


.347 


.355 


25.00 


12 


11.542 


.229 


36.260 


37.699 


104.629 


.7266 


113.098 


.7854 


.331 


.318 


.325 


28.50 


13 


12.524 


.233 


39.345 


40.841 


123.190 


.8555 


132.733 


.9217 


.305 


.294 


.300 


32.06 


14 


13.504 


.248 


42.424 


43.982 


143.224 


.9946 


153.938 


1.0690 


.283 


.273 


.278 


36.00 


15 


14.482 


.259 


45.497 


47.124 


164.721 


1.1439 


176.715 


1.2272 


.264 


.255 


260 


40.60 


16 


15.458 


.271 


48.563 


50.266 


187.671 


1.3033 


201.062 


1.3963 


.247 


.239 


.243 


45.20 


17 


16.432 


.284 


51.623 


53.407 


212.066 


1.4727 


226.981 


1.5763 


.232 


.225 


.229 


49.90 


18 


17.416 


.292 


54.714 


56.549 


238.225 


1.6543 


254.470 


1.7671 


.219 


.212 


.216 


54.82 


19 


18.400 


.300 


57.805 


59.690 


265.905 


1.8466 


283.529 


1.9690 


.208 


.201 


.205 


59.48 


20 


19.360 


.320 


60.821 


62.832 


294.375 


2.0443 


314.159 


2.1817 


.197 


.191 


.194 


66.77 


21 


20.320 


.340 


63.837 


65.974 


324.294 


2.2520 


346.361 


2.4053 


.188 


.182 


.185 


73.40 



surface in 
bes) is to 



In estimating the effective steam-heating or boiler surface of tubes, the su 
contact with air or gases of combustion (whether internal or external to the tu 
be taken. 

For heating liquids by steam, superheating steam, or transferring heat from one 
liquid or gas to another, the mean surface of the tubes is to be taken. 



RIVETED TROK PIPE. 



197 



To find the square feet of surface, S, in a tube of a given length, L, in feet, 
and diameter, d, in inches, multiply the length in feet by the diameter in 

inches and by .2618. Or, 8 - - L -- - = .2618dL. For the diameters in the 

table below, multiply the length in feet by the figures given opposite the 
diameter. 



Inches, 
Diameter. 


Square Feet 
per Foot 
Length. 


Inches, 
Diameter. 


Square Feet 
per Foot 
Length. 


Inches, 
Diameter. 


Square Feet 
per Foot 
Length. 


1 4 
2 4 


.0654 
.1309 
.1963 
.2618 
.3272 
.3927 
.4581 
.5236 


3 4 
4 4 


.5890 
.6545 
.7199 
.7854 
.8508 
.9163 
.9817 
1.0472 


5 
6 
7 
8 
9 
10 
11 
12 


1.3090 
1.5708 
1.8326 
2.0944 
2.3562 
2.6180 
2.8798 
3.1416 



RIVETED IRON PIPE. 

(Abendroth & Root Mfg. Co.) 

Sheets punched and rolled, ready for riveting, are packed in convenient 
form for shipment. The following table shows the iron and rivets required 
for punched and formed sheets. 



Number Square Feet of Iron 
required to make 100 Lineal 
Feet Punched and Formed 


I11H1 

^^^3c| 


Number Square Feet of Iron 
required to make 100 Lineal 
Feet Punched and Formed 




Sheets when put together. 




Sheets when put together. 


"el r S^"^ 3^ 




fijU^ & 




If jr | 














Diam- 
eter in 
Inches. 


Width of 
Lap in 
Inches. 


Square 
Feet. 




Diam-. 
eter in 
Inches. 


Width of 
Lap in 
Inches. 


Square 
Feet. 


oTS oS2fe oSCQ 


3 


1 


90 


1,600 


14 


JJX 


397 


2,800 


4 


1 


116 


1,700 


15 


l^i 


423 


2,900 


5 




150 


1.800 


16 


ji^j 


452 


3,000 


6 


ji/ 


178 


1,900 


18 


l/^ 


506 


3,200 


7 


]1Z 


206 


2,000 


20 


l/'ij 


562 


3,500 


8 


1^1 


234 


2,200 


22 


l/^ 


617 


3,700 


9 


JL 


258 


2,300 


24 


1^ 


670 


3,900 


10 


l^J 


289 


2,400 


26 


^/^ 


725 


4,100 


11 


1^1 


314 


2,500 


28 


1/^j 


779 


4,400 


12 


ji/ 


343 


2,600 


30 


1^3 


836 


4,600 


13 


% 


369 


2,700 


36 


% 


998 


5,200 



WEIGHT OF ONE SQUARE FOOT OF SHEET-IRON 
FOR RIVETED PIPE. 

Thickness by the Rirmiugliam Wire-Gauge. 



No. of 
Gauge. 


Thick- 
ness in 
Decimals 
of an 
Inch. 


Weight 
in Ibs., 
Black. 


Weight 
in Ibs., 
Galvan- 
ized. 


No. of 
Gauge. 


Thick- 
ness in 
Decimals 
of an 
Inch. 


Weight 
in Ibs., 
Black. 


Weight 
in Ibs., 
Galvan- 
ized. 


26 
24 
22 

20 


.018 
.022 
.028 
.035 


.80 
1.00 
1.25 
1.56 


.91 
1.16 
1.40 
1.67 


18 
16 
14 
12 


.049 

.065 
.083 
.109 


1.82 
2.50 J 
3.12 
4.37 


2.16 
2 67 
3.34 
4.73 



198 



MATERIALS. 



SPIRAL RIVETED PIPE. 

(Abendroth & Root Mfg. Co.) 



Thickness. 


Diam- 
eter, 
Inches. 


Approximate Weight 
in Ibs. per Foot in 
Length. 


Approximate Burst- 
ing Pressure in Ibs. 
per Square Inch. 


B. W. G. 

No. 


Inches. 


26 
24 
22 
20 
18 
16 
14 
12 


.018 
.022 
.028 
.035 
.049 
.065 
.083 
.109 


3 to 6 1 
3 to 12 
3 to 14 
3 to 24 
3to2i 
6 to 24 
8 to 24 
9 to 24 


bs.= 
= ^ofd 

= .4 
= .5 

= .6 
= .8 
= 1.1 
= 1.4 


iam. in 


ns. 


27001bs.-f-diam.inins. 
3600 " H- *' 

4800 " -T- " 
6400 " -*- " 
8000 " -*- " 



The above are black pipes. Galvanized weighs 10 to 30 % heavier. 
Double Galvanized Spiral Riveted Flanged Pressure Pipe, tested to 150 Ibs. 
hydraulic pressure. 



Inside diameters, inches.... 

Thickness, B. W. G 

Nominal wt. per foot, Ibs.. . 



2020 



71 8 

O 1C 



181818 



91011 



1816 

811 



13114151618202224 



16 16 14 



14 15' 20 22 24 29 34 4050 



1212 



DIMENSIONS OF SPIRAL PIPE FITTINGS. 











Diameter 




Inside 
Diameter. 


Outside 
Diameter 
Flanges. 


Number 
Bolt-holes. 


Diameter 
Bolt-holes.j 


Circles on 
which Bolt- 
holes are 


Sizes of 
Bolts. 










Drilled. 




ins. 


ins. 




ins. 


ins. 


ins. 


3 


6 


* 4 


% 


4% 


7/16 x 1% 


4 


7 


8 


i^ 


5 15/16 


7/16 x 1% 


5 


8 


8 


^ 


6 15/16 


7/16x1% 


6 


8% 


8 


% 


7% 


l^> x 1% 


7 


10 


8 


% 


9 


^6 x 1% 


8 


11 


8 


5^ 


10 


1^x2 


9 


13 


8 


% 


tt4 


Ux2 


10 


14 


8 


% 


1214 


>|x2 


11 


15 


12 


% 


13% 


^2*2 


12 


16 


12 


% 


i42 


1^x2 


13 


17 


12 


% 


15^4 


1^x2 


14 
15 


& 


12 
12 


i 


16*4 

17 7/16 


j|*2| 


16 


21 3/16 


12 


% 


19^ 


L x 2J-J2 


18 


23^ 


16 


11/16 


2ii| 


^ x 2^ 


20 


25^ 


16 


11/16 


23V 


^ x 2V& 


22 


28^ 


16 




26 


% X 2V^ 


24 


30 


16 


A 


27% 


%*^ 



SEAMLESS BRASS TUBE. IRON-PIPE SIZES. 
(For actual dimensions see tables of Wrought-iron Pipe.) 



Nominal 
Size. 


Weight 
pr Foot. 


Nom. 
Size. 


Weight 
per Foot. 


Nom. 
Size. 


Weight 
per Foot. 


Nom. 
Size. 


Weight 
per Foot. 


ins. 


Ibs. 
.25 
.43 
.62 
.90 


ins. 

f 

i 


Ibs. 
1.25 
1.70 
2.50 
3. 


ins. 
2 

P 

3^ 


Ibs. 
4.0 
5.75 
8.30 
10.90 


ins. 
4 

9* 

6 


Ibs. 
12.70 
13.90 
15.75 

18.31 



BRASS TUBING; COILED PIPES. 



199 



SEAMLESS DRAWN BRASS TUBING. 

(Randolph & Clowes, Waterbury, Conn.) 

Outside diameter 3/16 to 7% inches. Thickness of walls 8 to 
Gauge, length 12 feet. The following are the standard sizes: 



Outside 
Diam- 
eter. 


Length 
Feet. 


Stubbs' 
or Old 
Gauge. 


Outside 
Diam- 
eter. 


Length 
Feet. 


Stubbs' 
or Old 
Gauge. 


Outside 
Diam- 
eter. 


Length 
Feet. 


Stubbs 1 
or Old 
Gauge. 


H 


12 


20 


1% 


12 


14 


2% 


12 


11 


5-16 


12 


19 




12 


14 


2M 


12 


11 


% 


12 


19 


1% 


12 


13 


3 


12 


11 


l/; 


12 


18 


1% 


12 


13 


3*4 


12 


11 


% 


12 


18 


1 13-16 


12 


13 


31^1 


12 


11 


% 


12 


17 


m 


12 


12 


4 


10 to 12 


11 


13-16 


12 


17 


1 15-16 


12 


12 


5 


10 to 12 


11 


% 


12 


17 


2 


12 


12 


5/4 


10 to 12 


11 


15-16 


12 


17 


% 


12 


12 


51^3 


10 to 12 


11 


1 


12 


16 


2^4 


12 


12 


5M 


10 to 12 


11 




12 


16 




12 


12 


6 


10 to 12 


11 


1J4 


12 


15 


2J^ 


12 


11 









BENT AND COILED PIPES. 

(National Pipe Bending Co., New Haven, Conn.) 
COILS AND BENDS OF IRON AND STEEL PIPE. 



Size of pipe Inches 




8 6 


1-4 


$ 


1 


1M 


1^ 


2 


2U 


3 


Least outside diameter of 
coil Inches 


2 


01^ 


fti< 


4V 


fi 


8 


12 


16 


24 


32 
























Size of pipe Inches 


3U 


4 


41-6 


5 


Q 


7 


8 


9 


10 


12 


Least outside diameter of 
coil Inches 


40 


18 


50 


58 


66 


30 


92 


105 


130 


156 

























Lengths continuous welded up to 3-in. pipe or coupled as desired. 
COILS AND BENDS OF DRAWN BRASS AND COPPER TUBING. 



Size of tube, outside diameter Inches 
Least outside diameter of coil Inches 


1* 





2 * 





^ 


1 

4 


J 


JN 


Size of tube, outside diameter Inches 


m 


1** 


W 


^ 


2M 


23/ tf 


^ 


2 ^ 


Least outside diameter of coil Inches 


8 


9 


10 


12 


14 


16 


18 


20 



Lengths continuous brazed, soldered, or coupled as desired. 

90 BENDS. EXTRA-HEAVY WROUGHT-IRON PIPE. 



Diameter of pipe Inches 

Radius Inches 

Centre to end Inchei 



is 26 



24 



The radii given are for the centre of the pipe. *' Centre to end " means 
the perpendicular distance from the centre of one end of the bent pipe to a 
plane passing across the other end. Standard iron pipes of sizes 4 to 8 in. 
are bent to radii 8 in. larger than the radii in the above table; sizes 9 to 12 in. 
to radii 12 in. larger. 

Welded Solid I>rawn>steel Tubes, imported by P. S. Justice & 
Co., Philadelphia, are made in sizes from ^ to 4^ in. external diameter, 
varying by Vths, and with thickness of walls from 1/16 to 11/16 in. The 
maximum length is 15 feet. 



200 



MATERIALS. 



WEIGHT OF BRASS, COPPER, AND ZINC TUBING. 

Per Foot. 

Thickness by Brown & Sharpens Gauge. 



Brass, No. 17. 


Brass, No. 20. 


Copper, 
Lightning-rod Tube, 
No. 23. 


Inch. 

*\ 
& 
#> 

/& 
2 4 

$ 


Lbs. 
.107 
.157 
.185 
.234 
.266 
.318 
.333 
.377 
.462 
.542 
.675 
.740 
.915 
.980 
1.90 
1.506 
2.188 


Inch. 

HI 

3-16 
5-16 
$6 

! 
i 

m 
8? 


Lbs. 
.032 
.039 
.063 
.106 
.126 
.158 
.189 
.208 
.220 
.252 
.284 
.378 
.500 
.580 


Inch. 

& 

A 
g 


Lbs. 
.162 
.176 
.186 
.211 
.229 


Zinc, No. 20. 


r 

ijj 


.161 
.185 
.234 
.272 
311 
.380 
.452 



LEAD PIPE IN LENGTHS OF 10 FEET. 



In. 


3-8 Thick. 


5-16 Thick. 


M Thick. 


3-16 


Thick. 




Ib. 


oz. 


Ib. 


oz. 


Ib. 


oz. 


Ib. 


oz. 


2^ 


17 





14 





11 





8 





3 


20 





16 





12 





9 





3^3 


23 





18 





15 





9 


S 


4 


25 





21 





16 





12 


8 


4^ 










18 





14 





5 


31 









20 










LEAD WASTE-PIPE. 



in., 2 Ibs. per foot. 
"3 and 4 Ibs. per foot. 
" 3^6 and 5 Ibs. per foot. 



in., 4 Ibs. per foot. 
" 5, 6, and 8 Ibs. 
6 and 8 Ibs. 



5 in. 8, 10, and 12 Ibs. 

LEAD AND TIN TUBING. 

^ inch. J4 inch. 

SHEET LEAD. 



Weight per square foot, 2^, 3, 3*4 4, 
Other weights rolled to order. 



4 5, 6, 8, 9, 10 Ibs. and upwards. 



BLOCK-TIN PIPE. 



in., 4}4, 6}4, and 8 oz. per foot. 

" 6, 7^j, and 10 oz. " 

*' 8 and 10 oz. 

" 10 and 12 oz. " 



1 in., 15, and 18 oz. per foot. 
154 " 114 and lUlbs. " 
lj| " 2 and 2V Ibs. 

2 4i 2^ and 3 Ibs. " 



LEAD PIPE. 



201 



LEAD AND TIN-LINED LEAD PIPE. 

(Tatham & Bros., New York.) 









.s . 








a 


1 


1 


Weight per 
Foot and Rod. 


I 


1- 


I 


Weight per 
Foot and Rod. 


S5 
^ 


1 


S 




H 





I 






Mil. 


E 
D 


7 Ibs. per rod 
10 oz. per foot 


6 


1 u in. 


E 
D 


\y% Ibs. per foot 
2 " " 


10 

11 


** 


C 


12 " " 


8 


41 


C 


<>L " * 4 


14 


" 


B 


1 Ib. 


12 


M 


B 


31^ ** 


17 


** 


A 


1*4 " " 


16 


" 


A 


4 || || 


21 


" 


AA 


1*4 " " 


19 


'* 


AA 




24 


44 


AAA 


1% " 4 ' 


27 


4 ' 


AAA 


6 " '* 


30 


7-16 in. 




13 oz. " 




1*4 in. 


E 


2 " *' 


10 


" 




1 Ib. " 




4 


D 


gi^ 4 * 


12 


]/2 ' n 


E 


9 Ibs. per rod 


7 





C 


3 * ' 


14 




D 


% Ib. per foot 


9 


4 


B 


3% ' 


16 


44 


C 


1 44 4t 


11 




A 


4^4 ' * 


19 


" 


B 


1*4 " " 


13 





AA 


54 ' ' 


25 


44 




1*4 4t ** 




1 


AAA 


6M ' ' 




M 


A 


\%L 4< 4< 


16 


IHJn. 


E 


3 


12 


44 


AA 


2 " 


19 




D 


3*4 * 


14 








23 




C 




17 


* 


AAA 


3 " 


25 


41 


B 


5 | | 


19 


P& in. 


E 


12 " per rod 


8 


* 


A 




23 


44 


D 


1 ' per foot 


9 


44 


AA 


8 * ' 


27 


" 


C 


1*4 44 4t 


13 


44 


AAA 


9 * 




44 


B 


2 " " 


16 


1% in. 


C 


4 


13 


4 


A 


2*^3 " 4 ' 


20 


" 


B 


5 | | 


17 


' 


AA 


2% " " 


22 


44 


A 




21 


4 


AAA 


3*4 " " 


25 


44 


AA 


8Va ' 4t 


27 


% in. 


E 


1 " per foot 


8 


2 in. 


C 


4M * * l 


15 


1 


D 


1*4 " 


10 


41 


B 


6 4 


18 


1 


C 


jax *t 


12 


" 


A 


7 ' 


22 


" 


B 


0]X ' ** 


16 


44 


AA 


9 || ;; 


27 


44 


A 


3 ' " 


20 


* 


AAA 






44 


AA 


3^3 " 4 ' 


23 












AAA 


4% " 


30 











WEIGHT OF LEAD PIPE WHICH SHOULD BE USED 
FOR A OIVEN HEAD OF WATER. 

(Tatham & Bros., New York.) 



Head or 
Number 


Pressure 


Calibre and Weight per Foot. 


of Feet 
Fall. 


per 
sq. inch. 


Letter. 


%inch. 


*^inch. 


% inch. 


inch. 


1 inch. 


lJ4in. 


30 ft. 


15 Ibs. 


D 


10 oz. 


Ib, 


1 Ib. 


1*4 Ibs 


2 Ibs. 


2*4 Ibs. 


50ft. 


25 Ibs. 


C 


12 oz. 


1 Ib. 


1*4 Ibs. 


1% Ibs. 


2*4 Ibs. 


3 Ibs. 


75ft. 


38 Ibs. 


B 


1 Ib. 


1*4 Ibs. 


2 Ibs. 


2*4 Ibs. 


3*4 Ibs. 


3% Ibs. 


100 ft. 


50 Ibs. 


A 


1*4 Ibs. 


1M Ibs. 


2*4 Ibs. 


3 Ibs. 


4 Ibs. 


4% Ibs. 


150 ft. 


75 Ibs. 


AA 


1*4 Ibs. 


2 Ibs.i 2% Ibs. 


3*4 Ibs. 


4% Ibs. 


6 Ibs. 


200ft. 


100 Ibs. 


AAA 


1% Ibs. 


3 Ibs. ! 3*| Ibs. 

i 


4% Ibs. 


6 Ibs. 


6% Ibs. 



To find the thickness of lead pipe required when the 
head of water is given. (Chadwick Lead Works). 

RULE. Multiply the head in feet by size of pipe wanted, expressed deci- 
mally, and divide by 750; the quotient will give thickness required, in one- 
hundredths of an inch. 

EXAMPLE. Required thickness of half -inch pipe for a head of 25 feet. 

25 X 0.50 -?- 750 = 0.16 inch. 



202 



MATERIALS. 







O O O g? OO *H O? O5 C? O 00 O O Tj< T- O rH O 00 
OJ001>COCOOTjTj<COCOOOOJC<lOirH^THTH OJ 



0> 

^3 o5 ^ i- TH o* co co o r 

^O* 8, $S8S88ii3$3SSSg 

*" I 

^ tfj t^ O <> CO 


i - 

a 

SS f- 

| a^^, 

fiii52Siiilii|li > : , *J 

__ .~OOOOOOOOOOOOOOOOOOoS *r ri 
02 M 

|j 

Sc 
a 

35 " 

*i I 

6C02 P ^OQO^COr-iOOJOOI>0150TTfCQCo"(?iNNOJT 

_Q M^r-.^r-.rHrHTH 

^ 
fc^ 

5,< 

^ 

.fc^ CQ 

- $ 

O 

?*9Pr^^95Otr?5Q^ 

^W 

s& 
oi 



BOLT COPPER SHEET AND BAB BRASS. 



203 



WEIGHT OF ROUND BOLT COPPER, 

Per Foot. 



Inches. 


Pounds. 


Inches. 


Pounds. 


Inches. 


Pounds. 


1 


.425 

.756 
1.18 
1.70 
2.31 


1 


3.02 
3.83 
4.72 
5.72 
6.81 


2 


7.99 
9.27 
10.64 
12.10 



WEIGHT OF SHEET AND BAR BRASS. 



Thickness, 
Side or 
Diam. 


Sheets 
per 
sq. ft. 


Square 
Bars 1 
ft. long. 


Round 
Bars 1 
ft. long. 


Thickness, 
Side or 
Diam. 


Sheets 
per 
sq. ft. 


Square 
Bars 1 
ft. long. 


Round 
Bars 1 
ft. long. 


Inches. 








Inches. 








1-16 


2.72 


.014 


.011 


1 1-16 


46.32 


4.10 


3.22 


K 


5.45 


.056 


.045 


Ug 


49.05 


4.59 


3.61 


3-16 


8.17 


.128 


.100 


1 3-16 


51.77 


5.12 


4.02 


1 A 


10.90 


.227 


.178 


154 


54.50 


5.67 


4.45 


5-16 


13.62 


.355 


.278 


1 5-16 


57.22 


6.26 


4.91 


% 


16.35 


.510 


.401 


1% 


59.95 


6.86 


5.39 


7-16 


19.07 


.695 


.545 


1 7-16 


62.67 


7.50 


5.89 





21.80 


.907 


.712 


1*6 


65.40 


8.16 


6.41 


9-16 


24.52 


1.15 


.902 


1 9-16 


68.12 


8.86 


6.95 


% 


27.25 


1.42 


1.11 


m 


70.85 


9.59 


7.53 


11-16 


29 97 


1.72 


1.35 


1 11-16 


73.57 


10.34 


8.12 


H 

13-16 


3-2.70 
35.43 


2.04 
2.40 


1.60 

1.88 


1 fs-16 


76.30 
79.0-2 


11.12 
11.93 


8.73 
9.36 


7 /8 


38.15 


2.78 


2.18 


1% 


81.75 


12.76 


10.01 


15-16 


40.87 


3.19 


2.50 


1 15-16 


84.47 


13.63 


10.70 


1 


43.60 


3.63 


2.85 


2 


87.20 


14.52 


11.40 



COMPOSITION OF VARIOUS GRADES OF ROLLED 
BRASS, ETC. 



Trade Name. 


Copper 


Zinc. 


Tin. 


Lead. 


Nickel. 


Common high brass 


61.5 


38 5 








Yellow metal ... 


60 


40 








Cartridge brass 


66% 


33^ 








Low brass 


80 


20* 








Clock brass 


60 


40 




1L 




Drill rod 


60 


40 




\y> to 2 




Spring brass 




33V<* 


1V 






18 per cent German silver. . 


61J^ 


20^ 






18 



The above table was furnished by the superintendent of a mill in Connec- 
ticut in 1894. He says: While each mill has its own proportions for various 
mixtures, depending upon the purposes for which the product is intended, 
the figures given are about the average standard. Thus, between cartridge 
brass with 33J per cent zinc and common high brass with 38U per cent 
zinc, there are any number of different mixtures known generally as " high 
brass," or specifically as "spinning brass," "drawing brass," etc., wherein 
the amount of zinc is dependent upon the amount of scrap used in the mix- 
ture, the degree of working to which the metal is to be subjected, etc. 



204 



MATERIALS. 



AMERICAN STANDARD SIZES OF DROP-SHOT. 





Diameter. 


No. of Shot 
to the oz. 




Diameter. 


No. of Shot 
to the oz. 




Diam- 
eter. 


No. of Shot 
to the oz. I 


Fine Dust. 
Dust... . 
No. 12. . . 
" 11. . 
" 10.. . 
" 10. . . 
" 9. . . 
" 9.. . 


3-1 00" 
4-100 
5-100 
6-100 
Trap Shot 
7-100" 
Trap Shot 
8-100" 


10784 
4565 
2326 
1346 
1056 
848 
688 
568 


No. 8 
" 8 
7 
7 
6 
5 
4 
3 


Trap Shot 
9-100" 
Trap Shot 
10-100" 
11-100 
12-100 
13-100 
14-100 


472 

399 
338 
291 
218 
168 
132 
106 


No. 2... 
1.. . 
B... 
BB. 
BBB 
T... 
TT.. 
F.. 
FF.. 


15-100" 
16-100 
17-100 
18-100 
19-100 
20-100 
21-100 
22-100 
23-100 


86 
71 
59 
50 
42 
36 
31 
27 
24 



COMPRESSED BUCK-SHOT. 





Diameter. 


No. of Balls 
to the Ib. 




Diameter. 


No. of Balls 
to the Ib. 


No 3 . 


25-100" 


284 


No. 00.... ... 


34-100" 


115 


* 2 


27-100 


232 


" 000 


3-100 


98 


44 1 


30 100 


173 


Balls 


38-100 


85 


" 


32 100 


140 




44-100 


50 















SCREW-THREADS, SELLERS OR U. S. STANDARD. 

In 1864 a committee of the Franklin Institute recommended the adoption 
of the system of screw-threads and bolts which was devised by Mr. William 
Sellers, of Philadelphia. This same system was subsequently adopted as 
the standard by both the Army and Navy Departments of the United States, 
and by the Master Mechanics 1 and Master Car Builders' Associations, so 
that it may now be regarded, and in fact is called, the United States Stand- 
ard. 

The rule given by Mr. Sellers for proportioning the thread is as follows : 
Divide the pitch, or, what is the same thiug, the side of the thread, into 
eight equal parts; take off one part from the top and fill in one part in the 
bottom of the thread; then the flat top and bottom will equal one eighth of 
the pitch, the wearing surface will be three quarters of the pitch, and the 
diameter of screw at bottom of the thread will be expressed by the for 
mula 

1 299 
diameter of bolt - 



er inch' 
For a sharp V thread with angle of 60 the formula is 

1.733 

diameter of bolt -- -=. -. -- = - - r 
no. of threads per inch 

The angle of the thread in the Sellers system is 60. In the Whitworth or 
English system it is 55, and the point and root of the thread are rounded. 
Screw-Threads, United States Standard. 



5 



B 
3 
ft 



a 

Q 



5-16 



11-16 



20 
18 
16 
14 
13 
12 
11 
11 



13-16 
&6 
1 1-16 



2 13-16 
3 



3 5-16 

4 4 



TT. S. OR SELLERS SYSTEM OF SCREW-THREADS. 205 



Screw-Threads, Whltwortli (English) Standard. 






A 


g i 


^ 


j 


J 


- 


f 4 


LI 


,d 





3 




u 





w 





^ 





S 


5 


S 


O 


flH 


o 


S 


S 


K 


P 


S 


*4 


20 


& /8 


11 




8 


194 


5 


3 


3U 


l6 


18 


11-16 


11 


*6 


7 


1?6 


4*6 


3*4 


3*4 


*M6 


16 
14 


13-16 


10 
10 


n 


7 
6 


2 


4 S 


3% 


3^ 


I/ 


12 




9 


^ 


6 


2*6 


4 


4 


3 


9-16 


12 


15-16 


9 


% 


o 


2% 


3*6 







U. S. OR SI^I.I.KKS S VSTK1TI OF SCREW-THREADS. 



BOLTS AND THREADS. 


HEX. NUTS AND HEADS. 




i 

Q 


1 

3* 


CM 0) 


S 

*o 


%& 

.s 


Root of 
,d in Sq. 

s. 


5*1 


S:i 


*A 

5 g 3 


w 

00 ? 

sf 


cc 

ll 


1 


S 


* 
5 

~ 

EH 


P 


,g 

8 

5 


*% 

C60 

25 

^ 


"eg o>J 
c? 

k& 
-< 


ll 
s 


-w S3 
{->"-> 

ofe 


02 


f 


81 

e w 


I 1 


|l 


Ins. 




Ins. 


Ins. 






Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


*4 


20 


.185 


.0062 


.049 


.027 


y 


7-16 


37-64 


54 


3-16 


7-10 


5^16 


18 


.2^0 


.0074 


.077 


.045 


19-32 


17-32 


11-16 


5-16 


*4 


10-12 


96 


16 


.294 


.0078 


.110 


.068 


11-16 


% 


51-64 


% 


5-16 


63-64 


7-16 


14 


.344 


.0089 


.150 


.093 


25-32 


^3-3"- 


9-10 


7-16 


% 


1 7-64 


*6 


13 


.400 


.0096 


.196 


.126 


Vs 


13-16 


1 


V> 


7-16 


1 15-64 


9-16 


12 


.454 


.0104 


.249 


.162 


31-32 


29-32 


1*6 


9-16 


*6 


1 23-64 


% 


11 


.507 


.0113 


.307 


.202 


1 1-16 


1 


1 7-32 


% 


9-16 


]!/> 


M 


10 


.620 


.0125 


.442 


.302 


\y* 


13-16 


1 7-16 


% 


11-16 


1 49-64 


% 


9 


.731 


.0138 


.601 


.420 


1 7-16 


1% 


1 21-32 


% 


13-16 


21-32 


1 


8 


.837 


.0156 


.785 


.550 


!&/ 


1 916 


JT^ 


1 


15-16 


219-64 


1^ 


7 


.940 


.0178 


.994 


.694 


1 13-16 


1% 


2 3-32 


1*6 


1 1-16 


29-16 


IM 


7 


1.065 


.0178 


1 .227 


.893 


2 


1 15-16 


25-16 


1/4 


1 3-16 


253-64 


i^ 


6 


1.160 


.0208 


1.485 


1.057 


23-16 


2*6 


2 17-32 


18 


15-16 


33-32 


ji^j 


6 


1.284 


.0208 


1.767 


1.295 


2% 


25-16 


2M 




1 7-16 


3 23-64 


1% 


5*6 


1.389 


.0227 


2.074 


1.515 


29-16 


2*6 


2 31-32 


J 


19-16 


3% 


m 


5 
5 


1.491 
1.G16 


.0250 
.0250 


2.405 
2.761 


1.746 
2.051 


2% 
3 15-16 


211-16 
2% 


33-16 
313-32 


1 Jo 


1 11-16 
1 13-16 


3 57-64 
45-32 


2 


4*6 


1.712 


.0277 


3.142 


2.302 


3*6 


3 1-16 


3% 


2 


1 15-16 


427-64 


2)4 


4*6 


1.962 


.0277 


3.976 


3.023 




37-16 


41-16 


2*4 


23-16 


461-64 


2J^j 


4 


2.176 


.0312 


4.909 


3.719 


gl 


3 13-16 


4*6 


2U 


2 7-16 


5 31-64 


2M 


4 


2.426 


.0312 


5.940 


4.620 




43-16 


4 29-32 2% 


211-16 


6 


3 


3*6 


2.629 


.0357 


7.069 


5.428 


m 


49-16 


5% |3 


215-16 


6 17-32 


3)4 


3^ 


2.879 


.0357 


8.296 


6.510 


5 


415-16 


5 13-16 3U 


33-16 


7 1-16 


354 


3)4 


3.100 


.0384 


9.621 


7.548 


5% 


55-16 


67-64 


3*6 


37-16 


739 64 


3M 


3 


3.317 


.0413 


11.045 


8.641 




5 11-16 


6 21-32 3% 


3 11-16 


gi,^ 


4 


3 


3.567 


.0413 


12.566 


9.993 


6*1 


61-16 


7 3-32 14 


3 15-16 


8 41-64 


4)4 


2% 


3.798 


.0435 


14.186 


11.329 




67-16 


7 9-li? 4*4 


43-16 


93-16 


4*6 


2% 


4.028 


.0454 


15.904 


12.743 


(3 7^ 


6 13-16 


731-32,4*6 


47-16 


Q3/ 


4M 


2% 


4.256 


.0476 


17.721 


14.226 


7**4 


73-16 


8 13-324% 


411-16 


10*4 


5 


2^ 


4.480 


.0500 


19.635 


15.763 


<% 


79-16 


8 27-32 5 


4 15-16 


10 49-64 


5)4 


2*6 


4.730 


.0500 


21.648 


17.572 


8 


7 15-16 


9 9-32 ! 5*4 


53-16 


11 23-64 


5)^2 


2% 


4.953 


.0526 


23.758 


19.267 


8% 


85-16 


9 23-32 5*6 


5 7-16 


11% 


5%i 


2% 


5.203 


.0526 


25.967 


21.262 


854 


811-16 


105-32 ! 5% 


5 11-16 


12% 


6 




5.423 


.0555 


28.274 


23.098 


9*6 


9 1-16 


10 19-32 6 


515-16 


12 15-16 



LIMIT GAUGES FOR IRON FOR SCREW THREADS. 

In adopting the Sellers, or Franklin Institute, or United States Standard, 
as it is variously called, a difficulty arose from the fact that it is the habit 
of iron manufacturers to make iron over- size, and as there are no over-size 



206 



MATERIALS. 



screws in the Sellers system, if iron is too large it is necessary to cut it away 
with the dies. So great is this difficulty, that the practice of making taps 
and dies over-size has become very general. If the Sellers system is adopted 
it is essential that iron should be obtained of the correct size, or very nearly 
so. Of course no high degree of precision is possible in rolling iron, and 
when exact sizes were demanded, the question arose how much allowable 
variationjthere should be from the true size. It was proposed to make limit- 
gauges for inspecting iron with two openings, one larger and the other 
smaller than the standard size, and then specify that the iron should enter 
the large end and not enter the small one. The following table of dimen- 
sions for the limit-gauges was - commended by the Master Car-Builders' 
Association and adopted by letter ballot in 1883. 





Size of 


Size of 






Size of 


Size of 




Size of 


Large 


Small 


Differ- 


Size of 


Large 


Small 


Differ- 


Iron. 


End of 


End of 


ence. 


Iron. 


End of 


End of 


ence. 




Gauge. 


Gauge. 






Gauge. 


Gauge. 




Hin. 


0.2550 


0.2450 


0.010 


96 in. 


0.6330 


0.6170 


0.016 


5-16 


0.3180 


0.3070 


0.011 




0.7585 


0.7415 


0.017 


% 


0.3810 


0.3690 


0.012 


so 


0.8840 


0.8660 


0.018 


7l?6 


0.4440 


0.4310 


0.013 


\ 


1.0095 


0.9905 


0.019 


^ 


0.5070 


0.4930 


0.014 


i*i 


1.1350 


1.1150 


0.020 


9-16 


0.5700 


0.5550 


0.015 


m 


1.2605 


1.2395 


0.021 



Caliper gauges with the above dimensions, and standard reference gauges 
for testing them, are made by The Pratt & Whitney Co. 

THE MAXIMUM VARIATION IN SIZE OF ROUGH 
IRON FOR U. S. STANDARD BOLTS. 

Am. Mach., May 12, 1892. 

By the adoption of the Sellers or U. S. Standard thread taps and dies keep 
their size much longer in use when flatted in accordance with this system 
than when made sharp " V," though it has been found advisable in practice 
in most cases to make the taps of somewhat larger outside diameter than 
the nominal size, thus carrying the threads further towards the V -shape 
and giving corresponding clearance to *he tops of the threads when in the 
nuts or tapped holes. 

Makers of taps and dies often have calls for taps and dies, U. S. Standard, 
" for rough iron." 

An examination of rough iron will show that much of it is rolled out of 
round to an amount exceeding the limit of variation in size allowed. 

In view of this it may be desirable to know what the extreme variation in 
iron may be, consistent with the maintenance of U. S. Standard threads, i.e., 
threads which are standard when measured upon the angles, the only placo 
where it seems advisable to have them fit closely. Mr. Chas. A. Bauer, the 
general manager of the Warder, Bushnell & Glessner Co., at Springfield, 
Ohio, in 1884 adopted a plan which may be stated as follows : All bolts, 
whether cut from rough or finished stock, are standard size at the bottom 
and at the sides or angles cf the threads, the variation for fit of the nut and 
allowance for wear of taps being made in the machine taps. Nuts are 
punched with holes of such size as to give 85 per cent of a full thread, expe 
rience showing that the metal of wrought nuts will then crowd into the 
threads of the taps sufficiently to give practically a full thread, while if 
punched smaller some of the metal will be cut out by the tap at the bottom 
of the threads, which is of course undesirable. Machine taps are made 
enough larger than the nominal to bring the tops of the threads up sharp, 
plus the amount allowed for fit and wear of taps. This allows the iron to 
be enough above the nominal diameter to bring the threads up full (sharp) 
at top, while if i ia small the only effect is to give a flat at top of threads ; 
neither condition affecting the actual size of the thread at the point at which 
it is intended co bear. Limit gauges are furnished to the mills, by which the 
iron is rolled, the maximum size being shown in the third column of the 
table. The minimum diameter is not given, the tendency in rolling being 
nearly always to exceed the nominal diameter. 

In making the taps the threaded portion is turned to the size given in the 
eighth column of the table, which gives 6 to 7 thousandths of an inch allow- 
ance for fit and wear of tap. Just above the threaded portion of the tap a 



SIZES OF SCKEW-THEEADS FOE BOLTS AND TAPS. 207 



place is turned to the size given in the ninth column, these sizes being the 
same as those of the regular U. S. Standard bolt, at the bottom of the 
thread, plus the amount allowed for fit and wear of tap ; or, in other words, 
d' = U. S. Standard d + (D f D). Gauges like the one in the cut, Fig. 
72, are furnished for this sizing. In finishing the threads of the tap a tool 




FIG. 72. 

is used which has a removable cutter finished accurately to gauge by grind- 
ing, this tool being correct U. S. Standard as to angle, and flat at the point. 
It is fed in and the threads chased until the flat point just touches the por- 
tion of the tap which has been turned to size d'. Care having been taken 
with the form of the tool, with its grinding on the top face (a fixture being 
provided for this to insure its being ground properly), and also with the set- 
ting of the tool properly in the lathe, the result is that the threads of the tap 
are correctly sized without further attention. 

It is evident that one of the points of advantage of the Sellers-system is 
sacrificed, i.e., instead of the taps being flatted at the top of the} threads 
they are sharp, and are consequently not so durable as they otherwise would 
be ; but practically this disadvantage is not found to be serious, and is far 
overbalanced by the greater ease of getting iron within the prescribed 
limits ; while any rough bolt when reduced in size at the top of the threads, 
by filing or otherwise, will fit a hole tapped with the U. S. Standard hand 
taps, thus affording proof that the two kinds of bolts or screws made for the 
two different kinds of work are practically interchangeable. By this system 
\" iron can be .005" smaller or .0108" larger than the nominal diameter, or, 
in other words, it may have a total variation of .0158", while 1" iron can be 
.0105" smaller or .0309" larger than nominal a total variation of .0414" 
and within these limits it is found practicable to procure the iron. 
STANDARD SIZES OF SCREW-THREADS FOR BOLTS 
AND TAPS. 
(CHAS. A. BAUER.) 



1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


A 


n 


D 


d 


h 


/ 


D' -D 


D' 


d' 


H 






Inches. 


Inches 


Inches. 


Inches. 


Inches. 


Inches. 


Inches. 


Inches 


H 


20 


.2G08 


.1855 


.0379 


.0062 


.006 


.2668 


.1915 


.2024 


5-16 


18 


.3245 


.2403 


.0421 


.0070 


.006 


.3305 


.2463 


.2589 


% 


16 


.3885 


.2938 


.0474 


.0078 


.006 


.3945 


.2998 


.3139 


7-16 


14 


.4530 


.3447 


.0541 


.0089 


.006 


.4590 


.3507 


.3670 


M 


13 


.5166 


.4000 


.0582 


.0096 


.006 


.5223 


.4060 


.4236 


9-16 


12 


.5805 


.4543 


.0631 


.0104 


.007 


.5875 


.4613 


.4802 


% 


11 


.6447 


.5069 


.0689 


.0114 


.007 


.6517 


.5139 


.5346 


34 


10 


.7717 


.620! 


.0758 


.0125 


.007 


.7787 


.6271 


.6499 


% 


9 


.8991 


.7307 


.0842 


.0139 


.007 


.9061 


.7377 


.7630 


I 


8 


1.0271 


.8376 


.0947 


.0156 


.007 


1.0341 


.8446 


.8731 


V/B 


7 


1.1559 


.9394 


.1083 


.0179 


.007 


1.1629 


.9464 


.9789 


V/4, 


7 


1.2809 


1.0644 


.1083 


.0179 


.007 


1.2879 


1.0714 


1.1039 



A = nominal diameter of bolt. 
D = actual diameter of bolt. 

d = diameter of bolt at bottom of 

thread. 
n = number of threads per inch. 

/ = flat of bottom of thread. 

h depth of thread. 
D f and d' diameters of tap. 
H = hole in nut before tapping. 




208 



MATERIALS. 



STANDARD SET-SCREWS AND CAP-SCREWS. 

American, Hartford, and Worcester Machine-Screw Companies. 
(Compiled by W. S. Dix.) 





(A) 


(B) 


(C) 


(D) 


(E) 


(F) 


(G) 


Diameter of Screw. . . . 


K 


3-16 


/4 


5-16 


% 


7-16 


7& 


Threads per Inch 
Size of Tap Drill* 


40 
No. 43 


24 

No. 30 


No. 5 


18 
17-64 


16 
21-64 


14 


12 
27-64 




(H) 


(D 


(J) 


(K) 


(L) 


(M) 


(N) 


Diameter of Screw.. . . 


9-16 


Ys 


H 


% 


1 


1^6 


1J4 


Threads per Inch 


12 


11 


10 


9 


8 


7 


7 


Size of Tap Drill*.... 


31-64 


17-32 


21-32 


49-64 


% 


63-64 


*M 



Set Screws. 



Hex. Head Cap-screws. 



Sq. Head Cap-screws. 



Short 
Diam. 
of Head 



(C) < 

(D) 5-16 

$ & 
JM. 

(I) K 

(ft % 

(L) 1 
(M) 1L 
(N) 1M 



Long 
Diam. 
of Head 



.44 

.53 

.62 

.71 

.80 

.89 

1.06 

1.24 

1.42 

1.60 

1.77 



Lengths 
(under 
Head). 



Short 
Diam. 

of 
Head. 




7-16 



Long 
Diam, 

of 
Head. 



.51 

.58 

.65 

.72 

.87 

.94 

1.01 

1.15 

1.30 

1.45 

1.59 

1.73 



Lengths 
(under 
Head). 



Short 
Diam. 

of 
Head. 



to 3 




7-16 
9-16 



11-16 



Loug 
Diam. 

of 
Head. 



.53 
.62 

.71 



1.06 
1.24 
1.60 
1.77 
1.95 
2.13 



Lengths 
(under 
Head). 




Round and Filister Head 
Cap-screws. 



Diam. of 
Head. 



3-16 




Lengths 

(under 

Head). 




Flat Head Cap-screws. 



Button-head Cap- 
screws. 



Diam. of 
Head. 



Lengths 

(including 

Head). 



Diam. of 
Head. 




7-32 (.225) 
5-16 
7-16 
9-16 



13-16 

15-16 

1 



Lengths 
(under 
Head). 




* For cast iron. For numbers of twist-drills see p. 29. 

Threads are U. S. Standard. Cap-screws are threaded % length up to and 
including I" diam. x 4" long, and &j length above. Lengths increase by J4" 
each regular size between the limits given. Lengths of heads, except flat 
and button, equal diam. of screws. 

The angle of the cone of the flat-head screw is 76, the sides making angles 
of 52 with the top. 



STANDARD MACHINE SCREWS. 209 

STANDARD MACHINE SCREWS. 



No. 


Threads per 
Inch. 


Diam. of 
Body. 


Diam. 
of Flat 
Head. 


Diam. of 
Round 
Head. 


Diam. of 
Filister 
Head. 


Lengths. 


From 


To 


2 


56 


.0842 


.1631 


.1544 


.1332 


3-16 


2* 


3 


48 


.0973 


.1894 


.1786 


.1545 


3-16 


K 


4 


32, 36, 40 


,1105 


.2158 


.2028 


.1747 


3-16 


% 


5 


32, 36, 40 


.1236 


.2421 


.2270 


.1985 


3-16 


% 


6 


30, 32 


.1368 


.2684 


.2512 


.2175 


3-16 


1 


7 


30,32 


.1500 


.2947 


.2754 


.2392 


/4 


l/*6 


8 


30, 32 


.1631 


.3210 


.2936 


.2610 


/4 


1/4 


9 


24, 30, 32 


.1763 


.3474 


.3238 


.2805 


/4 


I&2 


10 


24, 30, 32 


.1894 


.3737 


.3480 


.3035 


/4 


ji^j 


12 


20,24 


.2158 


.4263 


.3922 


.3445 


% 


1% 


14 


20, 24 


.2421 


.4790 


.4364 


.3885 


% 


2 


16 


16, 18, 20 


.2684 


.5316 


.4866 


.4300 


% 


2^4 


18 


16, 18 


,2947 


.5842 


.5248 


.4710 


i^j 


2^J 


20 


16, 18 


.3210 


.6308 


.5690 


.5200 


8 


24 


22 


16,18 


.3474 


.6894 


.6106 


.5557 




3 


24 


14, 16 


.3737 


.7420 


.6522 


.6005 


/'is 


3 


26 


14, 16 


.4000 


.7420 


.6938 


.6425 


% 


3 


28 


14, 16 


.4263 


.7946 


.7354 


.6920 


% 


3 


30 


14, 16 


.4520 


.8473 


.7770 


.7240 


1 


3 



Lengths vary by 16ths from 3-16 to J^, by 8ths from ^ to 1J4 by 4ths from 
1^ to 3. 

SIZES AND WEIGHTS OF SQUARE AND 

HEXAGONAL NUTS. 

United States Standard Sizes. Chamfered and trimmed. 
Punched to suit U. S. Standard Taps. 



s 
s 



y 

5-16 
7?16 
9-?6 



2 

2^4 



I 
I 



2 15-16 



13-64 

1*9-64 

11-32 

25-64 

29-64 

33-64 

39-64 

47-64 

53-64 

59-64 

1-16 

5-32 

9-32 



1 13-32 



1 23-32 

1 15-16 

2 3-16 
2 7-16 
2% 



S3' 



11-16J 
13-161 



9-16 
11-16 
13-16 



7-16 

2' '1-16 11-16 
2 5-161 1% 
2 9-16 2 1-16 
2 13-16 2 5-16 



2 15-16 

3 3-16 



4 7-16 
4 15-16 



4 1-16 



4 15-16 

5 5-16 



Square. 


Hexagon. 


8 


fee 


8 


gj 


.sj 


jb 


. C 03 


8jg 




.s 


~f 


i* 


7270 
4700 


.0138 
.0281 


7615 
5200 


.0131 
.0192 


2350 


.0426 


3000 


.0333 


1630 


.0613 


2000 


.050 


1120 


.0893 


1430 


.070 


890 


.1124 


1100 


.091 


640 


.156 


740 


.135 


380 


.263 


450 


.222 


280 


.357 


309 


.324 


170 


.588 


216 


.463 


130 


.769 


148 


.676 


96 


1.04 


111 


.901 


70 


1.43 


85 


1.18 


58 


1.72 


68 


1.47 


44 


2.27 


56 


1.79 


34 


2.94 


40 


2.50 


30 


3.33 


37 


2.70 


23 


4.35 


29 


3.45 


19 


5.26 


21 


4.76 


12 


8.33 


15 


6.67 


9 


11.11 


11 


9.09 


IK 


13.64 


8^ 


11.76 



210 



MATERIALS. 



tt 

4 
- 

R 



H 

C 



o 
o 



vj I OD . . J>. r-< lO O ^J 1 Oi CO O* T-t C 

^ x .Q COTj< 1 i5'lOlOOCDI>-OOO 



rg 



* 



^ cc 
>^ o 



5 10 10 10 K 






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-1 itnOiCOt>-' iCOOrfQCS^l-^^w 



C'COCOCOCO'^ 1 ^9''^ 1 ^T^3'lOiOlOCOCDCOi-t^-i."-aDQOOSOSOSOOO 






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QOOOOO 



^aocooiio^-'J>cooco<?>co > *ocoGOT-iooT 
I ;i;oco;oi.~GcaoosooT-.i-,(?*cocoTfcoi--a 



SQOOOOCOCOCSWOOiOOlOpiqOOOO 



I _o t~ os rt" od o co* o io" oi oo i> I-H CD o c co" -<* ti TJ os ao co o co <M! o oo co' 10' -i< 

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^OiCO^C^OSCDCOOi.-^i-iGOlOC 



TRACK BOLTS. 

With United States Standard Hexagon Nuts, 



Rails used. 


Bolts. 


Nuts. 


No. in Keg, 
200 Ibs. 


Kegs per Mile. 


\ 


%x4^ 


H 


230 


6.3 




%x4 


1? 


240 


6. 


45to851bs.. J 


%x3% 


8 


254 
260 


5.7 
5.5 


1 


% x 3^ 


/4 


266 


5.4 


I 


%x3 


M 


283 


5.1 


r 


%x3^ 


1-16 


375 


4. 


30to401bs.. J 


%x3 


1-16 
1-16 


410 
435 


3.7 
3.3 


1 


^ x2 H 


1 1-16 


465 


3.1 


[ 


1^x3 


% 


715 


2. 


20to301bs..J 


^xl^ 


n 


760 

800 


2. 
2. 


I 


1^x2 


? 


820 


2. 



RIVETS TUIIKBUCKLES. 



CONK-HEAD BOILER RIVETS, WEIGHT PER 1OO. 

(Hoopes & Townsend.) 



Diam., in., 
Scant. 


1/2 


9/16 


5/8 


11/16 


X 


13/16 


% 


1 


i* 


W 


Length. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 


%inch 


8.75 


18.7 


16.20 
















H " 


9.35 


14.4 


17.22 
















i ;' 


10.00 


15.2 


18.25 


21.70 


26.55 














10.70 


16.0 


19.28 


23.10 


28.00 












/4 " 


11.40 


16.8 


20.31 


24.50 


29.45 


37.0 


46 


60 






% " 


12.10 


17.6 


21.34 


25.90 


30.90 


38.6 


48 


63 


95 




L " 


12.80 


18.4 


22.37 


27.30 


32.35 


40.2 


50 


65 


98 


133 


5X " 


13.50 


19.2 


23.40 


28.70 


33.80 


41.9 


52 


67 


101 


137 


M " 


14.20 


20.0 


24.43 


30.10 


35.25 


43.5 


54 


69 


104 


141 


y& " 


14.90 


20.8 


25.46 


31.50 


36.70 


45.2 


56 


71 


107 


145 


2 " 


15.60 


21.6 


26.49 


32.90 


38.15 


47.0 


58 


74 


110 


149 


jji^ ** 


16.30 


22.4 


27.52 


34.30 


39.60 


48.7 


60 


77 


114 


153 


2J4 " 


17.00 


23.2 


28.55 


35.70 


41.05 


50.3 


62 


80 


118 


157 


2% " 


17.70 


24.0 


29.58 


37.10 


42.50 


51.9 


64 


83 


121 


161 


2Vji> " 


18.40 


24.8 


30.61 


38.50 


43.95 


53.5 


66 


86 


124 


165 


2% " 


19.10 


25.6 


31.64 


39.90 


45.40 


55.1 


68 


89 


127 


169 


m " 


19.80 


26.4 


32.67 


41.30 


46.85 


56.8 


70 


92 


130 


173 


2% ' 


20.50 


27.2 


33.70 


42.70 


48.30 


58.4 


72 


95 


133 


177 


3 


21.20 


28.0 


34.73 


44.10 


49.75 


60.0 


74 


98 


137 


181 


3^4 ' 


22.60 


29.7 


36.79 


46.90 


52.65 


63.3 


78 


103 


144 


189 


3Vi* ' 


24.00 


31 5 


38.85 


49.70 


55.55 


66.5 


82 


108 


151 


197 


33 ' 


25.40 


33.3 


40.91 


52.50 


58.45 


69.8 


86 


113 


158 


205 


4 * 


26.80 


35.2 


42.97 


55.30 


61.35 


73.0 


90 


118 


165 


213 


4*4 ' 


28.20 


36.9 


45.03 


58.10 


64.25 


76.3 


94 


124 


172 


221 


4^5 ' 


29.60 


38.6 


47.09 


60.90 


67.15 


79.5 


98 


130 


179 


229 


4^4 ' 


31.00 


40.3 


49.15 


63.70 


70.05 


82.8 


102 


136 


186 


237 


5 


32.40 


42.0 


51.21 


66.50 


72.95 


86.0 


106 


142 


193 


245 


5/4 * 


33.80 


43.7 


53.27 


69.20 


75.85 


89.3 


no 


148 


200 


254 


5^ ' 


35.20 


45.4 


55.33 


72.00 


78.75 


92.5 


114 


154 


206 


263 


5M ' 


36.60 


47.1 


57.39 


74.80 


81.65 


95.7 


118 


160 


212 


272 


6 


38.00 


48.8 


59.45 


77.60 


84.55 


99.0 


122 


166 


218 


281 


6J4 ' 


40.80 


52.0 


63.57 


83.30 


90.35 


105.5 


130 


177 


231 


297 


7 ' 


43.60 


55.2 


67.69 


88.90 


96.15 


112.0 


138 


188 


245 


314 


Heads 


5.50 


8.40 


11.50 


13.20 


18.00 


23.0 


29.0 


38.0 


56.0 


77.5 



* These two sizes are calculated for exact diameter. 

Rivets with button heads weigh approximately the same as cone-head 
rivets. 

T URN BUCK LES. 

(Cleveland City Forge and Iron Co.) 

Standard sizes made with right and left threads. D = outside diameter 




of screw. A = length in clear between heads = 6 ins. for all sizes. B = 
length of tapped heads = l^D nearly. C = 6 ins. + 3D nearly. 



212 



MATERIALS. 



SIZES OF WASHERS. 



Diameter in 
inches. 


Size of Hole, in 
inches. 


Thickness, 
Birmingham 
Wire-gauge. 


Bolt in 
inches. 


No. in 100 Ibs. 


a. 


5-16 


No. 16 


M 


29,300 


a? 


H 


" 16 


516 


18,000 


1 


7-16 


** 14 


% 


7,600 




9-16 


" 11 


14 


3,300 


jijj? 


% 


44 11 


9-16 


2,180 


1^3 


11-16 


** 11 


% 


2,350 . 


1% 


13-16 


* 11 


% 


1,680 


2 


31-32 


* 10 


% 


1,140 


2^ 


1^ 


" 8 


1 


580 


252 


1J4 


* 8 


l/^ 


470 


3 


1% 


tt 7 


jix 


360 


3 


ig 


* 6 


ift 


860 



TRACK SPIKES* 



Rails used. 


Spikes. 


Number in Keg, 
200 Ibs. 


Kegs per Mile, 
Ties 24 in. 
between Centres. 


45 to 85 


5^x9-16 


880 


30 


40 " 52 


5 x9-16 


400 


27 


35 ** 40 


5 xU 


490 


22 


24 " 35 




550 


20 


24 " 30 


4J4 x 7-16 


725 


15 


18 " 24 


4 x7-16 


820 


13 


16 " 20 


8J4x% 


1250 


9 


14 " 16 


3 x % 


1350 


8 


8 " 12 


2^x% 


1550 


7 


8 " 10 


2^x5-16 


2200 


5 



STREET RAILWAY SPIKES. 



Spikes. 


Number in Keg, 200 Ibs. 


Kegs per Mile, Ties 24 in. 
between Centres. 


5^x9-16 
5 x^ 
4J^x7-16 


400 
575 
800 


30 
19 
13 



BOAT SPIKES. 

Number in Keg of 200 Ibs. 



Length. 


H 


5-16 


H 


H 


4 inch. 


2375 








5 " 


2050 


1230 


940 




6 ' 

7 " 


1825 


1175 

990 


800 
650 


450 
375 


8 " 




880 


600 


335 


9 






525 


300 


10 " 






475 


275 













SPIKES; CUT KAILS. 



213 



WROUGHT SPIKES. 

Number of Nails in Keg of 15O Founds. 



Size. 


Min. 


5-16 in. 


fcin. 


7-16 in. 


Kin. 


3 inches . 


2250 










3U " 


1890 


1208 








f" .. 


1650 


1135 








4U * 


1464 


1064 








5^ " 


1380 


930 


742 






6 " 


1292 


868 


570 






7 " .. . 
8 M 


1161 


662 
635 


482 
455 


445 

384 


306 
256 


9 




573 


424 


300 


240 


10 * 






391 


270 


222 


11 " 








249 


203 


IS * 








236 


180 















WIRE SPIKES. 



Size. 


Approx. Size 
of Wire Nails. 


Ap. No. 
in 1 Ib. 


Size. 


Approx. Size 
of Wire Nails. 


Ap. No. 
in 1 Ib. 


lOd Spike.... 


3 in. No. 7 


50 


60d Spike . . . 


6 in. No. 1 


10 


16d " 


3^ " " 6 


35 


6^ in. 44 .. . 


6^ *' ** 1 


9 


20d * 


4 "5 


26 


7 " " . . 


7 


7 


30d 


4^ " 4 


20 


8 " " .. . 


8 " " 00 


5 


40d " 


5 " "3 


15 


9 ** u 


9 " 00 


4 Hi 


50d * 


% " ** 2 


12 









LENGTH AND NUJttRER OF CUT NAILS TO THE 
POUND. 



Size. 


! 


Common. 


1 
'> 


i 

PR 


Finishing. 


| 

s 


Barrel. 


1 


03 

1 




Tobacco. 


Cut Spikes. 


M 


%in 












800 










7/| 














500 










2d 


1 


800 






1100 


1000 


376 










3d... 


WA 


480 






720 


760 


224 










4d 


u& 


288 






523 


368 


180 


398 








5d 


m 


200 






410 










130 




6d 


2 


168 


9^ 


84 


268 






224 


126 


96 




7d 




124 


74 


61 


188 








98 


82 




8d .... 


2V6 


88 


62 


48 


146 






J28 


75 


68 




9d 


9g/ 


70 


*)S 


8fi 


130 






110 


65 






lOd 


3 


58 


46 


30 


102 






91 


55 




28 


)2d. 


3V4 


44 


d 


0^ 


76 






71 


40 






16d....i.. 


l& 


34 


^S 





62 






54 


27 




99 


20d 


f~ 


23 


88 


16 


54 






40 






14i 


SOd 


VA 


18 













33 






12i/ 


40d 


5 


14 












27 






9vl 


50d 


51^ 


10 


















8 


60d 


6 /6 


8 


















6 



























214 



MATERIALS. 



S9ZJS 



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sil^N noraraoo 






"^^ ,_ ,_ ^ ^ TH - 1 _ 4 ciwwwwcocoTf ^o o 



APPROXIMATE NUMBER OF WIRE NAILS PER POUND. 215 



bT>!Ts 
co ^o ;.;;;;;;;;;; .M g 

^rio 
i 

loVooo-* j I 

<Ot>OOOiT-iOOlOOO 

^^Sc^S^o 

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x^f oc^^fooott-cOTHTt<' rfi?o(7't^JOr:oc^' 

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^ 

^ j i j i : : j : : i i : i : Jill 

I- 
g w 



2ib MATERIALS. 

SIZE, WEIGHT, LENGTH, AND STRENGTH OF IRON 
WIRE. 

(Trenton Iron Co.) 













Tensile Strength (Ap. 


No. by 
Wire 
Gauge. 


Diam. 
in Deci- 
mals of 
One 


Area of 
Section in 
Decimals of 
One Inch. 


Feet to 
the 
Pound. 


Weight of 
One Mile 
in pounds. 


proximate) of Charcoal 
Iron Wire in Pounds. 








Inch. 








Bright. 


Annealed. 


00000 


.450 


.15904 


1.863 


2833.248 


12598 


9449 


0000 


.400 


.12566 


2.358 


2238.878 


9955 


7466 


000 


.360 


.10179 


2.911 


1813.574 


8124 


6091 


00 


.330 


.08553 


3.465 


1523.861 


6880 


5160 





.305 


.07306 


4.057 


1301.678 


5926 


4445 


1 


.285 


.06379 


4.645 


1136.678 


5226 


3920 


2 


.265 


.05515 


5.374 


982 555 


- 4570 


3425 


3 


.245 


.04714 


6.286 


839.942 


3948 


2960 


4 


.225 


.03976 


7.454 


708.365 


3374 


2530 


5 


.205 


.03301 


8.976 


588.139 


2839 


2130 


6 


.190 


.02835 


10.453 


505.084 


2476 


1860 


7 


.175 


.02405 


12.322 


428.472 


2136 


1600 


8 


.160 


.02011 


14.736 


358.3008 


1813 


1360 


9 


.145 


.01651 


17.950 


294.1488 


1507 


1130 


10 


.130 


.01327 


22.333 


236.4384 


1233 


925 


11 


.1175 


.01084 


27.340 


193.1424 


1010 


758 


12 


.105 


.00866 


34.219 


154.2816 


810 


607 


13 


.0925 


.00672 


44.092 


119.7504 


631 


473 


14 


.080 


.00503 


58.916 


89.6016 


474 


356 


15 


.070 


.00385 


76.984 


68.5872 


372 


280 


16 


.061 


.00292 


101.488 


52.0080 


292 


220 


17 


.0525 


.00216 


137.174 


38.4912 


222 


165 


18 


.045 


.00159 


186.335 


28.3378 


169 


127 


19 


.040 


.0012566 


235.084 


22.3872 


137 


103 


20 


.035 


.0009621 


308.079 


17.1389 


107 


80 


21 


.031 


.0007547 


392.772 


13.4429 






22 


.028 


.0006157 


481.234 


10.9718 


' "22 2~ 


23 


.025 


.000*909 


603.863 


8.7437 


C?2iO <U~ ~ ~ 


24 


.0225 


.0003976 


745.710 


7.0805 


41 2*- 4 >a ^ 


25 


.020 


.0003142 


943.396 


5.5968 


5" i32S** ** 


26 


.018 


.0002545 


1164.689 


4.5334 


^J'S | ! d 


27 


.017 


.0002270 


1305.670 


4.0439 




28 


.016 


.0002011 


1476.869 


3.5819 


T3 ?J D-^ ol O * 2 


29 


.015 


.0001767 


1676.989 


3.1485 


43 S^'> -2''* tw fl 


30 


.014 


.0001539 


1925.321 


2.7424 


s S^li ^.s"*!^ ** 


31 


.013 


.0001327 


2232.653 


2.3649 


w -g SH .2S J^le o> ^ 


32 


.012 


.0001131 


2620.607 


2.0148 


2-2 ujS ^'j^^to O 


33 


.011 


.0000950 


3119.092 


1.6928 


id'~fi.h^ : ~~ M 


34 
35 


.010 
.0095 


.00007854 
.00007088 


3773.584 
4182.508 


1.3992 
1.2624 


1 


36 


.009 


.00006362 


4657.728 


1.1336 


O-^-rt ^^ t >o 


37 


.0085 


.00005675 


5222.035 


1.0111 


* w'ejf! '"SW jf3 j| 


88 


.008 


.00005027 


5896.147 


.89549 


d>.Q E_S < o'!5 ; S*^ *" 


39 


.0075 


.00004418 


6724.291 


.78672 




40 


.007 


00003848 


7698.253 


.68587 





TESTS OF 'TELEGRAPH WIRE. 



217 



GALVANIZED IRON WIRE FOR TELEGRAPH AND 
TELEPHONE LINES. 

(Trenton Iron Co.) 

WEIGHT PER MILE-OHM. This term is to be understood as distinguishing 
the resistance of material only, and means the weight of such material re- 
quired per mile to give the resistance of one ohm. To ascertain the mileage 
resistance of any wire, divide the " weight per mile-ohm " by the weight of 
the wire per mile. Thus in a grade of Extra Best Best, of which the weight 
per mile-ohm is 5000, the mileage resistance of No. 6 (weight per mile 525 
Ibs.) would be about 9J^ ohms; and No. 14 steel wire, 6500 ibs. weight per 
mile-ohm (95 Ibs. weight per mile), would show about 69 ohms. 

Sizes of \Virc used in Telegraph and Telephone Lines. 

No. 4. Has not been much used until recently; is now used on important 
lines where the multiplex systems are applied. 

No. 5. Little used in the United States. 

No. 6. Used for important circuits between cities. 

No. 8. Medium size for circuits of 400 miles or less. 

No. 9. For similar locations to No. 8, but on somewhat shorter circuits ; 
until lately was the size most largely used in this country. 

Nos. 10, 11. For shorter circuits, railway telegraphs, private lines, police 
and fire-alarm lines, etc. 

No. 12. For telephone lines, police and fire-alarm lines, etc. 

Nos. 13, 14. For telephone lines and short private lines: steel wire is used 
most generally in these sizes. 

The coating of telegraph wire with zinc as a protection against oxidation 
is now generally admitted to be the most efficacious method. 

The grades of line wire are generally known to the trade as " Extra Best 
Best " (E. B. B.), " Best Best " (B. B.). and "Steel." 

" Extra Best Best " is made of the very best iron, as nearly pure as any 
commercial iron, soft, tough, uniform, and of very high conductivity, its 
weight per mile-ohm being about 5000 Ibs. 

The " Best Best 11 is of iron, showing in mechanical tests almost as good 
results as the E. B. B., but not quite as soft, and being somewhat lower in 
conductivity; weight per mile-ohm about 5700 Ibs. 

The Trenton " Steel " wire is well suited for telephone or short telegraph 
lines, and the weight per mile-ohm is about 6500 Ibs. 

The following are (approximately) the weights per mile of various sizes of 
galvanized telegraph wire, drawn by Trenton Iron Co.'s gauge: 

No. 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. 
Lbs. 720, 610, 525, 450, 375, 310, 250, 200, 160, 125, 95. 

TESTS OF TELEGRAPH WIRE. 

The following data are taken from a table given by Mr. Prescott relating 
to tests of E. B. B. galvanized wire furnished the Western Union Telegraph 
Co.: 



Size 
of 
Wire. 


Diam. 
Parts of 
One 
Inch. 


Weight. 


Length. 
Feet 
per 
pound. 


Resistance. 
Temp. 75.8 Fahr. 


Ratio of 
Breaking 
Weight to 
Weight 
per mile. 


Grains, 
per foot. 


Pounds 
per mile. 


Feet 
per ohm. 


Ohms 
per mile. 


4 


.238 


1043.2 


886.6 


6.00 


958 


5.51 




5 


.220 


891.3 


673.0 


7.85 


727 


7.26 




6 


.203 


758.9 


572.2 


9.20 


618 


8.54 


3.05 


7 


.180 


596.7 


449.9 


11.70 


578 


10.86 


3.40 


8 


.165 


501.4 


378.1 


14.00 


409 


12.92 


3.07 


9 


.148 


403.4 


304.2 


17.4 


328 


16.10 


3.38 


10 


.134 


330.7 


249.4 


21.2 


269 


19.60 


3.37 


11 


.120 


265.2 


200.0 


26.4 


216 


24.42 


2.97 


12 


.109 


218.8 


165.0 


32.0 


179 


29.60 


3.43 


14 


.083 


126.9 


95.7 


55.2 


104 


51.00 


3.05 



JOINTS IN TELEGRAPH WIRES. The fewer the joints in a line the better. 
All joints should be carefully made and well soldered over, for a bad joint 
may cause as much resistance to the electric current as several miles of 
wire. 



218 



MATERIALS. 



ooi-ie< 



- i T-i 

-i eo IH T- >H U 



O O5 -* -^ 

>H o> m 



Ill 

15" 



11 



H^SSc5<N 



DIMENSIONS, WEIGHT, RESISTANCE OF OOPPEB WIRE. #19 



Op. 



& 



A 

o 

5 



s s 






oo 









320 



MATERIALS. 






^t-^ 



liil 



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jJTH^oiinNoJt-lto^sceooii-tiHrHO 



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HARD-DRAWK COPPER WIRE; INSULATED WIRE. 221 



HARD-DRAWN COPPER TELEGRAPH WIRE. 

(J. A. Roebling's Sons Co.) 
Furnished in half-mile coils, either bare or insulated. 



Size, B. & S. 
Gauge. 


Resistance in 
Ohms 
per Mile. 


Breaking . 
Strength. 


Weight 
per Mile. 


Approximate 
Size of E. B.B. 
Iron Wire 
equal to 










Copper. 


9 


4.30 


625 


209 


2 t? 


10 


5.40 


525 


166 


3 I 


11 


6.90 


420 


131 


4 I 


12 


8.70 


330 


104 


6 | 


13 


10.90 


270 


83 


6^3 


14 


13.70 


213 


66 


8 


15 


17.40 


170 


52 


9 


16 


22.10 


130 


41 


10 <g 

CD 



In handling this wire the greatest care should be observed to avoid kinks, 
bends, scratches, or cuts. Joints should be made only with Mclntire Con- 
nectors. 

On account of its conductivity being about five times that of Ex. B. B. 
Iron Wire, and its breaking strength over three times its weight per mile, 
copper maybe used of which the section is smaller and the weight less than 
an equivale&t iron wire, allowing a greater number of wires to be strung on 
the poles. 

Besides this advantage, the reduction of section materially decreases the 
electrostatic capacity, while its non-magnetic character lessens the self-in- 
duction of the line, both of which features tend to increase the possible 
speed of signalling in telegraphing, and to give greater clearness of enunci- 
ation over telephone lines, especially those of great length. 

INSULATED COPPER WIRE, WEATHERPROOF 
INSULATION. 





Double Braid. 


Triple Braid. 


Approximate 


Num- 










Weights, 


bers, 


Outside 


Weights, 


Outside 


Weights, 


Pounds. 


B. &S. 


Diame- 


Pounds. 


Diame- 


Pounds. 




Gauge. 


ters in 




ters in 








32ds 
Inch. 


1000 
Feet. 


Mile. 


Inch. 


1000 
Feet. 


Mile. 


Reel. 


Coil. 


0000 


20 


716 


3781 


24 


775 


4092 


2000 


250 


000 


18 


575 


3036 


22 


630 


3326 


2000 


250 


00 


17 


465 


2455 


18 


490 


2587 


500 


250 





15 


375 


1980 


17 


400 


2112 


500 


250 


1 


15 


285 


1505 


16 


306 


1616 


500 


250 


2 


14 


245 


1294 


15 


268 


1415 


500 


250 


I 


13 


190 


1003 


14 


210 


1109 


500 


250 




11 


152 


803 


12 


164 


866 


250 


125 


5 


10 


120 


G34 


11 


145 


766 


260 


130 


6 


9 


98 


518 


10 


112 


691 


275 


140 


8 


8 


66 


349 


9 


78 


412 


200 


100 


10 


7 


45 


238 


8 


55 


290 


200 


100 


12 


6 


30 


158 


7 


35 


185 


.... 


25 


14 


5 


20 


106 


6 


26 


137 




25 


16 


4 


14 


74 


5 


20 


106 


.... 


25 


18 


3 


10 


53 


4 


16 




.... 


25 



MATERIALS, 



Power Cables. Lead Incased, Jute or Paper Insulated* 

(John A. Roebling's Sons Co.) 



Nos,. 
B.&S.G. 


Circular 
Mils. 


Outside 
Diam. 
Inches. 


Weights, 
1000 feet. 
Pounds. 


Nos., 
B.&S.G. 


Circular 
Mils. 


Outside 
Diam. 
Inches. 


Weights, 
1000 feet. 
Pounds. 


1000000 
900000 
800000 
750000 
700000 
650000 
600000 
550000 
500000 
450000 
400000 
350000 


1 13/16 
1 23/32 
21/32 

19/32 
9/16 
17/32 

1/ie 

V 

5/16 


6685 
6228 
5773 
5543 
5315 
5088 
4857 
4630 
4278 
3923 
3619 
3416 




300000 
250000 
211600 
168100 
133225 
105625 
83521 
66564 
52441 
41616 
26244 


1M 

1 3/16 
1 3/32 
1 1/16 

15/16 
29/32 
% 
25/32 
% 
11/16 


3060 
2732 
2533 
2300 
2021 
1772 
1633 
1482 
1360 
1251 
1046 





0000 " ' 
000 
00 


1 

2 

3 
4 
6 






















Stranded Weather-proof Feed Wire. 



Circular 
Mils. 

1000000 
900000 
800000 
750000 
700000 
650000 
600000 


Outside 
Diam. 
Inches. 


Weights. 
Pounds. 


Approximate 
length on reels. 
Feet. 


Circular 
Mils. 


Outside 
Diam. 
Inches. 


Weights. 
Pounds. 


Approximate 1 
length on reels. 
Feet. 


1000 
feet. 


Mile. 


1000 
feet. 


Mile. 


1 13/32 
1 11/32 
1 5/16 
1 9/32 

H 

1 7/32 


3550 
3215 

2880 
2713 
2545 
2378 
2210 


18744 
16975 
15206 
14325 
13438 
12556 
11668 


800 

800 
850 
850 
900 
900 
1000 


550000 
500000 
450000 
400000 
350000 
300000 
250000 


3/16 

X 

3/32 

1/16 

15/16 
29/32 


2043 
1875 
1703 
1530 
1358 
1185 
1012 


10787 
9900 
8992 
8078 
7170 
6257 
5343 


1200 
1320 
1400 
1450 
1500 
1600 
1600 



The table is calculated for concentric strands. Rope-laid strands are 
larger. 
Approximate Rules for the Resistance of Copper Wire. 

The resistance of any copper wire at 20 C. or 66 F., according to Mat: 

thiessen's standard, is E = ^p in which E is the resistance in inter- 
national ohms, I the length of the wire in feet, and d its diameter in mils. 
(1 mil = 1/1000 inch.) 

A No. 10 Wire, A.W.G., .1019 in. diameter (practically 0.1 in.), 1000 ft. in 
length, has a resistance of 1 ohm at 68 F. and weighs 31.4 Ibs. 

If a wire of a given length and size by the American or Brown & Sharpc 
gauge has a certain resistance, a wire of the same length and three numbers 
higher has twice the resistance, six numbers higher four times the resist- 
ance, etc. 

Wire gauge, A.W.G. No 000 1 4 7 10 13 16 19 22 

Relative resistance 16 8 4 2 11/2 1/4 1/8 1/16 

section or weight.. 1/16 1/8 1/4 1/212 4 8 16 

Approximate rules for resistance at any temperature : 



v f : 



R = resistance at 0, R f = resistance at the temperature t C., I - 
in feet, d - diameter iii c mils. (See Copper Wire Table, p. 1034.) 



length 



STEEL WIRE CABLES. 



223 



GALVANIZED STEEL-WIRE STRAND. 

For Smokestack Guys, Signal Strand, etc. 

(J. A. Roebling's Sons Co.) 
This strand is composed of 7 wires, twisted together into a single strand. 





L 
4>.J 


br.~ 




LI 

S^" 


dg 5 ^ 




h 


eg'd 





^3? 


-S^bJC 


a> 




-2*2 "Ss 


qj 


^S 




I 

.1 


"So 

ss 


HI 


a> 
| 


5?o 


|ll 


(0 

I 


|| 


III 


p 


^ 


1 


O 




1 


, 3 


^ 


W 


in. 


Ibs. 


Ibs. 


in. 


Ibs. 


Ibs. 


in. 


Ibs. 


Ibs. 


\$ 


51 


8,320 


9/32 


18 


2,600 


5/32 


4J^ 


700 


15/32 


48 


7,500 


17/64 


15 


2,250 


9/64 


3Vi* 


525 


7/16 


37 
30 


6,000 
4,700 


7/32 


\\V 2 


1,750 
1,300 


3/32 


2^4 
2 


375 
320 


5^16 


21 


3,300 


3/16 


*y* 


1,000 









For special purposes these strands can be made of 50 to 100 per cent 
greater tensile strength. When used to run over sheaves or pulleys the use 
of soft-iron stock is advisable. 

FLEXIBLE STEEL-WIRE CABLES FOR VESSELS. 
(Trenton Iron Co., 1886.) 

With numerous disadvantages, the system of working ships' anchors with 
chain cables is still in vogue. A heavy chain cable contributes to the hold- 
ing-power of the anchor, and the facility of increasing that resistance by 
paying out the cable is prized as an advantage. The requisite holding- 
power is 'obtained, however, by the combined action of a comparatively 
light anchor and a corresponding^ 7 great mass of chain of little service in 
proportion to its weight or to the weight of the anchor. If the weight and 
size of the anchor were increased so as to give the greatest holding-power 
required, and it were attached by means of a light wire cable, the combined 
weight of the cable and anchor would be much less than the total weight of 
the chain and anchor, and the facility of handling would be much greater. 
English shipbuilders have taken the initiative in this direction, and many of 
the largest and most serviceable vessels afloat are fitted with steel- wire 
cables. They have given complete satisfaction. 

The Trenton Iron Co/s cables are made of crucible cast-steel wire, and 
guaranteed to fulfil Lloyd's requirements. They are composed of 72 wires 
subdivided into six strands of twelve wires each. In order to obtain great 
flexibility, hempen centres are introduced in the strands as well as in the 
completed cable. 

FLEXIBLE STEEL-WIRE HAWSERS. 

These hawsers are extensively used, They are made with six strands of 
twelve wires each, hemp centres being inserted in the individual strands as 
well as in the completed rope. The material employed is crucible cast steel, 
galvanized, and guaranteed to fulfil Lloyd's requirements. They are only 
one third the weight of hempen hawsers; and are sufficiently pliable to work 
round any bitts to which hempen rope of equivalent strength can be applied. 

13-inch tarred Russian hemp hawser weighs about 39 Ibs. per fathom. 

10-inch white manila hawser weighs about 20 Ibs. per fathom. 

1^-inch stud chain weighs about (58 Ibs. per fathom. 

4-inch galvanized steel hawser weighs about 12 /6s. per fathom. 

Each of the above named has about the same tensile strength. 



224 



MATERIALS. 



SPECIFICATIONS FOR GALVANIZED IRON WIRE. 

Issued by the British. Postal Telegraph Authorities. 



Weight per Mile. 


Diameter. 


Tests for Strength and 
Ductility. 


sk* 
























&* * * 


s 












* 


o 


-2 


g 


V) 


oS^lau 


Sg 


T3 








t^ 


^9 


a 




fl 


.2 


rti 5 


C 8 


-ed Standar 


Allowed. 


d Standard 


Allowed. 


Breakir 
Weigh 


d 


ing Weight 
ss than 


d ~ 


:ing Weight 
ss than 


0* 
ft 


Resistance 
of the S 
Size at 6 


t, being Sta 
t x Resista 


'o* 


g 


s 


i 


1 


| 


1 


5 


| 


8 




3 


3 


| 


& 


S 


a 


i 


I 


S 




1 


M 


i 


M 


a 


I 


g^ 




i 


1 




1 


M 


| 


| 


o 


"3 


o 


I 


N 

1 


O 


Ibs. 


Ibs. 


Ibs. 


mils. 


mils. 


mils. 


Ibs. 




Ibs. 




Ibs. 




ohms. 




800 


767 


833 


242 


237 


247 


2480 


15 


2550 


14 


2620 


13 


6.75 


5400 


600 


571 


629 


209 


204 


214 


1860 


17 


1910 


16 


1960 


15 


9.00 


5400 


450 


424 


477 


181 


176 


186 


1390 


19 


1425 


18 


1460 


17 


12.00 


5400 


400 


377 


424 


171 


166 


176 


1240 


21 


1270 


20 


1300 


19 


13.50 


5400 


200 


190 


213 


121 


118 


125 


620 


30 


638 


28 


655 


26 


27.00 


5400 



STRENGTH OF PIANO-WIRE. 

The average strength of English piano- wire is given as follows by Web 
ster, Horsfals & Lean: 



Numbers 


Equivalents 


Ultimate 


Numbers 


Equivalents 


Ultimate. 


in Music- 


in Fractions 


Tensile 


in Music- 


in Fractions 


Tensile 


wire 
Gauge. 


of Inches in 
Diameters. 


Strength in 
Pounds. 


wire 
Gauge. 


of inches in 
Diameters. 


Strength in 
Pounds. 


12 


.029 


225 


18 


.041 


395 


13 


.031 


250 


19 


.043 


425 


14 


.033 


285 


20 


.045 


500 


15 


.035 


305 


21 


.047 


540 


16 


.037 


340 


22 


.052 


650 


17 


.039 


360 


' 







These strengths range from 300,000 to 340,000 Ibs. per sq. in. The compo- 
sition of this wire is as follows: Carbon, 0.570; silicon, 0.090; sulphur, C Oil; 
phosphorus, 0.018; manganese, 0.425. 

" PLOUGH "-STEEI, 1IRE. 

The term "plough," given in England to steel wire of high quality, was 
derived from the fact that such wire is used for the construction of ropes 
used for ploughing purposes. It is to be hoped that the term will not be 
Tised in this country, as it tends to confusion of terms. Plough-steel is 
known here in some steel- works as the quality of plate steel used for the 
mould-boards of ploughs, for which a very ordinary grade is good enough. 

Experiments by Dr. Percy on the English plough-steel (so-called) gave the 
following results: Specific gravity, 7.814; carbon, 0.828 per cent; manga- 
nese, 0.587 per cent; silicon, 0.143 per cent; sulphur, 0.009 percent; phos- 
phorus, nil; copper, 0.030 per cent. No traces of chromium, titanium, or 
tungsten were found. The breaking strains of the wire were as follows: 

Diameter, inch 093 .132 .159 .191 

Pounds per sq. inch 344,960 257,600 224,000 201,600 

The elongation was only from 0.75 to 1.1 per cent. 



SPECIFICATIONS FOR HARD-DRAWH COPPER WIRE. 225 



WIRES OF DIFFERENT METALS AND ALLOYS. 

(J. Bucknall Smith's Treatise oil Wire.) 

Brass "Wire is commonly composed of an alloy of 1 3/4 to 2 parts of 
copper to 1 part of zinc. The tensile strength ranges from 20 to 40 tons per 
square inch, increasing with the percentage of zinc in the alloy. 

German or Nickel Silver, an alloy of copper, zinc, and nickel, ia 
practically brass whitened by the addition of nickel. It has been drawn into 
-wire as fine as .002" diam. 

Platinum wire may be drawn into the finest sizes. On account of its 
high price its use is practically confined to special scientific instruments and 
electrical appliances in which resistances to high temperature, oxygen, and 
acids are essential. It expands less than other metals when heated, which 
property permits its being- sealed in glass without fear of cracking. It is 
therefore used in incandescent electric lamps. 

Phosphor-bronze Wire contains from 2 to 6 per cent of tin and 
from 1/20 to 1/8 per cent of phosphorus. The presence of phosphorus is 
detrimental to electric conductivity. 

" Delta-metal " wire is made from an alloy of copper, iron, and zinc. 
Its strength ranges from 45 to 62 tons per square inch. It is used for some 
Mnds of wire rope, also for wire gauze. It is not subject to deposits of ver- 
digris. It has great toughness, even when its tensile strength is over 80 
tons per square inch. 

Aluminum Wire. Specific gravity .268. Tensile strength only 
about 10 tons per square inch. It has been drawn as fine as 11,400 yards to 
the ounce, or .042 grains per yard, 

Aluminum Bronze, 90 copper, 10 aluminum, has high strength and 
ductility; is inoxidizable, sonorous. Its electric conductivity is 12.6 percent. 

Silicon Bronze, patented in 1882 by L. Weiler of Paris, is made as 
follows : Fluosilicate or potash, pounded glass, chloride of sodium and cal- 
cium, carbonate of soda and lime, are heated in a plumbago crucible, and 
after the reaction takes place the contents are thrown into the molten 
bronze to be treated. Silicon-bronze wire has a conductivity of from 40 to 
98 per cent of that of copper wire and four times more than that of iron, 
while its tensile strength is nearly that of steel, or 28 to 55 tons per square 
inch of section. The conductivity decreases as the tensile strength in- 
creases. Wire whose conductivity equals 95 per cent of that of pure copper 
gives a tensile strength of 28 tons per square inch, but when its conductivity 
is 34 per cent of pure copper, its strength is 50 tons per square inch. It is 
being largely used for telegraph wires. It has great resistance to oxidation. 

Ordinary Drawn and Annealed Copper Wire has a strength 
of from 15 to 20 tons per square inch, 

SPECIFICATIONS FOR HARD-DRAWN COPPER 
WIRE. 

The British Post Office authorities require that hard-drawn copper wire 
supplied to them shall be of the lengths, sizes, weights, strengths, and con- 
ductivities as set forth in the annexed table. 



Weight per Statute 
Mile. 


Approximate Equiva- 
lent Diameter. 


! 


*l 


ill 


43 - 






wl 

a 

1^ 


gs 

gCO 

I.S 

G CQ 


:imum Re 
ice per Mi 
ire (when 
60 Fahr. 


5 8 g 

J> ij.H 

^E 

ill 


11 

3.3 

cr 1 fl 


a 

a 


I 

1 


1 
I 


a 
1 


| 

| 


I| 


a 

i 


1 

m 


a 

OQ 


& 
S 




% 


d 

a 


1 


3 ti& 08 
rt 


p* 


Ibs. 


Ibs. 


Ibs. 


mils. 


mils. 


mils. 


Ibs. 




ohms. 


Ibs. 


100 


97^ 


10^^> 


79 


78 


80 


330 


30 


9.10 


50 


150 


1461^ 


153M 


97 


95^ 


98 


490 


25 


6.05 


50 


200 


195 


205 


112 


110^ 


113J4 


650 


20 


4.53 


50 


400 


390 


410 


158 


155X8 


160^ 


1300 


10 


2.27 


50 



226 



, MATERIALS. 



WIRE ROPES. 

List adopted by manufacturers in 1892. See pamphlets of John A. 
Roebling's Sons Co., Trenton Iron Co., and other makers, 

Pliable Hoisting Rope, 

With 6 strands of 19 wires each. 
IRON. 



Trade Number. 


q 


Circumference in 
inched 


Weight per foot in 
pounds. Rope 
with Hemp Cen- 
tre. 


Breaking Strain, 
tons of 2000 Ibs. 


Proper Working 
Load in tons of 
2000 Ibs. 


Circumference of 
new Manila 
Rope of equal 
Strength. 


Min. Size of Drum 
[ or Sheave in- feet.j 


1 


SM 


6% 


8.00 


74 


15 


14 


13 


2 


2 


6 


6.30 


65 


13 


13 


12 


3 


1% 


5^ 


5.25 


54 


11 


12 


10 


4 


1% 


5 


4.10 


44 


9 


11 


8J4 


5 


Jl^ 


4M 


8.65 


39 


8 


10 




5^ 


1% 


4% 


3.00 


33 


6*4 




7 


6 


1/4 


4 


2.50 


27 


5^ 


8*2 


6^ 


7 


i/ij 


3^ 


2.00 


20 


4 


7/^ 


6 


8 


l 




1.58 


16 


8 


6^ 


5M 


9 




2M 


1.20 


11.50 


2^3 


5V<2 


XIX 


10 


'M 


2!4 


0.88 


8.64 


m 


494 


4 


10M 


% 


2 


0.60 


5.13 




m 




lOJ'i 


9-16 


1^ 


0.48 


4.27 


% 


v% 


25^ 


10% 


H 


]i,^ 


0.39 


3.48 


H 


3 


2M 


10a 


7-16 


3% 


0.29 


3.00 


n 


2% 


2 


10% 


% 


1J4 


0.23 


2.50 


* 


2^ 


1& 



CAST STEEL. 



1 


2J4 


6M 


8.00 


155 


31 




8U 


2 


2 


6 


6.30 


125 


25 




8 


3 




5V 


5.25 


106 


21 




714 


4 


^1 


5^ 


4.10 


86 


17 


15 




5 


/^3 


4% 


3.65 


77 


15 


14 


5% 


5^ 


% 


4% 


3.00 


63 


12 


13 


5^ 


6 


/4 


4 


2.50 


52 


10 


12 


5 


7 


1 


3^ 


2.00 


42 


8 


11 


4^ 


8 




3^/ 


1.58 


33 


6 


9^ 


4 


9 


% 


2% 


1.20 


25 


5 


w* 


3^ 


10 


% 


2*4 


0.88 


18 


3^ 


7 


3 


10)4 


% 


2 


0.60 


12 


gi^j 


5^ 


2H 


1014 


9-16 


1% 


0.48 


9 


1M 


5 


1% 


10% 


^ 


1^ 


0.39 


7 


ip 


4^j 


\\ 


10a 


7-16 


^% 


0.29 


5^ 




3% 


j^4 


10% 


% 


1M 


0.23 


4H 


% 





1 



Cable-Traction Ropes. 

According to English practice, cable-traction ropes, of about 3^ in. in 
circumference, are commonly constructed with six strands of seven or fif- 
teen wires, the lays in the strands varying from, say. 3 in. to 3^ in., and the 
lays in the ropes from, say, 7^ in. to 9 in. In the United States, however, 
strands of nineteen wires are generally preferred, as being more flexible; 
but, on the other hand, the smaller external wires wear out more rapidly. 
The Market-street Street Railway Company, San Francisco, has used ropes 
1J4 in. in diameter, composed of six strands of nineteen steel wires, weighing 
2^ Ibs. per foot, the longest continuous length being 24,125 ft. The Chicago 
City Railroad Company has employed cables of identical construction, the 
longest length being 27,700 ft. On the New York and Brooklyn Bridge cable- 
railway steel ropes of 11,500 ft.* long, containing 114 wires, have been used. 



WIRE ROPES. 



227 



Transmission and Standing Rope. 

With 6 strands of 7 wires each. 
IRON. 









.2 

Q.S 


a 




O 


11 


i 




i 


^O 

0*0 S 


^8 


11 


s 1 

ll d 


^.s 

O 1) 


fc 




u 

-M 
<t> 


M 

a 


jfl 


bJC^ 

1! 


l* 




II 

cc,a 


1 


s 


1 


fill 


M 
w* 3 


|ll 


Jill 


^ 

i 


11 


Lj 


4% 


3.37 


36 


9 


10 


13 


12 
13 


ft 


4% 
4 


8.77 
2.28 


30 

25 


pi 


9 


12 


14 


/^2 


31^ 


1.82 


20 


5 


71^2 


9^ 


15 




31^ 


1.50 


16 


4 


6J^ 


8^ 


16 


% 


2% 


1.12 


12.3 


3 


5% 


7^ 


17 

18 


11^16 


^ 


0.92 
0.70 


8.8 
7.6 


1^ 


Si 


6 4 


19 


% 


2 


0.57 


5.8 


JL^ 


4 


5^4 


20 


9-16 


Ja^ 


0.41 


4.1 


1 






21 


Ji 


J1Z 


0.31 


2.83 




2% 


4 


22 


7-16 


1% 


0.23 


2.13 


Hi 


31^ 


3^4 


23 


8^ 


1 V<< 


0.21 


1.65 




O1/J 


03/f 


24 


5-16 


1 


0.16 


1.38 




2 


2V^ 


25 


9-32 




0.125 


1.03 




1% 


2V4 



















CAST STEEL. 



11 


1^ 


m 


3.37 


62 


13 


13 


8^ 


12 


1% 


4% 


2.77 


52 


10 


12 


8 


13 


1^4 


4 


2.28 


44 


9 


11 




14 




31^ 


1.82 


36 


7L 


10 


6/4 


15 


1 


31^ 


1.50 


30 


6 


9 


5% 


16 


% 


2-M 


1.12 


22 


4/^ 


8 


5 


17 

18 


! 


2^ 


0.92 
0.70 


17 
14 


3 2 


7 
6 




19 


% 


2 8 


0.57 


11 


2*4 




3V 


20 


9-16 


1% 


0.41 


8 


1% 


4^4 


3 


21 


i^ 


JL/ 


0.31 


6 


ji^ 


4 




22 


7-16 


1% 


0.23 


4^ 


1/4 


31^ 


2^j 


23 




1^4 


0.21 


4 


1 


3/>4 


2 /V 


24 


5-16 


j 


0.16 


3 


^ 


2M 


1^< 


25 


9-32 


% 


0.12 


2 


*! 


2^ 


3i 



Plough-Steel Rope. 

Wire ropes of very high tensile strength, which are ordinarily called 
"Plough-steel Ropes," are made of a high grade of crucible steel, which, 
when put in the form of wire, will bear a strain of from 100 to 150 tons per 
square inch. 

Where it is necessary to use very long or very heavy ropes, a reduction of 
the dead weight of ropes becomes a matter of serious consideration. 

It is advisable to reduce all bends to a minimum, and to use somewhat 
larger drums or sheaves than are suitable for an ordinary crucible rope hav 
ing a strength of 60 to 80 tons per square inch. Before using Plough-stee 
Ropes it is best to have advice on the subject of adaptability. 



228 



MATERIALS. 



Plough-Steel Rope. 

With 6 strands of 19 wires each. 



Trade 
Number. 


Diameter in 
inches. 


Weight pel- 
foot in 
pounds. 


Breaking 
Strain in 
tons of 
2000 Ibs. 


Proper Work- 
ing Load. 


Min. Size of 
Drum or 
Sheave in 
feet. 


1 


& 


8.00 


240 


46 


9 


2 


2 \ 


6.30 


189 


37 


8 


3 


l^v 


5.25 


157 


31 


7J4 


4 


% 


4.10 


123 


25 


6 


5 


^ 


3.65 


110 


22 


51^8 


5^ 


% 


3.00 


90 


18 


5J4 


8 


J4 


2.50 


75 


15 


5 


7 


iHj 


2.00 


60 


12 


4^ 


8 


1 


1.58 


47 


9 


4J4 


9 


% 


1.20 


37 


7 


3-M 


10 


% 


0.88 


27 


5 


SH 


10M 


7 


0.60 


18 


m 


3 


10^ 


9-16 


0.44 


13 


2U 


2^ 


10% 


M 


0.39 


10 


2^ 


2 



With 7 Wires to the Strand. 



15 


1 


1.50 


45 


9 


% 


16 


7^ 


1.12 


33 


6^ 


5 


17 
18 


11-16 


0.92 
0.70 


25 
21 


5 
4 


4 

3^ 


19 


% 


0.57 


16 


3% 


3 


20 


9-16 


0.41 


12 




2M 


21 


K 


0.31 


9 


1% 


2^2 


22 


7-16 


0.23 


5 


IV 


2 


23 


% 


0.21 


4 


1 


1^ 



Galvanized Iron Wire Rope. 

For Ships' Rigging and Guys for Derricks. 
CHARCOAL ROPE. 



Circum- 
ference 
in inches. 



Weight 

per Fath- 

om in 

pounds. 



Cir. of 

new 
Manila 

Rope of 
equal 

Strength. 



Break- 
ing 
Strain 
in tons 
of 2000 
pounds 

43 
40 
35 
33 
30 
26 
23 
20 
16 
14 
12 
10 



Circum- 
ference 
in inches 



Weight 

per 
Fathom 



Cir. of 

new 

Manila 

Rope of 

pounds. j&$QL 



Break- 
ing 
Strain 
in tons 
of 2000 
pounds 



WIRE ROPES. 



229 



Galvanized Cast-steel Yacht Rigging. 



Circum- 
ference 
in inches. 


Weight 
per Fath- 
om in 
pounds. 


Cir. of 
new 
Manilla 
Rope of 
equal 
Strength. 


Break- 
ing 
Strain 
in tons 
of 2000 
pounds 


Circum- 
ference 
in inches 


Weight 
per 
Fathom 
in 
pounds. 


Cir. of 
new 
Manilla 
Rope of 
equal 
Strength. 


Break- 
ing 
Strain 
in tons 
of 2000 
pounds 


fa 

%& 

O1/ 

2*4 


M 

T 

4$ 


13 
11 

y^ 

8H 
8 

7 


66 
43 
32 

27 

22 
18 


2 

VA 

i 


2 2 

1% 

% 


3 4 


14 
10 

8 



Steel Hawsers. 

For Mooring, Sea, and Lake Towing. 







Size of 






Size of 


Circumfer- 


Breaking 


Manilla Haw- 


Circumfer- 


Breaking 


Manilla Haw- 


ence. 


Strength. 


ser of equal 
Strength. 


ence. 


Strength. 


ser of equal 
Strength. 


Inches. 


Tons. 


Inches. 


Inches. 


Tons. 


Inches. 


2^ 


15 


6^ 


3^ 


29 


9 


2% 


18 


7 


4 


35 


10 


3 


22 


8^ 









Steel Flat Ropes. 

(J. A. Roebling's Sons Co.) 

Steel-wire Flat Ropes are composed of a number of strands, alternately 
twisted to the right and left, laid alongside of each other, and sewed together 
with soft iron wires, These ropes are use'd at times in place of round ropes 
in the shafts of mines. They wind upon themselves on a narrow winding- 
drum, which takes up less room than one necessary for a round rope. The 
Soft-iron sewing-wires wear out sooner than the steel strands, and then it 
becomes necessary to sew the rope with new iron wires. 



Width and 
Thickness 
in inches. 


Weight per 
foot in 
pounds. 


Strength in 
pounds. 


Width and 
Thickness 
in inches. 


Weight per 
foot in 
pounds. 


Strength in 
pounds. 


%x2 


1.19 


35,700 


1^x3 


2.38 


71,400 


%x2}4 


1.86 


55,800 


^x3VS 


2.97 


89,000 


%x3 


2.00 


60,000 


^x4 


3.30 


99,000 


%x3J^ 


2.50 


75,000 


^x4^ 


4.00 


120,000 


%x4 


2.86 


85,800 


}^x 5 


4.27 


128,000 


%x4J4 


3.12 


93,600 


^x5^ 


4.82 


144,600 


%x5 


3.40 


100,000 


1^x6 


5.10 


153,000 


%x5^ 


3.90 


110,000 


1^x7 


5.90 


177,000 



For safe working load allow from one fifth to one seventh of the breaking 
stress. 

" Lang I*ay Rope. 

In wire rope, as ordinarily made, the component strands are laid up into 
rope in a direction opposite to that in which the wires are laid into strands; 
that is, if the wires in the strands are laid from right to left, the strands are 
laid into rope from left to right. In the " Lang Lay," sometimes known as 
01 Universal Lay," the wires are laid into strands and the strands into rope 
in the same direction ; that is, if the wire is laid in the strands from right to 
left, the strands are also laid into rope from right to left. Its use has been 
found desirable under certain conditions and for certain purposes, mostly 
for haulage plants, inclined planes, and street railway cables, although it 
has also been used for vertical hoists in mines, etc. Its advantages are that 



230 



MATERIALS. 



GALVANIZED STEEL CABLES* 
For Suspension Bridges. (Roebling's.) 



220 
200 
180 



13 

11.3 

10 



2 

m 



t 

II 



3 v 

il 



155 
110 
100 



8.64 
6.5 
5.8 



il 



95 
75 
65 






5.6 
4 35 
3.7 



COMPARATIVE STRENGTHS OF FLEXIBLE GAL- 
VANIZED STEEL-WIRE HAWSERS, 

With Chain Cable, Tarred Russian Hemp, and White 
Manila Ropes. 



Patent Flexible 

Steel-wire Hawsers 

and Cables. 



Chain Cable. 



Tarred Rus- 
sian Hemp 
Rope. 



White 
Manilla 
Ropes. 



5K 



11 



7 
9 
12 

15 
IS 
22 
20 
33 
39 
64 
74 
88 
102 
116 
130 
150 



9-16 
10-16 

11-16 
12-16 
13-16 
15-16 



1 17-32 



166 
1' 15-16 204 



2 1-16 

2 



3-16 256 
5-16 280 



14 



21 

30 101 

35 

4815' 

54 

68 
112 
1434 



23! 



1 

02 

"o 

2 

PH 





I 



107 1-10 

12014 

134^ 






51 

35^62 

42 -" 



22% 



NOTE. This is an old table, and its authority is uncertain. The figures in 
the fourth column are probably much too small for durability. 



WIRE ROPES. 231 

it is somewhat more flexible than rope of the same diameter and composed 
of the same number of wi^es laid up in the ordinary manner; and (especi- 
ally) that owing to the fact that the wires are laid more axially in the rope, 
longer surfaces of the wire are exposed to wear, and the endurance of the 
rope is thereby increased. (Trenton Iron Co.) 

Notes on the Use of Wire Rope. 
(J. A. Koebling's Sons Co.) 

Several kinds of wire rope are manufactured. The most pliable variety 
contains nineteen wires in the strand, and is generally used for hoisting and 
running rope. The ropes with twelve wires and seven wires in the strand 
are stiffer, and are better adapted for standing rope, guys, and rigging. Or- 
ders should state the use of the rope, and advice will be given. Ropes are 
made up to three inches in diameter, upon application. 

For safe working load, allow one fifth to one seventh of the ultimate 
strength, according to speed, so as to get good wear from the rope. When 
substituting wire rope for hemp rope, it is good economy to allow for the 
former the same weight per foot which experience has approved for the 
latter. 

Wire rope is as pliable as new hemp rope of the same strength; the for- 
mer will therefore run over the same-sized sheaves and pulleys as the latter. 
But the greater the diameter of the sheaves, pulleys, or drums, the longer 
wire rope will last. The minimum size of drum is given in the table. 

Experience has demonstrated that the wear increases with the speed. It 
is, therefore, better to increase the load than the speed. 

Wire rope is manufactured either wilh a wire or a hemp centre. The lat- 
ter is more pliable than the former, and will wear better where there is 
short bending. Orders should specify what kind of centre is wanted. 

Wire rope must not be coiled or uncoiled like hemp rope. 

When mounted on a reel, the latter should be mounted on a spindle or flat 
turn-table to pay off the rope. When forwarded in a small coil, without reel, 
roll it over the ground like a wheel, and run off the rope in that way. All 
untwisting or kinking must be avoided. 

To preserve wire rope, apply raw linseed-oil with a piece of sheepskin, 
wool inside; or mix the oil with equal parts of Spanish brown or lamp-black. 

To preserve wire rope under water or under ground, take mineral or vege- 
table tar, and add one bushel of fresh-slacked lime to one barrel 9f tar, 
which will neutralize the acid. Boil it well, and saturate the rope with the 
hot tar. To give the mixture body, add some sawdust. 

The grooves of cast-iron pulleys and sheaves should be filled with well- 
seasoned blocks of hard wood, set on end, to be renewed when worn out. 
This end-wood will save wear and increase adhesion. The smaller pulleys 
or rollers which support the ropes on inclined planes should be constructed 
on the same plan. When large sheaves run with very great velocity, the 
grooves should be lined with leather, set on end, or with India rubber. This 
is done in the case of sheaves used in the transmission of power between 
distant points by means of rope, which frequently runs at the rate of 4000 
feet per minute. 

Steel ropes are taking the place of iron ropes, where it is a special object 
to combine lightness with strength. 

But in substituting a steel rope for an iron running rope, the object in view 
should be to gain an increased wear from the rope rather than to reduce the 
size. 

Locked \Virc Rope. 

Fig. 74 shows what is known as the Patent Locked Wire Rope, made by 
the Trenton Iron Co. It is claimed to wear two to three times as long as an 




FIG. 74. 

ordinary wire rope of equal diameter and of like material. Sizes made are 
Irom y% to ly^ inches diameter. 



232 



MATERIALS. 



CRANE .CHAINS. 

(Bradlee & Co., Philadelphia.) 



11 D. B. G." Special Crane. 


Crane. 




i 


c 






0) 

be 


"S 




to 


"3 


"5s 


l 


Li 


1 


^ 


Is 

a fl 


tJ " 

M 


4J - 


a 

1 


^ of 


>$ 

W-S 


|1 


feij 


! 


<D M 


g 

PQft 


1^1 


w 3 


3 -3 


tj a5 
d^'O 
cc-gg 


O % 


P..FH 


^O o 


"3 rt 


O 


c 5 


t>i 53 o 





Jsr^ O 


hS o 












cs'S 




jr A 




2 c a 


& 


r 


I l 


O 




^ g 


r 


P-. 


8 ' 

1 


CC qj 

.So 

O 


y 


25-32 


Ys 


y 


1932 


3864 


1288 


1680 


3300 


1120 


5-16 


27-32 


1 


1 1-16 


2898 


5796 


1932 


2520 


5040 


1680 


% 


31-32 


17-10 


1H 


4186 


8372 


2790 


3640 


7280 


2427 


7-16 


1 5-32 


2 


1% 


5796 


11592 


3864 


5040 


10080 


3360 


Lj 


1 11-32 


31^ 


1 11-16 


7728 


15456 


5182 


6720 


13440 


4480 


9-16 


1 15-32 


3 -,'-10 


1% 


9660 


19320 


6440 


8400 


16800 


5600 


5 


1 23-32 




21-16 


11914 


23828 


7942 


10360 


20720 


6907 


11-16 


1 27-32 


5 8 


2^4 


14490 


28980 


9660 


12600 


25200 


8400 


H 


1 31-32 


5% 


2^ 


17388 


34776 


11592 


15120 


30240 


10080 


13-16 


23-32 


67-10 


2 fl-16 


20286 


40572 


13524 


17640 


35280 


11760 


% 


27-32 


8 


2% 


22484 


44968 


14989 


20440 


40880 


13627 


15-16 


215-32 


9 


31-16 


25872 


51744 


17248 


23520 


47040- 


15680 


1 


2 19-32 


10 7-10 


3J4 


29568 


59136 


19712 


26880 


53760 


17920 


1 1-16 


2 23-32 


11 2-10 


35-16 


33264 


66538 


22176 


30240 


60480 


20160 


u| 


2 27-32 




3-M 


37576 


75152 


25050 


34160 


68320 


22773 


1 3-16 


35-32 


13 7-10 


3% ' 


41888 


83776 


27925 


38080 


76160 


25387 


1^4 


37-32 


16 




46200 


92400 


30800 


42000 


84000 


28000 


1' 5-16 


3 15-32 




4% 


50512 


101024 


33674 


45920 


91840 


30613 


j^ 


3^Ha 


18 4 2 -10 


49-16 


55748 


111496 


37165 


50680 


101360 


33787 


1 7-16 


3 25-32 


19 7-10 


4% 


60368 


120736 


40245 


54880 


109760 


36587 


JJ 


3 31-32 


21 7-10 


5 


66528 


133056 


41352 


60480 


120960 


40320 



The distance from centre of one link to centre of next is equal to the in- 
side length of link, but in practice 1/32 inch is allowed for weld. This is ap- 
proximate, and where exactness is required, chain should be made so. 

FOR CHAIN SHEAVES. The diameter, if possible, should be not less than 
twenty times the diameter of chain used. 

EXAMPLE. For 1-inch chain use 20-inch sheaves. 

WEIGHTS OF LOOS, LUMBER, ETC. 
Weight of Green Logs to Scale 1,000 Feet, Board Measure. 

Yellow pine (Southern). . 8,000 to 10,000 Ibs, 

Norway pine (Michigan). 7,000 to 8,000 " 

Whit* ninp nVTiVhi>fln J off of Stump 6,000 to 7,000 " 

White pine (Michigan) ^ out Qf water _ 7,000 to 8,000 " 

White pine (Pennsylvania), bark off 5,000 to 6,000 " 

Hemlock (Pennsylvania), bark off 6,000 to 7,000 a 

Four acres of water are required to store 1,000,000 feet of logs. 
Weight of 1,OOO Feet of Lumber, Board Measure. 

Yellow or Norway pine Dry, 3,000 Ibs. Green, 5.000 Ibs. 

White pine 2,500 " 4,000 " 

Weight of 1 Cord of Seasoned Wood, 128 CuMc Feet per 

Cord. 

Hickory or sugar maple 4,500 Jbs 

White oak .- 3,850 " 

Beech, red oak or black oak 3,250 " 

Poplar, chestnut or elm 2,350 " 

Pine (white or Norway) 2,000 " 

Hemlock bark, dry 2,200 " 



SIZES OF FIKE-BRICK. 



233 




_ 

\ 
^ 



\ 






, 
thick x 4^ to 4 inches 



3 



iam. 




SIZES OF FIRE-BRICK, 



9-inch straight 9 x 4^ x 2^ inches. 

Soap 9 x 2J^ x 2J*j 

Checker 9x3 x3 " 

2-inch 9x4^x2 ** 

Split... 9x4 

Jamb 9 x 4 

No. Ikey 9x2; 

wide. 

113 bricks to circle 12 feet inside diam. 

No.2key ... 9x2^ thick x 4^ to 3 

inches wide. 

63 bricks to circle 6 ft. inside diam. 

No. 3 key 9x2^ thick x 4^ to 

inches wide. 

38 bricks to circle 3 ft. inside diam. 

No. 4 key / 9x2^ thick x 4^ to 2*4 

inches wide. 

25 bricks to circle 1^ ft. inside diam. 
No. 1 wedge (or bullhead). 9x4^ wide x 2*4 to 2 in. 
thick, tapering lengthwise. 

98 bricks to circle 5 ft. inside diam. 

No. 2 wedge 9 x 4*4 x 2^ to 1^ in. thick. 

60 bricks to circle 2J4 ft. inside diam. 

No. larch.., 9x4^x2^ to 2 in. thick, 

tapering breadthwise. 

72 bricks to circle 4 ft. inside diam. 

No.2arch 9x4^x ;_ 

42 bricks to circle 2 ft. inside 

No. 1 skew 9 to 7 x 

Bevel on one end. 

No. 2 skew 9x2^x4^ 

Equal bevel on both edges. 

No. 3skew 9x2^x4^ to 

Taper on one edge. 

24 inch circle 8*4 to 5J4 x 4V x 2>. 

Edges curved, 9 bricks line a 24-inch circle. 

36-inch circle 8% to 6^ x 4J4 x 2^. 

13 bricks line a 36-inch circle. 

48-inch circle 8% to 7J4 x 4^ x 2J4 

17 bricks line a 48-inch circle. 

inch straight 13^ x 2^ x 6. 

inch key No. 1 13^ x 2^ x 6 to 5 inch. 

90 bricks turn a 12-ft. circle. 

13i^-inch key No. 2 13^ x 2^ x 6 to 4% inch. 

52 bricks turn a 6-ft. circle. 

Bridge wall, No. 1 13x6^x6. 

Bridge wall, No. 2 13x6^x3. 

Mill tile 18,20,or24x6x3. 

Stock-hole tiles 18, 20, or 24 x 9 x 4. 

18-inch block 18x9x6. 

Flat back 9x6x2^. 

Flat back arch 9 x 6 x 314 to 2^. 

22-inch radius, 56 bricks to circle. 

Locomotive tile 32 x 10 x 3. 

34 x 10 x 3. 
34x 8x3. 
36 x 8x3. 
40x10x3. 

Tiles, slabs, and blocks, various sizes 12 to 30 inches 
long, 8 to 30 inches wide, 2 to 6 inches thick. 




, , 

Cupola brick, 4 and 6 inches high, 4 and 6 inches radial width, to line shells 
23 to 66 in diameter. 

A 9-inch straight brick weighs 7 Ibs. and contains 100 cubic inches. (=120 
Ibs. per cubic foot. Specific gravity 1.93.) 

One cubic foot of wall requires 17 9-inch bricks, one cubic yard requires 
460. Where keys, wedges, and other " shapes " are used, add 10 per cent in 
estimating the number required. 



234 



MATERIALS. 



One ton of fire-clay should be sufficient to lay 3000 ordinary bricks. To 
secure the best results, fire-bricks should be laid in the same clay from which 
they are manufactured. It should be used as a thin paste, and not as mor- 
tar. The thinner the joint the better the furnace wall. In ordering bricks 
the service for which they are required should be stated. 



NUMBER OF FIRE-BRICK REQUIRED FOR 
VARIOUS CIRCLES. 



g 

log 

ft 


KEY BRICKS. 


ARCH BRICKS. 


WEDGE BRICKS. 


^ 

d 
& 


w 
| 


OJ 

6 
fc 


6 
ft 


. I 


<?* 

6 
ft 


d 
fc 


OS 


a 

o 
E-t 


si 

6 
& 


0* 




O5 


1 


ft. in. 
1 6 
2 
2 6 
3 
3 6 
4 
4 6 
5 
5 6 
6 
6 6 
7 
7 6 
8 
8 6 
9 
9 6 
10 
10 6 
11 
11 6 
12 
12 6 


25 
17 
9 








25 
30 
34 
38 
42 
46 
51 
55 
59 
63 
67 
71 
76 
80 
84 
88 
92 
97 
101 
105 
109 
113 
117 


'42 
31 
21 
10 














13 
25 
38 
32 
25 
19 
13 
6 


10 
21 
32 
42 
53 
63 
58 
52 
47 
42 
37 
31 
26 
21 
16 
11 
5 


9 
19 
29 
38 
47 
57 
66 
76 
85 
94 
104 
113 
113 


'is' 

36 
54 
73 




42 
49 
57 
64 
72 










60 
48 
36 
24 


'26' 
40 
59 




6fl 
68 
76 
83 
91 
98 
106 
113 
121 
128 
136 
144 
151 
159 
166 
174 
181 
189 
196 
304 








72 
72 

72 
72 

72 
72 
72 

72 

72 
72 
72 
72 
72 
72 
72 


8 
15 
23 
30 
38 
45 
53 
60 
68 
75 
83 
90 
98 
105 
113 
121 


80 
87 
95 
102 
110 
117 
125 
132 
140 
147 
155 
162 
170 
177 
185 
193 


12 


79 

98 
98 
98 
98 
98 
98 
98 
98 
98 
98 
98 
98 
98 
98 
98 




"*8 

15 
23 
30 
38 
46 
53 
61 
68 
76 
83 
91 
98 
106 


































.... 


.... 







































For larger circles than 12 feet use 113 No. 1 Key, and as many 9-inch brick 
as may be needed in addition. 



ANALYSES OF MET. SAVAGE FIRE-CLAY. 



(1) 

1871 

Mass. 

Institute of 
Technology. -, 

50.457 
35.904 



0.133 
0.018 
trace 
12.744 



100.760 



(2) 

1877. 
Report on 



(8) 

1878. 



56.80 


*sey 
Silica 


Survey of 
Pennsylvania. 

44.395 


80.08 




,. 33.558 


1.15 


Titanic acid 


, i 530 


1 12 


Peroxide iron 


1 080 




Lime 






Magnesia 


0.108 


0.80 


Potash (alkalies). 


0.247 


10.50 


Water and inorg. 


matter. 14.575 



(4) 






100.450 



100.493 



56.15 
33.295 

*6".59" 
0.17 
0.115 

'9! ',68 
100.000 



MAGNESIA BRICKS. 230 

MAGNESIA BRICKS. 

** Foreign Abstracts " of the Institution of Civil Engineers, 1893, gives a 
paper by C. Bischof on the production of magnesia bricks. The material 
most in favor at present is the magnesite of Styria, which, although less 
pure considered as a source of magnesia than the Greek, has the property 
of fritting at a high temperature without melting. The composition of the 
two substances, in the natural and burnt states, is as follows: 

Magnesite. Styrian. Greek. 

Carbonate of magnesia 90.0 to 96.0# 94.46# 

" lime 0.5 to 2.0 4.49 

" " iron 3.0 to 6.0 FeO 0.08 

Silica 1.0 0.52 

Manganous oxide 0.5 Water 0.54 

Burnt Magnesite. 

Magnesia 77.6 82.46-95.36 

Lime 7.3 0.8310.92 

Alumina and ferric oxide 13.0 0.56 3.54 

Silica 1.2 0.737.98 

At a red heat magnesium carbonate is decomposed into carbonic acid and 
caustic magnesia, which resembles lime in becoming hydrated and recar- 
bonated when exposed to the air, and possesses a certain plasticity, so that 
it can be moulded when subjected to a heavy pressure. By long-continued 
or stronger heating the material becomes dead-burnt, giving a form of mag- 
nesia of high density, sp. gr. 3.8, as compared with 3.0 in the plastic form, 
which is unalterable in the air but devoid of plasticity. A mixture of two 
volumes of dead-burnt with one of plastic magnesia can be moulded into 
bricks which contract but little in firing. Other binding materials that have 
been used are: clay up to 10 or 15 per cent; gas -tar, perfectly freed from 
water, soda, silica, vinegar as a solution of magnesium acetate which is 
readily decomposed by heat, and carbolates of alkalies or lime. Among 
magnesium compounds a weak solution of magnesium chloride may also be 
used. For setting the bricks lightly burnt, caustic magnesia, with a small 
proportion of silica to render it less refractory, is recommended. The 
strength of the bricks may be increased by adding iron, either as oxide or 
silicate. If a porous product is required, sawdust or starch may be added 
to the mixture. When dead-burnt magnesia is used alone, soda is said to be 
the best binding material. 

See also papers by A. E. Hunt, Trans. A. I. M. E., xvi, 720, and by T. Egles- 
ton, Trans. A. I. M. E., xiv, 458. 

Asbestos. J. T. Donald, Eng. and M. Jour., June 27, 1891. 

ANALYSIS. 

Canadian. 

Italian. Broughton. Templeton. 

Silica 40.30# 40.57 40.52 

Magnesia 43.37 41.50 42.05 

Ferrous oxide 87 2.81 1.97 

Alumina 2.27 .90 2.10 

Water 13.72 13.55 13.46 



100.53 99.33 100.10 

Chemical analysis throws light upon an important point in connection 
with asbestos, i.e., the cause of the harshness of the fibre of some varieties. 
Asbestos is principally a hydrous silicate of magnesia, i.e., silicate of mag- 
nesia combined with water. When harsh fibre is analyzed it is found to 
contain less water than the soft fibre. In fibre of very fine quality from 
Black Lake analysis showed 14.38$ of water, while a harsh-fibred sample 
gave only 11.70#. If soft fibre be heated to a temperature that will drive off 
a portion of the combined water, there results a substance so brittle that it 
may be crumbled between thumb and finger. There is evidently some con- 
nection between the consistency of the fibre and the amount of water in its 
composition. 



236 STRENGTH OF MATERIALS. 



STRENGTH OP MATERIALS. 

Stress and Strain. There is much confusion among writers on 
strength of materials as to the definition of these terms. An external force 
applied to a body, so as to pull it apart, is resisted by an internal force, or 
resistance, and the action of these forces causes a displacement of the mole- 
cules, or deformation. By some writers the external force is called a stress, 
and the internal force a strain; others call the external force a strain, and 
the internal force a stress: this confusion of terms is not of importance, as 
the words stress and strain are quite commonly used synonymously, but the 
use of the word strain to mean molecular displacement, deformation, or dis- 
tortion, as is the custom of some, is a corruption of the language. See En- 
gineering News, June 23, 1892. Definitions by leading authorities are given 
below. 

Stress. A stress is a force which acts in the interior of a body, and re- 
sists the external forces which tend to change its shape. A deformation is 
the amount of change of shape of a body caused by the stress. The word 
strain is often used as synonymous with stress and sometimes it is also used 
to designate the deformation. (Merriman.) 

The force by which the molecules of a body resist a strain at any point is 
called the stress at that point. 

The summation of the displacements of the molecules of a body for a 
given point is called the distortion or strain at the point considered. (Burr). 

Stresses are the forces which are applied to bodies to bring into action 
their elastic and cohesive properties. These forces cause alterations of the 
forms of the bodies upon which they act. Strain is a name given to the 
kind of alteration produced by the stresses. The distinction between stress 
and strain is not always observed, one being used for the other. (Wood.) 

Stresses are of different kinds, viz. : tensile, compressive, transverse, tor- 
sional, and shearing stresses. 

A tensile stress, or pull, is a force tending to elongate a piece. A com- 
pressive stress, or push, is a force tending to shorten it. A transverse stress 
tends to bend it. A torsional stress tends to twist it. A shearing stress 
tends to force one part of it to slide over the adjacent part. 

Tensile, compressive, and shearing stresses are called simple stresses. 
Transverse stress is compounded of tensile and compressive stresses, and 
torsional of tensile and shearing stresses. 

To these five varieties of stresses might be added tearing stress, which is 
either tensile or shearing, but in which the resistance of different portions 
of the material are brought into play in detail, or one after the other, in- 
stead of simultaneously, as in the simple stresses. 

Effects of Stresses. The following general laws for cases of simple 
tension or compression have been established by experiment. (Merriman): 

1. When a small stress is applied to a body, a small deformation is pro- 
duced, and on the removal of the stress the body springs back to its original 
form. For small stresses, then, materials may be regarded as perfectly 
elastic. 

2. Under small stresses the deformations are approximately proportional 
to the forces or stresses which produce them, and also approximately pro- 
portional to the length of the bar or body. 

3. When the stress is great enough a deformation is produced which is 
partly permanent, that is, the body does not spring back entirely to its 
original form on removal of the stress. This permanent part is termed a 
set. In such cases the deformations are not proportional to the stress. 

4. When the stress is greater still the deformation rapidly increases and 
the body finally ruptures. 

5. A sudden stress, or shock, is more injurious than a steady stress or than 
a stress gradually applied. 

Elastic Limit. The elastic limit is defined as that point at which the 
deformations cease to be proportional to the stresses, or, the point at which 
the rate of stretch (or other deformation) begins to increase. It is also 
defined as the point at which the first permanent set becomes visible. The 
last definition is not considered as good as the first, as it is found that with 
some materials a set occurs with any load, no matter how small, and that 
with others a set which might be called permanent vanishes with lapse of 
time, and as it is impossible to get the point of first set without removing 



STRESS AKD STRAIN. 237 

the whole load after each increase of load, which is frequently inconvenient. 
The elastic limit, defined, however, as the point at which the extensions be- 
gin to increase at a higher ratio than the applied stresses, usually corresponds 
very nearly with the point of first measurable permanent set. 

Apparent Elastic lamit. Prof. J. B. Johnson (Materials of Con- 
struction, p. 19) defines the " apparent elastic limit " as "the point on the 
stress diagram [a plotted diagram in which the ordinates represent loads 
and the abscissas the corresponding elongations] at which the rate of de- 
formation is 50$ greater than it is at the origin," [the minimum rate]. An 
equivalent definition, proposed by the author, is that point at which the 
modulus of extension (length X increment of load per unit of section H- in- 
crement of elongation) is two thirds of the maximum. For steel, with a 
modulus of elasticity of 30,000,000, this is equivalent to that point at which 
the increase of elongation in an 8-inch specimen for 1000 Ibs. per sq. in. 
increase of load is 0.0004 in. 

Yield-point. The term yield-point has recently been introduced into 
the literature of the strength of materials. It is defined as that point at 
which the rate of stretch suddenly increases rapidly. The difference be- 
tween the elastic limit, strictly defined as the point at which the rate of 
stretch begins to increase, and the yield-point, at which the rate increases 
suddenly, may in some cases be considerable. This difference, however, will 
not be discovered in short test-pieces unless the readings of elongations are 

made by an exceedingly fine instrument, as a micrometer reading to 



of an inch. In using a coarser instrument, such as calipers reading to 1/100 
of an inch, the elastic limit and the yield-point will appear to be simultane- 
ous. Unfortunately for precision of language, the term yield-point was not 
introduced until long after the term elastic limit had been almost univer- 
sally adopted to signify the same physical fact which is now defined by the 
term yield-point, that is, not the point at which the first change in rate, 
observable tnly by a microscope, occurs, but that later point (more or less 
indefinite as to its precise position) at which the increase is great enough to 
be seen by the naked eye. A most convenient method of determining the 
point at which a sudden increase of rate of stretch occurs in short speci- 
mens, when a testing-machine in which the pulling is done by screws is 
used, is to note the weight on the beam at the instant that the beam " drops.' 1 
During the earlier portion of the test, as the extension is steadily increased 
by the uniform but slow rotation of the screws, the poise is moved steadily 
along the beam to keep it in equipoise; suddenly a point is reached at which 
the beam drops, and will not rise until the elongation has been considerably 
increased by the further rotation of the screws, the advancing of the poise 
meanwhile being suspended. This point corresponds practically to the point 
at which the rate of elongation suddenly increases, and to the point at 
which an appreciable permanent set is first found. It is also the point which 
has hitherto been called in practice and in text-books the elastic limit, and 
it will probably continue to be so called, although the use of the newer term 
"yield-point" for it, and the restriction of the term elastic limit to mean 
the earlier point at which the rate of stretch begins to increase, as determin- 
able only by micrometric measurements, is more precise and scientific. 

In tables of strength of materials hereafter given, the term elastic limit is 
used in its customary meaning, tue point at which the rate of stress has be- 
gun to increase, as observable by ordinary instruments or by the drop of 
the beam. With this definition it is practically synonymous with yield- 
point. 

Coefficient (or Modulus) of Elasticity. This is a term express- 
ing the relation between the amount of extension or compression of a mate- 
rial and the load producing that extension or compression. 

It is defined as the load per unit of section divided by the extension per 
uuit of length. 

Let P be the applied load, fc the sectional area of the piece, I the length of 
the part extended, A the amount of the extension, and E the coefficient of 
elasticity. Then P -f- fc = the load on a unit of section ; A -*- 1 = the elonga- 
tion of a unit of length. 



The coefficient of elasticity is sometimes defined as the figure expressing 
the load which would be necessary to elongate a piece of one square inch 
section to double its original length, provided the piece would not break, and 
the ratio of extension to the force producing it remained constant. This 
definition follows from the formula above given, thus: If fcssoiie square 
inch, I and t- each = one inch, then E = P. 

Within the elastic limit, when the deformations are proportional to the 



238 STRENGTH OF MATERIALS. 

stresses, the coefficient of elasticity is constant, but beyond the el&stic limit 
it decreases rapidly. 

In cast iron there is generally no apparent limit of elasticity, the deforma- 
tions increasing at a faster rate than the stresses, and a permanent set being 
produced by small loads. The coefficient of elasticity therefore is not con- 
stant during any portion of a test, but grows smaller as the load increases. 
The same is true in the case of timber. In wrought iron and steel, however, 
there is a well-defined elastic limit, and the coefficient of elasticity within 
that limit is nearly constant. 

Resilience, or Work of Resistance of a Material. Within 
the elastic limit, the resistance increasing uniformly from zero stress to the 
stress at the elastic limit, the work done by a load applied gradually is equal 
to one half the product of the final stress by the extension or other deforma- 
tion. Beyond the elastic limit, the extensions increasing more rapidly than 
the loads, and the strain diagram approximating a parabolic form, the work 
is approximately equal to two thirds the product of the maximum stress by 
the extension. 

The amount of work required to break a bar, measured usually in inch- 
pounds, is called its resilience; the work required to strain it to the elastic 
limit is called its elastic resilience. (See page 270.) 

Under a load applied suddenly the momentary elastic distortion is equal 
to twice that caused by the same load applied gradually. 

When a solid material is exposed to percussive stress, as when a weight 
falls upon a beam transversely, the work of resistance is measured by the 
product of the weight into the total fall. 

Elevation of Ultimate Resistance and Elastic Limit. It 
was first observed by Prof. R. H. Thurstqn, and Commander L. A. Beards 
lee, U. S. N., independently, in 1873, that if wrought iron be subjected to a 
stress beyond its elastic limit, but not beyond its ultimate resistance, and 
then allowed to "rest" for a definite interval of time, a considerable in- 
crease of elastic limit and ultimate resistance may be experienced. In other 
words, the application of stress and subsequent '* rest " increases the resist- 
ance of wrought iron. 

This " rest " may be an entire release from stress or a simple holding the 
test-piece at a given intensity of stress. 

Commander Beardslee prepared twelve specimens and subjected them to 
an intensity of stress equal to the ultimate resistance of the material, with- 
out breaking the specimens. These were then allowed to rest, entirely free 
from stress, from 24 to 30 hours, after which period they were again stressed 
until broken. The gain in ultimate resistance by the rest was found to vary 
from 4.4 to 17 per cent. 

This elevation of elastic and ultimate resistance appears to be peculiar to 
iron and steel: it has not been found in other metals. 

Relation of tlie Elastic Limit to Endurance under Re- 
peated Stresses (condensed from Engineering, August 7, 1891). 
When engineers first began to test materials, it was soon recognized that 
if a specimen was loaded beyond a certain point it did not recover its origi- 
nal dimensions on removing the load, but took a permanent set; this point 
was called the elastic limit. Since below this point a bar appeared to recover 
completely its original form and dimensions on removing the load, it ap 
peared obvious that it had not been injured by the load, and hence the work- 
ing load might be deduced from the elastic limit by using a small factor of 
safety. 

Experience showed, however, that in many cases a bar would not carry 
safely a stress anywhere near the elastic limit of the material as determined 
by these experiments, and the whole theory of any connection between the 
elastic limit of a bar and its working load became almost discredited, and 
engineers employed the ultimate strength only in deducing the safe working 
load to which their structures might be subjected. Still, as experience accu- 
mulated it was observed that a higher factor of safety was required for a live 
load than for a dead one. 

In 1871 Wohler published the results of a number of experiments on bars 
of iron and steel subjected to live loads. In these experiments the stresses 
were put on and removed from the specimens without impact, but it was, 
nevertheless, found that the breaking stress of the materials was in every 
case much below the statical breaking load. Thus, a bar of Krupp's axle 
steel having a tenacity of 49 tons per square inch broke with a stress of 28.6 
tons per square inch, when the load was completely removed and replaced 
without impact 170,000 times. These experiments were made on a large 



STRESS AXD STKAIS7 239 

number Of different brands of iron and steel, and the results were concor- 
dant in showing that a bar would break with an alternating stress of only, 
say, one third the statical breaking strength of the material, if the repetitions 
of stress were sufficiently numerous. At the same time, however, it ap- 
peared from the general trend of the experiments that a bar would stand an 
indefinite number of alternations of stress, provided the stress was kept 
below the limit. 

Prof. Bauschinger defines the elastic limit as the point at which stress 
ceases to be sensibly proportional to strain, the latter being measured with 

a mirror apparatus reading to r^th of a millimetre, or about </vinnA in. 

OUUU lUUUUU 

This limit is always below the yield-point, and may on occasion be zero. On 
loading a bar above the yield-point, this point rises with the stress, and the 
rise continues for weeks, months, and possibly for years if the bar is left at 
resl under its load. On the other hand, when a bar is loaded beyond its true 
elastic limit, but below its 3 ; ield-point, this limit rises, but reaches a maxi- 
mum as the yield-point, is approached, and then falls rapidly^ reaching even 
to zero. On leaving the bar at rest under a stress exceeding that of its 
primitive breaking-down point the elastic limit begins to rise again, and 
may, if left a sufficient time, rise to a point much exceeding its previous 
value. 

This property of the elastic limit of changing with the history of a bar has 
done more to discredit it than anything else, nevertheless it now seems as if 
it, owing to this very property, were once more to take its former place in 
the estimation of engineers, and this time with fixity of tenure. It had long 
been known that the limit of elasticity might be raised, as we have said, to 
almost any point within the breaking load of a bar. Thus, in some experi- 
ments by Professor Styffe, the elastic limit of a puddled-steel bar was raised 
16,000 Ibs. by subjecting the bar to a load exceeding its primitive elastic 
limit. 

A bar has two limits of elasticity, one for tension and one for compression. 
Bauschinger loaded a number of bars in tension until stress ceased to be 
sensibly proportional to strain. The load was then removed and the bar 
tested in compression until the elastic limit in this direction had been ex- 
ceeded. This process raises the elastic limit in compression, as would be 
found on testing the bar in compression a second time. In place of this, 
however, it was now again tested in tension, when it was found that the 
artificial raising of the limit in compression had lowered that in tension be- 
low its previous value. By repeating the process of alternately testing in 
tension and compression, the two limits took up points at equal distances 
from the line of no load, both in tension and compression. These limits 
Bauschinger calls natural elastic limits of the bar, which for wrought iron 
correspond to a stress of about 8^ tons per square inch, but this is practically 
the limiting load to which a bar of the same material can be strained alter- 
nately in tension and compression, without breaking when the loading is 
repeated sufficiently often, as determined by Wohler's method. 

As received from the rolls the elastic limit of the bar in tension is above 
the natural elastic limit of the bar as defined by Bauschinger, having been 
artificially raised by the deformations to which it has been subjected in the 
process of manufacture. Hence, when subjected to alternating stresses, 
the limit in tension is immediately lowered, while that in compression is 
raised until they both correspond to equal loads. Hence, in Wohler's ex- 
periments, in which the bars broke at loads nominally below the elastic 
limits of the material, there is every reason for concluding that the loads 
' were really greater than true elastic limits of the material. This is con- 
firmed by tests on the connecting-rods of engines, which of course work 
under alternating stresses of equal intensity. Careful experiments on old 
rods show that the elastic limit in compression is the same as that in ten- 
sion, and that both are far below the tension elastic limit of the material as 
received from the rolls. 

The common opinion that straining a metal beyond its elastic limit injures 
it appears to be untrue. It is not the mere straining of a metal beyond one 
elastic limit that injures it, but the straining, many times repeated, beyond 
its two elastic limits. Sir Benjamin Baker has shown that in bending a shell 
plate for a boiler the metal is of necessity strained bej^ond its elastic limit, 
so that stresses of as much as 7 tons to 15 tons per square inch may obtain 
in it as it comes from the rolls, and unless the plate is annealed, these 
stresses will still exist after it has been built into the boiler. In such a case, 
however, when exposed to the additional stress due to the pressure inside 



240 STRENGTH OF MATERIALS. 

the boiler, the overstrained portions of the plate will relieve themselves by 
stretching and taking a permanent set, so that probably after a year's work- 
ing very little difference could be detected in the stresses in a plate built in- 
to the boiler as it came from the bending rolls, and in one which had been 
annealed, before riveting into place, and the first, in spite of its having been 
strained beyond its elastic limits, and not subsequently annealed, would be 
as strong as the other. 

Resistance of Metals to Repeated Shocks, 

More than twelve years were spent by Wohler at the instance of the Prus- 
sian Government in experimenting upon the resistance of iron and steel to 
repeated stresses. The results of his experiments are expressed in what is 
known as Wohler's law, which is given in the following words in Dubois'S 
translation of Weyrauch: 

" Rupture may be caused not only by a steady load which exceeds the 
carrying strength, but also by repeated applications of stresses, none of 
which are equal to the carrying strength. The differences of these stresses 
are measures of the disturbance of continuity, in so far as by their increase 
the minimum stress which is still necessary for rupture diminishes." 

A practical illustration of the meaning of the first portion of this law may 
be given thus: If 50,000 pounds once applied will just break a bar of iron or 
steel, a stress very much less than 50,000 pounds will break it if repeated 
sufficiently often. 

This is fully confirmed by the experiments of Fairbairn and Spangenberg, 
as well as those of Wohler; and, as is remarked by Weyrauch, it may be 
considered as a long-known result of common experience. It partially ac- 
counts for what Mr. Holley has called the tl intrinsically ridiculous factor of 
safety of six." 

Another "long-known result of experience " is the fact that rupture may 
be caused by a succession of shocks or impacts, none of which alone would 
be sufficient to cause it. Iron axles, the piston-rods of steam hammers, and 
other pieces of metal subject to continuously repeated shocks, invariably 
break after a certain length of service. They have a "life " which is lim- 
ited. 

Several years ago Fairbairn wrote: * We know that in some cases wrought 
iron subjected to continuous vibration assumes a crystalline structure, and 
that the cohesive powers are much deteriorated, but we are ignorant of the 
causes of this change." We are still ignorant, not only of the causes of this 
change, but of the conditions under which it takes place. Who knows 
whether wrought iron subjected to very slight continuous vibration will en- 
dure forever? or whether to insure final rupture each of the- continuous small 
shocks must amount at least to a certain percentage of single heavy shock 
(both measured in foot-pounds), which would cause rupture with one applica- 
tion ? Wohler found in testing iron by repeated stresses (not impacts) that 
in one case 400,000 applications or a stress of 500 centners to the square inch 
caused rupture, while a similar bar remained sound after 48,000,000 applica- 
tions of a stress of 300 centners to the square inch (1 centner = 110.2 Ibs.). 

Who knows whether or not a similar law holds true in regard to repeated 
shocks ? Suppose that a bar of iron would break under a single impact of 
1000 foot-pounds, how many times would it be likely to bear the repetition 
of 100 foot-pounds, or would it be safe to allow it to remain for fifty years 
subjected to a continual succession of blows of even 10 foot-pounds each ? 

Mr. William Metcalf published in the Metallurgical Rev ieiv, Dec. 1877, the 
results of some tests of the life of steel of different percentages of carbon 
under impact. Some small steel pitmans were made, the specifications for . 
which required that the unloaded machine should run 4J4 hours at the rate 
of 1200 revolutions per minute before breaking. 

The steel was all of uniform quality, except as to carbon. Here are the 
results; The 

.30 C. ran 1 h. 21 m. Heated and bent before breaking. 

.490. ". Ih. 28m., " " " " * 

.43 C. " 4 h. 57 m. Broke without heating. 

.65 C. " 3 h. 50 m. Broke at weld where imperfect. 

.80 C. " 5h. 40m. 

.84 C. " 18 h. 

.87 C. Broke in weld near the end. 

.96 C. Ran 4.55 m., and the machine broke down. 

Some other experiments by Mr. Metcalf confirmed his conclusion, viz. 



STRESS AKB STRAIH. 241 

that high-carbon steel was better adapted to resist repeated shocks and vi- 
brations than low-carbon steel. 

These results, however, would scarcely be sufficient to induce any en- 
gineer to use .84 carbon steel in a car-axle or a bridge-rod. Further experi- 
ments are needed to confirm or overthrow them. 

(See description of proposed apparatus for such an investigation in the 
author's paper in Trans. A. I. M. E M vol. viii., p. 76, from which the above 
extract is taken.) 

Stresses Produced by Suddenly Applied Forces and 
Shocks. 

(Mansfield Merriman, R. R. & Eng. Jour., Dec. 1889.) 
Let P be the weight which is dropped from a height h upon the end of a 
bar, and let y be the maximum elongation which is produced. The work 
performed by the falling weight, then, is 

TF=P(fc + y), 

and this must equal the internal work of the resisting molecular stresses. 
The stress in the bar, which is at first 0, increases up to a certain limit Q t 
which is greater than P; and if the elastic limit be not exceeded the elonga- 
tion increases uniformly with the stress, so that the internal work is equaJ 
to the mean stress 1/2Q multiplied by the total elongation y, or 

W=l/2Qy. 
Whence, neglecting the work that may be dissipated in heat, 

l/2Qy=Ph + Py. 

If e be the elongation due to the static load P, within the elastic limit 
y p e\ whence 

........ (1) 



which gives the momentary maximum stress. Substituting this value of Q, 
there results 



which is the value of the momentary maximum elongation. 

A shock results when the force P, before its action on the bar, is moving 
with velocity, as is the case when a weight P falls from a height h. The 
above formulas show that this height h may be small if e is a small quan- 
tity, and yet very great stresses and deformations be produced. For in- 
stance, let h = 4e, then Q = 4P and y = 4e ; also let h = 12e, then Q = 6P 
and y = 6 e. Or take a wrought-iron bar 1 in. square and 5 ft. long: under a 
steady load of 5000 Ibs. this will be compressed about 0.012 in., supposing 
that no lateral flexure occurs; but if a weight of 5000 Ibs. drops upon its end 
from the small height of 0.048 in. there will be produced the stress of 20,000 
Ibs. 

A suddenly applied force is one which acts with the uniform intensity P 
upon the end of the bar, but which has no velocity before acting upon it. 
This corresponds to the case of h in the above formulas, and gives Q =s 
2P and y = 2e for the maximum stress and maximum deformation. Profoi I 
ably the action of a rapidly-moving train upon a bridge produces stressed 
of this character. 

Increasing the Tensile Strength of Iron Bars by Twist- 
ing them. Ernest L. Ransome of San Francisco has obtained an English 
Patent, No. 16221 of 1888, for an " improvement in strengthening and testing 
wrought metal and steel rods or bars, consisting in twisting the same in a 
cold state. . . . Any defect in the lamination of the metal which would 
otherwise be concealed is revealed by twisting, and imperfections are shown 
at once. The treatment may be applied to bolts, suspension-rods or bars 
subjected to tensile strength of any description." 

Results of tests of this process were reported by Lieutenant F. P. Gilmore, 
U. S. N., in a paper read before the Technical Society of the Pacific Coast, 
published in the Transactions of the Society for the month of December, 
1888. The experiments include inajs wuu unrty-nine bars, twenty-nine of 
which were variously twisted, from three-eighths of one turn to six turns per 
foot. The test-pieces were cut from one and the same bar, and accurately 



242 



STRENGTH OF MATERIALS. 



measured and numbered. From each lot two pieces without twist were 
tested for tensile strength and ductility. One group of each set was twister, 
until the pieces broke, as a guide for the amount of twist to be given those 
to be tested for tensile strain. 

The following is the result of one set of Lieut. Gilmore's tests, on iron 
bars 8 in. long, .719 in. diameter. 



No. of 
Bars. 


Conditions. 


Twists 
in 
Turns. 


Twists 
per ft. 


Tensile 
Strength. 


Tensile 
per sq. in. 


Gain per 
cent. 


2 


Not twisted. 








22,000 


54,180 




2 


Twisted cold. 


^ 


y 


23,900 


59,020 


9 


2 


" " 


1 


iJ2 


25,800 


63,500 


17 


2 




2 


3 


26,300 


64,750 


19 


1 




5 


3M 


26,400 


65,000 


20 



Tests that corroborated these results were made by the University of 
California in 1889 and by the Low Moor Iron Works, England, in 1890. 

TENSILE STRENGTH. 

The following data are usually obtained in testing by tension in a testing-- 
machine a sample of a material of construction : 

The load and the amount of extension at the elastic limit. 

The maximum load applied before rupture. 

The elongation of the piece, measured between gauge-marks placed a 
stated distance apart before the test; and the reduction of area at the 
point of fracture. 

The load at the elastic limit and the maximum load are recorded in pounds 
per square inch of the original area. The elongation is recorded as a per- 
centage of the stated length between the gauge-marks, and the reduction 
area as a percentage of the original area. The coefficient of elasticity is cal- 
culated from the ratio the extension within the elastic limit per inch of 
length bears to the load per square inch producing that extension. 

On account of the difficulty of making accurate measurements of the frac- 
tured area of a test-piece, and of the fact that elongation is more valuable 
than reduction of area as a measure of ductility and of resilience or work 
of resistance before rupture, modern experimenters are abandoning the 
custom of reporting reduction of area. The "strength per square inch of 
fractured section " formerly frequently used in reporting tests is now almost 
entirely abandoned. The data now calculated from the results of a tensile 
test for commercial purposes are: 1. Tensile strength in pounds per square 
inch of original area. 2. Elongation per cent of a stated length between 
gauge-marks, usually 8 inches. 3. Elastic limit in pounds per square inch 
of original area. 

The short or grooved test specimen gives with most metals, especially 
with wrought iron and steel, an appaient tensile strength much higher 
than the real strength. This form of test-piece is now almost entirely aban- 
doned. 

The following results of the tests of six specimens from the same 1*4" steel 
bar illustrate the apparent elevation of elastic limit and the changes in 
other properties due to change in length of stems which were turned down 
in each specimen to .798" diameter. (Jas. E. Howard, Eng. Congress 1893 
Section G.) 



Description of Stem. 


Elastic Limit, 
Lbs. per Sq. In. 


Tensile Strength, 
Lbs. per Sq. In. 


Contraction of 
Area, per cent. 


1.00" long... 


64,900 


94,400 


49.0 


.50 "..... .... 


65 320 


97,800 


43 4 




68,000 


102,420 


39.6 


Semicircular groove, 
A" radius. ... ... 


75 000 


116,380 


31.6 


Semicircular groove, 
%" radius 


80,000, about 


134,960 


23.0 


V-shaped groove 


90,000, about 


117,000 


Indeterminate. 



TENSILE STRENGTH. 



243 



Tests plate made by the author in 1879 of straight and grooved test-pieces 
Of boiler-plate steel cut from the same gave the following results : 
5 straight pieces, 56,605 to 59,012 Ibs. T. S. Aver. 57,566 Ibs. 
4 grooved " 64,341 to 67,400 " " ** 65,452 " 

Excess of the short or grooved specimen, 21 per cent, or 12,114 Ibs. 

Measurement of Elongation. In order to be able to compare 
records of elongation, it is necessary not only to have a uniform length of 
section between gauge-marks (say 8 inches), but to adopt a uniform method 
of measuring the elongation to compensate for the difference between the 
apparent elongation when the piece breaks near one of the gauge-marks, 
and when it breaks midway between them. The following method is rec- 
ommended (Trans. A. S. M. E., vol. xi., p. 622): 

Mark on the specimen divisions of 1/2 inch each. After fracture measure 
from the point of fracture the length of 8 of the marked spaces on each 
fractured portion (or 7 -}- on one side and 8 -f- on the other if the fracture is 
not at one of the marks). The sum of these measurements, less 8 inches, is 
the elongation of 8 inches of the original length. If the fracture is so 
near one end of the specimen that 7 + spaces are not left on the shorter 
portion, then take the measurement of as many spaces (with the fractional 
part next to the fracture) as are left, and for the spaces lacking add the 
measurement of as many corresponding spaces of the longer portion as are 
necessary to make the 7 + spaces. 

Shapes of Specimens for Tensile Tests. The shapes shown 
in Fig. 75 were recommended by the author in 1882 when he was connected 



No. 1. Square or flat bar, as 
rolled. 



No. 2. Round bar, as rolled. 



No. 3. Standard shape for 
flats or squares. Fillets % 
inch radius. 



No. 4. Standard shape for 
rounds. Fillets J^ in. radius. 

No. 5. Government, shape for 
marine boiler-plates of iron. 
Not recommended for other 
tests, as results are generally 
in error. 



r* 16 Vso" 




FIG. 75. 



with the Pittsburgh Testing Laboratory. They are now in most general 
use, the earlier forms, with 5 inches or less in length between shoulders, 
being almost entirely abandoned. 

Precautions Required in making Tensile Tests, The 
testing-machine itself should be tested, to determine whether its weighing 
apparatus is accurate, and whether it is so made and adjusted that in the 
test of a properly made specimen the line of strain of the testing-machine 
is absolutely in line with the axis of the specimen. 

The specimen should be so shaped that it will not give an incorrect record 
of strength. 

It should be of uniform minimum section for not less than five inches of 
its length. 

Regard must be had to the i/ira* occupied in making tests of certain mate- 
rials. Wrought iron and soft steel can be made to show a higher than their 
actual apparent strength by keeping them under strain for a great length 
of time. 

Tn testing soft alloys, copper, tin, zinc, and the like, which flow under con- 
stant strain their highest apparent strength is obtained by testing them 
rapidly. In recording tests or such materials the length of time occupied in 
the test should be stated. 



244 STRENGTH OF MATERIALS. 

For very accurate measurements of elongation, corresponding to incre- 
ments of load during the tests, the electric contact micrometer, described 
in Trans. A. S. M. E., vol. vi., p. 479, will be found convenient. When read- 
ings of elongation are then taken during the test, a strain diagram may be 
plotted from the reading, which is useful in comparing the qualities of dif- 
ferent specimens. Such strain diagrams are made automatically by the new 
Olsen testing-machine, described in Jour. Frank. Inst. 1891. 

The coefficient of elasticity should be deduced from measurement ob~ 
served between fixed increments of load per unit section, say between 2000 
and 12,000 pounds per square inch or between 1000 and 11,000 pounds instead 
of between and 10,000 pounds. 

COMPRESSIVE: STRENGTH. 

What is meant by the term "compressive strength " has not yet been 
settled by the authorities, and there exists more confusion in regard to this 
term than in regard to any other used by writers on strength of materials. 
The reason of this may be easily explained. The effect of a compressive 
stress upon a material varies with the nature of the material, and with the 
shape and size of the specimen tested. While the effect of a tensile stress is 
to produce rupture or separation of particles in the direction of the line of 
strain, the effect of a compressive stress on apiece of material may be either 
to cause it to fly into splinters, to separate into two or more wedge-shaped 
pieces and fly apart, to bulge, buckle, or bend, or to flatten out and utterly re- 
sist rupture or separation of particles. A piece of speculum metal under 
compressive stress will exhibit no change of appearance until rupture takes 
place, and then it will fly to pieces as suddenly as if blown apart by gun- 
powder. A piece of cast iron or of stone will generally split into wedge- 
shaped fragments. A piece of wrought iron will buckle or bend. A piece of 
wood or zinc may bulge, but its action will depend upon fts shape and si:;e. 
A piece of lead will flatten out and resist compression till the last degree; 
that is, the more it is compressed the greater becomes its resistance. 

Air and other gaseous bodies are compressible to any extent as long as 
they retain the gaseous condition. Water not confined in a vessel is com- 

ressed by its own weight to the thickness of a mere film, while when con- 
ned in a vessel it is almost incompressible. 

It is probable, although it has not been determined experimentally, that 
solid bodies when confined are at least as incompressible as water. When 
they are not confined, the effect of a compressive stress is not only to 
shorten them, but also to increase their lateral dimensions or bulge them. 
Lateral strains are therefore induced by compressive stresses. 

The weight per square inch of original section required to produce any 
given amount or percentage of shortening of any material is not a constant 
quantity, but varies with both the length and the sectional area, with the 
shape of this sectional area, and with the relation of the area to the length. 
The " compressive strength' 1 of a material, if this term be supposed to mean 
the weight in pounds per square inch necessary to cause rupture, may vary 
with every size and shape of specimen experimented upon. Still more diffi- 
cult would it be to state what is the 4t compressive strength " of a material 
which does not rupture at all, but flattens out. Suppose we are testing a 
cylinder of a soft metal like lead, two inches in length and one inch in diam- 
eter, a certain weight will shorten it one per cent, another weight ten per 
cent, another fifty per cent, but no weight that we can place upon it will 
rupture it, for it will flatten out to a thin sheet. What, then, is its compres- 
sive strength ? Again, a similar cylinder of soft wrought iron would prob- 
ably compress a few per cent, bulging evenly all around ; it would then com- 
mence to bend, but at first the bend would be imperceptible to the eye and 
too small to be measured. Soon this bend would be great enough to be 
noticed, and finally the piece might be bent nearly double, or otherwise dis- 
torted. What is the "compressive strength 1 ' of this piece of iron ? Is it 
the weight per square inch which compresses the piece one per cent or five 
per cent, that which causes the first bending (impossible to be discovered), 
or that which causes a perceptible bend ? 

As showing the confusion concerning the definitions of compressive 
strength, the following statements from different authorities on the strength 
of wrought iron are of interest. 

Wood's Resistance of Materials states, " comparatively few experiments 
have been made to determine how much wrought iron will sustain at the 
point of crushing. Hodgkinson gives 65,000, Rondulet 70,800, Weisbach 72,000 



COMPKESSIVE STKEKGTH. 245 

Rankine 30,000 to 40,000. It is generally assumed that wrought iron will resist 
about two thirds as much crushing as to tension, but the experiments fail 
to give a" very definite ratio." 

Mr. Whipple, in his treatise on bridge-building, states that a bar of good 
wrought iron will sustain a tensile strain of about 60,000 pounds per square 
inch, and a compressive strain, in pieces of a length not exceeding twice the 
least diameter, of about 90,000 pounds. 

The following values, said to be deduced from the experiments of Major 
Wade, Hodgkinson, and Capt. Meigs, are given by Haswell : 

American wrought iron 127,720 Ibs. 

" (mean) 85,500 " 

TTnHih " J 65 > 200 " 

knglisn -j 40j00 o . 

Stoney states that the strength of short pillars of any given material, all 
having the same diameter, does not vary much, provided the length of the 
piece is not less than one and does not exceed four or five diameters, and 
that the weight which will just crush a short prism whose base equals one 
square Inch, and whose height is not less than 1 to 1J^ and does not exceed 
4 or 5 diameters, is called the crushing strength of the material. It would 
be well if experimenters would all agree upon some such definition of the 
term " crushing strength, 1 ' and insist that all experiments which are made 
for the purpose of testing the relative values of different materials in com- 
pression be made on specimens of exactly the same shape and size. An 
arbitrary size and shape should be assumed and agreed upon for this pur- 
pose. The size mentioned by Stoney is definite as regards area of section, 
viz., one square inch, but is indefinite as regards length, viz., from one to 
five diameters. In some metals a specimen five diameters long would bend, 
and give a much lower apparent strength than a specimen having a length of 
one diameter. The words " will just crush " are also indefinite for ductile 
materials, in which the resistance increases without limit If the piece tested 
does not bend. In such cases the weight which causes a certain percentage 
of compression, as five, ten, or fifty per cent, should be assumed as the 
crushing strength. 

For future experiments On crushing strength three things are desirable : 
First, an arbitrary standard shape and size of test specimen for comparison 
of all materials. Secondly, a standard limit of compression for ductile 
materials, which shall be considered equivalent to fracture in brittle mate- 
rials. Thirdly, an accurate knowledge of the relation of the crushing 
strength of a specimen of standard shape and size to the crushing strength 
of specimens of all other shapes and sizes. The latter can only be 
secured by a very extensive and accurate series of experiments upon all 
kinds of materials, and on specimens of a great number of different shapes 
and sizes. 

The author proposes, as a standard shape and size, for a compressive test 
specimen for all metals, a cylinder one inch in length, and one half square 
inch in sectional area, or 0.798 inch diameter; and for the limit of compres- 
sion equivalent to fracture, ten per cent of the original length. The term 
"compressive strength," or "compressive strength of standard specimen," 
would then mean the weight per square inch required to fracture by com- 
pressive stress a cylinder one inch long and 0.798 inch diameter, or to 
reduce its length to 0.9 inch if fracture does not take place before that reduc- 
tion in length is reached. If such a standard, or any standard size whatever, 
had been used by the earlier authorities on the strength of materials, we 
never would have had such discrepancies in their statements in regard to 
the compressive strength of wrought iron as those given above. 

The reasons why this particular size is recommended are : that the sectional 
area, one-half square inch, is as large as can be taken in the ordinary test- 
ing-machines of 100,000 pounds capacity, to include all the ordinary metals 
of construction, cast and wrought iron, and the softer steels; and that the 
length, one inch, is convenient for calculation of percentage of compression. 
If the length were made two inches, many materials would bend in testing, 
and give incorrect results. Even in cast iron Hodgkinson found as the mean 
of several experiments on various grades, tested in specimens % inch in 
height, a compressive strength per square inch of 94,730 pounds, while the 
mean of the same number of specimens of the same irons tested in pieces 1J 
inches in height was only 88,800 pounds. The best size and shape of standard 
specimen should, however, be settled upon only after consultation and 
agreement among several authorities. 



246 



STllEKGTH OF MATERIALS. 



The Committee on Standard Tests 01 the American Society of Mechanical 
Engineers say (vol. xi., p. 624) : 

" Although compression tests have heretofore been made on diminutive 
sample pieces, it is highly desirable that tests be also made on long pieces 
from 10 to 20 diameters in length, corresponding more nearly with actual 
practice, in order that elastic strain and change of shape may be determined 
by using proper measuring apparatus. 

The elastic limit, modulus or coefficient of elasticity, maximum and ulti- 
mate resistances, should be determined, as well as the increase of section at 
various points, viz., at bearing surfaces and at crippling point. 

The use of long compression-test pieces is recommended, because the in- 
vestigation of short cubes or cylinders has led to no direct application of 
the constants obtained by their use in computation of actual structures, 
which have always been and are now designed according to empirical for- 
mulae obtained from a few tests of long columns." 

COLUMNS, PILLARS, OR STRUTS. 

Hodgkinson's Formula for Columns. 

P = crushing weight in pounds; d = exterior diameter in inches; d l = in- 
terior diameter in inches; L = length in feet. 



Kind of Column. 



Both ends rounded, the 
length of the column 
exceeding 15 times 
its diameter. 



P = 33,380 



- 



' = 95,850 



p- 



Both ends flat, the 
length of the column 
exceeding 30 times 
its diameter. 

(3.56 

>= 98,920-^ 



P = 99, 



^- 



Solid cylindrical col- ) 

umns of cast iron ) 

Hollow cylindrical col- ) 

umns of cast iron ) 

Solid cylindrical col- ) 

umns of wrought iron. > 
Solid square pillar of \ 

Dantzic oak (dry) ) 
Solid square pillar of ) 

red deal (dry) . . f 

The above formulae apply only in cases in which the length is so great that 
the column breaks by bending and not by simple crushing. If the column 
be shorter than that given in the table, and more than four or five times its 
diameter, the strength is found by the following formula : 

PCK 



P = 299,600- 
P = 24,540^ 



in which P= the value given by the preceding formulae, K= the transverse 
section of the column in square inches, C = the ultimate compressive resis- 
tance of the material, and W = the crushing strength of the column. 

Hodgkinson's experiments were made upon comparatively short columns, 
the greatest length of cast-iron columns being 60^ inches, of wrought iron 
90% inches. 

The following are some of his conclusions: 

1 In all long pillars of the same dimensions, when the force is applied m 
the direction of the axis, the strength of one which has flat ends is about 
three times as great as one with roun L ed ends. 

2 The strength of a pillar with ^ne nd rounded and the other flat is an 
arithmetical mean between the two given in the preceding case of the same 
dimensions. 

3. The strength of a pillar having both ends firmly fixed is the same as 
one of half the length with both ends rounded. 

4. The strength of a pillar is not increased more than one seventh by en- 
larging it at the middle. 



MOMENT OF INEKTIA AND RADIUS OF GYRATION. 247 

Gordon's formulae deduced from Hodgkinson's experiments are more 
generally used than Hodgkinson's own. They are: 

Columns with both ends fixed or flat, P = - ; 



fs 

Columns with one end flat, the other end round, P = 



' f sf 

Columns with both ends round, or hinged, P = - -; 



~ 



8 = area of cross-section in inches; 

P ultimate resistance of column, in pounds; 

/ = crushing strength of the material in Ibs. per square inch; 

. , _ Moment of inertia 
r least radius of gyration, in inches, ?- 2 = 

area of section ' 
I length of column in inches; 
a a coefficient depending upon the material; 

/and a are usually taken as constants; they are really empirical variables, 
dependent upon the dimensions and character of the column as well as upon 
the material. (Burr.) 

For solid wrought-iron columns, values commonly taken are: / = 36,000 to 
40,000; a = 1/36,000 to 1/40,000. 
For solid cast-iron columns, / = 80,000, a = 1/6400. 

80 non 
For hollow cast-iron columns, fixed ends, p - - -- , I length and 



ficients derived from Hodgkinson's experiments, for cast-iron columns is to 
he deprecated. See Strength of Cast-iron Columns, pp. 250, 251. 
Sir Benjamin Baker gives, 

For mild steel, / = 67,000 Ibs., a = 1/22,400. 
For strong steel, /= 114,000 Ibs., a = VH400 

Prof. Burr considers these only loose approximations for the ultimate 
resistances. See his formulae on p. 259. 

For dry timber Rankine gives/ = 7200 Ibs., a = 1/3000. 

MOMENT OF INERTIA AND RADIUS OF GYRATION. 

The moment of inertia of a section is the sum of the products of 
each elementary area of the section into the square of its distance from an 
assumed axis of rotation, as the neutral axis. 

The radius of gyration of the section equals the square root of the 
quotient of the moment of inertia divided by the area of the section. If 
E = radius of gyration, 1= moment of inertia and A area, 



The moments of inertia of various sections are as follows; 

d = diameter, or outside diameter; d } = inside diameter; 6 = breadth; 
h = depth; 6,, &, inside breadth and diameter; 

Solid rectangle I = l/126/i3; Hollow rectangle I = l/12(67i - Mi 3 ); 

Solid square 7= 1/126*; Hollow square 7= 1/12(6* - 6,*); 

Solid cylinder I- l/647rd 4 ; Hollow cylinder I- l/647r(d 4 - c^ 4 ). 

Moments of Inertia and Radius of Gyration for Various 
Sections, and their Use in the Formulas for Strength of 
Girders and Columns, The strength of sections to resist strains, 
either as girders or as columns, depends not only on the area but also on the 
form of the section, and the property of the section which forms the basis 
of the constants used in the formulas for strength of girders and columns 
to express the effect of the form, is its moment of inertia about its neutral 
axis. The modulus of resistance of any section to transverse bending is its 



248 STRENGTH OF MATERIALS. 

moment of inertia divided by the distance from the neutral axis to the 
fibres farthest removed from that axis; or 

Moment of inertia I 

= Distance of extreme fibre from axis* " y' 

Moment of resistance = section modulus X unit stress on extreme fibre. 

IHoment of Inertia of Compound Shapes. (Pencoyd Iron 
Works.) The moment of inertia of any section about any axis is equal to the 
I about a parallel axis passing through its centre of gravity -f (the area of 
the section X the square of the distance between the axes). 

By this rule, the moments of inertia or radii of gyration of any single sec- 
tions being known, corresponding values may be obtained for any combina- 
tion of these sections. 

Radius of Gyration of Compound Shapes, In the case of a 
pair of any shape without a web the value of R can always be found with- 
out considering the moment of inertia. 

The radius of gyration for any section around an axis parallel to another 
axis passing through its centre of gravity is found as follows: 

Let r = radius of gyration around axis through centre of gravity; R =t 
radius of gyration around another axis parallel to above; d = distance be- 
tween axes: R = Vd'* -f- r*. 

When r is small, R may be taken as equal to d without material error. 

Graphical Method for Finding Radius of Gyration. Ben j. 
F. La Hue, Eng. Neius, Feb. 2, 1893, gives a short graphical method for 
finding the radius of gyration of hollow, cylindrical, and rectangular col- 
umns, as follows: 

For cylindrical columns: 

Lay off to a scale of 4 (or 40) a right-angled triangle, in which the base 
equals the outer diameter, and the altitude equals the inner diameter of the 
column, or vice versa. The hypothenuse, measured to a scale of unity (or 
10), will be the radius of gyration sought. 

This depends upon the formula 

'Mom, oflnertia _ ^D* + d 2 

Area 4 

in which A = area and D = diameter of outer circle, a area and d = dia- 
meter of inner circle, and G = radius of gyration. ^D' 2 -f d 2 is the expres- 
sion for the hypothenuse of a right-angled triangle, in which D and d are the 
base and altitude. 

The sectional area of a hollow round column is .7854(D 2 d 2 ). By con- 
structing a right-angled triangle in which D equals the hypothenuse and d 
equals the altitude, the base will equal 4/D 2 d 2 . Calling the value of this 
expression for the base J5, the area will equal .78541? 2 . 

Value of G for square columns: 

Lay off as before, but using a scale of 10, a right-angled triangle of whicfc 
the base equals D or the side of the outer square, and the altitude equals d, 
the side of the inner square. With a scale of 3 measure the hypotheuuse, 
which will be, approximately, the radius of gyration. 

This process for square columns gives an excess of slightly more than 4#. 
By deducting 4% from the result, a close approximation will be obtained. 

A very close result is also obtained by measuring the hypothenuse with 
the same scale by which the base and altitude were laid off, and multiplying 
by the decimal 0.29; more exactly, the decimal is 0.28867. 

The formula is 



This may also be applied to any rectangular column by using the lesser 
diameters of an unsupported column, and the greater diameters if the col- 
umn is supported in the direction of its least dimensions. 

ELEMENTS OF USUAI, SECTIONS. 

Moments refer to horizontal axis through centre of gravity. This table is 
intended for convenient application where extreme accuracy is not impor- 
tant. Some of the terms are only approximate; those marked * are correct. 
Values for radius of gyration in flanged beams apply to standard minimum 
sections only; A = area of section; b ^ bjeadth; h = depth; D = diameter. 



ELEMENTS OF USUAL SECTIONS. 



249 



Shape of Section. 


Moment 
of Inertia. 


Section 
Modulus. 


Square of 
Least 
Radius of 
Gyration. 


Least 
Radius of 
Gyration. 


.._.... 


Solid Rect- 
angle. 


bh* * 

12 


~6~ 


(Least side)2* 


Least side * 


12 


3.46 




* 


Hollow Rect- 
angle. 


6W-Mi * 


bV-bfa** 


/ t a _f Ttja * 


MtM 


JJgli 


Vrb+ 


12 


6/1 


12 


4.89 


T 







Solid Circle. 


AD* * 
16 


AD* 

8 


D* * 

16 


"4 


(*- D 1 


Hollow Circle. 
A, area of 
large section ; 
a, area of 
small section. 


AD*-ad* 


AD* -ad* 


D2+ef2* 
16 


D + rf 


16 


SD 


5.64 


s 


Solid Triangle. 


bh* 

36 


bh* 
24 


The least of 
of the two: 

18 F 24 


The least of 
the two: 
h b 

4.24 r 4.9 


Even Angle. 


Ah* 
10.2 


Ah 

7.2 


b* 
25 


6 
5 


JE 


Uneven Angle. 


Ah* 


Ah 
6.5 


0*" 


hb 


9.5 


13(/i 2 + 62) 


2.6(71 -f 6) 


H8 


Even Cross. 


Ah* 

19 


Ah 
9.5 


M 

22.5 


& 

4.74 


i 


Even Tee. 


Ah* 
11.1 


Ah 
8 


62 
22.5 


6 
4.74 


^ 


I Beam. 


Ah* 
6.66 


Ah 
3.2 


b* 
21 


6 

4.58 


lirt^x] 


Channel. 


Ah* 

7.34 


Ah 
3.67 


12T5 


6 
3.54 


111 


Deck Beam. 


Ah* 
6.9 


Ah 
4 


62 

36.5 


b 

6 


Distance of base from centre of gravity, solid triangle, ^; even angle, -^-- 
6 6.6 

uneven angle, $-=; ven tee, ^-5? deck beam, - ; all other shapes given in 

o.o o.o 4.6 

the table, ~ or . 



250 STRENGTH OF MATERIALS. 

The Strength of Cast-iron Columns. 

Hodgkinson's experiments (first published in Phil. Trans. Royal Socy., 
1840, and condensed in Tredgold on Cast Iron, 4th ed., 1846), and Gordon's 
formula, based upon them, are still used (1898) in designing cast-iron col- 
umns. That they are entirely inadequate as a basis of a practical formula 
suitable to the present methods of casting columns will be evident from 
what follows. 

Hodgkinson's experiments were made on nine " long " pillars, about 7^ 
ft. long, whose external diameters ranged from 1.74 to 2.23 in., and average 
thickness from 0.29 to 0.35 in., the thickness of each column also varying, 
and on 18 "short " pillars, 0.733 ft. to 2.251 ft. long, with external diameters 
from 1.08 to 1.26 in., all of them less than J4 in. thick. The iron used was 
Low Moor, Yorkshire, No. 3, said to be a good iron, not very hard, earlier 
experiments on which had given a tensile strength of 14,535 and a crushing 
strength of 109,801 Ibs. per sq. in. The results of the experiments on the 
" long " pillars were reduced to the equivalent breaking weight of a solid 
pillar 1 in. diameter and of the same length, 714 ft., which ranged from 2969 
to 3587 Ibs. per sq. in., a range of over 12 per cent, although the pillars were 
made from the same iron and of nearly uniform dimensions. From the 13 
experiments on " short " pillars a formula was derived, and from it were 
obtained the " calculated " breaking weights, the actual breaking weights 
ranging from about 8 per cent above to about 8 per cent below the calcu- 
lated weights, a total range of about 16 per cent. Modern cast-iron columns, 
such as are used in the construction of buildings, are very different in size, 

S:oportions, and quality of iron from the slender " long" pillars used in 
odgkinson's experiments. There is usually no check, by actual tests or by 
disinterested inspection, upon the quality of the material. The tensile, com- 
pressive, and transverse strength of cast iron varies through a great range 
(the tensile strength ranging from less than 10,000 to over 40,000 Ibs. per sq. 
in.), with variations in the chemical composition of the iron, according to 
laws which are as yet very imperfectly understood, and with variations in 
the method of melting and of casting. There is also a wide variation in the 
strength of iron of the same melt when cast into bars of different thick- 
nesses. It is therefore impossible to predict even approximately, from the 
data given by Hodgkinson of the strength of columns of Low Moor iron in 
pillars 7% ft. long, 2 in. diam., and % in. thick, what will be the strength of 
a column made of American cast iron, of a quality not stated, in a column 
16 ft. long, 12 or 15 in. diam., and from % in. to 1^ in. thick. 

Another difficulty in obtaining a practical formula for the strength of cast- 
iron columns is due to the uncertainty of the quality of the casting, and the 
danger of hidden defects, such as internal stresses due to unequal cooling, 
cinder or dirt, blow-holes, u cold-shuts, 1 ' and cracks on the inner surface, 
which cannot be discovered by external inspection. Variation in thick- 
ness, due to rising of the core during casting, is also a common defect. 

In addition to^the above theoretical or a priori objections to the use of 
Gordon's formula, based on Hodgkinson's experiments, for cast-iron 
columns, we have the data of recent experiments on full-sized columns, 
made by the Building Department of New York City (Eng'g News, Jan. 13 
and 20, 1898). Ten columns in all were tested, six 15-inch, 190J inches long, 
two 8-inch, 160 inches long, and two 6-inch, 120 inches long. The tests were 
made on the large hydraulic machine of the Phoenix Bridge Co., of 2,000,000 
pounds capacity, which was calibrated for frictiorml error by the repeated 
testing within the elastic limit of a large Phoenix column, and the compari- 
son of these tests with others made on the government machine at the 
Watertown Arsenal. The average frictional error was calculated to be 
15.4 per cent, but Engineering Neivs, revising the data, makes it 17.1 per 
cent, with a variation of 3 per cent either way from the average with differ- 
ent loads. The results of the tests of the volumes are given on the opposite 
page. 

Column No. 6 was not broken at the highest load of the testing machine. 

Columns Nos. 3 and 4 were taken from the Ireland Building, which col- 
lapsed on August 8, 1895; the other four 15-inch columns were made from 
drawings prepared by the Building Department, as nearly as possible 
duplicates of Nos. 3 and 4. Nos. 1 and 2 were made by a foundry in New 
York with no knowledge of their ultimate use. Nos. 5 and 6 were made by 
a foundry in Brooklyn with the knowledge that they were to be tested. 
Nos. 7 to 10 were made from drawings furnished by the Department. 



THE STRENGTH OF CAST-IROK COLUMNS. 



251 



TESTS OF CAST-IRON COLUMNS. 







Thickness. 


Breaking Load. 


Niimhpi* 


Diam. 








Inches. 


Max. 


Min. 


Average. 


Pounds. 


Pounds 
per sq. in. 


1 


15 


1 


1 


1 


1,356,000 


80,830 


2 


15 


1 5/16 


1 


*6 


1,330,000 


27,700 


3 


15 


1*4 


1 


*6 


1,198,000 


24.900 


4 


15J^ 


1 7/32 


1 


*6 


1,246,000 


25,200 


5 


15 


1 11/16 


1 


11/64 


1,632,000 


32,100 


6 


15 


1*4 


1*6 


3/16 


2,082,000 + 


40,400 -f 


7 


7% to 8M 


1*4 


% 




651,000 


31,900 


8 


8 


1 3/32 


1 


3/61 


612,800 


26,800 


9 


61/16 


1 5/32 


1*6 


9/64 


400,000 


22,700 


10 


6 3/32 


1*6 


1 1/16 


7/64 


455,200 


26,300 



lying Gordon's formula, as used by the Building Department, 
^ ^, to these columns gives for the breaking strength per square 



inch of the 15-inch columns 57,143 pounds, for the 8-inch columns 40,000 
pounds, and for the 6-inch columns 40,000. The strength of columns Nos. 3 
and 4 as calculated is 128 per cent more than their actual strength; their 
actual strength is less than 44 per cent of their calculated strength; and the 
factor of safety, supposed to be 5 in the Building Law, is only 2.2 for central 
loading, no account being taken of the likelihood of eccentric loading. 

Prof. Lanza, in Jhis Applied Mechanics, p. 372, quotes the records of 14 
tests of cast-iron mill columns, made on the Watertown testing-machine in 
1887-88, the breaking strength per square inch ranging from 25,100 to 63,310 
pounds, and showing no relation between the breaking strength per square 
inch and the dimensions of the columns. Only 3 of the 14 columns had a 
strength exceeding 33,500 pounds per square inch. The average strength of 
the other 11 was 29,600 pounds per square inch. Prof. Lanza says that it is 
evident that in the case of such columns we cannot rely upon a crushing 
strength of greater than 25,000 or 30,000 pounds per square inch of area of 
section. 

He recommends a factor of safety of 5 or 6 with these figures for crush- 
ing strength, or 5000 pounds per square inch of area of section as the highest 
allowable safe load, and in addition makes the conditions that the length of 
the column shall not be greatly in excess of 20 times the diameter, that the 
thickness of the metal shall be such as to insure a good strong casting, and 
that the sectional area should be increased if necessary to insure that the 
extreme fibre stress due to probable eccentric loading shall not be greater 
than 5000 pounds per square inch. 

Prof. W. H. Burr (Eng'g News, June 30, 1898) gives a formula derived 
from plotting, the results of the Watertown and Phoenixville tests, above 
described, which represents the average strength of the columns in pounds 
per square inch. It isp = 30,500 - IQOl/d. It is to be noted that this is an 
average value, and that the actual strength of many of the columns was 
much lower. Prof. Burr says: " If cast-iron columns are designed with 
anything like a reasonable and real margin of safety, the amount of metal 
required dissipates any supposed economy over columns of mild steel." 

Transverse Strength of Cast-iron Water-pipe. (Technology 
Quarterly, Sept. 1897.) Tests of 31 cast-iron pipes by transverse stress 
gave a maximum outside fibre stress, calculated from maximum load, 
assuming each half of pipe as a beam fixed at the ends, ranging from 12,800 
Ibs. to 26,300 Ibs. per sq. in. 

Bars 2 in. wide cut from the pipes gave moduli of rupture ranging from 
28,400 to 51,400 Ibs. per sq. in. Four of the tests, bars and pipes: 

Moduli of rupture of bar 28,400 34,400 40,000 51 ,400 

Fibre stress of pipe ... 18,300 12,800 14,500 26,300 

These figures show a great variation in the strength of both bars and 
pipes, and also that the strength of the bar does not bear any definite rela- 
tion to the strength of the pipe. 



252 



STRENGTH OP MATERIALS. 



Safe Load, in Tons of 200O I/bs., for Round Cast-iron 
Columns, with Turned Capitals and Bases, 

Loads being not eccentric, and length of column not exceeding 20 times 
the diameter. Based on ultimate crushing strength of 25,000 Ibs. per sq. in. 
and a factor of safety of 5. (For eccentric loads see page 254.) 



Thick- 
ness, 
/nches. 


Diameter, inches. 


6 


7 


8 


9 


10 

54.5 
62.7 
70.7 
78.4 
85.9 
93.1 


11 


12 


13 


14 


15 


16 


18 


1 

IVii 

IK 

2 


26.4 
30.9 
35.2 

39.2 


31.3 
36.8 
42.1 
47.1 


42.7 

48.9 
55.0 
60.8 


48.6 
55.8 
62.8 
69.6 
76.1 


69.6 

78.5 
87.2 
95.7 
103.9 


76.5 
86.4 
96.1 
105.5 
114.7 
123.7 


94.2 
104.9 
115.3 
125.5 
135.5 


102.1 
113.8 
125.2 
136. 3 
147.8 
168.4 


110.0 
122.6 
135.0 
147.1 
159.0 
182.1 
204.2 


131.4 
144.8 
157.9 
170.8 
195.8 
219.9 


164.' 
179. ( 
194. < 
223.! 
251.; 






.... 





















































For lengths greater than 20 diameters the allowable loads should be 
decreased. How much they should be decreased is uncertain, since suf- 
ficient data of experiments on full-sized very long columns, from which 
a formula for the strength of such columns might be derived, are as yet 
lacking. There is, however, rarely, if ever, any need of proportioning cast* 
iron columns with a length exceeding 20 diameters. 

Safe Loads in Tons of 2000 Pounds for Cast-iron Columns^ 

(By the Building Laws of New York City, Boston, and Chicago, 1897.) 
New York. Boston. Chicago. 



8a 



5a 



5a 



Square columns 



Round columns. ., 



1 -f j: 
Sa 



! + i 



5a 




1 + 



I* 
400<2 a 



1 + 



800d a 



a = sectional area in square inches; I = unsupported length of column m 
inches; d = side of square column or thickness of round column in inches. 

The safe load of a 15-inch round column !$ inches diameter, 16 feet long, 
according to the laws of these cities would be, in New York, 361 tons; in 
Boston, 264 tons; in Chicago, 250 tons. 

The allowable stress per square inch of area of such a column would be, 
in New York, 11,350 pounds; in Boston, 8300 pounds; in Chicago, 7850 pounds. 
A safe stress of 5000 pounds per square inch would give for the safe load on 
the column 159 tons. 

Strengtn of Brackets on Cast-iron Columns, The columns 
tested by the New York Building Department referred to above had 
brackets cast upon them, each bracket consisting of a rectangular shelf 
supported by one or two triangular ribs. These were tested after the 
columns had been broken in the principal tests. In 17 out of 22 cases the 
brackets broke by tearing a hole in the body of the column, instead of by 
shearing or transverse breaking of the bracket itself. The results were 
surprisingly low and very irregular. Reducing them to strength per square 
inch of the total vertical section through the shelf and rib or ribs, they 
ranged from 2450 to 5600 Ibs., averaging 4200 Ibs., for a load concentrated 
at the end of the shelf, and 4100 to 10,900 Ibs., averaging 8000 Ibs., for a dis- 
tributed load. (Eng'g News, Jan. 20, 1898.) 



SAFE LOAD OF CAST-IROK COLUMKS. 



253 



Safe Loads, in Tons, for Round Cast Columns. 

In accordance with the Building Laws of Chicago.*) 



Diame 
ter in 
Inches 


Thick- 
ness in 
Inches. 


Unsupported Length in Feet. 


6 


8 


10 


12 


14 


16 


18 


20 


22 


24 


26 


28 


30 


/> ( 


% 


50 


43 


37 


32 


27 














KT, 




6 1 


H 


57 


5( 


42 


36 


31 






Formula: u 




; ~ ' Za ' 


aJ 


% 


62 


56 


49 


43 


38 


33 




1 -f- - 


7 i 


YB 


71 


64 


57 


49 


43 


38 




w = safe load in tons of 


i 


f 


8( 
97 


(59 
79 
89 


71 
81 


56 
64 

72 


50 
57 
63 


44 
50 
56 


39 
44 
50 


2000 pounds; 
a = cross-section of col- 
umn; 


\ 


7 A 


101 


94 


86 


78 


70 


63 


57 


I = unsupported length 


9 i 


1 


113 
126 


105 
117 


97 
107 


88 
97 


79 

88 


71 
79 


fr 

7] 


in inches; 
d = diameter in inches. 


r 


% 


116 


109 


101 


93 


85 


78 


71 


64 












i 


1 


130 


122 


114 


105 


96 


88 


80 


72 












1 


l^ 


145 


136 


126 


ir 


107 


97 


8 S 


80 












i 




158 


149 


139 


128 


117 


107 




88 












f 


1 


147 


139 


131 


122 


113 


104 


96 


88 


8 










11 j 


l^ 


163 


155 


146 


136 


126 


116 


100 


,97 


8 










1 


1/4 


179 


170 


160 


149 


138 


127 


119 


jo; 


9 










l 


1% 


195 


185 


174 


162 


150 


138 


12 r 




10 










r 


1^ 


181 


174 


165 


155 


145 


135 


125 


115 


10 


98 








19 j 


1M 


199 


191 


181 


170 


159 


148 


13* 




11 


108 








A 1 


1% 


217 


207 


197 


1ST 


173 


161 


149 


jl 


12 


117 








1 


1% 


234 


224 


212 


200 


187 


173 


161 




13 


12b 








f 


1/^j 


200 


192 


184 


174 


164 


154 


144 


134 


125 


116 


107 






13 


l^ 


219 


211 


202 


191 


180 


169 


158 


U7 




127 


117 








1% 


239 


230 


220 


208 


196 


184 


172 


160 


14 


138 


128 






1 


l^fj 


258 


248 


237 


225 


212 


199 


186 


173 


16 


149 


138 






r 


1J4 




232 


223 


213 


202 


191 


180 


168 


157 


147 


137 


128 






1% 




253 


243 


232 


220 


207 


195 


183 


17 


160 


149 


139 






l^j 




27 


263 


251 


238 


224 


211 


198 


185 


173 


161 


150 




I 


1% 




293 


282 


269 


255 


241! 22? 


212 


198 


185 


173 


161 




r 


1% 






266 


255 


243 


231 


219 


206 


194 


182 


171 


160 


150 


,J 


1^3 






287 


276 


263 


250 


236 


223 


210 


197 


185 


173 


162 


1 


1% 






309 


296 


283 


268 


254 


239 


225 


211 


198 


186 


174 


I 


1M 






329 


316 


301 


286 


271 


255 


240 


225 


211 


198 


185 


I 


1/^ 








301 


288 


275 


262 


248 


235 


222 


209 


197 


185 


16-\ 


j^ 








323 


310 


296 


282 


267 


253 


239 


225 


212 


199 


( 


1% 








345 


331 


316 


300 


285 


270 


254 


239 


225 


212 




l% 










366 


351 


337 


322 


307 


293 


279 


264 


251 


18-] 


1% 










391 


375 


360 


344 


328 


313 


298 


282 


268 


1 


1% 










415 


399 


383 


366 


349 


333 


317 


300 


285 


r 


1% 












435 


420 


404 


389 


373 


357 


341 


326 


on J 


1% 












463 


447 


431 


414 


397 


380 


363 


347 


^ V ^ 


2 ' 












490 


473 


456 


43S 


420 


402 


384 


367 


I 


2^j 












517 


499 


481 


462 


443 


425 


406 


387 


f 


1% 














480 


464 


448 


432 


416 


400 


384 


1 


l/'O 














511 


494 


478 


461 


443 


426 


409 


1 

















541 


524 


506 


488 


470 


452 


434 


1 


%}& 














581 


562 


543 


524 


504 


485 


465 


f 


2^ 
















626 


608 


589 


570 


550 


531 


oj 


2*4 
















668 


639 


620 


600 


579 


559 


** 1 


2% 
















691 


671 


650 


629 


608 


587 


i 


*" 
















724 


703 


681 


659 


637 


614 


From tables published by The Expanded Metal Co., Chicago, 1897.) 



254 STRENGTH OF MATERIALS. 

ECCENTRIC LOADING OF COLUMNS. 

In a given rectangular cross-section, such as a masonry joint under press- 
ure, the stress will be distributed uniformly over the section only when the 
resultant passes through the centre of the section ; any deviation from such 
a central position will bring a maximum unit pressure to one edge and a 
minimum to the other; when the distance of the resultant from one edge is 
one third of the entire width of the joint, the pressure at the nearer edge is 
twice the mean pressure, while that at the farther edge is zero, and that 
when the resultant approaches still nearer to the edge the pressure at the 
farther edge becomes less than zero; in fact, becomes a tension, if the 
material (mortar, etc., there is capable of resisting tension. Or, if, as usual 
in masonry joints, the material is practically incapable of resisting tension, 
the pressure at the nearer edge, when the resultant approaches it nearer 
than one third of the width, increases very rapidly and dangerously, becom- 
ing theoretically infinite when the resultant reaches the edge. 

With a given position of the resultant relatively to one edge of the joint or 
section, a similar redistribution of the pressures throughout the section may 
be brought about by simply adding to .or diminishing the width of the 
section. 

Let P = the total pressure on any section of a bar of uniform thickness. 

w = the width of that section area of the section, when thickness = 1. 

p = P/w the mean unit pressure on the section. 

M the maximum unit pressure on the section. 

m = the minimum unit pressure on the section. 

d = the eccentricity of the resultant = its distance from the centre of 
the section. 

ThenM = p (l+~ ) and m = p (l - ^). 
When d = - w then M = %p and m = 0. 



When d is greater than l/6w, the resultant in that case being less than 
one third of the width from one edge, p becomes negative. (J. C. Traut- 
wine, Jr., Engineering News, Nov. 23, 1893.) 

Eccentric Loading of Cast-iron Columns. Prof. Lanza 
writes the author as follows: The table on page 252 applies when the resultant 
of the loads upon the column acts along its central axis, i.e., passes through 
the centre of gravity of every section. In buildings and other construc- 
tions, however, cases frequently occur when the resultant load does not 
pass through the centre of gravity of the section ; and then the pressure is 
not evenly distributed over the section, but is greatest on the side where 
the resultant acts. (Examples occur when the loads on the floors are not 
uniformly distributed.) In these cases the outside fibre stresses of the 
column should be computed as follows, viz.: 
Let P = total pressure on the section; 

d = eccentricity of resultant = its distance from the centre of gravity 

of the section; 

A = area of the section, and Jt its moment of inertia about an axis in its 
plane, passing through its centre of gravity, and perpendicular 
to d (see page 26?) ; 
Cj = distance of most compressed and c a = that of least compressed 

fibre from above stated axis; 
*j = maximum and s a = minimum pressure per unit of area. Then 

*-+2& and H-Z 



Having assumed a certain trial section for the column to be designed, s l 
should be computed, and, if it exceed the proper safe value, a different 
section should be used for which Sj does not exceed this value. 

The proper safe value, in the case of cast-iron columns whose ratio of 
length to diameter does not greatly exceed 20, is 5000 pounds per square inch 
when the eccentricity used in the computation of s l is liable to occur fre- 
quently in the ordinary uses of the structure; but when it is one which can 
only occur in rare cases the value 8000 pounds per square inch may be used. 

A long cap on a column is more conducive to the production of eccen- 
tricity of loading than a short one, hence a long cap is a source of weakness 
in a column. 



ULTIMATE STRENGTH OF WROUGHT-IROK COLUMNS. 255 



ULTIMATE STRENGTH OF WROUGHT-IRON 

COL.UMNS. 
(Pottsville Iron and Steel Co.) 

Computed by Gordon's formula, p = 



14-0 

p = ultimate strength in Ibs. per square inch; 

I = length of column in inches; 

r least radius of gyration in inches; 

/= 40,000; 

C = 1/40,000 for square end-bearings; 1/30,000 for one pin and one square 
bearing; 1/20,000 for two pin-bearings. 

For safe working load on these columns use a factor of 4 when used in 
buildings, or when subjected to dead load only; but when used in bridges 
the factor should be 5. 



WROUGHT-IRON COLUMNS. 





Ultimate Strength in Ibs. 
per square inch. 




Safe Strength in Ibs. per 
square inch Factor of 5. 


I 




I 




r 








r 










Square 

Ends. 


Pin and 
Square 
End. 


Pin 

Ends. 




Square 
Ends. 


Pin and 
Square 
End. 


Pin 

Ends. 


10 


39944 


39866 


39800 


10 


7989 


7973 


7960 


15 


39776 


39702 


39554 


15 


7955 


7940 


7911 


20 


39604 


39472 


39214 


20 


7921 


7894 


7843 


25 


39384 


39182 


38788 


25 


7877 


7836 


7758 


30 


39118 


38834 


38278 


30 


7821 


7767 


7656 


35 


38810 


38430 


37690 


35 


7762 


7686 


7538 


40 


38460 


37974 


37036 


40 


7692 


7595 


7407 


45 


38072 


37470 


36322 


45 


7614 


7494 


7264 


50 


37646 


36928 


35525 


50 


7529 


7386 


7105 


55 


37186 


36336 


34744 


55 


7437 


7267 


6949 


60 


36697 


35714 


33898 


60 


7339 


7143 


6780 


65 


36182 


34478 


33024 


65 


7236 


6896 


6605 


TO 


35634 


34384 


32128 


70 


7127 


6877 


6426 


75 


35076 


33682 


31218 


75 


7015 


6736 


6244 


80 


34482 


32966 


30288 


80 


6896 


6593 


6058 


85 


33883 


32236 


29384 


85 


6777 


6447 


5877 


90 


33264 


31496 


28470 


90 


6653 


6299 


5694 


95 


32636 


30750 


27562 


95 


6527 


6150 


5512 


100 


32000 


30000 


26666 


100 


6400 


6000 


5333 


105 


31357 


29250 


25786 


105 


6271 


5850 


5-157 



Maximum Permissible Stresses in columns used in buildings. 
(Building Ordinances of City of Chicago, 1893.) 
For riveted or other forms of wrought-iron columns: 

# _ 12000a I = length of column in inches; 

, Z 2 r = least radius of gyration in inches; 

of column in f 



square inches. 



a = are; 
For riveted or other steel columns, if more than 60r in length: 

= 17,000 - . 
S = 13,500a. 

a = area of post in square inches ; 
~*ijleast side of rectangular post in inches; 
I =Tength v of post in inches; 

I 600 for" *w hite*x> r Jfl_r waw pine ; 
c = < 800 for oak ; ?~ \ 

( 900 for long-leaf yellow pine. 




256 



STRENGTH OF MATERIALS. 



BUILT COLUMNS. 



From experiments by T. D. Lovett, discussed by Burr, the values of / and 
a in several cases are determined, giving empirical forms of Gordon's for- 
mula as follows: p = pounds crushing strength per square inch of section, 
I = length of column in inches, r = radius of gyration in inches. 




Keystone 



Keystone 
Columns. 

39,500 



1-f 



~1 Z 2 

18,300 r 2 



(D 



36,000 



i(2) 




Flat Ends. 



Square 
Columns. 

39,000 



(4) 



Phoenix 
Columns. 

42,000 
1 Za 



Am. Dr. Co. 



American Bridge 
Co. Columns. 



(6) 



35,000 r 2 



50,000 r 2 



Flat Ends, Swelled* 



36,000 
1 Z 2 



(91 



46,000 ; 



36,000 

- * * ( 
r 15,000 r 



Pin Ends. 



39,000 



(5) 



42,000 



:(7) 



i . - 1 _ 1.1- _ 

^ 17,000 r 2 ^ 22,700 r 2 

Pin Ends, Swelled* 



Round Ends. 

42,000 



1 



12,500 r 



36,000 

1 J_ 
21,500ra 



(10) 



36,000 



1-f- 



1 Z 2 



(11) 



11,500 r 2 



With great variations of stress a factor of safety of as high as 6 or 8 may 
be used, or it may be as low as 3 or 4, if the condition of stress is uniform or 
essentially so. 

Burr gives the following general principles which govern the resistance of 
built columns : 

The material should be disposed as far as possible from the neutral axis 
of the cross-section, thereby increasing ?; 

There should be no initial' internal stress; 

The individual portions of the column should be mutually supporting; 

The individual portions of the column should be so firmly secured to each 
ofher that no relative motion can take place, in order that the column may 
fail as a whole, thus maintaining the original value of r. 

Stoney says: **When the length of a rectangular wrought- iron tubular 
column does not exceed 30 times its least breadth, it fails by the bulging or 
buckling of a short portion of the plates, not by the flexure of th6 pillar as a 
whole." 

In Trans. A. S. C. E., Oct. 1880, are given the following formulae for the 
ultimate resistance of wrought-iron columns designed by C. Shaler Smith : 



BUILT COLUMNS. 



257 



Flat Ends. 



- (! 

1 + 5820 d 



Phoenix 
Column. 

42,500 

1 Z 2 
1 j *_ j_ 

^4500 d 8 



American Bridge 
Co. Column. 



(15) 



36,500 



Common 
Column. 

36,500 



(1> 



-- 

3750 d* 



2700 d 



One Pin End. 



38,500 

14 i- -^ 
^3000 d a 



(13) 



40,000 - 

14- i- - 
^2250 d a 



(16) 



36,500 



(19) 



36,500 



(22) 



2250 
Two Fin Ends. 



ootn ^75 *"T 



1500 



36,600 



36,500 



1750 d 



36,500 

14- - 
^1200 d 



(23) 



The "common " column consists of two channels, opposite, with flanges 
outward, with a plate on one side and a lattice on the other. 

The formula for " square " columns may be used without much error for 
the common-chord section composed of two channel-bars and plates, with 
the axis of the pin passing through the centre of gravity of the cross- 
section. (Burr). 

Compression members composed of two channels connected by zigzag 
bracing may be treated by formulae 4 and 5, using / = 36,000 instead of 
89,000. 

Experiments on full-sized Phoenix columns in 1873 showed a close agree- 
ment of the results with formulae 6-8. Experiments on full-sized Phoenix 
columns on the Watertown testing-machine in 1881 showed considerable dis- 
crepancies when the value of I -*- r became comparatively small. The fol- 
lowing modified form of Gordon's formula gave tolerable results through 
the whole range of experiments : 



Phoenix columns, flat end, p 



40,000 ( 14-T-J 



14-50,000 r a 



(24) 



Plotting results of three series of experiments on Phcenix columns, a 
more simple formula than Gordon's is reached as follows : 

Phcenix columns, flat ends, p = 39,640 46-, when I -*- r is from 30 to 140; 
p = 64,700 - 4600 \/- when I -*- r is less than 30. 



Dimensions of Phoenix Columns* 

(Phoenix Iron Co.) 

The dimensions are subject to slight variations, which are unavoidable In 
rolling iron shapes. 

The weights of columns given are those of the 4, 6, or 8 segments of which 
they are composed. The rivet heads add from 2g to 5# to the weights given. 
Rivets are spaced 3, 4, or 6 in. apart from centre to centre, and somewhat 
more closely at the ends than towards the centre of the column. 

G columns have 8 segments, E columns 6 segments, C, 2 , B l , and A have 
4 segments. Least radius of gyration = D X .3636. 

The safe loads given are computed as being one-fourth of the breaking 
load, and as producing a maximum stress, in an axial direction, on a square- 
end column of not more than 14,000 Ibs, per sq. in, for lengths of 90 radii 
and under, 



258 



STRENGTH OF MATERIALS. 



Dimensions of Phoenix Steel Columns. 

(Least radius of gyration equals D x .3G?6.) 



One Segment. 


Diameters in Inches. 


One Column. 


S3 





jS 






. 


gj . 





S'a 


^? 


*M 


1-173 





9 


tt * 


202 


^TJ 


ijtj 


"~ i-> 


03 03 

Q) Q) 


.S 3 


1 


3 


> c? 


<*-, S CO 


&2 




"S o JS 


S * 


^3 ^H 


q 


p 


O J3$ 


o-2| 


SP 


5 * 


-v <t ~ > *> 


f 


II 







or^ 


03 O o 

Si2 


1 


o5 
J'S.S 


sjjj 


t^ 











< 


P 




02 


3/16 


9.7 




4 


6 1/16 


3.8 


12.9 


1.45 


18.2 


^4 


12.2 


A 


4/^ 


6 3/16 


4.8 


16.3 


1.50 


23.9 


5/16 


14.8 


3% 


4J4 


6 5/16 


5.8 


19.7 


1.55 


30.0 


% 


17.3 




4% 


6 7/16 


6.8 


23.1 


1.59 


35.9 


5/1 6 


16.3 
19.9 




f /B 


si/16 


6.4 
7.8 


21.8 
26.5 


1.95 
2.00 


36.4 
45.1 


% 


23.5 




5% 


8 5/16 


9.2 


31.3 


2.04 


54.4 


7/16 


27.0 


B.I 


5% 


8 7/16 


10.6 


36.0 


2.09 


63.9 


/^ 


30.6 


4/ 


5% 


gl^f. 


12.0 


40.8 


2.13 


73.3 


9/16 


34.2 




6 


8 9/16 


13.4 


45.6 


2.18 


83.2 


% 


37.7 




% 


8 11/16 


14.8 


50.3 


2.23 


93.1 


24 


18.9 




6 9/16 


91^ 


7.4 


25.2 


2.39 


48.3 


5/16 


22.9 




6 11/16 


9% 


9.0 


30.6 


2.43 


59.5 


% 


27.0 




6 13/16 


9 7/16 


10.6 


36.0 


2.48 


70 7 


7/16 


31.1 
35.2 


B.2 

6 1/16 


6 15/16 
7 1/16 




12.2 
13.8 


41.5 
46.9 


2.52 
2.57 


82.3 
93.9 


9/16 


39.3 




7 3/16 


^M 


15.4 


52.4 


2.61 


105.8 


% 


43.3 




7 5/16 


9 13/16 


17.0 


57.8 


2.66 


111.9 


M 


25^ 




7 13/16 


11 11/16 


10.0 


34.0 


2.84 


70.0 


5/16 


31 




7 15/16 


HM 


12.1 


41.3 


2.88 


85.1 


% 


36 




8 1/16 


11 13/16 


14.1 


48.0 


2.93 


98.8 


a 6 


41 




8 3/16 


11% 


16.0 


54.6 


2.97 


112.5 




46 




8 5/16 


11 15/16 


18.0 


61 3 


3.01 


126.3 


9/16 


51 




8 7/16 


12 


19.9 


68.0 


3.06 


140.0 


% 


56 


C 


8 9/16 


12 1/16 


21.9 


74.6 


3.11 


153.7 


11/16 


62 


7% 


8 11/16 


12 3/16 


24.3 


82.6 


3.16 


170.2 


M 


68 




8 13/16 


12 5/16 


26.6 


90.6 


3.20 


186.7 


13/16 


73 




8 15/16 


12 7/16 


28.6 


97.3 


3.24 


200.3 


% 


78 




9 1/16 


12^ 


30.6 


104.0 


3.29 


214.2 


1 


89 




9 5/16 


12% 


34.8 


118.6 


3.34 


244.3 


1% 


99 




9 9/16 


12 13/10 


38.8 


132.0 


3.48 


271.7 


1M 


109 




9 13/16 


13 


42.7 


145.3 


3.57 


299.2 


J4 


28 




11 9/16 


151/ 


' 16.5 


56.0 


4.20 


115.3 


5/16 


32^ 




11 11/16 


15% 


19.1 


65.0 


4.25 


133.8 


% 


37 




11 13/16 


15% 


21.7 


74.0 


4.29 


152.4 


7/16 


42 




11 15/16 


15% 


24.7 


84.0 


4.34 


173.0 




47 




12 1/16 


15 15/16 


27.6 


94.0 


4.38 


193 6 


9/16 


52 




12 3/16 


16 1/16 


30.6 


104.0 


4.43 


214.1 


% 


57 


E 


12 5/16 


16 3/16 


33.5 


114.0 


4.48 


234.7 


11/16 


62 


11 1/16 


12 7/16 


16 5/16 


36.4 


124.0 


4.52 


255.3 


% 


68 




12 9/16 


16 7/16 


40.0 


136.0 


4.56 


280.0 


13/16 


73 




12 11/16 


16 9/16 


43.0 


146.0 


4.61 


300 6 


% 


78 




12 13/16 


16 11/16 


45.9 


156.0 


4.66 


321.2 


1 


88 




13 1/16 


16 13/16 


51.7 


176.0 


4.73 


36',>.4 


1% 


98 




13 5/16 


17 1/16 


57.6 


196.0 


4.84 


403.6 


JJ4 


108 




13 9/16 


17 5/16 


63.5 


216.0 


4.93 


444.7 


5/16 


31 




15H 


P* 


24.2 


82.6 


5.54 


170.2 


% 


36 


G 


15% 




28.1 


96.0 


5.59 


197.7 


7/16 


41 


14% 


1514 




32.0 


109.3 


5.64 


225.1 


^ 


46 


i!5% 


1/16 


36.0 


122.6 


5.68 


252.6 



FORMULAE FOR IROK AtfD STEEL STRUTS. 



259 



One Segment. 


Diameters in Inches. 


One Column. 


II 
















a 


$ - 








V* 


K 


a 


C!A 


Thickness i 
Inches. 


Weight in L 
per Yard, 


d Inside. 


D Outside 


. 2 

bo 


5> h 


Area of Cro 
Section, S( 
Inches. 


Weight per 
in Pounds. 


' Least Radii 
of Gyratio 
in Inches. 


Safe Load i 
Tons for 11 
Lengths. 


9/16 


51 




15% 


]9 


39.9 


136.0 


5.73 


280.0 


?8 


56 




15% 


19% 


43.8 


149.3 


5.77 


307.4 


]1/16 


61 




16 


20 


47.7 


162.6 


5.82 


334.9 


M 


66 






20% 


51.7 


176.0 


5.88 


362.4 


13/16 


71 


G 


16^4 




55.6 


189.3 


5.91 


389.8 


7 


76 




16% 


20% 


59.6 


202.6 


5.95 


417.3 


1 


86 




16% 


20% 


67.4 


229.3 


6.04 


472.1 


1% 


96 




16% 


20% 


75.3 


256.0 


6.13 


527.3 


ig 


106 




17% 


21 


83.1 


282.6 


6.27 


582.0 


15i 


116 




17% 


21*4 


90.9 


309.3 


6.32 


636.9 



Working Formula; for Wr ought-iron and Steel Struts 

of various Forms. Burr gives the following practical formulae, which 
he believes to possess advantages over Gordon's: 

Pi = Working 

Strength = 

1/5 Ultimate, 

Ibs. per sq. 

in. of Section. 



p = Ultimate 

Strength, 

Ibs. per sq. in. 

of Section. 



Kind of Strut. 
Flat and flxed end iron angles and tees 44000 - 140 (1) 8800-28 ~ (2) 



Hinged-end iron angles and tees 46000-175 

r 



9200-35 
r 

I 



Flat-end iron channels and I beams.... 40000- 110 (5) 8000-22 (6) 









Flat-end mild-steel angles 52000-180 (7) 10400-36 (8) 

I 



Flat-end high-steel angles 76000- 290 (9) 

Pin-end solid wroughMron columns.. . .32000- 80 

" 1(11) 



15200-58- (10) 
6400-16-1 



32000-277 - [ 6400-55 | 

d) dJ 



Equations (1) to (4) are to be used only between = 40 and = 200 



(5) and (6) " "" " " " = 20 

(7) to (10) ' " " " = 40 

(11) and (12)" " = ^o 



=200 
=200 
=200 



or - = 6 and -- = 65 

d a 

rro, s ' P r operly made, of steel ranging in specimens from 65,000 to 
73,000 Ibs. per square inch should give a resistance 25 to 33 per cent in ex- 
cess of that of wrought-iron columns with the same value of I H- r, provided 
that ratio does not exceed 140. 

The ^ u on^ pport . ed Y idth of a P late in a compression member should not 
exceed 30 tunes its thickness. 

In built columns the transverse distance between centre lines of rivets 
securing plates to angles or channels, etc., should not exceed 35 times the 
elate thickness. If this width is exceeded, longitudinal buckling of the 



^60 



STRENGTH OF MATERIALS. 



plate takes place, and the column ceases to fail as a whole, but yields in 
detail. 

The same tests show that the thickness of the leg of an angle to which 
latticing is riveted should not be less than 1/9 of the length of that leg or 
side if the column is purely and wholly a compression member. The above 
limit may be passed somewhat in stiff ties and compression members de- 
signed to carry transverse loads. 

The panel points of latticing should not be separated by a greater distance 
than 60 times the thickness of the angle-leg to which the latticing is riveted, 
if the column is wholly a compression member. 

The rivet pitch should never exceed 16 times the thickness of the thinnest 
metal pierced by the rivet, and if the plates are very thick it should never 
nearly equal that value. 

Merrimaii's Rational Formula for Columns (Eng. News, 
July 19, 1894). 



(2) 



B = unit-load on the column = total load P-*-area of cross-section A\ 
C = maximum compressive unit-stress on the concave side of the column: 
I length of the column; r = least radius of gyration of the cross-section 
E = coefficient of elasticity of the material ; n = 1 for both ends round 
n = 4/9 for one end round and one fixed; n -* y\ for both ends fixed. Thift 
formula is for use with strains within the eristic limit only: it does not 
hold good when the strain C exceeds the elasUc limit. 

Prof. Merriman takes the mean value otEfot timber = 1,500,000, for cast 
iron = 15,000,000, for wrought-iron = 25,000,000, nud for steel = 30,000,000, 
and 7T 2 = 10 as a close enough approximation. With these values he com- 
putes the following tables from formula (1): 

I. Wrought-iron Columns wiftb fttonnd Ends. 



Unit- 
load. 


Maximum Compressive Unit-stress C. 


p 


1 = 20 


- = 40 


1 = 60 


1 = 80 


1 = 100 


i~l 


-1=140 


1=160 


A T ' 


r 


r 


r 


r 


r 


9* 


r 


r 


5,000 
6,000 


5,040 
6,055 


5,170 
6,240 


5,390 
6,560 


5,730 
7,090 


6,250 
7,890 


6,980 
9,0v>0 


8-2PO 
11,330 


10,250 
15,56(1 


7,000 


7,080 


7,330 


7,780 


8,530 


9,720 


11,610 


15,510 


24,720 


8,000 


8,100 


8,430 


9,040 


10,060 


11,660 


14,640 


21,460 




9000 


9 130 


9550 


10340 


11 690 


14,060 


18,380 






10,000 


10,160 


10,680 


11,680 


13,440 


16,670 


23,090 






11 000 


11 200 


11 750 


13070 


15 310 


19 640 








12000 


12 240 


13000 


14 500 


17320 


23080 








13,000 


13,280 


14,180 


15,990 


19,480 






' 











STRENGTH OF WROUGHT IROK AKD STEEL COLUMNS. 261 
II. Wrought-iron Column* with Fixed Ends* 



Unit- 
load. 


Maximum Compressive Unit-stress C. 


~orB. 

A 


i = 2 


1 = 40 


1 = 60 


1 = 80 


1 = 100 


l-o 


~ = 140 


1=160 


















6,000 
7,000 
8,000 
9,000 
10,000 
11,000 
12,000 
13,000 
14,000 


6,010 
7,020 
8,025 
9,030 
10,040 
11,050 
12,060 
13,070 
14,080 


6,060 
7,080 
8,100 
9,130 
10,160 
11,200 
12,240 
13,280 
14,320 


6,130 
7,180 
8,240 
9,300 
10,370 
11,450 
12,540 
13,640 
14,740 


6,240 
7,330 
8,430 
9,550 
10,710 
11,830 
13,000 
14,210 
15,380 


6,380 
7,530 
8,700 
9,890 
11,110 
12,360 
13,640 
14,940 
16,280 


6,570 
7,780 
9,040 
10,340 
11,680 
13,070 
14.510 
15,990 
17,530 


6,800 
8,- 110 
9,490 
10,930 
12,440 
14,020 
15,690 
17,440 
19,290 


7,090 
8,530 
10,060 
11,690 
13,440 
15,310 
17,320 
19,480 
21,820 



III. Steel Columns with Round Ends. 



Unit- 
load. 


Maximum Compressive Unit-stress O. 


5 rjB< 


i = 20 


7 = o 


1 = 60 
r 


1 = 80 
r 


1 = 100 

r 


1 =120 
r 


1 = 140 


7 = 160 


6,000 
7,000 
8,000 
9,000 
10,000 
11,000 
12,000 
13,000 
14,000 


6,050 
7,070 
8,090 
9,110 
10,130 
11,160 
12.200 
13,330 
14,250 


6,200 
7,270 
8,380 
9,450 
10,560 
11,690 
12,820 
13,970 
15,130 


6,470 
7,650 
8,770 
10,090 
11,360 
12,670 
14,020 
15,400 
16,830 


6,880 
8,230 
9,650 
11,140 
12,710 
14,370 
16,130 
18,000 
19,960 


7,500 
9,130 
10,870 
12,850 
15,000 
17,370 
20,000 
22,940 
26,250 


8,430 
10,540 
12,990 
15,850 
19,230 
23,300 
28,300 


9,870 
12,900 
16,760 
20,930 
28,850 


12,300 
17,400 
24,590 



























IV. Steel Columns with Fixed Ends. 



Unit- 
load. 


Maximum Compressive Unit-stress 01 


^or. 


1 = 20 


1 = 40 


1 = 60 

7,150 
8,200 
9,250 
10,310 
11,380 
12,450 
13,530 
14,610 
15,710 


7 = 80 


1 = 100 
r 


1=1*0 


1=140 


1=160 


7,000 
8,000 
9,000 
10,000 
11,000 
12,000 
13,000 
14,000 
15,000 


7,020 
8,020 
9,030 
10,030 
11,040 
12,050 
13,060 
14,070 
15,080 


7,070 
8,090 
9,110 
10,130 
11,160 
12,200 
13,230 
14,250 
15,310 


7,270 
8,380 
9,450 
10,560 
11,690 
12,820 
13,970 
15,130 
16,310 


7,430 
8,570 
9,730 
10,910 
12,110 
13,330 
14,580 
15,850 
17,140 


7,650 
8,770 
10,090 
11,360 
12,670 
14,020 
15,400 
16,830 
18,290 


7,900 
9,200 
10,550 
11,810 
13,410 
14,930 
16,500 
18,150 
19,870 


8,230 
9,650 
11,140 
12,710 
14,370 
16,130 
17,990 
19,960 
22,060 



The design of the cross-section of a column to carry a given load with 
maximum unit-stress C may be made by assuming dimensions, and then 



STRENGTH OF MATERIALS. 

computing C by formula (1). If the agreement between the specified and 
computed values is not sufficiently close, new dimensions must be chosen, 
and the computation be repeated. By the use of the above tables the work 
will be shortened. 

The formula (1) may be put in another form which in some cases will ab- 
breviate the numerical work. For B substitute its value P-^t4, and for 
Ar* write /, the least moment of inertia of the cross-section; then 



Jn which I and r 2 are to be determined. 

For example, let it be required to find the size of a square oak column 
with fixed ends when loaded with 24 000 Ibs. and 16 ft. long, so that the 
maximum compressive stress C shall be 1000 Ibs. per square inch. Here 
7 = 24,000, C = 1000, n = M, * 2 = 10, E = 1,500,000, I = 16 X 12, and (3) be- 
comes 

I - 24r = 14.75. 

Now let x be the side of the square; then 



so that the equation reduces to x* 24# 2 = 177, from which x* is found to be 
29.92 sq. in., and the side x = 5.47 in. Thus the unit-load B is about 802 
Ibs. per square inch. 

WORKING STRAINS ALLOWED IN BRIDGE 
MEMBERS. 

Theodore Cooper gives the following in his Bridge Specifications : 
Compression members shall be so proportioned that the maximum load 

shall in no case cause a greater strain than that determined by the follow- 

ing formula : 

8000 
P = - for square-end compression members ; 



P me - for compression members with one pin and one square end; 

1 ~*~ 30,000r 

8000 
P= for compression members^with pin-bearings; 

1 ~*~20,000r 

(These values may be increased in bridges over 150 ft. span. See Cooper's 
Specifications.) 
P = the allowed compression per square inch of cross-section; 



I = the length of compression member, in inches; 
r = the least radius of gyration < 



. f gyration of the section in inches. 

No compression member, however, shall have a length exceeding 45 times 
its least width. 

Tension Members. All parts of the structure shall be so proportioned 
that the maximum loads shall in no case cause a greater tension than the 
following (except in spans exceeding 150 feet) : 

Pounds per 
sq. in. 

On lateral bracing 15,000 

On solid rolled beams, used as cross floor-beams and stringers. 9,000 

On bottom chords and main diagonals (forged eye-bars) 10,000 

On bottom chords and main diagonals (plates or shapes), net 

section 8,000 

On counter rods anri long verticals (forged eye-bars) 8,000 

On counter and long verticals (plates or shapes), net section.. 6,500 

On bottom flange of riveted cross-girders, net section 8,000 

On bottom flange of riveted longitudinal plate girders over 

20ft. long, net section 8,000 



WORKING STRAINS ALLOWED IN BRIDGE MEMBERS. 263 

On bottom flange of riveted longitudinal plate girders under 

20 ft. long, net section ..'.'.. 7,000 

On floor-beam hangers, and other similar members liable to 

sudden loading (bar iron with forged ends) 6,000 

On floor-beam hangers, and other similar members liable to 

sudden loading (plates or shapes), net section 5,000 

Members subject to alternate strains of tension and compression shall be 
proportioned to resist each kind of strain. Both of the strains shall, how- 
ever, be considered as increased by an amount equal to 8/10 of the least of 
the two strains, for determining the sectional area by the above allowed 
strains. 

The Phoenix Bridge Co. (Standard Specifications, 1895) gives the follow- 
ing : 

The greatest working stresses in pounds per square inch shall be as fol- 
lows : 

Tension. 
Steel. Iron. 

P = OOP f 1 I Min ' Stress 1 Forbars P= 75o o ri i MiD - stress 1 
L Max. stressj forged ends. |_ Max. stress J 

P - 8 500 fl -I Min ' Stress 1 PIatesor p _ 700o r i , Mia, stress"] 
uu L T Max. stressj shapes net. J ~ r |/ T Max. stressj 

8,500 pounds. Floor-beam hangers, forged ends 7,000 pounds. 

7,500 Floor-beam hangers, plates or shapes, net 

section 6,000 " 

10,000 " Lower flanges of rolled beams. 8,000 " 

50,000 ** Outside fibres of pins 15,000 " 

30,000 " Pins for wind-bracing 22,500 " 

20,000 " Lateral bracing 15,000 " 

Shearing. 

9,000 pounds. Pins and rivets 7,500 pounds. 

Hand-driven rivets 20# less unit stresses. For 

bracing increase unit stresses 50%. 
6,000 pounds. Webs of plate girders 5,000 pounds. 

Bearing. 

16,000 pounds. Projection semi-intrados pins and rivets.. . . 12,000 pounds. 
Hand-driven rivets 20# less unit stresses. For 
bracing increase unit stresses 50#. 

Compression. 

Lengths less than forty times the least radius of gyration, P previously 
found. See Tension. 

Lengths more than forty times the least radius of gyration, P reduced by 
following formulae: 



For both ends fixed, b = 

For one end hinged, 
For both ends hinged, 




18,000 r 



P = permissible stress previously found (see Tension) ; b = allowable 
working stress per square inch; I = length of member in inches; r = least 
radius of gyration of section in inches. No compression member, how- 
ever, shall have a length exceeding 45 times its least width. 



io,ooo(i 



264 STRENGTH OF MATEEIAL8. 

Pounds per 
sq. in. 

In counter web members 10,500 

In long verticals 10,000 

In all main-web and lower-chord eye-bars 13,200 

In plate hangers (net section) 9,000 

In tension members of lateral and transverse bracing 19,000 

In steel-angle lateral ties (net section) 15,000 

For spans over 200 feet in length the greatest allowed working stresses 
per square inch, in lower-chord and end main-web eye-bars, shall be taken at 

min. total stress \ 
max. total stress J 

whenever this quantity exceeds 13,200. 

The greatest allowable stress in the main-web eye-bars nearest the centre 
of such spans shall be taken at 13,200 pounds per square inch ; and those 
for the intermediate eye-bars shall be found by direct interpolation between 
the preceding values. 

The greatest allowable working stresses in steel plate and lattice girders 
and rolled beams shall be taken as follows : 

Pounds per 
sq. in. 

Upper flange of plate girders (gross section) 10,000 

Lower flange of plate girders (net section) 10,000 

In counters and long verticals of lattice girders (net section) . . 9,000 
In lower chords and main diagonals of lattice girders (net 

section) " 10,000 

In bottom flanges of rolled beams 10,000 

In top flanges of rolled beams 10,000 

RESISTANCE OF HOLLOW CYLINDERS TO 
COLLAPSE. 

Fairbairn's empirical formula (Phil. Trans. 1858) is 

.i 
p = 9,675,600 '-rv-, . . , (1) 

Id 

where p = pressure in Ibs. per square inch, t = thickness of cylinder, d = 
diameter, and I = length, all in inches ; or, 

p = 806,300 ~^, if L is in feet (2) 

He recommends the simpler formula 

p = 9,675,600^ (3) 

as sufficiently accurate for practical purposes, for tubes of considerable 
diameter and length. 

The diameters of Fairbairn's experimental tubes were 4", 6", 8", 10", and 
12", and their lengths ; between the cast-iron ends, ranged between 19 inches 
and 60 inches. 

His formula (3) has been generally accepted as the basis of rules for 
ascertaining the strength of boiler-flues. In some cases, however, limits are 
fixed to its application by a supplementary formula. 

Lloyd's Register contains the following formula for the strength of circular 
boiler-flues, viz., 

89,600^ 

LA () 

The English Board of Trade prescribes the following formula for circular 
flues, when the longitudinal joints are welded, or made with riveted butt- 
straps, viz., 

- M.000< 



For lap-joints and for inferior workmanship the numerical factor may be 
reduced as low as 60,000. 



RESISTANCE OF HOLLOW CYLINDERS TO COLLAPSE. 265 

The rules of Lloyd's Register, as well as those of the Board of Trade, pre- 
scribe further, that in no case the value of P must exceed the amount given 
by the following equation, viz., 



In formulae (4), (5), (6) P is the highest working pressure in pounds per 
square inch, t and d are the thickness and diameter in inches, L is the 
length of the flue in feet measured between the strengthening rings, in case 
it is fitted with such. Formula (4) is the same as formula (3), with a factor 
of safety of 9. In formula (5) the length L is increased by 1 ; the influence 
which this addition has on the value of P is, of course, greater for short 
tubes than for long ones. 

Nystrom has deduced from Fairbairn's experiments the following formula 
for the collapsing strength of flues : 



............ 

where p, , and d have the same meaning as in formula (1), L is the length in 
feet, and Tis the tensile strength of the metal in pounds per square inch. 

If we assign to T the value 50,000, and express the length of the flue in 
inches, equation (7) assumes the following form, viz., 

p = 692,800 -. .......... (8) 

d yl 

Nystrom considers a factor of safety of 4 sufficient in applying his formula. 
(See "A New Treatise on Steam Engineering," by J. W. Nystrom, p. 106.) 

Formula (1), (4), and (8) have the common defect thai they make the 
collapsing pressure decrease indefinitely with increase of length, and vice 
versa. M. Love has deduced from Fairbairn's experiments an equation of 
a different form, which, reduced to English measures, is as follows, viz., 

p= 5,358,150 ^ + 41,906^+ 1323 j, ...... (9) 

where the notation is the same as in formula (1) . 

D. K. Clark, in his " Manual of Rules," etc., p. 696, gives the dimensions of 
six flues, selected from the reports of the Manchester Steam-Users Associa- 
tion, 1862-69, which collapsed while in actual use in boilers. These flues 
varied from 24 to 60 inches in diameter, and from 8-16 to % inch in thickness. 
They consisted of rings of plates riveted together, with one or two longitud- 
inal seams, but all of them unfortified by intermediate flanges or strength- 
ening rings. At the collapsing pressures the flues experienced compressions 
ranging from 1.53 to 2.17 tons, or a mean compression of 1.82 tons per square 
inch of section. From these data Clark deduced the following formula 
"for the average resisting force of common boiler-flues," viz., 



where p is the collapsing pressure in pounds per square inch, and d and t 
are the diameter and thickness expressed in inches. 

C. R. Roelker, in Tan Nostrand's Magazine, March, 1881, discussing f'e 
above and other formulae, shows that experimental data are as yet insuffi- 
cient to determine the value of any of the formulae. He says that Nystrom 's 
formula, (8), gives a closer agreement of the calculated with the actual col- 
lapsing pressures in experiments on flues of every description than any of 
the other formulae. 

Collapsing Pressure of Plain Iron Tubes or Flues. 

(Clark, S. E., vol. i. p. 643.) 

The resistance to collapse of plain-riveted flues is directly as the square of 
the thickness of the plate, and inversely as the square of the diameter. The 
support of the two ends of the flue does not practically extend over a length 
of tube greater than twice or three times the diameter. The collapsing 
pressure of long tubes is therefore practically independent of the length. 



266 STRENGTH OF MATERIALS. 

Instances of collapsed flues of Cornish and Lancashire boilers collated by 
Clark, showed that the resistance to collapse of flues of %-iuch plates, 18 to 
43 feet long, and 30 to 50 inches diameter, varied as the 1 75 power of the 
diameter. Thus, 

for diameters of ....................... 30 35 40 45 50 inches, 

the collapsing pressures were ......... 76 58 45 37 30 Ibs. per sq. in; 

for 7-16-inch plates the collapsing 

pressures were ........... ............ . . . 60 49 42 

For collapsing pressures of plain iron flue-tubes of Cornish and Lanca 
shirs steam-boilers, Clark gives: 

_ 200,000*2 



P = collapsing pressure, in pounds per square inch; 
t = thickness of the plates of the furnace tube, in inches. 
d = internal diameter of the furnace tube, in inches. 

For short lengths the longitudinal tensile resistance may be effective in 
augmenting the resistance to collapse. Flues efficiently fortified by flange= 
joints or hoops at intervals of 3 feet may be enabled to resist from 50 Ibs. 
to 60 Ibs. or 70 Ibs, pressure per square inch more than plain tubes, accord. 
ing to the thickness of the plates. 

Strength of Small Tubes. The collapsing resistance of solid- 
drawn tubes of small diameter, and from .134 inch to .109 inch in thickness, 
Has been tested experimentally by Messrs. J. Russell & Sons. The results 
lor wrought-iron tubes varied from 14.33 to 20.07 tons per square-inch sec- 
tion of the metal, averaging 18.20 tons, as against 17.57 to 24.28 tons, averag- 
ing 22.40 tons, for the bursting pressure. 

(For strength of Segmental Crowns of Furnaces and Cylinders see Clark, 
S. E., vol. i, pp. 649-651 and pp. 627, 628.) 

Formula for Corrugated Furnaces (Bng'g* July 24, 1891. p. 
102). As the result of a series of experiments on the resistance to collapse 
of Fox's corrugated furnaces, the Board of Trade and Lloyd's Registry 
altered their formulae for these furnaces in 1891 as follows: 

Board of Trade formula is altered from 



T = thickness in inches; 

D = mean diameter of furnace; 

WP = working pressure in pounds per square inch. 

Lloyd's formula is altered from 

1000 X (T - '!} = wp to 1S84XCT.-2) = WR 

T = thickness in sixteenths of an inch; 

D = greatest diameter of furnace; 

WP = working pressure in pounds per square inch. 

TRANSVERSE STRENGTH. 

In transverse tests the strength of bars of rectangular section is found to 
rary directly as the breadth of the specimen tested, as the square of its 
depth, and inversely as its length. The deflection under any load varies as 
the cube of the length, and inversely as the breadth and as the cube of the 
depth. Represented algebraically, if S = the strength and D the deflection, 
1 the length, 6 the breadth, and d the depth, 

7,,?3 19 

8 varies as -r- and D varies as ^. 

For the purpose of reducing the strength of pieces of various sizes to 
a common standard, the term modulus of rupture (represented by K) is 
used. Its value is obtained by experiment on a bar of rectangular section 



TRANSVERSE STRENGTH. 267 

supported at the ends and loaded in the middle and substituting numerical 
values in the following formula : 



to which P= the breaking load in pounds, I = the length in inches, b the 
breadth, and d the depth. 

The modulus of rupture is sometimes defined as the strain at the instant 
of rupture upon a unit of the section which is most remote from the neutral 
axis on the side which first ruptures. This definition, however, is based 
upon a theory which is yet in dispute among authorities, and it is better to 
define it as a numerical value, or experimental constant, found by the ap- 
plication of the formula above given. 

From the above formula, making I 12 inches, and b and d each 1 inch, it 
follows that the modulus of rupture is 18 times the load required to break a 
bar one inch square, supported at two points one foot apart, the load being 
applied in the middle. 

.. span in feet X load at middle in Ibs. 

Coefficient of transverse strength = ^^ in inches x (dep[h in ^^^ 

= th of the modulus of rupture. 

lo 

Fundamental Formulae for Flexure of Beams (Merriman). 

Resisting shear = vertical shear; 

Resisting moment =? bending moment; 

Sum of tensile stresses = sum of compressive stresses; 

Resisting shear = algebraic sum of all the vertical components of the in- 
ternal stresses at any section of the beam. 

Tf A be the area of the section and Ss the shearing unit stress, then resist- 
ing shear = ASs; and if the vertical shear = V, then V ASs. 

The vertical shear is the algebraic sum of all the external vertical forces 
on one side of the section considered. It is equal to the reaction of one sup- 
port, considered as a force acting upward, minus the sum of all the vertical 
downward forces acting between the support and the section. 

The resisting moment algebraic sum of all the moments of the inter- 
nal horizontal stresses at any section with reference to a point in that sec- 

or 

tion, = , in which 8 = the horizontal unit stress, tensile or compressive 

c 

as the case may be, upon the fibre most remote from the neutral axis, c = 
the shortest distance from that fibre to said axis, and / = the moment of 
inertia of the cross-section with reference to that axis. 

The bending moment M is the algebraic sum of the moment of the ex- 
ternal forces on one side of the section with reference to a point in that sec- 
tion moment of the reaction of one support minus sum of moments of 
loads between the support and the section considered. 



he bending moment is a compound quantity = product of a force by the 
(Distance of its point of application from the section considered, the distance 
being measured on a line drawn from the section perpendicular to the 
direction of the action of the force. 

Concerning the above formula, Prof. Merriman, Eng. News, July 21, 1894, 
says: The formula just quoted is true when the unit-stress <S on the part of 
the beam farthest from the neutral axis is within the elastic limit of the 
material. It is not true when this limit is exceeded, because then the neutral 
axis does not pass through the centre of gravity of the cross-section, and 
because also the different longitudinal stresses are not proportional to their 
distances from that axis, these two requirements being involved in the de- 
duction of the formula. But in all cases of design the permissible unit- 
stresses should not exceed the elastic limit, and hence the formula applies 
rationally, without regarding the ultimate strength of the material or any 
of the circumstances regarding rupture. Indeed so great reliance is placed 
upon this formula that the practice of testing beams by rupture has been 
almost entirely abandoned, and the allowable unit-stresses are mainly de- 
rived from tensile and compressive tests. 



268 



STRENGTH OF MATERIALS. 






+ ft, 18 



'1*1 MBS 



g 

1,5 

SjB 



II II 



~t 



~l8S 



i 



!! 

fe 
-h 
ftj 



J 


I 


I i 










: 


: 




^ 


> 


<D 


: ' 





ii 


3 


^ 


*--s tJ 







> 1 


1 

a 


s a 

"3 a 
* S 


3 


ii 


5 5 


1 


| 


g 


n. el 




APPROXIMATE SAFE LOADS IK LBS. OK STEEL BEAMS. 



Formulae for Transverse Strength of Beams* Referring to 

table on preceding page, 

P = load at middle; 

W= total load, distributed uniformly; 
I = length, 6 = breadth, d = depth, in inches; 

E =s modulus of elasticity; 

R = modulus of rupture, or stress per square inch of extreme fibre; 

/ =r moment of inertia; 

c = distance between neutral axis and extreme fibre. 

For breaking load of circular section, replace 5d 2 by 0.59d 9 . 

For good wrought iron the value of R is about 80,000, for steel about 120,000, 
the percentage of carbon apparently having no influence. (Thurston, Iron 
and Steel, p. 491), 

For cast iron the value of R varies greatly according to quality. Thurston 
found 45,740 and 67,980 in No. 2 and No. 4 cast iron, respectively. 

For beams fixed at both ends and loaded in the middle, Barlow, by experi- 
ment, found the maximum moment of stress = 1/6PI instead of 1&PI, the 
result given by theory. Prof. Wood (Resist. Matls. p. 155) says of this case: 
The phenomena are of too complex a character to admit of a thorough and 
exact analysis, and it is probably safer to accept the results of Mr. Barlow 
in practice than to depend upon theoretical results. 

APPROXIMATE: GREATEST SAFE LOADS IN LBS. ON 

STEEL BEAMS. (Pencoyd Iron Works.) 

Based on fibre strains of 16,000 Ibs. for steel. (For iron the loads should be 
one-eighth less, corresponding to a fibre strain of 14,000 Ibs. per square inch.) 
L = length in feet between supports; a = interior area in square 
A = sectional area of beam in square inches; 

inches; d = interior depth in inches. 

D = depth of beam in inches. w = working load in net tons. 



Shape of 
' Section. 


Greatest Safe Load in Pounds. 


Deflection in Inches. 


Load in 
Middle. 


Load 
Distributed. 


Load in 
Middle. 


Load 
Distributed. 


Solid Rect- 
angle. 


890.4D 


1 780.4 D 


wL* 
S2AD* 


tc3 


L 


L 


52AD* 


HollowRect- 
angle. 


890UD-orf) 


1780C4D-ad) 


wL* 


wL* 


L 


L 


32UZ)-ad 2 ) 


52UZ)2-ad a ) 


Solid Cylin- 
der. 


M7AD 


13334Z) 


wLs 
24AD* 


wl? 
3SAD* 


L 


L 


Hollow 
Cylinder. 


667UD-ad) 


1333(AD-ad) 


wL* 


wL* 


L 


L 


24(AD*-ad*) 


38(AD*-ad?) 


Even-legged 
Angle or 
Tee. 


S85AD 


1710AD 


wL* 


wL* 


L 


L 


32^Z> 


52AD* 


Channel or 
Zbar. 


1525AD 


3Q5QAD 


wL? 


wL* 
85AD* 


L 


L 


53^D 2 


Deck Beam. 


1380 AD 


2760 AD 


wL* 
504D* 


wL* 


L 


L 


SOAD* 


I Beam. 
I 


1695^1) 


mQAD 


mL* 


wL* 


L 


L 


5&AD* 


934Z>' 


II 


III 


IV 


V 



270 



STRENGTH OF MATERIALS. 



The above formulae for the strength and stiffness of rolled beams of va- 
rious sections are intended for convenient application in cases where 
strict accuracy is not required. 

The rules for rectangular and circular sections are correct, while those for 
the flanged sections are approximate, and limited in their application to the 
standard shapes as given in the Pencoyd tables. When the section of any 
beam is increased above the standard minimum dimensions, the flanges re- 
maining unaltered, and the web alone being thickened, the tendency will be 
for the load as found by the rules to be in excess of the actual; but within 
the limits that it is possible to vary any section in the rolling, the rules 
will apply without any serious inaccuracy. 

The calculated safe loads will be approximately one half of loads that 
would injure the elasticity of the materials. 

The rules for deflection apply to any load below the elastic limit, or less 
than double the greatest safe load by the rules. 

If the beams are long without lateral support, reduce the loads for the 
ratios of width to span as follows : 



Length of Beam. 

20 times flange width. 

30 " " " 

40 " ' ** 

50 ** * * 

60 " 

70 " " " 

These rules apply to beams supported at each end. For beams supported 
otherwise, alter the coefficients of the table as described below, referring to 
the respective columns indicated by number. 



Proportion of Calculated Load 

forming Greatest Safe Load. 

Whole calculated load. 

9-10 " " 

8-10 " " 

7-10 

6-10 " 

5-10 " " 



Changes of Coefficients for Special Forms of Beams. 



Kind of Beam. 


Coefficient for Safe 
Load. 


Coefficient for Deflec- 
tion. 


Fixed at one end, loaded 
at the other. 


One fourth of the coeffi- 
cient, col. II. 


One sixteenth of the co- 
efficient of col. IV. 


Fixed at one end, load 
evenly distributed. 


One fourth of the coeffi- 
cient of col. III. 


Five forty-eighths of the 
coefficient of col. V. 


Both ends rigidly fixed, 
or a continuous beam, 
with a load in middle. 


Twice the coefficient of 
col. II. 


Four times the coeffi- 
cient of col. IV. 


Both ends rigidly fixed, 
or a continuous beam, 
with load evenly dis- 
tributed. 


One and one-half times 
the coefficient of col. 
III. 


Five times the coefficient 
of col. V. 



ELASTIC RESILIENCE. 

In a rectangular beam tested by transverse stress, supported at the ends 
and loaded in the middle, 

2 Rbd* 

p -3-~T~ ; 
1 PJ3 



~lEbd* ' 

in which, if P is the load in pounds at the elastic limit, R = the modulus of 
transverse strength, or the strain on the extreme fibre, at the elastic limit, 
E= modulus of elasticity, A = deflection, I, 6, and d= length, breadth, and 
depth in inches. Substituting for P in (2) its value in (1), we have 

1 Rl* 

6 JEtT 



BEAMS OF UNIFORM STRENGTH THROUGHOUT LENGTH. 271 




The elastic resilience = half the product of the load and deflection 
and the elastic resilience per cubic inch 

_1 PA 

"~ 2 Ibd ' 

Substituting the values of P and A, this reduces to elastic resilience per 
cubic inch = jg^ which is independent of the dimensions; and therefore 

fhe elastic resilience per cubic inch for transverse strain may be used as a 
modulus expressing one valuable quality of a material. 
Similarly for tension: 

Let P = tensile stress in pounds per square inch at the elastic limit; 
e = elongation per unit of length at the elastic limit; 
E = modulus of elasticity = P -*- e\ whence e P-*- E. 

Then elastic resilience per cubic inch = y%Pe . 

2 E 

BEAMS OF UNIFORM STRENGTH THROUGHOUT 
THEIR LENGTH. 

The section is supposed in all cases to be rectangular throughout. The 
beams shown in plan are of uniform depth throughout. Those shown in 
elevation are of uniform breadth throughout. 

B iireadth of beam. D = depth of beam. 

Fixed at one end, loaded at the other; 
curve parabola, vertex at loaded end; BD Z 
proportional to distance from loaded end. 
The beam may be reversed, so that the up- 
per edge is parabolic, or both edges may be 
parabolic. 

Fixed at one end, loaded at the other; 
triangle, apex at loaded end; BD* propor- 
tional to the distance from the loaded end. 

Fixed at one end; load distributed; tri- 
angle, apex at unsupported end; BD'* pro- 
portional to square of distance from unsup- 
ported end. 

Fixed at one end; load 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 semi-ellipse; 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 deck-beams 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 
deck-beams, 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 deck-beams 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 inch-pounds 
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 roof-purlins. 

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 cross-section 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 


9-2.2 


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 



r-i I-H OO OO O OOOOOOOO 



300 00 r- 

CO W W TH TH TH TH TH TH fHl-tOOOOOO 



CCO OS CO I' 00 tOO OO <* W TH W CO 
* "DO COrH OS OOJ>tOOTf 

CO CO 



OJCO 30 t^CX5C>OOiO'*Tt<O 
O(?t OS t iOrf(>f 'OOSOO 



WOi O OS 00 1- CDO 1O OO^^-^COCOCO 



*^a ni 
sjaoddng 



taso t> oo cso T-ICJ co 



35 OO5OCO OSOOOOOSCOI.--COOSI 

w osjocoo i-ocoi-'Oaoi'iOTr 



GO O CO to i-i 

do'os'od 06 



OS?OOOS OS OS5OOOS 
OSt-t-l- OS <M->-HO 



J TH T-.OOOS 



THOOOSOSOJQOQOOOt- 



^99^ HI 

s^aoddng 



tCO-^O O 1>OOOJO TH (M CO ^ tf? O l> 00 OS O 

HrHTHTH TH TH TH TH O Oi W Gi Gt O* (?t <?i <H W CO 



tOTj<?tOOSaO{>COOTi<"^cOCOW 



WTHTHOO O OS OS OS 00 OO 



STHCOt-c^t^toosmcsooco^s^ 
COOSOOOTt<Ol-00?OOOOCO 



r t- co so t^ co w 



BSSSS 

- 1- to to o* 



ost~oocoi>-<*cso?ooocoono TP 
wc\no-^Tji-i-.ooiOTfTrin5Ooo T-. 



co w w w w oi 



Hww^ 



ooooooaej>osoeoosogc i o>io osooo-r-ios wo; 

OS OS 1O_ 1C Ci O tO OO TH L- CO TH OS OS OS TH W 10 J> TH ^ 

OTfWTHOSOOI>O-r COCOW-riO OOSOO O0t>t- 



s i 



t- O-T"CO??O OiOSOO t^- 

co cococococo www w 



^8^ Ul 

s^joddng 



STRENGTH OF MATERIALS. 



M 


10 


O OS CO 00 C* 


OOlOCJOOS 





CO 


o 


t- rf co w w 


^ ^ ^ o .... . . . 


Sw 
S 


H-i 


10 * 


1- 00 lO O OS 


(JJCOdOSCO ^CiTHOS 


2 ft 


^ 


*- 


St GO CO lO CO 


COOldTHTH T-HTHT-I 


1 1 


^ 


10 


CO CO 1O - ^ 


WCOCOTHCO COOGOtO Tj.COCtT-4 


1 M 


O 


o>- 


O "V O 00 CO 


lOTjICOCOG^ C*OiT-(TH l-(THT-(T-4 


*-" P 

3 


j_j 


& 


O 1O 00 TH CO 


OOTt<^COO ^-Ot-TK (NOSOOCO 


? 


^ 





CO 55 -l -! 


Z>COlOrt<T}' COCOWd O4r-(T-(T- 


i 1 

^.dw 


^ 


j 


(M t- lO CO CO 


THTH{>OCO OSCOOO^ HOOOCO 


8.1s 


r. 


lO 


rft O CJ t~ W 


THOSt-COlO ^t^COCO COddW 


-d-So 

e8 4) -t> 

o >- o 


^ 


S 


l> i O 1> t- 


OJlOlOOl- t-05COi> <WCO^^ 





6b 


oo 


(?> i i CO GO 
^ co c* - 


lOO?OOSJ>- COlOlO^ ^ CO CO CO 


0) 9^3 

.> i 


M 


CO 


10 05 !-l 10 05 


THCOOO5CO OOSOd COOCOOO 


gftO 


0, 


s 


8 s ^ 55 s; 


gJS^^JS ^^ l -^ wo^co 


5cc-S 

-rS 


P 0) 

5 is 


1 


1O CO t> 00 04 




1 

as'l 


.2 4> 


or* 

3 .2 






^<3 

-S5 

O-* 3 rt 


Hi 


JS 


-H Tf CO CO W 


007?1005 ^05100*05 COCOTHOS 


s M S 

> 0) 


^ 




00 lO CO 1-1 O 


OS 00t> CO 1O lO-^-'J"'^ CO COCOCOO4 


^oft 


s 


g 






6C<C c 




SjS 


CO l CO TH O 

co <*> os' t> 10 


COOOCOCO1> CSCOI>r-lI> COO5COCO 
CO'-OOSOO L-t>-COCOlO lO-^-^ 1 ^ 


111 


*""* 


CO" i 


ct 




'^v 


0* 


S3 


O> CO Tf CO J> 


10000*000 0OCOZ>TH COTHl>CO 


M C 

s ^S 




- g 

3d 


S?^^5JS 


COrtCOWO OSO500t>t- COCO1O1O 


**i 




1 


eo w TH os 10 


J^TtTj<t--<M OO5O5^-CO CCOiOO 
^ Sr^' *' r^ ^T^SS ^ ^OO^ 1 ^ 


-si 1 
-^ 

o- 5 


ni 


53 


TH CO O* 1O 00 


Ol-OJ>CO OSTfOOSOO 050COCO 






g"" 1 a. 
CO 


gsssss 


O CO ^ > i O5 1-COlOCOC-J TH . i O OS 


P! 




03 ^ 


CO O t^ CO O* 


0>C*COCOt- Tt<^COT-i> lOTjilOCO 


^^ 




il 


S S 3 


O5 Tji r-c CO 1O CO TH O5 00 CO lO ^ CO C< 


22 

<!) 


i-i 


^ 


10 00 T-, 00 


CO T-IT-I CO -^ lOOOTt<TH O5 O5OCS1O 




GO 





lO 1O CO T-I CO 
CO lO rf rr CO 


^^555 S^:Si2~ ^^^S 


^a 
ofA 




O3 


CO 00 l> lO I> 


OilOCOdCO COCOt--OlO THO5GOOS 


^feS' 


M 


i 


S S 2 J 5g 


%%% ^^5J8S 22 


g^fc 
^^s 


b 


v~ 


CO CO 00 10 TH 


T-"COCOTH1O COCOdOT- 1OOCO^< 


^ ;2 


G* 


l! 


OS 2 CO CO 


S^^SS 38688 S82S 


1! 


H-t 


I 

i 


3ate 


SSES^ S88SSS& SSSS 


Ill 


gg 
P 


lu 

Ife 


W CO Tf O CP 


. .. NS , sln<0 


s! 

sll 


5l 


I- 






EisO 

e 



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 


^ g5-g 


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 


0.61|2.81 


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 Peck-beams, 



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 10-in. 
beams, 4900 Ibs.; 9-in., 4500 Ibs.; 8-in., 4000 Ibs.; 7-in., 3400 Ibs., 6-in., 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 


"?< 


o 


^ 


"3 




^ 


oJ-S 


a 






S 


o 


|S| 


Is. 


|g| 


-d| 


!<* 


ii^l 


-2 

02 






2'i 


$$ . g. 


fsl 


ft S 


; oa2 


002 


{JOCQ 


^^"gpii 




g 


g 




a i 


||S 


.2o g 


cT^ 


ojd5 


.SQ^ 


*C"5^ 


G> 


<2 


o 


r> 


g"S;S 


rt5 


"cl' E <r H 

^ 6C&^ 


"S'S'^ 


^"to^ 


1^ 


& J^'o 


Sb 


1 


1 


(U 


|5 


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5 


M S 


III 


g|| 


lis^S) 


.. 


1 


<M 



Ss 

fl_te 


**| 


*>"SPH 


ajjl 


"o^'S 


n^' 
0.52 S 


g| 


igsflj 










S'x eg 


w> S . 


^ x 5 










N 


1 




.20 


o<i^ 




^^^ 










- O2 


P- 


<*! 


Q 


1 


,J 







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 X4Vj|j 


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 


4 


5 


6 


7 


8 


9 


10 


11 





















_ 


. 


03.2 


1 






o 


! . 




k- 


fei 


is! 


Is! 


oi-l 








<W -? 




"s 


G O be 


<S O</2 


go2 


,002 




1 


I 


.2 


^fl 
'o.fS 


-p b()pE4 


S |K 
^83 




^d* 

S^'s 

e bc^ 


o| 

fa* 




Shi 


1 




1 


$t 


g3 


0.52 "3 

33 X=3 


g 


s-s 

JUJ 


ill 


III 


|j2i 


& 


P 




0> 


CJ^^H 


8^ 


'oifl'^ 


2 


^^2 


o^2 


^cc . joo 


s 


I 





p 


||| 


If*? 


111 


ill 


||| 


ill 

,3 x o 


jis|| 


.s 

02 


1 


5 


lo 

Q 




|rf 


1^ 




iff 


iHo 


o 


in. 


TbT 


sq.in. 


in. 


/ 


8 


r 


l' 


S' r' 


C 


4 X2)4 


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 


O o 


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